REAUSTENITISATION FROM BAINITE IN STEELS
By
Manabu Takahashi
Darwin College
Cambridge
A dissertation submitted for the degree of
Doctor of Philosophy,
at the University of Cambridge
September 1992
To my parents, my wife Yukimi and my daughter Mayuko
PREFACE
This dissertation is submitted for the degree of Doctor of Philosophy at the University of
Cambridge. The work described was carried out under the supervision of Dr. H. K. D. H.
Bhadeshia in the Department of Materials Science and Metallurgy between October 1987 and
June 1989. Except where appropriately referenced, this work is, to my best knowledge, original
and contains nothing which is the outcome of collaboration. This dissertation has not been
submitted in whole or in part for a degree, diploma or other qualification. This dissertation
does not exceed 60,000 words in length.
~UJJ!
M. Takahashi
September 1992
i
ACKNOWLEDGMENTS
I would like to express my deepest gratitude to Dr. H. K. D. H. Bhadeshia for his pa-
tient guidance, helpful discussion and continued encouragement throughout this project. I am
also grateful to Professor D. Hull for the provision of laboratory facilities at the University of
Cambridge.
I would like to acknowledge the considerable help and encouragement which I received from
my colleagues in the Phase Transformation Group, particularly Dr. Jer-Ren Yang, Dr. Serdar
Atamert, Dr. Shahid Khan, Dr. Roger Reed and Dr. Suresh Babu. Dr. Roger C. Reed very
kindly agreed to proof-read the manuscript during his stay in Japan.
This work was supported by Nippon Steel Corporation to whom acknowledgement is also
made. I am also grateful to those who gave me the opportunity of this work at the University of
Cambridge, in particular, Professor T. Sakuma and Professor T. Kuroyanagi who kindly wrote
the recommendation to the University of Cambridge, S. Harada and K. Esaka who recommended
me as a candidate of the Foreign Student Scholarship of Nippon Steel Corporation.
Finally, I would like to thank my wife Yukimi for her moral support and encouragement
during the period of my research, and my daughter Mayuko who has joined recently in encour-
aging my writing.
11
REAUSTENITISATION FROM BAINITE IN STEELS
Manabu TAKAHASHI
Darwin College
Department of Materials Science and Metallurgy
SUMMARY
A large number of industrial steel manufacturing and processing methods involve heating
the alloy into a temperature regime where austenite is the stable phase. The characteristic and
subsequent transformation of this austenite during cooling determine the final properties of the
steel. The formation of austenite in steels has, therefore, been investigated both theoretically
and using a variety of experimental methods. Since the general problem of austenite formation
is very dependent on the initial microstructure, the work presented in this thesis concentrates
on bainitic starting microstructures. Bainitic steels are at the forefront of new and exciting
developments in steel technology.
The thesis begins with a literature review and critical assessment of the bainite reaction
and of austenite formation. As a consequence, two interesting and unresolved problems relating
to the bainitic transformation are first investigated. A quantitative model is developed which
is shown to be capable of predicting the transition from upper to lower bainite. This has
led to some unexpected predictions which explain a previously unnoticed phenomenon, that
lower bainite does not occur in plain carbon steels containing less than about 0.3 wt.% carbon,
and that upper bainite is not found when the carbon concentration exceeds about 0.4 wt.%.
The second problem tackled relates to the kinetics of the bainite reaction. A model has been
developed and applied to enable the prediction of time-temperature-transformation diagrams
for the bainite reaction.
The studies on bainite are followed by experimental work on the growth of austenite from
mixed microstructures of bainite and residual carbon enriched austenite. The need to nucleate
austenite has been avoided by using this starting microstructure, although comparative exper-
iments were also conducted on initial microstructures without any austenite. A detailed and
completely new theory has been developed to enable the interpretation of the experimental
data on austenite growth. The theory is based on the concept of local equilibrium at the trans-
formation interface, involves multicomponent diffusion in both the parent and product phases,
and makes predictions which are on the whole consistent with the experimental data. Non
equilibrium growth is also featured in the form of paraequilibrium growth of austenite.
UI
CONTENTS
CHAPTER 1:
REAUSTENITISATION IN STEELS
1.1 INTRODUCTION .
1.2 NUCLEATION OF AUSTENITE . . . . . . .
1.2.1 Initial microstructure: Ferrite and Pearlite
1.2.2 Initial microstructure: Ferrite and Cementite
1.2.3 Martensitic initial microstructure
1.2.4 Ferritic initial microstructure
1.2.5 Bainitic initial microstructure
1.2.6 Summary . . . . . . .
1.3 GROWTH OF AUSTENITE ...
1.3.1 Ferrite-Pearlite mixtures
1.3.2 Ferrite-Spheroidised Cementite mixture
1.3.3 Bainitic ferrite-Austenite mixture
1.3.4 Reaustenitisation from cementite and bainitic ferrite
1.4 OVERALL TRANSFORMATION KINETICS.
1.5 ANISOTHERMAL TRANSFORMATION
1.5.1 Continuous heating transformation
1.6 CRYSTALLOGRAPHY . . .
1.7 APPLICATIONS . . .
1.7.1 Ferrite-Martensite dual phase steels
1.7.2 Steels containing some retained austenite
1.7.3 Welding of steels . . . . . . . . . .
1.7.4 Initial austenite grain size . . . . . . .
1.8 TRANSFORMATION FROM AUSTENITE
1.8.1 Widmanstiitten ferrite formation in steels
1.8.2 Bainite transformation in steels ....
1.8.3 Martensitic transformation in steels
1.8.4 Reconstructive formation of ferrite in steels
1.9 PREDICTION OF MICROSTRUCTURE IN STEELS
1.9.1 Microstructure prediction in welding . . ..
1.9.2 Microstructure prediction of hot worked steels
1.10 SUMMARY . . . . . . . . . . . . . . . ..
CHAPTER 2:
TRANSITION FROM UPPER TO LOWER BAINITE
2.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 TIME REQUIRED TO DECARBURISE SUPERSATURATED FERRITE
2.3 TIME FOR THE PRECIPITATION OF CEMENTITE
2.3.1 An empirical method . . . . . . .
2.3.2 A n independent calculation ....
2.3.3 Parameters for the Avrami Equation
2.3.4 Calibration for t9 ....••..
2.4 DISCUSSION . . . . . . . . . . . . .
2.4.1 Comparison between calculated and observed Ls temperature
IV
1
2
3
3
4
5
5
6
6
6
7
8
13
14
15
16
17
18
18
18
19
19
19
20
20
21
22
22
22
23
26
37
38
39
39
40
41
41
42
42
2.4.2 Comparison with the tempering of martensite
2.4.3 Other differences between upper and lower bainite
2.5 CONCLUSIONS.
2.6 APPENDIX .
44
45
45
46
CHAPTER 3:
A MODEL FOR THE MICROSTRUCTURE OF SOME ADVANCED BAINITIC STEELS
3.1 INTRODUCTION . . . . . . . . . 63
3.2 CARBIDE-FREE BAINITIC STEELS 63
3.2.1 Thermodynamics . . . . . . . 64
3.2.2 Kinetics 65
3.3 CARBIDE PRECIPITATION FROM BAINITIC FERRITE 66
3.4 CEMENTITE PRECIPITATION FROM SUPERSATURATED AUSTENITE . 67
3.4.1 Effect of alloying elements on cementite growth ..... 67
3.5 STABILITY OF AUSTENITE AND EFFECT ON PROPERTIES 68
3.5.1 Ductility 70
3.6 SUMMARY. . . . . . . . . . . . . . . . . . . . . . . . 71
CHAPTER 4:
REAUSTENITISATION FROM A MIXTURE OF BAINITE AND AUSTENITE
4.1 INTRODUCTION .
4.2 EXPERIMENTAL PROCEDURE
4.3 BAINITE TRANSFORMATION
4.3.1 Bainite transformation in the Fe-O.3C-4.08Cr alloy
4.3.2 Bainite transformation in Fe-2.0Si-3.0Mn system
4.4 ISOTHERMAL REAUSTENITISATION FROM A MIXTURE
OF BAINITE AND AUSTENITE
4.4.1 Microstructural study . . . . . . . . . . . . . .
4.4.2 Dilatometry . . . . . . . . . . . . . . . . . .
4.4.3 Equilibrium study of the maximum degree of reaction
4.5 CONTINUOUS HEATING REAUSTENITISATION
4.5.1 Dilatometry . . . . . . . . . . . . .
4.5.2 Decomposition of austenite during heating
4.6 CONCLUSIONS .
CHAPTER 5:
PEARLITE TRANSFORMATION IN STEELS
5.1 INTRODUCTION .
5.2 DECOMPOSITION OF RESIDUAL AUSTENITE DURING HEATING
5.2.1 TTT curve calculation of untransformed austenite . . . . . .
5.2.2 Kinetic calculation of decomposition start temperature ....
5.3 FORMATION OF PEARLITE BELOW THE Bs TEMPERATURE.
5.4 GROWTH RATE OF PEARLITE . . . . . . . . . . . . .
5.5 CALCULATION OF THE INTERFACE COMPOSITIONS ..
5.6 CALCULATION OF THE GROWTH RATE OF PEARLITE .
v
80
80
81
82
85
87
87
88
89
93
93
93
96
127
127
127
129
129
130
133
136
5.7 CONCLUSIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
CHAPTER 6:
REAUSTENITISATION ACCOMPANIED BY NUCLEATION OF AUSTENITE
6.1 INTRODUCTION . . . . . . . . ..
6.2 EXPERIMENTAL PROCEDURE ...
6.3 ISOTHERMAL REAUSTENITISATION
6.2.1 Reaustenitisation from a martensitic microstructure
6.2.2 Reaustenitisation from mixtures of ferrite and carbide particles
6.3 CONTINUOUS HEATING REAUSTENITISATION
6.3.1 Dilatometry . . . . . . . . . . . .
6.3.2 Tempering of martensite during heating
6.4 CONCLUSIONS .
CHAPTER 7:
THERMODYNAMICS OF REAUSTENITISATION
7.1 INTRODUCTION .
7.2 RECONSTRUCTIVE FORMATION OF FERRITE FROM AUSTENITE
7.2.1 Reconstructive growth of ferrite in multi-component system
7.2.2 Calculation of tie lines in ternary systems . . . . .
7.3 THERMODYNAMICS OF REAUSTENITISATION .....
7.3.1 Local equilibrium conditions during reaustenitisation
7.3.2 Reconstructive growth of austenite under local equilibrium
7.3.3 Reconstrttctive growth of austenite under paraequilibrium
7.3.4 Special cases .... ..
7.4 CALCULATED EXAMPLES . . . . . . . . . . . . . . .
7.4.1 Thermodynamic parameters . . . . . . . . . . . . .
7.4.2 Growth of austenite from a mixture of bainitic ferrite and austenite
7.5 CONCLUSIONS.
7.6 APPENDIX .
CHAPTER 8:
TIME- TEMPERATURE- TRANSFORMATION CURVES
AND OVERALL REAUSTENITISATION IN STEELS
8.1 INTRODUCTION .
8.2 ASSESSMENT OF SCHEIL'S RULE IN REAUSTENITISATION
8.3 GROWTH OF AUSTENITE . . . . . . . . . . . . . . ..
8.4 AUSTENITE FORMATION INCLUDING NUCLEATION AND GROWTH
8.5 CONCLUSIONS.
8.6 APPENDIX
CHAPTER 9:
FURTHER WORK
Vi
157
157
157
157
162
163
163
164
164
196
196
196
197
199
199
201
204
205
210
210
211
214
217
232
232
232
234
236
237
246
NOMENCLATURE AND ABBREVIATIONS
Braces are used exclusively to denote functional relations; D{ x} thus implies that x is an
argument of the function D. The bases of log{ x} and In{ x} are 10 and e respectively.
a
a*,
~am
ACl
Ael
Ae3
Ae~
Ar3
Al
A2
b
bcc
Bs
Bl
B2
C
C
CN1-N4
CHT
Co
di
D
D
D!!OI.
'J
DB
fcc
9
~g:
~g1301.
G·,
Activation of carbon at a point A in an isothermal section of the phase diagram
Starting thickness of a plate of bainitic ferrite
Lattice parameters of austenite and ferrite at ambient temperature as a function
of chemical concentration
Lattice parameters of austenite and ferrite at the reaction temperature
before the reaction
Lattice parameter of austenite at reaction temperature at any stage of the
reaction
Volume of cementite unit cell which is equal to a9b9c9; a9, b9 and c9 are 4.51,
5.08 and 6.73 A at ambient temperature respectively
Minimum detectable increase in austenite thickness
Temperature at which transformation from ferrite begins during heating
Boundary separating the (1' and (1' + 'Y phase fields
Boundary separating the 'Y and (1' + 'Y phase fields
Paraequilibrium boundary separating the 'Y and (1' + 'Y phase fields
Temperature at which transformation from austenite begins during cooling
The ratio between x~' and xIOI.
The ratio between x~' and x;OI.
Coefficient of grain diameter in equation 1.27
Body centered cubic
Bainite-start temperature
Constant equal to (x;OI. - x~')/(xIOI. - x~')
The ratio between y~ and x;9
Initial length of bainitic ferrite plate
Carbon
Constants in equations 5.2 - 5.4
Con tinuous- heating- transformation diagram
Constant in equation 5.32
Average diameter of i- th phase
Diffusivity of carbon in austenite, D = DIl
Weighted average diffusivity of carbon in austenite
Diffusivity of i-th element in austenite or in ferrite
(i = 1 : carbon, i = 2 : substitutional alloying element)
Inter diffusion coefficient in austenite or in ferrite
Boundary diffusivity of a substitutional alloying element
Face centered cubic
Geometrical factor of pearlite equal to 0.72 in plain carbon steels
Activation energy of the interface process
Driving force for the transformation per atom
Free energy of each element, i = 0, 1 and 2 correspond to iron,
carbon and the substitutional alloying element
Free energy of cementite
Free energy of M3C type carbide
Chemical free energy change accompanying the formation of 1 mole
vu
b..G(}OI.
b..Gf
b..G~
b..G?OI.-+-Y,
h
Hi
Hp
Hv
H {t}
H {Dii}
Ho
b..H{t}
J(
kA,kAO
kokp
k(}
kc-y,kcOl.
kx-y, kxOl.
J(p
L'·,
Ls
LOI.' L-y
b..L/ L
b.. Li
b..Lm
I/L
Mp
Ms
of nucleating phase in a large amount of matrix phase
Maximum volume free energy change accompanying the formation
of a nucleus in a large amount of matrix phase
Driving force for nucleation of cementite from ferrite
Activation energy for nucleation of cementite
Driving force for nucleation of cementite
Standard free energy change between Q' and I for each element, Jmol-
l
Heating rate or alternatively Plank constant
Hardness of i-th phase
Hardness of tempered martensite just before the onset of recovery or coarsening
of cementi te particles
Micro-hardness of each phase
Hardness of martensite at time t during tempering
Kinetic function
Initial hardness of martensite
Change in hardness of martensite due to cementite precipitation during
tempering
Change in enthalpy between the parent and product phases
Iron (i = 0), carbon (i = 1) or substitutional alloying element X (i = 3)
Nucleation rate per unit volume
Flux of carbon (i = 1) or substitutional alloying element (i = 2)
Boltzmann's constant
Constant in equation 4.4
Constants in equation 4.11,
or alternatively k
2
is the ratio between alloy concentrations and iron concentration
Boundary segregation coefficient which is the ratio between alloying element
concentration in austenite near the boundary and that in the boundary
Rate constants in an Avrami type equation
Coefficients of El and I
p
in the equation of the yield strength of martensite
Constant in equation (4.4)
Constants of diffusion distance of car bon
Constants of diffusion distance of substitutional alloying element
Partition coefficient of substit utional alloying element between cementite
and ferrite in pearlite
Constant equal to 47rQ'~NoVl/3 in equation 8.13
Average distance between a particle and its two or three nearest neighbours
Diffusion distance of substitutional alloying element in ferrite and austenite
y coordinate of a point i in the temperature versus. relative length change
diagram
Temperature corrected x coordinate of a point i in the length change versus
temperature diagram
Lower bainite-start temperature
Diffusion distances of carbon in ferrite and austenite
Relative length change in dilatometry
The maximum relative length change without the temperature correction
The maximum relative length change with the temperature correction
Number of intercepts of austenite/ferrite interface per unit length of test line
Martensite-finish temperature
Martensite-start temperature
VUl
nNV
Ne
NPLE
p
P.,
PE
PLE
q
Q
QB
QQ,Q'
Q1
r
S,SQ,Se
se
5
5a,T'
5Ae3
5T,o
5v
51,52
Number of austenite plates per unit volume which can grow during
reaustenitisation from mixtures of bainite and austenite,
or alternatively time exponent of Avrami equation
Number of austenite particles in the starting microstructure
Decrease in the number of austenite particles which can grow due to impingement
Number of austenite particles
Initial nucleation site density for reaustenitisation from martensitic initial
microstructure
Number of atoms per unit volume which are on dislocation lines
Number of cementite particles at the completion of precipitation
Negligible partitioning local equilibrium
Peclet number, or alternatively exponent constant in equations=5.1 and 5.4
An increment of incubation period spent in a small period of time .6.t
Paraequilibrium
Partitioning under local equilibrium
Increase in the half-thickness of austenite
Effective activation energy for cementite precipitation
Constant in equation 5.32
Constants in equations 5.2 - 5.4
Activation enthalpy for diffusion
Average cementite particle radius
Critical radius for growth at which the concentration difference in the matrix
vanishes
Radius of curvature at the advancing tip of the plate
Average cementite particle radius before the onset of Ostwald ripening
Universal gas constant
Interlamellar spacing, respective thickness of ferrite and cementite lamellar
Critical spacing at which the growth rate becomes zero
Average carbon concentration in ferrite
Constants in equations 5.2 - 5.4
Slope of the Ae3 phase boundary
Slope of the T~ phase boundary
Effective austenite/ferrite interface area per unit volume
Functions of p as presented graphically in Trivedi's paper (ref. [46] in chapter 2),
or alternatively 51 is activation entropy for diffusion
Spike width of the substitutional alloying element in ferrite
Time
Time required for the carbon concentration in ferrite to drop to a specified level
Time required to decarburise a plate of bainitic ferrite
Time at which reaustenitisation starts on heating
Time required to obtain a detectable amount of epsilon carbide in bainitic
ferrite
Time required to obtain a detectable amount of cementite precipitation
in bainitic ferrite
Time interval spent by a sample at a temperature
Absolute temperature
Bainite or acicular ferrite transformation temperature
Temperature at which the decomposition of residual austenite starts
on continuous heating
lX
TA
TE
TS,TF
TTT
T-y1,-Y2
T-y
8T
t:.T
v
vd
VB
Vc
VE
VI
Vv
V9
V
V01,'1,9
VOl.'
V010,'10,90
Vo
WO,i
X~b,
X Ae3
X Ae~
XTo
XT~
X
OIe
I
Ambient temperature
Eutectoid temperature
Transformation-start and -finish temperatures during continuous heating
Time-temperature-transformation diagram
Reaustenitisation-start and 100% reaustenitisation-start temperatures for
bainite + austenite initial microstructure
Temperature at which austenite and ferrite of the same composition
have equal free energies
Temperature at which austenite and ferrite (with a stored energy of 400 J mol-I)
of the same composition have equal free energies
Austenitising temperature
Undercooling below the eutectoid temperature
An increment of temperature during continuous heating
Growth rate of precipitate phase
Rate of carbon concentration dilution in austenite
Boundary diffusion controlled growth rate of pearlite
Velocity of a flat interface which is controlled by interface kinetics only
Edgewise growth rate of cementite plates
Growth rate of pearlite controlled by the interface process
Volume diffusion controlled growth rate of pearlite
Volume of cementite
Partial molar volume fraction of solvent in cementite
Volume fraction of ferrite, austenite and cementite at any stage of the reaction
Volume fraction of martensite
Initial volume fraction of ferrite, austenite and cementite
Initial volume fraction of a pre-existing austenite plate with the thickness ao
and the length Co
Maximum volume fraction of bainite obtained at a temperature
Calculated volume fraction using To concept
Optically measured volume fraction
Volume fraction of film type retained austenite
Volume fraction of blocky type retained austenite
Maximum volume fraction of cementite at a temperature
Increase in volume fraction of austenite
Iron-X interaction coefficients
Iron (i = 0), carbon (i = 1) or the substitutional alloying element (i = 2)
concentration in phase a at an interface between phases a and b which are
in equilibrium or paraequilibrium
Iron (i = 0), carbon (i = 1) or the substitutional alloying element (i = 2)
concentration in ferrite, austenite or cementite
Average iron (i = 0), carbon (i = 2) or the substitutional alloying element
(i = 2) concentration in the steel
Average iron (i = 0), carbon (i = 2) or the substitutional alloying element
(i = 2) concentration in ferrite, austenite or cementite
Carbon concentration given by the Ae
3
curve
Carbon concentration given by the Ae~ curve
Carbon concentration given by the To curve
Carbon concentration given by the T~curve
Equilibrium carbon concentration in ferrite
x
x· .
',J
xT.
',J
x
Yi
Y
z
z*
0'
0"
f1
fij
".,
".,7''1
()
/La
/Li
/LFe3 C
/LM3C
V
V
1
ee
e {t}
Carbon (i = 1) or the substitutional alloying element (i = 2) concentration
on the transition line from PLE to NPLE regime in an isothermal section
of the phase diagram
Carbon (i = 1) or the substitutional alloying element (i = 2) concentration
of the steel at the j-th iteration to obtain a PLE tie line (see Fig. 7.4 b)
Carbon (i = 1) or the substitutional alloying element (i = 2) concentration
on the transition line from PLE to NPLE regime at the j-th iteration
to obtain a PLE tie line (see Fig. 7.4 b)
Substitutional alloying element
Normalized i- th element concentration in cementite
Mechanical property
Co-ordinate normal to transformation interface,
or alternatively constant in equations 5.2-5.4
Position of interface along co-ordinate z
Ferrite
Martensite
Bainitic ferrite
Lower bainite
Upper bainite
One dimensional parabolic thickening rate constant
Three dimensional parabolic rate constant
Aspect ratio of cementite plates,
or alternatively parabolic lengthening rate constant which is assumed to be
30'1 in the test
Thermal expansivities of ferrite, austenite and cementite
Austenite
Capillarity constant
Activity coefficient of i-th element
Boundary thickness
Epsilon carbide
Average transverse thickness of martensite cell structure
Wagner interaction parameters
".,-carbide
Growth rate constants for i-th element in ferrite or austenite
Cementite, or alternatively the maximum degree of reaction
Interface kinetics coefficient at the top of the plate in equation 2.26
Chemical potential of i-th element
Chemical potential of cementite
Chemical potential of M3C carbide
Characteristic frequency of the jump event of atoms at interfaces
Rate constant in the nucleation function
Extended volume fraction of precipitating phase
Volume fraction of precipitating phase at time t normalised by the
maximum degree of the reaction
Increase in e
Dislocation density
Either tensile or yield strength
Solid solution strengthening effect of carbon on the intrinsic strength of martensite
Yield strength of martensite
Xl
(To
(T~
(Too,(Toj3
Intrinsic strength of martensite
Residual strength equal to ((To - (Te)
Interface energies per unit area between two matrix grains and between
matrix and precipitating phase
Change in yield strength of martensite
Incu bation time prior to the formation of a detectable degree of transformation
product
Incu bation time at tern perat ure T for the C-curve representing a fraction
f of the reaction
Fraction of austenite in bainitic sheaf
Dimensionless supersaturation parameter
Xll
CHAPTER 1
REAUSTENITISATION IN STEELS
1.1 INTRODUCTION
A knowledge of the factors controlling the kinetics of austenite nucleation and growth is of
interest both from a fundamental point of view, and because most commercial processes rely
to some extent on heat treatments which cause the steel to revert into the austenitic condition.
The reverse transformation of low temperature (Cl') ferrite into austenite differs in many
ways from the more usual case where the parent phase, which is stable at elevated temper-
atures, transforms on cooling below the equilibrium temperature. This is highlighted by an
examination of the temperature dependence of kinetics for the two cases. The vast majority of
transformations exhibit classical C-curve kinetic behaviour, in which the overall transformation
rate goes through a peak as a function of supercooling below the equilibrium transformation
temperature. This can be interpreted to be a consequence of two opposing effects; diffusion
coefficients decrease with temperature, whereas the driving force for transformation increases as
the under cooling increases. The kinetics of austenite growth from ferrite depend on the degree
of superheating above an equilibrium transformation temperature. Both the diffusion coefficient
and the driving force increase with superheating, so that the overall rate of transformation rises
monotonically as the temperature is raised.
This in turn leads to several interesting effects. It is a common feature of normal recon-
structive transformations, that they can be suppressed by rapidly cooling into a temperature
regime where the rate of transformation becomes unreasonably small, when compared with the
service life of the component concerned. Hence, austenite can in many steels (e.g., 18/8 stain-
less steel) be retained by rapid cooling, even though it ceases to the thermodynamically stable
phase at ambient temperature. It should be impossible to retain similarly the ferrite phase
to high temperatures during a reaustenitisation experiment where only austenite is the stable
phase at the elevated temperature, since atomic mobility always increases with temperature.
In steels which are lightly alloyed, rapid cooling from the austenite phase field does not al-
ways lead to the retention of the austenite. However, its decomposition may then be suppressed
to very low temperatures so that any transformation cannot rely on substantial atomic mobil-
ity. The austenite may then decompose by a displacive mechanism, such as the martensitic
transformation. During reverse transformation however, the temperatures involved are usually
high enough to permit the rapid reconstructive transformation of austenite, and it is rare that
the austenite grows by a martensitic mechanism from the ferrite.
In compendiums of time-tempera tu re-transformation (TTT) diagrams for steel, the kinetics
of austenite decomposition are usually presented as a function of alloy chemistry and the amount
ofaustenite grain surface per unit volume (i.e. the austenite grain size). The number of variables
to be considered when presenting similar data for the reverse transformation to austenite is
much larger, since the initial microstructure can vary widely. It may consist of a mixture
of any of the usual transformation products of austenite. The degree of sophistication with
which it is necessary to specify the characteristics of the starting microstructure remains to
be determined, but factors such as particle size, the distribution and chemistry of individual
phases, homogeneity of the microstructure, the presence of nonmetallic inclusions, etc. should
all be significant.
1
From an industrial point of view, any attempt at the estimation of microstructure of
steels calls for a knowledge of the kinetics of reaustenitisation as a function of alloy chemistry,
thermomechanical processing and the starting microstructure. Since such information is rarely
available, there are few examples where the theory of phase transformations has been applied
quantitatively towards the design or optimisation of commercial processes.
It is possible to think of many examples where detailed knowledge on the reaustenitisation
process could be applied to considerable advantage. For example, one of the most difficult
and widely used processes is that of welding, where unlike most commercial routes, an opti-
mum microstructure is expected immediately after deposition from the liquid state, without
the luxury of homogenisation or thermomechanical treatments after deposition. Furthermore,
the welding process itself dissipates heat into the adjacent parent plates being joined, thereby
influencing its microstructure, often in a detrimental manner. Any attempt at the estimation
of microstructure in the fusion or heat-affected zones of such welds requires either a knowledge
of, or a method of computing the TTT diagram for the transformation of a specified initial
microstructure into austenite. It would be essential to have such information as a function of
the chemistry of both the weld deposit and of the parent plate, since they are unlikely to have
the same composition. A facility must also exist for converting the isothermal transformation
diagrams into continuous-heating transformation (CHT) diagrams, since most industrial pro-
cesses involve anisothermal heat-treatments. During welding, regions of the parent plate and of
the fusion zones of multirun welds undergo transient temperature rises, which sometimes take
the alloy into the Q' + 'Y phase field or beyond, causing profound changes in microstructure by
the time the weld has cooled again to ambient temperature. In multirun welds, each region can
be expected to undergo several such thermal cycles.
The purpose of this review is to collate and assess the work that has already been reported
in the literature on reaustenitisation. However, an attempt is also made to use the opportunity
to consider how the research can be extended in a systematic way, so as eventually to provide
a framework for the estimation of microstruct ure in steels.
1.2 NUCLEATION OF AUSTENITE
The formation of austenite in steels is well established to be a nucleation and growth process
[1]. The nucleation site, growth rate and morphology of the austenite that forms is sensitive to
the initial microstructure, as well as the chemical composition and reaction temperature. As
pointed out earlier, this contrasts with the decomposition reactions of austenite in steels, where
the initial austenite grain size and chemical composition are usually the only important factors
which need to be represented in TTT diagrams.
Much of the research on the role of the initial microstructure on the morphology of the
austenite during growth, points towards two main types; in one case, the austenite is said to
be "acicular", the term meaning needle-like, but implying a plate shape in three dimensions.
Acicular austenite is often said to form intragranularly with respect to the prior austenite grain
structure. The other common observation is of "globular" austenite in which the new phase
seems to grow more or less isotropically, the nucleation sites often being the prior austenite
grain boundaries. Most of these observations are based on reaustenitisation experiments in
which the initial microstructures consisted of pearlite, martensite or mixtures of ferrite and
carbide particles. There are very few data for initial microstructures which are completely
ferritic or bainitic.
2
1.2.1 Initial Microstructure: Ferrite and Pearlite
Reaustenitisation beginning with a mixture of ferrite and pearlite is important in the
production of dual phase steels which have a final microstructure of ferrite and about 20%
martensite. These steels have a good combination of strength and uniform ductility, and find
applications in the automobile industry.
When a fully pearlitic steel, or one containing pearlite and allotriomorphic ferrite is heated
to induce reverse transformation, the austenite nucleates heterogeneously at the junctions be-
tween pearlite colonies [1]' or at the interfaces between the grains of allotriomorphic ferrite and
pearlite colony boundaries [1-5]. Garcia and DeArdo [3] studied the formation of austenite in
a series of 1.5Mn wt.% steels which were annealed in and above the intercritical region; the
intercritical region is the part of the phase diagram where mixtures of austenite and ferrite can
be in thermodynamic equilibrium. The austenite was found to nucleate on cementite particles
located on either pearlite colony boundaries or boundaries separating pearlite colonies and fer-
rite grains. A similar result was reported by Speich et al. [2] for austenite formation in eutectoid
steels. This is in spite of the relatively large amount of inter-lamellar surfaces available within
the pearlite colonies, which seem to be much less effective as sites for the nucleation of austen-
ite [2]. Sharma et al. [6] have reported interlamellar spacings for pearlite in Fe-0.74C-OACr
wt.% alloy; using their data, we estimate that the interlamellar surface per unit volume is some
30-600 times larger than the amount of inter-pearlite-colony surface per unit volume. A full
explanation of why the latter kind of surface is more effective in nucleating austenite is not
yet available. It may be the case that the allotriomorphic ferrite/pearlite colony junctions, or
junctions between pearlite colonies represent relatively high energy interfaces, in which case the
free energy gained as such junctions are eliminated by the formation of nuclei would tend to
proportionately reduce the activation free energy for nucleation. Of course, the activation free
energy would have to be sufficiently small to overwhelm the higher site density associated with
the inter-lamellar boundaries.
1.2.2 Initial Microstructure: Ferrite and Cementite
For an initial microstructure of a ferrite mixed with isolated cementite particles, the parti-
cles are usually located either at ferrite/ferrite grain boundaries or within ferrite grains. When
low-carbon martensite is heavily tempered, the resulting cementite particles tend to locate
at the ferrite/ferrite grain boundaries. However, if a martensitic specimen is tempered, cold
worked and then recrystallised by tempering, then most of the cementite particles are dispersed
in the ferrite matrix away from the ferrite/ferrite grain boundaries [2]. The effects of these
different dispersions of cementite particles on the nucleation of austenite has been examined in
many investigations [2-4,7]. Austenite is found to nucleate predominantly at junctions between
cementite particles and ferrite/ferrite grain boundaries, rather than on the particles located
within the ferrite grains. Speich et al. [2] suggested that this is because an additional surface
free energy is available from the 0:/0: grain boundary when austenite nucleates on a cemen-
tite particle located at such a boundary. It is however, difficult to guess which of the three
different sites, i.e. Fe3C/0:/0: grain boundaries, grain edges and corners, is the most effective
nucleation site for austenite, given uncertainties in the values of the interface energy terms
for ferrite/carbide, austenite/carbide and ferrite/austenite boundaries. A knowledge of these
interface energies as a function of crystallography and chemical composition is also required for
a more complete description. Experimental data on three phase orientation relationships do
not seem to be available.
3
If the ferrite grain size is very large, the density of sites available at Fe3C/a/a junctions
can become very small compared with the density of cementite particles within the ferrite
grains, and nucleation on the latter sites might be expected to dominate even though the
activation energy is higher. Consistent with this, Lenel and Honeycombe [4] have reported the
intragranularly nucleation of austenite when the initial ferrite grain size is very large.
1.2.3 Martensitic Initial Microstructure
The formation of austenite from an initially martensitic microstructure is of obvious rel-
evance to high-strength steels, but also to dual phase steels with a microstructure which is
a mixture of martensite and allotriomorphic ferrite. Martensite is in general a very unstable
phase, especially so in iron alloys containing interstitials such as carbon or nitrogen. In such
alloys, the heating that is necessary to induce the reverse transformation may first lead to a
tempering of the martensite and eventually, to its decomposition into virtually carbon free fer-
rite and cementite or other carbide phases. The process of reaustenitisation from martensite in
steels can therefore be complex, and there are interesting differences that arise when the iron
alloy does not contain any interstitial alloying elements.
1.2.3.1 M adensite in steels
In experiments on reaustenitisation from martensite, the nucleation of austenite is found
to occur preferentially at the prior austenite grain boundaries [8-10]. Intragranular nucleation
with respect to the prior austenite grain boundaries, has been also reported, although it is not
clear whether such nucleation has been confirmed using stereological analysis [11-13]
The effect of alloy chemistry on the nucleation site of austenite has been studied by Plichta
and Aaronson [12] and Koo and Thomas [13]. Plichta and Aaronson [12] classified the sites in
two groups. In ternary steels containing Mo or Cr, the main nucleation site was found to be
the prior austenite grain boundaries. On the other hand, in ternary steels containing Mn, Ni,
Cu, Si, Al or Co, the martensite plate boundaries were found to be the the predominant sites
for the nucleation of austenite. There does not seem to be any explanation for this effect.
Martensite in steels tempers rather rapidly at elevated temperatures; carbon in iron even
has significant mobility at ambient temperatures [14,15]. Hence, when reaustenitisation is car-
ried out at high temperatures, the austenite forms from a spheroidised microstructure rather
than from martensite proper [8]. Another factor to consider is the possible existence of retained
austenite trapped between martensite laths, which may persist to the reaustenitisation tem-
perature if the heating rate to that temperature is high enough to prevent its decomposition
during heating [16]. Any retained austenite particles which survive to the reaustenitisation
tern peratures, can then grow, so that the nucleation of austenite is not required.
1.2.3.2 Martensite in interstitial-free iron alloys
The reverse transformation from marte'nsite in interstitial-free alloys can be classified fur-
ther into cases where the martensite behaves in a thermoelastic manner, and those in which
the transformation tends to be less reversible. The lack of reversibility in the latter cases is
usually attributed [17] to a loss of interface coherency during the growth of the martensite
itself, due to any plastic accommodation effects accompanying the martensitic transformation.
When the martensite is then heated, the plates do no shrink by the backwards movement of the
austenite/martensite interface, but small plates of austenite grow by a displacive transforma-
tion mechanism within the martensitic regions. This was demonstrated by Kessler and Pitsch
[18] using surface relief and light microscopy observations on an Fe-33Ni wt.% alloy. They also
4
showed that each martensite plate tends to transform into several different crystallographic
variants of austenite, so that the original austenite grain structure is not reproduced. On the
other hand, such a mechanism could be used to refine the austenite grain size, by repeated
cycling between the MF and TF temperatures.
For the austenitic transformation discussed above, it is evident that the reverse displacive
transformation of a single plate of austenite can lead to several orientations of austenite plates.
This contrasts with thermoelastic martensites, which are usually found in ordered alloys, the
ordering promoting crystallographic reversibility by making the reverse transformation path
unique [17]. Consequently, a single crystal of martensite would then be expected to trans-
form back to a unique orientation of austenite. The first thermoelastic ferrous martensite was
reported by Dunne and Wayman [19,20] in ordered Fe-Pt alloys, which in the disordered condi-
tion showed nonthermoelastic behaviour with considerable hysteresis between the Ts and Ms
temperatures.
The shape memory effect in thermoelastic martensites is a consequence of the reversibility
of the martensite/ austenite transformation. The martensite which forms when a single crystal
of the parent phase is cooled, tends to grow as clusters of self-accommodating groups. Under
the influence of stress, at temperatures below MF' the glissile inter-variant interfaces are able to
move in a way which accommodates the externally applied stress. Thus, some crystallographic
variants of martensite grow at the expense of others. Under suitable conditions, a single orienta-
tion of martensite may be left, which when heated, converts back into the original single crystal
of austenite [21]. The transformation back into the original single crystal of austenite is a con-
sequence of the fact that there is in these alloys, a unique reverse correspondence between the
martensite and austenite, even though there may be several forward lattice correspondences so
that each austenite grain can transform into several martensite crystals [22]. These arguments
can be extended to polycrystalline materials as well, although the details might be different
[22].
1.2.4 Ferritic Initial Microstructure
Little work has been done on reaustenitisation of specimens which are fully ferritic, because
of their extremely low harden ability. This makes it almost impossible to freeze the austenite
microstructure to ambient temperature, even in the form of martensite. Speich et al. [2] studied
the formation of austenite in a low carbon steel using a laser-pulse heating and a helium-water
droplet spray technique, which is capable of achieving a heating rate of '" 106QCS-l and a
cooling rate of", 105QCS-l. They showed that austenite initiated at the ferrite grain boundaries
and during the quench to ambient temperature, each austenite grain transformed back to ferrite
which had a much finer grain size when compared with the original ferritic microstructure.
1.2.5 Bainitic Initial Microstructure
The formation of austenite from bainitic microstructures such as bainite and acicular fer-
rite has been studied by Nehrenberg [23]' Matsuda and Okamura [9]' Law and Edmonds [10].
For a bainitic initial microstructure, Law and Edmonds [10] have reported that the nucle-
ation of austenite occurs primarily at the prior austenite grain boundaries. This contrasts
with a martensitic initial microstructure in the same steel, which exhibited the intragranular
nucleation of austenite during isothermal reaustenitisation at high temperatures, although the
intragranular nature of the nucleation was not established using stereological analysis.
5
1.2.6 Summary
The nucleation of austenite can, therefore, be usefully characterised by the nature of the
heterogeneous nucleation site. When the initial microstructure is either a mixture of ferrite and
cementite particles, or a mixture of ferrite and pearlite, the nucleation of austenite is found to
occur predominantly at ferrite/ferrite grain boundaries. At these grain boundaries, there may
be further preferential location of the nucleation site at positions where the boundaries have
junctions with carbide particles. These carbides may consist of isolated spheroidised cemen-
tite, or of the edges of the pearlitic-cementite lamellar where they impinge on ferrite/ferrite
boundaries. In pearlitic steels, sites where two colonies meet, giving rise to ferrite/ferrite grain
boundaries (with cementite lamellar in contact), are also known to be favourable positions for
the heterogeneous nucleation of austenite. It should be noted that when austenite forms from
martensite in steels, tempering during heating to the AC
1
temperature can make the reausteni-
tisation process identical to that for the ferrite and carbide starting microstructure. All these
results emphasise the role of high energy ferrite/ferrite grain boundaries, whereas the lower en-
ergy boundaries, such as those between martensite plates within the the prior austenite grains
do not seem to be as effective in nucleating austenite. It seems reasonable to assume that the
grain boundaries resulting from the impingement of ferrite grains growing from two different
adjacent austenite grains are likely to be of a higher energy, than those interfaces resulting from
the tempering of martensite plates formed within a single austenite grain.
Direct or indirect measurements of the nucleation rate or the nucleation density have been
carried out by Judd and Paxton [7]' Dirnfeld et al. [24] and Roosz et al. [25]. Roosz et al. [25]
measured the volume fraction of austenite formed in an eutectoid steel, and obtained a value four
for the time exponent n in an application of the Avrami equation. By expressing theoretically
the volume fraction and the amount of austenite/pearlite interfacial area per unit volume as a
function of the nucleation rate and the growth rate, and by comparison with experimental data,
they were able to determine the nucleation rate of austenite as a function of a superheating
from the Ae
1
temperature. Consequently, the nucleation and the growth rates were found to
be constant throughout the reaction.
Judd and Paxton [7] also measured the number of austenite particles formed from a mix-
ture of ferrite and cementite particles, and showed that the nucleation rate of austenite was
reasonably constant. They obtained the incubation periods for the reaction at different temper-
atures by extrapolating the linear relation between the number of nuclei and the holding time
back to zero nuclei. Dirnfeld et al. [24] also reported the change in the number of austenite
particles formed from a mixture of ferrite and cementite particles. Although the nucleation
rate at the very early stage of the reaction seems to be constant at a particular temperature,
it decreases with time when the volume fraction of austenite reaches about 0.2, because of site
saturation.
1.3 GROWTH OF AUSTENITE
1.3.1 Ferrite-Pearlite Mixtures
Speich et al. [26] studied the formation of austenite from a mixture of ferrite and pearlite,
and suggested that its growth could be divided into three stages. In the first stage, immedi-
ately after the nucleation of austenite at the ferrite-pearlite interface, the austenite grows into
pearlite until the latter is completely consumed. The growth rate during this stage is assumed
to be controlled by the rate of carbon diffusion in the austenite behind the advancing interface,
6
the diffusion path lying along the pearlite-austenite interface, and the diffusion distance being
related to the inter-lamellar spacing of the pearlite. Because of the short diffusion distance, the
growth rate during this step is expected to be rather high, with wider inter-lamellar spacings
tending to reduce growth rate; wider spacings are also known to lower the nucleation rate of
austenite [1]. When the dissolution of pearlite is complete, the austenite has a relatively high
carbon concentration but is not in equilibrium with the remaining ferrite. Consequently, in the
second stage, the austenite grows into the surrounding ferrite to achieve its equilibrium volume
fraction as specified by the lever rule in the two phase region, at a rather slow growth rate,
which at high temperatures is controlled by carbon diffusion in the austenite. In alloy steels the
growth involves either paraequilibrium or negligible-partitioning-Iocal equilibrium at the inter-
face. At relatively low temperatures, substitutional elements are required thermodynamically
to partition between the phases and growth may then occur under partitioning local equilibrium
conditions. In any event, as the austenite grows into the ferrite, the extent of the diffusion field
involved increases, causing the reaction to exhibit parabolic kinetics. Eventually, the diffusion
fields of different austenite grains may interfere, leading to soft impingement; in this, the final
stage, the growth rate is very slow indeed. The final stage may involve the homogenisation
of substitutional alloying element gradients in the austenite and ferrite, and diffusion of such
elements in austenite is particularly sluggish.
While the scheme described above is on the whole reasonable, Nemoto [27] has demon-
strated, using hot-stage electron microscopy, that as the austenite grows into the colonies of
pearlite, the lamellar of cementite may not dissolve completely at the advancing interface, but
may be engulfed by the austenite to dissolve later, as time permits. The ferritic part of pearlite
can therefore transform to austenite more rapidly when compared with pearlitic cementite.
Cementite dissolution in the austenite after the cementite has been engulfed by the austenite
has also been reported by Roberts and Mehl [1] and Nehrenberg [23].
1.3.2 Ferrite-Spheroidised Cementite Mixtures
When austenite nucleates at ferrite/ferrite grain boundaries in a mixed microstructure of
ferrite and cementite particles, the allotriomorphs of austenite eventually engulf any carbides ly-
ing on the grain boundary [2]. Subsequent growth can occur only by carbon diffusion within the
austenite envelope, from the position of the dissolving carbide to the advancing ferrite-austenite
boundary [2,4,7]. The rate of dissolution of cementite particles during reaustenitisation has been
investigated [7,28]. Judd and Paxton [7] assumed local equilibrium at all the interfaces, and
calculated the volume fraction of austenite during isothermal reaustenitisation using experi-
mentally determined incubation periods and a constant nucleation rate for the formation of
austenite. They also claimed that some of the carbides begin to dissolve in the ferrite during
the incubation period prior to austenite nucleation, thereby raising the carbon concentration
of the ferrite close to the concentration given by the extrapolated met astable phase boundaries
governing the equilibrium between ferrite and cementite. This leads to a reduction in the car-
bon concentration gradient within the ferrite to very low level, so that the flux of carbon from
isolated particles within the ferrite, to any growing austenite must be very small indeed.
Hillert et al. [28] categorised the process of austenite formation from Q' + Fe3C mixtures,
where the cementite is in the form of spherical particles isolated in a matrix of ferrite, into
five distinct regimes (Fig. 1.1). When the distance between particles of cementite is very
large (as in low carbon steels), the dissolution of each particle, after becoming engulfed by
austenite, will be controlled by the diffusion of carbon in the austenite shell (type 1). For more
7
realistic carbon concentrations, the distance between adjacent particles is not large enough
to neglect any flux arising from carbon transport through the ferrite matrix from isolated
cementite particles to austenite particles (type 2). This situation must become more prevalent
when the nucleation rate of austenite is low. At sufficiently high carbon contents, an austenite
grain containing an undissolved cementite particle may come into contact with neighbouring
cementite particles before the dissolution of the original cementite particle is complete (type
3). If the carbon content is much higher than the critical value at the Ae3 phase boundary, the
austenite continues to grow with the cementite particles inside the shell being partially dissolved,
and the dissolution continuing at some distance behind the austenite/ferrite interface, (type
4). At very high supersaturations, the formation of austenite can virtually reach completion
without the dissolution of the cementite particles, with subsequent slow cementite dissolution
occurring after the transformation of all the ferrite, (type 5). Hillert accounted for the effect
of substitutional alloying elements on the carbon flux through the austenite shell and ferrite
matrix assuming the local equilibrium at all the phase interfaces and linear gradients of chemical
composition in the two phases, and obtained a good agreement with the experimental data.
1.3.3 Bainitic Ferrite-Austenite Mixtures
When an iron-carbon alloy is heated to a temperature within the 0' + 'Y phase field until
equilibrium is established between allotriomorphic ferrite and austenite, a small rise or fall in
temperature leads to the growth or dissolution respectively, of the austenite until the volume
fractions once again satisfy the lever rule [29]. The transformation of austenite into allotriomor-
phic ferrite is in this sense reversible, and exhibits little or no hysteresis. On the other hand, for
martensite in steels, there is a large difference between the Ms and the austenite-start temper-
ature (Ts) recorded during heating. This is because the martensite tempers (or autotempers)
and because its formation does some work in the form of irreversible plastic deformation. It is
in this context that the results of reaustenitisation experiments on bainite can be interpreted.
If carbides precipitate from the austenite during the bainite reaction then a considerable
hysteresis is expected during reverse transformation which would require the renucleation of
austenite. On the other hand, a large hysteresis effect is not expected if reverse transformation
begins from an equilibrium mixture of just bainitic ferrite and austenite. Of course, if the
bainite forms by diffusionless transformation and the excess carbon is rejected into the residual
austenite subsequent to transformation, then the reaction would cease before an equilibrium
mixture of ferrite and austenite is reached, so that an increase in temperature would not lead
to an immediate reversal of transformation.
Recently, Yang [16), and Yang and Bhadeshia [30,31] have investigated the growth of
austenite in bainite or acicular ferrite microstructures. t The reaustenitisation experiments
involved the heating of mixtures of either bainitic ferrite and austenite or acicular ferrite and
austenite, obtained by isothermal transformation below the Bs temperature. The starting mi-
crostructures thus already contained austenite, which grew during the reaustenitisation process.
The problem can be studied best in steels which transform to bainite without any precip-
itation of carbides. In these circumstances, the microstructure obtained by isothermal trans-
t Acicular ferrite is similar in transformation mechanism to bainite, but because the plates of acicular ferrite
nucleate intragranularly on inclusions, the detailed morphology differs. Bainite sheaves consist of parallel platelets of
bainitic ferrite, whereas clusters of acicular ferrite consist essentially of platelets radiating outwards from the "point"
nucleation sites.
8
formation below Bs is a mixture of bainitic ferrite and carbon-enriched residual austenite. To
study the reverse transformation, the mixture can be heated to an isothermal reaustenitisation
temperature, so that the nucleation of austenite is unnecessary. Experiments like these have
shown that the reaustenitisation occurs by a diffusional process, and have established clearly
that there is indeed a large difference between the Bsand Ts temperatures. This is in spite of
the fact the starting microstructure exists in a metastable Q' + I phase field. Furthermore, the
Ts temperature is found to correspond approximately to the Ae3 temperature of the residual
austenite. The degree of reaustenitisation increases from zero at the Ts temperature, to 100% at
the austenite-finish or TF temperature, which is the Ae3 temperature of the alloy as a whole. It
should be emphasised that if bainite was simply the product of equilibrium or paraequilibrium
transformation like allotriomorphic ferrite, a rise in temperature above the isothermal bainite
transformation temperature should lead to a reversal of reaction with little hysteresis.
The observed reaustenitisation behaviour can be understood on the basis of the follow-
ing model [30,31]. In steels where carbide precipitation from austenite is relatively sluggish,
the formation of bainite ceases prematurely during isothermal transformation as the residual
austenite carbon concentration approaches the T~ curve on the phase diagram. The stage at
which reaction stops is when the carbon content of the residual austenite reaches the T~ curve
of the phase diagram (Fig. 1.2). It follows that the carbon concentration xi of the austenite
when the formation of acicular ferrite ceases at Tb' is given by:
(1.1)
as indicated by the point a Fig. 1.2. Furthermore, we note that:
(1.2)
where x Ae3 {T;} is marked b in Fig. 1.2.
Thus, although the formation of bainite ceases at Ti, because the carbon content of austen-
ite is far less than the equilibrium concentration (i. e. xi « x Ae3 {Td), the driving force for
austenite to transform reconstructively to ferrite is still negative.
This remains the case until the temperature T is high enough (i. e. T = Ts) to satisfy the
equation:
(1.3)
Hence, reaustenitisation will first occur at a temperature Ts, as indicated by the point c
III Fig. 1.2, and as observed experimentally. This is a consequence of the mechanism of the
bainite reaction, which does not allow the transformation to reach completion. If this were not
the case, then the lever rule demands that the temperature need only be raised infinitesimally
above Tb in order for the reverse Q' -+ I transformation to be thermodynamically possible.
The theory predicts that at any temperature Ty greater than Ts, the Q'b -+ I transformation
should cease as soon as the residual austenite carbon concentration (initially xT,{Tb}) reacheso
the Ae3 curve, with
xl = X Ae3 {T')'}
The equilibrium volume fraction of austenite at the temperature T')', is then given by:
9
(1.4)
(1.5)
assuming that the carbon concentration of ferrite is negligible and that x Ae3 {T-y} > Xl' When
x Ae3 {T-y} = xl' the alloy eventually becomes fully austenitic (point cl, Fig. 1.2), and if this
condition is satisfied at T-y = TF, then for all T-y > TF, the alloy transforms completely to
austenite.
This model explains why the degree of O'b -+ I transformation increases from approximately
zero at Ts (the Ae3 temperature of the residual austenite) to 100% at TF (the Ae3 temperature
of the alloy as a whole). The behaviour is a direct reflection of the fact that the composition
of the residual austenite after the bainite reaction has ceased is far below equilibrium. This
in turn provides further support for the incomplete reaction phenomenon and its implication
that the growth of bainite is diffusionless. Finally, it should be noted that the model discussed
above assumes that the carbon concentrations of both phases is uniform at all stages.
1.3.3.1 Kinetic theory: One-dimensional growth from a mixture of austenite and bainitic ferrite
When reaustenitisation is from a starting mixture of bainitic ferrite and austenite, the
transformation kinetics are relatively easy to interpret since austenite nucleation need not be
considered.
Since both bainite and acicular ferrite are in the form of thin plates, the movement of
the planar austenite/ferrite interfaces can, during the early stages of reverse transformation,
be modelled in terms of one-dimensional growth. For simplicity, it is assumed that growth is
diffusion-controlled. All the redistribution of carbon must occur within the austenite during its
growth, since the amount of carbon in the ferrite is negligible. Microanalysis experiments have
demonstrated that the reaustenitisation process involves the reconstructive growth of austenite
in the alloys studied by Yang and Bhadeshia [30,31], with substitutional elements partitioning
between the austenite and ferrite [30,31]. The extent of substitutional solute partitioning is
known to decrease with an increase in driving force (which in turn increases with T-y). The
microanalysis experiments reported were not of sufficient spatial resolution to identify the com-
positions at the transformation interface, but it is likely that local equilibrium exists at the
interface for low T-y with a tendency towards zero bulk partitioning, (i. e. negligible-partitioning
local equilibrium or paraequilibrium [32-40], as T-y increases to beyond the Ae3 temperature
of the alloy. This makes a full analysis impossible since it is not yet possible to determine
theoretically, which of these infinite possibilities, between the limits of local equilibrium and
paraequilibrium, the system chooses to adopt as a function of temperature. In other words, the
compositions of the phases at the interface cannot as yet be deduced theoretically.
If local equilibrium is achieved at least approximately at the transformation interface, then
the growth rate calculated assuming carbon diffusion-controlled motion of the I/O' interface,
using the equilibrium carbon concentrations may give a good guide to the factors influencing the
kinetics of transformation. This amounts to assuming that the effect of substitutional solute
gradients in influencing the flux of carbon is zero [36]. There is a further implicit assumption
that the tie-line (of the equilibrium phase diagram) which determines the interface cornpositions
passes through the bulk composition of the alloy. This is unlikely in substitutionally alloyed
steels [35,39], but may be a good approximation since the alloys considered here are dilute.
Finally, any effects due to soft impingement (overlap of diffusion fields) were not taken into
account, since it is only the early stages of transformation that are considered in the present
study. Hence, the austenite and ferrite are both in effect assumed to be semi-infinite in extent.
10
One-dimensional diffusion-controlled growth involves the parabolic thickening of layers of
austenite. The increase in the half-thickness of austenite can therefore be described as (see for
example, [40,41]):
q = 0'1 tl/2
dq = O.Sa
l
t-l/2dt
(1.6)
(1.7)
(1.8)
where q is the increase in the half-thickness of the austenite layer, of starting thickness a, and
0'1 is the one-dimensional parabolic thickening rate constant.
The geometry assumed for the thickening of austenite layers is based on the plate shape
of bainite or acicular ferrite. If c is the largest dimension of a bainitic ferrite plate, idealised as
a rectangular parallelepiped with sides of length a, band c, with c = b ~ a, then when both of
the sides of a ferrite plate are penetrated by the growing austenite, the total area of the, / a
interface which advances into the plate of ferrite is 2c2• This reduces the thickness of the plate
by tlam/2 from either side. If the minimum detectable change in volume fraction is tlV1" then
it follows that:
t::..a",,/2
tl V1' = 2nc2 J dq
o
where n is the initial number of particles of austenite per unit volume, and tlam is the minimum
detectable thickness increase.
On combining equations 1.7 and 1.8, we get:
T
tl V1' = 2nc2 J O.Sal el/2dt
o
where T is the time taken for the minimum detectable transformation.
After integration, this becomes:
so that
(
tlV )2
T- l'
- 20'1nc2
However,
2 -
2nc = Sv = 2/L
so that
(1.9)
(1.10)
(1.11)
(1.12)
T_(tlV1')2
alSV
where Sv is surface area of ,/ a boundary per unit volume, and 1/L is the number of intercepts
of ,/0' boundary per unit length of test line [42]. It is clear from equation 1.12, that the value
of T is dependent on not only the parabolic rate constant 0'1 but also the surface area of ,/0'
interface per unit volume Sv for a specific amount of reaustenitisation.
For the same starting microstructure and a specific amount of transformation, T decreases
rapidly as the isothermal reaustenitisation temperature increases due to the increase in al'
Equation 1.12 also indicates that the morphology of the starting microstructure will effect T,
because Sv must depend on the detailed nature of the initial microstructure. This explains the
11
different rates at which the O'b + 'Y and O'a + 'Y reaustenitise; the distribution of plates in an
acicular ferrite microstructure is such that Sv is smaller than that of a bainitic microstructure,
so that it reaustenitises at a slower rate. The analysis shows that for a specified amount of
reaustenitisation, and a fixed initial microstructure,
(1.13)
1.3.3.2 Estimation of the Parabolic Thickening Rate Constant
The parabolic rate constant 0'1 can be deduced by analogy with already existing theory
for the 'Y ---+ 0' transformation [40,43,44]. Fig. 1.3 shows the carbon concentration profiles in 0'
and 'Y before reaustenitisation and during austenite growth; the austenite must become more
dilute in carbon as it grows, the rate of interface motion being determined by the diffusion of
carbon in the austenite behind the interface. In Fig. 1.3, the xl is the carbon concentration
in the austenite before the start of reaustenitisation; it is given by xl = xT'; for Tb = 4600 C,
o
xT' = 0.01235 mole fraction of carbon (Fig. 1.3). In Fig. 1.3, the carbon concentration of 'Y at
o
'Y/0' interface during reaustenitisation is xf:>t, and the carbon concentration of'Y far away from
the interface remains xl. It is assumed that the carbon concentration of 0' remains the same,
x~'Y, before and during reaustenitisation. The coordinate z is defined normal to 'Y / 0' interface.
During reaustenitisation the flux of carbon in the austenite, towards the 'Y /0' interface, at
the position of interface can be expressed as:
J = _D{x'Ya} (8xI)
1 8z
(1.14)
where the use of braces implies a functional relationship, i. e. D{ x la} implies that the function
D is evaluated at the concentration xla. The diffusion coefficient of carbon in austenite,
D, is known to be strongly concentration dependent [45-49]. We assume that a weighted
average diffusivity, D, can adequately represent the effective diffusivity of carbon [50] in the
concentration gradient; it is given by:
X "''I
I
D = J Ddx1/(X~'Y - xi)
-'I
XI
The rate at which carbon concentration of austenite is diluted can then be written as:
(1.15)
(1.16)
where v is the velocity of interface.
Given that the position z* of the interface along the coordinate z is defined by the equation:
z* - 0' t1/2- 1 ,
it follows that:
(1.17)
Combining equations 1.16 and 1.17, the rate at which carbon concentration of'Y is diluted can
be expressed as:
(1.18)
12
Conservation of mass at the interface requires that (i.e., combining equations 1.14 and 1.18):
(1.19)
The concentration gradient axIlaz* in equation 1.19 is evaluated at the position of the interface,
i. e. at z = z*. Equation 1.19 simply states that the rate of dilution of the austenite, per unit
of time, equals the carbon flux towards the I/O' interface. From Fick's laws, the differential
equation for the matrix is given by:
ax 'Y
__ 1 _
at (1.20)
subject to the boundary condition xi = xil> at z = z{ t}, and xi = xi at t = 0, and equation
1.19. It can be solved [40,43,44,51] to give an implicit relation for 0'1 as a solution of the form:
(1.21)
(1.22)
where
H{D} = (O.~~)0.50'1 [erfc{ O;o~;}]exp{ :L}
Finally the parabolic rate constant 0'1 can be calculated using equations 1.21 and 1.22 with
the diffusivity calculated following Bhadeshia [49].
1.3.3.3 The Relation between the Parabolic Rate Constant and TTT Curves
Consistent with the theory discussed earlier, Fig. 1.4 and Fig. 1.5 [18] show that log{ T} is
found experimentally to be proportional to log{ 0'12}, the linear correlations in all cases being
extremely good. It is, however, noted that the slope, which is expected to be the unity from
the theory discussed earlier [40,41], appears not to be equal to unity.
1.3.4 Reaustenitisation from Cementite and Bainitic Ferrite
The earliest reported work on reaustenitisation from bainite containing cementite seems
to be that of Nehrenberg [23]; the morphology of austenite which grew from high-temperature
transformation products, such as pearlite, was found to be more or less equiaxed in shape.
On the other hand, the austenite particles formed by the transformation of martensite or
bainite, was found to assume an "acicular" morphology. The acicular morphology seemed to
be generated by the growth of austenite between bainite or martensite platelets (i. e. along
plate boundaries). By contrast, later work on the reaustenitisation of a Fe-1V-0.2 wt.% carbon
steel found that with martensite and bainite as the starting microstructure, the austenite forms
predominantly at the prior austenite grain boundaries [10]. This latter study also indicated
that the nucleation rate of austenite tends to increase in the order of martensite, bainite and
allotriomorphic ferrite as the starting microstructures.
Whether the austenite grows in an equiaxed or an acicular morophology is of practical
importance, not because of the detailed difference in morphology, but because in the latter
case, the steel exhibits a memory effect in which the original austenite grain structure is re-
generated (both with respect to shape and crystallography) when the reaustenitisation process
is completed [52,53]. When the memory effect operates, the austenite grain structure cannot
13
be refined by repeated cycling into the austenite phase field followed by transformation. This
can be a disadvantage in many commercial applications. The creep ductility of bainitic and
martensitic steels of the type used in the power generation industry is improved by grain re-
finement [53]. t The memory effect prevents the achievement of a fine austenite grain structure
even when the austenitising temperature used is relatively low.
The memory effect has been shown to be a direct consequence of the existence of retained
austenite in the starting bainitic or martensitic microstructure [53]. The reaustenitisation
heat treatment causes the films of austenite to grow, and those originating from the same prior
austenite grain then coalesce to regenerate the prior austenite grain structure (Fig. 1.6). In these
circumstances, the reaustenitisation process does not require the nucleation of new austenite
grains, although if the superheating is large enough, then the nucleation of new grains may
follow in addition to the growth of the retained austenite.
The memory effect vanishes if the bainitic microstructure is annealed at a sufficiently high
temperature to remove the retained austenite, and then reheated into the austenite phase field
(Fig. 1.6). Furthermore, the austenite then grows with a more or less equiaxed morphology.
The austenite may also decompose during heating to the reaustenitisation temperature, so that
slow heating from ambient temperature also destroys the memory effect. Very rapid heating to
the reaustenitisation temperature can reduce the memory effect by inducing the nucleation of
new austenite grains [53].
It is interesting to note that the memory effect does not exist when the starting microstruc-
ture is allotriomorphic ferrite [53]. This is probably because the ferrite allotriomorphs usually
grow into both the adjacent austenite grains, thereby destroying the prior austenite grain struc-
ture. With martensite and bainite, the plate growth is entirely restricted to the grain in which
they nucleate, so that there exist sharp discontinuities in crystallographic orientation at the
position of the prior austenite grain boundaries. Indeed, it is for this reason that the prior
austenite grain boundaries are good sites for the nucleation of new grains of austenite when the
initial microstructure is bainitic or martensitic. If the steel contains residual impurities such
as arsenic, phosphorus or tin, which tend to segregate to the prior austenite grain boundaries,
then the memory effect is enhanced [53]' presumably because the segregation reduces the grain
boundary energy, thereby making heterogeneous nucleation less likely.
1.4 OVERALL TRANSFORMATION KINETICS
The overall transformation kinetics can be characterised by TTT (time temperature trans-
formation) curves for different volume fractions of a phase formed by the transformation. Once
TTT curves are obtained, it becomes possible to calculate the evolution of the transforma-
tion during any heat treatment assuming the additivity of the transformation, although this
assumption is not generally correct as discussed later [41].
During decomposition process of austenite in steels, the driving force for the transformation
increases when the reaction temperature is reduced. However, the mobility of atoms decreases
with temperature. Therefore, the well known C-shaped TTT curves are obtained for this process
(Fig. 1.7). In the case of reaustenitisation, on the other hand, both the driving force and the
mobility of atoms increases with the reaction temperature. As the result, the time required for
reverse transformation decreases monotonically with temperature (Fig. 1.7).
t Note that creep in these materials, for typical circumstances, is not controlled by grain boundary diffusion or
sliding.
14
The volume fraction of austenite normalised by its equilibrium volume fraction, e, trans-
formed during isothermal holding at a temperature T can be expressed by the Avrami type
equation.
(1.23)
As was pointed out by Christian [41]' a kinetics investigation limited to the establishment of
the value of n in the Avrami equation most appropriate to the assumed growth law, does not
give sufficient information for the growth mechanism to be deduced. Nevertheless, this method
is one of the shortest ways of obtaining information of the overall reaction. For investigating
the overall transformation kinetics of austenite, attempts have been made at fitting the Avrami
equation to extra data [2,24,25]. Speich et al. [2] have made optical microscopic measurements
of the volume fraction of austenite in hypereutectoid steels transformed during isothermal hold-
ing at different reaction temperatures, and obtained a value of three for the power of time n in
the Avrami equation. Rooz et al. [25] have also measured the volume fraction of austenite in
an eutectoid steel, and declared that the slope was four instead of three. They have concluded
from this n value and the optical studies which show that the austenite nuclei appear at the
edges of the pearlite colonies, that reaustenitisation from pearlitic microstructures occurs at a
constant growth rate and the constant nucleation rate without the occurrence of site saturation
throughout the reaction [25]. In the case of the formation of austenite from a mixture of ferrite
and spheroidised cementite particles, Dirnfeld et al. [24] reported that the slope n depends on
the reaction temperature and changes in the vicinity of about 0.05 and 0.4 of volume fractions
transformed. Although all these experiments seem to be valid for the particular system consid-
ered, they cannot directly be applied to other systems having different initial conditions and
chemical compositions.
The constant kA in equation (1.23) was studied in a special case where site saturation
occurs [24]. The value kA calculated for the grain boundary, edge and corner nucleations using
the observed grain diameter and growth rate of austenite. The observed kA value at 40% of
transformation is almost constant which is close to the values of the grain boundary nucleation
at a low temperature (i.e. 760°C), the edge nucleation at medium temperatures (i.e. 780°C
and 800°C) and the corner nucleation at a high temperature (i.e. 820°C).
There have been several attempts to predict the formation of austenite theoretically [7,28,
54,55]. They all assume the existence of the local equilibrium at the interface between austenite
and ferrite matrix. The calculations solve the diffusion equation either analytically or numer-
ically. All of them, therefore, dealt with the growth rate of austenite into a ferrite matrix
controlled either by the diffusion of carbon in austenite and/or ferrite or by the diffusion of
both carbon and the substitutional alloying element in austenite and/or ferrite. Although they
managed to calculate the growth rate of austenite in each case, it was necessary to assume or
neglect the nucleation rate of austenite since there was no theory available. In the case of the
formation of austenite from a mixture of ferrite and austenite, since the nucleation of austenite
is essentially unnecessary, the calculation of the growth rate of austenite can be converted to
the evolution of the reaction directly.
1.5 ANISOTHERMAL TRANSFORMATION
In a real heat treatment which involves reaustenitisation, the transformation does not take
place isothermally but in a non-isothermal manner. As it is widely used for the calculation of the
transformation process in a non-isothermal process from isothermal data such as a TTT curve,
15
the additivity of the transformation is usually assumed for the calculation of reaustenitisation
during a non-isothermal heat treatment. However, as discussed by Christian [41]' the additivity
of the reaction is valid only when the reaction is isokinetic [56], in which the fraction transformed
at a fixed temperature is dependent only on the time and on a single function of the temperature.
In general, the reaction is not isokinetic because of the different temperature dependence of the
growth rate and nucleation rate of the transforming phase. However, when the site saturation
of the reaction is maintained at the early stage of the reaction, the reaction will be considered
to be isokinetic and the additivity of the transformation can be used [41].
In the case of the formation of austenite from a mixture of ferrite and austenite, the
nucleation of austenite is not necessary since austenite already exist before the onset of the
reaction. The increase in volume fraction of austenite is then conducted by the growth of pre-
existing austenite, and therefore the reaction can be treated as isokinetic allowing the use of
the additivity of the reaction.
If T{T} is the time taken to produce a fixed amount of transformation, an additive reaction
implies that the total time t to reach a specified stage of the transformation is obtained [41]
from
i
t dtl
o T{T} = 1
where T is now a function of time. The calculation can be done practically by replacing the
non-isothermal integration by the summation of a set of sufficiently small steps of isothermal
reactions.
1.5.1 Continuous Heating Transformation
The behaviour during continuous heating should be related to the isothermal transforma-
tion kinetics. For example, the continuous heating curve could be treated as a series of small
isothermal steps, each occurring at a successively higher temperature, with a time interval t;
associated with each step (where i is the subscript identifying the step number). If the time
necessary to reach a specified increment of transformation is written as T; for the isothermal
transformation at temperature T;, then the simplest approximation is to assume Scheil's rule
[70]. In this, the specified increment of transformation is achieved during continuous heating
when
n
""' t·L~=1.
T·
;=1 '
( 1.24)
An application of this rule to the T values listed in Table 1.1, for the O'a +1" starting microstruc-
t ure, showed that during continuous heating (0.06 QCS-l) of that microstructure, reausteniti-
sation should begin at a temperature of ~ 685 QC. It is however evident from Fig. 1.8 that for
the same heating rate, the as-deposited welds begin to transform to austenite at a much lower
temperature of about 630 QC.
Of course, the incubation time data of Table 1.1 refer to reaustenitisation from an initial
microstructure generated by isothermal transformation to acicular ferrite or bainite at 460 QC.
On the other hand, the primary weld microstructure arises during continuous cooling of the weld
to ambient temperature. The carbon concentration xl of the austenite in the weld may then be
approximated by xT' evaluated at the martensite-start Ms temperature of the alloy concerned.
Q
This is because the weld can be assumed to continue transforming to acicular ferrite until the
Ms temperature is reached. Unfortunately, the composition ofthe residual austenite is expected
16
to change as acicular ferrite forms, so that its Ms temperature is not easy to evaluate, especially
if the carbon is inhomogeneously distributed in the austenite [57-59]. Nonetheless, the values of
xT' evaluated at the Ms temperature of the untransformed alloy must provide a lower limit to
o
the carbon concentration of the residual austenite. Since this is larger than the corresponding
value at 4600 C, it is not surprising that the welds begin the reaustenitisation process at lower
temperatures and at faster rates when compared with isothermal reaustenitisation in which the
initial microstructure was generated by transformation at 460 a C.
In fact, the rate at which the reaustenitisation-start temperature Ts is expected to decrease
as the temperature Tb for the isothermal formation of acicular ferrite decreases, is given by
(1.25)
where ST~ and 5Ae3 refer to the slopes of the T~ and Ae3 curves of the phase diagram re-
spectively. For the alloy (Table 1.1), the ratio ST~/SAe3 is found to be 2.36 (Fig. 1.9) for the
carbon concentration range 0 ---+ 0.04 mole fraction on the phase diagram. Since the difference
in Ts {0.06 a Cs} between the samples isothermally transformed at 460 a C and the as-welded
microstructure is 685 - 630 ac, the effective value of Tb for the as-welded microstructure is
estimated to be 436 ac.
1.6 CRYSTALLOGRAPHY
There are few crystallographic studies on partially reaustenitised microstructures when
compared with those on the microstructures obtained by the decomposition of austenite.
The orientation relationship between austenite formed during heating and ferrite matrix
was studied by D'Yachenko and Fedorov [60] in a Fe-0.6C wt.% steel using X-ray diffraction at
high temperatures. They observed a relation (111)1' 11 (110)0 when a quenched specimen was
heated slowly. However there were no general features of the orientation relations when the
specimens were annealed before the heat treatment, or when a higher heating rate was used,
and grain refinement was observed in these cases. A consistent orientation relationship between
austenite and the ferrite matrix has also been reported by other researchers. Fong and Glover
[61] reported orientation relationships between austenite precipitates formed during nit riding
in a Fe-1.93Mn wt.% steel, which were close to a Kurdjumov-Sachs (K-S). This observation was
supported by Matsuda and Okamura [9]' Koo and Thomas [13]' Law and Edmonds [10] and
Lenel and Honeycombe [4].
Matsuda and Okamura [9] showed that the acicular austenite grains formed in a martensitic
initial microstructure nearly all had the same orientation, with a K-S orientation relationship
between the austenite and the ferrite matrix. They concluded from the results that the mecha-
nism of formation of acicular austenite is martensitic. However, it is known that this orientation
relationship can also be obtained even for reconstructive transformation [40]. Hence different
evidence such as surface relief will be required to confirm this conclusion.
Law and Edmonds [10] studied the crystallographic relationship between grain boundary
nucleated austenite grains and adjacent ferrite grains. They found that austenite nodules were
K-S related to the ferrite grains into which they did not grow but not with the ferrite grain
into which they grew. However Lenel and Honeycombe [4] studied the orientation relationships
between austenite and ferrite matrix in the case of the formation of austenite from a mixture
of ferrite and pearlite, showing that the austenite can grow into one or both ferrite grains to
17
which the austenite grain was K-S or Nishiyama- Wasserman (N- W) related. Further studies
are necessary to understand the direction of the growth of grain boundary nucleated austenite.
1. 7 APPLICATIONS
There are many examples in which reaustenitisation plays an important role in the evolu-
tion of the ultimate microstructures, and thus the final mechanical properties of products.
1.7.1 Ferrite-Martensite Dual Phase Steels
Ferrite-martensite dual phase steel, which gives good strength-ductility combinations, is
one ofthe major successful applications of heat treatment in the manufacture of steels [62]. Such
steels are now used widely in the automobile industry and contribute to the major reduction in
automobile weight achieved in the past two decades. This has been one of the most desirable
targets in the automobile industry from not only an economical point of view after the oil crisis
but an environmental point of view. There are two different approaches in the production of
dual phase steels [63]. One of them involves a combination of the accelerated cooling and a
slower cooling of austenite at around the ferrite transformation-start temperature, Ar3, after
hot rolling at rather low temperatures. This method provides as-hot-rolled dual phase steels
which are cheaper than those produced after cold rolling followed by intercritical annealing at
temperatures between Ae1 and Ae3. In this latter heat treatment, a certain volume fraction of
austenite, which has been produced by the intercritical annealing, is surrounded by soft ferrite
matrix; the austenite then transforms into martensite during a final quenching process. In both
cases, the amount, hardness and distribution of martensite, and the ferrite grain size determine
the ultimate mechanical properties of the dual phase steels. When a steel is annealed at an
intercritical temperature, the amount, size, distribution and chemical concentration of austenite
particles formed during the annealing are determined by the reaction temperature, holding time
at the temperature and initial microstructure of the steel. In order to obtain the microstructure
including an appropriate amount of austenite with its optimum size and distribution, one needs
to select these conditions carefully.
1.7.2 Steels containing some Retained Austenite
Using a similar process as the production of dual phase steels, Sawai et al. [64] and Mat-
sumura et al. [65,66] reported very high strength steels with an excellent ductility. These steel
were reported to contain less than 25 % retained austenite. In high carbon steels, this phe-
nomenon is known as transformation induced plasticity (TRIP after Zackay [67]). The heat
treatment conducted here was more complicated than that to get the dual phase steel mentioned
earlier. Steels are heated to an intercritical temperature for a short period (e.g. 1.5 min). The
steels are then cooled down to a bainite transformation temperature followed by an isothermal
holding at the temperature for less than 30 min, thus allowing bainite transformation at the
temperature to be completed. The steels quenched after the bainite treatment have microstruc-
tures containing ferrite, bainite, martensite and retained austenite. In order to make austenite
particles stable even at ambient temperature, bainite transformation process was added to the
process for the production of conventional dual phase steels. In this process, not only are the
prediction of the amount, size, distribution and chemical content of austenite particles formed
during the intercritical annealing important, but also the bainite transformation following the
intercritical annealing must be predicted. This low carbon retained austenite steel is expected
to be used for automobiles.
18
1.7.3 Welding of Steels
Welding is, as well recognised, a complicated heat treatment of an inhomogeneous material.
The welded part is usually divided into two different regions; the weld deposit, which is the
melted region, and the heat affected zone, which may have been heated to various temperatures
during welding. Especially in multirun welds, the layers deposited initially are reheated by the
deposition of subsequent layers and experience a complicated thermal cycle which results in
several modifications of microstruct ure. The weld deposit and heat affected zone sometimes can
act the weakest part of toughness because of the coarse microstructure developed in these regions
during welding. The fine microstructure which has been achieved by special technique such as a
controlled rolling and an accelerated cooling, can be easily broken during welding. A theoretical
model to predict the microstructure of weld deposits has been developed by Bhadeshia et
al. [40]. However, the heat affected zone can contained regions of complete reaustenitisation,
partially reaustenitisation, recrystallisation and tempering. Therefore the prediction of the
reverse transformation is essential to complete the prediction of the microstructure of the heat
affected zone.
1.7.4 Initial Austenite Grain Size
Obtaining a fine grain microstructure is known to be almost the only way of improving
the toughness and ductility without sacrificing the strength of steels. It is for this purpose that
micro-alloying and controlled-rolling technology have become so prominent in steel manufac-
tures. In addition to this, there is no doubt that the initial austenite grain dimensions can
affect the scale of the final microstructure in steels.
Even in the case of hot working, the larger the initial austenite grain diameter the larger
the scale of final microstructure in those cases where the total reduction in thickness is not
large. It is also useful to note that rapid heating and cooling techniques are used to get
the fine microstructures [68]. After the completion of reverse transformation, austenite grows
by reducing its total grain boundary energy, and the rate of growth depends on the metal's
composition, the temperature and on the initial grain diameter (see for example [69]). Normal
grain growth usually stops long before a metal has become converted into a single crystal and
there is, in practice, a maximum attained grain size. The magnitude of this maximum grain size
usually depends on the composition and the annealing temperature. When there exist dispersed
particles in the specimen, a maximum grain size beyond which grains cannot be expected to
grow is determined by the ratio of the mean radius and volume fraction of the particles (see
for example [69]). Therefore the austenite grain diameter will be determined by the reverse
transformation and the grain growth after the completion of the transformation. In the cases,
where the initial austenite grain diameter can influence the scale of the final microstructure,
not only reverse transformation kinetics bu t also grain growth mechanism after transformation
is important and needs to be investigated.
1.8 TRANSFORMATION FROM AUSTENITE
The main aim of this work is to investigate the formation of austenite from different initial
microstructures as a function of chemical composition and temperature. Since the reausteniti-
sation process is strongly affected by the initial conditions such as microstructure and chemical
distribution, it is essential to understand the transformation from austenite to ferrite.
In this section, the Widmanstatten, bainitic, martensitic transformation and diffusional
formation of ferrite are summarised. The key characteristics of phase transformations in steels
19
have been rationalised by Bhadeshia ([70]' Table 1.2). When austenite is rapidly cooled to
a very low temperature, there may not be enough time or atomic mobility to facilitate the
diffusional formation of ferrite. Under the circumstance, Widmanstatten ferrite, bainite or
martensite can form depending on a level of the under-cooling. In contrast, when specimens
are cooled to a relatively high temperature below the Ae3 temperature, the austenite phase can
undergo complete reconstruction into the ferrite phase. Allotriomorphic ferrite, idiomorphic
ferrite and pearlite are considered to be in this category.
1.8.1 Widmanstiitten ferrite formation in steels
Widmanstatten ferrite can form at low under-coolings below the Ae3 temperature where the
driving force for transformation is small, so that the partitioning of carbon during transforma-
tion is thermodynamically necessary. The formation of Widmanstatten ferrite is accompanied
by a change in the shape of the transformed region; the shape change due to a single wedge
of Widmanstatten ferrite consists of two adjacent and opposing invariant-plane strain defor-
mations which allow an elastically accommodated strain energy accompanying plate formation
to be rather small, of the order of 50 Jmol-1 [70]. These invariant-plane strain deformations
imply the existence of an atomic correspondence between parent and product phases as far
as the iron and substitutional solute atoms are concerned, although carbon atoms can diffuse
during the growth. When Widmanstatten ferrite nucleates from grain boundary allotriomorphs
of ferrite, it is called a "Widmanstatten ferrite side-plate" but when it nucleates directly from
austenite grains, it is referred to as a "Widmanstatten ferrite primary side-plate". The growth
rate of Widmanstatten ferrite has been reported to be in a good agreement with the calculated
edgewise growth of a plate or a needle. The growth of Widmanstatten ferrite is controlled by
the carbon diffusion in the matrix ahead of the moving interface.
1.8.2 Bainite transformation in steels
The bainite transformation has been summarised recently by Christian et al. [71] and
Bhadeshia [70]. Bainite forms from austenite in steels in a temperature between that in which
pearlite is produced and the martensitic transformation. Although the pearlitic and bainitic
transformation temperature ranges overlap each other in low alloy steels, and this makes the
interpretation of microstructure and kinetics difficult, two separate C-curves can be detected in
medium alloy steels in the isothermal time-temperature-transformation (TTT) diagrams. The
upper C-curve represents the time taken for the initiation of diffusional transformations whereas
the lower C-curve for the initiation of the Widmanstatten ferrite or bainite transformation which
are considered to be conducted by a shear mechanism rather then a diffusional mechanism.
The lower C-curve usually exhibits the flat top corresponding to the bainite-start tem-
perature B s' and this relates to the nature of bainite transformation in steels. The bainite
transformation exhibits the classical kinetics of nucleation and growth, but ceases well before
the completion of the decomposition of residual austenite has occurred. The volume fraction
of bainite isothermally transformed is a function of the reaction temperature, and increases
with decreasing reaction temperature. The point which corresponds to 0% transformation can
define the Bs temperature. The termination of bainite transformation occurs when the carbon
content of residual austenite reaches the value where ferrite, whose free energy is raised by a
stored energy term associated with the strain of the transformation, and austenite of identical
composition have the same free energy [70]; this is called the incomplete reaction phenomenon.
Unlike the prod ucts of diffusional transformation, a platelet in a sheaf grows to a limiting
20
size which is typically about 10 p,m long with a thickness of about 0.2 p,m. The bainite trans-
formation causes a change in the shape [70] of the transformed region, which is known to be an
invariant-plane strain. In the invariant-plane strain shape change, an atomic correspondence
between the parent and product phases exists at least for the iron and substitutional alloying
elements. As a consequence of the shape deformation, which is identical for each platelet within
a sheaf, bainitic ferrite has a stored energy of about 400 Jmol-1 [72]. The platelets within a
sheaf have a small spread of orientations so that where they impinge on one another only "low
angle" boundaries are formed. At relatively high temperatures where bainite forms, any excess
carbon in ferrite can rapidly partition into the residual austenite since the diffusivity of carbon
at the temperatures is high when it is compared to the rate of carbide precipitation from a
supersaturated ferrite. At lower temperatures, on the other hand, the carbide precipitation
can occur prior to the partitioning of carbon atoms from the supersaturated ferrite into the
surrounding untransformed austenite,
Bainite is usually classified into upper bainite and lower bainite. The difference of these
phases is the existence of cementite particles within the ferrite matrix in the case of lower
bainite but not in the case of upper bainite. Both upper and lower bainite tend to form as
aggregates (sheaves) of small lenticular platelets of ferrite separated by regions of austenite,
martensite and/or cementite, When the time to decarburise the ferrite is small relative to the
time required to relieve the carbon supersaturation by the precipitation of carbides within the
ferrite, then upper bainite is obtained; otherwise, lower bainite forms [73].
The orientation relationship between bainitic ferrite and austenite is close to either the
Kurdjumov-Sachs or the Nishiyama- Wassermann orientation relationships. Though the relative
orientations of the cementite and austenite is not known, the ferrite and cementite are relatively
oriented in a variant of the Bagaryatskii relationship commonly observed for the precipitation
of cementite in tempered martensite or quench-aged ferrite [74].
1.8.3 Ma7'tensitic transformation in steels
Martensitic transformation has been studied initially because of its technological impor-
tance in the hardening of steels and recently because of its special characteristic of shape
memory. Martensitic transformations have recently been summarised by Nishiyama [75,76].
The martensite transformation is a phase transformation that occurs by the cooperative atomic
movements without any diffusion of atoms. The time taken to form a martensite crystal in
steels is in some cases said to be of the order of 10-7 sec. A necessary condition for the oc-
currence of martensitic transformation is that the free energy of martensite be lower than that
of austenite [76]. Moreover, since additional energy, such as that due to surface energy and
transformation strain energy, is necessary for the transformation to take place, the difference
between the free energies of austenite and martensite must exceed the required additional en-
ergy. Therefore the austenite to martensite transformation cannot occur until the specimen is
cooled to a particular temperature below the value where the free energy difference between
austenite and martensite is zero. This temperature is called martensite-start temperature Ms,
and varies with the chemical composition of steels.
It is well known that martensite crystals produced in an austenite crystal have definite
crystallographic relations to those of the untransformed part of the austenite crystal, which are
designated the Kurdjumov-Sachs or Nishiyama- Wassarmann orientation relationships.
21
1.8.4 Reconstructive formation of ferrite in steels
The reconstructive transformation of austenite to ferrite in steels has been summarised by
Bhadeshia [77]. When transformation is controlled totally by diffusion in the matrix ahead of
the interface, a reasonable approximation for the growth of ferrite is that the compositions of
the phases in contact at the interface are in equilibrium; this is referred to as local equilibrium
at the interface. The diffusional formation of ferrite in Fe-X-C (X indicates a substitutional
alloying element) alloys can be governed by a variety of possible growth modes [77], because of
the large difference in the diffusivities of carbon and substitutional atoms in austenite.
At a low supersaturation, considerable partitioning and long-range diffusion of substitu-
tional alloying elements can occur, so that the driving force for carbon diffusion will be reduced
to a level which allows the substitutional element flux to keep pace with the carbon flux at the
interface. This state is called the partitioning under local equilibrium (or PLE) [39]. However
at higher supersaturations, the partitioning of substitutional atoms can be negligible, so that
it causes a very large gradient of the substitutional element at the interface, which increases
the driving force for X diffusion in austenite and allows the flux of X to keep up with the long
range diffusion of carbon in austenite. In this situation, the diffusion of substitutional atoms
is limited to an extremely short range. This state is referred to as the negligible partitioning
under local equilibrium (or NPLE) [39]. In contrast to the two equilibrium states where the
local equilibrium exists at the interfaces and both carbon and substitutional element diffuse
during transformation, there exists a state where the local equilibrium cannot be satisfied at
the interfaces because of the rapid reaction. In this case, at very high supersaturations, the
diffusion of substitutional atoms can be completely negligible and the two adjoining phases
have identical X/Fe atom ratio, even though carbon atoms diffuse during the reaction keeping
the chemical potential identical in both the phases at the interface. Therefore, the rate of the
reaction is controlled by the diffusion of carbon in the austenite. This state is designated as
paraequilibrium [32].
1.9 PREDICTION OF MICROSTRUCTURE IN STEELS
Theories of phase transformations can be effectively used in the prediction of the mI-
crostructure of steels. In turn, these could be applied to the design of new alloys or new
processes. Although all of the kinetics of phase transformations in steels have not yet been es-
tablished, even partial knowledge of the phase transformations can be applied to this purpose.
1.9.1 Microstructure prediction in welding
Welding is a process where an extended range of temperature is experienced by a material
used in the process. Therefore there are various different reactions happen during welding such
as reaustenitisation, melting, solidification, delta ferrite to austenite transformation, formation
of ferrite from austenite and dissolution and formation of various kinds of precipitations. In
addition to this, a repeated heat input in a multi-run welding makes the problem more compli-
cated. The phases formed in a deposition during cooling may be heated again by the subsequent
deposition on it, which may make the position experience additional phase transformations and
leads to a completely different microstructure from the beginning.
Bhadeshia and his coworkers [40] have reported a model to predict the microstructure of
a weld deposit from the alloy chemistry and thermal history of the deposition.
The model consists of the TTT curve prediction and the kinetics calculation of phase trans-
formations from austenite. They assumed the initial austenite microstructure as a columnar
22
which is usually observed in solidification. The growth offerrite allotriomorph at austenite grain
boundaries is calculated by first evaluating the one-dimensional parabolic thickening controlled
by carbon diffusion in austenite. The paraequilibrium is assumed in the calculation. When the
temperature reaches the Ws temperature, which can be calculated from the alloy chemistry, the
formation of Widmanstatten ferrite takes place. Since Widmanstatten ferrite can grow through
the whole grain, hard impingement, which is an impingement by a structural contact of two
adjacent growing phases, is expected to occur. They took into account of the effect and added
the formation of acicular ferrite, which is well known as a phase which improves the toughness
of the weld, within the austenite grains. Since the model is based on thermodynamics, they
managed to take into account the effect of alloy additions on the microstructural development
during the heat cycle such as Si, Mn, Ni, Mo, Cr, V. Although the TTT calculation model
does not deal with the effect of the initial dimension of austenite grains, the subject has been
recently tackled by Reed and Bhadeshia [78] with a model for predicting heat cycles which is
experienced by any part of a weld during multirun welding from welding conditions and the
weld geometry.
1.9.2 Microstructure prediction of hot worked steels
Computer control systems has been widely used in hot working processes mainly to control
the sizes and shapes of products. However, there has been a growing demand for numerical
models which make it possible to predict mechanical properties of hot worked steels via the
prediction of microstructural development during and after hot working. This sort of model
is desired not only from the industrial point of view but also from its fertile theoretical inter-
est. Once a model is obtained, it could be used and contribute to a remarkable reduction in
the amount of mechanical testing carried out to guarantee the properties of products, to an
extension of the possibility of producing various levels of mechanical properties out of a steel
using a wide range of prod uction conditions which may lead a new concept of an alloy design
in industries, and to a decrease in time on research and development by providing a flexible
simulation model on computers.
When one intends to predict a microstructural development of hot worked steels, one may
consider the effect of hot working; in other words the effect of defects introduced by the hot
working, on phase transformations which occur on cooling after the hot working and determine
the ultimate microstructure in the products. Especially when it is essential to obtain a very
fine grain structure in the final stage of the production in order to maintain a high toughness
or ductility without tolerating its strength, a combination of hot working at relatively low
temperatures of austenite single phase region, and rapid cooling after the hot working has
been commonly used; this is referred as the controlled rolling technique. In this case, the
transformation occurs from work-hardened austenite which is very different from that from
undeformed austenite in terms of the overall reaction rate.
A phenomenological study on microstructural development of austenite associated with
hot working led to the construction of an empirical model which deals with recrystallisation,
recovery and grain growth processes of austenite (for example [79]). Sellars [79] collated the
knowledge on recrystallisation and grain growth of austenite and proposed an empirical model
which can calculate dynamically the development of austenitic microstructure during and after
hot working. The change in the austenitic microstructure during and after hot working in
a austenite single phase region consists of two phenomenologically different recrystallisation
processes as well as the grain growth and recovery.
23
Recrystallisation which occurs during hot working is called the dynamic recrystallisation.
The minimum required strain for the onset of the dynamic recrystallisation is a function of
strain rate, temperature, initial austenite grain size and alloy chemistry. The larger the initial
grain size, the lower the hot working temperature and the larger the strain rate of the working,
the larger is the strain required for the onset of the dynamic recrystallisation. In addition to
this, the minimum required strain for the onset of the dynamic recrystallisation is increased
by additions of alloying elements such as niobium and titanium. The average grain diameter
of recrystallised grains relates only on the strain rate and reaction temperature but not on the
strain nor on the initial austenite grain diameter.
When a hot worked steel which has not recrystallised dynamically, is held at a high tem-
perature after the hot working, the static recrystallisation may take place. In this case, the
minimum required strain for the onset of the static recrystallisation is dependent on the initial
austenite grain size, reaction temperature and alloying compositions but not on the strain rate.
The average grain diameter of statically recrysatallised grains is determined only by the initial
austenite grain diameter and strain but not by the reaction temperature nor the strain rate.
Therefore static recrystallisation is believed to play an important role in the case of high
speed hot workings such as continuous hot rolling processes whereas the dynamic recrystallisa-
tion in low speed hot workings such as thick plate hot rolling processes.
Dynamic recovery occurs on hot working and determines the final dislocation density at
the end of the hot working. The higher the strain rate and the lower the reaction temperature,
the higher the dislocation density. The static recovery, on the other hand, occurs after the hot
working by a diffusion of atoms.
Considering all of these factors as well as the grain growth of austenite, one can predict the
microstructure of austenite before the onset of phase transformations during cooling after hot
working as a function of hot working conditions such as temperature, time, strain and strain
rate, and chemical compositions of steels. An example of the calculation flow chart can be seen
in Fig. 1.10.
When hot working is carried out at a temperature at which work-hardened austenite can
recrystallise in a short period, the initial austenite microstructure of phase transformations
during the following cooling process does not differ significantly from that of the reheated
austenite despite the fact that the average grain diameter of austenite is finer in the former
case than in the later case. In the case, therefore, phase transformation could be considered
independently from the hot working except the effect of the initial austenite grain diameter,
which can be taken into account as the difference in the nucleation site density. However, when
a hot working is conducted at lower temperatures, partially recrystallised or unrecrystallised
austenite microstructures is expected to dominate the microstructure before the temperature
of the steel reaches the point at which the formation of ferrite can occur. It has been well
established that finer microstructure can be obtained in the case than the cases with hot working
at higher temperatures. Because the controlled rolling technique has been very important from
industrial point of view, the effect of hot working at lower temperatures on phase transformation
has been studied although most of this work is empirical.
Umemoto et al. [80] studied the formation of pearlite from work-hardened austenite and
managed to model the effect of hot working on the pearlite formation. The effect of hot working
consists of three factors when a steel was hot worked at a temperature where no recrystallisation
was observed, which are; 1) an increase in the austenite grain surface area per unit volume due
24
to an elongation of austenite grains, 2) an increase in the nucleation rate per unit area of
austenite grain surface, and 3) an formation of additional nucleation sites such as deformation
bands and deformed twin boundaries. They considered the effect of the elongation of austenite
grains from geometrical calculation; i. e. a spherical austenite grain becomes an ellipsoid after
rolling. The effect of the increase in the nucleation rate per unit area and the increase in the
additional nucleation sites (deformation bands) were assumed to be proportional, respectively,
to the true strain and a square of the true strain based on previous experimental results [81,82].
Using experimentally obtained constants for these factors, they compared the effect of the three
factors mentioned above on the acceleration of the formation of pearlite. When the austenite
grain diameter is relatively small (less than 100 JLm ), the acceleration of the formation of
pearlite is mainly due to the increase in the nucleation rate per unit area of austenite grain
surface. When the initial austenite grain diameter is larger, on the other hand, the formation
of additional nucleation site rather than austenite grain surface becomes dominant [81,82].
Numerical models which deal with the hot working and cooling process have been reported
independently [83,84,85]. They adopted the Avrami type expressions for the kinetics ofthe for-
mation of ferri te, pearlite and bainite, and independently obtained coefficients in the expression
by a least square fitting to the experimental data. Once one can calculate the microstructure
such as volume fraction, strength and diameter of each phases, mechanical properties can then
be predictable through an empirical relations between microstructure and the mechanical prop-
erties. The relationship between microstructure and mechanical properties in multi-phase steels
are, however, less understood. Even for a dual phase structure, a full stress-strain curve (S-S
curve) is not reproducible from S-S curves of each phases. Empirically observed relations which
have been checked to have good correlations to experiment have a form as follows.
(1.26)
where Y is a mechanical property and Vi, Hi and di are respectively volume fraction, hardness
and diameter of i-th phase. It is rather well known that the strength of a ferritic steel can be
expressed by the form, namely;
(1.27)
where u is either tensile or yield strength, da average ferrite grain diameter, and uO' ai and b
are constants to be determined from experiments.
Esaka et al. [83] have extended the idea to ductilities of steels without any justification,
and succeeded in expressing the mechanical properties of hot worked steels with various kind of
microstructure including ferrite, acicular ferrite, pearlite, bainite and martensite from chemical
content of the steel, hot working conditions and cooling conditions.
Although all of the models are not physically based, it must be emphasised that the
accumulated knowledge of recrystallisation and phase transformation in steels is now at a stage
of being applied to predict the ultimate microstructure obtained by heat treatment and thermo-
mechanical treatment. The work has not, however, been extended to processes including reverse
transformation which may become important either when the initial microstructure plays an
important role for the development of the final microstructure or when full or partial reverse
transformation is introduced during the process.
25
1.10 SUMMARY
It appears that in practice, austenite grows by a reconstructive transformation mechanism
because diffusion rates are significant at the elevated temperatures where the reaction normally
occurs. The exception to this is iron based shape memory alloys, and cases involving remarkably
large heating rates.
The reconstructive growth process may involve a variety of conditions at the transformation
front, including paraequilibrium, local equilibrium and a variety of intermediate states. Which
of these operates in practice is a challenging subject for research.
There is very little known about the nucleation mechanism of austenite, apart from some
general data on the types of nucleation sites as a function of the starting microstructure. How-
ever, a first approximation in the derivation of overall transformation kinetics might be to ignore
nucleation on the grounds that it is likely to be rapid at high temperatures. There are inter-
esting features of the overall transformation kinetics; both the driving force and diffusivities
increase with superheat, making austenite formation distinct from the usual C-curve kinetics
of classical TTT diagrams. In fact, the formation of austenite should always become easier as
the degree of superheating is increased.
REFERENCES
1. R. A. Roberts and R. F. Mehl: Trans. ASM, 1943,31, 613.
2. G. R. Speich and A. Szirmae: Trans. TMS-AIME, 1969, 245, 1063.
3. C. I. Garcia and A. J. DeArdo: Metall. Trans., 1981, 12A, 521.
4. U. R. Lenel and R. W. K. Honeycombe: Metal Science, 1984a 18, 201.
5. X-L. Cai, A. J. Garratt-R and W. S. Owen: Me tall. Trans., 1985, 16A, 543.
6. R. C. Sharma, G. R. Purdy and J. S.Kirkaldy: Metall. Trans., 1979 lOA, 1129.
7. R. R. Judd and H. W. Paxton: TMS AIME, 1968, 242, 206.
8. M. Baeyertz: Trans. ASM, 1942, 30, 458.
9. S. Matsuda and Y. Okamura: Trans. ISIl, 1974a, 14, 363.
10. N. C. Law and D. V. Edmonds: Me tall. Trans., 1980, HA, 33.
11. S. Kinoshita and T. Ueda: Tetsu-to-Hagane, 1973, 59, 55.
12. M. R. Plichita and H. I. Aaronson: Metall. Trans., 1974, 5, 2611.
13. J-Y. Koo and G. Thomas: Metall. Trans., 1977, I8A, 525.
14. P. G. Winchell and M. Cohen: Trans. ASM, 1962, 55, 347.
15. P. G. Winchell and M. Cohen: Electron Microscopy and Strength of Crystals, Interscience,
New York, 1963, 995.
16. J-R. Yang: Ph. D. Thesis, University of Cambridge 1988.
17. C. M. Wayman: Proceedings of ICOMAT 89., 1989.
18. H. Kessler and W. Pitsch: Acta Me tall, 1965, 13, 871.
19. D. Dunne and C. M. Wayman: Metall. Trans., 1973a, 4, 137.
20. D. Dunne and C. M. Wayman: Metall. Trans., 1973b, 4, 147.
21. C. M. Wayman: "Phase Transformations", York Conference, Institute of Metals, London,
1979, 1, IV 1.
22. C. M. Wayman: "Phase Transformations in Solids", ed. T. Tsakalakos, Materials Research
Society, North Holland, 1983,21, 657.
26
23. A. E. Nehrenberg: Trans. AIME J. of Metals, 1950, 188, 162.
24. S. F. Dirnfeld, B. M. Korevaar and F. B. Spijker: Metall. Trans., 1976,15, 1437.
25. A. Roosz, Z. Gacsi and E. G. Fuchs: Acta Metall., 1983, 31, 509.
26. G. R. Speich, V. A. Demarest and R. L. Miller: Metall. Trans., 1981, 12, 1419.
27. M. Nemoto: Metall. Trans., 1977, I8A, 431.
28. M. Hillert, K. Nilsson and L-E. Torndahl: JISI, 1971,209,49.
29. J. W. Christian: "The Theory of Transformations in Metals and Alloys", Part 1, 2nd. ed.,
Pergamon Press, Oxford, 1988.
30. J-R. Yang and H. K. D. H. Bhadeshia: "Welding Metallurgy of Structural Steels", TMS-
AIME, Warren dale, Ohio, ed. J. Y. Koo, 1987, 549.
31. J-R. Yang and H. K. D. H. Bhadeshia: "Phase Transformations 87", The Institute of Metals,
London, ed. G. W. Lorimer, 1988, 203.
32. A. Hultgren: Jernkontorets Ann., 1951, 135, 403.
33. E. Rudberg: Jernkontorets Ann., 1952, 136, 91.
34. M. Hillert: Jernkontorets Ann., 1952, 136, 25.
35. M. Hillert: Internal Report, Swedish Institute of Metals Research. 1953.
36. J. S. Kirkaldy: Canad. J. Phys., 1958, 36, 907.
37. G. R. Purdy, D. H. Weichert and J. S. Kirkaldy: Trans. Met. Soc. A. 1. M. E., 1964, 230,
1025.
38. H. 1. Aaronson, H. A. Domian and G. M. Pound: Trans. Met. Soc. A. 1. M. E., 1966, 236,
768.
39. D. E. Coates: Me tall. Trans., 1973,4, 2313.
40. H. K. D. H. Bhadeshia, L-E. Svensson and B. Gretoft: Acta Metall., 1985, 33, 1271.
41. J. W. Christian: "The Theory of Transformations in Metals and Alloys", Part 1, 2nd. ed.,
Pergamon Press, Oxford, 1988.
42. R. T. DeHoff and F. N. Rhines, eds.: "Quantitative Microscopy", McGraw-Hill Book Com-
pany, New York. 1968,
43. C. Zener: J. App. Phys., 1949, 20, 950.
44. C. A. DuM: Ph.D. Thesis, Carnegie Inst. of Tech. 1948.
45. R. P. Smith: J. Amer. Chem. Soc., 1946, 68, 1163.
46. C. Wells, W. Batz and R. F. Mehl: Trans. Met. Soc. A. 1. M. E., 1950, 188, 533.
47. R. H. Siller and R. B. McLellan: Trans. Met. Soc. A. 1. M. E., 1969, 245, 697.
48. R. H. Siller and R. B. McLellan: Metall. Trans., 1970, 1, 985.
49. H. K. D. H. Bhadeshia: Metal Science, 1981, 15,477.
50. R. Trivedi and G. M. Pound: J. Appl. Phys., 1967,38, 3569.
51. C. Atkinson: Acta Metall., 1967, 15, 1207.
52. V. D. Sadovskii: C. E. Trans., 1956. 7648.
53. S. T. Kimmins and D. J. Gooch: Metal Science 1983, 17,519.
54. A. D. Roming and R. Salzbrenner: "Solid-Solid Transformation '81", ed. H. 1. Aaronson,
AIME, 1981, 849. .
55. J. Agren: Mat. Sci. and Eng., 1982, 55, 135.
56. Avrami, M.: J. Chem. Phys., 1939,7, 1103.
57. A. Schrader and F. Wever: Arch Eisenhuttenwesen, 1952, 23, 489.
58. S. J. Matas and R. F. Hehemann: Trans. Met. Soc. A. 1. M. E., 1961, 221, 179.
27
59. H. K. D. H. Bhadeshia and A. R. Waugh: Acta Me tall. , 1982, 30, 775.
60. S. S. D'Yaehenko and G. V. Fedorov: Fiz. Metal. Metalloved., 1964,18, 73.
61. H. S. Fong and S. G. Glover: Trans. JIM, 1975, 16, 115.
62. S. Hayami and T. Furukawa: Proc. of "Microalloying '75", Union Carbide Corp., New York,
1975, 56.
63. S. Hayami, T. Furukawa, H. Gondoh and H. Takeehi: "Formable HSLA and Dual Phase
Steels", ed. A. T. Davenport, The Metall. Society of AIME, 1979, 167.
64. 1. Sawai, A. Ueda and E. Kamisaka: Tetsu-to-Hagane, 1985, S1292.
65. O. Matsumura, Y. Sakuma and H. Takeehi: Trans. ISIl, 1987,27, 570.
66. O. Matsumura, Y. Sakuma and H. Takeehi: Scripta Metall., 1987,21, 1301.
67. V. F. Zaekay, E. R. Parker, D. FaIn and R. Buseh: Trans. ASM, 1967,60, 252.
68. R. A. Grange: Trans. ASM, 1966, 59, 26.
69. Cotterill, P. and Mould, P. R.: "Recrystallization and grain growth In metals", Surrey
University Press, London. 1976.
70. H. K. D. H. Bhadeshia: "Phase Transformation '87", G. W. Lorimer ed., Institute of Metals,
London, 1988, 309.
71. J. W. Christian and D. V. Edmonds: "Phase transformation in Ferrous Alloys", A. R.
Marder and J. I. Goldstein ed., The Metall. Soc. of AIME, 1983, 293.
72. H. K. D. H. Bhadeshia: Acta Metall., 1981,29, 1117.
73. S. J. Matas and R. F. Hehemann: Trans. AIME, 1961, 221, 179.
74. H. K. D. H. Bhadeshia and J. W. Christian: Metall. Trans. A, 1990, 21A, 767.
75. Z. Nishiyama: "Some Old and New Views of the Nature of Martensites", Proc. of 1st. JIM
Int. Symp. on New Aspects of Martensitic Transformation, Kobe, Japan, 1976, 1.
76. Z. Nishiyama: "Martensitic Transformation", ed. M. E. Fine, 1978.
77. H. K. D. H. Bhadeshia: ((Diffusional Formation of FelTite in Iron and its Alloys", Progress
in Materials Science, 1986, 29, 321.
78. R. C. Reed and H. K. D. H. Bhadeshia: Proc. Int. Conf. on "Trends in Welding Research",
1990.
79. C. M. Sellars: Sheffield Int. Conf. on "Hot working and Processes", 1979, 7, 3.
80. M. Umemoto, H. Ohtsuka and 1. Tamura: Trans. ISIl, 1983, 23, 775.
81. 1. Kozasu, C. Ouehi, T. Sampei and T. Okita: Proc. of "Micro Alloying '75", Union Carbide
Corp., New York, 1975, 100.
82. K. Esaka, J. Wakita, M. Takahashi, O. Kawano and S. Itarada: Seitetsu-Kenkyu, 1988,321,
92.
83. Y. Saito: Tetsu-to-Hagane, 1988, 74, 609.
84. M. Suehiro, K. Sato, Y. Tsukano, H. Yada, T. Senuma and Y. Matsumura: Trans. ISIl,
1987,27, 438.
28
12
3
4
5
ea eae
a. a.
0 1
1\
0Ifcem cem
y. :y a.
0
cem
y. ':i a.
-0
cem
y. y.
_.
Fig. 1.] Different mechanisms of dissolution of cementite during the formation of austenite from
the mixture of ferrite and spheroidised carbide [28].
800
d
Q)
'-::J
•••
ctl
'-
Q)
0.
E
~ 400
300
o 0 .02 0.04 0 .06 0.08 o. 10 o. 12 o. 14 o. 16
Mole fraction of carbon
Fig. 1.2 Calculated phase diagram showing the Ae3,Ae~ and T~ curves for a Fe-O.27Si-1.84Mn-
2.48Ni-O.201\fo wt.% alloy.
29
a)
X1 ---------------------------
c
0 y
.•....
CO
"-.•....
c
Q)
u
C ay
0 X1U a
C
Zo0 b)
.0 X1"-
CO
0
y
X
ya --------------------
1
·I
ay ·X1 ··------f I
a ,
*Z Zo
Interface position z
Fig. 1.3 Schematic illustration ofthe carbon concentration profile (a) before and (b) during reausteni-
tisation. The interface position z· is defined normal to the interface. Xl is the initial carbon
concentration in austenite, xi() and x~'" are carbon concentration in ferrite and austenite
at the interface which are in equilibrium.
30
Inltlel structure CXa +"( I nltlel .tructu~ CXb+"Y
VI 10
10
10
~ 10
0
VI
~ 0
VI z
"i"e 0td
u VI
N.••...••..,09 "i"~
'0
9
u
I -
~ ••...••..
N
I
~
8
10
8
'0
10 100 10
TIME, 11 SECONDS
TIME, 11 SECONDS
Fig. 1.4 Relation between the time taken for the smallest detectable amount of reaustenitisation
and a;2 (a) in the case of an acicular ferrite + austenite initial microstructure and (b) a
bainite + austenite initial microstructure.
Initial slruclure CXa+"(
Vl 10
~, 0
o
u•...•
Vl
'"'E
u
8
'0 50
10
10
VI
0
z:
0
::::l
Vl
'"
, 09
'E
u
••...••..
N
I -
~
100 200 500
10
8
30
TIME,l 1 SECONDS
70 100 200
TIME,1. 1 SECONDS
300
Fig. 1.5 Relation between the time taken for 0.05 volume fraction of reaustenitisation ana a;2 (a)
in the case of an acicular ferrite + austenite initial microstructure and (b) a bainite +
austenite initial microstructure.
31
a)
Time
b)
Time
c
a
Bainitic ferrite + austenite
A
a
Bainitic ferrite + austenite
c
of austenite layers
B
8
8
Ferrite + cementite (tempered)
c
8
Nucleation and growth
Fig. 1.6 illustrations of the formation of austenite from two different starting microstructures. (a)
A mixture of bainitic ferrite and austenite is generated by isothermal transformation at a
temperature below Bs· When this microstructure is reheated to an elevated temperature,
the growth of preexisting austenite occurs. (b) The mixture of bainitic ferrite and austenite
is tempered at an intermediate temperature to remove any allstenite left llntransforlllcd
after the bainite transformation. Reheating to an elcvated temperature thcn requires the
austenite to nucleate before it can grow.
~
t
>. ~
+'" ~
> <l
Cl)
~::::J
t.•....•...
~
~
Cl
<l
-----------------------:~:::::-.~---_._ .•. -'"
... -"'-.--...
,..
,.
I
I,.
---,--------------------------,
\
"
""
" "
--".1 - a
'.
Q)
L.
::::J
+'"
~ Ae
3
--
Q)
a.
E
Q).-
Time
Fig. 1.7 TTT curves for 1~ l1' and l1' ~ I'
700
680
• HWB
• WB
LJ
o
Q) 660
'-::J
d
~ 6 40
E
Q)
I-
6 20
10
1
100 10-1
Heating Rate (oC/S)
Fig. 1.8 Reaustenitisation-start temperature during continuous heating m the steel III Table 1.1
(HWB) with an as deposited weld microstructure.
33
Table 1.1 Time taken for the smallest detectable amOllnt of reaustenitisation from a mixture of
acicular ferrite and austeni te.
C Si Un Ni Mo Cr V Ti Nb
0.060 0.27 1.84 2.48 0.20 0.05 0.01 0.02 0.01
Temperature, ·c Time, seconds
680 250
690 90
700 30
710 10
720 5
wt% Fe-0'060C - O'27Si -l84Mn- 2-48Ni-0·20t1:>
ClJStalitisgj
• 1200°C,30 min.
D 950°C,10 min. I
700
u 600
o
w
~500
~
E5
Cl..
~400
300
0·00 0·02 0·04 0·06 0·08 0'10 0'12
MOLE FRKTION OF CARBON
0·14 0'16
Fig. 1.9 Calculated phase diagram of the steel in Table 1.1
3·1
Table 1.2 Key characteristics of transformations in steels [70]. a' is martensite, a/b is lower bainite,
aub is upper bainite, aa is acicular ferrite, alV is \Vidmanstiitten ferrite, a if allotriomorphic
ferrite, Pis pearlite, and X denotes suhstitutional alloying elements. Comments all each
characteristics are (=) consistent, (#) inconsistent and (<:9) only sometimes consistent with
transformation.
Comment a
,
a'b aub a. a •. Pa ai
Nucleation and growth reaction
Plate morphology '!' '!' '!'
IPS shape change with shear component '!' '!' '!'
Diffusionless nucleation '!' '!' '!' '!' '!' '!' '!'
Only carbon diffuses during nucleation '!' '!' '!' '!'
Reconstructive diffusion during nucleation '!' '!' '!' '!' '!'
Often nucleates intragranularly on defects '!' '!' '!' '!' '!'
Diffusionless growth '!' '!' '!' '!'
Reconstructive diffusion during growth '!' '!' '!'
Atomic correspondence (all atoms) during growth '!' '!' '!' '!'
Atomic correspondence. during growth.
for atoms in substitutional sites '!' '!' '!'
Bulk: redistribution of X atoms during growth '!' '!' '!' '!' '!' ® ® ®
Local equilibrium at interface during growth '!' '!' '!' '!' '!' ® ® ®
Local paraequilibrium at interface during growth '!' '!' '!' '!' ® ® '!'
Diffusion of carbon during transformation '!' '!' '!' '!'
Carbon diffusion-controlled growth '!' '!' '!' '!' ® ® ®
Cooperative growth of ferrite and cementite '!' '!' '!' '!' '!' '!' '!'
High dislocation density ® '!' '!' '!'
Incomplete-reaction phenomenon '!' '!' '!' '!' '!'
Necessarily has a glissile interface '!' '!' '!'
Always has an orientation within the Bain region '!' '!' '!'
Grows across austenite grain boundaries ..J.r
High interface mobility at low temperatures '!' '!' '!'
Displacive transformation mechanism '!' '!' '!'
Reconstructive transformation mechanism
35
Initial grain
size of FM
••
Calculation
of
coefficients
Reheating
condition
Conditions of RM
Conditions of FM
recovery : k, Tk
static rex: S, Ts
dynamic rex: m,Cs, Cc
grain growth: K, A
Recrystall izati on
l-fo l-fo-fs fN
fc fs ( )
(dynamic rex.) (static rex.) non rex.
t 5
[
fo=l- exp{-( c- _cc )m} [fs=l-eXP{-( Ts)} [fN=l-fo-fs
do=DZ-o•t55 Cs C ds= C .donc-2. dN=do.exp(-~)•. ~
(grain growth) (grain growth)
(J)
c
o
+J
-0
C,..-.....
SW
m<1
::: 1-0-
c '--/
Calculation of average grain size)
Calculation of residual strain
d=l/( fo 2 + fS
2
+ ~2 )'1/2
[
d dyn. d st. d non.
6c=(fo"cc+fN"E) "exp{-(~k )k}
Fig. 1.10 Flow chart for the calculation of microstructural development of allstenite during and after
hot working.
36
CHAPTER 2
A MODEL FOR THE TRANSITION FROM UPPER TO LOWER BAINITE
2.1 INTRODUCTION
Bainite can be regarded as a non-lamellar mixture of ferrite and carbides, but within this
broad description, it is possible to identify two classical morphologies, traditionally called upper
and lower bainite (see for example, [1-4]). Lower bainite is obtained usually by transformation at
lower temperatures, although both phases can sometimes be found in the same microstructure.
Both upper and lower bainite tend to form as aggregates (sheaves) of small platelets or laths
(sub-units) of ferrite. The essential difference between upper and lower bainite is with respect
to the carbide precipitates. In upper bainite, the bainitic ferrite is free of precipitation, any
carbides growing from the regions of carbon-enriched residual austenite which are trapped
between the sub-units of ferrite. By contrast, lower bainitic ferrite contains a fine dispersion of
plate-like car bides (e.g. J E-carbide or cementite) within the bainitic ferrite plates.
The transition between upper and lower bainite is generally believed to occur over a narrow
range of temperatures. There are circumstances where both phases can form simultaneously
during isothermal transformation near the transition temperature [5]. The first clear indication
of the mechanism of the transition emerged from the work of Matas and Hehemann [6]' whose
experiments on several steels (containing 0.38-1.0 wt.% C) indicated a narrow transition tem-
perature range centered around 350 DC, irrespective of steel composition. They suggested that
the difference between upper and lower bainite is related to the kinetics of carbide precipita-
tion from ferrite. In their model, both upper and lower bainite form with a supersaturation of
carbon, but with the former, almost all of the excess carbon is rejected into the residual austen-
ite; with lower bainite, carbon precipitates rapidly in the supersaturated ferrite, so that the
amount that diffuses into the residual austenite is reduced. The relatively constant transition
temperature was explained by suggesting that E-carbide will, for some reason, not precipitate
from ferrite at temperatures above"" 350 DC. The E-carbide was envisaged as a precursor to
the formation of cementite.
While the Matas and Hehemann model is intuitively reasonable, their belief that the tran-
sition temperature, Ls, is constant for all steels is not consistent with other experimental results
[5,7-10]. The transition temperature can be as high as 500DC and is found to vary with the
carbon concentration. The model also requires the carbide in lower bainite to be E-carbide,
which might then convert to cementite on further tempering. Later work has shown that it
is possible to obtain lower bainite containing the appropriate cementite particles, without any
E-carbide as a precursor [11]. Franetovic et al. [12,13] have also reported lower bainite con-
taining 77-carbide (Fe2C) in a high-silicon cast iron and there is no reason to suppose that
precipitation temperature and behaviour of 77-carbide should be similar to that of E-carbide.
It is also difficult to explain why E-carbide should not precipitate from ferrite at temperatures
above 350 DC.
Pickering [5] found that Ls rises initially and then decreases to "" 350 DC, becoming inde-
pendent of the carbon concentration beyond"" 0.8wt. %C. His explanation of the transition is
essentially the same as that of Matas and Hehemann [6]' that the transition to lower bainite
occurs when the rate of carbon diffusion from ferrite is slow, so that the carbides have an oppor-
tunity to precipitate. The model does not, however, account for any changes in the kinetics of
37
(2.1 )
carbide precipitation as a function of carbon concentration. Similar results have been obtained
for more heavily alloyed steels, where the peak in the experimental transition curve is found to
shift to lower carbon concentrations [8]. Pickering suggested that for high carbon steels (where
the transition was claimed to be insensitive to carbon concentration), the cementite precipitates
directly from the austenite, as its carbon concentration xl exceeds the concentration xt~ which
is given by the extrapolated 1'/ er + Fe3C) phase boundary; this does not explain the formation
of carbides within the bainitic ferrite.
To summarise, a plausible model for the transition from upper to lower bainite could be
constructed from the assumption that there is no fundamental difference in transformation
mechanism between these two forms of bainite, if the bainitic ferrite is, when it forms, super-
saturated with carbon. The excess carbon may partition eventually into the residual austenite
or precipitate from the ferrite in the form of carbides. If the latter process is dominant, then
lower bainite is obtained. Upper bainite is obtained only when the carbon partitions relatively
rapidly into the residual austenite, before the carbides have an opportunity of precipitate. This
essentially amounts to the Matas and Hehemann model [6]' but without the constraint that the
transition temperature is limited to a narrow temperature range around 350 QC. The model is
illustrated schematic ally in Fig. 2.1. The purpose of this chapter is to test as far as is possible,
the quantitative form of the hypothesis summarised above, and to assess any predictions in the
context of available experimental data. The work utilises recent results on the theory for the
time required to decarburise supersaturated ferrite; cementite precipitation kinetics are treated
approximately using information from martensite tempering data.
2.2 TIME REQUIRED TO DECARBURISE SUPERSATURATED FERRITE
If it is assumed that the diffusivity of carbon in ferrite is very high when compared with
that in austenite, and that local paraequilibrium is established during the partitioning of carbon
between the austenite and ferrite, then the time td required to decarburise a supersaturated
bainitic ferrite plate of thickness w is given by [4]:
a21r(x _ XQ"Y)2
t - I I
d - 16D(xr - xl)
where xl is the average carbon concentration in the steel as a whole. x~"Y and xIQ are the carbon
concentrations in ferrite and austenite respectively, when the two phases are in paraequilibrium.
The diffusivity D of carbon in austenite is very sensitive to the carbon concentration [14-17].
Hence, when dealing with concentration gradients, it is necessary to consider instead a weighted
average diffusivity [18]' D, given by
(2.2)
This procedure is valid strictly for the situation where the concentration profile does not change
with time, but is recognised to be a good approximation for non-steady state conditions, as
exist during the partitioning of carbon from the supersaturated ferrite. In the present work, D
was calculated as discussed in [19].
The calculated kinetics of partitioning are illustrated for three steels with different carbon
concentrations with calculated [20,21] martensite-start (Ms) and bainite-start (Bs) tempera-
tures, in Fig. 2.2. The calculations assume that a = 0.2 JLm, and this is the approximation used
38
throughout this work. For each steel, the time td goes through a mInImUm as a function of
transformation temperature. The minimum arises because the diffusion coefficient of carbon
decreases with temperature (leading to an increase in td), while at the same time, the amount of
carbon that the austenite can tolerate, xl''', rises with falling temperature; xII:> was calculated
as in [22]. The decarburisation time also increases as the average carbon concentration of the
steel rises; given that an increase in carbon concentration should accelerate the precipitation
of carbides in the ferrite, it should lead to an increase in Ls' as reported by Pickering [5] and
Llopis and Parker [8] for low carbon concentrations.
2.3 TIME FOR THE PRECIPITATION OF CEMENTITE
Information on the kinetics of the cementite precipitation from supersaturated ferrite is
not available in sufficient depth to enable the first principles calculation of the volume fraction
of cementite as a function of time, temperature and chemical composition. An attempt is made
here to derive the overall kinetics of cementite precipitation using published data [23,24] on
hardness changes observed during the initial stages of isothermal tempering of martensite.
2.3.1 An Empirical Method
When martensite contains an excess concentration of carbon in solid solution, the carbon
will tend to precipitate in the form of carbides during tempering. Prolonged annealing can also
lead to recovery, recrystallisation and the coarsening of cementite precipitates. For the present
purposes, it is consequently important to focus on the early stages tempering, which should
represent solely, the effects of precipitation from supersaturated ferrite.
Speich [23] reported that the change in hardness of martensite in plain carbon steels after
an hour of tempering at temperatures above 3200 C, includes significant contributions from
recovery, recrystallisation and coarsening of cementite particles (Fig. 2.3). Hence, the data
representing hardness changes during tempering below 3200 C were utilised to obtain a function
which expresses the change in the volume fraction of cementite precipitation as a function of
time and temperature. An Avrami type equation [25] was used for the purpose:
(2.3)
(2.4)
where ~{t} is the volume fraction of cementite normalised by its equilibrium volume fraction
at the reaction temperature, t is the time, and k A and n are rate constants determined from
the experimental data. It is assumed that ~{t} is related at any time t to the hardness of the
martensite, H {t} as follows,
~{t} = Ho - H{t}.
Ho - HF
Ho is the hardness of the as-quenched virgin martensite, H F is its hardness when all the
carbon has precipitated, but before any significant recovery, recrystallisation or coarsening has
occurred. Implicit in this relation is the assumption that the amount of carbon precipitated is
linearly related to the change in hardness during the early stages of tempering.
Using the values of hardness for plain carbon martensite tempered for 1 hour at 3200 C,
reported by Speich [23]' HF was expressed empirically as a function of the initial hardness and
average carbon concentration x~ (mole fraction), as follows:
(2.5)
39
This equation is valid for plain carbon steels containing less than 0.4 wt.% carbon (xr < 0.0186),
the value of H F becoming constant thereafter. The hardness Ho of plain carbon martensite
before tempering can be also be deduced from the data reported by Speich [23]:
Ho = 1267xr 0.9 + 240 (2.6)
where the hardness of martensite in pure iron is 240 Hv [24]. This equation reflects empirically,
the hardness of virgin martensite in plain carbon steels as a function of carbon in solid solution;
there is however, evidence to suggest that the effect of carbon tends to saturate, so that Ho
should not exceed a maximum value of about 800 Hv irrespective of carbon concentration [26].
Consequently, the maximum value of Ho permitted in the present analysis is taken to be 800
Hv.
2.3.2 An Independent Calculation
There are more elaborate theories available for the change in the strength of low-carbon
martensite due to the precipitation of cementite, so that the difference (Ho - H F) can be
evaluated independently from the empirical approach discussed above. The change can be
expressed in terms of the decrease in solid solution strengthening as carbon is absorbed during
the growth of cementite, and an increase in strength as the cementite particles precipitation
harden the martensite. Thus, the yield strength of martensite, (T y' is expressed as a combination
of the intrinsic yield strength, the effect of the dislocation cell structure, and precipitation
hardening by cementite [27]:
(2.7)
where (To is the intrinsic strength of martensite, El is the average transverse thickness of the
cell structure, and lp is the average distance between a particle and its two or three nearest
neighbours. The latter is given by lp = 1.18r( 32~8)' where r is the average particle radius
measured in rn, and Ve is volume fraction of the particles. k£ and kp are constants, with
k
p
= 0.519Ve
1
/
2
, MPam-1. The value of k£ is not relevant for the present work since it is the
changes in hardness prior to recovery that are of interest.
The intrinsic strength of martensite can be factorised into the solid solution strengthening
effect of carbon, (Te' and the residual strength (Tb:
(2.8)
From the work of Speich and Warlimont [28]'
(2.9)
Since the hardness H {t} relates to the yield strength as follows [28]'
(Ty = 2.59H {t} - 78.20 , MPa,
the change in hardness of martensite due to cementite precipitation can be written
(2.10)
40
where f}.uy is the change in yield strength (measured in units of MPa), due to cementite
precipitation in the early stages of tempering. Equations to calculate the number of cementite
particles and its volume fraction will be given in a later section. The calculated hardness
changes of martensite in 0.1, 0.2 and 0.4 wt.% carbon steel, in terms of the decrease in solute
carbon and the increase in cementite precipitates, are shown in Fig. 2.4. Although the relation
between hardness and the amount of the precipitation (thus the decrease in solute carbon) is
not linear, the predicted changes in hardness are remarkably consistent with those indicated by
Speich [23] (Fig. 2.5).
2.3.3 Parameters for the Avrami Equation
The tempering data can now be used to 0btain the parameters of the Avrami relation given
in equation 2.3. For the case where martensite is tempered for one hour, the rate constant k
can be calculated using the relationship:
kA = -In{l - ~{1 hr}} (hours)-n (2.11)
where ~{lhr} = (Ho - H{lhr})/(Ho - HF). The calculated k values for different temper-
ing temperatures for data from Speich [23] could then be used to express k as a function of
temperature:
(2.12)
giving kAO = 4.07 X 104XIO.635hours-n, Q = 33598 Jmol-I. R is the universal gas constant.
The time exponent n in the Avrami equation can be obtained by plotting logln{l/(l -
~{t})} against log{t}. The data reported by Speich [23]' which show the changes in hardness
during the tempering of martensite in 0.18 and 0.097 wt.% carbon steels, were used and n was
found to be 0.62. It follows that
(2.13)
This equation can be used to estimate the time necessary to obtain a specified degree of trans-
formation as a function of temperature and the carbon concentration of the steel.
The formation of cementite is known to be exceptionally slow in steels containing large
amounts of silicon. Bhadeshia and Edmonds [26] measured the change in hardness of martensite
by tempering in Fe-0.43C-2.0Si-3.0Mn (wt.%) system at different temperatures. Although their
data are not extensive enough to reveal all the constants needed in the Avrami equation, k AO
was, for that steel, evaluated by assuming that Q and n are the same as for the plain carbon
steels considered previously:
kA = 550exp{-33589/RT}, hours-no (2.14)
2.3.4 Calibration for tf)
The method used to estimate the upper to lower bainite transition temperature, involves
a comparison of the time td required to decarburise a plate of ferrite, with the time interval tf)
necessary to obtain a "detectable" amount of cementite precipitation in the ferrite. If td ~ tf)
then it may be assumed that upper bainite is obtained, and vice versa (Fig. 2.6). Of course, tf)
is a function of ~, and instead of choosing a detectable value of ~ in an arbitrary way, the value
was fixed by comparison with experimental data on Ls' The Fe-0.43C-2.0Si-3.0Mn (wt.%)
41
system is ideal for this purpose since the Bs, Ms and Ls temperatures are well characterised
[20]. Calculated values of td and te (the latter for e = 0.01, 0.02 and 0.05) are plotted, together
with Bs, Ms and Ls temperatures in Fig. 2.7. As expected, the td curves exhibit minima, while
te was, over the temperature range of interest, found to increase as the reaction temperature
decreased. For temperatures below the calculated Bs temperatures, where it is possible for
bainite to grow, any intersections between the td and te curves are of relevance to the location
of the transition temperatures. Ls is defined as the highest temperature at which te < td.
On comparing the te and td curves for the Fe-0.43C-2.0Si-3.0Mn wt.% alloy, it was found that
the experimental Ls of around 320
0 C could be predicted fairly accurately is the "detectable"
volume fraction of cementite is set as e = 0.01. Consequently, for all subsequent calculations,
the Ls temperature is defined by the point where te{e = 0.01} < td'
The reason why it is necessary to consider only a very small amount of precipitation
(1%) to explain the onset of lower bainite may be that the relation between the hardness and
the amount of carbon atom used up for the precipitation is not linear (Fig. 2.3); thus, the
calculated 1% precipitation may in reality correspond to a larger degree of precipitation. The
time for decarburisation may also be increased by soft-impingement of the diffusion fields of
neighbouring sub-units of bainitic ferrite in a sheaf of bainite. The partitioning of carbon may
also be retarded by the precipitation of carbides within the ferrite, since the net flux towards
the austenite/ferrite interface would be reduced.
The calculated td and te for plain carbon steels with carbon concentrations of 0.1,0.2,0.3,
0.4 and 0.5 wt.% are shown in Fig. 2.8 as a function of reaction temperature. As expected, the
higher the carbon concentration the longer is the time required to decarburise the plates. On the
other hand, the driving force for cementite precipitation increases with carbon supersaturation,
so that te is found to decrease with Xl' The calculated Ms and Bs temperatures are also
plotted in the Fig. 2.4.
The calculated Ls temperatures are plotted against the carbon content of steels in Fig. 2.9.
According to the calculations, lower bainite should not be observed in plain carbon steels with
carbon concentrations less than 0.32 wt. %. Furthermore, only lower bainite (i. e. no upper
bainite) should be found in steels with carbon content more than 0.4 wt.%. Steels containing
between 0.32 and 0.4 wt.% of carbon should exhibit both both upper and lower bainite, de-
pending on the reaction temperatures. Finally, it should be noted that at low temperatures
where te and td both become very large, the times required for precipitation or redistribution
of carbon exceed that to complete transformation, consistent with the fact that untempered
martensite can be obtained at temperatures near Ms, with the degree of autotempering of the
martensite decreasing as Ms is reduced.
2.4 DISCUSSION
2.4.1 Comparison between calculated and observed Ls temperatures
The general behaviour indicated by the calculations for plain carbon steels, is found to
be that observed experimentally. Some very recent interesting work by aka and Okamoto [10]
(Fig. 2.10) proves that there is no upper bainite in plain carbon steels with more than 0.8 wt.%
of carbon; the only bainite observed was classical lower bainite at all temperatures above the
Ms temperature. This is consistent with the present calculations.
Ohmori and Honeycombe [29]' in a study of plain carbon steels, showed that during isother-
mal transformation above the Ms temperature, only upper bainite could be obtained in samples
42
containing less than OAC wt.% (Fig. 2.11). This is consistent with the calculations presented
earlier, although their observation that upper bainite can be obtained in steels with a carbon
concentration up to '" 0.85C wt.% is not consistent with the theory, nor with the data reported
by Oka and Okamoto [10]. Their diagram additionally indicates a constant Ls temperature
of around 350°C, which is also inconsistent with the theory and with the Oka and Okamoto
results. These particular discrepancies and contradictory experimental results cannot be ex-
plained at this moment, and further research is called for. Although their published diagram
[29] is based on experimental data, the actual data points are not presented and it is therefore
difficult to assess the validity of some of the boundaries illustrated.
The change in the transition temperature from upper to lower bainite has also been reported
by Pickering [5] and Llopis and Parker [8] for a variety of alloyed steels. Their observations
(Fig. 2.12) show that the Ls temperature increases with carbon concentration in lower carbon
region and then goes through a maximum value as the carbon concentration is increased further.
After the maximum, Ls is found to stabilise at a constant value. This tendency is quite
consistent with the present calculations, although the reported constant value of Ls at higher
carbon concentrations is not. A possible explanation arises from the fact that the calculated
Bs temperatures [21] for the alloys concerned turns out to be in very good agreement with
L s temperatures reported by Pickering [5] for carbon concentrations above 004 wt.%. The
implication is that the high carbon data become consistent with the work of Oka and Okamoto,
if it is assumed that no upper bainite is obtained in those steels, the so called transition
temperature corresponding to the Bs temperature. Furthermore, if we focus on Pickering's
data (excluding some of the other points he plotted from unspecified published research), then
the Ls plateau shown in Fig. 2.12 may simply be an artifact of plotting since a smooth sloping
curve can be fitted through all the high carbon data.
According to the present calculations, only lower bainite is expected in steels with more
than 0.32 wt.% of bulk carbon content. However, the calculations are for ferrite plates whose
carbon concentration is initially identical to that of bulk alloy, since the model assumes that
bainite growth is diffusionless, with carbon redistribution occurring after the growth event. As
a consequence of the redistribution, which is expected to be substantial when t
d
is smaller
than to, there is an enrichment in the carbon concentration of residual austenite as the bainite
transformation proceeds. Consequently, any bainite which forms from enriched austenite will
itself have a higher than bulk concentration of carbon. This leads to the possibility of the
transformation beginning with the growth of upper bainite, but with the enriched austenite
then decomposing to lower bainite at the later stages of transformation. There is then a real
possibility of obtaining a mixture of upper and lower bainite in steels containing less than 0.32
wt.% carbon, especially if carbide precipitation from the austenite is relatively sluggish, and
therefore does not act to relieve any carbon enrichment in the austenite.
The maximum carbon concentration that can be tolerated in residual austenite before
the bainite reaction ceases is expressed approximately by the T~ curve [3,4,22]. Therefore if
the carbon concentration in residual austenite at the T~ curve (i.e. xT' ) is greater than 0.32o
wt.%, lower bainite can be expected to form during the later stages of reaction. However, the
formation of cementite from the residual austenite also becomes possible if xT' > xl
o, where
o
xt is a point on the 'Ylb + B) phase boundary (calculated as in [30,31]), since the austenite
will then be supersaturated with respect to the cementite. The fact that a curve showing the
carbon concentration in austenite which is in equilibrium with cementite in plain carbon steels
43
crosses the T~ curve at 0.4 wt.% of carbon concentration (560 DC), leads to the identification
of three regimes for bainite on the Fe-C phase diagram (Fig. 2.13). In steels with more than
0.4 wt.% of the initial bulk carbon content (region B), lower bainite is to be expected from the
earliest stages of transformation. For steels whose composition lies in region A, lower bainite is
expected to be absent during isothermal transformation at all temperatures above Ms, and this
behaviour is valid for any stage of transformation since the austenite cannot be supersaturated
with cementite as far as regime A is concerned. The behaviour in the region marked C should
be more complex. The residual austenite for these steels (region C) may at some stage of
transformation contain enough carbon to precipitate cementite. If the kinetics of cementite
precipitation from austenite are rapid, then lower bainite may not be obtained in steels with
an average carbon concentration less than 0.32 wt.%, but otherwise, a mixed microstructure of
upper and lower bainite might arise.
2.4.2 Comparison with the tempering of martensite
In the present model for the upper to lower bainite transition, the microstructure of lower
bainite in effect arises due to the "autotempering" of supersaturated plates of bainitic ferrite.
The lower bainite should consequently exhibit many of the characteristics of tempered marten-
site. When high-carbon martensite is tempered, the first carbide to form is usually a transition
carbide such as (-carbide, which is replaced eventually by the thermodynamically more stable
cementite. Similarly, when lower bainite forms in high carbon steels, (-carbide forms first, and
transforms subsequently into cementite during prolonged holding at the isothermal transforma-
tion temperature [6].
The chances of obtaining (-carbide (instead of cementite) in lower bainite increase as the
transformation temperature is reduced for the same steel (see Table Il, [6]). As the transfor-
mation temperature is reduced, and the time required to decarburise a supersaturated plate
of bainite increases, a high carbon concentration can persist in the ferritic matrix for a time
period long enough to allow the formation of (-carbide, which does not form if the carbon
concentration is less than about 0.25wt.%, [32]. This effect also explains the result that a
medium carbon Fe-0.43C-3Mn-2Si wt.% steel transforms to lower bainite containing cementite
particles [33]' although when quenched to martensite, gives (-carbide on tempering [26] Some
of the carbon is in the former case, lost to the austenite by diffusion, thereby preventing the
formation of (-carbide.
The ideas discussed here can in principle be predicted using the present model. Fig. 2.14 il-
lustrates calculation for a Fe-0.6C wt% alloy, for transformation temperatures where only lower
bainite is obtained. The continuous curves represent the time required for the carbon concen-
tration in the bainitic ferrite to drop to a specified level, and the vertical line represents the time
tc taken for this concentration to reach 0.25 wt.%, the level below which (-carbide should not
form [32]. The bainitic ferrite is assumed to have an initial carbon concentration of 0.6 wt.%.
The curves are calculated using equation 2.1, but by replacing x~..,with xr, which represents
the amount of carbon in the bainitic ferrite at any instant of time, with x~'" :s; xr :s; Xl' The
dashed curves are schematic, and represent the time (tJ required to precipitate a detectable
volume fraction of (-carbide; there is as yet no theory which can predict these curves, nor
are there suitable experimental data which can be used to estimate the curves empirically. Al-
though the curves are schematic, their form is based on the corresponding curves for cementite,
as used in the earlier analysis. If te < tc, then the lower bainite should contain (-carbide
rather than cementite, and vice versa. It is evident that it is possible to envisage circumstances
44
where a lowering of the transformation temperature can lead to a transition from lower bainite
containing cementite, to lower bainite containing (-carbide. A similar diagram could be used to
rationalise the observation that in a medium carbon steel, the lower bainite is found to contain
cementite, while the tempering of martensite in the same steel leads to (-carbide formation.
2.4.3 Other Differences between Upper and Lower Bainite
The fact that lower bainite forms at a higher undercooling below Bs when compared with
upper bainite has other implications. Sub-unit growth during the bainite transformation ceases
when the interface is blocked by plastic accommodation induced defects [34]. For a given defect
density, lower-bainite sub-units should be longer than those of upper bainite, since the driving
force for transformation increases with undercooling. At lower transformation temperatures the
matrix is able to support higher strains without plastic deformation so that the defect density in
the matrix itself would be lower. Step quenching experiments in which an alloy is first partially
transformed to lower bainite and then up-quenched into the upper bainite transformation range
are consistent with this since they show that the growth of lower bainite ceases following the
up-quench [35]. This also appears to be the case when specimens partially transformed to lower
bainite experience an increase in temperature within the lower bainite transformation range
[36].
2.5 CONCLUSIONS
A model, based on an idea by Matas and Hehemann, has been developed to enable the
estimation of the temperature at which the upper bainite reaction gives way to the formation
of lower bainite. The model involves a comparison between the times required to decarburise
supersaturated ferrite plates with the time required to precipitate cementite within the plates.
If the decarburisation process dominates, upper bainite is predicted whereas relatively rapid
carbide precipitation within the ferrite leads to the formation of lower bainite.
Some of the predictions of the theory are in agreement with reported experimental data.
Consistent with the results of Ohmori and Honeycombe, it is found that lower bainite cannot
form in plain carbon steels containing less than,..; 0.3 wt.% carbon. Upper bainite is predicted
to be absent in plain carbon steels containing more than 0.4 wt.% carbon; this is in agreement
with the results of Oka and Okamoto, although contradictory results have been reported by
Ohmori and Honeycombe, who were able to obtain both upper and lower bainite in high carbon
Fe-C alloys. The maximum in the curve of transition temperature versus carbon concentration,
reported by Pickering and Llopis & Parker is also consistent with the theory.
To summarise, more experimental work is needed to verify some of the detailed predictions
of the model, and to resolve some of the discrepancies between experimental data reported in the
li terature. More work is also needed from a theoretical point of view, to develop fully the kinetics
of cementite precipitation from supersaturated ferrite, and to couple the processes of cementite
precipitation with the simultaneous redistribution of carbon into the residual austenite. In the
mean time, the current model seems to provide a rational basis for the transition temperature.
45
(2.15)
APPENDIX
- Further Modelling of the Kinetics of Cementite Precipitation in Ferrite -
In this section, we consider alternative models for the kinetics of cementite precipitation
in ferrite, with the aim of examining the possibility of more fundamental theory compared with
the rather empirical martensite tempering data based methods used earlier. Such models could
be useful in taking account of alloying element effects on the kinetics of cementite precipitation
from supersaturated ferrite, and hence permit an easy extension of the transition work to alloy
steels.
It is assumed here that the growth of cementite platelets is controlled by the diffusion
of carbon in the supersaturated ferrite, and that it involves the one-dimensional advance of
interfaces parallel to the habit plane of each cementite particle. A one-dimensional parabolic
thickening rate constant ((}'l) for this process can be calculated using the following equation
[25]:
Xl - xre = ~(}' exp{~} [1 _ erf{ (}'l }]
x~a - xre V 4Dfl 1 4Dfl J4Dfl
where x~a and xre are the equilibrium carbon concentrations in cementite and ferrite respec-
tively, at the interface between cementite and ferrite, and Dfl is the diffusion coefficient of
carbon in ferrite [37].
The lengthening rate of the "allotriomorphs" of cementite is correspondingly given by a
rate constant with is taken to be (}'3 ::::= 3 x (}'l; this seems somewhat arbitrary, but gives a similar
aspect ratio to that obtained for allotriomorphic ferrite [38]. The volume of a cementite plate,
Ve, is then given approximately by:
Ve = ((}'ltl/2)((}'3tl/2)((}'3tl/2)
= 9(}'~t3/2
(2.16)
Venugopalan and Kirkaldy [39] reported the average grain size of cementite fa before the
onset of the Ostwald ripening, as a function of tempering temperature of martensite.
(2.17)
where Ve is the volume fraction of cementite, T is the tempering temperature in K, and fa is
the average grain size of the cementite particles in p.m. The number of cementite particles per
unit volume (Ne) can therefore be described by
(2.18)
where vee is the maximum volume fraction of cementite obtained at the temperature concerned,
and can be calculated from the initial carbon concentration in the ferrite Xl (mole fraction) by
vee ::::= 1.0065l~~~1 ' taking account of the difference in ferrite and cementite densities in respect
of the unit cell.
Assuming now, that there is initially a number Ve of sites available for the growth of
cementite, and that no new sites are formed subsequently, the precipitation process can be
described simply in terms of the growth of cementite. Thus, using the extended volume method
46
of Avrami to take account of impingement between growing particles, the volume fraction of
cementite normalised by the maximum volume fraction vee is expressed by
(2.19)
where ee is an extended fraction, given by
(2.20)
For cases where it may not be justified to start the transformation from a fixed number
of growth centres, it is necessary to have some kind of a nucleation rate function. If it is
assumed that nucleation always occurs heterogeneously on dislocations, the nucleation rate per
unit volume, I, on dislocations can be written as follows [25,40]:
(2.21)
(2.22)
where h is the Planck constant, NV the number of atoms per unit volume which are on dis-
location lines, tlG~ the activation free energy of nucleus formation on a dislocation, and tlG~
the activation energy for the transfer of atoms across the nucleus/matrix interface. p is the
dislocation density.
Therefore the volume fraction of cementite normalised by the maximum volume fraction
of cementite is in these circumstances expressed by:
with
1 it 18Ia3e = -I(9a3) (t - r)3/2dr = - _1 t5/2
e vee 1 ° 5 vee
where r is the incu bation period.
The activation energy for nucleation, tlG~, should decrease as the inverse square of the
driving force for nucleation of cementite from ferrite:
(2.23)
so that tlG~should tend to become small relative to tlG~at high carbon supersaturations. In
such circumstances, tlG~may be ignored and the nucleation rate may be expected to decrease
with undercooling. This is consistent with the data from the empirical analysis discussed earlier,
where it is found that te increases monotonically with a decrease in temperature, rather than
showing a C-curve behaviour. Consequently, it seems justified to ignore the tlG~ term for the
present analysis, where it is assumed that tlG~« tlG~. The value of tlG~ is not known,
but the activation energy for the self diffusion in ferromagnetic iron is 240 kJ mol-1 [41]'
and since the a/ () interface has a relatively high energy, tlG~is expected to be less than 240
kJ mol-1. In order to "derive" its value, an attempt was made to match the values of Ne
obtained as discussed earlier, with the number of particles per unit volume, to be expected
using the nucleation function in equation 2.21.
Assuming that the dislocations all lie along < 111 > directions, NV = p/[V} X 2.8664 X
10-1°]. The dislocation density in bainitic ferrite formed at different temperatures has been
47
measured by Smith [42] and Fondekar et al. [43]. Because these data are not on their own
adequate to obtain an expression of the dislocation density as a function of the reaction tem-
perature, similar data for martensite, reported by Kehoe and Kelly [44]' were included in the
analysis, to yield the following empirical relation for the dislocation density in ferrite (Fig. 2.15):
6880.73
log p = 9.28480 + T
1780360
T2
(2.24)
where p is the dislocation density in m -2, and T is the reaction temperature in K. For the
martensite, the transformation temperature was taken to be the Ms temperature. Although
dislocation densities of martensite measured by Norstrom [45] are also plot ted in the figure, those
data were not used in deriving the above expression because of uncertainties in the method used
to assess the thickness of the thin foil samples used. The curve plotted in Fig. 2.14 does not
therefore take account of these data.
To find the most appropriate value of ~G~, the number of particles per unit volume Ne at
the completion of precipitation was calculated as follows:
00
Ne = J 1[1 - ~{t}] dt
o
(2.25)
and Fig. 2.16 shows that a reasonable fit with the empirical data of Venugopalan and Kirkaldy
[39] could be obtained by setting ~G~ = 190 kJ mol-
1
. This was the value used in all subsequent
calculations.
Another possibility of the calculation of the volume fraction of cementite is based on the
theory of diffusion-controlled growth of plate shaped particles [46]. The edgewise growth of a
precipitate plate from a matrix which is initially at a uniform solute concentration Xl' can be
obtained from following equation.
(2.26)
with
vEr",
p = 2Da
11
xaer
1 D
re = ae -
Xl - Xl
( ae -)Vc = /Lo Xl - Xl
oq -
n - Xl - Xl
0- xa-y _ xea
1 1
W here Vc is the velocity of a fiat interface which is controlled by interface kinetics only, r", is
the radius of curvature at the advancing tip of the plate, re is the critical radius for growth at
which the concentration difference in the matrix vanishes in absence of the interface kinetics,
and vE is the edgewise growth rate of the plate. The functions Sl and S2 have been presented
graphically in [46]. The terms xre, x~a are the carbon concentrations at the interface in ferrite
and in cementite, rD is the capillarity constant, given by
48
where (J'a(J is the alO interfacial free energy per unit area, V is the molar volume of cementite,
and Drl is the diffusivity of carbon in ferrite.
If a disc shape is assumed for a cementite plate, whose height and radius are respectively
c(J and r, the volume v(J per cementite plate is
(2.27)
where f3 is an aspect ratio of cementite plates. Therefore, using the extended volume method
of Avrami, the volume fraction of cementite normalised by the maximum volume fraction of
cementite V(Je is found to be
where
e{t} = 1- exp{-ee} (2.28)
(2.29)
and N(J is the number of cementite particles per unit volume.
In Fig. 2.17, the calculated times required for e = 0.05 of cementite precipitation in a Fe-
OAC wt.% in terms of only the parabolic growth, the combination of nucleation and parabolic
growth, and plate growth are compared with the calculated times using the empirical methods
used earlier. The calculations used (J'a(J = 0.7J m-2 [47), x~a = 0.25, and the equilibrium carbon
concentration in ferrite as presented in [48]. The aspect ratio f3 in the plate growth model is
assumed either to be three, which is to consistent to the parabolic growth model, or fifteen,
which was as observed approximately in Fe-OA3C-2.0Si-3.0Mn wt.% .
It is evident that the models based on parabolic thickening both indicate much faster
transformation kinetics relative to the empirical results, although the result from the nucleation
and growth model approaches the empirical result at lower temperatures. The plate growth
model, while differing in an absolute sense from the empirical data, gives a similar trend as a
function of temperature. It could in principle be adapted (for example, by reducing the number
of nucleation sites available per unit length of dislocation line) to better fit the experimental
data. An appropriate selection of the nucleation function for cementite precipitation, and the
use of the plate growth theory could, therefore, give closure with the empirical result and allow
the estimation of the lower bainite transformation temperature in alloyed steels as well as in
plain carbon steels. However, it is essential to investigate the early stages of the cementite
precipi tation in detail before any further modifications to the theory for the transition. Note
also that the models ignore any precursor reactions such as the precipitation of (-carbide,
which may influence the overall kinetics of cementite formation.
REFERENCES
1. R. F. Mehl: "Hardenability of Alloy Steels", A. S. M., Ohio, 1939, 1.
2. R. F. Hehemann: "Phase Transformations", 1970, 397.
3. J. W. Christian and D. V. Edmonds: "Phase Transformations in Ferrous Alloys", eds. A. R.
Marder and J. I. Goldstein, A. S. M., Ohio, 1984, 293.
4. H. K. D. H. Bhadeshia: "Phase Transformations '87", ed. G. W. Lorimer, Institute of Metals,
London, 1988, 309.
5. F. B. Pickering: Transformation and Hardenability in Steels, Climax Molybdenum Co, Ann
Arbor, 1967, 109.
49
6. S. J. Matas and R. F. Hehemann: Trans. AIME, 1961,221,179.
7. S. Matsuda: Tetsu-to-Hagane, 1970, 56, 1428.
8. A. M. Llopis, referred to in E. R. Parker: Metall. Trans., 1977, 8A, 1025.
9. H. K. D. H. Bhadeshia: Acta Metall., 1980, 28, 1103.
10. M. Oka and H. Okamoto: Proc. Int. Conf. Martensitic Transformations '86, The Japan
Institute of Metals, 1986, 271.
11. H. K. D. H. Bhadeshia and D. V. Edmonds: Metal Science, 1979, 13, 325.
12. V. Franetovic, A. K. Sachdev and E. F. Ryntz: Metallography, 1987, 20, 15.
13. V. Franetovic, M. M. Shec and E. F. Ryntz: Materials Science and Engineering, 1987, 96,
231.
14. R. P. Smith: Acta Metall., 1953, 1, 578.
15. C. Wells, W. Batz and R. F. Mehl: Trans. Met. Soc. A. I. M. E., 1950, 188, 533.
16. R. H. Siller and R. B. McLellan: Trans. Met. Soc. A. I. M. E., 1969, 245, 697.
17. R. H. Siller and R. B. McLellan: Metall. Trans., 1970, 1, 985.
18. R. Trivedi and G. M. Pound: J. Appl. Phys., 1967, 38, 3569.
19. H. K. D. H. Bhadeshia: Metal Science, 1981, 15, 477.
20. H. K. D. H. Bhadeshia and D. V. Edmonds: Acta Metall., 1980, 28, 1265.
21. H. K. D. H. Bhadeshia: Metal Science, 1981, 15, 178.
22. H. K. D. H. Bhadeshia: Acta Metall., 1981, 29, 1117.
23. G. R. Speich: Trans. AIME, 1969, 245, 2553.
24. W. C. Leslie: Mc Graw-Hill, Inc, New York, USA., 1981.
25. J. W. Christian: "The Theory of Transformations in Metals and Alloys", 2nd edition, Part
1, Pergamon Press, Oxfm'd, 1975.
26. H. K. D. H. Bhadeshia and D. V. Edmonds: Metal Science, 1983,17,411.
27. J. Daigne, M. Guttmann, and J. P. Naylor: Materials Science and Engineering, 1982,56, 1.
28. G. R. Speich and H. Warlimont: JISI, April, 1968, 385.
29. Y. Ohmori and R. W. K. Honeycombe: Trans. ISIl, 1971, 11, 1160.
30. C. Zener: Trans. AIME, 1969, 169, 513.
31. W. A. West: Trans. AIME, 1969, 169, 535.
32. C. S. Roberts, B. L. Averbach and M. Cohen: Trans. A. S. M., 1957,45, 576.
33. H. K. D. H. Bhadeshia and D. V. Edmonds: Metal Science, 1979, 13,325.
34. H. K. D. H. Bhadeshia and D. V. Edmonds: Metall. Trans. A, 1979, lOA, 895.
35. R. H. Goodenow and R. F. Hehemann: Trans. AIME, 1965, 233, 1777.
36. J. S. White and W. Owen: J. I. S. I., 1961, 197,241.
37. R. B. McLellan, M. 1. Rudee and T. Ishibachi: Trans. Met. Soc. A. I. M. E., 1965, 233,
1938.
38. J. R. Bradley and H. 1. Aaronson: Metall. Trans. A, 1977, 8A, 317.
39. D. Venugopalan and J. S. Kirkaldy: Hardenability Concept with Applications to Steel, ed D.
V. Doane and J. S. Kirkaldy, 1978, 249.
40. J. W. Cahn: Acta Metall., 1956, 4, 572.
41. J. Fridberg, L. -E. Torndahl and M. Hillert: Jernkont. Ann., 1969, 153, 263.
42. G. M. Smith: Ph. D. Thesis, University of Cambridge, 1984.
43. M. K. Fondekar, A. M. Rao and A. K. Mallik: Metall. Trans., 1970, 1, 885.
50
44. M. Kehoe and P. W. Kelly: Scripta Metall., 1970,4, 473.
45. 1. -A. Norstrom: Scandinavian Journal of Metall., 1976,5, 159.
46. R. Trivedi: Metall. Trans., 1970, 1, 921.
47. J. J. Kramer, G. M. Pound and R. F. Mehl: Acta Metall., 1958, 6, 763.
48. H. K. D. H. Bhadeshia: Metal Science, 1982, 16, 167.
51
TRANSITION FROM
UPPER TO LOWER BAINITE
SUPERSATURATED FERRITE
/
.......
.. . .
. .
, ,
:..~/~J ;};)" "-:.'.
" ~ ~ I 1/ , .
.. :. -.---- .... : ..
-
UPPER BAINITE LOWER BAINITE
Fig. 2.1 Schematic illustration of the transition from upper to lower bainite. Lower bainite is oh-
tained when the time required for excess carbon to partition from supersaturated ferrite
into the residual austenite becomes large relative to the time required to precipitate cemen-
tite within the bainitic ferrite. Note that any carbon-enrichment of the residual austenite
may eventually lead to the precipitation of further carbides, as illustrated above.
52
III
1.0E+00
1.0E-01
1.0E-02
1.0E-03
1.0E-04
300 400 500
Temperature, °c
600 700
Fig. 2.2 Calculated time for the decarburisation of supersaturated ferrite plates (of thickness 0.2
JLm ) in plain carbon steels with 0.1, 0.2 and 0.4 wt.% carbon respectively. The calculated
martensite-start and bainite-start temperatures are also indicated.
800
600
>
I
tIi
III
CD
C
-0 400•...
0
I
200
...•~
Recovery etc.
o
o 100 200 300 400 500 600
Temperature,OC
Fig. 2.3 Hardness curves for iron-carbon martensite samples which were tempered for 1 hour at
the temperatures indicated; data due to Speich, [23]. The data to the left of the vertical
line lar&e1y represent changes due to the precipitation of carbides, rather than recovery or
coarselllllg processes.
53
o-100
>
I
vi
III
(I)
C
"0
L-
a
-200..c
c
(I)
Cl
c
a
..c
U
-300
-400
0.0 0.2 0.4 0.6 0.8 1.0
Normalised volume fraction of cementite
Fig. 2.4 Calculated changes in the hardness of martensite, due to cementite precipitation in 0.],
0.2 and 0.4 wt.% plain carbon steels. Note that the horizontal axis represents the volume
fraction of cementite normalised with respect to its equilibrium volume fraction.
400300200100o
o
400
•
>
I
vi 300
III
(I)
C
"0
L-
a
..c
c
(I)
200
Cl
c
a
..c
u
"0
.2
a
:J 100u
a
U
Observed change in hardness, HV
Fig. 2.5 Comparison of calculated changes in the hardness of plain carbon martensite, during tem-
pering which leads to the precipitation of excess carbon in the form of cementite, with data
reported by Speich [23].
5·1
10
3
10
2
.
1
j10
10'0
10-
1
1
00 200
No LB
'.
'.
300 400
Temperatur •• oC
te
500 600 200
LB ! UB
300 400
r.mp.ratur •. OC
500 600 200
No UB
300 400
Temp.ratura.oC
500 600
U?
U?
Fig. 2.6 Schematic illustration of how differences in the relative behaviours of the td and te curves
can lead to: (a) a steel which is incapable of transforming to lower bainite; (b) a steel which
should under appropriate transformation conditions be able to transform to upper or lower
bainite; (c) a steel in which bainitic transformation always leads to the formation of lower
bainite. The abbreviations LB and UB refer to lower and upper bainite respectively.
·· .. 0.02
0.01
10-1
100 200 300 400 500 600
Temperature,OC
Fig. 2.7 Calculated decarburisation time (td : solid line) and the time required for cementite pre-
cipitation (to: dashed line) as discussed in the text, for a Fe-0.43C-2.0Si-3.01\fn, wt.%,
alloy. to was calculated for 0.01, 0.02 and 0.05 of cementite precipitation.
56
600500400300200100
10-3
o
10-2
10+3
Fe-C system
10+2
O.2wt% C
10+1 0.3wt% C
l/)
- 10°Q)
E
l-
10-1
Temperature,OC
'"
'"
'"
'"
'"
'"
'"
'"
' ..
'"
,"
10+1
0.5wt% C
0.6wt% C
l/)
- 10°Q)
E
l-
10-1
Fe-C system
100 200 300 400 500 600
Temperature,OC
Fig. 2.8 Calculated t
d
and t
9
curves for plain carbon steels.
57
700
600
500
0
0
,;
400•....
:J--0
•....
CD
a.
300E
CD
I-
200
100
Fe-C system
-.----- .------.
Ss .- •...•..•.
cf·'·
•...•+ I -----____________ a • ----
+__________ I Ls
Ms +.
+~+....
! +. +. + ----------- +
. "-.0
a
~~
~;
;,P"';
0 " .
,.
o
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Carbon wt.%
Fig. 2.9 Calculated lower bainite-start temperatures Ls for plain carbon steels, as a function of the
transformation temperature. !Ifs and Bs are, respectively, the calculated martensite-start
and bainite-start temperatures.
500
•
B~"'"..
400 '.'. • • •'.'.
FP...•.
'.
a '. • • •'.
'.
0
0 '.'.
300 a 0 '. • •,; '.
'.•.... ,.
:J '.-- Ms···.... a a a •0•....
CD
". LBa. '.
E 200 '. 0 a a
CD '.
I- a
'.'.
'. a 0
'.
100
M
o
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Carbon content, wtJ.
Fig, 2.10 Experimental data (Oka and Okamoto, [10]) illustrating the temperatures at which fine
nodules of pearlite (FP), classical lower bainite (LB) and martensite (M) were obtained
by isothermal transformation of plain carbon steels. The lines represent our calculated
bainite-start and martensite-start temperatures.
58
LP
us
8s
0.8
Ms
0.4 0.6
Carbon content, wt%
0.2
900
800
700
0
0
a> 600L..
:J-C
L..
Cl)
a.
500
E
Cl)
I-
400
300
200
0.0
Fig. 2.11 The effect of carbon concentration on the temperature range where each microstructure
is formed, after Ohmori and Honeycombe [29]. (A : austenite, F : ferrite, LP : lamellar
pearlite, DP : degenerate pearlite, DB : upper bainite, LB : lower bainite, M : martensite,
Bs : bainite-start temperature, Ms : martensite-start temperature.)
600
·.8s
o550
500
0 •0
a> rL..:J /- .,.- Upper 8ainitec 450 o • ••• •L..
// • \ L10pisCl) •a.
E •
Cl) \I-
400 •
.\
350
Lower 8ainite
0 0-
300
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Carbon content, wt.%
Fig. 2.12 Effect of carbon concentration on the temperature of change from upper to lower bainite,
in alloy steels. After Pickering, [5]' and Llopis and Parker, [8]. The dashed line represents
our values of calculated bainite-start temperatures.
59
800
700
u 600
o
CD
L..
:J.•...
~ 500
CD
a.
E
CD
t-
400
300
200
0.0 0.2 0.4 0.6
Carbon content, wtl.
0.8
Fig. 2.13 Identification of regimes (A, B, C) in which the progress of isothermal transformation can
lead to changes in the nature of the transformation product. The line marked xI8 is the
calculated I I( 1+ FeaC) phase boundary.
1.0
0.8
0.6
Cl)
qj
E
i=
0.4
0.2
0.0
0.000 0.005 0.010 0.015 0.020 0.025 0.030
Carbon mole fraction in ferrite
Fig. 2.1-1 TIlustration of a possible explanation for the transition from lower bainite containing epsilon
carbide, to lower bainite containing cementite
60
1016
~Q
N
'E
>.
:= •1Il
C
Q)
-0
1015c
0
- •0
0
0
.!::! •Cl ••
10
14
200 300 400 500 600 700
Fig. 2.]5
Temperature, QC
Changes in the dislocation density of bainitic ferrite (open circles) and martensite (solid
circles) as a function of the reaction temperature. The cluster of points that lie below the
curve are due to Norstrom [45]. The other data are due to Smith [42]' Fondekar et al. ['13],
and Kehoe and Kelly [44].
1.0E+23
1.0E+22
C'?
'E
IIi"
Q)
.~
1:
1.0E+210
Cl.-0
...
Q)
..c
E
:J
1.0E+20z
_ - -..
1.0E+19
200 250 300 350 400 450 500
Temperature, DC
Fig. 2.]6 Calculated number of cementite particles at the completion of precipitation NB (line ])
compared with the values (line 2) estimated from the empirical data of Venugopalan and
Kirkaldy [39].
6]
500
"'"
450400
.••.•..•••• "tl.... •.•
".0 ~ .
"-
350300
3.~
0.. "'.•
-.. .'
".. ,
.. :......•.
".' ...... ..
....~' ',6
.,:::~:":':"'"
4 ", ''0',.',
'0
250
1.0E-03
200
1.0E+04
1.0E+03
III
c: 1.0E+02
0
:;:
0
:=
a.
1.0E+Ol
u
Q)
L-
a.
Il)
1.0E+000
0
L-
a••...
Q) 1.0E-Ol
E
t=
1.0E-02
Temperature, °c
Fig. 2.17 Time for ~ = 0.05 cementite precipitation calculated from an empirical method (line ]), a
parabolic growth model (line 2), a nucleation and parabolic growth model with ~G~ = ]90
J mol-1 (line 3) and with ~G~ = 240 J mol-1 (line 4), and a plate growth model with
the aspect ratio of 3 (line 5) and ] 5 (line 6) discussed in the text, as a function of the
reaction temperature.
62
CHAPTER 3
A MODEL FOR THE MICROSTRUCTURE OF SOME ADVANCED BAINITIC STEELS
3.1 INTRODUCTION
Bainitic steels are now at the forefront of some potentially exciting developments in the
steel industry, especially when the steels are destined for high-technology applications [1-15].
This, combined with recent advances in phase transformations theory, provides a unique op-
portunity for the development of fundamental alloy design procedures which could be used in
the optimisation of candidate steels before they are fully commercialised.
In discussing bainitic steels, it is necessary to distinguish between two microstructural
classes of bainitic steels. Although bainite is generally recognised to be a non-lamellar mix-
ture of ferrite and carbides, the carbide precipitation reaction often lags behind the growth
of bainitic ferrite, the time lag sometimes being so long that carbides are simply not found
in the microstructure obtained by transformation within the bainite temperature range - the
microstructure then consists of just bainitic ferrite and carbon-enriched residual austenite (to-
gether with any martensite which might form as the residual austenite is cooled to ambient
temperature) [16-18]. A large number of bainitic alloys are used in a condition where carbides
are not found in the microstructure, and this forms one of the classes of bainitic steels; the
other is the more conventional microstructure in which carbide particles are found between the
bainitic ferrite platelets, and in the case of lower bainite, within the platelets as well. The
theoretical problem which is the aim of the present work, is also therefore two fold: to predict
mixed microstructures of bainitic ferrite, austenite and martensite, and to predict those con-
taining mostly mixtures of bainitic ferrite and carbides. The complete problem is in fact quite
formidable (Fig. 3.1) and only some aspects of it are addressed in this chapter.
3.2 CARBIDE-FREE BAINITIC STEELS
It is an experimental fact that in steels where other reactions do not interfere (or occur
simultaneously) with the growth of bainitic ferrite, the maximum volume fraction of ferrite
that forms on prolonged holding at the isothermal transformation temperature is far below that
expected on the basis of equilibrium or paraequilibrium transformation [19-22]. The important
characteristic of this incomplete reaction phenomenon is that the reaction stops well before the
austenite achieves its paraequilibrium carbon concentration as given by the Ae~ curve on the
phase diagram [17,18]. In fact, it stops when the carbon concentration of the residual austenite
approaches the To curve, which describes the locus of all points on the phase diagram where
austenite and ferrite of the same chemical composition have identical free energies. For bainite,
whose growth is accompanied by an invariant-plane strain shape deformation, the strain energy
of transformation is about 400 Jmol-1 [23]' and the To curve as modified to account for this
stored energy is called the T~ curve.
A transformation in which carbon partitions during growth, and in which the mechanism
of transformation is reconstructive (so that the product phase is not limited by austenite grain
boundaries, and can grow to any size), can continue until the austenite achieves its equilibrium
or paraequilibrium composition. Such a mechanism cannot therefore explain the observations
63
described above. On the other hand, the incomplete reaction phenomenon can be understood
if it is assumed that the bainitic ferrite grows withou t diffusion, with the carbon being parti-
tioned into the residual austenite immediately after the growth event. As the austenite becomes
progressively enriched with carbon, a stage is eventually reached when it is thermodynamically
impossible for further bainite to grow by diffusionless transformation. At this point, the compo-
sition of the austenite is given by the T~curve of the phase diagram. This also explains why the
degree of transformation to bainite is zero at the bainite-start (B s) temperature and increases
with under cooling below Bs' The T~ curve has a negative slope on a temperature/carbon con-
centration plot; the austenite can therefore tolerate more carbon as the temperature is reduced,
before diffusionless transformation becomes impossible. The To and T~ curves are in fact only
slightly different in carbon concentration, and given that the carbon is often inhomogeneously
distributed in the residual austenite [24-27], it is a reasonable assumption to neglect the strain
energy term in estimating the austenite composition when isothermal bainitic ferrite growth
ceases.
3.2.1 Thermodynamics
As emphasised above, the incomplete reaction phenomenon can only be assessed quantita-
tively in steels where other reactions do not overlap with the formation of bainitic ferrite. These
steels must obviously have a large enough bainitic hardenability and in addition should con-
tain elements such as AI, Si, or er which retard the precipitation of cementite [28-33]. Many
commercial alloys satisfy these conditions: ultrahigh-strength alloys such as "300M" steel,
austempered ductile cast irons, forging steels, bainitic dual phase steels, etc. In all cases, their
microstructures should consist of mixtures of bainitic ferrite, retained austenite and martensite.
An established method of estimating the limiting volume fractions in such microstructures
is based on the To concept [18]. This assumes that isothermal reaction is permitted for a time
period te which is long enough to permit the volume fraction of bainitic ferrite to reach its
limiting value. The effect of kinetic limitations is considered later.
The microstructure after isothermal transformation (at Tb) in the upper bainite trans-
formation range followed by cooling to ambient temperature (TA) consists of bainitic ferrite,
carbon-enriched retained austenite and untempered martensite which forms during cooling from
Tb -+ TA' At Tb' the bainite reaction will cease when the carbon concentration of the residual
austenite (xi) approaches the To curve:
(3.1 )
If the small difference in density between the ferrite and austenite is ignored, then the
maximum volume fraction of bainitic ferrite (i.e. vbe) is given by a lever rule applied to the
Ae~ and To curves:
(3.2)
where xr'Y is the paraequilibrium carbon concentration of the bainitic ferrite, Xl the average
carbon concentration of the alloys and 5 is the amount of carbon which is tied up as carbides
within the bainitic ferrite.
On cooling the sample to ambient temperature, some of the residual austenite may trans-
form to martensite with the remainder being retained. The martensite-start tern perature (Ms)
64
of the residual austenite can be estimated by assuming that its carbon concentration is xTo' so
that the amount of martensite (V",/) that forms is given by [34]:
(3.3)
Note that when dealing with upper bainite, the method permits the calculation of the
volume fractions of all the phases and also their detailed chemical compositions. There is
unfortunately, no method for predicting S, so that similar calculations for lower bainite are not
yet possible. The method also assumes that the partitioned carbon is distributed homogeneously
within the residual austenite. No account is taken of the fact that many industrial alloys are
chemically heterogeneous.
Some new published data are available for comparison against the To criterion. The lattice
parameters of austenite retained in Fe-0.2C-1.5Si wt.% alloys containing varying concentrations
of manganese have been measured using X-ray diffraction by Usui et al. [35]. Since carbon
in solid solution causes an expansion of the austenite lattice parameter, these data can be used
to deduce the carbon concentration of the retained austenite. Taking the relationship between
the lattice parameter and xI to be given by [36]:
a-yo = 0.3573 + 0.00075xi, nm (3.4)
It is found that the calculated compositions of the retained austenite agree well with the
calculated To curves (Fig. 3.2), and the effect of manganese is also well represented by the
thermodynamically calculated To curves.
3.2.2 Kinetics
Many industrial heat-treatments involve isothermal transformation for time periods less
than those required to allow the bainite reaction to reach completion. It is often the case
that the heat-treatment utilised is not isothermal. The thermodynamic approach described
above fails in both of these circumstances. To correctly treat such cases, it is necessary to
be able to predict the appropriate time-temperature-transformation (TTT) and continuous-
cooling-transformation diagrams for bainite.
Bhadeshia has presented a method [37]' based on Russell's expression for incubation time
during nucleation [38,39], for estimating the TTT diagram of multicom ponen t steels. The
TTT diagram is considered to consist essentially of two 'C'-curves, the high temperature curve
representing reconstructive reactions such as allotriomorphic ferrite and pearlite, the lower
temperature curve representing displacive transformations such as Widmanstatten ferrite and
bainite. Most of the features of published TTT diagrams for steels can be understood via
the different effects of alloying elements on these two C-curves. The original paper however,
dealt with just a prediction of the incubation period prior to the onset of transformation at each
temperature (i.e. the curves representing the initiation of a detectable degree oftransformation).
This is of course inadequate for the present requirements but the method can be adapted to
predict the whole family of C-curves representing the progress of bainitic reaction in a particular
steel, as described below.
In the original model, the incubation time T for the beginning of transformation is given
by:
65
(3.5)
where p, z, Q' and C
4
are constants obtained by fitting to experimental data, b..Gm represents
the maximum driving force available for nucleation and T represents the incubation time prior
to the formation of a detectable degree of transformation product. As the bainite reaction
proceeds beyond this initial stage, the austenite becomes enriched with carbon. The degree
of enrichment can easily be calculated using the mass balance condition. In the present work,
bainite C-curves representing further degrees of reaction are calculated by applying equation
(3.5) to enriched austenite, with b..Gm depending on the new composition of the austenite.
Scheil's rule [39] is then used to convert the TTT diagram into a CCT diagram. It is
therefore assumed that the additivity principle applies to each of the C-curves in the TTT
diagram, with the specified degree of transformation being achieved if
(3.6)
where Tj{T} is the incubation time at temperature T for the C-curve representing a fraction f
of reaction, b..t is the time interval spent by the sample at the temperature T, and t is the time
defined to be zero above the bainite-start temperature.
Typical calculations, for a series of low alloy silicon-rich steels are presented in Fig. 3.3.
Note that they do not allow for the formation of lower bainite, for the solidification-induced
chemical segregation that is inevitably present in commercial alloys, the effects on inhomoge-
neous carbon distributions, etc. The calculations are in this sense, unrealistic, but the trends
that they indicate should be correct.
3.3 CARBIDE PRECIPITATION FROM BAINITIC FRRITE
As discussed earlier, the growth of bainite is probably diffusionless, but any excess carbon in
the supersaturated ferrite soon afterwards partitions into the residual austenite or precipitates
within the bainitic ferrite in the form of carbides. When the process of carbon partitioning
into the residual austenite is rapid relative to that of car bide precipitation, the transformation
product is called "upper bainite", whereas "lower bainite" is obtained when some of the carbon
supersaturation is relieved by precipitation within the bainitic ferrite.
The transition from upper to lower bainite can therefore be estimated by comparing the
time td required to partition excess carbon into austenite, with the time tc necessary to achieve
a detectable degree of carbide precipitation within the ferrite [40]. The diffusion time is given
by:
(3.7)
where Xl is the average carbon concentration in the steel as a whole, x~'"1and xr' are the carbon
concentrations in the ferrite and austenite which are in paraequilibrium, and D is a weighted
average carbon diffusivity in austenite [41,42].
66
Although there is as yet no rigorous model capable of predicting the kinetics of carbide pre-
cipitation from supersaturated ferrite, it is possible to fit Avrami type relationships to marten-
site tempering data and estimate the time period te [43]' and consequently, by comparing td
and te, the upper to lower bainite transition temperature L s' Using this method, it has been
demonstrated that there is a maximum in Ls as a function of carbon concentration at around
0.4 wt.% carbon (Fig. 3.4) in Fe-C alloys. It is further predicted that in plain carbon steels,
only upper bainite can be obtained before the onset of martensitic transformation, followed
by a narrow range of carbon in which both upper and lower bainite are possible, and as the
carbon concentration is increased (> 0.4 wt.% C), only lower bainite and martensite can be
obtained during heat treatment at temperatures below that at which pearlite forms [43]. These
predictions, together with the position of the peak and the shape of the transition curve are in
fact consistent with published experimental data on plain carbon steels [43] (Fig. 3.5).
On applying this model [43] to experimental data on Fe-Mn-Mo-C alloys [44]' without
taking account of any effect of the substitutional alloying elements on carbide precipitation
kinetics, good agreement between theory and experiment is once again obtained (Fig. 3.6).
3.4 CEMENTITE PRECIPITATION FROM SUPERSATURATED AUSTENITE
Austenite is supersaturated with respect to cementite (0) precipitation when xl > xt; for
the bainite reaction, this means that xTo > xI
8
since the growth of bainitic ferrite stops when
xl reaches the value xTo given by the To curve of the phase diagram. A consequence of the
precipitation of cementite from austenite is that its carbon concentration drops below xTo' so
that the growth of bainitic ferrite can continue to an extent larger than would be otherwise
possible. It is therefore important to be able to predict the kinetics of cementite precipitation
from the residual austenite, and the problem is one of nucleation and growth of cementite.
Assuming that the cementite forms by reconstructive transformation (the actual mecha-
nism has yet to be established), the incubation time T for nucleation can be estimated using
Russell's model [37,38]. The incubation time should decrease as the magnitude of the maximum
driving force for nucleation, !:i.Gm, increases (see equation 3.4). Our calculations based on the
assumption that the nuclei form with the equilibrium composition, indicate that the retardation
of cementite precipitation by silicon cannot be related to the nucleation stage, since silicon is
actually found to increase !:i.Gm (Fig. 3.7).
3.4.1 Effect of Alloying Elements on Cementite Growth
The growth rate is here estimated using the theory of diffusion-controlled interfacial motion.
Since the cementite particles found in bainitic microstructures are plate shaped and usually
have large aspect ratios, it may be a good approximation to represent the formation of these
plates using a one-dimensional parabolic thickening rate constant (a1), so that the plates are
essentially treated here as allotriomorphs. It is further assumed that any concentration gradients
can be approximated as being constant (after Zener) and the problem is studied for ternary
Fe-C-X alloys, where "X" denotes a substitutional solute; C and X are in the equations that
follow, identified using the subscripts 1 and 2 respectively. When local equilibrium is assumed to
exist at the transformation interface, it can be demonstrated using the methods of Kirkaldy and
Coates [45,46] that the parabolic thickening rate constant can be obtained from the simultaneous
solution of the following equations:
67
(3.8)
(3.9)
(3.10)
(-'Y 'Y9)2
2 D'Y X2 - X2
0'1 = 22( 9'Y -'Y)( 9'Y 'Y9)·X2 - X2 X2 - x2
where xl and x; represent the average composition of the austenite, and X~:2 and xI~2 the
equilibrium compositions of the cementite and austenite respectively. 15J.l is the weighted
average diffusivity of car bon in the austenite, DI2 represents the dependence of the carbon flux
in austenite on the concentration gradient of the substitutional solute, and D;2 the diffusivity
of the substitutional solute in the austenite. The diffusion coefficient DI2 is given by
'Y
D'Y - D'Y f12Xl
12 - 11 1+ f X 'Y
11 1
where the f terms are the Wagner interaction parameters which arise in dilute solid solution
models, as discussed elsewhere [47]. It is assumed that the composition of the growing cementite
is uniform everywhere.
It is also possible that the cementite grows with paraequilibrium, in which case the iron
atom to substitutional solute atom ratio should be constant everywhere. The rate constant is
then given by:
(-'Y 'Y9)2
0'2 _ If' Xl - Xl
1 - 11 (9'Y -'Y)( 9'Y 'Y9) (3.11)
Xl - Xl Xl - Xl
Note that the concentration terms x~'Y and xI
9
now represent paraequilibrium rather than
equilibrium compositions of the two phases.
Fig. 3.8 shows the results of growth rate calculations, carried out assuming paraequilibrium
growth of cementite from austenite. It is evident that the growth of cementite is substantially
retarded by silicon.
3.5 STABILITY OF AUSTENITE AND EFFECT ON PROPERTIES
The mixture of bainitic ferrite and austenite is in principle an ideal combination from
many points of view [11-15]. Most modern high-strength steels are clean in the sense that
they are largely free from nonmetallic inclusions. Those destined for critical applications are
usually vacuum arc refined prior to fabrication and heat-treatment. As a consequence, it is the
components of the intrinsic microstructure, such as particles of cementite, which are responsible
for damage initiation. The upper bainitic ferrite and austenite mixture is however, free from
cleavage and void nucleating cementite. The ferrite also has a very low interstitial content, since
much of the excess carbon is partitioned into the residual austenite; the toughness of ferrite is
known to deteriorate rapidly with an increasing concentration of carbon in solid solution.
The microstructure derives its strength from the ultrafine grain size which results from the
displacive mechanism of ferrite growth, giving an effective grain size which is much less than
1 tt m. Such a small grain size cannot be achieved by any commercial process other than me-
chanical alloying (powder metallurgical process). A fine grain structure is an optimum method
for improving strength since unlike most other strengthening mechanisms, the improvement in
68
strength is also accompanied by an improvement in toughness. The intimately dispersed and
ductile fcc austenite films between the ferrite platelets can be expected at the very least to
have a crack blunting effect, and could also make increase the work of fracture by undergoing
martensitic transformation under the influence of the stress field of the propagating crack (i.e.
the TRIP, or transformation induced plasticity effect [48,49]). The diffusivity of hydrogen in
austenite is relatively sluggish [50]' so that its presence can in some circumstances enhance
stress corrosion resistance [51,52]. And all these potential benefits can be achieved by cre-
ating a duplex microstructure with the cheapest austenite stabiliser available, carbon, whose
concentration in the austenite is enhanced during transformation, so that the average carbon
concentration of the steel need not be large.
In spite of all these potential advantages, the bainitic ferritejaustenite microstructure has
on many occasions failed to live up to its promise [11-15, 53]' primarily because of the instability
of relatively large or blocky regions of austenite which become trapped between sheaves of
bainite. The blocks of austenite tend to transform to high-carbon, untempered martensite
under the influence of small stresses and consequently have an embrittling effect. The films
of austenite that are trapped between the platelets of ferrite in a sheaf are much more stable,
partly because of their higher carbon concentration, and also because of the physical constraint
to transformation due to the close proximity of plates in all directions.
If it is assumed that a fraction 4J of a sheaf is consists of films of austenite, then it can be
demonstrated that the ratio of the volume fractions of film and blocky austenite (prior to any
martensitic transformation) is given by:
(3.12)
where V: and V-yB are the volume fractions of film and blocky type retained austenite re-
spectively, and Va and V-y the total volume fractions of bainitic ferrite and residual austenite
respectively. It is found experimentally that high strength and good toughness can be obtained
by maintaining the above ratio to a value greater than 0.9 [11,12]. The question then arises
as to the factors which control this ratio. There are in fact three different ways of minimising
the volume fraction of blocky austenite, each involving an increase in the volume fraction of
bainite.
Lowering the transformation temperature permits the bainite reaction to proceed to a
greater extent but there is a limit to the minimum transformation temperature since the lower
bainite and martensite reactions eventually set in. An increase in the extent of reaction can
also be achieved by reducing the overall carbon concentration of the steel, so that the austenite
reaches its limiting composition at a later stage of the reaction. The To curves of the phase
diagram, which determine the composition of the austenite at the point where the reaction
stops, can also be shifted to higher carbon concentrations by altering the substitutional solute
concentration of the steel. The effect on toughness in reducing the amount of blocky austenite
is very pronounced, with large reductions in the impact transition temperatures as the ratio
of film to blocky austenite is increased in the manner just described. Note that for a duplex
(}'+ 'Y microstructure, the strength actually increases as the fraction of bainitic ferrite increases,
so that the better toughness is obtained without sacrificing strength. Typical compositions of
high-strength steels which show good toughness are given in Table 3.1. Fig. 3.9 shows how the
69
mechanical properties compare with quenched and tempered steels. It is evident that in some
cases, the properties match those obtained from much more expensive maraging steels.
Table 3.1 Chemical compositions (wt. %) of some successful alloys based on a mixed microstruc-
ture of bainitic ferrite and austenite.
C
0.22
0.40
Si
2.0
2.0
Mn
3.0
Ni
4.0
The properties of these steels improve only slightly when tempered at temperatures not
much higher than the transformation temperature at which the original bainite formed. How-
ever, annealing at elevated temperatures or prolonged periods at low temperatures can lead
to the decomposition of the austenite into ferrite and carbides, with a simultaneous drop in
strength and toughness, especially the upper shelf energy. The latter effect can be attributed
directly to the void nucleating propensity of carbide particles in the tempered microstructure,
as illustrated by the much smaller void size evident in the fracture surface of the tempered
sample.
The mechanical property data on these high silicon steels, especially those steels designed
using the phase transformation theory discussed earlier, look extremely promising. It is how-
ever, unlikely that the experimental steels represent the optimum compositions and further
development work could lead to even better properties. It is also necessary to carry out a
comprehensive assessment of properties such as stress corrosion resistance, fatigue etc.
3.5.1 Ductility
The influence of retained austenite on ductility has been studied mainly in steels contain-
ing a high silicon concentration, where cementite formation can be prevented, and consequently
large quantities of carbon-enriched austenite can be retained. Ductility as measured by tensile
elongation, reaches a peak (optimum) value as a function of the volume fraction of retained
austenite, when the amount of austenite is varied by altering the volume fraction of isother-
mal transformation to bainite [54]. Furthermore, the uniform elongation behaves in a similar
way to total elongation when plotted against the volume fraction of retained austenite. The
difference between the uniform and total elongation decreases as the optimum volume fraction
of retained austenite is reached; beyond the optimum value, tensile failure occurs before the
necking instability so that the difference between uniform and total elongation vanishes.
It seems that the best elongation behaviour is observed when the retained austenite is
present mainly in the form of films between the sub-units of bainite, rather than as blocky
regions between the sheaves of bainite [54]. Hence, the optimum retained austenite content
increases as the transformation temperature decreases, because the sub-unit thickness decreases,
permitting more of the austenite to be in the film morphology for a given volume fraction
of transformation to bainite. For the same reason, the elongation becomes less sensitive to
retained austenite content as the transformation temperature is reduced. While mechanically
70
unstable austenite, i.e. the austenite which decomposes to deformation induced martensite, is
recognised to cause a deterioration in toughness for bainitic steels [11,12,53], this is not the
case for ductility, presumably because of the TRIP effect and the lower strain rates involved in
conventional tensile tests.
It must be emphasised that all these results are very difficult to interpret quantitatively.
Changes in retained austenite content cannot easily be made without altering other factors
such as the tensile strength and the distribution of the austenite. For example, Miihkinen and
Edmonds [14] have reported a monotonic increase in uniform and total ductility with retained
austenite content. The latter was varied by altering the transformation temperature, so that
the strength increased as the austenite content decreased.
3.6 SUMMARY
A start has been made on the complicated problem of predicting the microstructure of
bainitic steels, by developing an approximate method of calculating the time-temperature-
transformation diagram for the formation of bainite. There remain significant difficulties: future
work will have to address the problem of chemical segregation, nonuniform distribution of
carbon in any residual austenite, more complete models for carbide precipitation during bainitic
transformation, and other problems highlighted in Fig. 3.1. It should nevertheless be possible
to give good hints on the expected changes of microstructure and mechanical properties as alloy
chemistry and thermal treatments are varied. Steady progress is also being made in evaluating
the relationships between microstructure and mechanical properties, although more research is
needed to establish these on a quantitative basis.
REFERENCES
1. O. Matsumura, Y. Sakuma and H. Takechi: Trans. ISIl, 1987, 27, 570.
2. O. Matsumura, Y. Sakuma and H. Takechi: Scripta Me tall. , 1987, 21, 1301.
3. S. Bandoh, O. Matsumura and Y. Sakuma: Trans. ISIl, 1988, 28, 569.
4. M. Imagumbai, R. Chijiiwa, N. Aikawa, M. Nagumo, H. Homma, S. Matsuda and H. Mimura:
((HSLA Steels: Metallurgy and Applications", eds. J. M. Gray, T. Ko, Z. Shouhua, W. Baorong
and X. Xishan, ASM International, Ohio, 1985, 557.
5. K. Yamamoto, S. Matsuda, T. Haze, R. Chijiiwa and H. Mimura: ((Residual and Unspecified
Elements in Steel", ASM International, Ohio, USA, 1987, 1.
6. R. Chijiiwa, H. Tamehiro, M. Hirai, H. Matsida and H. Mimura: ((Offshore Mechanics and
Arctic Engineering Conference (OMAE)",Houston, Texas, 1988, 1.
7. K. Nishioka and H. Tamehiro: ((Microalloying '88: International Symposium on Applications
of HSLA Steel", Chicago, illinois, USA, September 1988, 1.
8. H. K. D. H. Bhadeshia: ((Steel Technology International", ed. P. H. Scholes, Sterling Publi-
cations International Ltd., London, 1989, 289.
9. Y. Tomita and K. Okabayashi: Metall. Trans. A, 1983, 14A, 485.
10. Y. Tomita and K. Okabayashi: Metall. Trans. A, 1985, 16A 73.
11. H. K. D. H. Bhadeshia and D. V. Edmonds: Metal Science, 1983,17,411.
12. H. K. D. H. Bhadeshia and D. V. Edmonds: Metal Science, 1983,17,420.
13. V. T. T. Miihkinen and D. V. Edmonds: Materials Science and Technology, 1987, 3, 422.
71
14. V. T. T. Miihkinen and D. V. Edmonds: Materials Science and Technology, 1987, 3, 432.
15. V. T. T. Miihkinen and D. V. Edmonds: Materials Science and Technology, 1987,3, 441.
16. R. F. Hehemann: Phase Transformations, ASM, Metals Park, OH, USA, 1970, 397.
17. J. W. Christian and D. V. Edmonds: 1nl. Conf. on ((Phase Transformations in Ferrous
Alloys", A. R. Marder and J. 1. Goldstein eds., TMS-AIME, Warrendale, PA, USA, 1984, 293.
18. H. K. D. H. Bhadeshia and J. W. Christian: Metall. Trans. A, 1990, 21A, 767.
19. A. Hultgren: Trans. ASM, 1947, 39, 915.
20. A. Hultgren: Jernkontorets Ann. 1951, 135, 403.
21. M. Hillert: Jernkontorets Ann. 1951, 135, 25.
22. E. Rudberg: Jernkontorets Ann. 1952, 136, 91.
23. H. K. D. H. Bhadeshia: Acta Me tall. , 1981,29, 1117.
24. S. J. Matas and R. F. Hehemann: Trans. TMS-A1ME, 1961,221,179.
25. A. Schrader and F. Wever: Arch. Eisenhiittenwesen, 1952, 23, 489.
26. H. K. D. H. Bhadeshia and A. R. Waugh: Acta Metall., 1982, 30, 775.
27. 1. Stark, G. D. W. Smith and H. K. D. H. Bhadeshia: Phase Transformations '87 G. W.
Lorimer ed., Institute of Metals, London, 1988, 211.
28. E. C. Bain: ((Alloying Elements in Steel", ASM, Cleaveland, OH, USA" 1939.
29. W. S. Owen: Trans. ASM, 1954, 46, 812.
30. J. Gordine and 1. Codd: JIS1, 1969,207.1,461.
31. R. M. Hobbs, G. W. Lorimer, N. Ridley: JIS!, 1972,210.2, 757.
32. A. G. Alten and P. Payson: Trans. ASM, 1953,45, 498.
33. E. E. Langer: Metal Science Journal, 1968, 2, 59.
34. P. P. Koistinen and R. E. Marburger: Acta Metall., 1959, 7,59.
35. N. Usui, K. Sugimoto, E. Nishida, M. Kobayashi and S. Hashimoto: CA MP-1SIJ, 1990, 3,
2013.
36. H. K. D. H. Bhadeshia: Metal Science 1982, 16, 159.
37. K. C. Russell: Acta Metall. 1968, 16, 761.
38. K. C. Russell: Acta Me tall., 1969, 17, 1123.
39. J. W. Christian: ((The Theory of Transformations in Metals and Alloys", Part I, 2nd edition,
Pergamon Press, Oxford, 1975.
40. H. K. D. H. Bhadeshia: ((Phase Transformation '87", G. W. Lorimer ed., Institute of Metals,
London, , 1988, 309.
41. R. H. Siller and R. B. McLellan: Metall. Trans. A, 1970, 1, 985.
42. H. K. D. H. Bhadeshia: Metal Science, 1981, 15, 477.
43. M. Takahashi and H. K. D. H. Bhadeshia: Materials Science and Technology 1990, 6, 592.
44. F. B. Pickering: ((Transformation and hardenability in steels", Ann Arbor, MI, Climax
Molybdenum, 1967, 109.
45. J. S. Kirkaldy: Canadian Journal of Physics, 1958, 36, 899.
46. D. E. Coates: Metall. Trans. 1972, 3, 1203.
47. C. Wagner: ((Thermodynamics of Alloys", Addison-Wesley, Reading, Massachusetts, USA,
1952, 53.
48. W. W. Gerberich, G. Thomas, E. R. Parker and V. F. Zackay: 2nd 1nl. Conf. ((Strength of
Metals and Alloys", ASM, Metals Park, Ohio, USA, 1970, 3, 894.
72
49. E. R. Parker and V. F. Zackay: Engineering Fracture Mechanics, 1975, 7, 371.
50. J. H. Shively, R. F. Hehemann and A. R. Troiano: Corrosion, 1966,22, 253.
51. R. Kerr, F. Solana, 1. M. Bernstein and A. W. Thompson: Metall. Trans. A, 1987, 18A
1011.
52. F. Solana, C. Takamatate, 1. M. Bernstein and A. W. Thompson: Metall. Trans. A, 1987,
18A, 1023.
53. R. M. Horn and R. O. Ritchie: Metall. Trans. A, 1978, 9A, 1039.
54. B. P. J. Sandvik and H. P. Nevalainen: Metals Technology, 1981, 13, 213.
73
MODELLING OF BAINITIC MICROSTRUCTURES
I
(carbide-free bainite J (Conventional bainiteJ
I
Bainitic ferrite nucleation and
growth kinetics
I
Effect of solidification induced
chemical segregation
I
Nonuniform distribution
of carbon
I
Intragranular nucleation on
inclusions
I
Martensitic decomposition of
carbon-enriched austenite
Carbide precipitation
from austenite
Carbide precipitation
from supersaturated
ferrite
Transition from upper
to lower bainite
Fig. 3.1 Flow chart summarising the aspects of transformation which need to be addressed in order
to be able to generally predict the microstructure of bainitic steels.
74
Fe-O.2C-1.5SI-Mn mass 0/0
1 10 100
COOLING RATE I K·s·'
1.0Mn
1.5Mn
2.0Mn
2.5Mn
w 1.0
!:::
a:
a:
w
0.8u..
u
E
z
0.6cl:
lXl
Z
0
0.4i=
u
cl:
a:
u..
w 0.2
~
:>
..J
0
0.0>
.1
-..-
-a--··--...-
-- .. -.
(a)
- ...•..,,,,,
'1
\
\
\
\
\
\
\
\
\
Fe·O.2C-1.5SI·Mn mass %
w
!::: 0.12
z
w
I-
III
0.10:>
cl:
0
w
0.08z
cl:
I-
w
0.06a:
z
0
i= 0.04
u
cl:
a:
u..
0.02
w
~
:>
..J
0.000
> .1 1
(b) -..-
-a--·--...-
-- .. -.
10
1.0Mn
1.5Mn
2.0Mn
2.5Mn
100
COOLING RATE I K·s·1
Fe-O.2C-1.5SI-Mn mass %
1.0 .--w ------.- 1.0Mn (C) II- Iin -_ ..•...... 1.5Mn Iz
0.8
Iw --...- 2.0Mn
I- I
a: -- ..-. 2.5Mn Icl: I~
I
u.. 0.6 I
0 I
Z •0 ",",
i= 0.4
",
",
U ",
cl: ",
a: A
u..
w 0.2~
:>
..J
0
>
0.0
.1 1 1 0 100
COOLING RATE I K·s
-1
Fig. 3.3 Typical calculated microstructure for a series of Fe-Mn-Si-C alloys.
75
Fig. 3.2
3
Ae3' --- .....•._- ..•.
;::g --- ..0
(J)
(J)
co
E 2
C1> 1~ ~- 0c 2~
0C1> •- 0en 1 To:::::s - •et •
c
c at 673K
0
0.0~
ctl 0 1 2 3u
Manganese Concentration (mass %)
An analysis of published data [35] on the carbon concentration of retained austenite in
a mixed microstructure of bainitic ferrite, martensite and retained austenite. The cal-
culations of the Ae~ and To curves were carried out as described in [11,12]. Usui et al.
[35] measured the lattice parameter of austenite and used a relationship between the pa-
rameter and carbon concentration to deduce the data illustrated by curve 1. A better
parameter/carbon relation is used here to generate curve 2. It is evident that there is good
agreement of the austenite composition with that predicted by the To calculations.
0.4 0.8 1.2
Carbon Concentration / (mass %)
700
u
6000-
E-c
500
.•..
Q,)
J,..,
:::s 400..•...
~
J,..,
Q,)
ic..
300 ,.E "..........
Q,)
..........
"E-c •• 'to••••••••••••• ,
200
.............
LBs
100
u.O
Bs
Fig. 3.4 Prediction of the upper to lower bainite transition temperature as a function of the carbon
concentration for a series of plain carbon steels [43]. Bs, L s and NI s represent the bainite-
start, lower bainite-start and martensite-start temperatures. Both calculation and tlleory
seem to indicate that only upper bainite is obtained with the carbon concentration for
Fe-C alloys is less than about 0.4 wt.%, and only lower bainite when that concentration is
exceeded.
76
600
c B
U 500
B :B
~f0-E-c 400 b
.I'"
~ d
Q) B B
l-o
='~ 300~ .
l-o
.
Ms
.
Q)
..
c.. ...
E ..
Q) 200 a.' LBsE-c
Data due to Pickering
100
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Carbon Concentration / (mass %)
Fig. 3.5 Comparison between calculated and observed upper to lower bainite transition temperature
in Fe-Mn-Mo-C alloys. Observed data (circle) is after Pickering [44]. On applying the model
[43] to the data, the effect of substitutional alloying elements on carbide precipitation
kinetics is not taken into account .
Fe-O.8C
Fe-O.8C-1 Mn
Fe-O.8C-1 Si
Fe-O.8C-1 Ni
o
300 400 500 600 700 800
Temperature, T / 0 C
1000
2000
3000
.,-
,
-
Jo..
0
0 E-
>< J
CU
E -
CJ ><
ca- E0
CJ
Q,)
"0 ~
:J C.•... 0
c .....
O)CU
CU~
:2 0
:J
Z
Fig. 3.6 Calculated maximum free energy change (G
max
) due to the nucleation of cementite from
austenite in Fe-O.8C-X wt.% ternary alloys, assuming that the cementite nuclei have the
equilibrium composition.
77
160
\0
Cl..
L:
'- 120
u
~-
~
C/)
C/)
l.LJ
Z
:I:
C!) 80
::J
o
I-
l.LJ
et:
::J
I-
U
< 40
et:
L.1..
• Ni Alloy
• Mn Alloy
• Mn Alloy
1000 1400 1800 2200
ULTIMATE TENSILE STRENGTH a / MPll
I
Fig. 3.8 Comparison of the mechanical properties of mixed microstructure of bainitic ferrite and
austenite, versus those of quenched and tempered martensitic alloys [15].
79
Paraequilibrium Growth at 673K
3
Fe-D.8C-Si
Fe-D.6C-Si
Fe-1 C-Mn
Fe-D.8C-Mn
Fe-1 C-Si
21
-----r------ •
10.10
o
~
•.. cs
o
- -ca
C _ •••
o C ~
'(ij ~ 'c
C en (1)
(1)C-
E 0 ~
CU<c
ci> ~ E
C ca 0
oa:';:
Mn or Si concentration (mass %)
lfl
0 10-9
en
E
10-10
~
CS
Cl
.!:: J::.
c-
(1) ~
.lI: 0
o •..
:cC-'
f- (1)-0;:
.- C
"0 (1)
.0 E
ca (1)
~u
a..
10-11
10-12
0.2
Paraequilibrium Growth at 673K
--0- Fe-C
--- •. -- Fe-1Mn-C
• Fe-1 Si-C
0.4 0.6 0.8 1.0
Carbon Concentration in
Austenite (mass %)
1.2
Fig. 3.7 Parabolic thickening rate constants for the paraequilibrium growth of cementite allotri-
omorphs from austenite.
78
CHAPTER 4
REAUSTENITISATION FROM A MIXTURE OF BAINITE AND AUSTENITE
4.1 INTRODUCTION
In this chapter, reaustenitisation from a mixture of ferrite and austenite is studied since it
may then proceed only by the growth of pre-existing austenite. The nucleation of austenite may
not therefore be a controlling factor. For this purpose, a mixture of bainitic ferrite and residual
austenite was selected as the initial microstructure. Prior to reaustenitisation experiments from
a mixture of bainitic ferrite and austenite, the bainite transformation itself was studied to better
understand the mechanism of transformation. Fe-C-Cr and Fe-C-Si-Mn alloys were used for
this purpose.
4.2 EXPERIMENTAL PROCEDURE
A Fe-0.30C-4.08Cr wt.% ternary alloy was used to study reaustenitisation from a mixture
of a bainitic ferrite and austenite. The same material has been studied separately by Bhadeshia
[1,2] .
All dilatometry was performed on a Theta industries high speed dilatometer, which has a
water cooled radio-frequency furnace of essentially zero thermal mass, since it is only the speci-
men which undergoes the programmed thermal cycle. The length transducer on the dilatometer
was calibrated using a pure platinum specimen of known thermal expansion characteristics. The
dilatometer has been specially interfaced with a BBCj Acorn microcomputer so that length, time
and temperature information can be recorded at microsecond intervals during the heat cycle,
and the data are stored on a floppy disc. The information is then transferred to a mainframe
IBM3084 computer for further analysis. Specimens for dilatometry were machined in the form
of 3 mm diameter rods with about 20 mm of length. To avoid surface nucleation and surface
degradation, they were plated with nickel. This nickel plating process is in two stages; nickel
striking and nickel plating. Striking was carried out in a solution made up of 250g nickel sul-
phate, 27 ml concentrated sulphic acid and distilled water, amounting to one litre in all, at 50 °C
with a current density of 7.75 mAmm-2 for three minutes. The plating solution consisted of
l40g nickel sulphate, l40g anhydrous sodium sulphate, 15g ammonium chloride and 20g boric
acid, was made up to one litre with distilled water. The plating was carried out at 50 QCwith
a current density of 0.4 mA mm-2 for fifteen minutes. This two processes finally give a plate
thickness of approximately 0.08 mm.
Isothermal and continuous heating reaustenitisation experiments from a mixt ure of bainitic
ferrite and austenite have been carried out on the dilatometer. All specimens used in the
experiments were heated at 1l00°C for 30 min and quenched in water with ice, to get a
martensitic microstructure prior to the actual heat treatments.
In order to obtain a mixture of bainite and austenite, specimens were reaustenitised at
1000 °C for 10 min and quenched to selected bainite transformation temperatures, 420, 448,
472 °C and held there for 30 min. The quenching from the reaustenitising temperature to the
bainite transformation temperatures was conducted in the dilatometer using helium gas giving
about 30 °C S-l of cooling rate between 800 °C and 500 °C, where the diffusional transforma-
tion of austenite was expected to occur. The calculated [3] time-temperature-transformation
80
(TTT) curve for the Fe-0.3C-4.08Cr wt.% alloy is shown in Fig. 4.1. According to this calcula-
tion, the equilibrium transformation temperature from austenite to ferrite Ae3, paraequilibrium
transformation temperature from austenite to ferrite Ae~, bainite-start temperature Bs and
martensite-start temperature Ms are 781, 730, 492, 348 °C respectively. As it can be seen in
the figure, the depressed diffusional C-curve (upper C-curve) makes the formation ofbainite di-
rectly from austenite easy. In fact, no evidence of transformation during helium quenching was
observed from the length change of the specimens on quenching to the bainite transformation
temperatures.
Following to isothermal transformation to a mixture of bainite and residual austenite, the
specimens were heated directly to isothermal reaustenitisation temperatures of interest with-
out being cooled below the bainite transformation temperatures. Isothermal reaustenitisation
experiments were carried out at temperatures between 720 °C and 820 °C with a rapid heat-
ing from the bainite transformation temperatures at around 500 °C S-l. Continuous heating
reaustenitisation experiments at heating rates between 0.1 and 11.0 °C S-l were also carried
out to study the formation of austenite on heating. We emphasize again that the reausteniti-
sation treatments were started directly from the bainite transformation temperatures after 30
min of isothermal holding without cooling below it (Fig. 4.2).
After the isothermal reaustenitisation at each reaction temperature, the specimens have
been helium quenched to ambient temperature to freeze the completely or partially transformed
austenite into untempered martensite for metallographic study and other experiments.
Prior to the experiments, all specimens were homogenised at 1250°C for 3 days while
sealed in quartz tubes containing a partial pressure of pure argon, and they were quenched into
iced-water to get fully martensitic microstructure.
Optical microscopy was carried out on an Olympus microscope and photographs were taken
by using an Olympus camera fitted to the microscope. Specimens were hot mounted in acrylic
plastic, and ground on silicon carbide paper to a sufficient depth to remove any unrepresentative
surface, and then mechanically ground down to 1200 grade emery paper and finally polished
with 6 and 1 /lm diamond pastes. Before optical microscopy, specimens were etched using 2%
nital.
Microhardness measurements were done on a Leitz hardness measuring digital eyepiece
with the option of Vickers hardness tester to which a computer-count er-printer is attached.
Specimens were polished and etched by 2% nital before the measurements. The indentation
load applied was selected either 0.0981 N or 98.1 N in each case.
Thin foil specimens were prepared for transmission electron microscopy from 0.25 mm thick
discs slit from specimens used in the dilatometry. The discs were ground to about 0.04 mm
by abrasion on 1200 grade emery paper and then electro-polished in a twin jet electro-polisher
using a 5% perchloric acid, 25% glycerol and 70% ethanol mixture solution either at ambient
temperature or at around 0 cC, and 50 V. The microscopy was conducted on either a Philips
EM300 or a Philips EM400T transmission electron microscopes operated at 100 or 120 kV.
4.3 BAINITE TRANSFORMATION
Isothermal experiments were conducted to examine the nature of bainite transformation
which is used to obtain the initial microstructure for reaustenitisation experiments. Fe-2.0Si-
3.0Mn wt.% systems with four different carbon contents; 0.059, 0.12, 0.22 and 0.43 wt%, were
studied as well as the Fe-0.3C-4.08Cr wt.% alloy.
81
4.3.1 Bainite transformation in the Fe-O.3C-4.08Cr wt. % alloy
Three bainite transformation temperatures were selected to obtain three different mixtures
of bainite and residual austenite. The relative length changes during isothermal bainite trans-
formation at these temperatures are shown in Fig. 4.3. It can be seen that 30 min of holding at
each bainite transformation temperature is long enough to allow the specimen to stop reacting.
In interpreting the experimental data, it is usually assumed that the dimensional change
observed during isothermal reaction is proportional to the volume fraction of transformation,
and it is sometimes assumed that the point where dimensions cease to change represents a 100%
of transformation. When reaction ceases before the parent phase has completely transformed,
it is useful to be able to calculate the volume fraction of product phase that has been obtained.
For the transformation of austenite er) into a mixture of bainitic ferrite and carbon-enriched
austenite, Bhadeshia [3] has shown that the relative length change can be related to the volume
fraction of ferrite (Va) by the equation:
with
!:i.L
L
2Vaa~ + (1 - Va)a~3 - a~
3a3-y
(4.1)
where
aa = aao(1 + 13a(T - 300))
a-y = a-yo{x1, xl}[1 + 13-y(T - 300)]
a; = a-yo{ xi, xl} [1 + 13-y(T - 300)]
(4.2)
: length change due to transformation per unit length,
: linear thermal expansion coefficients of ferrite and austenite respectively, K-1,
average carbon content in the steel, mole fraction,
alloy concentrations in austenite, mole fraction; i's denote different alloying element,
carbon content of residual austenite at any stage of the reaction, mole fraction,
lattice parameter of austenite at ambient temperature as a function of chemical concen-
trations in the austenite,
: lattice parameter of austenite at the reaction temperature before the reaction,
: lattice parameter of austenite at reaction temperature at any stage of the reaction,
: lattice parameter of ferrite at ambient temperature (300 K),
aa : lattice parameter of ferrite at reaction temperature.
In equation (4.1), it is assumed that ~L is a third of the relative volume change a:; this
is a very good approximation since the changes in density during transformations in steels are
small. For the same reason, the implicit assumption that mass fractions are identical is also
justified. If this last assumption is avoided, we obtain [4]:
{ [ ] }
1/3
!:i.L 1 2a3 a*3
_ - 1+ - a -y _ a3 - 1
L - a~ Vaa~3 + 2(1 - Va)a~ -y
However the difference between the results from equation (4.1) and (4.2) is found to be
negligible. The factor of two in the numerator of equation (4.1) arises because the unit cell of
ferrite contains two iron atoms whereas that of austenite has four.
82
Note that equation (4.1) can be used for all situations where austenite decomposes into a
mixture of carbon-enriched residual austenite and ferrite, and furthermore, the ferrite mayor
may not be supersaturated with respect to carbon, since the effect of excess carbon is manifested
in an alteration of the lattice parameter of the ferrite. Using a similar method, the following
equation can be derived for the case where cementite precipitation occurs in conjunction with
ferrite formation from austenite [4]:
!:i.L
L
2Vaa~ + Vea~/3 + (1 - Va - Ve)a;3 - a~
3a3
i
(4.3)
where
Ve volume fraction of cementite,
a~ : aebece,
ae, be, ce : lattice parameters of cementite, which are 4.51,5.08 and 6.73 A at ambient temperature
respectively.
The volume fraction of cementite, where the total carbon content in ferrite including any
cementite particles is S, can be obtained as follows assuming all carbon atoms in the ferritic
matrix are locked in cementite:
with
v: __ l_
e - 1+ k
e
(4.4)
(4.5)
where xg is iron mole fraction in ferrite.
The dependence of the lattice parameter of austenite on alloying elements was reported by
Ridley et al. [5] and Dyson and Holmes [6]' giving,
aiD =3.573 + 0.0065C + 0.0010Mn - 0.0002Ni + 0.0006Cr + 0.0056N + 0.0028AI
- 0.0004Co + 0.0014Cu + 0.0053Mo + 0.0079Nb + 0.0032Ti
+ 0.0017V + 0.0057W
where chemical composition and aiD are measured in at% and A respectively.
The carbon concentration of resid ual austenite xl is related to the volume fraction Va of
ferrite or ferrite and cementite transformed, and expressed by :
i xl - VaS
Xl = ----
1- Va
Therefore the lattice parameter of austenite in the equations changes with the volume fraction
of ferrite or ferrite and cementite.
Fig. 4.4 represents the relations between volume fractions transformed and corresponding
relative length changes during transformation at 4200 C to ferrite with 0.03 wt. % carbon, to
supersaturated ferrite with 0.2 wt.% carbon and to a mixture of carbon free ferrite and cementite
with 0.2 wt. % of carbon locked in the transformed phases altogether respectively. In these
83
calculations, the expansion coefficients of ferrite and austenite are taken to be 1.1826 x 10-5
and 1.8431 X 1O-5K-1 [3), and the linear expansivity of cementite is assumed to be the same
as that of ferrite. It is evident that the linearity between volume fraction of transformation
products and corresponding relative length change is preserved, at least, up to 0.7 of the volume
fraction.
It is emphasized however in equations (4.1), (4.2) and (4.3) that it is never justified that the
maximum length change observed during isothermal transformation corresponds to the com-
plete transformation of austenite [4]. Bainite transformation in steels is considered to be a good
example of this. Although the growth of bainitic ferrite is displacive and diffusionless, carbon
redistribution can occur after the reaction because bainite transformation usually takes place
at higher temperatures where carbon atoms can still move very quickly. As a consequence,
the carbon enrichment of residual austenite occurs. The following formation of bainitic ferrite
occurs from the carbon-enriched residual austenite causing further enrichment of carbon in the
residual austenite. When the carbon concentration in residual austenite reaches the T~ curve
(where ferrite, whose free energy has been raised by a stored energy term (400 Jmol-1 [7])
associated with the transformation strain, and austenite of identical composition have the same
free energy) further diffusionless formation of ferrite becomes thermodynamically impossible
since it requires a positive free energy change as shown in Fig. 4.5: this is referred to as the in-
complete reaction phenomenon. Therefore even when the reaction is observed to have stopped,
carbon enriched austenite can still exist if other competing processes such as cementite precipi-
tation directly from the carbon enriched austenite and reconstructive formation of ferrite, such
as pearlite formation do not overlap the reaction. This can actually be seen in Fig. 4.6 where a
relative length change during helium quenching after the completion of bainite transformation
at 420 QC is presented. There is an expansion which corresponds to martensitic transformation
of untransformed austenite at 420 QC on cooling.
The calculated carbon concentrations of residual austenite after 30 min of isothermal hold-
ing at 420, 448 and 472 QC in the Fe-0.3C-4.08Cr wt.% alloy were plotted in Fig. 4.7, where the
calculated Ae3, Ae~, To and T~ curves were also drawn. The calculation of the Ae3 curve is
based on the work reported by Gilmour et al. [8] and Kirkaldy et al. [9), and the other values
are calculated according to Bhadeshia's work [10]. The error bars correspond to the values of
the carbon content offerrite, 5, between 0.03 and 0.2 wt.% [3]. The calculated carbon contents
of residual austenite from the maximum relative length change during the isothermal bainite
transformation are in good agreement with the T~ curve. This shows that bainite transfor-
mation ceases well before the equilibrium condition is achieved, thus establishing that bainitic
ferrite grows by a diffusionless transformation mechanism with carbon redistribution after the
initial growth event.
The microstructure of a specimen which was helium quenched after 30 min of isothermal
bainite transformation at the temperature consists of martensite, which was austenite before
the helium quench, and some retained austenite and bainite (Fig. 4.8), as it was expected from
the TTT curve of the material used (see Fig. 4.1) and its thermal history. Micro-hardness
measurements show 802 Hv{10g} for martensite which is the white etched area in the picture
and 550 Hv{10g} for bainite.
As it can be seen in a TEM micrograph of a specimen quenched after the termination of bai-
nite transformation at 420 QC (Fig. 4.9), the initial microstructure shows a typical lower bainitic
microstructure. An average thickness of the residual austenite films was about 0.04JLm whereas
84
the average thickness ofbainitic ferrite plates was found to be 0.3JLm. The ferrite matrix (which
is plate or lath in shape) contains cementite particles as it can be seen in the figure. As discussed
in Chapter 2, the cementite particles in lower bainite are considered to have precipitated directly
from carbon supersaturated bainitic ferrite. The cementite particles which can be seen within
bainitic ferrite seem to have a very similar orientation as it can be seen in the TEM micrograph.
It is usually said that cementite particles which precipitate from carbon supersaturated bainitic
ferrite show a single variant; this can be seen in Fig. 4.9. This is sometimes taken as evidence
of cementite precipitation at interfaces between bainitic ferrite and austenite matrix, instead
of within the bainitic ferrite. It is not understood why cementite particles in lower bainite
have a single variant. However, a multi-variant cementite precipitation is also observed in some
cases although the possibility of a sectioning effect has not been assessed. Fig. 4.10 shows a
TEM micrograph of the Fe-0.3C-4.08Cr wt. % alloy isothermally held at 478 a C for 23 days after
austenitisation at 1000 ac. The microstructure is still a mixture of bainite and austenite, and
the bainitic ferrite seems to contain multi-variant cementite precipitation.
4.3.2 Bainite transformation in Fe-2.0Si-3.0Mn system
In order to investigate the formation of ferrite without carbide, Fe-Si-Mn alloys with dif-
ferent levels of carbon content were examined. A Mn addition can suppress the upper (recon-
structive) C-curve, and provides a remarkable difference in incubation periods for the displacive
and reconstructive formation of ferrite below the Bs temperature. Since the solubility of Si in
cementite is extremely small, a Si addition can retard remarkedly the cementite precipitation
from untransformed carbon enriched austenite. Therefore, the alloys are ideal for examining
the formation of ferrite from austenite without any carbide precipitation throughout a wide
range of reaction temperatures. The materials used in the present experiment were Fe-2.0Si-
3.0Mn wt.% alloys with 0.059, 0.12, 0.22 and 0.43 wt% of carbon respectively. All specimens
were homogenised at 1250 a C for 3 days prior to the experiments. After being heated to 950 a C
for 10 min, specimens were helium quenched to and held at various ferrite formation tempera-
tures for an extended period of time in the dilatometer which allows the specimens to complete
the reaction. First the bainite transformation temperature for each alloy was selected as such
to give the identical free energy change due to the transformation from austenite to ferrite:
-615 Jmol-
1
calculated as discussed by Bhadeshia [11]. The bainite transformation temper-
atures used in the experiment were listed in Table 4.1 for each alloy with its bainite-start
and martensite-start temperatures. In addition to these temperatures, a bainite transforma-
tion temperature, 270 a C for Fe-0.43C-2.0Si-3.0Mn wt. % alloy at which austenite is expected
to transform to lower bainite instead of to upper bainite, and temperatures between 400 and
500 ac for the Fe-0.12C-2.0Si-3.0Mn wt.% alloy were used to examine the incomplete reaction
phenomenon in bainite transformation.
The relative length changes obtained during the bainite transformation are shown in
Fig. 4.11. It can be seen that the holding time is long enough to allow the reaction to cease
during the isothermal holding at each reaction temperature. Typical TEM micrographs for
upper and lower bainite obtained in the present experiments in the Fe-0.43C-2.0Si-3.0Mn wt.%
alloy are shown in Fig. 4.12. The relative length changes were then converted to the volume
fractions of ferrite transformed during the isothermal holding and the carbon concentrations in
untransformed austenite using equation (4.1). The calculated results can be seen in Table 4.2.
These calculated carbon concentrations at the end of the bainite transformation were plotted
85
Carbon content wt% Tb' ° C Bs Ms
0.059 495 545 422
0.12 470 515 388
0.22 435 475 337
0.43 350 398 227
Table 4.1: Bainite transformation temperature Tb for each alloy used in the experiment at which
the free energy change due to ferrite transformation is identical; i.e. -615 Jmol-i. Calculated
B s' and Ms temperatures are also listed.
in a calculated phase diagram of Fe-2.0Si-3.0Mn wt.% system (Fig. 4.11). It can be clearly
seen that the bainite transformation has terminated far before the residual austenite reaches
its equilibrium in the system, and that the carbon concentrations in the residual austenite are
close to the T~ curve even in the case of lower bainite transformation at 270 °C in the Fe-O.43C-
2.0Si-3.0Mn wt.% alloy. The calculated result for 270 °C in Table 4.2 did not take into account
of cementite precipitation in ferrite. When the total carbon content in ferrite including any
cementite is altered up to 0.2 wt.%, the calculated xl varies between 0.028 and 0.030 mole
fraction.
Carbon content, wt% Tb' ° C
0.059 495
0.12 470
0.22 435
0.43 350
0.43 270
t::.L/L
0.00318
0.00334
0.00226
0.00343
0.00299
x I, mole fraction
0.004
0.010
0.014
0.035
0.030
Table 4.2: Carbon concentration in residual austenite calculated by equation (4.1).
A TEM micrograph of a specimen which was allowed to transform partially to bainite
at 350 °C shows that a bainite sheaf consists of many small units of bainitic ferrite plates
which form side by side. An extension of bainite sheaf seems to stop at a prior austenite grain
boundary or when it encounters one of the neighbouring bainite sheaves as can be seen in
Fig. 4.13. Although it is not clear from the micrograph, the two bainite sheaves which touch
may have started from different points in the same austenite grain, since an identical origin of
the start of the extension of a bainite sheaf may have one direction of propagation.
Fig. 4.14 shows a lower bainite microstructure obtained at 270 °C. Untransformed austenite
is trapped between two adjacent bainitic ferrite plates.
As clarified by isothermal bainite transformation experiments in the Fe-2.0Si-3.0Mn and
Fe-0.3C-4.08Cr wt.% alloys, transformation stops when the average carbon concentration in
austenite reaches the To or the T~ curves. This means that the microstructure at the completion
of bainite transformation consists of bainitic ferrite and austenite as long as no additional
reaction, such as cementite precipitation from untransformed austenite or pearlite formation,
overlaps with the bainite reaction. Therefore a mixture of bainitic ferrite and austenite can be
obtained easily for use as an initial microstructure for reaustenitisation experiments.
86
4.4 ISOTHERMAL REAUSTENITISATION FROM A MIXTURE OF
BAINITE AND AUSTENITE
Isothermal reaustenitisation from a starting microstructure of bainitic ferrite and residual
austenite was studied in the Fe-0.3C-4.08Cr wt. % alloy (heat cycle 3 in Fig 4.2). The nucleation
of austenite is therefore unnecessary during reheating experiments thereby allowing growth
effects to be studied in isolation.
4.4.1 Microstructural study
A bainite transformation temperature of 4200 C was used to obtain a mixture of bainite
and austenite as the initial microstructure for the experiments.
Optical and transmission electron microscopy was carried out on specimens quenched after
30 min of isothermal reaustenitisation at different reaustenitisation temperatures. No transfor-
mation except martensitic transformation was detected during the helium quench to ambient
temperature.
Optical micrographs of isothermally reaustenitised specimens are presented in Fig. 4.15.
The white areas correspond to martensite, which was austenite at the reaction temperature, and
the dark etched area is tempered bainite which remained untransformed during austenitisation.
The volume fraction of austenite increases with the reaction temperature and reaches unity
above 805 °C (Fig. 4.15). At a temperature slightly below 8050 C, the martensitic microstructure
shows traces of a residual bainitic microstructure indicating the existence of some chemical
heterogeneity.
Although the volume fraction of austenite changes significantly with the reaction tem-
perature, the morphology of the untransformed bainite does not show large differences. This
suggests that reaustenitisation proceeds by the dissolution of the bainitic ferrite and there was
no evidence of independent nucleation of austenite. The pre-existing austenite simply grows by
the motion of the original Q:bh interfaces as discussed by Yang and Bhadeshia [12-14].
A typical TEM micrograph of a specimen quenched after isothermal reaustenitisation is
shown in Fig. 4.16. A bainitic structure still remains after 30 min of isothermal reaustenitisa-
tion at 781 0 C. Since cementite is not stable at that temperature, the lower bainitic cementite
precipitates have disappeared completely, and new precipitates, larger in size and located rather
randomly in the tempered ferrite matrix were observed. These precipitates are expected under
equilibrium conditions to be M7C3 chromium rich carbides confirmed by a convergent beam
diffraction pattern (Fig. 4.17). Fig. 4.17 also shows that the average thickness of austenite is
around OApm, which is far larger than that in the initial microstructure. This shows that the
reaustenitisation progresses by the thickening of pre-existing austenite films at the expense of
ferrite.
A TEM micrograph of a specimen reaustenitised at 778 °C is presented in Fig. 4.18. A
grain boundary between the austenite and ferrite matrix can be seen in the micrograph. The
dislocation density in the ferrite was found to be very low because it has been tempered at
778 °C for 30 min during the austenitisation heat treatment. The M7C3 precipitates were also
found in the specimen. However, the size of the precipitates found in the tempered ferrite
matrix was larger than those in austenite. The average particle size in the ferrite matrix was
about 0.3pm compared to around 0.06pm in the austenite region. This may be understood as
follows. Since the solubility of chromium in austenite is higher than that in ferrite, the particles
which have precipitated in ferrite dissolved in austenite after being engulfed by the austenite.
As a result, fine precipitates, which may have been dissolving during the reaction, but have not
87
had enough time to dissolve completely, can be found in the austenite. This is also supported
by the fact that the advancing interface in Fig. 4.18 seems to have been growing towards the
particles which locate in ferrite matrix. The same feature can be seen in Fig. 4.19 where no
particles were observed in austenite whereas precipitates, about 0.2 p,m in diameter, were found
in ferrite.
The micro-hardness measurements (Fig. 4.20), show an increase in hardness with the re-
action temperature. Since the hardness of martensite depends mainly on the carbon concen-
tration, the hardness of martensite should be higher at a lower reaustenitisation temperature,
where a higher carbon concentration in austenite is expected as long as there is no phase other
than ferrite and austenite. There is, however, M7C3 carbide at these temperatures which may
reduce the carbon concentration in austenite. As it is discussed in Chapter 6, the equilibrium
carbon concentration in austenite decreases with superheat in the Q' + 'Y + M7C3 region. In
the 'Y + M7C3 region, on the other hand, the carbon concentration increases with temperature.
Chromium concentration, however, increases mono tonically with temperature. As a result, the
hardenability of austenite seems to increase with austenitisation temperature.
4.4.2 Dilatometry
A contraction of specimens during reaustenitisation is expected to occur due to the dif-
ference in the density between austenite and ferrite. Typical relative length changes obtained
during isothermal reaustenitisation are shown in Fig. 4.21.
In the case of high reaustenitisation temperatures, reaction starts during heating in spite
of the high heating rate. Therefore in order to obtain the total degree of transformation at the
reaction temperature, the relative length change should be corrected as follows [15]. Since the
length change during up-quenching was recorded by the computer, the correction can be done
easily as shown in Fig. 4.22. If there is no transformation during heating, the length of the
specimen will vary approximately linearly with temperature due to the constant expansivity
of the initial phase. Therefore the deviation from the straight line which corresponds to the
linear thermal expansion of the initial microstructure must due to the transformation which
occurred on heating. If the low temperature part of the curve is extrapolated to the reaction
temperature, the vertical difference between the extrapolated line and the act uallength change
curve gives the true length change due to the transformation, as if no reaction had occurred
during heating to the isothermal transformation temperature. As a result, the maximum relative
length change due to the whole transformation should be t1Lm instead of t1Li illustrated in
the figure. This correction can be regarded as a temperature correction of the data. Consider
a point i in a length versus temperature plot, with coordinates Li and Ti. The contributions
to the term Li arises from thermal expansion and dimensional changes due to transformation
during heating to the isothermal reaustenitisation temperature T-y. We need to remove the
effect of the transformation during heating to T-y. This can be done by adding to Li, the length
increment associated with heating from Ti to T-y, so that the corrected length Li is given by;
(4.6)
where l3i indicates the thermal expansivity of the initial microstructure obtained from the
up-quenching part of the data. The corrected relative length changes for the data shown
in Fig. 4.22 are plotted in Fig. 4.23. The maximum relative length change t1Lm is also
indicated in the figure. The relative length changes during isothermal reaustenitisation at
88
different reaction temperatures are compared in Fig. 4.24; these data have been temperature
corrected using equation (4.6). At each reaction temperature, the relative length decreases
rapidly at the beginning of the reaction. Then the reaction rate becomes sluggish and reaches
a saturation value. Not only the maximum relative length change but also the reaction rate
increases with reaction temperature. The maximum length changes b.Lm due to the isothermal
reaustenitisation at different reaction temperatures are plotted in Fig. 4.25. The b.Lm increases
with the reaction temperature and reaches the maximum value at around 790°C.
Any further increase in the reaction temperature did not result in an increase in relative
length change: a slight decrease in the relative length change is observed between 805 and
820 °C. This can be understood as follows. The relative length change due to the 100% trans-
formation can be obtained as the difference in the mean atom spacing in ferrite and in austenite
at the temperature: the mean atom spacing in austenite is smaller than that of in ferrite. How-
ever the thermal expansivity of austenite is larger than that of ferrite, the higher the reaction
temperature the smaller the difference in the mean atom spacing is, hence the smaller relative
length change due to the 100% transformation at the temperature is obtained (Fig. 4.26). While
the maximum degree of the transformation increases with the reaction temperature, the relative
length change due to the reaction increases until the reaction temperature is raised to the point
above which the 100% reaustenitisation can occur. A further increase in the reaction temper-
ature, however, causes the decrease in the relative length change due to the reaustenitisation
because of the difference in the thermal expansivities of ferrite and austenite. Therefore the
maximum in the relative length change is obtained at a certain temperature which corresponds
to the Ae3 temperature of the system.
Martensite transformation occurs during helium quenching after 30 min of isothermal
reaustenitisation. The relative length changes during quenching are shown in Fig. 4.27. The
higher temperature part in each graph shows a straight line corresponding to thermal con-
traction of the microstructure obtained at the end of the isothermal reaustenitisation. The
relative length change due to transformation during quenching (Fig. 4.27 e) is due to marten-
sitic transformation. Fig. 4.27 f shows that the relative length changes are almost the same at
the reaction temperatures 805°C and 815 °C corresponding to martensitic transformation from
fully austenitic microstructures, consistent with the results discussed above.
4.4.3 Equilibrium study of the maximum degree of the reaction
As discussed by Yang and Bhadeshia [12,13], reaustenitisation from a mixture of austenite
and bainite or acicular ferrite starts when the reaction temperature is raised to the Ae3 tem-
perature T"(l of residual austenite in the initial microstructure, whose carbon concentration is
given by the T~curve at the bainite or acicular ferrite transformation temperature. And partial
reaustenitisation is expected to occur between T"(l and T"(2 (the Ae3 temperature of the steel).
The temperatures, T"(l' and T"(2' above which the completion of reaustenitisation occur in
the Fe-0.3C-4.08Cr wt.% alloy can be then obtained from the phase diagram shown in Fig. 4.28.
For the material used in the present experiment, the carbon content of residual austenite after
the cessation of bainite transformation at 420 °C was calculated to be 0.00228 in mole fraction
which is xT' at 420°C. Paraequilibrium is assumed here for the calculation at 420°C. Theo
calculated Ae3 temperatures of austenite with its carbon concentration identical to 0.00228
and the bulk carbon concentration are used to determine the temperatures T"(l and T"(2' The
temperature range of partial reaustenitisation was found to be between T"(l = 751 °C and
T"(2 = 781 ° C. The equilibrium volume fraction of austenite at a temperature between these
89
two temperatures can be calculated from Ae3 curve of the system as discussed earlier [12-14].
In the case where the carbon content of ferrite is not negligible, however, the volume fraction
of austenite should also be a function of carbon concentration in ferrite transformed
(4.7)
where S is carbon content of ferrite (the carbon may be in solution or in the form of precipitates
in the ferrite). The calculated equilibrium volume fraction of austenite V-y is listed in Table 4.3,
where the value S has been assumed to be 0.2 wt%.
Temperature, 0 C XAe3' at.% V-y
750 2.30 0.33
755 2.16 0.38
760 2.01 0.42
765 1.86 0.48
770 1.72 0.57
775 1.57 0.70
780 1.42 0.91
Table 4.3: Equilibrium volume fractions of austenite at different temperatures calculated from
equation (4.7)
To calculate the relative length change from the equilibrium volume fraction of austenite,
three different reactions were assessed:
(1) Q' + I'(x-y = XT~) ---+ I'(x-y = XAe3) + Q'
(2) Q' + B + I'(x-y = XT~) ---+ I'(x-y = X Ae3) + Q' + B
(3) Q' + B + I'(x-y = XT~) ---+ I'(x-y = X AeJ + Q' + B no dissolution of B
In case (1), the starting microstructure assumed to be a mixture of ferrite (Q'), whose carbon
content is negligibly small, and austenite h) whose carbon concentration can be determined by
the T~ curve at the bainite transformation temperature Tb' and the final structure of a lower
quantity of ferrite and austenite with its carbon concentration is identical to that given by the
Ae3 curve at the reaustenitisation temperature. For case (2), the starting structure is assumed
to be a mixture of ferrite with zero carbon content, cementite (B) and austenite with its carbon
content identical to the value at the T~ curve. The final stage, in this case, is assumed to be a
mixture of ferrite with zero carbon, cementite and austenite with the carbon content identical
to the equilibrium value. The ratio of the volumes between ferrite and cementite is assumed
to be the same during the reaction. In the last case, (3), the same initial structure as in the
case (2) and a mixture of ferrite, cementite and austenite with the carbon content identical to
the equilibrium value for the final structure are assumed. So that, in this case, no dissolution
of cementite in austenite is assumed. The conversion of the equilibrium volume fraction of
austenite into relative length change can be done using the following equations:
For case (1);
t:..L !a;3V-y + 2a~(1- V-y) - a~V-yo - 2a~(1- V-yo)
L 3 a~ V-yo + 2a~ (1 - V-yo)
90
(4.8)
For case (2);
(4.9)
For case (3);
with
!:::.L
L
1 a;3V, + 2a~(1- V, - Veo) - a~V,o - 2a~(1- V,o - Veo)
3 a~V,o + 2a~(1- V,o - Veo)+ ~a~Veo
aa = aao(1 + f3a(T - 25))
a, = a,o(1 + f3,(T - 25))
a~ = a~o(1+ 1,(T - 25))
(4.10)
where
V,o = volume fraction of austenite in the initial microstructure,
V, = volume fraction of austenite at any stage of the reaction,
Veo = volume fraction of cementite in the initial microstructure,
The volume fraction of cementite in the initial microstructure, where the average car bon content
in ferrite including cementite particles is 5, can be obtained as follows assuming all carbon atoms
in ferrite are locked in as cementite particles:
1
Veo= --k-(l - V,o)
1+ e
with
a3(1 - 3~)a x"
ke = 31 S D
a !:I 2 xo
The relative length change due to complete reaustenitisation at each temperature can be then
calculated by setting V, = 1 in the equations.
The calculated relative length changes for each case are drawn in Fig. 4.29, where aa =
2.866 A (after Bhadeshia [3]), f3a" = 1.244x1O-5, 2.065x10-5 K-1 (obtained in the present
work from continuous heating experiments with the starting structure of martensite) and the
same expression as the previous section for the thermal expansivity of austenite as a function
of alloying element compositions was used. The thermal expansion coefficient of cementite was
reported to increase with temperature [16]. Using data published by Stuart and Ridley [16] the
expression of the mean linear expansion coefficient as a function of temperature was obtained
as follows:
f3e = 5.4872 x 10-6 + 3.6450 x 1O-9T + 9.2833 X 1O-12T2
where T is temperature in DC. Although the data were available only up to 700 DC, this ex-
pression was assumed without any justification to be applicable at higher temperatures than
91
7000 C. It may be worth noting that the thermal expansivity of cementite is very close to that
of ferrite at temperatures where reaustenitisation occurs.
For cases (1) and (2), there are rather big discrepancies between the experimental data
and calculated values. The value obtained for the maximum relative length change at around
8000 C differs significantly from that given by the two calculations. The calculated slope for the
partially reaustenitised part, however, seems very similar to that obtained in the experiments
although there is about 10 or 150 C difference between the calculations and experimental results.
The maximum relative length changes which correspond to full transformation into austenite;
i.e. the cases reaustenitised at 805 and 815 °C, are well expressed by case (3), where the cemen-
tite is assumed not to dissolve in austenite. The slope for the region where the transformation
to austenite is incomplete, on the other hand, does not agree with the experimental results.
When the reaction temperature is very high, where alloy carbides are not thermodynamically
stable, the dissolution of cementite (which has existed in the initial ferrite matrix) will occur
after the completion of reaustenitisation as discussed by Hillert et al. [17]. Therefore, in this
case, the relative length change due to the isothermal reaustenitisation can be calculated by
equation (4.10). However, when the reaction temperature is not high enough for the effect of
dissolution of cementite to be neglected, the calculation in case (3) should be modified. This
might be one of the reasons that the slope of partially transformed part is not in good agreement
with the data.
Another reason of the discrepancy could be due to the existence of alloy carbide precipita-
tion at the reaction temperatures. The equilibrium phase diagram of a steel which has similar
chemical composition; Fe-5 wt.% Cr, is shown in Fig. 4.30. The calculated equilibrium phase di-
agram for Fe-C-Cr ternary alloy [18] at different temperatures is also shown in Fig. 4.30. M7C3
is stable below about 820°C both in the ferrite and in austenite phases. Thus, the theory
which has been proposed by Yang and Bhadeshia [12-14] should be modified. The precipitation
reduces the carbon content ofaustenite below the a/I equilibrium carbon content, XAe3, which
is obtained assuming that there is no effect of precipitation on the equilibrium. Hence further
transformation is expected to occur after the precipitation; transformation from austenite to
ferrite might occur. When the reaction temperatures are not high enough, the precipitation
can occur before or during the formation of austenite. In this case, the Ae3 temperature should
be calculated under the ferrite, austenite and alloy carbide three phase equilibrium condition.
Although this calculation has not been tackled, the existence of stable precipitation will raise
the Ae3 temperature, so that the Ty1 and T-y2 temperatures will be calculated higher than in
the absence of carbide precipitation. This may Oexplain the difference between the theory and
experimental data in the temperature range of partial reaustenitisation.
The calculated temperature range of partial reaustenitisation in the Fe-0.3C-4.08Cr wt. %
system by the theory proposed by Yang and Bhadeshia [12-14] was between 751 °C and 781 °C
and the value expected from the experimental results was, on the other hand, between 7600 C
and 7950 C.
92
4.5 CONTINUOUS HEATING REAUSTENITISATION FROM A MIXTURE OF
BAINITE AND AUSTENITE
Although isothermal study of transformation is always necessary to understand the ki-
netics of the reaction, heat-treatments in practice are rarely isothermal. Continuous heating
reaustenitisation is therefore studied in this section.
4.5.1 Dilatometry
Experiments have been carried out on the specimens with three different starting mixtures
of bainite and austenite. The relative length changes during continuous heating are shown in
Fig. 4.31. The upper part of the figure shows changes in the relative length and temperature
during heating as a function of time, and the lower part is the plot of the relative length
change against temperature. It can be seen that the change in temperature of the specimen is
controlled to give a constant heating rate throughout continuous heating up to 1000 °C although
a constant rate is of course not necessary in any fundamental sense. The transformation-
start and -finish temperatures can be determined easily by plotting the relative length change
against temperature as can be seen in Fig. 4.31. The lower temperature part of the relative
length change curve shows a straight line corresponding to a linear thermal expansion of the
starting microstructure (a mixture of bainite and austenite). The linear thermal expansion
coefficient of the initial microstructure increases with the bainite transformation temperature
(Fig. 4.32). When a lower bainite transformation temperature is selected, a specimen contains
larger amount of austenite whose thermal expansion coefficient is larger than ferrite. The higher
temperature part of the curve, on the other hand, is due to the linear thermal expansion of
austenite since the specimen is then fully austenitic. Therefore the points where the actual
relative length curve deviates from each straight lines give the transformation-start, Ts, and
-finish, TF, temperatures respectively.
The transformation-start and -finish temperatures determined from the relative length
change during continuous heating at different heating rates are plotted in Fig. 4.33. Both
increase with the heating rate. The effect of the initial austenite volume fraction and the carbon
content of residual austenite, i.e. bainite transformation temperature, on the transformation-
start and -finish temperatures can be also seen in the figure. When the heating rate is low,
1.1 °C s-t, the higher the bainite transformation temperature the lower the transformation-
start and -finish temperatures. At 5°C S-l of heating rate, in contrast, this tendency has been
inverted. The normalized austenite volume fraction transformed on heating was calculated from
the relative length change (Fig. 4.34).
With relatively low heating rates, however, an expansion ofthe specimen before the onset of
reaustenitisation has been detected (Fig.4.31). This expansion of specimen has been pointed out
as the decomposition process of residual austenite by Yang [15]. Therefore the transformation-
start temperatures which have been observed in low heating rates are not the proper Tc for
the starting microstructure of the mixture of bainite and austenite obtained at each bainite
transformation temperature. However the data obtained with higher heating rates can be
treated as proper transformation-start and -finish temperatures, because no transformation of
specimens has been observed before the start of reaustenitisation during continuous heating.
4.5.2 Decomposition of austenite during heating
As it has been discussed by Yang and Bhadeshia [12-14], the transformation-start tem-
perature during continuous heating from a mixture of austenite and either bainite or acicular
93
ferrite is restricted by the carbon concentration of residual austenite just before the start of
continuous heating directly from the bainite or acicular ferrite transformation temperatures.
When the bainite or acicular ferrite transformation temperature is lower, the carbon content of
residual austenite will be higher because of the negative slope of the T~ curve. Higher bainite
transformation temperatures give correspondingly higher reaustenitisation-start temperatures
during continuous heating.
However, in the present experiments, the transformation-start temperature during the
continuous heating at 1.1 °C S-l is lower when the bainite transformation temperature Tb was
higher. The situation corrects itself when the heating rate is raised to 5 °C S-l , with the order
of the transformation-start temperature being in agreement with the theory. The reason for
the disagreement at the low heating rate, is the decomposition of austenite during heating. For
a heating rate of 1.1 °Cs-I, austenite decomposition during heating was observed for all the
bainite transformation temperatures used (420, 448 and 472 °C). For a heating rate of 5 °Cs-I,
in contrast, the decomposition of austenite could not be detected during heating.
The temperature at which the relative length change versus temperature first indicates
the decomposition of austenite is designated Td •. When heating rate was 1.1 °Cs-I, the Td•
temperatures determined experimentally were found to be as shown in Fig. 4.35. Td• is found
to decrease with increasing Tb' This order can be understood as follows. When Tb is lower,
the carbon concentration in residual austenite is higher because of the negative slope of the T~
curve. Thus reconstructive formation of ferrite from the residual austenite becomes sluggish.
This corresponds to to the fact that the upper C-curve in the TTT diagram is shifted to a
longer time direction by increasing the carbon concentration in austenite. As a result, the lower
the bainite transformation temperature, the higher the temperature at which a reconstructive
formation of ferrite from the residual austenite on heating starts.
In order to investigate the decomposition process during continuous heating, a specimen
was helium quenched from 730 °C after heating at 1.1 °C S-l after 30 min of isothermal bainite
transformation at 448 °C. The microstructure of the specimen (Fig. 4.36) consists of three
phases; white, black and a dark etched areas. The micro-hardness of the white areas correspond
to that of martensite (austenite before quenching) at 753 Hv (lOg). The dark etched areas are
expected to be bainite, with a hardness of 444 Hv (lOg). The hardness of the third phase was
400 Hv (lOg), and a TEM micrograph confirmed (Fig. 4.37) that it is pearlite.
The volume fraction of pearlite formed by the decomposition of austenite during continuous
heating can be calculated from the relative length change obtained in the experiments. A
relative length change due to decomposition of residual austenite to pearlite during heating was
expressed by:
b.L
L
with
~ V,a;3 + (1- V, - V9)2a~ + V9~a~ - V,oa~ - (1- V,o - V9o)2a~ - V90~a~
3 V,oa~ + (1 - V,o - V9o)2a~ + V90~a~
94
(4.11)
The lattice parameters of austenite all> a,2' ferrite aa' and cementite a~ = aebece were calcu-
lated as described earlier. The carbon content of austenite has been assumed to be constant
during the decomposition process. The calculated volume fractions of austenite (now marten-
site), bainite and pearlite are listed in Table 4.4. The calculations are for the data obtained at
730 QC. According to the calculated results, the volume fraction of pearlite is larger when the
bainite transformation temperature was 448 QC than both the cases of the transformations at
420 and 472 QC. This can be understood as follows. When the bainite transformation tempera-
ture is lower, decomposition starts at a lower temperature. So a higher degree of decomposition
may be expected than in the case of a higher bainite transformation temperatures giving a
higher amount of normalised volume fraction of decomposition. It is, however, important to
note that the initial amount of austenite which has the potential to decompose is smaller in lower
bainite transformation temperatures. As a result, the amount of austenite which decomposes
on heating to 730 QC reaches a peak at the bainite transformation temperature.
Temperature, QC xl, at.%
420
448
472
2.4
1.6
1.4
0.30
0.67
0.94
0.03
0.08
0.59
0.27
0.59
0.35
Table 4.4: Calculated volume fractions of the phases obtained by the reaction.
Phase Vopt Veal
Bainite 0.58 0.50
Martensite 0.24 0.23
Pearlite 0.18 0.27
Hv(10g)
444
753
400
Table 4.5: Volume fractions derived from dilatometry, of the specimen helium quenched during
heating at 1.1 QCS-l. Starting microstructure is obtained by bainite transformation at 445 QC.
The volume fractions of the three phases observed in the specimen helium quenched from
730 QC on heating at 1.1 QCS-l after the 30 min of isothermal bainite transformation at 448Q C
can also be calculated from the relative length change obtained. The total relative length
change due to the decomposition of austenite during heating was measured as 6.64xlO-4. The
calculated volume fraction and carbon content of residual austenite after 30 min of isothermal
bainite transformation, which showed 2.24x 10-3 ofrelative length change, was 0.5 and 1.7 at.%
95
respectively. From these values, the volume fraction of pearlite transformed from austenite
during heating was calculated (designated as Veal in Table 4.5). The results of the quantitative
optical measurements were also listed (designated as Vopt in Table 4.5). The dilatometric results
are in reasonable agreement with the optically measured volume fraction of each phase.
4.6 CONCLUSIONS
Bainite transformation behaviour has been studied in a Fe-0.3C-4.08Cr wt.% alloy, and
III a Fe-2.0Si-3.0Mn steel with 0.06, 0.12, 0.22 and 0.43 wt%C. Bainite transformation was
confirmed to cease prematurely when the carbon concentration in the residual austenite reaches
the To or T~ curve rather than the paraequilibrium concentration. This suggests that bainite
transformation in steels proceeds by diffusionless growth of bainitic ferrite.
Bainite transformation was used to obtain mixtures of bainitic ferrite and austenite, so
that the growth of austenite layers could be studied. The following results were obtained.
From isothermal reaustenitisation experiments:
1) Reaustenitisation does not start until the temperature is raised to the Ae3 temperature of
the austenite in the initial microstructure.
2) 100% reaustenitisation occurs at temperatures above the Ae3 temperatures of the bulk
carbon content.
3) The maximum relative length change due to 100% reaustenitisation shows a slight decrease
with increasing temperature since the difference in thermal expansivities of ferrite and
austenite decreases as the temperature is raised.
4) From an equilibrium analysis of the relative length change for reaustenitisation, the reaction
at higher temperatures seems not to be accompanied by dissolution of cementite whereas
some dissolution does occur at partial reaustenitisation temperatures.
5) Reaustenitisation seems to proceed by the thickening of pre-existing austenite in the start- -
ing microstructure.
6) M7C3 chromium rich carbide can be found after 30 min of isothermal reaustenitisation at
temperatures below 8150 C.
From continuous heating reaustenitisation experiments:
7) A lower reaustenitisation-start temperature was obtained for a lower bainite transformation
temperature.
8) Decomposition of pre-existing austenite to pearlite occurs on heating when the heating
rate is less than 5 °c s-1.
9) The decomposition-start temperature can be understood by the TTT curves for the austen-
ite in the initial microstructure.
10) This supports that the concept of the incomplete reaction phenomenon, which was also
suggested by isothermal bainite transformation.
96
REFERENCES
1. H. K. D. H. Bhadeshia: Acta Metall., 1980, 28, 1103.
2. H. K. D. H. Bhadeshia: Proc. Int. Conf. "Solid-Solid transformations", 1981, 1041.
3. H. K. D. H. Bhadeshia: Metal Science, 1982, 16, 159.
4. M. Takahashi and H. K. D. H. Bhadeshia: J. of Materials Science Letter, 1989, 8, 477.
5. N. Ridley, H. Stuart and L. Zwell: Trans. AIME, 1969, 245, 1834.
6. D. J. Dyson and B. Holmes: JISI, 1970, May, 469.
7. H. K. D. H. Bhadeshia: Acta Metall., 1981, 29, 1117.
8. J. G. Gilmour, G. R. Purdy and J. S. Kirkaldy: Metall. Trans., 1972, 3, 1455.
9. J. S. Kirkaldy, B. A. Thomson and E. A. Baganis: ((Hardenability Concepts with Applications
to Steel", ed. D. V. Doane and J. S. Kirkaldy, AIME, 1978, 82.
10. H. K. D. H. Bhadeshia and D. V. Edmonds: Acta Metall., 1980, 28, 1265.
11. H. K. D. H. Bhadeshia: Metal Science, 1980, 14, 230.
12. J-R. Yang and H. K. D. H. Bhadeshia: Proc. Int. Conf. on ((Welding Metallurgy of Structural
Steels", ed. J. Y. Koo, AIME, Warrendale, Pennsylvania, 1987, 549.
13. J-R. Yang and H. K. D. H. Bhadeshia: Proc. Int. Conf. "Phase Transformations '87",
Institute of Metals, London, ed. G. W. Lorimer, 1987b 203.
14. J-R. Yang and H. K. D. H. Bhadeshia: Proc. Int. Conf. on ((Welding Metall. of Structural
Steels", ed. by J. Y. Koo, The Metall. Society of the AIME, 1987, 549.
15. J-R. Yang: Ph.D thesis in the University of Cambridge, 1988.
16. H. Stuart and N. Ridley: JISI, 1966, July, 711.
17. M. Hillert, K. Nilsson and L-E. Torndahl: JISI, 1971,209,49.
18. B. Uhrenius: "Hardenability Concepts with Applications to Steel", ed. D. V. Doane and J.
S. Kirkaldy, AIME, 1978, 28.
97
900
Fe-O.3C-4.08Cr wt. %
800 Ae3
Ae'3..._---- _---- _--- __ _- _------. __ ._- __ _---_ _-- ..__ ._- _--- " _-- .
700
u
0
a> 600L-
:J
-+-
0
L-
a>
Q.
500E
a>~
400
Ms
300
200
1.0E+00 1.0E+02 1.0E+04
Time, S
Fig .4.1 Calculated time-temperature-transformation curve for the material used in the experiments
[3].
ID
\-
:J 1000 ·C X 10m in.•...
~ Ae3Q.)
Q.
E
Q)
~ Ss .
Ms .
Fig. 4.2 Schematic illustration of heat treatments.
98
( 1 )
H.Q H.Q.
Time
H.Q.
0.005
0.004
et .420oC
Cl
0)
c: 0.0030~
0
==0)
c:
0.002 et .448°C~
Cl
.:!:
Ci
Cl
0:: 0.001
et .472°C
0.000
0 500 1000 1500 2000
TIme, S
Fig. 4.3 Relative length change during isothermal bainite transformation at 420 °C, 448 °C and
472 °C after austenitisation at 1000 °C for 10 min.
1.0
0.007
b
0.006
'" 0.005'"c
0
..c:
u
..c: 0.004
'"c
.!!•. 0.003
~
0•.
0:: 0.002
0.001
0.000
1.0 0.0 0.2 0.4 0.6 0.80.8
0.006
a
0.005
•.
'" 0.004c
0
..c:
u
..c:
rn 0.003c
.!!•.
~~ 0.002•.
0::
0.001
0.000
0.0 0.2 0.4 0.6
Volume fradion Volume fraction
Fig. 4.4 Length change as a function of volume fraction of ferrite during isothermal transformation
of austenite to ferrite which is supersaturated with 0.2 wt.% carbon (a), and to a mixture
of ferrite and cementite (b) at 420°C [4].
99
>.
0.0
L..
(1)
C
W
(1)
(1)
L..
LL
(1)
L..:J
+J T,co
L..
(1)
n-
E
(1)
l-
at temperature T,
a
-------
To
Carbon Content
y
Fig. 4.5 Schematic illustration of the free energy curves of austenite and ferrite.
100
0.007
0.006
On cooling after bainite
transformation at 420
Q
C
(1)
0)
c
~ 0.005
..c
+-'
0)
c
(1) 0.004
500400300
Ms
200100
0.003
0.002 o
(1)
>
+-'ro
(1)
a:
Temperature QC
Fig. 4.6 Relative length change during helium quenching after the cessation of bainite transforma-
tion at 4200 C showing a martensi tic transformation.
Fe-O.3C-4.08Cr wt.%
900
800
P 700
~
Q)
•.....
:J 600+-'ro•.....
Q)
a. 500
E
Q)....
400
300
200
0.00 0.05 0.10 0.15 0.20
Carbon content, mole fraction
Fig. 4.7 Calculated phase diagram of the Fe-0.3C-4.08Cr wt.% alloy; plots correspond to the calcu-
lated carbon concentration of residual austenite after 30 min of isothermal bainite trans-
formation at each reaction temperatures.
101
Fig. 4.8 Optical microstructure of a specimen helium quenched from 4200 C after bainite transfor-
mation.
102
/0.2,urn
I I
;..~
~
~
t ,..~
-
-,• \", •.
i ~
t
.:~
~,.
.~
"1 1
Fig. 4.9 TEM bright field image of the initial lower bainitic microstructure and a dark field image
of cementite particles.
103
Fig. 4.10 TEM bright field image of lower bainite after 23 days of isothermal holding at 4780C.
104
2500200015001000500
0.000 o
0.004
wt'l.
O.06C O,43C
O.22C
Q) 0.003 O.12C
Cl
c
0..c
u
..c-Cl 0.002c
Q)
Q)
>
0
Q)
c:::
0.001
Time, sec
3aa
••
To' '.
2aa
a.0a a.05 Iil.!a 0.15 a.20 0.25
9aa
Fe-2.0Si-3.0Mn wt.%
aaa
• O.06C wt.%'Y-+
.O.12C
P
7aa .•. O.22C
.•. O.43C
Q)•...
:J b0a
+-'
('lj ,•... Ae3Q)
a.
E 50a
Q)
I-
4aa
Carbon content. mole fraction
Fig. 4.11 Relative len~th change during isothermal bainite transformation at each reaction tem-
peratures (a) and calculated phase diagram of the Fe-2.0Si-3.0Mll system with plots of
calculated carbon concentration of residual austenite (b).
105
ab
Fig. 4.12 Typical TEM micrographs of upper and lower bainite obtained in the Fe-0.43C-2.0Si-3.0Mn
alloy at a) 350°C and b) 270°C respectively.
106
Fig. 4.]3 TEM bright field image of upper bainite microstucture obtained at 350°C in the Fe-0.43C-
2.0Si-3.0Mn alloy.
107
a b
c d ~
""
",(
/ \
t( \
\ \
\ • \ ITo
\ •• Q, -
• \ / 111 ..,
Fig. 4.14 (a) TEM bright field image of lower bainite obtained at 270 QC in the Fe-0.43C-2.0Si-3.0Mn
alloy and (b) a dark field image of residual austenite trapped in between two adjacent
bainitic ferrite plates. (c) is a selected area diffraction pattern.
108
a) Reaustenitised at 778 QC
b) Reaustenitised at 785 QC
. ~
\~1l'1\ • ~ . ~.,"
.' .
- • t
Fig. 4.15 Optical micrographs of isothermally reaustenitised specimens at a) 778 QC, b) 785 QC, c)
790 QC and d) 805 QC.
109
c) Reaustenitised at 790 °C
'.
d) Reaustenitised at 805°C
50 ~m
I I
110
Fig. 4.16 TEM bright field image of a specimen quenched from 785 QC after 30 min of isothermal
reaustenitisation at the temperature.
111
Fig. 4.17 TEM bright field image of precipitates with a selected area electron diffraction pattern and
a convergent beam electron diffraction pattern of one of the particles.
112
Fig. 4.18 TEM bright field image of a specimen reaustenitised at 778 QC.
Fig. 4.19 TEM bright field image of a specimen reaustenitised at 778 QC.
113
" .
to
'.'
..•..., ..
'~~0 3 m
I • J.L I
...., 1 11
850
Starting structure
...--. 8ainite + AusteniteCl
0•... 800
'-'>:r: •
~
VI
C
Cl) 750 •.-•....
Cl
E- •0
VI
VI 700Cl) •C
"tJ•....
Cl
.c
I
0
650•....u •~
600
760 780 800 820 840
Reaustenitisation temperature,OC
Fig. 4.20 Change in hardness of martensite with reaustenitisation temperature.
114
0.014
a)
"Ol
C
o
.r::
u
.r:: 0.010
Ol
c
"
".~oc; 0.006
a:O.OOB
0.010 •
0.012 ~.u ••"'•••••••••••••••••• 1••• ' •••••••••••• u •••••••••••••.•••••
Reouslenlllsed 01 776°C
"Ol
C
o
.r::
u
.r::
Ol
c~
">
]
"a:
0.006
0.006
o 20 40 60 Ba 100 o 20 40 60 BD 100
Timet S
Time, s
0.006
"Ol
C
o
.r::
u
.r:: 0.010 •;;,
c
.!!
Reouslenitised 01 B 15°C
e)
O.OOB
.\
0.012 : \.. ••••, ., .•, ,•....•.,•,••..•.,_, ••.._ ••..•....•..•••,• •••..••, . .•.•__11•••••••'M"
0.014
"'"co
.r::
u
.r:: 0.010
;;,
c
.!!.,
~
oc;
a:
Reouslenilised at 7BSoC
~ ....•..........................•......•..............•.........
b)
0.012
0.014
0.006 0.006
o 20 40 60 BD 100 o 20 40 60 60 100
Time. s
0.014
c)
Roouslonillsed 01 790°C
"Ol
C
o
.r::
u
.r::;;,
c
.!!.,
~
.!!
"a:
0.012
0.006
V, I, •••••••.••
0.006
o 20 40 60 BD 100
Time, s
Fig. 4.21 Relative length changes during isothermal reaustenitisation from a mixture of bainite and
austellite at a) 778 QC, b) 785 QC, c) 790 QC, d) 805 QC and e) 815 QC.
115
0.015
0.014
CD
(J)
c
0.0130
.c
U
.c
+-
(J)
0.012c
CD
-.J
CD.~
+-
0.0110
CD
a:::
0.010
0.009
400 500 600 700 800 900
Temperature,OC
Fig. 4.22 Change in relative length during up-quenching and isothermal holding at 815 QC. A method
of the temperature correction is illustrated in the figure.
corrected
0.015
0.014
CD
(J)
§ 0.013
.c
u
.c
+-
g> 0.012
CD
-.J
CD
>
o 0.011
CD
a:::
0.010
ITemp","'u,"
-c6l2>~o~-------------------,---
+
~Original data
+
+
+
0.009 o 5 10
Time, S
15 20
Fig. 4.23 Temperature corrected relative length change of the data in Fig. 4.22.
116
10080604020
0.0005
0.0000
Q) -0.0005C1l
c
0
.c
u
.c -0.0010-C1l
C
Q)
Q) -0.0015
>
0
Q)
-0.0020a::
-0.0025
-0.0030
0
Time, S
Fig. 4.24 Relative length changes during isothermal reaustenitisation from the mixture of bainite
and austenite at different reaction temperatures.
0.003
(1,) Fe-O.3C-4.08Cr (wt%)
C)
cca
.t:
0
.t:- 0.002C)
C
(1,)
(1,)
>-ca
(1,) 0.001:l-
E
:J
E
><ca
~
0.000
720 740 760 780 800 820 840
Temperature, QC
Fig. 4.25 Maximum relative length changes due to isothermal reaustenitisation from the mixture of
bainite and austenite at different reaction temperatures.
117
0.014
Cl)
Cl
C
0
.s::.
u 0.012 I
I.s::. I
0+- I
Cl I
C 1
~ 1
I
Cl) 0.010
I
> 1
0+- I.
0 I
Q) I
Cl:: II
I
0.008
I
I
I
I
I
I
I
1
I
0.006
0 200 400 600 800 1000 1200
Temperature, GC
0.016
Fig. 4.26 Relative length change as a function of temperature during heating. Difference in thermal
expansion coefficients of ferrite and austenite causes a change in the maximum relative
length change due to 100 % of transformation.
118
0.014 0.014
a) b)
Reauslenlllsed 01 778"C 0.012 Reaustenlllsed at 78S'C0.012
Cl 0.010 '" 0.010'" '"c c
0 0
r. r.
u U
r. 0.008 r. 0.008
0> '"C C
'" Cl.J .J
0.006 Cl 0.006Cl
~ .~"0~ c;
Cl a: 0.004a: 0.004
0.002 0.002
0.000 0.000
0 200 400 600 800 0 200 400 600 800
Temperature, 'C Temperature, 'C
0.014 0.014
C) d)
Reauslenlllsed 01 790'C 0.012 Reauslenlllsed 01 80S'C0.012
'" 0.010
Cl
0>
'" CC 0
0 r.r. uu 0.0080.008 r.r.
0. 0>C
C '".. .J.J 0.0060.006 '"Cl .~~
"0 ~
c; Cla: 0.004a: 0.004
0.002 0.002
0.000 0.000
0 200 400 600 800 0 200 400 600 800
Temperature, ·c Temperalure, 'C
800
••tr..::
Reaction Temp.:C
--- 778
785
790
805
815
400 600
Temperature. ·C
200
"-"' ....,
....... "-"
~.,\-
"', .."~).~
" ..'~\
..\
• A~\
~-............. \
..•.......... .\.- ~;..
......~~..
0.010
f)
-0.002 o
<i 0.008
'"0>c
0.006°r.
U
r.
0>
0.004c
'".J
'"~
° 0.002C;
0:
0.000
800600400
Temperature, 'C
200
0.000 o
0.014
e)
0.012 Reauslenltlsed aI8IS'C
Cl 0.010
0>
C
a
r.
u
:: 0.008
0>
c
Cl
.J
Cl 0.006.~
"0
c;
a: 0.004
0.002
Fig. 4.27 Relative length changes during helium quenching after 30 min of isothermal reaustenitisa-
tion from the mixture of bainite and austcnite at a) 778°C, b) 785 QC, c) 790°C cl) 805 QC
and e) 815 QC. Rclative lcngth changes 61 clue to Inartcllsitic transformation are cOlnpared
in f).
119
900
Fe-O.3C-4.0BCr wt.%
800
700
Ae3
u
0
Q)
600L-:J--0
L-
Q)
CL 500E I
Q) I.- I I
--+-
400 I I
I I
I I
I I
300
I ITo'
I I
I I
I I
I I
200
0.15 0.200.00 0.05 0.10
Carbon content, mole fraction
Fig. 4.28 Calculated phase diagram of the Fe-0.3C-4.08Cr alloy, which illustrates the partially reausteni-
tised temperature range.
0.003
Fe-O.3C-4.08Cr (wt%)
~.•...
Cl 0.002
c
Q)
Q)
>.•...
ca
Q)~
E
:J
E
><ca
~
0.001
0.000
720 740 760 780
1
800 820 840
Temperature, GC
Fig. 4.29 Calculated maximum relative length changes for equilibrium isothermal reaustenitisatioll.
Line 1 is calculated for case (1), line 2 for case (2) and line 3 for case (3). Plols are
experimenlally observed maximum relative length changes.
120
a
'C
1600
I~OO
Q
c
L
5~ chromium
I
4.0 4.~ ~.O
b
d
CHY
•." t." •." '0 "
"" ,.U 0." D.n 0," •." '.n ton •. " t.n
N[ 1~wn [ARlaw
Fig. 4.30 (a) Equilibrium phase diagram of a Fe-5 w1.% Cr alloy (Metals Handbook) and calculated
equilibrium phase diagrams for Fe-C-Cr ternary alloy at (b) 750QC, (c) 800QC and (d)
850 QC [18].
121
0.020
a) During hcating at 1.10 C/scc from 4200 C
0.018
· 0.016 T.mperolur.,·C (UI rlghl) 1000~a
-:;
s; O.OH
C. 750
~· 0.012>;;
Relollv. length.. 500
"" Change (u. lelt)
0.006
200 300 .00 500 6000 100
Time. s
0.016
0.015
· O.OH'"c:a
0.013-:;
s;
C.
0.012~·> 0.011;;..
"" 0.010
0.009
500 600 700 800 900 1000 tlOO
TemperolufI.·C
0.020 0.020
b) During hcating at 5.00 C/scc from 4200 C c) During hcating at 1J 0 C/scc from 4200 C
0.018 0.018
Temperalure.·C (see rlghl)
· Temp.rolur.,·C (see right) 1000 · 0.016 10000.016 DO'" c:c: a
~ s;u
0.01. s; 0.014.r; 0.C. 750 750
~ ~· 0.012· >> ;;;;.. R.lallve lenglh 500 .. 0.010 500
"" ""Chango ( ••• loft)
0.008
0.006
150 200
0.006 0
20 40 60 80 lOO0 50 100
Time. s Time, S
0.016 0.0.4
O.OIJ
0.014
0.012·'"c:. a
'" -:; 0.011c: 0.0120 ~-:;
s; ~
C. ·~ 0.010 > 0.009;;. ..
>
"";; 0.008..
"" 0.008
0.007
0.006
600 700 800 900 1000 tlOO 500 600 700 800400 500 900 1000 tlOO
Temperature, ·C Temporalur •• ·C
Fig. 4.31 Change in relative length and temperature during continuous heating reaustenitisation
from the mixture of bainite and austenite at a) 1.1 °Cs-1 from 420°C, b) 5.0°Cs-1 from
420°C, c) 11 °CS-l from 420°C, d) 1.1°Cs-l from 44BoC, e) 5.0°Cs-1 from HBoC,
f) 1.1 °Cs-1 from 472°C and g) 5.0°Cs-1 from 472°C.
122
0.020 0.020
d) During healing al 1.10 C/sec from 4180 C c) During hcaling ill 5.00 C/scc from 4480 C
0.018 0.018
0.016 Temp.relur. (se. right) 1000 · 0.016 \000· '"'" cc aa -:;-:; 0.014 0.014~ 750 t 750a. ii 0.012 · 0.012· >> ;;;;
0.010 Reloti .•.• longlh 500
.. Relollvl lenglh - 500..
""
"" Chang. (s •• 1.1I) Chang. ( ••• I,")
0.008 0.008
0.006 0.006
0 100 200 JOO 400 500 600 0 SO 100 ISO 200
Time, S Time, s
0.016 0.016
0.014 0.014
· ·'"'" cc aa -:;~ 0.012u 0.012 ~~ a.a.
i i· · 0.010> 0.010 >;; ;;....
""""
0.008 0.008
0.006 0.006
400 500 600 700 800 900 1000 1100 400 500 600 700 800 900 1000 1100
Temperature, ·C T.mp •...olur •. ·C
0.020 0.020
f) During hcating at 1.10 C/scc from 4720 C g) During heating at 5.00 C/sec from 4720 C
0.016 0.018
· 0.016 Temp.rolur. (st. right) 1000 · 0.016 1000'" '"c ca ~-:;
0.014 0.014
t 750 t 750c ~~ 0.012 0.012· ·> >;; "0.. 0.010 Relative length 500 .. 0.010 R.latl .•.e length 500
"" ""Chang. ( ••• 1.1I) Change (s •• left)
0.008 0.008
0.006 0.0060 100 200 JOO 400 500 600 0 SO 100 \50 200
TIme, S Time, s
0.016 0.016
0.0\4 0.014
· ·'" '"c 0.012 c 0.012a 0-:; ~u
t ~0.010 0.~ ~ 0.0\0· ·> >"0 0.008 "0 0.008.. ..
"" ""
0.006 0.006
0.004 0.004400 500 600 700 800 900 1000 1100 400 500 600 700 600 900 1000 1100
Temp.,olur •. ·C
remp.roture, ·C
Fig. 4.31 (continued).
123
--
o
•
I
.
B
•
•
900 I
u
° Tb,oCcD•...
:l 0420.•...
0•... 044BQ)
0-
E ~ 472
2 B50 f-.c.~ •c
.•.... !
"U
C
0
t
0
Vi
c BOO-
Q
:;: 0
0
E•...
Q.•....
III 0
C
0 ••...
I-
750
I
1.0[+00
I
1.0[+01
Heating rate,oC/s
Fig. 4.32 Change in the thermal expansion coefficient of the initial microstructure of mixtures of
bainite and austenite as a function of bainite transformation temperature.
2.5E-05
.•...
s::
Q)
u
.•....
2.0E-05.•....CD 0
Q
u
0
c
0
III
C
0
0-
X
CD
0 1.5E-05E•... 0
CD
.c
I-
1.0E-05
400 420 440 460 4BO 500
Temperature,oC
Fig. 4.33 Transformation-start (open) and -finish (solid) temperatures during continuous heating
reaustenitisation from the mixtures of bainite and austenite obtained at Tb.
124
1.0 Bainite transformationtemperature,QC • •.
+ 420
2 448
'c 0.8 • 472
2
l/l
::J
«
JI.- 0.60 ·tr +c ·f +.2 +"0 «#x +
0 0.4 :j: +•.. +- +Q) , ,[' tE 'xx.~ +::J 4: x" +0 0.2 -.-, +t>
•••• l-' ++i
• •••• ~If t +
•.:• J
0.0 ~~-.t
760 780 800 820 840
Temperature, QC
Fig. 4.34 Normalised volume fraction of austenite transformed on heating as a function of tempera-
ture.
700
U
Q
680rD•..
::J
0+-
0•..
CD
a.
E 660CD
0+-
t •
0
"in
c
0 640
0+-
l/l
0
a. •
E
0 •u
620CD
Cl
600
400 420 440 460 480 500
8ainite transformation temperature.QC
Fig. 4.35 Decomposition-start temperatures during continuous heating plotted against the bainite
transformation tern perat ure.
125
Fig. 4.36 Optical micrograph of a specimen helium quenched during the proceeding of decomposition
of residual austenite on the way of the continuous heating reaustenitisation experiment at
1.1 QCS-l after 30 min of isothermal bainite transformation treatment at 445 QC.
Fig. 4.37 TEM bright field image of the specimen in Fig. 4.36.
126
CHAPTER 5
PEARLITE TRANSFORMATION IN STEELS
5.1 INTRODUCTION
The reconstructive formation of pearlite in steels is studied in this chapter. Since the
bainite transformation stops before equilibrium is achieved, the untransformed austenite can
decompose to ferrite and cementite by a reconstructive mechanism. This may happen during a
prolonged isothermal holding at a bainite transformation temperature and during heating after
bainitic transformation. The later case can be found, for example, in multirun welding where
the successive deposition of weld metal heats up the underlying layers.
5.2 DECOMPOSITION OF RESIDUAL AUSTENITE DURING HEATING
As discussed in the previous chapter, the austenite left untransformed after the completion
of bainite transformation (i.e. residual austenite) decomposes to pearlite during slow heating.
The decomposition-start temperature Td• increases with the heating rate and no detectable
decomposition of austenite occurs at higher heating rates (Fig. 5.1). The lower the bainite
transformation temperature, the higher the decomposition-start temperature (Fig. 5.1).
An attempt to calculate the temperature Td• at which the decomposition of residual austen-
ite starts on heating, from an initial microstructure which is a mixture of bainitic ferrite and
residual austenite, is made in this section.
The temperature Td• for the reconstructive formation of ferrite from austenite on heating
can be calculated by using the TTT curve for austenite which has the same chemical compo-
sition as the residual austenite. Scheil's rule may then be applicable for the calculation of the
reconstructive formation of ferrite on heating, although there is no proof of that the reaction is
isokinetic [1].
5.2.1 TTT curve calculation of untransformed austenite
Bainite transformation ceases prematurely when the carbon concentration of the residual
austenite reaches the T~curve (xT')' because the diffusionless formation of ferrite from austeniteo
with its carbon content of more than xT' at a reaction temperature will cause a positive freeo
energy change, which is thermodynamically impossible. However, since the xT' value is faro
less than the equilibrium carbon concentration of austenite, reconstructive transformations can
occur even after the bainite reaction ceases. This decomposition may be expressed by the upper
C-curve of the TTT diagram, which will be used here for the estimation of Td •.
The calculation of TTT diagrams for the materials with different chemical compositions
were carried out using a method reported by Bhadeshia [2]. He assumed an expression for
incubation periods proposed by Russell [3]' given by:
T
T<X ----
(6.G':r,)p D (5.1)
where
T : absolute temperature,
D : an effective diffusion coefficient related to boundary or volume diffusion, depending on
the coherency state of the nucleus concerned,
127
~G~ : the maximum volume free energy change accompanying the formation of a nucleus in a
large amount of matrix phase,
p : an exponent whose magnitude is a function of the nature of the nucleus. Russell [3]
obtained p = 2 for a coherent nucleus and p = 3 for an incoherent one.
On substituting an Arrhenius expression for diffusion coefficients:
(5.2)
where 51 is the activation entropy for diffusion and Q1 is the activation enthalpy for diffusion.
Bhadeshia [2] obtained the following relation:
(5.3)
where ~Gm is the chemical free energy change accompanying the formation of one mole of
nucleating phase in a large amount of matrix phase and CN1 is a constant. He then expressed
Q1 and 51 as a function of temperature; i.e. Q1 = QO+CN2(T-T') and 51 = 50+CN3In(T-T'),
where So, Qo, CN2, CN3 and T' are constants, and finally obtained the following expression [2]:
(5.4)
with constants z, Q' and CN4. By maximizing the correlation coefficient when the equation is
compared with experimentally observed TTT curves, Bhadeshia [2] obtained Q' = 0.2432 X 10-6
Jmol-1, CN4 = -0.0135, P = 5 and z = 20 for shear transformation, and Q' = 0.6031 X 10-6
Jmol-1, CN4 = -0.01905, P = 4 and z = 20 for diffusional transformation respectively. These
values were used to calculate the transformation-start temperature of austenite during heating
in the present work.
The carbon content of residual austenite xi {Td and the volume fraction of untransformed
austenite VI have been calculated from the relative length change ~L/ L during isothermal
bainite transformation at each temperature Tb using equation 4.1, and the results are listed in
Table 5.1. The calculated TTT curves for the austenite which has the carbon concentration
identical to each values listed in Table 5.1 can be seen in Fig. 5.2. The incubation periods for
both diffusional and displacive transformations become large with lowering the bainite trans-
formation temperature. The lower the bainite transformation temperature, the lower the Bs
and Ms temperatures of the austenite because of an increase in carbon concentration in the
untransformed austenite.
420
448
472
~L/L xi {Td, at.% VI
3.86xlO-3 2.4 0.299
1.76 X 10-3 1.8 0.669
2.90xlO-4 1.4 0.944
Table 5.1: Volume fractions and carbon contents of residual austenite in the initial microstruc-
ture
128
5.2.2 Kinetic calculation of decomposition-start temperature
A point on an upper C-curve representing the minimum detectable reconstructive formation
offerrite at a temperature T; is now designated as an incubation period T{T;}. When 6.T refers
to a small temperature step in a small time period 6.t, ~~ gives a heating (or cooling) rate
at a temperature. At a step between T; to T; + 6.T, austenite spends a part of the incubation
period;
(5.5)
with
1
T; = 2( T{T;} + T{T; + 6.T} ).
When the summation of P; becomes unity, the transformation is assumed to start. In order
to calculate the position where the austenite is on the TTT diagram during heating after the
completion of bainite transformation, the following equation is used (Fig. 5.3).
(5.6)
where
T : T; + ~6.T;
tdT} :instantaneous time on the TTT diagram at a temperature T on heating.
The calculations were carried out for continuous heating at constant rates between 0.01
and lOQCs-1, for three materials whose carbon contents are identical to the residual austenite
left untransformed after the cessation of the bainite transformation at 420, 448 and 472 QC
in the Fe-0.3C-4.08Cr wt.% alloy respectively (Fig. 5.4). Since the formation of bainitic fer-
rite is thermodynamically impossible above the temperature at which the bainite treatment
was conducted, and since the incubation period of reconstructive formation of ferrite at tem-
peratures below the B s temperature is usually extremely large, the calculation was started
from the Bs temperature. In Fig. 5.5, the transformation-start temperatures for each material
are drawn with the Td• temperatures obtained in the present experiments. Each calculated
transformation-start curve corresponding to the different carbon content increases linearly with
heating rate at low heating rates, and rapidly increases at higher heating rates, reaching to
the Ae~ temperature; above this heating rate, transformation is not expected to occur during
heating. Although the calculation treats the austenite in isolation (i. e. ignore the presence of
the bainitic ferrite), rather good agreement was obtained with experimental results (Fig. 5.5).
The fact that almost no decomposition has been observed in the cases of the heating at more
than 5 QCS-l is also explained as the austenite can be heated up above the Ae~ temperature
without meeting the diffusional C-curve on heating (Fig. 5.5).
The fact that the change in the decomposition-start temperature of austenite during con-
tinuous heating can be understood by using the To concept, is further evidence in support of
the incomplete reaction phenomenon.
5.3 FORMATION OF PEARLITE BELOW THE Bs TEMPERATURE
Pearlite formation at very low temperatures may become important under circumstances
where austenite is held at low temperatures for an extended period of time without being cooled
129
below the Ms temperature of the alloy. This may occur, for example, in the heat affected zone
of a multirun weld or in the very slow cooling of hot rolled steel strip after coiling.
Bhadeshia [4,5] reported a diffusional perturbation of the interface between bainitic ferrite
and residual austenite in the Fe-0.3C-4.08Cr wt. % alloy after an extended period of isothermal
holding at 478°C, just below the Bs temperature 492°C of the alloy. The growth rate of
the perturbation of the interface was found to be far slower than the calculated growth rate
of ferrite controlled by the carbon diffusion [5]. Although bainite transformation ceases when
the carbon concentration in residual austenite becomes xT" reconstructive transformation too
ferrite is still possible since xT' is far smaller than the equilibrium carbon concentration of theo
austenite. A relative length change due to bainite transformation at 472 °C; which is very close
to the reaction temperature used by Bhadeshia [5]' in the Fe-0.3C-4.08Cr wt.% alloy can be
seen in Fig. 4.3, showing that the bainite transformation completed within about 30 min of
isothermal holding at the temperature.
Specimens with the identical alloy chemistry as [5] were heated to 1100 °C in a furnace and
transfered to a different furnace at 478°C. The specimens were then quenched into water after
300, 600, 241200 (67 hours), 576000 (160 hours), 1987200 (23 days), and 3715200 (43 days)
seconds. The volume fraction of bainite increases with time and seems to remain constant
beyond 30 min [5] as it can be seen in optical micrographs (Fig. 5.6). After 160 hours of
isothermal holding, the formation of a new phase can be found at the austenite grain boundaries
(Fig. 5.6). The new phase then grows into one of the austenite grains separated by the grain
boundaries at which the nucleation takes place. From a TEM micrograph, one can see a
lamellar structure of carbide and ferrite (Fig. 5.7). The structure and the temperature of
the transformation suggest that the new phase is pearlite nucleated at the austenite grain
boundaries.
The calculated upper C-curve which corresponds to reconstructive formation of austenite,
was extrapolated to temperatures lower than the Bs temperature of the alloy. The extrapolated
C-curve shows the time required to obtain the minimum detectable amount of reconstructive
ferrite is about 150 hours, close to the value observed optically in the present work. This may
mean that the austenite after the completion of bainite transformation behaves like an isolated
austenite for reconstructive formation of ferrite.
The maximum thickness of pearlite at each holding time were measured and plotted in
Fig. 5.8, giving the growth rate of pear lite in the alloy at 478°C equal to about 1.0 X10-9 cm S-I.
5.4 GROWTH RATE OF PEARLITE
The pearlite discussed above seems to consist of a mixture of alloy carbide (M7Ca) and
ferrite as mentioned by Bhadeshia [5]. However, it is established that the formation of conven-
tional pearlite can form at a temperature below the Bs temperature after the completion of
bainite transformation [6]. Therefore it is important to investigate the formation of pearlite at
very low temperatures as well as at temperatures close to the Ae1 temperature. In this section,
the growth theories of conventional pearlite which have been published so far are studied.
The formation of pearlite is expected to occur when austenite is cooled to temperatures at
which both ferrite and cementite are thermodynamically stable. Pearlite is a lamellar mixture of
iron and iron carbide which is very common constituent of a wide variety of steels contributing
substantially to strength. As a result, it has been investigated intensively so far. The kinetics
of pearlite transformation have been reviewed recently (for example [7,8]).
130
Pearlite transformation in steels is reconstructive and known to show a constant growth
rate because the composition of untransformed matrix remains unchanged except near the
transformation front [1]. The growth rate of pearlite is believed to be controlled by either
volume diffusion of carbon [9,10] or by boundary diffusion of substitutional alloying element
[11-13]. The two growth theories are summarized as follows.
When the growth rate of pearlite is controlled by the bulk diffusion of atoms in austenite
ahead of the interface, the diffusion of carbon may play a more important role than that of
substitutional alloying elements, since the diffusivity of the substitutional alloying elements in
austenite is far smaller than that of carbon. As a result, the substitutional alloying elements
may not diffuse a long distance during the reaction. Therefore the equilibrium condition which
is maintained at the interface between austenite and pearlite may be either the negligible
partitioning local equilibrium (NPLE) or paraequilibrium (PE). The growth rate of pearlite is
expressed as follows [10]:
(5.7)
where
Vv volume diffusion controlled growth rate of pearlite,
D volume diffusion coefficient for carbon in austenite,
g geometric factor equal to 0.72 in plain carbon steels,
xf" carbon concentration at the 'Y/ a interface in 'Y, under either local equilibrium or parae-
quilibrium with a,xr~: carbon concentration at the 'Y/8 interface in 'Y, under either local equilibrium or parae-
quilibrium with 8,
x~1' : carbon concentration at the 8/'Y interface in 8, under either local equilibrium or parae-
quilibrium with 'Y,
x~1' : carbon concentration at the a/'Y interface in a, under either local equilibrium or parae-
quilibrium with 'Y,
s interlamellar spacing,
se critical spacing at which the growth rate becomes zero,
sa' Se respective thickness of ferrite and cementite lamellar.
with the maximum velocity criterion, the relation between S and se can be obtained by setting
the first derivative of equation (5.7) equal to zero, which leads to (9),
S = 2se
where
with
(Tae surface energy of ferrite/cementite interface,
6.H change in enthalpy between the parent and product phases,
TE eutectoid temperature,
fiT undercooling below the eutectoid temperature.
131
(5.8)
(5.9)
The critical spacing se can be compared to observed interlamellar spacing assuming the relation
(5.8). Interlamellar spacings in Fe-C, Fe-C-Cr, Fe-Ni and Fe-C-Mn alloys [8,15-17] were used
to obtain an empirical expression for the interlamellar spacing as a function of the temperature
and the alloy contents. Using the data (Table 5.2), the following relation was obtained for Fe-C,
Fe-C-Mn, Fe-C-Ni and Fe-C-Cr steels.
T -T
log{ s} = -2.2358 + 0.09863 x Mn - 0.05427 x Cr + 0.03367 x Ni -log{ E }TE
(5.10)
where s is measured in J.Lm and Mn, Cr and Ni in wt.%. It is noted that an addition of Cr
or Ni decreases the interlamellar spacing as it is observed in many other alloys, whereas a Mn
addition causes an increase in the lamellar spacing. A comparison between observed and calcu-
lated interlamellar spacing can be seen in Fig. 5.9. The expression for the interlamellar spacing
was applied to a material which contains both Mn and Cr. The observed and calculated inter-
lamellar spacing of Fe-1.02Mn-1.05Cr wt.% alloy [18] are plotted in Fig. 5.10. The expression
of interlamellar spacing obtained in the present work seems to represent experimental results
very well even in alloys which contain both Mn and Cr.
Comparing equations (5.9) and (5.10) assuming the relation (5.8), the following relation
should be maintained.
4u 9 .
log{ .6.H } = -2.2358 + 0.09863 x Mn - 0.05427 x Cr + 0.03367 x NI
If coefficients reported by PuIs and Kirkaldy [7] are adopted, the left hand side becomes
-2.404 in plain carbon steels which is very close to the value -2.236 obtained in the present
analysis .
.6.H =6.09 x 109 erg cm-3
ua9 =600 ergcm-2
TE =1000 K
It is, however, worth noting that using a more advanced theory proposed by Hashiguchi and
Kirkaldy [19]' the ratio si se varies between 1 and 2.
When the partitioning of the substitutional alloying elements is substantial during the
growth event of pearlite, boundary diffusion of the alloying elements may control the growth
rate of pearlite, since the boundary diffusivity of the substitutional alloying elements may
become comparable to the bulk diffusivity of carbon in austenite, and, as a result, a long range
diffusion of the substitutional alloying elements may become possible during the reaction. The
growth rate, in the case, is expressed as follows [13].
(5.11)
where
vB : boundary diffusion controlled growth rate,
K : boundary segregation coefficient which is the ratio between alloying element concentration
in austenite near the boundary and that in the boundary,
DB boundary diffusion coefficient of substitutional alloying element,
132
o : thickness of the boundary,
x~a : substitutional alloying element concentration at the 'Y Ia interface in 'Y which is under
either local equilibrium or paraequilibrium with e,
x~e : substitutional alloying element concentration at the 'Yle interface in 'Y which is under
either local equilibrium or paraequilibrium with e,
x
2
: average substitutional alloying element concentration in the alloy concerned.
Sharma et al. [14] determined the factor K DBo for Cr steels with from 0.9 to 1.8 wt.% of Cr,
gIvmg:
(5.12)
These models have been applied to calculate theoretically the growth rate of pearlite
[7,14,20,21]. Although the calculated results show reasonable agreement with experimentally
observed growth rates of pearlite in plain carbon steels [7]' it seems to be more difficult to
predict the growth rate of pearlite in alloyed steels. At higher temperatures, where lower su-
persaturations are expected to exist, boundary diffusion of the substitutional alloying element
seems to control the growth of pear lite [7,14,21]. At higher supersaturations, on the other hand,
the reaction seems to be controlled by volume diffusion under local equilibrium [14,19,22]. It is,
however, worth pointing out that experimentally observed growth rate of pearlite tends to have
the maximum velocity at a certain temperature. This was observed in Fe-C-Cr alloys [14,21],
in Fe-C-Mn-Cr alloys [18]' in Fe-C-Mn alloys [16,23] and Fe-C alloys [15,24], whereas the the-
oretical calculations have not succeeded in reproducing the position of the peak and the slope
below the peak, except a work of Hashiguchi and Kirkaldy [19] who have combined volume and
boundary diffusion controlled growth models and managed to obtain an excellent agreement
with experimentally observed growth rates of pearlite in a Fe-C binary system, although they
had to adopt a few adjustable parameters. Therefore it can be said that the calculation of the
growth rate of pearlite at lower temperature is not yet established.
5.5 CALCULATION OF THE INTERFACE COMPOSITIONS
The growth rate of pearlite as discussed in the previous section depends upon the interface
compositions of ferrite, cementite and austenite. In order to determine the full set of the
interface compositions, we need to know the 'Yla, 'Yle and ale interface compositions of carbon
and the alloying element in Fe-C-X ternary alloys (X denotes the third substitutional element
in the alloy). If the heterogeneity of the chemical concentration in ferrite and cementite is
. d h . h' f . . -ya -ya a-y a-y -ye -ye e-y d e-y bIgnore , t e eIg t mter ace compOSItIOns Xl , X2 ,Xl 'X2 , Xl , X2 ,Xl an X2 are to e
determined.
An approximate method which has been reported by Kirkaldy and his coworkers [25-27]
will be used for the calculation of the interface compositions under either paraequilibrium or
local equilibrium. The equilibrium condition can be expressed by the equality of the chemical
potentials of each element in both phases at the interface. The chemical potentials of carbon,
X and iron in ferrite, austenite and cementite are expressed as follows; the numbers 0, 1 and 2
denote, respectively, iron, carbon and the substitutional alloying element X.
133
In austenite and ferrite;
(5.13)
(5.14)
(5.15)
and in cementite;
(5.16)
(5.17)
where yf = ~xt~ are the i-th element concentration in cementite, w's are Fe-X interaction
coefficients and €'s are Wagner's interaction coefficients.
In the case of paraequilibrium, the interface compositions can be determined from the
following equations.
-y9( 4 -y 1 -Y) -y9(4 -y 1 -Y)_
Xo "iJLFe3C - JLo - "iJL! + x2 "iJLM3C - JL2 - "iJL! - 0
with the relation expressing the definition of the paraequilibrium,
Cl!"( -ya. -y9 9 -
x2 x2 x2 Y2 x2----ci'f = ----::ya = -9 = 9 = =- = k2
Xo Xo X6 Yo Xo
and the constant carbon concentration in cementite, x~-y = 0.25.
In the case of the local equilibrium at the interface, on the other hand,
conditions can be expressed as follows.
JLf = JLl ,(i = Oto2)
(5.18)
(5.19)
(5.20)
(5.21)
the equilibrium
(5.22)
(5.23)
(5.24)
As long as a dilute solution is concerned, the following expressions can be adopted for the cal-
culation of the interface compositions under either the paraequilibrium or the local equilibrium.
For the 1/ ex equilibrium:
(5.25)
134
(5.26)
and for the I/O equilibrium:
with
exp(.0.G~1"/ RT + fl1xI")A
1
= ----------
1+ fr1xI" exp(.0.G~1" / RT)
exp(.0.G~1"/ RT + flixI")
A2 = 1+ friXI" exp(.0.G~1"/ RT)
(5.27)
(5.28)
(5.29)
(5.30)
When the growth rate of pearlite is controlled by the diffusion of carbon in austenite
ahead of the interface between pearlite and austenite, it is natural to assume that there is no
redistribution of X between matrix and precipitate phases. Therefore, the interface composition
may be determined by the paraequilibrium condition.
When the redistribution of the third element X can occur during the growth event of
pearlite, on the other hand, the local equilibrium condition is maintained at the interfaces.
Since the diffusion of X is the controlling process in this case, the flux of car bon in the parent
phase should be negligible when it is compared with that of X in the interfaces. This condition
is referred to as 'partitioning under local equilibrium' (PLE) as it is well established in the
case of the formation of ferrite from austenite [28,29]. The diffusivity of X, in this case, is
expected to be much faster than that in the matrix (i.e. austenite) [30]. The bulk diffusion
coefficient of Cr in austenite at 600°C is calculated to be 2.5 x 10-18 cm2 S-l whereas the
boundary diffusivity is 9.6 x 10-10 cm2 S-l [30]' assuming that the thickness of the interface
is 3 X 10-8 cm2 S-l, which is comparable to the diffusion coefficient of carbon in austenite
calculated to be 1.7 x 10-9 cm2 S-l [31]. When the growth rate of pearlite is controlled by
the diffusion of X atoms within the interface region, the partitioning under local equilibrium
(PLE) may exist at the interfaces, under which the activities of carbon in austenite at the I/a
and I/O interfaces are identical so that the flux of carbon ahead of the interfaces is negligible,
thus allowing the diffusion of X atoms catch up with that of carbon. Therefore the interface
compositions can be determined by two tie lines AB and CD in Fig. 5.11, where the line AC is the
isoactivity line of carbon in austenite. The PLE condition maintains until the supersaturation
becomes large enough to make the flux of X atom comparable to that of carbon by producing
a sharp spike at the vicinity of the interface. This condition should be designated as 'negligible
partitioning under local equilibrium' (NPLE). Since there are two interfaces, I/a and I/O, four
different combinations of equilibrium at the interfaces are possible to exist; (1) PLE at the
both interfaces, (2) NPLE at the both interfaces, (3) PLE at the 1/ a interface and NPLE at
the I/O interface, and (4) the vice versa. The transition from the PLE to the NPLE condition
occurs when x~1" = x~ for the 1/ a interface and under the PLE condition x~1"= x~ for the I/O
interface (Fig. 5.12).
135
At the final stage of the reaction, ferrite and cementite should be in equilibrium such that
the compositions are determined by the tie line passing through the bulk composition of the
steel in the ferrite/cementite two phase region. In the real situation, however, the partitioning
of the substitutional alloying element becomes small as the temperature decreases.
Calculated phase diagrams for a Fe-1.08Mn-C wt.% and a Fe-1.41Cr-C wt.% alloys at
different temperatures are presented in Fig. 5.13.
5.6 CALCULATION OF THE GROWTH RATE OF PEARLITE
Using the method described in the previous section for the calculation of the interface
compositions, the growth rates of pearlite in Fe-C-X (X = Mn and Cr) ternary alloys were cal-
culated. The two alloying elements were selected because fine experimental results are available
from the published literature.
Calculation of paraequilibrium growth rate of pearlite is straightforward; i. e. the interface
compositions calculated as discussed in the previous section will be used in equation (5.7) with
the expressions for sand sc. The diffusion coefficient of carbon in austenite is calculated as
discussed by Trivedi and Pound [32]. The growth rate under the local equilibrium, on the
other hand, is not as simple as the paraequilibrium case. In order to calculate the growth rate
of pearlite under the local equilibrium, the constants K, DB and 8 in equation (5.11) should
be determined. It is, however, less clear how to derive the constants. Therefore an empirical
method which was used by Sharma et al. [14] is adopted here.
As mentioned earlier, the diffusion of substitutional alloying elements plays an important
role when the supersaturation is low. As a result, the growth rate of pearlite is considered to
be controlled by the interface diffusion of the substitutional alloying element at higher tem-
peratures, whereas by the bulk diffusion of carbon at lower temperatures. However, as it is
well established, the partitioning of chromium is observed even at temperatures where bainite
transformation takes place [21]. Therefore interface diffusion may play an important role on
governing the growth rate of pearlite even at low temperatures. Although Sharma et al. [14]
obtained equation (5.12) using the data of the growth rate of pearlite at low supersaturations,
it may be possible to modify the equation to include the data at higher supersaturations.
Therefore the factor K DB8 is estimated using the calculated interface compositions and the
interlamellar spacing for eutectoid steels containing Mn or Cr using the following equation.
[
2 (-yo. -ye) ]
K D 6 = v / 12_s_ x2 ~ x2 1 (1 _ Sc)
B B S S X S2 S
0. e 2
(5.31)
(5.32)
where the ratio between So. and se is assumed to be 7. The values Sc and s are calculated
using equations (5.8) and (5.9) respectively. The data used to obtain the equation are listed in
Table 5.3.
Assuming the same expression of K 8DB as equation (5.12), we now obtain a relation as
follows (Fig. 5.14).
K DB 8 = Co exp ( - ~~ )
where constants Co and QB are 2.223 x 10-8 cm3 S-l and 140940 J mol-
1 for the 1.1 wt.%
Mn steel and 1.052 x 10-8 cm3 S-l and 148320 J mol-1 for the Cr steels with between 0.9
and 1.8 wt.% of Cr. Assuming that the boundary diffusivities of Mn and Cr are same as
136
T, 0 C Xl' wt.% V cm2 S-l 1/ s, cm-1 ,0: ,e KD {; cm3s-1E' x2 - x2 E'
720 Cr=1.8 0.35E-04 0.78E+05 0.0102 8.90E-17
720 Cr=1.3 0.18E-04 0.58E+05 0.0074 8.94E-17
710 Cr=1.8 0.70E-04 0.97E+05 0.0186 7.94E-17
710 Cr=1.3 0.40E-04 0.76E+05 0.0157 7.00E-17
710 Cr=0.9 0.38E-04 0.58E+05 0.0133 9.41E-17
700 Cr=1.8 1.20E-04 1.07E+05 0.0269 7.87E-17
700 Cr=1.3 0.84E-04 0.94E+05 0.0238 6.35E-17
690 Cr=I.8 1.80E-04 1.36E+05 0.0353 5.67E-17
690 Cr=I.3 1.40E-04 1.12E+05 0.0316 5.65E-17
680 Cr=1.8 2.15E-04 1.55E+05 0.0441 4.22E-17
670 Cr=1.8 2.20E-04 1.75E+05 0.0530 2.78E-17
660 Cr=l.4 5.00E-04 1.94E+05 0.0504 3.94E-17
650 Cr=1.4 4.50E-04 2.14E+05 0.0504 2.56E-17
640 Cr=1.4 3.70E-04 2.33E+05 0.0592 1.57E-17
670 Mn=1.1 0.82E-04 0.56E+05 0.0220 1.16E-16
660 Mn=I.1 1.45E-04 0.72E+05 0.0266 1.01E-16
640 Mn=1.1 5.22E-04 1.04E+05 0.0378 1.21E-16
620 Mn=1.1 11.3E-04 1.36E+05 0.0529 1.09E-16
580 Mn=1.1 1l.3E-04 2.01E+05 0.0827 3.25E-17
550 Mn=I.1 9.22E-04 2.49E+05 0.1050 1.37E-17
530 Mn=1.1 4.59E-04 2.81E+05 0.1200 4.73E-18
Table 5.3: Calculation of K DE{; from published pearlite growth data in Fe-C-Cr and Fe-C-Mn
alloys. The first eleven data are after Sharma et al. [14), the next three data after Chance and
Ridley [21] and the last seven data after Razik [16]. x~O: - x~e is measured in mole fraction.
the self boundary diffusion coefficient of iron in austenite, DE{; can be expressed as 5.4 X
10-8 exp( -155500/ RT) cm2 S-l [30]. This gives expressions for the boundary segregation co-
efficients K = 0.41 exp(14560/ RT) and K = 0.20 exp(7180/ RT) for Mn and Cr respectively.
The boundary segregation coefficients of Mn and Cr are evaluated to be 3.0 and 0.54 at 6000 C
respectively.
Ridley [8] reported the growth rate of pearlite, interlamellar spacing and partition coeffi-
cient in 1.08 wt% Mn and 1.41 wt.% Cr eutectoid steels (Fig. 5.15). The growth rate of pearlite
has a peak in both cases whereas the partition coefficient shows a monotonous decrease towards
the unity at lower temperatures. It is worth noting, however, the change in the partition coef-
ficient as a function of reaction temperature in the Cr steel is more moderate than that in the
Mn steel (Fig. 5.15).
The calculation of the interface compositions under local equilibrium was carried out using
the method described in the previous section. The calculated interface compositions x~' and
x~' were then used to derive the partition coefficient Kp of X using the following expression.
(5.33)
At higher temperatures, the partition coefficient is expected to have a larger value since the
supersaturation of austenite is small at those temperatures whereas the mobility of X atom is
high. The value Kp decreases with temperature and reaches the unity when the NPLE condition
maintains both at the 'Y/ Q' and at the 'Y/ () interfaces. A change in the Kp value in eutectoid
steels with either 1.08% Mn or 1.41% Cr as a function of the transformation temperature is
plotted in Fig. 5.16 and Fig. 5.17. Experimentally observed partition coefficients [8] were
137
also plotted in the figures. The calculation was conducted for 0.8, 0.7 and 0.6 wt.% alloys to
clarify the effect of supersaturation on the partition coefficient in the case of the Mn steel. As
it can be seen from the figures, the calculation seems to express the change in the Kp value
successfully except at higher temperatures where an effect of the level of supersaturation is
large. The difference in the change in the Kp values between the Mn and Cr steels was also
reproduced successfully by the present model. The Kp in the Fe-C-Cr alloy stays higher even at
low temperatures when it is compared with the Fe-C-Mn alloy. The PLE condition maintains
even at 600QC in the Fe-C-Cr alloy whereas the NPLE condition takes over at around 600QC
in the Fe-C-Mn alloy.
The calculated growth rates of pearlite are compared with the experimental values reported
by Ridley [8] in Fig. 5.18. The growth rate calculated assuming paraequilibrium diffusion of
carbon in austenite shows a good agreement with the observed growth rates except at lower
temperatures than the temperature at which the growth rate becomes the maximum. The
growth rate of pearlite under local equilibrium also shows a peak which is consistent with the
experimental results (Fig. 5.18). Since the constants in equation (5.11) were determined by
fitting the equation to the data for the Fe-C-Mn alloy, the growth rate of pearlite calculated
under the local equilibrium is naturally consistent to the experimental values.
The paraequilibrium growth rate is retarded by Mn additions as shown in Fig. 5.19. Since
the Ae1 temperature is affected strongly by Mn addition, the growth rate at higher temper-
ature is more sensitive to Mn addition than the lower temperatures. In the case of the local
equilibrium, the effect of the Mn addition is more substantial (Fig. 5.20). In the both cases,
the growth rate and the peak temperature are reduced by the Mn addition.
The growth rate calculation was also carried out for the Fe-1.41 wt% Cr eutectoid alloy.
Calculated growth rates of pearlite under paraequilibrium or local equilibrium conditions are
compared against observations in [8] (Fig. 5.21). Although the agreement is not as good as
for the Mn steel, the temperature dependence of the growth rate seems to be well expressed
especially by the local equilibrium calculation.
Although the equations derived in this work can only be appropriate for the steels concerned
above, the calculation was extended to a Fe-0.3C-4.08Cr wt.% in which the growth rate of
pearlite at 478 QC was measured to be around 1.0 x 10-9 cm S-1. The paraequilibrium growth
rate of pearlite at 478 QC was calculated to be 1.9 x 10-3 cm S-1 which is far larger than the
observed one. The local equilibrium calculation gave the growth rate of pearlite equal to
6.8 x 10-5 cm S-1 at 478 QC which is much smaller than the paraequilibrium growth rate but
still orders of larger than the experimental value. However, it should be noted that the theory
discussed here is for conventional pearlite whereas the pearlite observed in the Fe-0.3C-4.08Cr
wt. % alloy is alloy pearlite. Although the theory of the growth of alloy pearlite has not been
developed, this result may suggest, for the first approximation, that the growth of pearlite at
very low temperatures is not a diffusion controlled reaction but rather an interface controlled
process. As a movement of an interface is controlled both by diffusions of atoms, which provide
the chemical concentration differences within each phase, and by a sluggish transfer of atoms
across the interface, which therefore does not give rise to any large concentration differences.
The former case is referred to as a diffusion-controlled process, and the latter as an interface-
controlled process [33]. It is not difficult to imagine that transformations from austenite to
ferrite are controlled by the interface process more strongly than the diffusion of atoms. The
rate of reaction will then be smaller than that expected by the local equilibrium mechanism
138
[33].
The rate of reaction controlled by the interface process is given by the following equation
[1]:
_~ g* _~g{3(){
vI = 8vexp( RT )[1 - exp( RT )] (5.34)
where 8 is the width of the interface, ~ag* and ~g{3(){ are the activation energy and the driving
force for the transformation per atom, v is a characteristic frequency which is given by the
value kTfh where k and hare Boltzmann's and Plank's constants respectively.
~g{3(){ is calculated as the free energy change due to the formation of pearlite from austenite
as follows [34].
(5.35)
where
~G8 = 29325 - 28.74T J mol-1
~G};(){ = -5024 + 9.86 x 1O-3T2 - 6.44 X 1O-6T3 J mol-1
Assuming that 8 being the lattice parameter of austenite, v being kTfh = 1.57 x 1013
S-l, the growth rate controlled by the interface process can be calculated using equation (5.34)
as a function of the activation energy ~ag*. Since the growth rate of pearlite at 4780 C was
measured to be 1.0 x 10-9 cm S-l, the activation energy ~ag* can be calculated from the
equation. ~G8,~q;(){ are 7743.5 and -1551 Jmol-1, hence ~g{3(){ = -1496 Jmol-1. Using
the data 8 = 3.573 X 10-10 m and v = ktf h = 1.57 X 1013 s-1, ~ag* is calculated to be
202 kJ mol-I, which is close to the activation energies of the self diffusion of iron in austenite
and ferrite; i.e. 286 and 240 kJ mol-I. Although the activation energy calculated from the
observed growth rate at 4780 C is close to that of the self diffusion of iron, it is still uncertain if
the reaction is controlled by the interface process. Further work is necessary together with the
study of the growth rate of alloy pearlite instead of conventional pearlite to reach a decisive
conclusion.
5.7 CONCLUSIONS
Although the bainite transformation terminates when the carbon concentration of the
untransformed austenite reaches the To or T~ curve, the austenite can nevertheless decompose to
ferrite and cementite by reconstructive transformation. A case is the formation of pearlite after
the cessation of the bainite transformation. In this chapter, pearlite transformation kinetics at
low temperatures and the formation of pearlite during heating after the termination of bainite
transformation were studied and following conclusions were obtained.
1) The formation of pearlite during heating from a bainite transformation temperature after
the cessation of the bainite reaction can be understood by the TTT curve for untransformed
austenite with the carbon concentration of xT' at the bainite transformation temperature,
o
and by Scheil's rule.
2) The result of the analysis is evidence for the incomplete reaction phenomenon during the
bainite transformation.
139
3) The incubation period for pearlite transformation observed at 478 QC is reproducible by
the upper C-curve of the TTT curve for the alloy.
4) The growth rate of pearlite was studied theoretically and the local equilibrium calculation
of the interface compositions gave the partition coefficient of the substitutional alloying
element which agrees reasonably well with the published results.
5) The growth rate of pear lite observed at 478QC (1.0 x 1O-9cms-1) is far smaller than the
calculated diffusion controlled growth rate. Although the pearlite observed in this work is
alloy pearlite, this discrepancy in the growth rate may mean that the formation of pearlite
at very low temperatures is controlled by the interface process instead of the long range
diffusion of atoms.
REFERENCES
1. J. W. Christian: "The Theory of Transformations in Metals and Alloys Part 1 ", Pergamon
Press, Oxford, 1975.
2. H. K. D. H. Bhadeshia: Metal Science, 1982, 16, 159.
3. K. C. Russell: Acta Metall., 1969, 17, 1123.
4. H. K. D. H. Bhadeshia: Acta Metall., 1980, 28, 1103.
5. H. K. D. H. Bhadeshia: Proc. 1nl. Conf. "Solid-Solid Phase Transformations", 1981, 1041.
6. H. K. D. H. Bhadeshia and D. V. Edmonds: Metall. Trans. A, 1979, lOA, 895.
7. M. P. PuIs and J. S. Kirkaldy: Metal. Trans., 1972,3, 2777.
8. N. Ridley: Metal. Trans., 1984, 15A, 1019.
9. C. Zener: Trans. AIME, 1946, 167, 550.
10. M. Hillert: Jernkont Ann., 1957, 141, 757.
11. J. M. Shapio and J. S. Kirkaldy: Acta Metall., 1968,16, 579.
12. B. E. Sandquist: Acta Metall., 1968, 16, 1413.
13. M. Hillert: "The Mechanism of Phase Transformations in Crystalline Solids", Institute of
Metals, London, 1969, 231.
14. R. C. Sharma, G. R. Purdy and J. S. Kirkaldy: Metall. Trans., 1979, lOA, 1129.
15. D. Brown and N. Ridley: JISI, 1966,204, 811.
16. N. A. Razik, W. Lorimer and N. Ridley: Acta Metall., 1974,22, 1249.
17. J. W. Cahn and W. C. Hagel: "Decomposition of Austenite by Diffusional Processes", ed. by
V. F. Zackay and H. 1. Aaronson, Interscience Publishers, New York, NY, 1962, 131.
18. S. A. AI-Salman, G. W. Lorimer and N. Ridley: Acta Me tall. , 1979,27, 1391.
19. K. Hashiguchi and J. S. Kirkaldy: Scandinavian Journal of Metall., 1984, 13, 240.
20. B. E. Sandquist: Acta Metall., 1969, 17, 967.
21. J. Chance and N. Ridley: Metall. Trans., 1981, 12A, 1205.
22. M. Hillert: Proc. 1nl. Conf. on "Solid-Solid Phase Transformations", 1981, 789.
23. M. L. Picklesimer, D. L. McElroy, T. M. Kegley, Jr., E. E. Stansbury and J. H. Frye, Jr.:
Trans. AIME, 1960, 218, 473.
24. J. H. Frye, Jr, E. E. Stansbury and D. L. McElroy: Trans. AIME, 1953, February, 219.
25. J. S. Kirkaldy, B. A. Thomson and E. A. Baganis: "Hardenability Concepts with Applications
to Steel", ed D. V. Doane and J. S. Kirkaldy, AIME, 1978, 82.
26. J. G. Gilmour, G. R. Purdy and J. S. Kirkaldy: Metal/. Trans., 1972,3, 1455.
140
27. K. Hashiguchi and J. S. Kirkaldy: CA LPHAD, 1984,8,173.
28. M. Hillert: Internal Report, Swedish Institute of Metals Research, 1953.
29. J. S. Kirkaldy: Canadian J. Phys., 1958,36, 907.
30. J. Fridberg, L-E. Torndahl and M. Hillert: Jernkont. Ann., 1969, 153, 263.
31. J. .Agren: Acta Metall., 1982, 30, 841.
32. R. Trivedi and G. M. Pound: J. Appl. Phys., 1969, 38, 3569.
33. M. Hillert: Metal/. Trans. A, 1975, 6A, 5.
34. 1. Kaufman and S. V. Radcliffe: "Decomposition of Austenite by Diffusional Processes", ed.
by V. F. Zackay and H. 1. Aaronson, Interscience Publishers, New York, NY, 1962, 313.
141
~ 700
Cl)~
::J-ca~
Cl)
c.
E
Cl)-
Tb
o 42cfc
!:::.. 44lfc
o 47~C o
!:::..o
-~ca-en 600
c
o-en
o
c.
E
oo
Cl)
C 500
.01 . 1 1
Heating rate, °C/s
1 0
Fig. 5.1 Effect of heating rate on decomposition-start temperature of austenite remaining untrans-
formed after the completion of the bainite transformation.
800
TTT curves for residual austenlte
Ms
Ss
~ •..r - ---------- -_..-------_.. _.-. -_._.. :!h ""'-.
•... _- --- --
(,)
°
600
Cl)~
::J-ca soo~
Cl)
c.
E
Cl) 400
••••
700
862
200
o 4
log{Time, s}
Fig. 5.2 Calculated TTT curves of austenite with chemical composition identical to the residual
austenite after the formation of bainitic ferrite at 420°C, 4'18 °C and 472 °C respectively.
142
900
a)
900
b)
500
800
Ms
Ae"3._-----_._---------------------_ ............ ",'...---:./,-........; .
//;('<'( 'j/'__~.J1.",lJ../) ..j
uo
~ 600
.2
o
L-
lD
E 500
lD
l-
300
400
700
Ms
300
u
o
~ 600
.2
o
L-
lD
0-
E
lD
l-
400
200
1.0E-Ol 1.0E+Ol 1.0E+03
200
1.0E+05 1.0E-Ol 1.0E+Ol 1.0E+03 1.0E+05
TIme, S Time, S
900
c)
BOO
500
400
300 Ms
Ae'3
700 ·-------j,-r----------·--··---·
--:~:>-
//"//-
/1//1
I{ ({f
););) ))
_i./ ,-~!/.// B!
u
o
~ 600
.2
o
L-
lD
0-
E
lD
l-
200
1.0E+00 1.0E+02 1.0E+04 1.0E+06 1.0E+08
Time, s
Fig. 5.3 Calculated TTT curves of residual austenite at (a) 420 QC (x1'O = 2.4 at.%), (b) 448 QC
(x1'O = 1.8 at.%) and (c) 472 QC (x1'O = 1.4 at.%) (solid line), and the position of austellite
in the TTT curves during continuous heating at constant heating rates between 0.01 and
10 QCs-l(dashed lines).
143
750
2.4 at. %
1.6
1.4
Tb,OC
o 420
•• ----- 1::. 448
----0472
,
o ....
: '
..·l~··
.,< '0........:....•.' .•-'<•.•.•........ ...' .
-.-.-."-,-,-,-,-,-,-'-,-,-'-,-,-,-,":'-;.-,-,":-:,-,,,.._....;,
u
°Q)
700I-
::l
-+-
0
l-
Q)
0...
E
Q) 650-+-
-+-
I-
0
-+-
(I)
C
Q 600:;:
0
E
l-
Q
't-
(I)
c 550
0
l-
f-
500
-2.0 -1.0 0.0
log(Heating rate, °C/s )
1.0
Fig. 5.4 Comparison between observed and calculated decomposition-start temperatures during the
heating of austenite which is left untransformed after the bainite transformation steps.
144
ab
.
~.
I,
/
/
, I. '/
. ~"
I
4
~,
'"' ., ~ /
,,' ,,-'r
"1)'
"')' • .!t..,
-,
\ I
\1,.
'{
,I
:\
/
Fig, 5.5 Optical micro9raphs of specimens isothermally held at 478 QC for (a) 300 sec, (b) 600 sec,
(c) 67 hours, (d) 160 hours, (e) 23 days and (f) 43 days,
145
Fig. 5.5 (continued)
146
ef
Fig. 5.5 (continued)
147
Fig. 5.6 TEM bright field image of alloy pearlite.
40
12001000800400 600
Time, S
200
30
20
E
::L..
o
o
E 10
:J
E
><eu
~
.c-Cl
C
(1)
(1)-
-o
Fig. 5.7 Change in the maximum thickness of pearlite with isothermal holding time at 478 cC.
148
0.40.1 0.2 0.3
Observed spacing / ,..m
0.0
0.0
0.4
)( Fe-C
11 Fe-C-l.98NI
E • Fe-C-3.00NI::l...
0 Fe-C-l.08Mn0.3 11- • Fe-C-l.80Mn
Cl 0 Fe-C-o.4Cr •c • Fe-C-o.9Cr0ca • Fe-C-l.8Cra. 0.2
UJ
'0
(1)-ca
::s 0.10
ca
(,)
Fig. 5.8 Comparison between observed and calculated interlamellar spacing in Fe-C-Mn, F-C-Ni
and Fe-C-Cr alloys.
o
Fe-l.02Mn-l.05Cr alloy
0.3
0.4
E
::1.
'0 0.1
(1)-ca
::s
o
ca
(,) 0.0
0.0 0.1 0.2 0.3 0.4
Observed interlamellar spacing / J.lm
a..
(1)-C
-
(1)
. E 0.2
ca
Fig. 5.9 Comparison between observed [18J and calculated interlamellar spacing in a Fe-l.02Mn-
1.05Cr alloy.
H9
...........
co
:;::;
CO
L..•...
C
Q)
()
co
()
~o
c:{
Carbon concentration
o
()
Fig. 5.10 Schematic illustration of the phase diagram showing the interface compositions during the
growth of pearlite under local equilibrium.
co
:;::;
CO
L..•...
C
Q)
()
Co
()
~o
c:{
Carbon concentration
Fig. 5.11 Schematic illustration of the phase diagram showing the interface compositions during the
growth of pearlite under the negligible partitioning under local equilibrium.
150
0.080,02 0.04 0.06
Carbon mole fraction
aly ylB
~0.00
0.00
0.05
Fe-C-1.4ICr (wt%) alloy
ylaat 700°C
0.04
c
0
.•.... 0.03u
ro
L.•.... ylB
C1l aly
0 0.02
E
L
u
0.01
0.00
0.00 0.01 0.02 0.03 0.04 0.05
Carbon mole fractIon
0.10
Fe-C-1.41Cr (wt%) alloy
at 660°C
0.08
c
0
.•.... 0.06u
ro
L.•....
C1l
0 0.04
E
L
u
0.02
0.014
Fe-C-l.08Mn (wt%) alloy
0.012 at 700°C ylB
c 0.010
0
.•....
u
ro 0.006
L...-
C1l
0 0.006
E
c
L 0.004 aly
0.002
0.000
0.00 0.0\ 0.02 0.03 0.04 0.05
Carbon mole fraction
0.04
Fe-C-1.08Mn (wt%) alloy
at 660°C yla
0.03
ylB
c
0
.•....
u
ro
L 0.02...-
C1l
0
E
c
L 0.01
aly
0.00
0.00 0.01 0.02 0.03 0,04 0.05 0.06
Carbon mo 1e fract ion
0.06
Fe-C-I.08Mn (wt%) alloy
at 600°C
0.06
c
ylB0
.•....
u
ro
L
0.04...-
C1l
0
E
c
L 0.02
aly
0.00
0.00 0.02 0.04 0.06 0.08
Carbon mole fractIon
Fig. 5.12 Calculated phase diagrams for a Fe-l.08Mn-C wt.% and a Fe-l.41-C wt.% alloys.
151
1.3e-31.2e-31.1 e-3
10 -18
1.0e-3
10 -15
0 Fe-C-Mn alloys
• Fe-C-Cr alloys
Cl)
M- 10 -16
E
()
to
co
0 10 -17
~
1IT 1/K
Fig. 5.13 Comparison between observed and calculated J( DB8 values in Fe-C-Cr and Fe-C-Mn alloys.
700
u650
o
::J
'-:J~
~600
ell
Co
E
ell
t---
550
-0_
0 _
o~
o
)
"I
10-3
Gro •••.th
760 ~' _
141%(r Tg-6~0(
ou720 .......,~o
ClJ
L..
.2680
'"'-ClJ
c.
E
~640
600
5 10 15 20
K~m((r)
Fig. 5.14 Growth r~te of pearlite ~nd the partitiOl~in9 coefficient of substitutional alloying element
as a functIOn of the reactIOn temperature III ta) a Fe-l.08Mn-C wt.% and (b) a Fe-1.41Cr-C
wt.% alloys [8].
152
::
~
~ 4
0
..•..
::
Q,).-u.-~~
Q,)
0 2u
::
0.-..•...-..•..
~
~
~
0
550
o
Fe-0.8C-1.08Mn (ca le.)
Fe-0.7C-1.08Mn (calc.)
Fe-0.6C-1.08Mn (calc.)
Fe-1.08Mn (obs.)
600 650
Temperature, T/ CC
700
Fig. 5.15 Calculated partition coefficient of Mn in a Fe-l.08Mn-C wt.% alloy. Plots are after Ridley
[8] .
Fe-C-1.41 Cr wt. % alloy
750600 650 700
Temperature, TrC
--- Calculated
.•. Observed
20
~
U
~
0 15..•..
::
Q,).-
U.-~
10~
Q,)
0
u
::
0.- 5..•...-..•..
~
~
~
0
550
Fig. 5.16 Calculated partition coefficient of Cr in a Fe-1.41Cr-C wt.% alloy. Plots are after Ridley
[8] .
153
800
~--
Q,)
~
=••• 600~
~
Q,)
~
8
Q,)
E-4
400
Fe-0.8C-1.08Mn alloy
--- Local equilibrium (calc.)
------- Paraequilibrium (calc.)
o observed
-------------------------------
----------- .•.
o
10-5 5 10-4 5 10-3
Growth rate of pearlite / cm/s
Fig. 5.17 Comparison between observed and calculated growth rate of pearlite in a Fe-1.08Mn-C
wt.% alloy.
800
--- 1.08Mn wt. % alloy
------- 2.0 Mn wt. % alloy
---- 3.0 Mn wt. % alloy
~ ----- .•.-- ...•.... - .•.. ,----..... ......................... '
Q,) 600 " '" \~ " \= \ "\ I••• , l~ I '~ I :
Q,) I I
~ It'
8 /,,'1/Q,) I'E-4 1/I'400 1/
10-6 10-5 10-4 10-3
Growth rate of pearlite / cm/s
Fig. 5.18 Effect of the amount of Mn addition on the growth rate of pearlite under paraequilibrium.
15-1
800
1.08Mn wt. % alloy
------- 2.0 Mn wt. % alloy
---- 3.0 Mn wt. % alloy
~
---
(1) 600~
=:1..•..
Cl:l
~
(1)
~
E
(1)
~
400
----- ------ -------------------- --- .... ---- .•.....•. ---- ......•......••.•••
.............•.... "
"" '.\ I
\ I, :
J /
/ -'
/ '
/ "1/ /'
II ,/
// ;,,/
,/ .",,"
10-6 10-5 10-4 10-3
Growth rate of pearlite / cm/s
Fig. 5.19 Effect of the amount of Mn addition on the growth rate of pearlite under local equilibrium.
720
700
Local equilibrium
Paraequi 1ibrium
0
0 680 a
Q)~
:3-ca 660~
Q)
a.
E
Q) 640
I-
620 Observed a
600
10.5 10.4
Growth rate of
1 O· 3
pearlite, cm/s
1 O· 2
Fig. 5.20 Comparison between observed and calculated growth rate ofpearlite in a Fe-l.41Cr-C alloy.
155
Table 5.2 Interlamellar spacing data for Fe-C, Fe-Mn-C, Fe-Ni-C and Fe-Cr-C alloys reported by (a)
Brown and llidley []5], (b) Razik et al. [16]' (c) llidley [8] and (d) Calm and Hagel [17].
Temp. ·C Alloy wt% S IJ,m ref. Temp. ·C Alloy wt% S IJ,m reI.
700 0.221 a) 672 1.98N i 0.652 a)
680 O. 130 a) 667 1.98N i 0.379 a)
660 0.092 a) 660 1.98N i 0.227 a)
640 0.071 a) 656 1.98N i 0.201 a)
620 0.057 a) 652 1.98N i O. 145 a)
690 1.08/11n 0.398 b) 647 1.98N i O. 129 a)
685 1.08/11n 0.299 b) 641 1.98N i O. 112 a)
680 1.08/11n 0.241 b) 629 1.98Ni 0.088 a)
670 1.08/11n O. 196 b) 620 1.98N i 0.071 a)
660 1.08/11n O. 156 b) 648 3. OON i 0.649 a)
640 1.08/11n O. 100 b) 644 3.00Ni 0.463 a)
620 1.08/11n 0.080 b) 640 3.00Ni 0.227 a)
670 1.80/lln 0.373 b) 633 3. OON i O. 168 a)
660 1.80/lln 0.218 b) 625 3. OON i O. 130 a)
650 1.80/lln O. 164 b) 616 3.00Ni O. 103 a)
640 1.80/lln O. 131 b) 605 3. DON i 0.079 a)
630 1.80Mn O. 119 b)
620 1.80/lln O. 104 b)
610 1.80Mn 0.077 b)
600 1.80/lln 0.065 b)
Temp. ·C Alloy wt% S IJ,m ref. Temp. ·C Alloy wt% S IJ,m reI.
720 1.80Cr O. 135 c) 706 0.40Cr 0.240 c)
705 1.80Cr 0.092 c) 696 0.40Cr O. 174 c)
690 1.80Cr 0.069 c) 680 0.40Cr O. 109 c)
670 1.80Cr 0.060 c) 660 0.40Cr 0.079 c)
650 1.80er 0.046 c) 637 0.40Cr 0.065 c)
630 1.80Cr 0.041 c) 620 0.40Cr 0.055 c)
620 1.80Cr 0.037 c) 790 9.00Cr 0.200 d)
710 0.90Cr O. 178 c) 750 9.00Cr 0.075 d)
700 0.90Cr O. 128 c) 700 9.00Cr 0.050 d)
680 0.90Cr 0.088 c) 675 9.00Cr 0.040 d)
660 0.90Cr 0.072 c)
640 0.90Cr 0.057 c)
618 0.90Cr 0.046 c)
]56
CHAPTER 6
REAUSTENITISATION ACCOMPANIED BY NUCLEATION OF AUSTENITE
6.1 INTRODUCTION
The aim of this chapter is to study the effect of the initial microstructure on reaustenitisa-
tion, in circumstances where the nucleation of austenite is required. Martensitic microstructures
and mixtures of ferrite and carbide particles, are studied as the starting microstructures. It
is essential for austenite to nucleate in these cases unlike the mixture of bainite and austenite
which was discussed in Chapter 4. Differences in the overall transformation kinetics between
reaustenitisations with and without the need to nucleate austenite may provide insight into the
effect of nucleation on reaustenitisation.
6.2 EXPERIMENTAL PROCEDURE
All materials used in the experiments were homogenised at 1250 aC for 3 days before
the heat treatment to obtain the appropriate initial microstructures. A Fe-0.3C-4.08Cr wt.%
and a Fe-0.12C-2.0Si-3.0Mn wt.% alloys were heated to either 1250aC or 1100aC and were
quenched into iced water to give martensitic microstructures. Mixtures of carbide particles
and ferrite were obtained by tempering specimens of the martensitic microstructure either at
500 aC or at 700 aC. After these heat treatments, the specimens were "up-quenched" to the
reaustenitisation temperatures at the fastest heating rate available (about 500 ac S-l) in order
to investigate isothermal reaustenitisation, or were heated continuously at a variety of rates to
study anisothermal reaustenitisation. The specimens were helium quenched in the dilatometer
after the heat treatment. The cooling rate during quenching was about 30 ac S-l between 800
and 500 ac. The same methods were used for the nickel plating to prevent surface degradation
and for the sealing of specimens into a quartz tube while they were heat treated in a furnace
as mentioned in Chapter 4. After the experiments, specimens were examined as discussed in
Chapter 4.
6.3 ISOTHERMAL REAUSTENITISATION
The isothermal reaustenitisation behaviour of ferrite in which nucleation of austenite is
required, is described in this section.
6.3.1 Reaustenitisation from a martensitic microstructure
Martensitic initial microstructure in the Fe - a.SC - 4.a8Cr wt.% alloy
First, reaustenitisation from martensite in the Fe-0.3C-4.08Cr wt.% alloy, obtained by
quenching from 1100 aC, was studied. An optical micrograph of the initial microstructure is
shown in Fig. 6.1. A TEM micrograph of the specimen shows a lath martensitic microstructure
(Fig. 6.2). The hardness of the initial microstructure was measured to be Hy{98.1N} = 551,
which is very close to the value 582 calculated from the chemical composition and the cooling
rate using the following equation (concentrations expressed in wt.%) [1].
Hy = 127 + 949C + 27Si + llMn + 8Ni + 16Cr + 21 x log{cooling rate,(aCs-1)} (6.1)
157
The martensitic samples were then up-quenched to elevated temperatures to study isother-
mal reaustenitisation. Optical micrographs of the specimens helium quenched after 30 min of
isothermal reaustenitisation at each reaction temperature are shown in Fig. 6.3. Austenite was
not observed after reaction at 750°C (Fig. 6.3 a). When the reaction temperature was raised
to 780°C, isolated islands of austenite (white regions, Fig. 6.3 b) were found. The preferential
nucleation site of austenite is the prior austenite grain boundaries. The austenite particles
are globular in shape or slightly elongated along the prior austenite grain boundaries. The
volume fraction of austenite increased with the reaction temperature (Fig. 6.3 b,c,d) and a
fully austenitic microstructure (fully martensitic on quenching to ambient temperature) can be
obtained above 805°C (Fig. 6.3 e).
The austenite particles which nucleated at the prior austenite grain boundaries tended to
grow only into one of the two adjacent prior austenite grains. At some grain boundaries, all
of the austenite particles were found to grow into the same prior austenite grain (indicated
by arrows in Fig. 6.3 c). A magnified micrograph shows detail of the structure of austenite
nucleated at a prior austenite grain boundary (Fig. 6.4). Most of the austenite particles seem
to grow into one prior austenite grain and a coalescence between adjacent austenite particles
seems to occur at the later stage of transformation.
Similar results have been reported by Speich et al. [2] and Law and Edmonds [3] for the
formation of austenite from a mixture of ferrite and spheroidised cementite particles. These
austenite particles were found to grow into a grain to which the austenite particles are not
orientation related.
At relatively high temperatures (790 °C for 20 seconds), the prior austenite grain bound-
aries first became covered completely with newly formed austenite particles (Fig. 6.5). The
new austenite particles coalesced to form an allotriomorph of austenite at the prior austen-
ite grain boundaries. After the occurrence of the site saturation at the prior austenite grain
boundaries, nucleation of austenite within the prior austenite grains seems to become more
prominent (Fig. 6.6; 790°C for 50 seconds). The austenite particles are rather acicular in shape
instead of the globular grains which form at the prior austenite grain boundaries. The austenite
plates found within a given martensite block were parallel to each other. This suggests that the
austenite plates are influenced by the ferrite/ferrite interfaces. They grew along these interfaces
as discussed by Plichta and Aaronson [4].
These microstructural features of reaustenitisation from martensite were also obtained
when the prior austenite grain size was large. Specimens heated to 1250 °C for three days and
quenched in iced water were reaustenitised. The amount of austenite which forms at 778 °C
was found to be very small (Fig. 6.7) with individual austenite grains separated at the prior
austenite grain boundaries. Small particles of austenite appears at a prior austenite grain
boundary at an early stage of the reaction, which seem to grow along the boundary and bulge
into one of the grains which are separated by the boundary.
Martensitic initial microstructure in the Fe - 0.12C - 2.05i - 3.0Mn wt.% alloy
When a martensitic starting microstructure in the Fe-0.12C-2.0Si-3.0Mn wt.% alloy was
heated, intragranular austenite was found to form as discussed by Plichta and Aaronson [4]. The
prior austenite grain boundaries are no longer the most preferred nucleation sites as shown in
Fig. 6.8, which illustrates partially reaustenitised microstructures. The austenite particles are
acicular and there is almost no nucleation observed at the prior austenite grain boundaries. The
158
is, therefore, expected to be higher in ferrite than in austenite causing the difference in particle
sizes in the two phases.
In some part of tempered region, dislocation networks can be observed. Fig. 6.12 a) shows
an example of this in a specimen isothermally held at 790 QC for 20 seconds and quenched.
Dislocations are found to be tied up by carbide particles in the tempered region. Interaction
between the particles and dislocations can be clearly seen in another position of the specimen
heat treated at 790 QC (Fig. 6.12 b). Fig. 6.13 shows a bright field image of a recovered
ferrite microstructure where ferrite grains whose interfaces seems to be pinned by particles, are
found. However, a dark field image of the area shows that those grains have almost the same
orientations, suggesting that those grains are subgrains separated by dislocation networks.
The particles observed in the tempered region show a similar orientation which can be seen
in Fig. 6.14, together with a bright and a corresponding dark field images. This may indicate
that these particles have precipitated in ferrite with a reproducible orientation relationship with
that phase.
Twinned martensite is often observed in austenite regions (Fig. 6.15), which suggests that
the carbon concentration there is high. Microhardness measurements near interfaces between
tempered ferrite and austenite were conducted with the indentation load of 0.0981 N. Since
the microstructure which is to be examined is martensite with more than 0.3 wt.% of carbon,
the hardness measurement with the load smaller than 0.0981 N causes a substantial scatter.
A decrease in hardness near the interfaces was observed (Fig. 6.16). This may correspond
to the carbon profile in austenite which is considered to be maintained while austenite grows
(Fig. 6.17). There is, however, the possibility of interference from surrounding soft ferrite phase
near the interfaces.
Dilatometry
Typical relative length changes during isothermal reaustenitisation from martensitic initial
microstructure obtained by water quenching from llOOQC and from 1250QC, are shown in
Fig. 6.18 and Fig. 6.19. Since reaustenitisation happens during heating at high temperatures
even at heating rates as high as 500 QC s- 1, the temperature correction which was discussed
in Chapter 4 was used here again. The temperature corrected relative length changes are
plotted in Fig. 6.20. The rate of the reaction increases with temperature. The maximum
relative length change at each temperature increases with temperature and reaches a maximum
value. These features of isothermal reaustenitisation are very similar to those in the case
of reaustenitisation from a mixture of bainitic ferrite and austenite. The maximum relative
length changes were plotted against the reaction temperature in Fig. 6.21. As was observed for
the initial microstructure of bainitic ferrite and austenite, the maximum relative length change
shows a maximum value at around 790 QC which is identical to the value observed in the bainite
+ austenite initial microstructure within the accuracy of the experiments.
These two sets of the maximum relative length changes are compared in Fig. 6.22. The
maximum relative length change at temperatures higher than 790QC (100% austenite) is larger
for the martensitic initial microstructure since the amount offerrite available for transformation
is larger. The I).L/ L values at 790 QCfor the two different initial microstructures are consistent
with the initial amount of ferrite which is 1.0 in the martensitic initial microstructure and
0.7 in the other. The maximum relative length change above the peak temperature shows a
linear decrease which is parallel to that in the bainite + austenite starting microstructure. This
160
feature can be attributed to the difference in the thermal expansivities of ferrite and austenite
as discussed in Chapter 4.
The thermal expansion coefficient during cooling after isothermal reaustenitisation at each
reaction temperatures are plotted in Fig. 6.23 with the data obtained for reaustenitisation
from the mixture of bainite and austenite in the Fe-0.3C-4.08Cr wt.% alloy. The thermal
expansion coefficients during cooling depend on the microstructure at the end of isothermal
holding and vary with the volume fraction of austenite and ferrite; i.e. 1.244 x 1O-5oC-1, when
the volume fraction of austenite is zero, to that of austenite; i.e. 2.065 x 1O-5oc-1, when the
microstructure is fully austenitic. It can be said that the fully austenitic microstructure is
obtained above 790°C, consistent with the results obtained from dilatometry and from optical
micrographs.
When the microstructure at the end of isothermal reaustenitisation contains austenite,
decomposition of that austenite can be observed during cooling. The relative length changes
during helium quenching after 30 min of isothermal reaustenitisation at each temperatures can
be seen in Fig. 6.24. The temperatures at which the decomposition of austenite is first observed
were plotted against the reaustenitisation temperature (Fig. 6.25). The decomposition-start
temperature decreases with increasing the reaustenitisation temperature independent of the
initial microstructure. Since the average carbon concentration in austenite is higher at lower
reaustenitisation temperatures, decomposition of austenite is expected to be retarded more
when the reaustenitisation temperature is lowered. The experimental result contradicts this.
To understand this, the phase equilibrium calculation was conducted using "Thermo-Calc"
for the Fe-0.3C-4.08Cr wt.% alloy, although calculated equilibrium phase boundaries are not
completely in agreement with the experimental results obtained in the present work. "Thermo-
Calc" gives phase boundaries as shown in Fig. 6.26 a in which fee, bee, M7C3 and cementite
were taken into account. As expected, the carbon concentration in austenite decreases with
increasing temperature in the intercritical temperature range as shown in Fig. 6.26 b, but
increases again with temperature when ferrite phase disappears. The chromium concentration,
on the contrary, increases mono tonically with temperature. This means that the kinetics of
the formation of ferrite from austenite formed at an intercritical temperature can be retarded
by higher concentration of chromium when the temperature is raised. Four temperatures,
which are indicated as points A, B, C and D in Fig. 6.26 b,c, are selected to assess the
effect of reaustenitisation temperature on the TTT curve for the formation of ferrite in this
alloy. The TTT curve calculation method proposed by Bhadeshia [6] was used here. Using
the equilibrium chemical compositions at these four temperatures, the TTT curves for these
austenite compositions were calculated (Fig. 6.27). Both of the upper and lower C-curves
are shifted to the right so the formation of ferrite is retarded by raising the reaustenitisation
temperature. In addition to this, the lower the reaustenitisation temperature is, the smaller
the austenite grain is expected, which also accelerates the reaction rate of ferrite formation.
161
6.3.2 Reaustenitisation from mixtures of ferrite and carbide particles
Two different mixtures of ferrite and carbide particles were studied. These initial mi-
crostructures were obtained by tempering the martensitic microstructure which was studied in
the previous section at either 700 QC or at 500 QC.
Microstructure tempered at 500 QC
An optical micrograph of a specimen tempered at 500 QC for 17 hours is shown in Fig. 6.28.
A TEM bright field image (Fig. 6.29) clearly shows a tempered microstructure. There are two
types of carbides in this microstructure, one of which is rather small and elongated, and the
other is larger and globular in shape. TEM diffraction patterns (Fig. 6.29) show that the
former one is cementite and the latter M7C3 carbide. As shown in Chapter 4, cementite
precipitation occurs from carbon supersaturated bainitic ferrite below the Bs temperature.
Therefore cementite precipitation is expected to occur at 500QC (which is just above the Bs
temperature) at least at the early stages of tempering. A prolonged holding at the temperature,
however, may have allowed the M7C3 carbide, which is thermodynamically more stable than
cementite, to form. The specimens were then up-quenched to intercritical temperatures to study
isothermal reaustenitisation. A typical microstructure obtained by 30 min at 784 QCis presented
in Fig. 6.30. Austenite particles can be seen not only at the prior austenite grain boundaries
but also within the grains, before all of the prior austenite grain boundaries are covered with
newly formed austenite. The intragranular austenite as well as that nucleated at the grain
boundaries were found to be globular in shape, in contrast to the formation of austenite from
a martensitic microstructure. When the latter is heated, the prior austenite grain boundaries
are first covered with austenite and intragranular transformation occurs afterwards. The shape
of the austenite is acicular along the ferrite/ferrite interfaces in the tempered microstructure.
Clearly, a nearly tempered microstructure contains ferrite/ferrite interfaces within the prior
austenite grains, which are preferential nucleation sites for austenite. There are some prior
austenite grain boundaries which are free from austenite formation as indicated by arrows in
Fig. 6.30 even when austenite particles are found within the prior austenite grains.
Microstructure tempered at 700 QC
When a martensitic specimen was tempered at 700 QC for 51 hours, a slightly different
microstructure was obtained. Fig. 6.31 shows carbide particles both at prior austenite grain
boundaries and within the grains. The particles at the prior austenite grain boundaries are
coarse and elongated along the boundaries, showing a "stitched" morphology. The particles at
the boundaries appear disc shaped, of about 5 /Lm diameter. The particles within the grains
are small in comparison. When this microstructure is heated to 785 QC for 30 min, austenite
particles nucleate both at the prior austenite grain boundaries and within the grains (Fig. 6.32).
The preferred nucleation site is intragranular. Austenite particles can be found to grow across
the boundaries as illustrated by the arrows in Fig. 6.32. In some regions, the growth of austenite
is blocked by elongated carbide particles at the boundaries. This starting microstructure gives
a more homogeneous dispersion of austenite than previously discussed.
During reaustenitisation from both the tempered martensitic microstructures, austenite
nucleated within the prior austenite grains exhibited irregular (or wavy) interfaces. This may
162
be a result of a irregular substructure of the matrix which contains a lath martensitic mi-
crostructure, but is broken by prolonged isothermal holding at elevated temperatures. This
substructure is not polygonal but acicular, and a sheaf like structure can also be seen which
may provide various different direction of the growth of austenite causing wavy interfaces of
the growing austenite.
A TEM micrograph of a specimen quenched after 30 min of isothermal reaustenitisation
at 785 °C from the microstructure tempered at 700 °C is shown in Fig. 6.33. The micrograph
contains an interface between tempered ferrite and austenite (now martensite). The carbide
particles are found in both phases as discussed in the previous section. The particles are,
however, almost of the same size in both phases. If the dissolution of carbide particles occurs
in austenite after being engulfed, the particles found in austenite region are expected to be
smaller than that in tempered ferrite region as shown in the starting microstructures of the
bainite + austenite (Chapter 4) or martensite. This assumption, however, cannot explain the
fact observed in the present case. The other possibility is due to the difference in the growth
rate of carbide in each phase as discussed in the previous section. This assumption seems to be
applicable for all of the cases. When the starting microstructure does not contain alloy car bide,
the alloy carbide is required to nucleate and grow at the reaction temperature. The difference
in the growth rate of carbide in each phase, as a result of the lower diffusivities of atoms in
austenite than in ferrite, may play the most important role in determining the particle sizes
in both phases; i.e. particles in austenite are smaller than those in ferrite. When the starting
microstructure contains these carbide particles, these particles may grow or dissolve in both
phases. The growth or dissolution of the particles may not be significant in the present case
since the particles in the starting microstructure formed at 700°C which is only 85°C below
the reaction temperature. As a result, the particles found in both phases are of almost the
same Size.
Dilatometry
Relative length changes during reaustenitisation from a microstructure tempered at 500°C
are presented in Fig. 6.34. The maximum relative length change which can be obtained
from these data after the temperature correction (Chapter 4) can be seen in Fig. 6.35. The
changes obtained in the present work; i.e. reaustenitisation from mixtures ofbainite and austen-
ite, martensitic microstructures and tempered martensitic microstructures, are compared in
Fig. 6.36. No significant difference can be found for these different initial microstructures ex-
cept for the bainite and austenite mixture where the maximum relative length change for 100%
of reaustenitisation is smaller than others, as discussed in the previous section.
6.4 CONTINUOUS HEATING REAUSTENITISATION
6.4.1 Dilatometry
The relative length changes during continuous heating are shown in Fig. 6.37. The overall
reaustenitisation behaviour can also be seen in Fig. 6.38. The transformation-start and -finish
temperatures can be determined from the relative length change during continuous heating as
points at which the relative length deviate from the constant thermal expansion of the initial
and final microstructures. The transformation-start and -finish temperatures are plotted against
heating rate in Fig. 6.39. Both temperatures increase with heating rate, consistent with the
data on reaustenitisation from a mixture of bainite and austenite.
163
6.4.2 Tempering of marten site during heating
When the starting microstructure is supersaturated martensite, tempering can occur during
heating to the austenitisation temperatures. The importance of this has been pointed out
by Baeyertz [7]. When a specimen is heated at a slow heating rate, the tempering can be
detected using dilatometry. The relative length changes obtained during continuous heating
of a martensitic microstructure are presented in Fig. 6.37. A small deviation from the normal
thermal expansion effect is seen to occur before the onset of reaustenitisation, at around 3000 C,
which is designated as the "tempering-start temperature" (marked by arrows in Fig. 6.37), As
expected, the tempering-start temperature increases with the heating rate (Fig. 6.40).
The expected relative length change during the tempering of martensite can be calculated
from the difference in volume between carbon supersaturated martensite and a mixture of ferrite
and cementite:
6.L
L
(6.2)
where a(J(' and COl' are lattice parameters of martensite, a(J( is that of ferrite and a~ is the
volume of cementite per unit cell, respectively. The volume fraction of cementite formed by
tempering of carbon supersaturated martensite can be calculated from the total amount of
carbon in the martensite. The thermal expansion coefficient of martensite was assumed to be
identical to that of ferrite. The constants discussed in Chapter 4 were used here for the lattice
parameters and the thermal expansion coefficients. The relative length change in equation 6.2
was calculated to be -0.00077. This decrease in relative length change is consistent with the
experimentally observed one during heating and also during isothermal holding at 5000 C for
30 min (Fig. 6.41). To compare this value with the experimentally observed contraction due to
tempering, relative length changes at each heating rates were measured as differences between
the two linear expansion lines before and after the tempering effect. The average of these values
gives -0.00036. The smaller observed magnitude can be attributed to incomplete tempering.
6.5 CONCLUSIONS
Reaustenitisation from either martensite or mixtures of ferrite and carbide particles has
been studied, in circumstances where new nucleation of austenite is necessary.
1) Prior austenite grain boundaries are the preferred nucleation sites for reaustenitisation
from a martensitic initial microstructure, but are less prominent when the starting mi-
crostructure is tempered martensite in a Fe-0.3C-4.08Cr wt.% alloy. Almost no nucleation
of austenite was observed at the prior austenite grain boundaries during reaustenitisation
from martensite in a Fe-0.12C-2.0Si-3.0Mn wt.% alloy. The reason for this is not clear.
2) Formation of austenite within prior austenite grains became possible in the Fe-0.3C-4.08C
wt. % alloy after the prior austenite grain boundary sites became saturated.
3) Austenite particles found at the prior austenite grain boundaries were globular whereas
those formed within the grains are acicular in the case of reaustenitisation from marten-
sitic initial microstructures. Globular austenite can, however, be found within the prior
austenite grains when tempered martensite (a mixture of ferrite and carbide particles) is
reaustenitised.
4) M7C3 particles observed in the untransformed ferrite matrix are larger than those in the
austenite region when the starting microstructure is martensite, whereas no significant
164
difference in the particle size was observed for the starting microstructure tempered at
700 cC. This suggests that the difference in the particle size can be attributed to the
difference in the growth rate of the particles in both phases.
5) When martensitic initial microstructure is heated slowly, tempering can occur on heating,
and the starting temperature increases with heating rate. Even in the case of high heating
rates carbide precipitation seems to occur at the very early stages of the reaction.
REFERENCES
1. R. Blondeau, Ph. Maynier, J. Dollet and B. Vieillard-Baron: "Heat Treatment '76", Metals
Society, London, 1976, 189.
2. G. R. Speich and A. Szirmae: Trans. TMS-AIME, 1969, 245, 1063.
3. N. C. Law and D. V. Edmonds: Metal/. Trans., 1980, llA, 33.
4. M. R. Plichita and H. I. Aaronson: Metal/. Trans., 1974, 5, 2611.
5. J. Fridberg, L-E. Torndahl and M. Hillert: Jernkont. Ann., 1969, 153, 263.
6. H. K. D. H. Bhadeshia: Metal Sci., 1982, 16, 159.
7. M. Baeyertz: Trans. ASM, 1942, 30, 458.
165
Fig. 6.1 Optical micrograph of martensite obtained by water quenching from 11000 C in the Fe-
0.3C-4.08Cr wt.% alloy.
Fig. 6.2 TEM bright field image of martensite obtained by water quenching from 11000 C in the
Fe-0.3C-4.08Cr wt.% alloy.
166
ab
c
Fig. 6.3 Optical micrographs of specimens helium quenched after 30 min of isothermal reausteniti-
sation at a) 750oe, b) 780oe, c) 785°e, d) 8000e and e) 805°e.
167
Fig. 6.3 (continued)
168
Fig. 6.4 Optical micrograph of a specimen helium quenched after 30 min of isothermal reausteniti-
sation at 800 QC.
Fig. 6.5 Optical micrograph ofa specimen helium quenched after 20 seconds of isothermal reausteni-
tisation at 790 QC showing a polycrystalline layer of austenite at the prior austenite grain
boundaries.
169
Fig. 6.6 Optical micrograph of a specimen helium quenched after 50 seconds of isothermal reausteni-
tisation at 790 DC.
170
ab
Fig. 6.7 Optical micrograph of a specimen helium quenched after 30 min of isothermal reausteni-
tisation at 7780 C. The initial microstructure is martensite obtained by water quenching
from 12500 C.
171
a b
Fig. 6.8 Optical micrograph of a specimen helium quenched after 30 min of isothermal reausteni-
tisation at a) 770QC and b) 790QC. The initial microstructure is martensite obtained by
water quenching from 1250QC in the Fe-0.12C-2.0Si-3.0Mn wt.% alloy. (A: austenite, F:
ferrite)
Fig. 6.9 TEM bright field image of a specimen helium quenched after 20 seconds of isothermal
reaustenitisation from martensite at 790 QC in the Fe-0.3C-4.08Cr wt.% alloy.
172
ab
\~. • ".J ,» '.- ,......•.•. . . <, -:""' .," \ .•.. ~ ~
4
, ..., .~ •
.•.. ,...•.~.. ,..
..•.
.;. tf,
~.•.... ~.... (
""l.
I
Fig. 6.10 TEM bright field images ofa specimen helium quenched after 30 min of isothermal reaustelli-
tisation from martensite at 8000 C. a) represents an untransformed part and b) an austeni-
tised area in the Fe-0.3C-4.08Cr wt.% alloy.
173
••
Fig. 6.11 TEM bright field image of a specimen helium quenched after 30 min of isothermal reausteni-
tisation from martensite at 785°C in the Fe-0.3C-4.08Cr wt.% alloy.
174
ab
Fig. 6.12 Dislocation networks which are found in an untransformed region tempered during isother-
mal reaustenitisation for 20 seconds at 7900 C in the Fe-0.3C-4.08Cr wt. % alloy.
175
ab
Fig. 6.13 TEM bright and dark field images showing a subgrain structure in a tempered martensite
region after reaustenitisation at 790 QC in the Fe-0.3C-4.08Cr wt.% alloy.
176
ab
Fig. 6.14 TEM bright and dark field images of carbide particles found in the tempered martensite
region in the Fe-O.3C-4.08Cr wt.% alloy.
177
ab
Fig. 6.15 Twinned martensite observed near the interface between austenite and untransformed fer-
ritic matrix and a selected area electron diffraction pattern showing a twin relation.
178
800
600
Z.•..
eo
en
0
400
....'~)'
0
>
• •:I: 20 15 1
200
o
o 5 1 0
Position
1 5 20
Fig. 6.]6 Microhardness of phases measured by a O.098N indentation across the austenite layer at
prior austenite grain boundaries.
a)
Xi ---------------------------
c
o
+-'eo
L-
+-'
C
CD
()
c
o
()
c
o
..0
L-
eo
()
y
b)
Xi ---------------------------:----
y
z* zo
Interface position z
Fig. 6.]7 illustrative carbon profile in austenite near interface during the growth event of the austen-
ite which exists in the initial microstructure of a mixture of bainite and austenite.
]79
0.018 0.018
a R.oustenlt1s.d al 750·C d Reaustenltlsed at 790'C
0.016 0.016
•• ••
'" 0.014 0> 0.014<: <:
0 0
.s;;; .s;;;
u u
£ 0.012 .s;;; 0.012
0> C.
<: <:
.! .!
0.010
••
0.010
••.~ .~
;; ;;
""i 0.008 Ii 0.008
0: 0:
0.006 0.006
0.004 0.004
0 100 200 300 400 500 600 0 100 200 300 400 500 600
Time, s Time, s
0.018 I I I 0.018
b Reou.tenllt •• d al· 780'C e Reauslenltlsed at 800'C
0.016 0.016
•• r--.. ••0> 0.014 '" 0.014<: <:0 0
.s;;; .s;;;
u u
.s;;; 0.012 - .s;;; 0.012
0, C.
<: <:
.! .!
0.010••
0.010
••> .~
~ ;;
""i 0.008 ""i 0.0080: 0:
0.006 0.006
I I
0.0040.004
300 400 500 6000 100 200 300 400 500 600 0 100 200
Time, s Time, s
0.018 0.018
C Reouslenlllsed ot 785°C Reaustenltlsod at 815'C
0.016 0.016
•• ••
0> 0.014 '" 0.014<: <:
0 0
.s;;; .s;;;
u u
.s;;; 0~12 .s;;; 0.012
C. 0,
<: <:
.! .!
0.010 •• 0.010••.~ .~
;; ;;
""i 0.008 ""i 0.0080: 0:
0.006 0.006
0.004 0.004
0 100 200 300 400 500 600 0 100 200 300 400 500 600
Time, s Time, s
Fig. 6.18 Relative length changes obtained during isothermal reaustenitisation from martensite,
which was obtained by quenching from 1100 QC, at a) 750 QC, b) 780 QC, c) 785 QC, cl)
790 QC, e) 800 QC and f) 815 QC in the Fe-0.3C-4.08Cr wt.% alloy.
180
0.018
a Raauslanlllsed al 780'C
0.018
e RalullanfUsad II 795'C
0.008 0.008
0.016
..
g' 0.014
o
L
U
L 0.012
0.
c:
.!
•• 0.010
.~
"0..
0:
0.016
..
go 0.014
o
L
U
L 0.012
0.
c:
~ 0.010
.~
"0..
0:
0.006 0.006
0.004
o 100 200 300 400 500 600 o 100 200 300 400 500 600
Time, • Time, •
0.018
b Rl8uslanlllsad II 783'C
0.018
f RaluslanlUsad al 800'C
0.012
0.008
0.012
0.008
0.016
0.006
..
go 0.014
o
L
U
L
0.
c:
.!
•• 0.010
.~
"0.•
0:
..
g' 0.014
o
L
U
L
0.
c:
~ 0.010
.~
"0..
0:
0.006
0.016
0.004
o 100 200 300 400 500 600
0.004
o 100 200 300 400 500 600
Time •• Time, •
0.018
c R8Iuslanlllsed II 786'c
0.018
9 RalustenlUsed II 810'C
0.008 0.008
0.016
..
go 0.014
o
L
U
L 0.012
0.
c:
.!
.~ 0.010
"0.•
0:
0.016
..
go 0.014
o
L
U
L 0.012
0.
c:
.!
.~ 0.010
"0
'i
0:
0.006 0.006
0.004
o 100 200 300 400 500 600
0.004
o 100 200 300 400 500 600
Time. s TIme, S
0.018
d R8Iustenlllsed at 790'C
0.018
hi
Rea~slanIUsa~ al 820'C
I
0.008 0.008
0.016
..
go 0.014
o
L
U
L 0.012
0.
c:
.!
.~ 0.010
"0.•
0:
0.016
..
go 0.014
o
L
U
£ 0.012
en
c:
.!
•• 0.010
.~
"0.•
0:
0.006 0.006
0.004
o 100 200 300 400 500 600
0.004
o 100
I
200 300
I
400 500 600
TIme. s TIme, S
Fig. 6.19 Relative length changes obtained during isothermal reaustenitisation from ma.rtensite,
which was obtained by quenching from 1250 cC, at a) 780 cC, b) 783 cC, c) 786 cC, d)
790 cC, e) 795 cC, f) 800 cC, g) 810 cC and h) 820 cC in the Fe-0.3C-4.08Cr wt.% alloy.
181
10080604020
0.0005
0.0000
" -0.0005'"c
0~
u
~ -0.0010
0.
c
~
"
-0.0015
.~
'0
•• -0.0020
'"
-0.0025
-0.0030
0
Time, S
Fig. 6.20 Effect of the reaction temperature on the temperature corrected relative length change
during the isotherm al reausteni tisation from m artensite.
0.003
Cl)
Cl
c:
III
.c
o 0.002
o Martensile (125c1'C)
• Martensile (110c1'C)
•
o
o
•
o
•
•
.c-Cl
c:
~
Cl)
>
i;j
Cl)
c:
0.001
•
0.000
725 750 775 800 825 850
Temperature, ·c
Fig. 6.21 Maximum relative length changes during 30 min of isothermal reaustenitisation from
martensite which was obtained by quenching from either 1100 QC or 1250 QC in the Fe-
0.3C-4.08Cr wt.% alloy.
0.003
0 Bainile .auslenite •0 Martensile (125c1'C) 0
Martensile (11OC1'C)
0 •• •
0 0
0.002 tit 0
~§
0
Cl)
Cl
c:
III
.c
o
.c-Cl
c:
Cl)
650
o
•o
o %
750
o 0
0.000
725
Cl)
>
- 0.001
III
Cl)
c:
775 800 825
Temperature, ·C
Fig. 6.22 Maximum relative length changes during 30 min of isothermal reaustenitisation from
martensite and from the mixture of bainite and austenite in the Fe-0.3C-4.08Cr wt.%
alloy.
182
1.0L.----....L...---.-..-----''----- .•.....--- ....•
760 780 800 820 840 860
Reaustenitisation temperature, °c
Initial microstructure
():Bainite+Austenite
• :Uartens ite
o
d,
1.5
2.0
2.5.------, ---------:---------,
,
,
X 10-5 :
,,
I
I
I 0 .: I
------- ---4t--- -------~-------~-------. :,
,
I
I
I,
,
I
I I ,
- - - - - --.- - - - - - -~- - - - - - - ~- - - - - - - -:- - - - - - - -
, ,
I I
I
I
I
I
I
I
I,
't-
't-
(1)
o
u
c:
o
Cl)
c:
~
X
(1)
Fig. 6.23 Thermal expansion coefficient during cooling after 30 min of isothermal reaustenitisation
from martensite as a function of reaction temperature.
350
760 780 800
Reaustenitisation
860820 840
temperature, °c
Initial microstructure
():Bainite+Austenite
• :Martens ite
--- ----1-- ---- -1- -- - ---
I I
I I
I I
I I
I ,
I I
, I
_______! J _
I I
I I
I I
I ,
I I
I I
, I
I I
-:0-~-------~-------
,
,
,
-------!._-
I
I
I
I
I
I,
I I
_______ l i _
I I
, I
I I
I I
I I
I ,
, I, ,
I
I
I
I
I
I
I,
-------1-------
I
I
t 450
as
+'
Cl)
c:
a
+' 400
as
E
L-
a
'+-
Cl)
c:
as
L-
t-
(.) 550
°
-
(1)
L-
:::J
+'as 500L-
(1)
0-
E
(1)
+'
Fig. 6.25 Decomposition-start temperature during cooling after 30 min of isothermal reaustenitisa-
tion from martensite as a function of reaction temperature.
183

a 1.0
0.9 J:T-273.J5,NPtCC A2)
2:T-273.J5,NP FCC Al)
0.8 3:T-273.l5,NP M7C3) 2
c:: 0.7
0....•
-' 0.6CJ
CIl
l-<
0.5•....
Cl)
VJ 0.4
CIl
~
Cl-. 0.3
0.2
0.1
0
A 740 760 780 800 820 840 860
Temperature, ·C
b 30
28
Cl)
-'....•
26c::
Cl)
-'VJ
24;:j
CIl
c:: 22....•
1
~ 20
1
1
-'
~
c:: 18
I I
0
, I '.D 16l-<
CIl
U
I I14
E-4
c 42
40
Cl) 38
-'....•
c::
Cl) 36
-'
VJ
;:j
34CIl
c::
32....•
~
-' 30;t
l-<
28u
26
E-3
24
A 740 760 780 800 820 840 860
Temperature, ·C
Fig. 6.26 Calculated phase boundaries and chemical compositions in austenite formed at intercritical
temperatures. Calculation was carried 011 t using "Thermo-Calc".
185
800
Fe-O.3C-4.08Cr (wt%)
.--- .----
700 point A
0 0
0
~ point B
Cl)
10-
~ 600-res
10-
•................
Cl)
c. --.
E
500
Cl)
I-
400
300
1 00 1 01
Time, S
Fig. 6.27 Calculated TTT curves for austenite formed at points A, B, C and D in Fig. 6.28.
Fig. 6.28 Optical micrograph of a specimen water quenched after tempering of martensite at 5000 C
for 17 hours in the Fc-0.3C-4.08Cr wt.% alloy.
186
Fig. 6.29 TEM bright field images of a specimen water quenched after tempering of martensite at
500 QC for 17 hours in the Fe-0.3C-4.08Cr wt.% alloy.
187
Fig. 6.30 Optical micrograph of a specimen helium quenched after 30 min of isothermal reausteniti-
sation from tempered martensite at 784 QC in the Fe-0.3C-4.08Cr wt. % alloy.
Fig. 6.31 Optical micrograph of a specimen water quenched after tempering of martensite at 700 QC
for 51 hours.
188
Fig. 6.32 Optical micrograph of a specimen helium quenches after 30 min of isothermal reausteniti-
sation from tempered martensite, which was obtained at 700 QC, at 785 QC.
Fig. 6.33 TEM bright field image of a specimen helium quenches after 30 min of isothermal reausteni-
tisation from tempered martensite, which was obtained at 700QC, at 785QC.
189
0.018 0.018
a Aeauslenlllsed al 760'C d Aeauslenlllsed al 795·C
0.016 0.016
Cl Cl
'" 0.014
Cl 0.014
I: I:
0 0
.r. .r.
u u
.r. 0.012 .r. 0.012
0, 0,
I: I:
.!!'. .!!'.
0.010 Cl 0.010Cl
.~ .~
0 0
0; 0.008
c; 0.008
a: a:
0.006 0.006
0.004 0.004
0 100 200 300 400 500 600 0 100 200 300 400 500 600
TIme, S Time. S
0.018 0.018
b Aeauslenlllsed al 775·C e Reauslenlllsed at 815'C
0.016 0.016
Cl
Cl
'" 0.014
Cl 0.014
I: I:
0 0
.r. .r.
u u
.r. 0.012 .r. 0.012
0, 0,
I: I:
.!!'. .!!'.
0.010 Cl 0.010Cl
.~ .~
0 0
c; 0.008 c; 0.008a: a:
0.006 0.006
0.004 0.004
0 100 200 300 400 500 600 0 100 200 300 400 500 600
TIme, S Time, S
0.018 0.018
C Reauslenlllsed al 787·C Reauslenillsed at 823'C
0.016 0.016
Cl Cl
Cl 0.014 - '" 0.014I: I:
0 0
.r. .r.
u u
.r. 0.012
==
0.012
0, 0>
I: I:
.!!'. .!!'.
.~
0.010
Cl
0.010
.~
0 0
c; 0.008 c; 0.008a: a:
0.006 0.006
0.004 0.004
0 100 200 300 400 500 600 0 100 200 300 400 500 600
Time, S Time, S
Fig. 6.34 Relative length change during 30 min of isothermal reaustenitisation from tempered marten-
site which was obtained at 500°C.
]90
0.003 "T"""---------------------,
(1)
Cl
C
ca
.c: 0.002 -
0
.c:-Cl
C
(1)
(1)
0.001 ->-ca
(1)
CC
0.000
725
A AA
750 775 800 825 850
Temperature, °C
Fig. 6.35 Maximum relative length changes during 30 min of isothermal reaustenitisation from tem-
pered martensite which was obtained at 500 QC.
850
•
•
o
o
o
I:i.
D•
I:i. ° 0
•
o
•o
~
oo
750
o 8ainite+austenite
o Martensite (1250°C)
• Martensite (11ocfC)
I:i. Tempered martensite
0.000
725
0.003
(1)
Cl
C
ca
.c: 0.002
0
.c:-Cl
C
(1)
(1)
0.001>-ca
(1)
CC
775 800 825
Temperature, °C
Fig. 6.36 Maximum relative length changes during 30 min of isothermal reaustenitisation from
martensite, from the mixture of bainite and austenite, and from tempered martcnsite
at 500 QC in the Fe-0.3C-4.08Cr wt.% alloy.
191
0.020 0.020
a) Heating rate is 0.5· C/sec b) Heating rate is 1.1· C/scc
0.018 0.018
. 1000 . 0.016 - 1000
0.016 '"DO Cc Relative length 0 Relative lenglh
0
-5-5 change (see 1ell) change (se8 left)
0.01~ .c 0.01..c
0. 750 0. 750
~ ~
0.012
.~
0.012 -
1
;; ;;.. 500 .. 0.010 5000.010 IXIX
Temp.rolun,·e (see right) Temperature,'e (sea right)
0.008 0.008
0.006 0 2000
0.006
.00 600 800 1000500 1000 1500 0 200
TIme. s Time. s
0.018
0.016
0.016 0.01.. ~
DO C
C O.OU 00 .c
.c u 0.012
u
7-.c0.
0.012
c
~ .!. .~ 0.010
> ;;
;;
0.010 ....
IX
IX
0.008
0.008
0.006
200 ~OO 600 800 1000 1200 0 200 .00 600 800 1000 1200
T amporofur.,·e T~mperolur •• 'e
0.020 0.020
c) Heating rate is 2.0· C/sec d) Heating rale is 5.0· C/sec
0.018 0.018
· 0.016 1000 . 0.016 1000DO DOC C
0 Relallve lenglh 0.c
-5
change (u. left)
u
0.01. .c 0.01.
7- 0. 750
~ ~
0.012
.~
0.012
1
;; ;;..
500
..
0.010 500
IX 0.010 IX
Temperalure,'e (see right)
0.008 0.008
0.006 0.006
0 100 200 JOO .00 0 50 lOO 150 200
Tl?le, s Tima. s
0.016 0.018
0.01.
· .DO ..•c
0.012 c0 0
-5 -5
.c .c. 0.012
0.
0.010
0.
c
~.!
· .~>;;
0.008
;;.. ..
QC IX 0.008
0.006
0.006
0.00. 0.00.
0 200 ~OO 600 800 1000 0 200 .00 600 800 1000 1200
Temperalure,'e Tomp.rolur.,·e
Fig. 6.37 Relative length changes during continuous heating at constant heating rates from the
martensitic initial microstructure.
192
e) Heating rale is 11.0· Clsec
I
Relative lenglh ~
chong' (s •• I,ll) / )
~ ••rn." •••••.~ COO, ,""I
1000
500
750
605040302010
~
Relallva length f". /
chong, (s •• lell) ",1' 7"
/' '
,If
",/'
/' Temperature.·e (see right)
/
C) Heating rate is 18.6· Clsec
0.020
0.018
· 0.0160>c
~
.c O.Ot 4 -
I
· 0.012>;;
v
0.0\0er
0.008
0,006
0
750
500
1000
1008060~O20
0,020
0,0\8
.
0,016Do
c
0
.c
u
.c 0.01~
C.
i
!
0,012
;;..
0,0\0er
0,008
0,006
0
Time, S Time. s
0,018
0.016
.
Do
c 0,01 ~0
.c
u
.c
C.
0,012
i.
>
;;..
er
1200
0.018
0.016
· 0.01~Doc
0
.c
u
.c 0,0\2
C.
c
.!
· 0,010>;;..
er
0.004
o 200 ~OO 600 800 1000 1200
Temperature. 'e Temperalure, 'e
Fig. 6.37 (continued)
..0.0
1.0 Heatin~ Rate
+*~~
+ 0.5 C/s Vv
1.1 i,..'k if •.•
• 2.0 t •Q) +"t.~ ,.- • 5.0c 0.8 +x *l ·
Q) ·11.1 t: .. v- v 18.3 +. .. •.t/) + ,:l
<( 1j. •
+ •- 0.6 •0 + • •+ ., v
C
+0
+
.. •..- ..u + •0 0.4 +. .'L . ••- +
Q) + •.
E
• "
:l
0 0.2
>
780 800 820 840 860
Temperature,OC
Fig. 6.38 Overall reaustenitisation behaviour during heating at constant heating rate from the marten-
sitic initial microstructure.
193
900 I I
StarlIng structure
o 8alnlte+Austenlte
o Marlenslte •
•
• •
• •
850 f- • -0
0
•Ql •l...
:J-0
l...
Ql
c..
E
Ql 0
I-
0
800 - 0 0 -0
0 0
0 0
750
1.0E-01
I
1.0E+00
I
1.0E+01 1.0E+02
Heating Rate,Oe/s
Fig. 6.39 Effect of heating rate on transformation-start and -finish temperatures during continuous
heating at constant heating rates from the martensitic initial microstructure.
400 ...--------,-I-----',r------
u
° 350 f- -re•..
:J-0 ••..
CD
c..
E
~
300 - •1:
.2 •Cl'
•
Cl
C
'C
CD •c..
E 250 l- • -
CD
I-
200
1.0E-Ol
I
1.0E+00
I
1.0E+Ol 1.0E+02
Heating rate. °e/ s
Fig. 6.40 Effect of heating rate on temperature at which tempering of martensite starts during
continuous heating reaustenitisation from martensite.
194
0.013
Fe-0.3C-4.08Cr (wt%)
0.012
at 500 'C
<tl
0>
c _______________L.C\S
0.011L:- --- --
(,)
1L:-
T-'
0.0100>
c
<tl
<tl 0.009
>
T-'
C\S
0.008
<tl
c::
0.007
0 500 1000 1500 2000
Time, s
Fig. 6.41 Relative length change during 30 min of isothermal tempering of martensite at 500 QC.
195
CHAPTER 7
THERMODYNAMICS OF REAUSTENITISATION IN TERNARY ALLOYS
7.1 INTRODUCTION
The conditions which exist at the transformation interface during the growth of ferrite
from austenite in steels have been discussed in much more detail than is the case for the
reverse transformation from ferrite to austenite. The thermodynamic theory which has been
developed for the decomposition of austenite should in principle be adaptable for the reverse
transformation, but as will be seen later, there are some important differences which need to
be taken into account.
The thermodynamics of the reconstructive formation of ferrite from austenite will be sum-
marised first in order to summarise the techniques available for handling transformation in
ternary steels, and an attempt will then be made to deduce a similar framework for the reverse
transformation processes.
7.2 RECONSTRUCTIVE FORMATION OF FERRITE FROM AUSTENITE
Ferrite formation is expected to occur when austenite is cooled down below the equilibrium
, /( a + ,) transformation temperature Ae3. It is well established that it is possible to obtain
various kinds of transformation products, whose mechanism of transformation may be differ-
ent, depending on a level of supercooling below the equilibrium transformation temperature.
The key characteristics of these transformation products in steels have been summarised by
Bhadeshia [1].
7.2.1 Reconstructive growth of ferrite in multi-component systems
Since the main interest here is the reconstructive formation of austenite, we begin with a
summary of the thermodynamics dealing with the reconstructive transformation of austenite to
ferrite in multi-component systems, with a view to using some of the principles for the reverse
case.
In multi-component steels, a variety of different modes of equilibrium can be maintained
locally at the moving transformation interface because of the existence of a remarkable differ-
ence in the diffusivities of interstitial atoms and substitu tional alloying elements in iron [2-13]
These modes can be classified into three categories; 1) partitioning local equilibrium (PLE) 2)
negligible local equilibrium (NPLE) and 3) paraequilibrium (PE).
In a system such as Fe-C-X, the diffusion rate of carbon in the austenite may be as much as
107 - 109 times greater than that of a substitutional atom in the temperature range of interest.
These very different rates of atomic migration mean that true equilibrium segregation with
regard to all components may not be produced at a migrating interface. It is, however, possible
to envisage growth under diffusion control with local equilibrium at the interface in the sense
that the compositions of two phases at the interface are related by a tie line of the equilibrium
diagram, even though this tie line may not pass through the point representing the initial (or
average) composition of the alloy. When these kinetic restrictions apply, the two phases may
differ either significantly or negligibly in substitutional solute content. A qualitative analysis
for ternary steels was first developed by Hillert [5]' and a simplified quantitative theory in which
diffusion cross terms are neglected was developed by Kirkaldy [6]' Purdy et al. [7] and Coates
196
[la]; the effect of the cross terms was later examined by Coates [11,12]. The simple theory shows
that for the diffusion-controlled growth of ferrite from austenite in an Fe-C-X alloy with initial
composition near the 'Y/ (a + 'Y) phase boundary (i. e. with a small supersaturation), the tie line
selected will have the carbon composition of austenite at the interface almost equal to that of
the bulk alloy so that the activity of carbon is nearly constant in the austenite, thus reducing the
driving force for carbon diffusion almost to zero. There will be a concentration gradient of the
substitutional solute ahead of the interface, resulting in appreciable partition, and the relatively
slow growth rate will be determined by the diffusion rate of this solute (Fig. 7.1a). For large
supersaturations, the tie line will have the Fe+ X composition of the ferrite virtually identical
with that of the bulk austenite, and partitioning of the substitutional solute will be extremely
small, with a relatively fast growth rate (Fig. 7.1b). These are referred to as the partitioning
local equilibrium (PLE) and negligible partitioning local equilibrium (NPLE) growth modes
[7,11 ,12], and for a high ratio of the diffusivities, the theory predicts an abrupt transition from
one to the other as the supersaturation increases.
In the NPLE mode, the concentration of the substitutional alloying element is uniform
except for a small "spike" in the parent phase adjacent to the interface. As the ratio of inter-
stitial and substitutional diffusion rates increases, the width of this spike decreases, and when
it becomes of the order of atomic dimension, the concept of local equilibrium at the interface
is invalid and has to be replaced (assuming the growth is nevertheless diffusion-controlled) by
that of paraequilibrium [5,2,3,8,9]. In conditions of paraequilibrium, there is no redistribution
of Fe + X atoms between the phases, the Fe/X ratio remaining uniform right up to the in-
terface. One interpretation of the paraequilibrium limit is that reconstructive transformation
occurs with all displacements of the Fe+X atoms taking place in the incoherent interface; an-
other interpretation might be that only displacive transformation can occur. In either case,
to quote from Coates [10]' "the slow diffuser and the solvent participate only in the change of
crystal structure". Paraequilibrium implies that the growth rate is controlled by the interstitial
diffusivity, the interface compositions now being given by the tie lines of the pseudo equilibrium
between the two phases under the constraint of a constant Fe/X ratio.
7.2.2 Calculation of tie lines for reconstructive formation of ferrite in ternary systems
For simplification, a ternary system Fe-C-X, where C and X denote respectively carbon
and a su bstit utional alloying element, will be discussed in this section to demonstrate briefly
the way to calculate tie lines and the growth rate of the product phase in different equilibrium
modes. The bulk composition of austenite is designated (xl ,x;), and the interface compositions,
which will be determined in the manner discussed below, are xi"', x~'Y, xr", and x~'Y, where the
subscripts denote carbon and the substitutional alloying element concentrations respectively,
and the superscripts 'Ya and a'Y represent respectively the interface compositions in austenite
('Y) and in ferrite (a) which are in equilibrium.
Because the reaction is now controlled by the diffusion of both carbon and the substi-
tutional alloying element, Fick's Law for both species has to be solved simultaneously. The
one-dimensional diffusion equations in austenite are expressed as follows. (Note that the con-
cept can easily be extended to two and three dimensional problems: see for example [11,12]):
oxI _ ~ (D'Y OXI) + ~ (D'Y OX;)
ot - OZ 11 OZ OZ 12 OZ
197
(7.1)
8x; = ~ (D'Y 8X;) + ~ (D'Y 8xi)
8t 8z 22 8z 8z 21 8z
(7.2)
where z is the space co-ordinate normal to the transformation interface, Dil and D~2 are the
interdiffusion coefficients of carbon and X in austenite respectively, and Di2 and D~l are the
cross interdiffusion coefficients in the ternary system taking account of the dependence of the
flux of one alloy element on the concentration gradient of another. Since the product phase is
here assumed to grow with a constant composition, it is necessary only to consider the diffusion
equations within the parent austenite phase.
The requirement of mass conservation at the interface is expressed by the following two
equations
(
"la a'Y) - J I
Xl - Xl V-I z=z.
(
"la Ol"() - J I
X2 - X2 V - 2 z=z.
(7.3)
(7.4)
where V is the growth rate of the prod uct phase, z* is the position of the moving interface along
the space co-ordinate z, and the fluxes of carbon and X at the moving interface, Jllz=z. and
J2Iz=z.' are expressed by
(7.5)
(7.6)
where the differentiations should be taken at the interface.
The four interface compositions in the mass conservation equations are the components of
two ends of the tie line in an isothermal section of the ternary equilibrium phase diagram at the
reaction temperature of the system concerned. As has been discussed (for example see [11,12]),
only one of them is independent given equilibrium since the chemical potentials of the species
have to be identical in the phases concerned, giving the additional equations:
JLf =JLJ;i=O,1,2 (7.7)
where i's (i=O,l and 2) correspond to iron, carbon and X respectively. Therefore the two
mass conservation equations contain only two independent variables, one of which represents
compositions at the interface, and the other the growth rate of the product phase. A unique
solution can therefore be obtained from these equations.
When the supersaturation is low (i. e. at higher temperatures in the a + , two phase field
where the bulk concentration is close to the, phase field), the PLE condition will be maintained
at the interface (see Fig. 7.1a). In order to get a reduced driving force for carbon diffusion in
austenite, the interface compositions in austenite are determined by the intersection between
the carbon isoactivity line passing through the bulk composition (point A in Fig. 7.1a) and the
,/b + a) phase boundary. The other end of the tie line is then obtained via equations 7.7.
On the other hand at relatively higher supersaturations (NPLE) where the bulk alloy
composition is close to the a phase field, the tie line is fixed by the intersection between a
horizontal line (x = x;) which passes through the bulk composition (point B in Fig. 7.1b) and
198
the al( a + I) phase boundary, making the extent of partitioning of X effectively negligible
and hence increasing its gradient in the austenite and allowing its flux to keep up with that of
carbon. The other end of the tie line is again obtained from equations 7.7.
The intersection of two lines, one of which is the horizontal line passing through the ferrite
end of the tie line, and the other is the carbon isoactivity line in austenite passing through the
austenite end of the tie line, form the transition line between the PLE to the NPLE modes
of growth. When the bulk composition is on the left hand side of the transition line, the
NPLE condition will be maintained, whereas the PLE condition will otherwise be achieved
(Fig. 7.1a, b).
Although the method discussed above is convenient for discussion, the required rigorous
solution (i.e. the interface compositions and the growth rate) can be obtained only when the
five equations, two of which are the mass conservation conditions at the moving interface and
the others the equilibrium equations, are solved simultaneously.
7.3 THERMODYNAMICS OF REAUSTENITISATION
Austenite can be expected to form when steels which contain ferrite, carbide, austenite
and their combinations, are heated to a temperature in the a +1two phase region or into the
austenite single phase field. When the reaction temperature is raised; i.e. the supersaturation
is increased, the mobility of atoms also increases. This contrasts with transformation to ferrite
from austenite, in which atomic mobility decreases as supercooling increases. Consequently,
during austenite formation, the mechanism of transformation is likely to be reconstructive,
especially for low alloy steels, whereas the tendency to form ferrite by displacive transformation
increases as the undercooling is increased.
7.3.1 Local equilibrium conditions during reaustenitisation
In the case of reaustenitisation in steels, local equilibrium has been usually assumed to
exist at the interface between austenite precipitate and ferrite matrix (see for example [14-16]),
although Agren suggested the possibility of partitionless transformation of ferrite to austenite
at very high temperatures (higher than 8470 C) in Fe-1.5 wt.% Mn system [16].
Assuming local equilibrium, when the reaction temperature is low (low supersaturation),
the growth of austenite may be expected to occur by a PLE mechanism, and by the NPLE
mechanism at relatively higher supersaturations. The procedure has been discussed qualita-
tively by Hillert [17]; to calculate the operative tie lines requires a model for the ferrite and
austenite solid solution and the dilute solution, the Wagner interaction parameter method is
adopted below. The transition line from the PLE to the NPLE condition in an isothermal
section of the ternary equilibrium phase diagram is expressed by the line RQ instead of PS
(Fig. 7.2), which is the transition line for the decomposition process of austenite [17].
When a steel, which is initially fully ferritic (i.e. a low carbon steel), is superheated to
an intercritical temperature, it might be a good assumption that local equilibrium will be
maintained at the interfaces between austenite and ferrite matrix, because of the high mobility
of alloying elements at that temperature. In order to simplify the discussion, attention is focused
initially on the reconstructive formation of austenite under conditions oflocal equilibrium, in a
ternary alloy Fe-C-X. When the bulk composition of the steel (see point I{xu x2} in Fig. 7.3)
at the temperature T is in the NPLE condition (region RQS in Fig. 7.3), the operative tie line
can be obtained as follows. In the NPLE mode, the gradient of X in the parent phase must
be large: this can be achieved by allowing xr~ -+ x
2
' One end of the tie line is, therefore,
199
obtained as the intersection of the ,/(0' + ,) phase boundary and a horizontal x2 = x2 line,
which gives the point {xr,x;a} designated "A" in Fig. 7.3. The other end of the tie line
follows automatically as the point B{x~',x~'} (Fig. 7.3), from the fact that for a tie line,
11<:' = 11 fa . i = 0 1 2
t'"'" t-"'t' ".
(7.8)
(7.9)
Kirkaldy and his co-workers [18-20] have used dilute solution methods to calculate the tie lines
in multi-component systems. The requirement of identical chemical potentials both in austenite
and in ferrite at the interface, leads to the approximation [18].
"G?a--+, ~ f~
u. _ I x. I ' .. - 0 1 2---- - n ----::y + n f' ,z - , ,
RT xi i
where ~G?a--+, is the standard free energy change between 0' and, for each element, and fi
are the activity coefficients for each of the elements, given by,
lnfl = EllXl + E12X2
In f 2 = E12Xl+ E22X2
Ell 2 E22 2
lnfa = --Xl - E12X1X2- -x2'
2 2
(7.10)
(7.11)
(7.12)
where Eij are empirical coefficients known as the Wagner interaction parameters. Therefore,
from the fact that the alloying element concentration in austenite at the interface is identical
to that of the bulk, it is possible to calculate the interfacial compositions (points A and B in
Fig. 7.3) which satisfy the thermodynamic equations listed above, simultaneously. Then one
can obtain the point M{xf,x~} on the transition line RQ (Fig. 7.3), which is the intersection
of the isoactivity line of carbon in ferrite passing through the point B, and the constant substi-
tutional alloying element line lA. t The activity of carbon at the points B and M (aB and aM
respectively) in a dilute ternary system are given as follows, ignoring the higher-order terms of
the expansion [21]:
I c I a, + a, + a,naB = nXl EllXl E12x2
In ak = In xi + Ell xi + E12x~ .
Using these two equations, and by ignoring the small difference between the products Ell x~'
and Ell xi, we obtain
(7.13)
On replacing x~ by X2' xi can be calculated given the interface compositions xr and x~'.
Because the bulk composition is now in the NPLE region, it should be noted that xi < Xl'
In the case of the PLE condition, in contrast, it is expected that xi > Xl' The following
method is used in order to determine the tie line in this situation. When the bulk composition
of ferrite is in the PLE region (see point K{xl, x2} in Fig. 7.4), it is necessary to ensure that the
gradient of carbon in the parent phase is reduced to a level consistent with the sluggishness of X
element diffusion. This can be done by ensuring that the carbon isoactivity line in ferrite passes
t Note that lA is not strictly parallel to the carbon axis since it is really the ratio of Fe/X atoms is constant along
lA.
200
through the bulk composition K; the intersection of that isoactivity line with the 0:/(0: + 'Y)
phase boundary (see point B{ x~1', x~1'} in Fig. 7.4), then defines the operative tie line.
However, when the phase boundary is to be determined simultaneously with the tie line,
it is possible to use a more convenient method to calculate these values. It is noted that the
operative tie line AB for the bulk composition K is the NPLE tie line for an alloy whose
bulk composition is same as the point M in Fig. 7.4, which is an intersection between the
transition line and the isoactivity line of carbon in ferrite passing through the bulk composition
K. Therefore the operative tie line AB can be obtained by finding the point M and then
calculating the NPLE tie line for the point.
First, one can calculate the point G{ xi, x~1'} corresponding to the point K using equa-
tion 7.13 and the NPLE interface compositions determined for the bulk composition K; i.e.
under the condition of x;a = x
2
' Since the bulk composition is now in the PLE region, it
is expected that xi > Xl' We now rename these points for simplicity: the bulk composition
and the corresponding point G are now called KO{xl,o' x2,o} and Go{xI,o' x[o} respectively
(Fig. 7.4b). In order to find the point M effectively, a point Kl {Xl 1, X2 l} on the isoactivity line, ,
of carbon in ferrite (i.e. , line BK), which has a slightly higher alloying element composition
than that of the bulk; i. e. x2,1 > x2,o, is selected. And the corresponding point Gl (XI,l' xL)
on the transition line is determined as mentioned above. Now one can estimate the point
G2(XI,2' X[2) which may be close to the point M using points Ko, K l' Go and Gl by the
following equations:
(7.14)
and
(7.15)
The point M is obtained by repeating this calculation until the difference between Xl,i+! and
XL+l becomes smaller than the accuracy required. Then one can use exactly the same method
to determine the tie line corresponding the point M as was used for the determination of the
tie line for the NPLE condition. This tie line is, in fact, the operative tie line for the bulk
composition K in Fig. 7.4.
7.3.2 Reconstructive growth of austenite under the local equilibrium
- starting from a mixture of ferrite and austenite of compositions Xi and xl
When the initial microstructure is a mixture of bainitic ferrite and austenite whose carbon
concentration was determined by the T~ curve at the bainite transformation temperature, tie
lines for the local equilibrium mode can, in this case, also be determined using the method
discussed above. The chemical distributions in both phases are, however, different from those
in the case of reaustenitisation from a fully ferritic sample. Since the initial carbon and X
concentrations in austenite are different from those at the interface, fluxes of carbon and X
in austenite are also expected to exist. Illustrative chemical distributions are shown with an
isothermal section of the equilibrium phase diagram in Fig. 7.5 and Fig. 7.6.
As mentioned in the previous section, rigorous solutions of the interface compositions and
the growth rate of the product phase (i.e. austenite), can be obtained only when the mass
conservation equations are solved giving the growth rate of the product phase which satisfies
the equilibrium conditions at the interface.
201
Because there are now diffusional fields both in the matrix (i. e. ferrite) and in the product
(i.e. austenite) phases, it is necessary to handle the diffusion equations in both phases, in a way
which satisfies the mass conservation conditions at the interface. The one-dimensional diffusion
equations in the ferrite matrix and austenite precipitate phase are expressed as follows.
axI = !- (D'Y axI) + !- (D'Y ax;)
at az 11 az az 12 az
ax; = !- (D'Y ax;) + !- (D'Y axI)
at az 22 az az 21 az
axr = !- (Da axr) + !- (Da ax~)
at az 11 az az 12 az
ax~ = !- (Da ax~) + !- (Da axr)
at az 22 az az 21 az
(7.16)
(7.17)
(7.18)
(7.19)
The requirement for conservation of mass at the interface gives the following two equations:
(7.20)
(7.21)
where v is the growth rate of the product phase. We emphasise again that the fluxes in both the
matrix and product phases contribute to mass conservation at the interface, and thus, to the
growth rate of the product phase. The fluxes of carbon and the substitutional alloying element
at the moving interface are given by,
"I I - "I axI I "I ax; I
J1 z=z. - -Dl1 ~ -D12~
uZ z=z· uZ z=z·
"I I - "I ax; I "I axI I
J2 z=z· - -D22~ -D21 ~
uZ z=z* uZ z=z·
Jal Da axr I Da ax~ I
1 z=z· = - 11~ - 12 ~
uZ z=z· uZ z=z·
Jal - _Da ax~ I _Da axr I
2 z=z· - 22 az z=z. 21 az z=z.
(7.22)
(7.23)
(7.24)
(7.25)
where the differentials are taken in the respective phases at the interface.
Because analytical equations of the concentration distributions in both matrix and product
phases (for the cases where diffusion fields in both phases should be considered simultaneously)
are not available, linear chemical concentration gradients are assumed in both phases for the
calculation of the growth rate of the product phase; this is the so-called Zener approximation.
As it is illustrated in Fig. 7.7, the diffusion distances of carbon and the substitutional
alloying element in ferrite and in austenite are respectively, L'Y' La' 1"1 and la. The fluxes of
carbon and the substitutional alloying element at the moving boundary are then given by,
(7.26)
(7.27)
202
(7.28)
(7.29)
where :tt'''' and x~'''''are average concentrations of carbon and the substitutional alloying element
in each phase, and z* is the position of the moving interface. The conservation conditions for
carbon and the substitutional alloying element in the system as a whole provide the following
equations:
(L..,. - z*)(xi -~) = ~..,.[(xia -~) + (xi - ~)] - L2a (~ - x~"")
la (X" _ xa..,.) = I..,. (x..,.a _ x"") .
2 2 2 2 2 2
(7.30)
(7.31)
In addition to these equations, because the decrease in the amount of carbon (or X) in ferrite
is equal to the integrated flux of carbon (or X) at the interface from t = 0 to t = t, one can
introduce two further equations given below:
(7.32)
(7.33)
As discussed in Appendix, the interface position z* and the diffusion distances L..,., La' 1..,.,and
la are proportional to t1/2:
z* = 0'1t1/2
L = k t1/2..,. C..,.
L - k 1/2a - Cat ,
1 = k t1/2..,. X..,.
1 = k t1/2a Xa
and, therefore, the growth rate of the prod uet phase is given by:
(7.34)
where 0'1 is one-dimensional parabolic thickening rate constant. Equations (7.32) and (7.33)
give the following relations for the proportionality constants for the diffusion distances in ferrite:
kXa = -0'1 + JO't+ 4D22
kCa = -ka + Jk~ + 4D~\
with
From equations 7.30 and 7.31, kc..,.and kx..,.are expressed by 0'1' kCa' and kXa' giving,
203
xa - X°'"l
kx'"l = kxo;o -=-'"1X
2
- X
2
X'"l - xa xa - X°'"l
k -2 1 1 k 1 1
C'"I - 0'1_'"1 '"10 - CO-'"I '"10
Xl - Xl Xl - Xl
Now, these four diffusion distances and the interface position can be substituted into the mass
conservation equations 7.30 and 7.31 to obtain the interface compositions and the growth rate
of the product phase, assuming that the cross interdiffusion coefficient D21 is negligible [21]:
with
O'i - C10'1 - C2 = 0
O'i - C3 = 0
(7.35)
(7.36)
One can then calculate the interface compositions and the growth rate of austenite by solving
equations 7.35 and 7.36 and three thermodynamic equations 7.8 simultaneously.
It is worth noting that the flux in austenite may be negligible in the case of the formation
of austenite from a fully ferritic initial microstructure by analogy with the reconstructive ferrite
formation from austenite in which the ferrite is usually assumed to grow with a constant com-
position. Therefore the growth of austenite, in this case, is controlled only by diffusion of atoms
in ferrite instead of in austenite. This is the same situation as the formation of ferrite from
austenite despite the fact that the growth of ferrite is in later case controlled by diffusion in
austenite. For the formation of austenite either from a mixture of ferrite and carbide particles,
or from a mixture of bainitic ferrite and austenite, on the other hand, concentration gradients
of car bon and X in austenite are expected to exist, and therefore, the growth of austenite will
depend on diffusion in austenite as well as that in ferrite.
7.3.3 Reconstructive growth of austenite under paraequilibrium
When the supersaturation of ferrite is very high; i. e. at very high temperatures, paraequi-
librium might be the condition which governs the growth of austenite.
According to the definition of the paraequilibrium, an identical X/Fe atoms ratio is now
expected both in the matrix and product phases:
(7.37)
204
Although the chemical potential of the substitutional alloying element or Hon are no longer
identical in the two phases at the interface, subject to that constraint, the carbon at the
interface achieves equality of chemical potentials in both phases [18), giving
ACOa ....•1' a ra
w. 1 Xl 1
RT = In X I + In q .
The following equation is used in order to define the paraequilibrium tie line [18].
(7.38)
(7.39)
ACOa ....•1' a ra 1 ACOa ....•1' a ra
w. 2 X2 2 w. ° Xo °----In- -In - = -------In - -In-
RT x~ q k2 RT x6 f6
These three equations allow the interface compositions to be calculated independently from the
mass conservation equation at the moving interface. Therefore one can calculate the interface
compositions first, and then put them into the mass conservation equation to obtain the growth
rate of the product phase at the moving interface. Because of the definition of paraequilibrium,
no flux of X atoms needs to be considered in this case. As a result, it is necessary only to set the
mass conservation for carbon at the interface. The growth of the product phase is, therefore,
controlled by carbon diffusion in both ferrite and austenite phases. The mass conservation
equation for carbon, in this case, can be simplified as follows:
(7.40)
Fig. 7.8 and Fig. 7.9 represent illustrative phase diagrams for reaustenitisation from a ferritic
initial microstructure, and from an initial microstructure which is a mixture of bainitic ferrite
and austenite respectively.
7.3.4 Special cases
As mentioned earlier, with some exceptions, it is necessary to consider diffusion in both the
ferrite matrix and austenite precipitate during reaustenitisation. However, there might exist
circumstances where diffusion in just one of the two phases largely controls the growth of the
product phase. The mass conservation equations derived in the previous section can then be
simplified and also, more rigorous solu tions for the chemical distributions in the phase are then
available [6,11,12].
When a fully ferritic initial microstructure is heated into the 0' + , two phase region, it
may be a good approximation that there is no flux in the precipitate phase (i.e. austenite).
For ferrite growth from austenite, the approximation is often very good because of the low
solubility of carbon and in ferrite. On the other hand, in the case of the austenite growth from
ferrite, the product phase usually has a high solubility for carbon and the substitutional alloying
element. Therefore the assumption of no flux in austenite should be examined experimentally.
In addition to this, fluxes in austenite may exist when soft impingement occurs: overlapping of
diffusion fields can lead to a change in the boundary conditions.
Reaustenitisation from a mixture of bainitic ferrite and austenite or from a mixture offerrite
and carbides shows rather different features. The carbon concentration of the initial austenite
at the reaustenitisation temperature is usually expected to be higher than the equilibrium
205
concentration at the interface at the reaction temperature [22-25], leading to the fact that
the supersaturation level in carbon could be larger in austenite than that in ferrite. As a
consequence, the contribution of fluxes in ferrite may be negligible [25] to the overall problem.
In the case of reaustenitisation from an initial microstructure of ferrite and cementite, the
austenite is found to nucleate preferentially at the junctions between ferrite/ferrite grain bound-
aries and carbide particles [14,26-29]. The dissolution of carbide particles happens essentially
happens after becoming engulfed by austenite. The early stages of the growth of austenite is
then controlled by carbon diffusion through the austenite envelope [14,15]. Because the equi-
librium carbon concentration in austenite at the interface between austenite and cementite is
much larger than that at the interface between austenite and ferrite, the carbon concentration
gradient in austenite is expected to be correspondingly larger than that in ferrite. Hence any
flux in the ferrite may be considered negligible [14,15].
Under the circumstances where one of the two phases is not important in terms of the
diffusion field which controls the growth of the product phase, the mass conservation equations
relate to the first one of the two phases, giving:
(7.41)
(7.42)
where J1Iz=z. and J2Iz=z. are fluxes either in ferrite or in austenite depending on the phase
which dominates as far as diffusion is concerned.
Growth of austenite from a ferritie sample
When the initial microstructure is ferritic, the growth of austenite might be controlled by
fluxes of carbon and X in ferrite instead of in austenite as mentioned earlier. Assuming the
approximation of the linear gradients of chemical compositions, the following equations for the
conservation of mass at the interface (see Fig. 7.10) are obtained:
v(x'YOI _ xOl'Y) = D?l(xa _ xOl'Y) + D?2(xa _ xOl'Y)
11 L 11 122
01 01
v(x'YOI _ xOl'Y) _ D~2(xa _ xOl'Y)
22 -/22'
01
(7.43 )
(7.44)
(7.46)
(7.45)
The mass conservation conditions for carbon and X for the whole system give the following two
equations (Fig. 7.10):
Z*(x'YOI _ xa) - LOI (xa _ xOl'Y)
1 1 - 2 1 1
*( 'YOI =01) _ 101(=01 OI'Y)Z x2 - x2 - - x2 - x2 .
2
The one-dimensional parabolic thickening rate constant for the growth of austenite is therefore
given by the simultaneous solution of:
(7.47)
(7.48)
206
For paraequilibrium austenite growth, the interface compositions are calculated indepen-
dently from the thermodynamic equations as discussed earlier. The parabolic rate constant of
austenite in that case is given by:
(7.49)
Growth of austenite from a mixture of ferrite and austenite
When the initial microstructure contains ferrite and austenite of equilibrium chemical
composition, the growth of the pre-existing austenite into ferrite is mainly controlled by the
diffusion fields in austenite if the superheating is high (Fig. 7.11). For that case, the mass
conservation of carbon and the su bstitu tional alloying element at the interface, and in the
whole system give the following equations (see Fig. 7.11):
v(x'YOI. _ xOl.'Y)= Dll(x'Y _ x'YOI.)+ DI2(x'Y _ x'YOI.)
11 L 11 122
'Y 'Y
v(x'YOI. _ xOl.'Y) _ D;2(x'Y _ x'YOI.)
2 2 - I 2 2
'Y
L
(L'Y - z*)(xI - x~'Y) = -t[(xI - x~'Y) + (xIOI. - x~'Y)]
(1'Y - z*)(x; - x~'Y) = I;[(x; - x~'Y) + (x;OI. - x~'Y)] .
(7.50)
(7.51)
(7.52)
(7.53)
The one-dimensional parabolic thickening rate constant of austenite is therefore given by the
simultaneous solution of:
(7.54)
(7.55)
For the case of paraequilibrium, the rate constant is given by:
(7.56)
Growth of austenite controlled by dissolution of cementite
In the case of reaustenitisation from a mixture of ferrite and cementite particles, the nucle-
ation of austenite occurs preferentially at the junctions between ferrite/ferrite grain boundaries
and cementite particles which locate on the grain boundaries. The austenite particles, then,
engulf the cementite particles. The growth of austenite particles after this stage is mainly con-
trolled by the dissolution of cementite particles within the austenite, if the supersaturation is
reasonably low, and the effect of the dissolution of cementite particles within the ferrite matrix
is small (see for example [15]). Assuming linear gradients of chemical composition in austen-
ite (Fig. 7.12), we can obtain the following equations to satisfy the mass conservation at the
interface and for the whole system:
207
(7.57)
(7.58)
(7.59)
(7.60)
where le is the dissolution thickness of cementite (see Fig. 7.12), xe.." x..,e are respectively carbon
(or X) concentrations at the interface in cementite and in austenite. From these equations, we
can obtain the following two expressions which on simultaneous solution can yield the one-
dimensional parabolic thickening rate constant:
For the paraequilibrium growth of austenite, it follows that:
(7.63)
The chemical compositions at the interface between austenite and cementite also need to be
determined. A method of this has been discussed by Hashiguchi and Kirkaldy [20].
A nalytical solutions
When only one of the two phases is important in terms of the control of the growth of the
product phase, analytical expressions for chemical distributions are available. The equations for
the concentration distributions in each phase (i.e. ferrite and austenite) can be obtained from
the diffusion equations [6] independently when no interaction between the fiuxes in two phases
exists. If the cross interdiffusion coefficient D21 is negligible [21]' it is possible to simplify the
general equation [6] as follows [10]:
(7.64)
208
(7.65)
(7.67)
(7.66)
erfc{ z }
a _ =<le a"l =<le) 2J D';2 t
X2 - X2 + (X2 - X2 erfc{ z· }
2JD ';2 t
erfc{ z..,}
"I _ -"I ("la _"I ) 2.jD'f;t
X2 - X2 + X2 - X2 .(' { z· }
erLC /n"Y;
2yD22t
It should be noted that these equations are not applicable if the diffusion fields in the two
phases interact with each other; this is expected to happen in most real circumstances. The
interface position is expressed by the diffusion coefficients and growth rate constants 11i (see for
example [ID)):
or,
• - "I ~D"I t - "I ~D"I t - a ~a t - a ~a t
Z - 111V Llll' - 112V Ll22' - 111 V Llll' - 112 V Ll22' (7.68)
(7.69)
where Q'1 is an one-dimensional parabolic thickening rate constant. The growth rate v is then
given by:
v = dz· = 11i
D
ii = ~Q'1 r1/2 (7.70)
dt 2 ~ 2V.IJiiL
Substituting the derivatives of the distributions of chemical composition and the growth
rate of the product phase into the mass conservation equations at the interface, one can obtain
the relations between the interface compositions and the growth rate of the product phase.
When the growth of austenite is controlled by the diffusion field in austenite, the relations
are given as follows [10]:
(7.71)
(7.72)
where
H {Did = J D7r Q'l[erfc{ ~}] exp{ Q'~ }
4 11 4D·· 4D··
It U
"la a"l
X
2
- X
2
B1 = x"Ia _ Xa"l
1 1
On the other hand, when the growth of austenite is controlled by diffusion in ferrite only,
the following equation applies:
(7.73)
(7.74)
209
7.4 CALCULATED EXAMPLES
In order to demonstrate the calculation of tie lines under different equilibrium conditions
as discussed in the previous section, reaustenitisation in a Fe-C-Cr ternary system will be
discussed.
7.4.1 Thermodynamic parameters
The thermodynamic parameters for the system which are used for the calculation have
represented by I<irkaldy and his co-workers [19,20]. Those parameters for the Fe-C-Cr system
are listed below, with some correction of typographical errors.
~GgQ-" = 8933 - 14.406T + 12.083 x lO-3T2 - 11.51 X 1O-6T3
+ 5.23 X 1O-9T4 , J mol-1 (T < lOOOI<)
= 71659 - 216.84T + 24.773 x lO-2T2 - 12.661 X lO-5T3
+ 24.397 X lO-9T4 , J mol-1 (T > 1000K)
~G~-" = -65562 + 32.949T , J mol-1
~GgQ-+1' = -1534 - 19.472T + 2.749TlnT , Jmol-1
£~1 = 1.3
£11 = 4.786 + 5066/T
£~2 = 2.819 - 6039/T
£;2 = 7.655 - 3154/T - 0.661ln T
£12 = 14.19 - 30210/T
In order to calculate the growth rate of product phase and interface compositions satisfying
the mass conservations at the moving interface and the diffusion equations in both phases, it is
essential to know the diffusion coefficients of carbon and alloying elements both in ferrite and
in austenite.
The diffusivity of carbon in austenite is known to vary significantly with carbon concentra-
tion. Therefore the diffusion equations should be solved with a diffusion coefficient which is a
function of carbon concentration; thus it varies from place to place in austenite. However, it has
also been shown that a weighted average diffusion coefficient of carbon in austenite, in which the
average is taken between the minimum and maximum carbon concentrations in austenite, can
be used in the calculation giving reasonable accuracy. Hence, in the present work, a weighted
average diffusion coefficient of carbon in austenite will be used for further calculations. The
average diffusion coefficient is expressed as follows [30]:
- l,,'Ya Ddx
D= ---
xi (x1'Q - xi)
(7.75)
This procedure is valid strictly for the situation where the concentration profile does not change
with time, but is recognised to be a good approximation for non-steady state conditions, as
exist during the growth of austenite into ferrite. For simplicity and consistency, we use a symbol
DI1 as D. As discussed by Brown and Kirkaldy [21]' the cross coefficients are expressed by,
210
(7.76)
at"'Y at""
'" '1 '" '1 E12 Xl
D12 = Dd. "','1 "','1'
1+ En Xl
The diffusion coefficient of carbon in ferrite is expressed as a function of temperature [16]'
glvmg,
{
10115} { [2 15309 ]}Dr1 = 0.02 exp ----;y- exp 0.5898 1 + ; arctan(1.4985 - ---;y-) , cm2 S-l (7.77)
The diffusivities of alloying element in ferrite and austenite were discussed by Fridberg et
al. [17]. On reprod ucing the data for Cr of figure 14 in their paper [17]' the diffusion coefficients
of Cr in ferrite and austenite are expressed as follows:
'1 { 286000}
D22 = 3.325 exp - RT '
'" { 240000}
D22 = 4.288 exp - RT '
{
240000}
= 1.340 exp - RT '
cm2 S-l (paramagnetic ferrite)
cm2 S-l (ferromagnetic ferrite)
(7.78)
(7.79)
Calculated isothermal sections of the phase diagram of a Fe-0.3C-4.08 wt.% alloy are pre-
sented in Fig. 7.13.
7.4.2 Growth of austenite from a mixture of bainite and austenite
When the initial microstructure contains a certain amount of austenite, nucleation of
austenite might be unnecessary during reaustenitisation. Especially when the initial micro
structure is a mixture of bainitic ferrite and residual austenite, the plate shape of the two
phases may allow us to treat reaustenitisation process as an one-dimensional parabolic growth
of the residual austenite plates.
Specimens are assumed to be heated up to the austenite phase field, and then quenched
to and held at a bainite transformation temperature for a sufficient length of time to allow
the specimens to complete the bainite transformation. Because of the nature of bainite trans-
formation, the reaction stops when the carbon concentration in residual austenite reaches the
point where the diffusionless formation of ferrite from austenite becomes thermodynamically
impossible: this if referred to as the incomplete reaction phenomenon. Therefore, if the other
reactions such as carbide precipitations and reconstructive formations of ferrite are sluggish, one
can obtain a mixture of bainitic ferrite plates and residual austenite trapped in between those
bainitic ferrite plates. The carbon concentration in the residual austenite has been reported to
be in a good agreement with T~ curve in the phase diagram of the system, where ferrite, whose
free energy has been raised by a stored energy term (i. e. 400 J mol-I, after Bhadeshia and Ed-
monds [31]) associated with the transformation strain, and austeni te of identical composition
have the same free energy. Since the decarburisation of bainitic ferrite occurs soon after the
reaction, the carbon concentration in ferrite may be close to its equilibrium level. There is no
partitioning of substitutional alloying elements during bainite transformation [32-35], the alloy-
ing element compositions in the two phases are identical as long as there is no reconstructive
formation of fe.rrite. Thus, if one chooses a bainite reaction temperature, one can obtain the
average chemical compositions in ferrite and residual austenite.
211
The specimens are then heated directly to a reaustenitisation temperature without being
cooled below the bainite transformation temperature Tb' So the initial microstructure is a
mixture of bainitic ferrite with the equilibrium carbon concentration at the bainite reaction
temperature; xr = x~'Y {Td, and residual austenite with carbon concentration at T~ at the
bainite transformation temperature; xl = xT' {Td·
°One-dimensional parabolic thickening rate constants in the Fe-0.3C-4.08Cr wt.% alloy cal-
culated for the paraequilibrium and the local equilibrium conditions are shown in Fig. 7.14. In
order to show the effect of the initial carbon concentration on the rate constant, three different
bainite transformation temperatures; 360, 420 and 480 °C, were selected. The calculated initial
carbon concentrations in residual austenite and in ferrite are listed below [37,38].
360
420
480
0.000553
0.000623
0.000692
XT' {Td, mole fraction
°
0.0265
0.0228
0.0163
Table 7.1: Carbon concentrations in bainitic ferrite and residual austenite at the end of bainite
transformation at each temperature in Fe-0.3C-4.08Cr wt.% alloy.
It can be seen that the growth rate of austenite increases mono tonically with temperature.
It should be noted that the rate constant under local equilibrium is always larger than that
under paraequilibrium. When the one dimensional parabolic rate constant for the growth of
ferrite from austenite is calculated, it has been found that the rate constant under paraequilib-
rium is larger than that under local equilibrium at lower temperatures whereas the order swaps
at higher temperatures. This could be understood as follows. Although the driving force for
the formation of ferrite is always higher in local equilibrium than that under paraequilibrium,
the diffusivity of the substitutional alloying element is remarkably low at low temperatures; this
causes the retardation of the growth of austenite. Therefore the rate constant for the growth
of ferrite under local equilibrium at lower temperatures is smaller than that under paraequi-
librium (for example see [40]). At higher temperatures, on the other hand, the diffusivity of
the substitutional alloying element becomes high and the driving force for the transformation
under paraequilibrium decreases rapidly: in fact the driving force for the growth offerrite under
paraequilibrium becomes zero at the Ae~ temperature. As a consequence, the rate constant
under local equilibrium becomes greater than that under paraequilibrium [40]. However it
should be noted that a spike of the substitutional alloying element in austenite adjacent to the
transformation interface, which is under local equilibrium, becomes sharper with decreasing
temperature, and little difference will be expected for the rate constants calculated from local
equilibrium and paraequilibrium conditions as can be seen in Enomoto's calculations [40].
In the case of reaustenitisation, however, the diffusivity of the substitutional alloying el-
ement is very high because of the elevated temperatures for reaustenitisation, leading to the
fact that the rate constant for the growth of austenite under local equilibrium always remains
higher than that under paraequilibrium.
The rate constants for the growth of austenite under paraequilibrium for the three different
mixtures of bainitic ferrite and residual austenite are compared in Fig. 7.15. The rate constants
decrease gently with temperature at higher temperatures, but fall down steeply to zero at lower
212
temperature showing the lowest possible temperature where austenite can actually grow. It can
also be seen that the rate constant at a temperature is larger when the bainite transformation
temperature is lower. This can be understood qualitatively from the fact that xT,{Tb} is
o
larger in lower bainite transformation temperature, which provides larger supersaturation of
carbon in austenite at a reaction temperature and hence a larger growth rate of austenite. The
reaustenitisation-start temperature under paraequilibrium; i.e. the lowest temperature where
the growth of austenite can occur, also varies with the bainite transformation temperature.
The reaustenitisation-start temperature from a mixture of bainitic ferrite and austenite can be
understood in terms of the carbon concentration in residual austenite and the negative slope of
the Ae3 (or Ae~) curve [22-25].
However it should be noted that Yang and Bhadeshia ignored the effect of diffusion of
carbon in ferrite [22-25], which might have a substantial effect on the reaustenitisation-start
temperature. This was examined by comparing the rate constant for the growth of austenite
under paraequilibrium calculated by equation 7.40; in which the diffusion of carbon in both
phases are taken into account, or equation 7.56; where the diffusion field exists only in austenite.
Calculated rate constants for the growth of austenite from two different mixtures of bainitic
ferrite and austenite are shown in Fig. 7.16. The calculated rate constant assuming the diffusion
field only in austenite is larger than that calculated for the case where the diffusion fields exist
in both phases at lower temperatures, whereas the order changes at higher temperatures. This
crossover in the rate constants can be interpreted as follows. At a temperature where carbon
concentration in residual austenite reaches the Ae~ (note that this is not Ae3) (see Fig. 7.17a),
the growth of austenite can start if the diffusion of carbon in ferrite can be ignored. However, the
growth of austenite is not yet possible in the case where the diffusion of carbon in both phases
should be taken into account, since the negative gradient of carbon exists at the transformation
interface in ferrite. This circumstance remains until the temperature is raised to a point at
which the fluxes of carbon in ferrite and in austenite cancel each other out (Fig. 7.17b). Above
this temperature, the growth of austenite can proceed even in the case where the diffusion
of carbon in both phases is considered. In this circumstance, the growth rate of austenite
calculated with carbon fluxes in both phases is smaller than that with the carbon diffusion only
in austenite because of the negative slope of carbon in ferrite at the transformation interface.
When the temperature reaches a point where the carbon concentration in the ferrite matrix is
equal to that on the a/(a + ,) phase boundary (see Fig. 7.17c), the rate constant calculated
by the two different methods becomes identical since there is no flux of carbon in ferrite at the
temperature. Above this temperature, the rate constant calculated with diffusion of the carbon
in both phases becomes higher than that with the diffusion of carbon only in austenite since
the positive flux of carbon in ferrite accelerates the movement of the transformation interface
(Fig. 7.17d) .
It is also suggested that the smaller the initial carbon concentration in ferrite the larger the
difference in reaustenitisation temperatures calculated by the two different methods since more
superheating is required to reach the temperature shown in Fig. 7.17b. In fact the difference in
reaustenitisation-start temperatures calculated by the two different methods for the case of the
bainite transformation at 3600 C was larger than the case of the 4200 C bainite transformation
temperature, since in the former case the carbon concentration in the initial ferrite is lower
than that of the later (see Table 7.1).
The calculated interface carbon concentrations during reaustenitisation under local equi-
213
The theory was applied to reaustenitisation from a mixture of bainitic ferrite and austenite,
where nucleation of austenite may not be necessary since the initial microstructure contains
austenite.
REFERENCES
1. H. K. D. H. Bhadeshia: "Phase Transformations '87", Institute of Metals, London, ed. G. W.
Lorimer, 1987, 309.
2. A. Hultgren: Jernkontorets Ann., 1951, 135, 403.
3. E. Rudberg: Jernkontorets Ann., 1952, 136, 91.
4. M. Hillert: Jernkontorets Ann., 1952, 136, 25.
5. M. Hillert: Internal Report, Swedish Institute of Metals Research. 1953.
6. J. S. Kirkaldy: Can. J. Phys., 1958,36,907.
7. G. R. Purdy, D. H. Weichert and J. S. Kirkaldy: TMS-AIME, 1964, 230, 1025.
8. H. 1. Aaronson, H. A. Domian and G. M. Pound: TMS-AIME, 1966 236, 753.
9. H. 1. Aaronson, H. A. Domian and G. M. Pound: TMS-AIME, 1966 236, 768.
10. D. E. Coates: Metall. Trans., 1972, 3, 1203.
11. D. E. Coates: Metall. Trans., 1973, 4, 1077.
12. D. E. Coates: Metall. Trans., 1973, 4, 2313.
13. H. K. D. H. Bhadeshia: Progress in Materials Science, 1985, 29, 321.
14. R. R. Judd and H. W. Paxton: TMS-AIME, 1968,242, 206.
15. M. Hillert, K. Nilsson and L-E. Torndahl: JISI, 1971 209,49.
16. J . .Agren: Mat. Sci. and Eng., 1982, 55, 135.
17. M. Hillert: "Mechanism of Phase Transformations In Crystalline Solid", Monograph 13,
Institute of Metals, London, 1969, 231.
18. J. B. Gilmour, G. R. Purdy and J. S. Kirkaldy: Metall. Trans., 1972, 3, 1455.
19. J. S. Kirkaldy, B. A. Thomson and E. A. Baganis: (Wardenability Concepts with Applications
to Steel", ed. D. V. Doane and J. S. Kirkaldy, AIME, 1978, 82.
20. K. Hashiguchi and J. S. Kirkaldy: CA LPHAD, 1984,8, 173.
21. L. C. Brown and J. S. Kirkaldy: Trans. A/ME, 1964, 230, 223.
22. J-R. Yang and H. K. D. H. Bhadeshia: ((Welding Metallurgy of Structural Steels", TMS-
AIME, Warrendate, Ohio, ed. J. Y. Koo, 1987, 549.
23. J-R. Yang and H. K. D. H. Bhadeshia: "Phase Transformations '87", The Institute of Metals,
London, ed. G. W. Lorimer, 1988, 203.
24. J-R. Yang and H. K. D. H. Bhadeshia: Mat. Sci. and Eng., 1989, III press.
25. J-R. Yang: Ph.D. Thesis, University of Cambridge, 1988.
26. G. R. Speich and A. Szirmae: TMS-AIME, 1969, 245, 1063.
27. C. 1. Garcia and A. J. DeArdo: Me t all. Trans., 1981, 12A, 521.
28. U. R. Lenel and R .W. K. Honeycombe: Metal Science, 1984,18,201.
29. U. R. Lenel and R .W. K. Honeycombe: Metal Science, 1984, 18, 503.
30. R. Trivedi and G. M. Pound: J. Appl. Phys., 1967, 38, 3569.
31. H. K. D. H. Bhadeshia: Acta Metall., 1981,29, 1117.
32. H. K. D. H. Bhadeshia and A. R. Waugh: Proc. of Int. Conf. on "Solid/Solid Phase
Transformations", Pittsburgh, ASM, Ohio, U.S.A., 1981, 1041.
215
33. H. K. D. H. Bhadeshia and A. R. Waugh: Acta Me tall., 1982, 30, 775.
34. 1. Stark, G. D. W. Smith and H. K. D. H. Bhadeshia: Proc. of Int. Conf. on "Solid/Solid
Phase Transformations", Institute of Metals, London, 1989, in press.
35. 1. Stark, G. D. W. Smith and H. K. D. H. Bhadeshia: Proc. of Int. Conf. on "The Bainite
Transformation", Chicago, U.S.A, A.S.M., 1988, in press.
36. B. Josefsson and H. O. Andren: Proc. of the 35th Int. Conf. on "Field Emission Symp. ",
Oak Ridge, Tennessee, U.S.A. 18-22 July, 1988.
37. H. K. D. H. Bhadeshia and D. V. Edmonds: Acta Metall., 1980,28, 1265.
38. H. K. D. H. Bhadeshia: Metal Science, 1982, 16, 167.
39. J. W. Christian: "The Theory of Transformation in Metals and Alloys", second ed. Part 1,
Pergamon, Oxford, 1975.
40. M. Enomoto: 1988, Trans. lSIJ, 28, 826.
41. K. R. Kinsman and H. 1. Aaronson: Metall. Trans., 1973,4, 959.
216
APPENDIX
The conservation of mass for the substitutional alloying element in a ternary system is
expressed by the following three equations as discussed in the text:
v(x'YOt _ xOt'Y) = _ D;2(x'YOt _ x ) + D~2(X _ xOt'Y)
22 122 122
'Y Ot
lOt (xa _ xOt'Y) = I'Y (x 'YOt _ x'Y)
2 2 2 2 2 2
t D~2(~ - x~'Y) dt = 1(lOt + 2z*)(~ _ x~'Y) .
la Ot
(7.81)
(7.82)
(7.83)
(7.84)
On eliminating I'Y from equations 7.81 and 7.82, a relation between v and lOt can be obtained:
D'Y ('YOt -)2 DOt (- Ot'Y)I - _ 22 x2 - x2 22 x2 - x2
V Ot - (x'YOt _ xOt'Y)(x _ xOt'Y) + x'YOt _ xOt'Y
2 2 2 2 2 2
The right hand side of equation 7.84 is independent on time as long as soft impingement does
not occur. Equation 7.84 can therefore be rewritten as follows regarding the fact that v = dz/dt:
dz
lOt dt = Cl (7.85)
where Cl is a constant which is equal to the right hand side of equation 7.84. In equation 7.85
lOt and z are both functions of time. Since lOt = 0 at t = 0, it might be reasonable to assume
the expression for lOt as follows:
where Cl and a are constants. Thus equation 7.85 becomes:
dz Cl_a
-=-t
dt C
2
leading the expression for z:
(7.86)
(7.87)
(7.88)
where C3 = C2( ~a+1) is a constant. On substituting equations 7.86 and 7.88 into 7.83, one can
obtain the following expression:
(7.89)
Since equation 7.89 should be satisfied for any value of time t, the time exponent have to be
equal to zero:
As a result, a = 1/2 is obtained.
-2a + 1 = 0
217
(7.90)
xx
ISooctlVlty line
--~
c
C Isooctivity line
c
Fig. 7.1 Schematic isothermal section of a Fe-C-X ternary system, illustrating ferrite growth occur-
ring with local equilibrium at the oIy interface. (a) Growth at low supersaturations (PLE)
with bulk composition of X, and (b) growth at high supersaturations (NPLE) with negli-
gible partitioning of X during transformation. The bulk alloy compositions are designated
"A" and "B" respectively.
218
xc
Fig. 7.2 Isothermal section of a ternary Fe-C-X phase diagram, with the (0' + "y) phase field lying
within the region PQSR. The decomposition of austenite by the NPLE mechanism can
occur if the bulk composition lies in the region QPS, whereas the growth of austenite by
the NPLE mechanism can only occur if the ferrite is superheated into the region QRS.
x NPLE
Fig. 7.3 The NPLE growth of austenite as a consequence of the superheating of a fully ferritic
sam pIe of bulk corn posi tion "I" to a tem perat 11 re T.
219
ax PLE
c
b
a/a+y
boundary
transition line
Fig. 7.4 The PLE growth of austenite as a consequence of the superheating of a fully ferritic sample
of bulk composition "l(" to a temperature T.
x NPLE
Fig. 7.5 The NPLE growth of austenite as a consequence of the superheating of a sample with a
mixture of bainitic ferrite and residual austenite initial microstructure of bulk composition
"I" to a temperature T.
220
x PLE
Fig. 7.6 The PLE growth of austenite as a consequence of the superheating of a sample with a
mixture of bainitic ferrite and residual austenite initial microstructure of bulk composition
"K" to a temperature T.
zo Z"
C
L..,
•••
)
xl
..,0
I x)
I
I
I
I
I
I
I
1,.,I
I ~I
I
xfI
0..,
x) I
I I
I
I
I
I
..,0
X
X1
xj xl'
Fig. 7.7 Schematic diagram of carbon and X profiles during reconstructive growth of austenite
from a mixture of bainitic ferrite and austenite assuming linear gradients. z· designate the
position of the transformation interface along a space coordinate normal to the interface.
221
x PE
c
Fig. 7.8 The paraequilibrium growth of austenite as a consequence of the superheating of a fully
ferritic sample of bulk composition "L" to a temperature T.
x PE
c
Fig. 7.9 The paraequilibrium growth of austenite as a consequence of the superheating of a sam-
ple with a mixture of bainitic ferrite and residual austenite initial microstructure of bulk
composition "L" to a temperature T.
222
cx
L.,
< ~
X°"I I
I I
I I
I., I
~
- Ix., I
1 I
I I
I I
I
I
I• .,0
Xl
z
Fig. 7.10 Schematic diagram of carbon and X profiles during the growth of austenite from a fully
ferritic initial microstructure assuming linear gradients which exist only in ferrite matrix.
z* designate the position of the transformation interface along a space coordinate normal
to the interface.
c
.,0
XI
I
I
I
LoI
I < )
I
I
I
I
I Zo
I I
I
0"
%~ I
I I
I
I
I
X
I I
I I
I ! z.,.o
1
10
z~
z
Fig. 7.11 Schematic diagram of carbon and X profiles during the growth of austenite from a mixture
of ferrite and allstcnite assuming linear gradients which exist only in austenite. z* designatc
position of the transformation interface along a space coordinate normal to thc interface.
223
c()
X ""(8
1
x
z
Fig. 7.12 Schematic diagram of carbon and X profiles during the growth of austenite from a mixture
of ferrite and cementite particles assuming linear gradients which exist only in austenite.
z· designates the position of the transformation interface along a space coordinate normal
to the interface.
224
0.10
Phcse Dicgrcm cl 750'C in r.-C-Cr $y$l.m
0.08
'02 \
°6~800 0.005 0.010 0.015 0.C20 :;).025 0.030
C-ccntent. mole Ircclio"
I _ I
Phose Dicgrcm ct 760'C In re-C-Cr sy$tem
\
0.00
0.000 0.005 0.010 0.015 0.020 0.025 0.030
c ::: r:
oS!
u
0
0.06i.:Cl
'0
E •r
C f.
11 0.0,(
C
0
U,
<3
0.02
C-ccn!enl, mol. Ircction
0.10
I I I I
Phcse Dicgrcm ct 770'C i" r e-C-Cr s)'Slem
O.Oil ~
0.00 \
0.000 0.005 0.010 0.015 0.020 0.025 0.030
C-content, mol. I,cctio"
Fig. 7.13 Calculated isothermal sections of the phase diagram in the Fe-0.3C-4.08Cr wt.% alloy at
750,760 and 770°C respectively. Dashed lines represent the transition lines from NPLE
to PLE condition.
225
1.0E-02 a •
Beinile trensformetion ot 480 C
1.0E-02 b
Beinile trensformetlon et 420·C
1.0E-03
~- ..
"-E
u
C
1.0E-04c
••c
0
u
.!
c
'"
1.0E-05
1.0E-03
~- ..
"-E
u
C
1.0E-04c
••c
0
u
.!
c
'"
1.0E-05
1.0E-06
700 750 800
Tempereture, ·C
850 900
1.0E-06
700 750 800
Tempereture, ·C
850 900
1.0E-02 C
Beini!e trensformetion et 360·C
1.0E-03
~ ..
"-E
u
C
1.0E-04c..
c
0
u
.•
C
'"
1.0E-05
1.0E-06
700 750 800
Tempereture, ·C
850 900
Fig. 7.14 Calculated one dimensional parabolic thickening rate constant of austenite as a function
of the reaction temperature during reaustenitisation from a mixture of bainitic ferrite and
residual austenite. The bainitc transformation treatment was carried out at (a) 480
Q
C, (b)
420 QC and (c) 360 QC. Solid lines are the rate constant for the local equilibrium growth of
austenite and dashed lines for the paraequilibrium growth of austenite.
226
Fig. 7.15
1.0E-02
Para.qullibrium
1.0E-03
-;f!.
,~ ...
'- ......E /'//<...u
C
1.0E-04a
;;; I
c
/ :....480·C0
u
.. / .
"0 I0::
I1.0E-05
I
I420·C
1.0E-06
850 900700 750 800
Tomp.ratur., ·C
Calculated one dimensional parabolic thickening rate constant of austenite as a function
of the reaction temperature during reaustenitisation under paraequilibrium from mixtures
of bainitic ferrite and residual austenite obtained at three different bainite transformation
temperatures.
1.0E-02
a Paraequllibrlum
Balnit. transformation at 420·C
1.0E-02
b Paraequillbrlum
Balnll. transformation at 360·C
1.0E-03
~ ..
'-E
u
C
1.0E-04a
;;;
c
0
u..
"0
0::
1.0E-05
Diffusion
In aust.nit. :.
Diffusion In
forrit. and aust.nil.
1.0E-03
$••
'-E
u
Diffusion
C
1.0E-04 In aust.nit.c
;;;
c
a
u..
"0
0::
1.0E-05 Diffusion In
ferrlt. and aust.nit.
I,OE-06
750 800 850 900
1.0E-06
750 800 850 900
Tomp.ratur., ·C Temp.rature, ·C
Fig. 7.16 Calculated one-dimensional parabolic thickening rate constant of austenite during reausteni-
tisation under paraequilibrium from a mixture of bainitic ferrite and residual austenite ob-
tained at (a) 4200 C and (b) 3600 C. Solid line shows the rate constant as a function of the
reaction temperature for the case where diffusion fields exist both in ferrite and austellitc,
whereas dashed line is for the case with diffusion fields only in austenite.
227
ax
'Y
c
b
X
c
c
x
c
d
X
c
Fig. 7.17 Schematic isothermal sections of the phase diagrams at different temperatures with the
paraequilibrium a/a + I and I/a + I phase boundaries, in which the initial carbon con-
centrations of ferrite and austenite are indicated by open and solid circles respectively.
228
0.0015 r--
a
-)----"'T1------,I------r--I------,
Bainit. transformation at 420·C
-
-
PE
bulk carbon content
--- .._ ..-._-----_._-_ .._ .._._._-_._-_._-_ .._._--_._.-.-._------_ .._._ .._._._._-_._._.-.
~~
o
u
ID
"0
E 0.0005 f-
.: 0.0010
c:
~
U
o
.t
0.000900
I
750
I
800
I
850 900
Temperatur., ·C
900850
PE
800
. bulk carbon content
--_._--------":---------_._----_._-_._-_._----_._-_. -._.
LE
750
b)
Balnit. transformation at 420·C
0.04
§ 0.01
oD•..
o
u
••
"0
E
ID
'2
.!
~ 0.03
o
.:
c:
~
] 0.02
.t
Temperature, ·C
Fig. 7.18 Calculated interface carbon concentrations in ferrite (a) and austenite (b) for the parac-
quilibrium (solid line) and the local equilibrium (dashed line) growth of austenite from a
mixture of bainitic ferrite and austenite formed at 420 cC. Initial carbon concentration in
the both phases are also shown.
229
1.0(-02 ~a----r----'---.---'I---I
Bainile Iransformation at 480 C
1.0(-02 b •
Bainite Iransformatlan at 420 C
1.0£-0;
1.0(-03
~
.(:! ~•.~•. -.....
-..... E
E u
u
C PE
C
1.0(-04
1.0(-04 -
0
0 Vi
Vi c
c 0
0 :;--r uu •••. 0'0 0::
0:: - 1.0(-05
1.0(-05
1.0(-06
700
I
750 800
Temperature, ·C
I
850 900
1.0(-06
700 750 800
Temperature, ·C
850 900
1.0(-02 C I I
Bainite Irons formation 01 360·C
1.0(-03 f-
~•.
-.....
E
u
~ 1.0E-04
Vi
c
o
u•.
o
0::
1.0(-05 f-
~
;' PE
-
1.0(-06
700
i
750
I
800
Temperature, ·C
850 900
Fig. 7.19 Calculated one dimensional parabolic thickening rate constant of austenite as a function
of the reaction temperature during reaustenitisation from a mixture of bainitic ferrite and
residual austenite. The bainite transformation treatment was carried out at (a) 480°C,
(b) 420°C and (c) 360°C. Solid lines are the rate constant for the local equilibrium growth
of austenite and dashed lines for the paraequilibrium growth of austenite. The carbon
concentration at the interface in austenite was replaced by the bulk carbon content when
the calculated xiQ becomes smaller than the bulk carbon content.
230
900850800
,
\ '.
\" 480'C
...."..... '. 420'C
"~~'/
,.....<.:~:~<.:..
.....~~
1.0E-05
1.0E-06
E
u
{
1.0E-07
.!
.E
.s:
:<;
"i 1.0E-08
ID
""ii
III
1.0E-09
1.0E-10
750
Temperature, ·c
Fig. 7.20 Spike width of the substitutional alloying element calculated by equation 7.80 in the text,
as a function of temperature and the initial condition.
1.0E-02
Bainite transformation at 420'C
Para equilibrium
1.0E-03
/'
~ .. /"-
E .'
u
/
,
C
1.0E-04 //a;;
c:
0
u
I•.
I;;
cc I
lCr:
1.0E-05 2Crj
I
I
Temperature, ·c
1.0E-06
750 800 850 900
Fig. 7.21 Effect of Cr content on the one dimensional parabolic thickening rate constant of the
paraequilibrium growth of austenite. The bainite transformation was carried out at 420 QC.
231
CHAPTER 8
TIME-TEMPERATURE-TRANSFORMATION CURVES
AND OVERALL REAUSTENITISATION IN STEELS
8.1 INTRODUCTION
As mentioned in Chapter 1, the prediction of the microstructural development during
cooling can be carried out using calculated time-temperature-transformation (TTT) curves for
ferrite formation and an assumption of the additivity rule. For many applications, it is just as
important to determine TTT curves for reaustenitisation from the initial microstructure of in-
terest. Once TTT curves for various degrees of reaustenitisation are obtained, reaustenitisation
during continuous heating, can be calculated adopting Scheil's rule [1] along any heat cycle.
This is the major goal of the present work.
8.2 ASSESSMENT OF SCHEIL'S RULE IN REAUSTENITISATION
If the incubation period T is obtained as a function of temperature, reaustenitisation will
start when an integration of T along a heating pass reaches unity (i.e. Scheil's rule)
(5
la T{T} -ldt = 1 (8.1)
where the reaction temperature T is now a function of time t, and reaustenitisation starts at
time ts on heating.
The TTT curve for reaustenitisation in a Fe-C-Mn-Ni-Mo alloy reported by Yang [2]
(Fig. 8.1) is reanalysed and continuous heating reaustenitisation experiments were conducted
on the same alloy as Yang in order to assess the application of Scheil's rule.
Specimens were heated to 1000 QCfor 10 min followed by quenching to 460 QCfor 30 min al-
lowing the bainite transformation completed. The specimens were then heated directly from the
temperature at four different constant heating rates, 5, 10, 20 and 50 QCS-l. Reaustenitisation-
start temperatures were determined by deviations of the relative length change from a linear
thermal expansion curve. The observed results were plotted against heating rate in Fig. 8.2.
An empirical expression for T was obtained by regression analysis using Yang's isothermal
data for the smallest detectable reaustenitisation from the mixture of bainite and residual
austenite:
log T = -57.737logT + 173.42 (8.2)
where T and T are measured in sand J( respectively.
The reaustenitisation start temperatures during continuous heating were calculated using
equation 8.2 and are presented in Fig. 8.2. The agreement between experiment and calculation
is reasonably good.
8.3 GROWTH OF AUSTENITE
Reaustenitisation in steels proceeds by a nucleation and growth mechanism. When the
retained austenite is present in the initial microstructure, it may not be necessary to nucleate
new austenite since the reaction can proceed by the growth of the retained austenite.
232
Assuming one-dimensional growth, and if the austenite/ferrite interfacial area per unit
volume is Sv, the increase in volume fraction of austenite ~ V-y during heat treatment is given
by:
(8.3)
Adopting the extended volume concept (see Appendix), ~ the volume fraction of newly trans-
formed austenite normalised by the maximum degree of reaction ()= V-ye- V-yo (v-ye and V-yo are
the equilibrium and initial volume fractions of austenite), is expressed as follows:
(8.4)
When reaustenitisation proceeds not only by the one-dimensional thickening of austenite
plates but also by lengthening of the plates, ~{t} should be then expressed by the parabolic
thickening rate constants in two directions. Assuming that the initial volume of an austenite
plate in the starting microstructure is aoc5, and that at time t is ac2, the volume fraction change
~ V-y at time t during isothermal reaustenitisation can be expressed as follows:
~V-y = n(ac
2
- aoc~)
= n{(ao + O'1t1/2)(CO + f3t
1/2)2 - aoc~}
= n{(6aoco + C5)O'1t1/2 + (9ao + 6co)O'it + 9O'rt3/2}
(8.5)
where f3 is the parabolic lengthening rate constant which was assumed to be 30'1 after [3]' and
n is the number of growing austenite plates per unit volume. The volume fraction of newly
transformed austenite normalised by the maximum degree of reaction () is thus given by:
Yang and Bhadeshia [4] derived the time required to obtain a small increase in the volume
fraction of austenite T using equation 8.4 as follows:
(8.7)
Therefore the incubation period T is proportional to O'~2. If this theory is acceptable, the plot
of log{ T} against log{O'~2} should show a linear correlation as shown by Yang [2]. It should be
noted, however, that there are poor linear correlations between T and O'~2 as shown in Fig. 8.3 a.
The slopes of the log-log plot are 1.56 for the smallest detectable reaustenitisation and 1.80 for
the 0.05 of increase in the volume fraction of austenite (Fig. 8.3 b), which are expected to be
unity from the theory. When log{-ln{l -~}} is plotted against log{t}, the slope of the plot
should be 1/2 in Yang's theory at the early stage of reaustenitisation. The slope of the plot
is, however, not always 1/2 as shown in Fig. 8.4. These results may indicate that equation 8.6
could be more appropriate for an expression of the kinetics of reaustenitisation from a mixture
of bainitic ferrite and austenite than equation 8.4. For simplification, the initial dimension of
the retained austenite is assumed here to be ao = 0.2 /Lm and Co = 5 /Lm . Equation 8.6 becomes
as follows:
233
e{t} = 1 - exp[- 31;(1'1 t1/2{1 + 1.03(1'1t1/2 + 0.29(1'it}]. (8.8)
The slope of the plot of log{-ln{1 - e}} against log{t} varies with (1'1as shown in Fig. 8.5.
The slope should then vary with temperature since (1'1depends on the reaction temperature.
The slope of the plot oflog{-ln{l- O} against log{t} varies from 0.51 to 1.42 when (1'1is
altered between 0.01 and 10 /Lm S-1/2. When (1'1becomes 1.0 /Lm S-1/2 (1 X 10-4 cm S-1/2),
the slope reaches around unity.
Using calculated parabolic thickening rate constants for Yang's alloy [2] and for the Fe-
0.3C-4.08Cr wt. % alloy, the slopes of the plot of log{ -In {I - O} against log{ t} calculated as
discussed above were compared with the observed slopes at 680, 700 and 710 QC for Yang's
alloy [2] and at 778, 785 and 805 QCfor the Fe-0.3C-4.08Cr wt.% alloy (Fig. 8.6). A reasonable
agreement between experiment and calculated results was obtained. This suggests that equa-
tion 8.6 is applicable to the kinetic calculation of reaustenitisation from mixtures of bainitic
ferrite and austenite where the initial austenite is plate in shape and grows in two directions;
i. e. lengthening and thickening.
The effective number of austenite plates per unit volume which can grow during reausteni-
tisation; i.e. the number n in equation 8.6, can be evaluated from the observed data discussed
above. The 50% reaustenitisation time (i.e. the time for e = 0.5) was used. The calculated n
values are 1.30 x 10-4 at 680 QC, 2.91 X 10-5 at 700 QC and 5.49 x 10-4 at 710 QC for Yang's
alloy, and 4.30 x 10-4 at 778 QC, 8.31 x 10-4 at 785 QC and 5.33 X 10-4 at 805 QC for the
Fe-0.3C-4.08Cr wt.% alloy. Although there are significant scatter in the calculated n values,
especially in Yang's alloy, the average value for each alloy is used for further analysis since the
same starting microstructure was used for each alloy.
The TTT curve for a 0.05 increase in the volume fraction of austenite (i.e. ~ V1' = 0.05)
was calculated and compared with the observed data [2] in Fig. 8.7. A reasonable agreement
was obtained at high temperatures but not at temperatures below 690 QC. This may because
the theory does not take into account of the effect of soft impingement. Since the maximum
degree of reaction at low temperatures is not very large in comparison with 0.05, the growth
rate of austenite may decrease significantly by soft impingement before ~ V1' reaches 0.05.
The effect of reaction temperature on the increase in the volume fraction of austenite was
calculated using equation 8.6 for the Fe-0.3C-4.08Cr wt.% alloy with bainite + 0.3 of retained
austenite starting microstructure (Fig. 8.8). The calculated results were rearranged to show the
TTT curves for different degrees of ~ V1' (Fig. 8.9). The calculated TTT curve for ~ V1' = 0.05
was compared with the observed values in the present experiments (Fig. 8.10). The agreement
is reasonably good. It should, however, noted that the formation of austenite was observed
during heating when the reaction temperature is higher than 785 QC. The observed values in
Fig. 8.10 may represent overestimates.
8.4 AUSTENITE FORMATION INCLUDING NUCLEATION AND GROWTH
Examples of the nucleation and growth reaustenitisation processes include the reheating
of martensite or mixtures of ferrite and carbides. Although the morphology of the austenite
formed depends strongly on the initial microstructure (Chapter 6), the simplest assumption
is the spherical austenite model. The growth is then three-dimensional and the appropriate
parabolic rate constant (1'3is used. The formation of austenite is assumed to occur from a
carbon supersaturated ferrite matrix. The paraequilibrium carbon concentration at 500 QCwas
234
used, without any proof, as the matrix carbon concentration after tempering during heating,
since there is no knowledge about the amount of carbon left in the matrix before the reaction.
The rate constant 0'3 is given by [5]:
(8.9)
(8.10)
where
0'2 1r 0'2 0'2
H {D?:} = _3 [1 _ (_)0.50' exp{_3 }erfc{(-3 )0.5}]
3 It 2D<:'. 4D<:'. 3 4D<:'. 4D<:'.
It n U It
o.-y -yo.
X
2
- X
2
BI = o.-y -yo.
Xl - Xl
where Dfl and D~2 are diffusivities of carbon and substitutional alloying element in ferrite, and
the x's are chemical compositions at the interface.
Using the extended volume concept with the nucleation rate I and the growth rate 0'3' the
volume fraction of austenite e at time t normalised by the maximum degree of reaction () (note
that () here is equal to V-ye) is given by:
t 4
e{t} = 1 - exp[- lo I 3() 1rO'i(t - t')3/2dt']
41rO'
31t= 1- exp(- __ 3 I(t - t')3/2dt'].
3() °
The nucleation rate I is a function of temperature, driving force for nucleation, site density and
time. To simplify the calculation, I is assumed to have the following form:
(8.11)
(8.12)
where No is the initial nucleation site density and VI is the rate constant for nucleation. Sub-
stituting the expression of I into equation 8.10, the following relation is obtained:
41rO'3N v t /
e{ t} = 1 - exp(- ;() ° I lo exp{ -VI t'}(t - t'? 2dt'].
The total number of particles N at time t can be calculated from equation 8.11, namely;
(8.13)
Therefore, the time required to exhaust 95% of the nucleation sites is roughly equal to 3/v
l
.
At a temperature T, the following relation can be obtained from equation 8.12 and be used
to determine the values VI and No.
(8.14)
where KT = 41rO'~Novd3(). By altering the value VI in the second term, the slope of the
plot of the both sides of the equation against log t can be made identical. Then the value VI
obtained here will be substituted into the expression of KT giving the value No at temperature
235
T using the calculated 0'3 from thermodynamic knowledge as discussed earlier. Once the values
in equation 8.12 are obtained, the time required to obtain the increase in the volume fraction
of austenite ~ V-y can be obtained.
When the rate constant v1 in the nucleation function is altered, the slope of the plot of
log{ -In {1-0 } against log{ t} decreases with v1 as it can be seen in Fig. 8.11. Since the slopes of
the plot which were obtained in the present experiments (Fig. 8.12) were close to unity between
778 and 805 cC, the rate constant v
1
is said to be about 2.0s-1 in this temperature range. Using
the normalised volume fraction, v1 = 2.0 S-1 and 0'3 (Fig. 8.13) which was calculated using
equation 8.9 with the thermodynamic equations discussed in Chapter 7, the initial nucleation
site density No was obtained as a function of the reaction temperature (Fig. 8.14). No increases
slightly with reaction temperature, which may assist in increasing the rate of reaustenitisation
with reaction temperature. Using No, v1 and 0'3' the effect of temperature on the kinetics of
reaustenitisation was calculated (Fig. 8.15).
8.5 CONCLUSIONS
The diffusional growth theory discussed in the previous chapter was applied to overall
reaustenitisation either from mixtures of bainite and austenite where growth of the pre-existing
austenite dominates the reaction, or from a martensitic initial microstructure where the nucle-
ation of austenite is necessary. The analysis leads the following concluding remarks.
1) Reaustenitisation from mixtures of bainite and austenite can be expressed by the parabolic
growth theory of the pre-existing austenite films.
2) Reaustenitisation from the mixture of bainite and austenite seems to proceed by the growth
of the pre-existing austenite films in two directions; lengthening and thickening.
3) Observed time-temperature-transformation curves for reaustenitisation from the mixtures
of bainite and austenite can be reproduced successfully by the growth theory.
4) The formation of austenite including nucleation and growth was modelled using a simple
expression for nucleation and the three-dimensional parabolic rate constant.
5) This analysis showed that the initial nucleation site density of austenite from the marten-
sitic initial microstructure increases with reaction temperature.
REFERENCES
1. J. W. Christian: "The Theory of Transformation in Metals and Alloys", 2nd ed. Part 1,
Pergamon, Oxford, 1981.
2. J-R. Yang: Ph. D. Thesis, University of Cambridge 1988.
3. J. R. Bradley, J. M. Rigsbee and H. 1. Aaronson: Metall. Trans. A, 1977, 8A, 323.
4. J-R. Yang and H. K. D. H. Bhadeshia: "Welding Metallurgy of Structural Steels", TMS-AIME,
Warrendale, Ohio, ed. J. Y. Koo, 1987, 549.
5. D. E. Coates: Metall. Trans., 1973,4, 1077.
236
8.6 APPENDIX
When there are ne interfaces between 0' and I per unit volume at the beginning of the
reaction, and these thicken parabolically, the actual volume fraction of newly formed austenite
is V)' and the volume fraction of newly formed austenite assuming that there is no impingement
is v)'e, the increase in V)' and v)'e; ;. e. dV)' and dv)'e, can be expressed by the increase in the
thickness of austenite dq as follows:
dV)' = ndq
where n is the number of interfaces per unit volume which can grow at time t. Since the thickness
offerrite plates has a distribution, a thinner ferrite plate can be exhausted by growing austenite
earlier than a thicker one. So the number of interfaces which has already been encountered by
the adjacent interface is b.n = ne - n. Although the distribution function of the thickness of
ferrite plates is not known, the simplest assumption is b.n = ;r.ne, where () = 1 - V)'O with the
initial volume fraction of austenite given by V)'O. Therefore the relation between dV)' and dv)'e
is as follows:
which leads to the relation:
Therefore,
V)' v)'e
0- = 1 - exp{ - 0-}
which is the same as the extended volume concept.
237
a) 730
For the minimum detectable reaustenltlsatlon
o
o
~
Q)
~
:]-tU~
Q)
a.
E
Q)
I-
720
710
700
690
680
670
0 1 0 20 30 40
Time, s
b)
730
For 0.05 of reausten Itlsatlon
720
0
0
710
~
Q)
~
:]
700-tU~
Q)
a.
E 690
Q)
I-
680
400300100
670
o 200
Time, S
Fig. 8.1 TTT curves for reaustenitisation from a mixture of bainite and austenite in a Fe-C-Mn-
Ni-Mo alloy (after Yang [2]). a) is for the minimum detectable reaustenitisation and b) for
0.05 of increase in the volume fraction of austenite.
238
800
Calculated
-0- Observed
(.)
7500
C1>
~
:J-ctI~
C1>
Co
E 700
C1>
I-
650
1 1 1 0 100
Heating rate, °e/s
Fig. 8.2 Calculated (line) and observed (plots) reaustenitisation-start temperatures during contin-
uous heating at constant heating rates from a mixture of bainite and austenite in the
Fe-C-Mn-Ni-Mo alloy as a function of heating rate.
:open:detectabla
sol id:d.05%increase
Cl)
N
I
E
U
.•...••.
CXl
10
1I
0
X
"'~
~
.•...••.
600500400
/ s
300
Time, r
starting IIlcr~atructura 0: ~.+ y :
: : 0: ~b+ Y :
---.------,-- ------.------.--
I I I I
I I I I
I I I I
I I I I
I , I I
----r------ --r------r------
, , ,
, , ,
, , ,
, , ,
I I, I
-----r--- -----r------r------r------
I I I I
I I I I
I I I I
I I I t
I I I I I
-----~ -~---_._~----.- ------~------
I I I I
I I I I
" I
" ,
" ,
-----~------~------
120
Cl)
100
N
I
E
U
80
.•...••.
CXl
I
600
X
"'~ 40
~
.•...••.
20
0
Fig. 8.3 Relation between T and Q'~2 plotted (a) III linear scale and (b) III logarithmic scale.
239
o• at 680°C
o at 700°C
--
I,...-c
I-C)
o
-1
o
·2
1 00
1 021 01
1/2
a J.l m/ s slope
o 0.01 0.51
~ o 0.1 0.62 ./
~ 0.3 0.77 /• 0.8 0.99
" • 1.0 1.04
/' ~.-:::::-• 10.0 1.42
-:
/ .~
• ~ A--
,,/ ~I A---------O-
/ ~D---
~~~
~~~~e~b-;;;;;;;;
"
104
- 103
J.J.,JI
I
102,...-C
I
101
100
Time, S
Fig. 8.4 log{ -In{l - e}} V.s. log{t} plots in the Fe-C-Mn-Ni-Mo alloy (after Yang [4]).
105
Time, S
Fig, 8.5 Change in the slope of the plot of log{ -In{l - O} against log{ t} with the parabolic
thickening rate constant Q'l'
240
1.6
Q,)
a.
o
en
1.4
1.2
1.0
0.8
0.6
Fe-0.3C-4.08Cr (wt%) alloy
~O
o
o ~after Yang
Parabolic rate constant,
0.4
.01 . 1 1 10 100
/
1 I 2
JJrn s
Fig. 8.6 Comparison between observed and calculated slopes of the plot oflog{ -In{l- O} against
log{ t}.
Calculated
o Observed
o
D.Vy=O.05
680 o
o 200 400
Time, s
Fig. 8.7 Comparison between calculated (line) and observed (plots [4]) TTT curves for reausteniti-
sation from the mixture of bainite and austenite in the Fe-C-Mn-Ni-Mo alloy.
241
100
at 7700C
at 780
at 785
at 790
at 805
at 815
50105
Vyo=0.3
/ .
./'
./
./
/ ..--------------_ .
/. --•..•.-•..•.. ....."'"
./ ////
/' ..",// -----------------------------
..--...-' -- -----
--=-.:~~::=-:::-.:-----------
1
Q.l..•..
=Q.l.....
'"=eu
•...
Cl
= 0.5Cl
..•..
u
eu
s..•...
Q.l
e
=-Cl
;,
0
1
Time, s
Fig. 8.8 Calculated reaustenitisation behaviour showing the effect of the reaction temperature on
the kinetics of reaustenitisation.
Fc-0.3C-4.08Cr wt.%
----- -------
LlVy=0.05
LlVy=0.10
LlVy=0.20
LlVy=0.50
LlV y=0.65
.~
,'\~
\ ~-
\ .~
" ~.-~
.', "'-.. '--
"', --'-- ---\, '" .--",""'" --._~~-===~---
...... ......_----- ---.-- .
.......•......•.•. ~ ----
- --
.•._--- .•...•. - ..•. - --------------------
----'- ------
780
820
o 10 15
Time, s
Fig. 8.9 Calculated TTT curves for ~ V.., = 0.05,0.10,0.20,0.50 and 0.65 of reaustenitisation from
the bainite + 30% of austenite obtained at 420°C in the Fe-0.3C-4.08Cr wt.% alloy.
242
810
0
800
fr
~
I..
::l
~ 790
I..
Cl.I
Q.
E
Cl.I
t-<
780
Fe-O.3C-4.08Cr wt. %
L!>.V y=O.05
o Observed
-- Calculated
770
o 2 4 8
Time, s
10
Fig. 8.10 Comparison between calculated (line) and observed (plots) TTT curves for AV.,. = 0.05
from the mixture of bainite and austenite obtained at 420 DC in the Fe-0.3C-4.08Cr wt.%
alloy.
1.6
1.4
1.2
Cl)
C-
O- 1.0en
0.8
0.6
0.4
o 2 4
Nucleation
6 8
rate constant,
1 0
- 1
S
1 2
Fig. 8.11 Change in the slope of the plot of log {- In {1 - e}} against log {t} with the nucleation rate
constant VI.
243
- 7- 0 at 7800C- ./- 0 at 7SSOC
- ~ at 790°C
V ./ ..~.
- • at80SOC ./ In ~.~ ~ ~~ V7 f7 .",
.1.-
/}~ f
~
/
V
-:LJ "7\J
" "r/ -/
~
./ L.- P
/ )) r/
/ • / l/ L-'~)
/ £"
./
/
/ rV'
~
2
/]
/ re-O.3 ~-4 .O~ C ~ I (~:.~ol0/
I
1 01
Time, s
Fig. 8.12 log{ -In{l - O} V.s. log{ t} plots for reaustenitisation from martensitic initial microstruc-
ture in the Fe-0.3C-4.08Cr wt.% alloy.
10
5
"!
<:>
I
Vl
E
::i.
;,
lj
2
Fc-O.3C-4.08Cr w\. %
1
740 760 780
Temperature, QC
800 820
Fig. 8.13 Calculated three-dimensional parabolic rate constant 0"3 of austenite formed in a carbon su-
persaturated ferrite matrix whose carbon concentration is the same as the paraequilibrium
car bon concentration at 5000 C.
2<14
Fc-O.3C-4.08Cr wt. %
5
Q
Z
5
o
o
o
o
760 780 800
Temperature,OC
820
Fig. 8.14 Change in the initial nucleation site density for reaustenitisation No from martensitic initial
microstructure with temperature.
1
/
•...
/",
/
/
I
/ at 770°C
I
I at 780
QJ /
I
at 790..... , /
/
/ at 805c:
/
/ 815QJ at..... /
Cl> /
:2 I
/
/
t':l
/
/
/•... /
0
/
/, /
c:
/
/
0 0.5
/
/
..... /
u /
t':l
,
/
/
s..
/
/•... /
I
QJ
/
/
E
I
/
:2
I
/
/
0 / -----------------------------
;> / ---
/ ///"/"/ --- --
--/ ------ ---
a ----- ----- ---
I 5 10 50 100
Time, s
Fig. 8.15 Calculated isothermal reaustenitisation from martensitic initial microstructure.
245
CHAPTER 9
FURTHER WORK
The ultimate goal of this work is to develop a complete computer model which enables
the calculation of microstructural development during austenitisation as a function of alloy
chemistry and starting microstructure. Such a model would form a powerful combination with
an appropriate computer model for the decomposition of austenite to transformation products
such as allotriomorphic ferrite, pearlite, Widmanstatten ferrite, acicular ferrite, bainite and
martensite. A grand scheme like this could be applied to a variety of engineering applications,
and for the optimisation of material and process designs. Although this final goal is still out
of reach, the work presented in this thesis shows that substantial progress can be made for the
formation of austenite from well defined initial microstructures.
Since the general problem of austenite formation is strongly dependent on the initial mi-
crostructure, it is essential to understand the decomposition products of austenite in steels.
The models developed in chapter 2, 3 and 5 illustrate how the details of the mechanism of
the bainite and pearlite transformation allow the formation of austenite to be characterised
theoretically. It is, however, important to take into account the effect of nucleation if these
models are ever to be capable of generally useful application. The diffusional growth theory
discussed in chapter 7 can be combined with any nucleation theories which may be developed
in the future.
Apart from nucleation theory, the major problems highlighted by the present work, and
which need further research are as follows:
1) It is necessary to be able to model the dissolution/precipitation of carbides during austenite
formation and during heating to the test temperature.
2) It is convenient to assume that conditions of local equilibrium or paraequilibrium exist
at the transformation interface. However, non-equilibrium growth in which none of the
elements achieve equality of chemical potential, is also possible and needs to be investi-
gated. In fact, it would be very useful to experimentally investigate the conditions at the
transformation interface using an atom-probe.
246
COMPUTER PROGRAMS
C Program for the calculation of the time required to decarburise a plate of bainitic ferrite and time for
C a certain amount of cementite.
C
C Typical dataset:
C 1 No. of alloys,
C 0.2 (C) 0.2 (Si) 0.3 (Mn) 0.4(Ni) O.5(Mo) 0.6 (Cr) 0.1 (V) wt.%
C
IMPLICIT REAL*8(A-H,K-Y), JNTEGER(1,J2)
DOUBLE PRECISION DIFF(300),CARB(300),C(8),A TIME(40),ATEMP(40)
C HH=PLANCKS CONST JOULES/SEC,KK=BOL TZMANNS CONST.JOULESIDEGREE KELVlN
C D=DIFFUS1VITY OF CARBON IN AUSTENITE
C Z=COORDINA TION OF JNTERSTIAL SITE
C TIME=Time to decarburise a plate of ferrite, seconds
C THICK= thickness of ferrite plate, meters
C PSI==COMPOSmON DEPENDENCE OF DIFFUSION COEFFICIENT
C THET A=NO. C ATOMS/ NO. FE ATOMS
C ACTIV=ACTIVITY OF CARBON IN AUSTENITE
C R=GAS CONSTANT
C X=MOLE FRACTION OF CARBON
C T=ABSOLUTE TEMPERATURE
C SJGMA=SITE EXCLUSION PROBABLITY
C W==CARBONCARBON INTERACTION ENERGY IN AUSTENITE
C IJK1 gives the number of steels, 16 the number of data points
C
HH=6.6262D-34
KK=1.38062D-23
READ(S,*) IJK1
DO 1008 1122= l,IJK 1
CALL OMEGAO(W ,xBAR,C)
Z=12
AS=l.OO+OO
R=8.314320+00
RADIUS=O.O
VMAX=O.O
M1=O.OO
16=12
DO 222 17=1,16
T=SO.00+00*17 +SO.OO+OO
THICK=O.2D-06
CfEMP=T
STRAJN=400.0D+OO
CALL AA3(C,CfEMP ,XMAX,STRAIN)
T=T +273.000+00
ll2--o
ICRIT=O
XAU>HA=XALPH(T)
A6=(XMAX -XBAR)/290.0D+OO
IF(A6 .LT. O.OD+OO)GOTO 222
DASH=(KK*T/HH)*DEXP( -(21230.0D+OO/T)*DEXP( -31.84D+OO)
DO 999 ll=l,300
CARB(l)=XBAR
IF (ll.GT. l)GOTO 1000
Garo 1001
1000 CARB(II)==CARB(JI-1)+A6
IF (CARB(ll) .GT. XMAX) GOTO 1002
1001 X=CARB(ll)
ll2=1l2+1
THET A=XI(AS-X)
ACTIV==CG(X,T,W,R)
ACTIV=DEXP(ACI1V)
DACTIV=DCG(X,T,W,R)
DACTIV=DACTIV*ACTIV
DACTIV=DACTIV* AS/«AS+ THET A)**2)
SIGMA=AS-DEXP« -(W»/(R *n)
PSJ=ACTIV*(AS+Z*«AS+ THET A)/(AS-(AS+Z!2)*THET A+(Z!2)*(AS+7J2)*
&(AS-SIGMA)*THET A*THET A»)+(AS+ THETA)*DACTIV
DIFF(JI)=DASH*PSI
IF(DIFF(1I) .LE. O.OD+OO.AND. ICRIT .EQ. 0) THEN
ICRIT=ll
ENDIF
999 CONTINUE
1002 IF(lCRIT .EQ. 0) GOTO 1012
DO 13331=1,112
IF(I.LT. ICRIT-lO) GOTO 1333
A-I
DIFF(I)=DIFF(ICRIT-10)
1333 CONTINUE
1012 ll3=0
CALL DOlGAF(CARB,DIFF,II2,ANS,ERROR,ll3)
ANS=ANS*I.OD-04/(XMAX-XBAR)
TlME=(THICK *(XBAR-XALPHA)*DSQRT(3.14 IS9D+OO)/( 4.0D+OO
&* ANS**O.S*(XMAX -XBAR»)**2.0D+OO
ATIME(l7)= TIME
ATEMP(I7)=CfEMP
222 CONTINUE
WRITE(6,*)' TEMP td,s to.O\',
&' to.02 to.OS to.lO to.SO·
DO 3 JJJ=1,40
IF(A TEMP(JJJ) .EQ. 0) GOTO 3
TCOO1=TC(ATEMP(JJJ),C,O.OID+OO)
TCOO2=TC(ATEMP(JJJ),C,O.02D+OO)
TCOOS=TC(ATEMP(JJJ),C,O.OSD+OO)
TC010=TC(ATEMP(JJJ),C,O.lOD+OO)
TC050= TC(A TEMP(JJJ),C,O.50D+OO)
WRITE(6,4) ATEMP(JJJ),A TIME(JJJ),TCOO I,TC002,TC005,TCO IO,TCOSO
ATIME(JJJ)=O.OD+OO
ATEMP(JJJ)=O.OD+OO
4 FORMAT(IH ,F8.2,6DI2.4)
3 CUNTINUE
1008 CONTINUE
END
C
C----------------------------------------------------------------
SUBROUTINE OMEGAO(W ,XBAR,D)
C SUBROUTINE TO CALCULATE THE CARBON CARBON INTERACTION ENERGY IN
C AUSTENITE, AS A FUNCTION OF ALLOY COMPOSmON. BASED ON MUCG 18
C THE ANSWER IS IN JOULES PER MOL. **7 ocrOBER 1981**
DOUBLE PRECISION C(8),D(8),W,P(8),B I ,B2,Y(8),TI O,T20,B3,XBAR
INTEGER B5,I,U,B4
READ (5,*) C(1),C(2),C(3),C(4),C(5),C(6),C(7)
D(1)=C(l)
D(2)=C(2)
D(3)=C(3)
D(4)=C(4)
D(5)=C(5)
D(6)=C(6)
D(7)=C(7)
D(8)= 1OO.O-C(1)-C(2 )-C(3 )-C( 4 )-C(5)-C( 6)-C(7)
WRITE(6,*) •.
WRITE(6,261) (C(B4),B4=I,7)
B3=0.OD+OO
C(8 )=C( 1)+C(2)+C(3 )+C( 4)+C(5)+C( 6)+C(7)
C(8)=100.0D+OO-C(8)
C(8)=C(8)/55.84D+OO
C(1)=C(I)/12.0115D+OO
C(2)=C(2)/28.09D+OO
C(3)=C(3)/54.94D+OO
C(4)=C(4)/58.7ID+OO
C(5)=C(5)195.94D+OO
C(6)=C(6)/52.0D+OO
C(7)=C(7)/50.94D-tOO
B 1=C( 1)+C(2)+C(3)+C( 4 )+C(5)+C(6)+C(7)+C(8)
DO 107 U=2,7
Y(U)=C(U)/C(8)
107 CONTINUE
DO 106 U=1,8
C(U)=C(U)/B 1
106 CONTINUE
XBAR=C(I)
XBAR=DINT(lOOOO.OD+OO*XBAR)
XBAR=XBAR/I  oo
B2=O.OD+OO
T1O=Y(2)*(-3)+ Y(3)*2+ Y(4)*12+ Y(5)*(-9)+ Y(6)*( -1)+ Y(7)*( -12)
T20=-3*Y(2)-37.5*Y(3)-6*Y(4)-26*Y(5)-19*Y(6)-44*Y(7)
P(2)=2013.0341 +763.8167*C(2)+4S802.87*C(2)**2-280061.63*C(2)**3
&+ 3.864 D+06 *C(2)* *4- 2.4233D+07 *C(2)* *5+6. 9S4 7D+07*C(2)** 6
P(3)=20 12.067 -1764.09S*C(3)+6287 .S2*C(3)**2-21647 .96*C(3 )**3-
&2.0119D+06*C(3)**4+3.1716D+07*C(3)**S-I.388SD+08*C(3)**6
P(4 )=2006.8017 +2330.2424*C(4)-54915.32*C(4 )**2+ 1.6216D+06*C(4 )**3
A-2
&-2.49680+07*C( 4)**4+ 1.88380+08*C( 4)**S-S.SS31 0+08*C(4 )**6
P(S)=2006.834-2997 .314*C(S)-37906.61*C(S)**2+1.03280+06*C(S)**3
&-1.33060+07*C(S)**4+8.4110+07*C(S)**S-2.08260+08*C(S)**6
P(6)=20 12.367 -9224.26SS*C(6)+ 336S7.8*C(6)**2-S66827 .83 *C(6)* *3
&+8.S 6760+06*C( 6)**4-6. 74820+07 *C( 6)** S +2.083 70+08*C( 6)* *6
P(7)=2011.9996-6247.9118*C(7)+S411.7S66*C(7)**2
&+2S0118.108S*C(7)**3-4.16760+06*C(7)**4
DO 108 U=2,7
B3=B3+P(U)*Y(U)
B2=B2+Y(U)
108 CONTINUE
IF (B2 .EQ. 0.00+00) GOTO 4SS
W=(B3/B2)*4.1S7
GOT0456
455 W=SOS4.0
4S6 WRITE (6,261)(C(B4),B4=I,7)
261 FORMAT (4H C=,F6.4,4H SI=,F6.4,4H MN=,F6.4,
&4H NI=,F6.4,4H MO=,F6.4,4H CR=,F6.4,4H V=,F6.4)
RETURN
END
C
C--- -- ------ -------- ---- ------ ------------- -------- ------- --- ----
SUBROUTINE AA3(CW ,AE3T XEQ,STRAIN)
C AE3T is read in centigrade
C Matrix C represents chemical composition in weight percent
C STRAIN is in Joules per mole
C XEQ is the paraequilibrium carbon concentration in gamma at AE3T
C
DOUBLE PRECISION X,Xl,T,R,A,AI,AFE,AIFE,OAl,OA2,OAl FE,H,Hl,S,SI
INTEGER T1 J,NO,U,B4,J99,C99
DOUBLE PREOSION 011,STRAIN
1OEQ,ETEQ,T1 O,T20XA,AFEQ,AEQ,ETEQ2,TEQ,
lJ,J1,O,Ol,W ,W I,F,TEST,ERROR,T4,XEQ,FPRO
1,C(S),B I,B2,P(7),Y(7),B3,AE3T ,CW(S),Z
Z--o.OO+oo
DO 104 1=1,7
C(I)=CW(I)
Z=Z+CW(I)
104 CONTINUE
CW(S)= 1OO.OO+oo-Z
B3=0.00+00
C(8)=(l OO.O-C(1)-C(2)-C(3)-C(4 )-C(S)-C(6)-(C7»/SS.84
C(1)=C(1)/12.0115
C(2)=C(2)/2S.09
C(3)=C(3)/54.94
C(4)=C(4)/SS.71
C(5)=C(S)/9S.94
C(6)=C(6)/S2.0
C(7)=C(7)/SO.94
B 1=C( 1)+C(2)+C(3 )+C( 4 )+C(S)+C(6)+C(7)+C(8)
DO 107 U=2,7
Y(U)=C(U)/C(8)
107 CONTINUE
DO 106 U=I,7
C(U)=C(U)/B 1
106 CONTINUE
B2=O.O
T1 0= Y(2)*( -3)+ Y(3)*2+ Y(4)*12+ Y(S)*( -9)+ Y(6)*( -1)+ Y(7)*( -12)
T20=-3*Y(2)-37 .S*Y(3)-6*Y(4 )-26*Y(S)-19*Y(6)-44*Y(7)
P(2)=20 13 .0341 +763.8167*C(2)+45802.S7*C(2)**2-280061.63 *C(2)**3
1+3.864 0+06*C(2)* *4- 2.4233 0+07*C(2)** 5+6. 9S4 70+07 *C(2)* *6
P(3)=2012.067 -1764.09S*C(3)+6287 .52 *C(3 )**2-21647 .96*C(3)** 3-
12.01190+06*C(3)**4+3.17160+07*C(3)**S-I.388S0+08*C(3)**6
P(4 )=2006.8017 +2330.2424 *C( 4 )-S491S .32*C(4 )**2+ 1.62160+06*C(4)* *3
1-2.49680+07*C(4)**4+ 1.88380+08*C(4)**S-S.SS31 O+08*C(4)**6
P(S)=2006.834-2997 .314 *C(S )-3 7906.61*C(S)* *2+ 1.03 280+06*C(S )**3
1-1.33060+07*C(S)**4+8.41I O+07*C(S)**S-2.08260+08*C(S)**6
P(6)=2012.367 -9224.26SS*C(6 )+336S7 .8*C(6)**2-S66827 .83*C(6)* *3
1+8.S6 760+06*C( 6)**4 -6.74820+07 *C( 6)** S +2.0837D+08 *C( 6)* *6
P(7)=2011.9996-6247.911S*C(7)+S411.7S66*C(7)**2
1+2S011S.1 OSS*C(7)**3-4.16760+06*C(7)**4
DO 10S U=2,7
B3=B3+P(U)*Y(U)
B2=B2+Y(U)
A-3
108 CONTINUE
IF(B2 .EQ. 0.0) GOTO 455
W=(B3/B2)*4.187
GOT0456
455 W=8054.0
456 CONTINUE
Xl=C(I)
R=8.31432
Wl=48570.0
H=38575.0
S=13.48
201 XEQ=O.1
T=AE3T +273.0
IF (T .LE. 1000) GOTO 20
Hl=105525
SI=45.34521
G0T019
20 Hl=111918
SI=51.44
19 F=ENERGY(T,TlO,T20)+STRAIN
J= I·DEXP( -W/(R*T»
51 DEQ=DSQRT(I-2*(l +2* J)*XEQ+(1 +8* J)*XEQ*XEQ)
TEQ=5*DLOG«I-XEQ)/(l·2*XEQ»
TEQ= TEQ+DLOG«( 1-2*J+(4*J-l )*XEQ-DEQ)/(2* J*(2*XEQ-l »)**6)
TEQ=TEQ*R*T-F
IF (DABS(TEQ) .LT. 1.0) GOTO 50
ETEQ=5*«I/(XEQ-l »+21(1.2*XEQ»
ETEQ2=6*«4*J-l-(0.5/DEQ)*( -2-4*J+2*XEQ+ 16*XEQ* J»/(1-2* J+(4* J
1-1)*XEQ-DEQ»+6*(4*J/(2* J*(2*XEQ-l »)
ETEQ=(ETEQ+ETEQ2)*R *T
XEQ=XEQ TEQ/ETEQ
G0T051
50 IF (XEQ .LT. 0.001) GOTO 2444
2444 CONTINUE
802 RETURN
END
C
C----- ---- ---- ------ ------ -- ------ ----------- ---------- ----------
DOUBLE PRECISION FUNCTION TC(T,C,x)
DOUBLE PRECISION K, Q, R, T, KO,N, C, X, TT
INTEGER I
R=1.987D+OO
Q=8030.0D+00
K0=36.2D+OO*C**0.635
N=O.62D+OO
TT=T +273.0D+OO
K=KO*DEXP( -Q/R/TT)
TC=( -DLOG( 1.0D+OO-X)!K)**( 1.0D+OOIN)
RETURN
END
C------------- ------------------ --------- ---------- --------------
A-4
C Program to calculate the parabolic thickening rate constants
C (I, 2, and 3 dimensional) and the time required to obtain a certain
C amount of cementite ignoring the effect of nucleation of cementite.
C
C 25 October 1988
C
C Typical data set
C 0.1 (C) 0.2 (Si) 0.2 (Mn) 0.2 (Ni) 0.2 (Mo) 0.2 (Cr) 0.1 (V)
C 420 --- temperature in deg. C
C 500
C
C
IMPUCIT REAL+8(A-H,K-Y), INTEGER(I,J,Z)
DOUBLE PRECISION S13AAF,CTEMP,T ,XALPHA,XGAG,T5,DlFF ,BO,B I,B2,A2,DIS
&,DIFFO,T50
EXTERNALS13AAF
CALL OMEGA(W ,XONE)
READ (5,·, END=2) CTEMP
T=CTEMP+273.000
XALPHA={).OOO
XGAG={).25
BO=9.28480435985404817D+00
B 1=6880. 73079777086696D+00
B2=-1780359.64568034932D+OO
DIFF0=4.0D-3·DEXP( -19200.0001l.987DOff)·I.OD-04
DIFF=DF(T)
CALL AL(XONE,XALPHA,XGAG,DIFFO,CTEMP ,ALPHA)
A2= 1.0D+OO/ALPHA **2
DIS=BO+B Iff +B2/Tff
DIS=IO.OOO**DIS
WRITE(6,·) 'DISLOCATION DENS., M··-2 = ',DIS
WRITE(6,·) 'DIFFUSION COEFF.H, M··2/S = ',DIFF
WRITE(6,·) 'DIFFUSION COEFF.E, M**2/S = ',DIFFO
YE= 1.037D-03·XONE
DISO=IO.0D-06
T5=(0.0500·VE/(DIS/DISO·9 .0· ALPHA· ALPHA· ALPHA»*·(2.0/3.0)
T50=(0.500·VE/(DIS/DISO·9.0· ALPHA· ALPHA· ALPHA»**(2.0/3.0)
WRITE(6,·)' TEMP ALPHA 1/ALPHA··2'
&: t(0.05) t(0.5)'
WRITE(6,1229) CTEMP, ALPHA, A2, T5, T50
WRITE(6,·) , ,
1229 FORMAT(F8.1,4D12.4)
G01Ul
2 STOP
END
c····················································· .
C FUNCTION GIVING VO
DOUBLE PRECISION FUNCTION VO(T,ALPHA)
HLEN=IO.OD-04
HWID={).2D-04
BET A=3.0D+OO· ALPHA
VO={).05-«(2·BET NHLEN)+(ALPHNHWlD»·T*·0.5 -
&«BET NHLEN)··2 + 2· ALPHA· ALPHN(HLEN·HWID»·T +
& ALPHA ·BET A·B ETA ·T··l.5/(HLEN·HLEN·HWID»
RETIJRN
END
c····················································· .
C FUNCTION GIVING LFG LN(ACTIVITY) OF CARBON IN AUSTENITE
DOUBLE PRECISION FUNCTION CG(X,T,W,R)
DOUBLE PRECISION J,DG,DUMMY,T J{,W,X
J=I-DEXP(-W/(R·n)
DG=DSQRT( 1-2·( I+2· J)·X+(1 +8· J)*X·X)
DUMMY =5·DLOO« 1-2·X)IX)+6·W I(R ·T)+«38575 .0)-(
&13.48)·T)/(R·T)
CG=DUMMY +DLOG«(DG-I +3·X)/(DG+ 1-3·X»··6)
RETIJRN
END
C····················································· .
C FUNCTION GIVING DIFFERENTIAL OF LN(ACTIVITY) OF CARBON IN AUSTENITE, LFG
C DIFFERENTIAL IS WITH RESPECT TO X
DOUBLE PRECISION FUNCTION DCG(X,T,W,R)
OOUBLE PRECISION J,OO,DDG,X,T,W,R
J=I-DEXP(-W/(R·n)
DG=DSQRT(I-2·(1 +2·J)·X+(1 +8·J)·X·X)
A-5
DDG=(O.5jDG)*( -2-4 *J+ 2*X + 16*J*X)
OCG=-« 1O/( 1-2*X»+(5/X»+6*«Doo+ 3)/(00-1 +3*X
&)-(Doo-3)/(00+ 1-3*X»
REruRN
END
C*******************************************************************************
SUBROUTINE OMEGA(W,XONE)
C SUBROtJTINE TO CALCULATE THE CARBON CARBON INTERACTION ENERGY IN
C AUSTENITE, AS A FUNCTION OF ALWY COMPOSmON. BASED ON MUCG 18
C THE ANSWER IS IN JOULES PER MOL. **7 OCTOBER 1981**
OOUBLE PRECISION C(8),W,P(8),B 1,B2,Y(8),Tl O,T20,B3,XONE
INTEGER B5,I,U,B4
READ (5,*) C(l),C(2),C(3),C(4),C(5),C(6),C(7)
WRITE(6,261) (C(B4),B4=I,7)
B3=0.OD+OO
C(8)=C( 1)+C(2)+C(3)+C( 4)+C(5)+C(6)+C(7)
C(8)=loo.0D+OO-C(8)
C(8)=C(8)/55.84D+OO
C(1)=C(l)/12.0115D+00
C(2)=C(2)/28.09D+OO
C(3)=C(3)/54.94D+OO
C(4)=C(4)/58.7ID+OO
C(5)=C(5)/95.94D+OO
C(6)=C(6)/52.0D+OO
C(7)=C(7)/50.94D+OO
B 1=C( 1)+C(2)+C(3)+C( 4 )+C(5)+C(6)+C(7)+C(8)
00 107 U=2,7
Y(U)=C(U)/C(8)
107 CONI1NUE
00 106 U=I,8
C(u)=c(U)/B 1
106 CONI1NUE
XONE--c(l)
XONE=DINT(IOOOO.OD+OO*XONE)
XONE=XONE/l   oo
B2=O.OD+OO
TIO= Y(2)*( -3)+ Y(3)*2+ Y(4)*12+ Y(5)*( -9)+ Y(6)*(-I)+ Y(7)*(-12)
T20=-3* Y(2)-37 .5*Y(3)-6* Y(4 )-26*Y(5)-19*Y( 6)-44 *Y(7)
P(2)=20 13.0341 + 763.8167*C(2)+45802.87*C(2)**2-28oo61.63*C(2)* *3
&+ 3.864 D+06 *C(2)* *4- 2.4233 D+07 *C(2)* *5+6.954 7D+07*C(2)** 6
P(3)=2012.067 -1764.095*C(3)+6287 .52*C(3)**2-21647 .96*C(3 )**3-
&2.0119D+06*C(3)**4+3.1716D+07*C(3)**5-1.3885D+08*C(3)**6
P(4)=2006.8017+2330.2424*C(4)-54915.32*C(4)**2+ 1.6216D+06*C(4)**3
&-2.4968D+07*C(4)**4+ 1.8838D+08*C( 4 )**5-5.5531 D+08*C(4)**6
P(5)=2OO6.834-2997 .314*C(5)-37906.61 *C(5)**2+1.0328D+06*C(5)**3
&-1.3306D+07*C(5)**4+8.411 D+07*C(5)**5-2.0826D+08*C(5)**6
P(6)=2012.367 -9224.2655*C(6)+33657.8*C(6)**2-566827 .83 *C(6)* *3
&+8.5676D+06*C( 6)**4-6. 7482D+07*C( 6)* *5 +2.083 7D+08*C( 6)* *6
P(7)=20 11.9996-6247 .9118*C(7)+5411.7566*C(7)**2
&+250118.1085*C(7)**3-4.1676D+06*C(7)**4
00 108 U=2,7
B3=B3+P(U)*Y(U)
B2=B2+Y(U)
108 CONI1NUE
IF (B2 .EQ. O.OD+OO)GOTO 455
W=(B3/B2)*4.187
GOT0456
455 W=8054.0
456 WRITE (6,261)(C(B4),B4=I,7)
WRITE(6,loo6)W
1006 FORMATC CARBON-CARBON lNTERACfION ENERGY IN GAMMA, J/MOL=',F9.4)
261 FORMAT (4H C=,F6.4,4H SI=,F6.4,4H MN=,F6.4,
&4H NI=,F6.4,4H MO=,F6.4,4H CR=,F6.4,4H V=,F6.4)
RE11JRN
END
C ******************************************************************************
C FUNCfION GIVING THE EQUILIBRIUM MOL.FRAC. CARBON IN ALPHA
C BASED ON MY PAPER ON FIRST ORDER QUASICHEMICAL THEORY,METSCI
OOUBLE PRECISION FUNCTION XALPH(T)
OOUBLE PRECISION T,crEMP
CTEMP=(T -273.0D+OO)/9oo.0D+00
XALPH=0.1528D-02-0.8816D-02*CTEMP+O.2450D-Ol *CTEMP*CTEMP
&-0.24170-01 *CTEMP*CTEMP*CTEMP+
&0.6966D-02*CTEMP*CTEMP*CTEMP*CTEMP
A-6
REruRN
END
c····················································· .
c····················································· .
SUBROUTINE AL(XGAGXAGAXONE,DIFF,CfEMP,ALPHA)
DOUBLE PRECISION DIFF,ALPHAXAGAXGAGXONE,CfEMP,DUMMYl
&,DER,FUN2,ALPH,DUMMY2,DUMMY3
III=O
ALPHA=DSQRT(DIFF)
DUMMY2=ALPHA
C ABOVE IS A GUESSED VALUE OF ALPHA
4 CALL FUNN(DUMMYl,ALPHAXGAGXONEXAGA,DIFF,CfEMP)
ITI=III+l
C WRlTE(6,20)ALPHA,DUMMYl,DER
46 IF(III.GT. 100) GOTO 3
IF (DABS(DUMMY1) .GT. 0.00010-06) GOTO 2
G0T03
20 FORMAT(3DI2.4)
2 DUMMY2=DUMMYl
ALPH=ALPHA ·1. 0ooooo 1D+OO
CALL FUNN(FUN2,ALPHXGAGXONEXAGA,DIFF,CfEMP)
DER=(DUMMY l-FUN2Y(ALPHA-ALPH)
ALPHA=ALPHA-(DUMMY I/DER)·O.5DO
GOT04
3 WRITE(6,23)CfEMP, ALPHA,DUMMYl
WRITE(6,·) 'ALPHNSQRT.DIFF = ',(ALPHA/DSQRT(DIFF)
23 FORMATC CTEMP=',F8.2,' ALPHA (M PER SEC··.5)=',DI2.5,
&' DUMMY1=',DI2.5)
WRITE(6,·) 'Fi=', «XGAG-XAGAY(XONE-XAGA»
REruRN
END
C····················································· .
SUBROUTINE FUNN(FUN ,ALPHAXGAGXONEXAGA,DIFF,CfEMP)
C 'FUN' COMPUTES EQ.2 OF KINSMAN AND AARONSONS TRANSFDRMA nON AND
C HARDENABILITY PAPER. DIFF~rNTEGRA TED AVERAGE DIFFUSNITY OF C IN
C GAMMA, ALPHA = PARABOLIC RATE CONSTANT.
DOUBLE PRECISION DIFF,ALPHA,XAGAXGAGXONE,FUN
FUN=DSQRT(3.141593D+OO/DIFF)/2.OD+OO
FUN=FUN· ALPHA ·DEXP(ALPHA· ALPHN(4.0D+OO·DIFF»
FUN=FUN·( 1.OD+OO-DERF(ALPHN(2.0D+OO· DSQRT(DIFF»»
FUN=(XGAG-XAGAY(XONE-XAGA)-FUN
REruRN
END
C····················································· .
SUBROUTINE FUNNl(FUN,ALPHAXGAGXONEXAGA,DIFF,crEMP)
DOUBLE PRECISION DIFF,ALPHAXAGAXGAGXONE,FUN
&,S13AAF
EXTERNAL S13AAF
FUN=ALPHA· ALPHN4.0D+OO/DIFF·DEXP(ALPHA· ALPHN( 4.0D+OO·DIFF)
FUN=FUN· S 13AAF(ALPHA· ALPHN( 4.0D+OO·DIFF), 0)
FUN=(XGAG-XAGAY(XONE-XAGA)-FUN
RETIJRN
END
C····················································· .
SUBROUTINE FUNN2(FUN,ALPHAXGAGXONEXAGA,DIFF,CTEMP)
DOUBLE PRECISION DIFF,ALPHAXAGAXGAGXONE,FUN
FUN=ALPHA· ALPHN(2.0D+OO·DIFF)·( 1.OD+OO-DSQRT(3 .141592D+OO/DIFF)·
&ALPHA/2.0D+OO·DEXP(ALPHA· ALPHN( 4.0D+OO·DIFF»
&.( 1.0D+OO-DERF(ALPHN(2.0D+OO·DSQRT(DIFF»»)
FUN=(XGAG-XAGAY(XONE-XAGA)-FUN
RETIJRN
END
C····················································· .
DOUBLE PRECISION FUNCTION DF(KTEMP)
DOUBLE PRECISION R,KTEMP 'pHI,DOTO,DTT ,F
R=8.3143D+OO
PHI= 1.0D+OO-I.0/(0.5D+OO·DEXP(7 .2D+03·4.184/(R ·KTEMP»
&·DEXP(4.4D+OO)+ 1.OD+OO)
DOT0=3.3D-07·DEXP( -19 .3D+03·4.184D+OO/(R ·KTEMP»
DTT =3 .0D-04 ·DEXP( -14. 7D+03·4.184D+oo/(R ·KTEMP»
F=0.86D+OO
DF=PHI·DOTO+(1.0D+OO-PHI)·F*DTT +(1.0D+OO-PHI)
&·(l.OD+OO-F)·DOTO
C DIFFUSION OF CARBON IN FERRITE, M·M/S
A-7
C MCLELLAN ET AL., TRANS. MET. SOC. AIME, VOL.233 (1965) 1938
C R = UNIVERSAL GAS CONSTANT. J/MOUK
C KTEMP = ABSOLlITE TEMPERA TIJRE
RElURN
END
C
C***********************************************************************
A-8
C fTVSCLR PROGRAM=%H% OAT A=&O NAG
C Program to calculate cementite precipitation from a supersaturated
C ferrite based on a nucleation and growth theory. Nucleation occurs
C on the dislocation. The growth of cementite plate is calculated
C using the one-dimensional parabolic thickening rate constant (ALPHA).
C
IMPLICIT REAL*8(A-H,K-Z), INTEGER(l,J)
DOUBLE PRECISION X(200),Y(200)
C
C------- ------ -- ---- ------ ----- -------- ------ ---------- ----------
C Initial carbon mole fraction in ferrite (XMOL) and the maximum
C possible volume fraction of cementite (YE).
C
XMOL--o.0183
YE: 1.006450+OO*4.00+OO*XMOl)( 1.00+OO-4.00+00*XMOL)
C
C --- ---- ---- ------ ---- ------------ ----------------------- --------
C ALPHA: one demensional parabolic rate constant, m/secuO.5
COO: activation energy of the nucleation of cementite, J/mole
C
C Typical data set
C 450:Temperature deg.C, 0.12340-06: Rate constant m/secu5
C 500 0.34560-06
C
1 READ(5,*,END=333) T,ALPHA
00=18  oo.00+00
WRITE(6,*) 'G* = ',00
R=8.3140+OO
TT=T+273.00+00
C
C----- ------ -- ---- ------ ---- ------ ------- -------- ------------ ----
C Calculation of the dislocation density in ferrite.
C
Al =9.284804359854048170+00
A2=6880.730797770866960+00
A3=-1780359.645680349320+00
ROU=Al +A2/TT +A.31TT1TT
ROU= 1O.OO+oouROU
RO=((2.260+OO-6.40-03*TT +4.60-06*TT*TT)*VE/l 000.0)
& U(1.0/3.0)* 1.00-06
C
C----------------------------------------------------------------
C NO : Number of particles per unit volume
C
NO=YE/( 4.0/3.0*3.141593*RO*RO*RO)
C
C ----- ---- ---- ---- ---- ------ ---- --------- -------- ---------- ------
C Coefficients in the nucleation function.
C see Christian "The Theory of Transformations in Metals and Alloys"
C
A=2.8664 0-1 0*3 .00+oou0.5/2.00+00
K=(R OUt A)u(l.0/3.0)*2.080+ 1O*TT*OEXP( -OGIRITT)*ROU
VNC=(ROU/ A)U( 1.0/3.0)*ROU
C
C ----- ------ ---- -- ------ ---- ------ --------- ------ ---------- ------
WRITE(6,98) T,YE,RMAX,ALPHA,oo,ROU,NO,VNC,K
98 FORMAT(f
&' TEMPERATURE, OEG = ',FIO.4/
&' EQUILffiRIUM V-THETA = ',012.4/
&' R-MAX (FROM V-THETA), urn = ',012.4/
&' ALPHA, M/SEcuO.5 = ',012.4/
&' OELTA-G, J/MOLE = ',012.4/
&' DISLOCATION OENSITY, 1/M**2 = ',012.4/
&' EATA (NO. OF PARTICLE),Mu-3= ',012.4/
&' Nv= ROU*N = ',012.4/
&' NUCLEATION RATE I, = ',012.4)
C
C------- ---- ---- ---- ------ ---- --------------- -------- -------- ----
C Calculation of time required for 0.05 cementite precipitation
C (TIME), and the total number of particles at the end of the
C reaction (mOT).
C
Z=-OLOG(l.00+00-0.050+OO)
TIME=(Z*5 .0/18.0* VE/K/ ALPHA u3 .0)U(2.0/5.0)
A-9
30 WRITE(6,99) TIME
99 FORMAT(lH ,'TIME FOR 0.05 PRECIPITATION = ',012.4)
TC=(30.0*5 .0/18.0*YE/KI ALPHA **3.0)** (2.0/5 .0)
DTC=TC/200.0
00 20 1=1,200
X(I)=DTC*(I-l)
Y(I)=DEXP( -18.0/5.0*K* ALPHA **3.0/YE*X(I)**(5.0/2.0»
20 CONTINUE
II3=O
CALL DOIGAF(X,Y,200,ANS,ERRORJI3)
N1UT=K*ANS
WRITE(6,911) NTOT
911 FORMAT(lH ,'NUMBER OF PARTICLE, M**-3 = ',DI2.4)
GOTOl
333 STOP
END
C
C----- ---- ---- ------ ------ -------- ------- ------ ---- ------ --------
A-lO
C FfYSCLR PROORAM=%H% NAG DATA=&DATA
C H. K. D. H. Bhadcshia and M. Takahashi
C
C Program to calculate the growth rate based on a plate growth theory
C and the time required to obtain 0.05 cementite precipitation.
C
C Typical dataset:
C 1 40 No. of centigrade-mole fraction carbon pairs. No. of alloys
C 0.2 (C) 0.2 (Si) 0.3 (Mn) O.4(Ni) O.5(Mo) 0.6 (Cr) 0.1 (V) wt.%
C 600 -------- temperature in deg.C
C
IMPUCIT REAL*8(A-H,K- Y), INTEGER(l,J,Z)
DOUBLE PRECISION DIFF(3OO),CARB(3OO)
&, BO,B 1,B2,T5,DIS
C
C XMAXR IS THE EQUILffiRIUM CONC AT PLATE TIP OF RADIUS R. IN GAMl\1A
C HH=PLANCKS CONST JOULES/SEC,KK=BOL TZMANNS CONST.JOULESlDEGREE KELVIN
C
HH=6.6262D-34
KK=1.38062D-23
READ(5,·)I6,IJKl
DO 1008 1122= I,IJK 1
CALL OMEGA(W )(BAR)
Z=12
A5=1.0D-tOO
R=8.31432D-tOO
C
C D=DIFFUSIVITY OF CARBON IN AUSTENITE
C Z=COORDINA TION OF INTERSTIAL SITE
C R=GAS CONSTANT
C X=MOLE FRACTION OF CARBON
C T=ABSOLUTE TEMPERATURE
C SIGMA=SITE EXCLUSION PROBABLITY
C W=CARBON CARBON INTERACTION ENERGY IN AUSTENITE
C
RADIUS=O.O
VMAX=O.O
Ml=O.oo
WRITE(6,IOO9)
1009 FORMAT(························ ····/5H
DO 22217=1,16
READ(5,·)T
XTH=O.25
XMAX=XTH
CfEMP=T
T=T +273.000+00
112=0
XAlPHA=XALPH(f)
WRITE(6.1 005)T ,CfEMP ,xBAR.XALPHA
CALL RRAD(RADIUS,xMAX,xALPHA)(BAR.T ,R,xMAXR,W)
A6=(XM AXR -XBAR)(29D.OD-tOO
1005 FORMAT(' ABSOLUTE TEMPERATURE. DEGREES KELVIN =',F8.1/
&' TEMPERATURE IN DEGREES CENTIGRADE =·.F8.I/
&. MOL FRAC CARBON IN ALLOY = ·,F8.4/
&. EQUTIlBRIUM MOL FRAC OF C IN FERRITE=·,D12.4)
DlFFO=DF(f)
CALL YEL(YMAX,DIFFO.RADIUSXMAX,xBAR,xALPHA)
CALL YEL3(YMAX.DIFFO,RADIUS,xMAX)(BAR,xALPHA)
CALL YEL2(YMAX.DIFFO,RADIUSXMAX)(BAR,xALPHA)
CALL YEL4(YMAX,DIFFO,RADIUS,xMAX)(BAR,xALPHA)
B0=9.05839817855180618DO
B 1=7528.72136707062509DO
B2=-21 07436.07233 760227DO
DIS=BO+B lIT +B2{f/T
DIS=10.0DO**DIS
WRlTE(6,902) DIS
902 FORMAT(lH ,'DISLOCATION DENS., M"-2 = ·.DI5.6)
VE=I.00645D+00·4.0D-tOO·XBARJ(I.OD-tOO-4.0D-tOO*XBAR)
RO=( (2.26D+00-6.4D-03*T +4.6D-06*T*T)* VEl 1000.0)
& **(1.0/3.0)*\.OD-06
NO: VE/( 4.0/3.0*3 .14159265D+OO*RO*RO* RO)/DIS
NfOT=NO*DIS
ASP=I5.00+00
WRITE(6,903) YE
A-I1
903 FORMAT(1H ,'MAXIMUM VOLUME FRACfION OF CEMENTITE = ',015.6)
WRlTE(6,904) ASP
904 FORMAT(lH ,'ASPECT RATIO OF CEMENTITE = ',FI0.4)
WRITE(6,905) RO
905 FORMAT(IH ,'AVERAGE PARTICLE SIZE, M = ',015.6)
WRlTE(6,906) NTOT
906 FORMAT(lH ,'NUMBER OF PARTICLES, M**-3= ',015.6)
T5=(VE*0.050+00/(3.141592650+OO
& *VMAX*VMAX*VMAX*DlS/ ASP*NO»**( 1.00+00/3.00+(0)
WRITE(6,901) VMAX,T5
WRITE(6,*) , ,
WRITE(6,*)' 0
901 FORMAT(IH ,'MAX. VELOCITY, MlSEC = ',015.41
&' TIME FOR 0.05 PRECIPITATION OF CEMENTITE, SEC = ',015.4)
222 CONTINUE
lOOS CONTINUE
END
C
C--- -- ------ --- ------- ------ ---- --------- ------ -------- ------- ---
SUBROUTINE OMEGA(W )(BAR)
C SUBROUTINE TO CALCULATE THE CARBON CARBON INTERACTION ENERGY IN
C AUSTENlTE, AS A FUNCTION OF ALLOY COMPOSITION. BASED ON .MUCGI8
C THE ANSWER IS IN JOULES PER MOL. **7 OCTOBER 1981**
DOUBLE PRECISION C(S),W,P(8),B I,B2,Y(8),Tl O,T20,B3,XBAR
INTEGER BS,I,U,B4
READ (S,*) C( 1),C(2),C(3),C(4 ),C(S),C(6),C(7)
WRlTE(6,261) (C(B4),B4=I,7)
B3=0.00+OO
C(S)=C( 1)+C(2)+C(3)+C( 4)+C(S)+C(6)+C(7)
C(S)= l00.00+OO-C(8)
C(S)=C(S)/SS.840+OO
C(l)=C(l )/12.011SO+OO
C(2)=C(2)!2S.090+OO
C(3)=C(3)/S4.940+OO
C(4)=C(4)/SS.71D+OO
c(S)=C(S)/9S.940+OO
C(6)=C(6)/S2.00+OO
C(7)=C(7)/SO.940+OO
B 1=C( 1)+C(2)+C(3 )+C( 4 )+C(S)+C(6)+C(7)+C(8)
DO 107 U=2,7
Y(U)=C(U)/C(S)
107 CONTINUE
DO 106 U=I,S
C(U)=C(U)/B 1
106 CONTINUE
XBAR=C(l)
XBAR=OINT(I0000.00+OO*XBAR)
XBAR=XBAR/l   oo
B2=O.OO+OO
T1 0= Y(2)*( -3)+ Y(3)*2+ Y(4)*12+ Y(S)*( -9)+ Y(6)*( -1)+ Y(7)*( -12)
T20=-3*Y(2)-37 .5*Y(3)-6*Y( 4 )-26*Y(S)-19*Y( 6)-44 *Y(7)
P(2)=20 13 .0341 +763.S167*C(2)+4S802.87*C(2)**2-28OO61.63 *C(2)* *3
&+ 3 .8640+06*C(2)* *4- 2.42330+07 *C(2)* *S+6. 954 70+07*C(2)** 6
P(3)=2012.067-1764.09S*C(3)+6287 .S2*C(3)**2-21647 .96*C(3)**3-
&2.01190+06*C(3)**4+3.17160+07*C(3)**S-I.38850+08*C(3)**6
P(4 )=2006.8017+2330.2424*C(4)-S491S.32*C(4 )**2+ 1.62160+06*C(4 )**3
&-2.49680+07*C( 4)**4+ 1.88380+08*C( 4)**S-5 .SS31 0+08*C( 4)**6
P(S)=2OO6.834-2997 .314*C(S)-37906.61*C(S)**2+ 1.032S0+06*C(S)**3
&-1.33060+07*C(S)**4+8.411 0+07*C(S)**S-2.08260+08*C(5)**6
P(6)=2012.367 -9224.26SS*C(6)+336S7 .8*C(6)* *2-566S27 .83 *C(6)**3
&+S.56760+06*C( 6)* *4-6.7 4820+07*C( 6)** S +2.083 70+08*C(6)* *6
P(7)=20 11.9996-6247 .9118*C(7)+5411.7S66*C(7)**2
&+2S0 IIS.1 OSS*C(7)**3-4.16760+06*C(7)**4
DO 108 U=2,7
B3=B3+P(U)*Y(U)
B2=B2+Y(U)
10S CONTINUE
IF (B2 .EQ. 0.00+(0) GOTO 4SS
W=(B3/B2)*4.1S7
GOT04S6
4S5 W=SOS4.0
456 WRITE (6,261)(C(B4),B4=I,7)
WRITE(6,IOO6)W
1006 FORMATC CARBON·CARBON INTERACTION ENERGY IN GAMMA, J/MOL=',F9.4)
A-12
261 FORMAT (4H C=,F7.4,4H SI=,F7.4,4H MN=,F7.4,
&4H NI=,F7.4,4H MO=,F7.4,4H CR=,F7.4,4H V=,F7.4)
RETIJRN
END
C
C----- -------- ------------------ ------- -------- ------------------
SUBROUTINE RRAD(RADIUS,XMAX,XALPHA,XBAR,T,R,XMAXR,W)
DOUBLE PRECISION RADIUS,XMAX,XBAR,T,R,SIG,MOLVOL,XMAXR
&,XALPHA,RAD,OMEGA,CAPCON,EPSI
SIG=O.7
C SIG=INTERFAClAL ENERGY, JOULES PER METRE SQUARED
MOL VOL=5.8033D-06*( 1.0D+OO+3.549D-05*(T -298.00+00»
C MOLVOL = MOLAR VOLUME OF FERRITE
C RADIUS IS THE CRITICAL RADIUS fDR ZERO GROWTH
C RAD IS THE RATIO OF THE ACTUAL RADIUS TO THE CRITICAL RADIUS
EPSI=XALPHA *DF(T)
C CAPCON=(SIG*MOL VOl)(R *T»*« 1.0D+OO-XALPHA)/(XMAX -XALPHA»
C &/EPSI
CAPCON=(SIG*MOL VOl)(R *T)Y(XMAX-XALPHA)
RADIUS=CAPCON*XAl1'HN(XMAX-XALPHA)
OMEGA=(XALPHA-XBARY(XALPHA-XMAX)
RAD=0.2026D+OI-0.1917D+O 1*OMEGA-0.8953D+OO*OMEGA *OMEGA
&+0.36700+01 *OMEGA*OMEGA *OMEGA-0.2519D+O 1*OMEGA *OMEGA *OMEGA *OMEGA
RAD=10.00D+00**RAD
RAD=RADIUS*RAD
XMAXR=XMAX*( 1.0D+OO+(CAPCONjRAD»
WRITE(6,1 )SIG ,MOL VOL,RADIUS,XMAXR,CAPCON ,EPSI
FORMAT( INTERFACIAL ENERGY=',F8.4,' JOULES/METERS SQUARED'!
&' MOLAR VOLUME OF CEMENTITE(METERS CUBED PER MOL)=',015.6/
&' GIBBS THOMPSON CRITICAL RADIUS(METERS)=',DI5.6/
&' EQUILIBRIUM CONC AT PLATE TIP, MOL FRAC, XMAXR=',DlS.6/
&' CAPILLARITY CONSTANT CAPCON,=',DI5.6/
&' NON-IDEALITY PARAMETER EPSI=',D1S,6)
RETURN
END
C
C----- ------ ---- ---- ---- ------ ---- ------- ---------- ---------- ----
SUBROUTINE RRA01(RADIUS,XMAX,XALPHA,XBAR,T ,R,XMAXR,W)
DOUBLE PRECISION RADIUS,XMAX,XBAR,T,R,SIG,MOL VOL,XMAXR
&,XALPHA,RAD,OMEGA,CAPCON,EPSI
C RADIUS IS THE CRITICAL RADIUS fDR ZERO GROWTH
XMAXR=XMAX
RADIUS=6.5D-08
WRITE(6,9) XMAX,RADIUS
9 FORMAT(l H ,'CARBON IN CEMENTITE, MOLE FRACTION = ',FI0.4/
&'CRmCAL RADIUS, M = ',015,4)
RETURN
END
C
C--- -- ---- ---- ---- ------ ------ ------ ----- ---------- -------- ------
SUBROUTINE VEL(VMAX,ANS,RADIUS,XMAX,XBAR,XALPHA)
DOUBLE PRECISION OMEGA,ANS,RADIUS,P, VMAX,XMAX,XBAR
&,XALPHA
OMEGA=(XALPHA-XBARY(XALPHA-XMAX)
P=( 1.0D+OO/8.0D+OO)*(OMEGN( 1.0D+OO-OMEGA»
VMAX=(ANS* 1,00-04 )*P/RADIUS
WRITE(6,1 )VMAX
FORMAT(' MAXIMUM GROWTH RATE (EQ.9,MET.TRANS,V6A,1975,P7,=',
&DlO.4,'METERS PER SECOND')
RETURN
END
C
C--- ---- -- ---- ---- ---- ---- ---- --------- -------- ------ ------------
SUBROUTINE VEL2(VMAX,ANS,RADIUS,XMAX,XBAR,XALPHA)
DOUBLE PRECISION OMEGA,ANS,RADIUS,P,VMAX,XMAX,XBAR
&,XALPHA
OMEGA=(XALPHA-XBARY(XALPHA-XMAX)
P=( I,OD+OO/4.0D+OO)*(OMEGN( 1.0D+OO-OMEGA»
VMAX=«1 0.00+00)**( -2.50+00*( I,OD+OO-OMEGA»)
&*(ANS* 1.0D-04)*PjRADIUS
WRlTE(6,1 )VMAX
FORMAT(' MAXIMUM GROWTH RATE (EQ.14,MET.TRANS,V6A,1975,P7,=',
&01 0.4,'METERS PER SECOND')
RETURN
A-13
END
C
C--------- ---- -------- ------------ --------- ------------ ----------
SUBROUTINE VEL3(VMAX,ANS,RADIUS,xMAXXBARXALPHA)
DOUBLE PRECISION OMEGA,ANS,RADIUS,P,VMAX,xMAX,xBAR
&XALPHA
OMEGA=(XALPHA-XBARY(XALPHA-XMAX)
P=«OMEGN(I.OD+00-(2.0D+00/3.14159D+00)*OMEGA-
&( 1.0D+OO/(2.0D+OO*3.14159D+OO»*(OMEGA *OMEGA»)**3)
P=P*27.0D+OO/(256.0D+00*3.14159D+00)
VMAX=(ANS* 1.0D-04)*P/RADIUS
WRITE(6,1 )VMAX
FORMAT(' MAXIMUM GROWTH RATE (EQ.13,MET.TRANS,V6A,1975,P7,=',
&DlO.4,'METERS PER SECOND')
RETURN
END
C
C----- ---- ---------- -- -------------- ----- -------------- ----------
SUBROUTINE VEL4(VMAX,ANS,RADIUS,xMAXXBAR,xALPHA)
DOUBLE PRECISION OMEGA,ANS,RADIUS,PECLET XMAXXBAR,XALPHA,PI
&,VDUM,DUMMY,VMAX,DD,S2,RAD,DDD
INTEGER Il,I2,I3
WRITE(6,4)
4 FORMAT(f DIFFUSION CONTROLLED GROWTH, TRIVEDI ANALYSIS)
12=100
DD=O.OI
VDUM=DD*VMAX
VMAX=O.5*VMAX
ANS=ANS* 1.0D-04
WRITE(6,44)
44 FORMAT(' OMEGA DUMMY VMAX(M/S) PECLET RAD(M)')
OMEGA=(XALPHA-XBARY(XALPHA-XMAX)
RAD=0.2026D+OI -0. I9170+0 1*OMEGA-0.8953D+OO*OMEGA *OMEGA
&+O.3670D+OI *OMEGA *OMEGA *OMEGA-0.2519D+Ol *OMEGA *OMEGA *OMEGA *OMEGA
RAD=IO.OD+OO**RAD
RAD=RADIUS*RAD
DO 1 Il=I,I2
PECLET =VMAX*RAD/(2.0D+OO* ANS)
S2=-1.0730 19925D+00*(DLOG 10(PECLET»-0.273767575 D+OO
S2=IO.OD+OO**S2
PI=3.14159D+00
DUMMY =(DSQRT(PI*PECLET»*(DEXP(PECLET»*(DERFC(DSQRT(pECLET»)
&*(1.0D+OO+{RADIUS/RAD)*OMEGA *S2)
DDD=DABS(1 oo.OD+OO*(DUMMY -OMEGA»
IF (DDD .GT. 0.10+00) GOTO 55
WRITE(6,2)OMEGA,DUMMY,VMAX,PECLET ,RAD
2 FORMAT(7DI2.4)
IF (DDD .LT. 0.010+00) GOTO 56
55 VMAX=VMAX+VDUM
1 CONTINUE
56 RETURN
END
C
C----- -------- ---- ---------- ------ --------- ----------------------
C FUNCTION GIVING THE EQUILIDRIUM MOL.FRAC. CARBON IN ALPHA
C BASED ON THE PAPER ON FIRST ORDER QUASICHEMICAL THEROY,METSCI
DOUBLE PRECISION FUNCTION XALPH(T)
DOUBLE PRECISION T,CTEMP
CTEMP=(T -273.0D+OO)/9oo.0D+00
XALPH=0.1528D-02-0.8816D-02*CTEMP+O.2450D-0 1*CTEMP*CTEMP
&-0.2417D-0 1*CTEMP*CTEMP*CTEMP+
&0.6966D-02*CTEMP*CTEMP*CTEMP*CTEMP
RETURN
END
C
C----------------------------------------------------------------
DOUBLE PRECISION FUNCTION DF(KTEMP)
DOUBLE PRECISION R,KTEMP 'pHI,DOTO,DTT,F
R=8.3143D+OO
PHI=I.0D+OO-l.0/(0.5D+OO*DEXP(7.2D+03*4.184/(R *KTEMP»
&*DEXP(4.4D+OO)+ 1.0D+OO)
DOTO=3.3D-07*DEXP( -19.3D+03*4.184D+OO/(R *KTEMP»
DTT =3 .0D-04 *DEXP( -14. 70+03*4. 184D+OO/(R *KTEMP»
F=0.86D+OO
A-14
DF=PHI*DOTO+(I.0D+OO-PHI)*F*DIT +(1.0D+OO-PHI)
&*(I.0D+OO-F)*DOTO
C DIFFUSION OF CARBON IN FERRITE, M*MjS
C MCLELLAN ET AL., TRANS. MET. SOC. AlME, VOL.233 (1965) 1938
C R = UNIVERSAL GAS CONSTANT, J/MOl)K
C KTEMP = ABSOLUTE TEMPERA nJRE
RE1URN
END
C
C----------------------------------------------------------------
A-I5
C PROGRAM lD CALCULATE THE GROWTII RATE OF CEMENTITE ASSUMlNG
C ONE DIMENSIONAL PARABOUC TIIlCKENING OF CEMENTITE FROM AUSTENITE.
C KfRKALDY'S METHOD IS USED TO CALCULATE EQUILIBRIUM CONDmONS.
C
C SEE HASHIGUCHl ET AL., CALPHAD VOL.S NO.2(l984) PP173-1S6
C
C THE LINEAR APPROXIMA nON IS USED FOR THE CHEMICAL PROFllE IN
C AUSTENITE AHEAD OF THE INTERFACE.
C
C M. TAKAHASHl,19.1.1990
C
IMPLICIT REAL*S (A-H,K-Z)
OOUBLE PRECISION C(S),Y(S),K(S),X(S),CO(S),WO(S),CB(3),DUM(3)
&,G(S ),E(S ,S),WW (S),G2(S) ,LFX (2,S), GM(S) ,AC( S)
COMMONrrRANS/CO,WO
COMMON/COEFF/G,E,WW,GFE3C
COMMON/HILI)G2,LFX,LCV,GM,AC,DDG,GFEC
READ(S")ID
IID=O
27 IID=IID+I
IF (lID .GT. ID) GOlD 26
READ(S,*) (WO(l)J=I,7)
READ(S") n,OCf,TF
READ(S") SECS,DSEC,SECFJN
CALL CONV(l,WO,CO)
IM=O
WRITE(6") 'INITIAL COMPOsmONS'
WRITE(6,19S)
WRITE(6,199) (WO(l)J=I,7)
WRITE(6,199) (CO(l),I=I,7)
199 FORMAT(lH ,7F9.S)
19S FORMAT(lH,' C SI MN NI'
&,' CR MO CU ')
WRITE(6")
CALLOMEGA(W)
CC=O.OD+OO
00 1 1=1,7
CC=CC+CO(l)
CONTINUE
CO(S)=1.0D+OO-CC
KK=O.OD+OO
0021=2,7
K(l)=CO(I)/CO(S)
KK=KK+K(l)
IF (CO(l) .EQ. 0.00+00) GOTO 2
IM=I
2 OONTINUE
YY=O.OD+OO
0031=2,7
Y(I)=K(l)/( 1.0D+OO+KK)
YY=YY+Y(l)
3 OONTINUE
Y(S)=l.OD+OO-YY
crEMP=TI
20 CTEMP=CTEMP+OCf
T=CTEMP+273.0D+OO
IF (CTEMP .GT. TF) GOTO 2S
C
C THERMODYNAMIC PARAMETERS
C IN ORDER OF SI,MN,NI,CR,MO,CU.
C
DG=461S0.0D+OO/3.0·19.205D+OO*T/3.0
GFE3C=I.332D+04·64.71S*T +7.4S1 *T*DLOG(f)-OO
G(2)=2SS3S.0D+OO·OO
G(3)=-14263 .OD+OO+1O.OD+OO*T-OO
C G(3)=-13S32.0D+OO-OO
G(4 )=2033S.0D+OO-2.36SD+OO*T-DG
G(S)=-2441S.0D+OO+ 16.61 D+OO*T-2.749D+OO*T*DLOG(T)-DG
G(6)=-19644.0D+OO-0.62S*T-DG
G(7)=2SS3S.0D+OO-OO
C
E( 1,1)=4.7SS9D+OO+S066.0D+oorr
C E(l,2)=4.84D+OO-7370.0D+oorr
E(l,2)=1479S.0D+oorr
A-16
CE(l,3)=-4811.0D+oorr
E(I ,4)=-2.2D+OO+ 7600.0D+oorr
E( 1,5)=24.4-38400.0D+oorr
E(l,6)=3.855D+OO-17870.0D+OOrr
E(l,7)=4200.0D+oorr
C
E(2,2)=26048.0D+oorr
E(3,3)=2.406D+OO-175.6D+OOrr
C E(3,3)=O.2D+OO
E(4,4 )=-721. 7D+oorr
E(5,5)= 7 .655D+OO-3154.0D+OOrr -0.661 D+OO*DLOG(T)
E(6,6)=-2330.0D+OOrr
E(7 ,7)=-0.161 D+00-7834.0D+oorr
WW(2)=0.OD+OO
WW(3)=8351.0D+OO-15.188D+OO*T
WW(4)=0.OD+OO
WW(5)=1791.0D+OO
WW(6)=0.OD+OO
WW(7)=0.OD+OO
C------- ---- -- ------ ---- ------ ------ ----- ------ ------ -------- ----
G2(2)=123000D+00
G2(3)=-48500D+oo
G2(4)=46000D+OO
G2(5)=-251160D+OO+118.0D+OO*T
G2(6)=-267200D+OO
G2(7)=-46000 .OD+OO+55. OD+OO*T
LFX( 1,2)=-1 08280.0D+00
LFX( 1,3)= 730.0D+OO-l O.OD+OO*T
LFX(l,4)=-14600.0D+OO
LFX(l,5)=1311O.0D+OO-31.82D+OO*T +2.748D+OO*T*DLOG(T)
LFX(l,6)=9686.0D+OO
LFX(l, 7)=49752.0D+OO-9 .431 D+OO*T
LFX(2,2)=0.OD+OO
LFX(2,3)=0.OD+OO
LFX(2,4 )=8800.0D+OO
LFX(2,5)=0.0D+OO
LFX(2,6)=0.OD+OO
LFX(2,7)=-8594.0D+OO+ 5.05D+OO*T
LCV=-21058.0D+OO-l1.581D+OO*T
C GM(2)=28535.0D+OO
GM(2)=O.0D+OO
GM(3)=-13532.0D+OO
GM(4)= 14540. 0D+OO-2.367D+OO*T
GM(5)=-850.0D+OO-14.58D+OO*T
GM(6)=502.0D+OO
GM(7)=28535.0D+OO
AC(2)=O.OD+OO
AC(3)=8350.0D+OO-15.2D+OO*T
AC(4)=O.0D+OO
AC(5)=1790.0D+OO
AC(6)=O.0D+OO
AC(7)=O.0D+OO
GFEC=1.332D+04-64.718*T +7.481 *T*DLOG(T)
DOO=46150.0D+OO-19 .221 D+OO*T
C----- ---------------------- ------------- ------ ------------------
C
C CALCULATION OF THE MAXIMUM FREE ENERGY CHANGE
CALL GCM(T,GMAX,xCEM)
WRITE(6,*) 'GMAX=',GMAX
C DIFFUSIVITY OF X IN GAMMA
D22=DIFX(1)
WRITE(6,*) 'D22=',D22
CFE-CCASE
C CALL FUNCFC(T ,xGA)
CALL FUNCFC2(T ,xGA)
CALL ALP(T,ALPHA,xGA,Dl1 ,W)
WRITE(6,195) CfEMP ,xGA,ALPHA
195 FORMAT(lH ,'FE-C : AT TEMP =',F8.2,' C-EQ=',F8.6,
& 'ALPHA=',DI2.6)
C
IF (IM .EQ. 0) GOTO 24
C PARAEQUlLIBRIUM CASE
C CALL FUNCP(T ,xGA,IM)
A-17
CALL FUNCP2(f,XGA,IM)
CALL ALP(f,ALPHA,XGA,011,W)
WRITE(6, 194) CfEMP,XGA,ALPHA
194 FORMAT(lH ;PARA: AT TEMP =',F8.2; C.EQ=',F8.6,
& 'ALPHA=' ,012.6)
C
C DETERMlNA TION OF THE FINAL EQUILIBRIUM
CALL FUNCO(f,CO,X,Y,IM)
C CALL FUNCOR(T,CO,X,Y,IM)
XEQl=X(I)
XEQ2=X(lM)
YEQl=Y(l)
YEQ2=Y(lM)
WRITE(6,196) CTEMP
196 FORMAT(lH ;AT TEMPERATURE DEG C = ',F8.2)
WRlTE(6,*)' EQUILmRIUM COMPOSmONS'
WRlTE(6,91) XEQl,XEQ2,YEQl,YEQ2
91 FORMAT(lH; CGC=',F9.s; MGC=',F9.s; CCG=',F9.s,
&' MCG=',F9.s)
89 FORMAT(lH; CO(IM)·X(1M) = ',DI2.s)
IF (X(l) .GT. CO(l» GOTO 25
CB(3)=(9 .0*YEQ2+C0(IM»/1 O.OD+OO
CB(3)=C0(1)+ 1.0D·Os
C CB(3)=O.24D+OO
C CB(3)=CB(1M)
DEI.,--eB(3)
OCB=1.0D-1O
C
C
C
CDETERMlNATIONOFATIEUNE
C
C
C
C IF(YEQ2 .GT. CO(lM» THEN
C DMAX=YEQ2
C DMIN=CO(lM)
C ELSE
C DMAX=CO(lM)
C DMIN=YEQ2
C ENDIF
DMAX=O.2sD+OO
DMIN=CO(l)
DMIN=O.OD+OO
Y2=',F9.s)
C
C
C
C
188
C
C CALCULATION OF THE GROWTH RATE, ALPHA CMjSEC**O.s
C
C
10 CB(I)=CB(3)-OCB
CB (2)=CB(3)+OCB
C IF(CB(3) .LT. CO(IM)+2.0D-1O) GOTO 30
DOS II=I,3
DO 23 13=1,8
C(13)=O.OD+OO
23 CONTINUE
C C(lM)=CB(Il)
C C(I)=O.24D+OO
C(IM)=CO(lM)
C( 1)=CB(Il)
C(8)= 1.0D+OO-C(IM)-C(I)
C WRlTE(6, *)'Cl,CIM=' ,C( 1),C(IM)
CALL FUNCO(f,C,X,Y,IM)
CALL FUNCOR(f,C,X,Y,IM)
WRITE(6,*) 'Cl,CIM=',C(l),C(1M)
WRlTE(6,*) 'CO,CGC="CO(l),X(l)
WRITE(6,188) II'x(l ),X(1M),Y(IM)
FORMAT(IH ,12; Xl=',F9.s; X2=',F9.s;
CALL AL(f ,X,Y,ALPHA,DUM(Il),W,011,IM)
5 CONTINUE
C WRITE(6,*) 'DUM 1 23=',DUM(l),DUM(2),DUM(3)
IF (ABS(DUM(3» .LT. ALPHNl.0D+02) GOTO 30
IF (ABS(0.2SD+OO.CB(3» .LT. 1.0D·04) THEN
WRlTE(6,*) 'ENTER THE NPLE REGIME'
GOT030
ENDIF
DEI.,--cB(3)- 2.0* DCB *DUM (3)/(D UM(2)- DUM (1»
A-I8
C WRITE(6,*) 'CB3,DEL=',CB(3),DEL
IF (DEL .LT. DMlN) THEN
DEL=(CB(3)+DMIN)/2.0D+OO
ENDIF
IF (DEL .GT. DMAX) THEN
DEL=(CB(3)+DMAX)/2.0D+OO
ENDIF
C WRITE(6,*) 'CB3,DEL=',CB(3),DEL
C WRITE(6,*) 'ALPHA,DUM=',ALPHA,DUM(3)
IF (DUM(3) .GT. 0.00+(0) THEN
CBUI..,--eB(3)
CB(3)=DEL
ELSE
CB(3 )=(CB (3)+CB UL)/2.0
ENDIF
GOTOlO
30 WRITE(6,*) 'LOCAL EQUILIBRIUM'
WRlTE(6,*) , IN AUSTENITE'
WRITE(6,198)
WRITE(6,199) (X(I)J=I,7)
WRITE(6,*) , IN CEMENTITE'
WRITE(6,199) (Y(I)J=I,7)
WRITE(6, 193) CTEMP X(l ),ALPHA
193 FORMAT(IH ,'L-EQ: AT TEMP =',F8.2,' C-EQ=',F8.6,
& 'ALPHA=',012.6)
WRITE(6,*) , ,
C
C CALCULA nON OF NUCLEA nON
IF (IN .EQ. 1) GOTO 24
CALL NuqCO,011,n
C
C
24 GOT020
25 GOT027
26 STOP
END
C
C------------------------------------ ------------ ----------------
C CALCULA nON OF NUCLEATION RATE AND VOLUME FRACTION OF
C CEMENTITE ASSUMING A PILLBOX TYPE NUCLEUS
C
C----------------------------------------------------------------
C CONSTANTS IN NUCLEA nON FUNCTION
SUBROUTINE NuqCO,011,n
IMPUCIT REAL*8 (A-H,K-Z)
OOUBLE PRECISION C0(8)
CTEMP=T -273.00+00
N=1.0D+15
SV=5.0D+08
NA=6.022D+23
V=I.285D-23
A4=8.268D-30
KK=8.314D+OO*1.0D+07!NNV
PAI=3.141592654D+OO
EP=1.0D+OO
SIGE=1.0D+OO
AR=3.0D+OO
PHAI=GMAX*I.0D+07!NNV
CJ=SV*N*(2.0D+OO*CO(l )*Dll*V*EP**0.5)
&I(A4*(3 .00+00* KK*T)* *0.5)
CJ=CJ*DEXP( -4.0D+OO*P AI* SIGE*SIGE*EP
&I(PHAI*PHAI*KK*T»
CJ1=12.0D+00*KK*P A4*SIGE/(Dll*CO(l)*V*PHAI*PHAI)
VTET A=4.0D+OO*CO( 1)
CJO=AR *AR *ALPHA *ALPHA *ALPHNVTET A*CJ
WRITE(6,*) 'CJO=',CJO
WRITE(6,*) 'CJ1=',CJ1
WRITE(6,*) '011=',011
C ----- ---- ---- ---- ------ ------ --- --- ----- -------- ----------------
C CALCULA nON OF VOLUME FRACTION OF CEMENTITE
SEC=SECS
YE=O.OD+OO
YO=O.OD+OO
200 SEC=SEC+DSEC
A-19
IF(SEC .GT. SECF) GOTO 202
XMIN=O.OD+OO
XMAX=SEC
XX=XMAX-XM]N
DO 201 11=1,150
IO=II*2.0D+OO-l.0D+OO
IE=II*2.0D+OO
XO=XXJ300.00+00*IO+XMIN
XE=XXJ3 00.00+00* IE+XMIN
IF (LOG(CJ1/XO) .GT. 3.5D+OO) GOTO 201
YO= YO+(XMAX-XO)** 1.5*DEXP( -Cl I/XO)
IF (XMAX .LE. XE) GOTO 201
YE: YE+(XMAX-XE)** l.5*DEXP( -Cl I/XE)
201 CONTINUE
XE=CJO*XXl300.0D+00*( 4.0/3.0*YE+2.0/3.0*YO)
XXE=1.OO+OO-DEXP(-XE)
WRITE(6.299) CTEMP.SEC)CXE
IF (XXE .GT. 0.950+00) GOTO 202
299 FORMAT(lH ,TEMP='.F6.1: TIME=',DI2.4: V-TETA=',D12.4)
GOT0200
202 RETURN
END
C
C--- -- ---- -------- ---- ------ ---- --------- ------------ ------------
C SUBROUTINE TO CALCULATE ONE DIMENSIONAL RATE CONSTANT
C ASSUMING THE LINEAR GRADIENT OF CHEMISTRY
C ALPHA = CM/SEC **0.5
C----------------------------------------------------------------
C
SUBROUTINE AL(T ,x,Y,ALPHA,DUM,W,D11,IM)
IMPLICIT REAL*8(A-H,K-Z)
DOUBLE PRECISION C0(8),WO(8),X(8),Y(8),G(8),E(8,8),WW(8)
COMMON/TRANS/CO,WO
COMMON/COEFF/G.E,WW,GFE3C
CCG=O.25D+OO
D22=DIFX(T)
D11=DIFFG(T.CO(l),x(l),W)
C WRITE(6,*) 'Dll='.Dll
C WRITE(6,*) 'D22='.D22
DI2=D11 *E(l ,IM)*CO(l)/(l.OD+OO+E(l ,1)*CO(l»
C D12=O.OD+OO
ALPHAC=D11 *(CO(l )-X(l »*(CO(l )-X(I»
&I(CCG-CO( I»)/(CCG-X( I»
ALPHAC=ALPHAC+D 12*(CO(IM)-X(IM»*(CO(IM)-X(IM»
&I(Y (IM)-CO(IM) )/(CCG-X( 1»
IF(ALPHAC .LT. 0.00+00) THEN
ALPHAC=O.OD+OO
G0T0330
ENDIF
ALPHAC=ALPHAC**0.5D+OO
330 ALPHAX=D22 *(CO(IM)-X(IM»* (CO(IM)- X(IM»
&I(Y (IM)-CO(IM) )/(Y (IM)- X(IM»
IF(ALPHAX .LT. 0.00+00) THEN
DUM=O.OO+OO
G0T0333
ENDIF
ALPHAX=ALPHAX**0.50+00
C
DUM=ALPHAC-ALPHAX
IF (DUM .GT. O.OD+OO)THEN
ALPHA=ALPHAC
ELSE
ALPHA=ALPHAX
ENDIF
C WRITE (6,*) 'ALPHA,DUM='.ALPHA.DUM
333 RETURN
END
C
C------- ---- -- ------ ---------- ------- ---. -- -------- ---------- ----
C SUBROUTINE TO CALCULATE ONE DIMENSIONAL RATE CONSTANT
C ASSUMING THE LINEAR GRADIENT OF CHEMISTR Y
C ALPHA = CM/SEC **0.5 ****PARAEQUILffiRIUM CONDmON****
C ----- ---- ---- --' ----- --.- -------- ------- ------ ---------. --------
C
A-20
SUBROUTINE ALP(T,ALPHA,XGA,Dll,W)
IMPLICIT REAU8(A-H,K-Z)
ooUBLE PRECISION C0(8),W0(8),G(8),E(8,8),WW(8)
COMMONffRANS/CO,WO
COMMON/COEFF/G,E,WW,GFE3C
CCG=O.25D+OO
D1 I=DIFFG(T,CO(I),XGA,W)
ALPHAC=Dll·(CO(l)-XGA)·(CO(l)-XGA)
&/(CCG-CO( 1»)/(CCG-XGA)
ALPHAC=ALPHAC··O.5D+OO
AlPHA=AlPHAC
C WRITE (6,·) 'ALPHA,DUM=',ALPHA,DUM
333 RETURN
FND
C
C--------- ------ ---- ---- ---- ------------------- ---- --------------
C SUBROUTINE TO CALCULATE THE EQUILIBRIUM INTERFACE CHEMISTR Y
C----- ---- ---- -- ------ ---------- --------- ---- -------- ------------
C
SUBROUTINE FUNCOR(T,CX,Y,IM)
IMPLICIT REAU8 (A-H,K-Z)
ooUBLE PRECISION X(8),Y(8),E(7,7),WW(7),C0(8),C(8),G(7)
&,YY(3),FUN(3)
COMMONffRANS/CO,WO
C
C INITIAL GUESS OF X2 AND Y2
C
C
XI=X(l)
X2=X(IM)
Y2= Y(IM)·4.0/3.0
DELTA=1.0D-1O
C
499 YY(l)=Y2-DELTA
YY(2)= Y2+DELT A
YY(3)=Y2
00 101=1,3
Y2=YY(l)
C
500 FF=FXY(T,CXIX2,Y2JM,l)
MM=FXY(T,CX I,X2,Y2,IM,O)
DFX=(FXY(T,C,X 1+DELT A,X2,Y2,IM,1 )-FXY(T,CXI-DELT A,X2, Y2,IM,I»
& /2.0/DEL TA
DFY =(FXY(T,C,X I,X2+DELT A,Y2,IM,1 )-FXY(T,CXl X2-DELT A,Y2,IM,I»
& /2.0/DELT A
DMX=(FXY(T,CXl +DELT AX2,Y2,IM,O)-FXY(T,C,XI-DELT AX2,Y2,IM,O»
& /2.0/DELTA
DMY =(FXY(T,CX 1X2+DELT A,Y2,IM,O)-FXY(T,C,Xl X2-DELT A,Y2JM,O»
& /2.0/DELTA
C
IF (ABS(FF) .LT. 1.0D-06.AND.
& ABS(MM) .LT. 1.0D-06) GOTO 502
DENO=1.0D+OO
DET=DFX·DMY-DMX·DFY
DXl =(-FF*DMY +MM·DFYYDET
DX2=(-MM·DFX+FF·DMXYDET
IF (Xl/DENO+DXl .LT. O.OD+OO)THEN
Xl=Xl/2.0/DENO
ELSE
Xl=Xl/DENO+DXl
FNDIF
IF (X2/DENO+DX2 .LT. O.OD+OO)THEN
X2=X2/2.0/DENO
ELSE
X2=X2/DENO+DX2
ENDIF
C
C WRITE(6,·) X2,Y2,FF
G0T0500
502 FUN(I)=1.0D+00-(C(IM)-X2)/(C(I)-Xl)
& /(3.0·Y2-4.0·X2)·(l.OD+00-4.0·X 1)
10 CONTINUE
COEFF=(C(IM)-X2)/(C( 1)-Xl)
IF (ABS(FUN(3» .LT. 1.0D-02) GOTO 503
A-21
DUMMY= Y2-2.0*DELT A*FUN(3)/(FUN(2)-FUN(l»
IF (DUMMY .LT. O.ODtOO)THEN
DUMMY=Y2/2.0D+OO
ENDIF
Y2=DUMMY
001U499
503 X(1)=XI
X(IM)=X2
Y(IM)= Y2*3.0/4.0
RE1URN
FND
C
C
C--- -- ------ ---- ------------ ------ ------- ---- ---- -------- --------
C FUNcnON TO CALCULATE EQUILIBRIUM FUNcnON F AND M
C IMF=I: F(X,Y) IMF=O: M(X,Y)
C
DOUBLE PRECISION FUNCTION FXYO(f,C,XI,x2,Y2]M]MF)
IMPUCIT REAL*8 (A-H,K-Z)
DOUBLE PRECISION E(8,8),WW(8),G(8),CO(8),C(8)
COMMON{fRANSlCO,WO
COMMON/COEFF/G,E,WW,GFE3C
R=8.314D+OO
C
C
CG=XI
X I=(C( I )*(3.0*Y2-4.0*X2)-(C(IM)-X2»/(3 .0* Y2-4 .O*C(IM»
X8=1.0D+OO-XI-X2
Y8= 1.0D+OO-Y2
C
MFE3C=R *T*DLOG(Y8)
MFE=R*T*DLOG(X8)-R *T/2.0*E(I,1 )*XI *XI
MC=R *T/3.0*DLOG(X 1)+R *T/3.0*E(l, 1)*X I
MFE3C=MFE3C+(I- Y8)*WW(IM)*Y2
MFE=MFE-R *T/2.0*E(IM,IM)*X2*X2-R *T*X I *E( I,IM)*X2
MC=MC+R *T/3.0*E( I,IM)*X2
MX= R*T*DLOG(X2)
& +R*T*(E(1,IM)*Xl+E(IM]M)*X2)
MM3C= R*T*DLOG(Y2)
& +Y8*WW(IM)
MM3C=MM3C- Y8*WW(IM)*Y2
C
IF (IMF .EQ. I) THEN
FXYO=GFE3C+MFE3C-MFE-MC
ELSE
FXYO=G(IM}+MM3C-MX-MC
ENDIF
C
RE1URN
FND
C
C
C----- -------- ---- ------ ---- ------ ----- ------------ --------------
C SUBROUTINE TO CALCULATE THE EQUILIBRIUM INTERFACE CHEMlSTR Y
C----------------------------------------------------------------
C
SUBROUTINE FUNCO(f,C,x,Y]M)
IMPUCIT REAL*8 (A-H,K-Z)
DOUBLE PRECISION X(8),Y(8),E(8,8),WW(8),x1(3),K(8)
& ,FUN (3 ),G(8),MM 3C(8 ),MX (8 ),CO(8) ,C(8)
COMMON{fRANSlCO,WO
COMMON/COEFF/G,E,WW,GFE3C
R=8.314D+OO
C
C
XGA=1.0D-I0
DUMMY = 1.0D-12
lOO XI(1)=XGA-DUMMY
XI (2)=XGA+DUMMY
XI (3)=XGA
DO 101=1,3
XX=O.OD+OO
YY=O.OD+OO
FUN(I)=O.OD+OO
A-22
CB=DEXP«GFE3C-G(lM)- WW(lM»/RIT +E(I,lM)"'XI (l»
X(IM)=C(lM)*(4.0"'XI (1)-1.00+00)/
& (3.0"'B"'(XI (1)-C(1»+4.0"'C(1 )-1.0D+OO)
Y(lM)=B"'X(lM)
XX=XX+X(lM)
YY=YY+Y(lM)
X(l)=X1 (l)
X(8)= 1.0D+OO-X(I )-XX
IF (X(8) .LT. O.OD+OO)THEN
WRITE( 6, "') 'X(8),x( 1),XX=' ,x(8),x( 1),xX
ENDIF
Y(8)=1.0D+OO- YY
C
C
MFE3C=R*T*DLOG(Y(8»
MFE=R"'T*DLOG(X(8»-R"'T/2.0"'E(I ,1)"'X(I )"'X(I)
MC=R "'T/3.0"'DLOG(X( 1»+R "'T/3.0"'E(l, 1)"'X( I)
M FE3C=MFE3C+( 1-Y(8»"'WW(lM)"'Y(lM)
MFE=MFE-R "'T/2.0"'E(lM ,lM)"'X(lM)"'X(lM)-R "'T*X( 1)"'E( I,lM)"'X(lM)
MC=MC+R "'T/3.0*E( l,lM)"'X(lM)
MX(lM)= R"'T*DLOG(X(lM»
& +R*T"'(E(l,lM)*X(l)+E(lM,lM)"'X(lM»
MM3C(lM)= R"'T"'DLOG(Y(lM»
& +Y(8)"'WW(lM)
MM3C(lM)=MM3C(lM)- Y(8)"'WW (lM)"'Y(lM)
C
FUN(l)=GFE3C+MFE3C-MFE-MC
C
10 CONTINUE
IF(ABS(FUN(3» .LT. 1.0D-06) GOTO 20
XGA=X 1(3)- 2.0"'DUMMY'" FUN (3 )/(FUN(2)-FUN (I»
IF(XGA .LE. 0.00+00) THEN
XGA=X1(3)/2.0D+OO
ENDlF
GOTO 100
20 DO 301=1,8
X(l)=X(l)
Y(l)=3 .OD+OO/4.0D+OO'"Y(l)
30 CONTINUE
Y(l)=0.25D+OO
RETURN
END
C
C--- -- ---- ------ -- ---- ------ -- ----------- -------- ---------- --------
C SUBROUTINE TO CALCULATE THE PARAEQUlLIBRlUM INTERFACE CHEMISTRY
C------------------------------------------------------------------
C
SUBROUTINE FUNCP(f ,xGA,lM)
lMPUCIT REAL"'8 (A-H,K-Z)
DOUBLE PRECISION X(8),Y(8),E(8,8),WW(8),xl(3)
&, FUN (3 ),G(8) ,CO(8),WO(8)
COMMON/TRANS/CO,WO
COMMON/COEFF/G,E,WW,GFE3C
R=8.314D+OO
XGA=O.OO5D+OO
C
C
DUMMY=1.0D-08
100 X1(l)=XGA-DUMMY
Xl (2)=XGA+DUMMY
X1(3)=XGA
DO 101=1,3
XX=O.OD+OO
FUN(I)=O.OO+OO
K=C0(lM)/CO(8)
Y(lM)=K!(l +K)
X(lM)=( 1-X1(1»*Y(lM)
X(1)=X1(l)
X(8)= 1.0D+OO-X(l )-X(lM)
Y(8)= 1.0D+00- Y(lM)
DENO=1.0D+OO
C
MFE3C=R *T*DLOG(Y(8»
A-23
MFE=R *T*DLOG(X(8)/DENO)-R *T/2.0*E(l,1 )*X(l )*X(I )/DENO/DENO
MC=R *T/3.0*DLOG(X(l )/DENO)+R *T/3.0*E( 1,1)*X(l )/DENO
MFE3C=MFE3C+(I- Y(8»*WW(lM)*Y(lM)
MFE=MFE-R *T/2.0*E(lM ,lM)*X(lM)*X(IM)/DENO/DENO
MFE=MFE-R *T*X(l )*E(l,IM)*X(IM)/DENO/DENO
MC=MC+R *T/3.0*E( 1,IM)*X(lM)/DENO
MX= R*T*DLOG(X(lMYJ)ENO)
& +R *T*(E(I JM)*X(I )+E(lMJM)*X(lM»/DENO
MM3C= R*T*DLOG(Y(IM»
& +Y(8)*WW(IM)-Y(8)*WW(IM)*Y(IM)
C
FUN(I)=X(8)*(GFE3C+MFE3C-MFE-MC)
FUN(I)=FUN(I)+X(IM)*(G(IM)+MM3C-MX-MC)
C
C WRITE(6,*) IM,E(I,I),E(lJM),E(lM,lM)
C
10 CONTINUE
IF(ABS(FUN(3» .LT. 1.0D-1O) GOTO 20
XG A=X I(3)- 2.0*DUMM Y*FUN (3)/(FUN(2)-FUN (I»
ooTO lOO
20 XGA=XI(3)
RETURN
END
C
C----------------------------------------------------------------
C SUBROUTINE TO CALCULATE THE EQun..mRIUM INTERFACE CONPOSrnON
C IN FE-C SYSTEMS
C----- ---- ---- ---- ---- ------ ---- --------- --------- --- ------------
C
SUBROUTINE FUNCFC(T XGA)
IMPUCIT REAL*8 (A-H,K-Z)
ooUBLE PRECISION FUN(3),X(3),G(8),E(8,8),WW(8)
COMMON/COEFF/G,E,WW,GFE3C
R=8.314D+OO
XGA=O.OOSD+OO
DUMMY = I.OD-08
10 X(l)=XGA-DUMMY
X(2)=XGA+DUMMY
X(3)=XGA
00 20 1=1,3
X8=I.OD+OO-X(I)
MFE=R *T*DLOG(X8)-R *T/2.0*E(l,I)*X(I)*X(I)
MC=R *T/3.0*DLOG(X(I»+R *T/3.0*E( 1,1)*X(I)
FUN(I)=GFE3C-MFE-MC
20 CONTINUE
IF(ABS(FUN(3» .LT. 1.0D-06) GOTO 30
XGA=X(3)-2.0*DUMMY*FUN(3)/(FUN(2)-FUN( I»
GOTOlO
30 XGA=X(3)
RElURN
END
C
C
C----------------------------------------------------------------
C SUBROUTINE 1UCONVERT Wf% 1UMOLE FRACTION
C OR MOLE FRACTION 1UWf%.
C ----- ---- ------ ---- -- ------ ------ ------- ------ ---------- --------
C
SUBROUTINE CONV(N,Wf,C)
IMPUCIT REAL*8 (A-H,O-Z)
ooUBLE PRECISION AN(8), Wf(8), C(8), A(8)
AN(I)=12.01ISD+OO
AN(2)=28.09D+OO
AN(3)=54.94D+OO
AN(4)=S8.71D+OO
AN(6)=9S.94D+OO
AN(S)=S2.00D+OO
AN(7)=63.SSD+OO
AN(8)=SS.84D+00
IF (N .EQ. 0) GOTO I
WT(8)=IOO.OD+OO-WT(I)- WT(2)-WT(3)- WT(4)- WT(S)-WT(6)- Wf(7)
AT=O.OD+OO
00 2 1=1,8
A(I)=WT(1)/AN(I)
A-24
AT=AT+A(I)
2 OONTINUE
0031=1,8
C(I)=WT(IY AN(I)/AT
3 OONTINUE
GOT04
C(8)= 1.0D-tOO-C( 1)-C(2)-C(3)-C( 4 )-C(5)-C( 6)-C(7)
AT =O.OD-tOO
00 51=1,8
A(I)=C(I)· AN(I)
AT=AT+A(I)
5 OONTINUE
00 6 1=1,8
WT(I)=C(W AN(I)/ AT· 1oo.OD+OO
6 OONTINUE
4 RETIJRN
END
C
C----- -- ---- ---- ---- ------ ------ --------------- ---------- --------
C FUNCTION GIVING THE CARBON DIFFUSIVITY IN FERRITE BASED ON
C AGREN'S WORK
C OCA : CM"2/SEC
C
DOUBLE PRECISION FUNCTION OCA(T)
ooUBLE PRECISION T ,PAI
PAI=3.141592654D+OO
OCA=O.02D+OO·DEXP( -10115.OD+OO/T)
OCA=DCA ·DEXP(0.5898D+OO*(1.0D+OO+2.0jP AI
&·DAT AN(1.4985D+OO-15309.0D+OO/T)))
RETIJRN
END
C
C----- -- ---- ---- ---- ---- ------ ------ --------- -------- -------- ----
C FUNCTION GIVING THE CARBON DIFFUSIVIlY IN AUSTENITE
C
ooUBLE PRECISION FUNCTION DIFFG(T ,Xl,X2,W)
IMPLICIT REAL*8(A-H,K- Y),INTEGER(I,12)
ooUBLE PRECISION DlFF(3OO),CARB(3OO)
C HH=PLANCK CONSTJ/S, KK=BOLTZMAN CON ST. IlK
HH=6.6262D-34
KK= 1.38062D-23
Z=12
AS=l.OD-tOO
R=8.31432D-tOO
C
C DlFF=DlFFUSIVlTY OF CARBON IN AUSTENITE, CM**/SEC
C Z=COORDINA nON OF INTERSTIAL SITE
C PSI=COMPOsmON DEPENDENCE OF DIFFUSION COEFFICIENT
C THET A=NO. C ATOMS/ NO. FE ATOMS
C ACTIV=ACTIVlTY OF CARBON IN AUSTENITE
C R=GAS CONSTANT
C X=MOLE FRACTION OF CARBON
C T=ABSOLUTE TEMPERATURE
C SIGMA=SITE EXCLUSION PROBABLITY
C W=CARBON CARBON INTERACTION ENERGY IN AUSTENITE
C
DASH=(KK*T/HH)*DEXP( -(21230.0D+OO/T»*DEXP( -31.84D-tOO)
XINCR=(XI-X2)/3oo.0D-tOO
00 11=1,300
CARB(I)=X2+(I-l)·XINCR
X=CARB(I)
TIIET A=XI(AS-X)
ACTIV=CG(X,T,W,R)
ACTIV=DEXP(ACTIV)
DACTIV=OCG(X,T,W,R)
DACTIV=DACTIV*ACTIV
DACTIV =DACTIV* AS/«A5+ THET A)**2)
SIGMA=AS-DEXP« -(W»/(R *n)
PSI=ACTIV*(A5+Z*«A5+ THET A)/(A5-(A5+Z!2)*THET A+(Z!2)*(AS+Zl2)*
&(AS-SIGMA)*THET A*THET A»)+(AS+ THET A)*DACTIV
DIFF(I)=DASH*PSI
OONTINUE
ll3=O
CALL DQSES(XINCR,DIFF,300,ANS,ERROR)
A-25
DIFFG=ANS/(XI-X2)
RETIJRN
END
C
C------- ---- ---- -------- ---- ---- --------- ---------- ---------- ----
C FUNcnON GIVING THE CARBON DIFFUSIVITY IN AUSTENITE.
C
OOUBLE PRECISION FUNCfION DIFFG 1(f X,W)
IMPLICIT REAU8(A-H,K- Y), INTEGER(l,J,Z)
C HH=PLANCK CONST.J/S, KK=BOLTZMAN CONST. JIK
HH=6.6262D-34
KK=I.38062D-23
Z=I2
A5=I.OD+OO
R=8.3I432D+OO
C
C DlFF=DlFFUSMTY OF CARBON IN AUSTENITE, CM**2/SEC
C Z=COORDINA TION OF INTERSTIAL SITE
C PSI=COMPOsrnON DEPENDENCE OF DIFFUSION COEFFICIENT
C THET A=NO. C ATOMS/ NO. FE ATOMS
C ACTIV=ACITVITY OF CARBON IN AUSTENITE
C R=GAS CONSTANT
C X=MOLE FRACfION OF CARBON
C T=ABSOLUTE TEMPERATIJRE
C SIGMA=SITE EXCLUSION PROBABLITY
C W=CARBON CARBON INTERACTION ENERGY IN AUSTENITE
C
DASH=(KK *TIHH)*DEXP( -(2I230.0D+D0/T) )*DEXP( -31.84D+OO)
THETA=XI(A5-X)
ACITV=CG(X,T,W,R)
ACTIV=DEXP(ACTIV)
DACTIV=DCG(X,T,W,R)
DACTIV=DACTIV*ACTIV
DACfIV =DACTIV* A5/«A5+ THET A)**2)
SIGMA=A5-DEXP« -(W»/(R *n)
PSI=ACTIV*(A5+Z*«A5+ THET A)/(A5-(A5+ZI2)*THET A+(ZI2)*(A5+Zl2)*
&(A5-SIGMA)*THETA*THETA»)+(A5+ THET A)*DACTIV
DIFFG I=DASH*PSI
RETIJRN
END
C
C--- ---- ---- -- ---- ---- ---- ------ ----------------- ---- ------------
C FUNCfION GIVING LFG LN(ACTIVITY) OF CARBON IN AUSTENITE.
C
DOUBLE PRECISION FUNCfION CG(X,T,W,R)
DOUBLE PRECISION J,DG,DUMMY,T ,R,WX
J=I-DEXP(-W/(R*n)
DG=DSQRT(I-2*(1 +2*J)*X+(I +8*J)*X*X)
DUMMY =5*DLOO«(1-2*X)/X)+6*W/(R *T)+«38575.0)-(
& 13.48)*T)/(R *T)
CG=DUMMY +DLOO«(DG-I +3*X)/(DG+ I-3*X»**6)
RETIJRN
END
C
C------- -- ------ -- ------ ---- ------ ----- ------------ ---- ---- ------
C FUNCTION GIVING DIFFERENTIAL OF LN(ACTIVITY) OF CARBON
C IN AUSTENlTE.
C DIFFERENTIAL IS WITH RESPECf TO X.
C
OOUBLE PRECISION FUNCTION DCG(X,T,W,R)
OOUBLE PRECISION J,DG,DDGX,T,W,R
J=I-DEXP(-W/(R*n)
DG=DSQRT( 1-2*(1 +2*J)*X+(1 +8* J)*X*X)
DDG=(0.5/OG)*( -2-4 *J+2*X + 16*J*X)
OCG=-«IO/( I-2*X»+(5/X»+6*«DDG+ 3)/(DG-I +3*X
&)-(DDG-3)/(DG+ I-3*X»
RETIJRN
END
C
C----------------------------------------------------------------
SUBROUTINE OMEGA(W)
C SUBROUTINE TO CALCULATE THE CARBON CARBON INTERACTION ENERGY IN
C AUSTENITE, AS A FUNCTION OF ALLOY COMPOsmON. BASED ON MUCGI8
C THE ANSWER IS IN JOULES PER MOL. **7 ocrOBER 1981**
A-26
COMMONrrRANS/CO,WO
DOUBLE PRECISION qS),W,P(S),B 1,B2,Y(S),TlO,T20,B3,XONE
&,CO(S),WO(S)
INTEGER B5,I,U,B4
DO 1 1=1,8
C(I)=WO(I)
CONTINUE
B3=0.OD+OO
qS)=C( 1}tC(2}tC(3 )+C( 4)+C(5}tC(6)+C(7)
qS)= loo.00+00-C(S)
q8)=C(S)/55.84D+OO
ql)=C(I)/12.0115D+00
q2)=C(2)/2S.09D+OO
q3)=C(3)/54.940+OO
q4)=C(4)/5S.71D+OO
q5)=C(5)/95.94D+OO
q6)=q6)/52.0D+OO
q7)=C(7)/50.940+OO
B 1=C( 1)+C(2}tC(3 )+C( 4}tC(5)+q6)+C(7}tC(S)
DO 107 U=2,7
Y(U)=C(U)/qS)
107 CONTINUE
DO 106 U=I,S
qU)=C(U)/B 1
106 CONTINUE
XONE--c(1)
C XONE=OINT(lOOOO.OD+OO*XONE)
C XONE=XONE/l0000
B2=O.OO+OO
TlO=Y(2)*( -3)+ Y(3)*2+ Y(4)* 12+Y(5)*( -9)+ Y(6)*( -1)+ Y(7)*( -12)
T20=-3*Y(2)-37 .5*Y(3)-6*Y(4 )-26*Y(5)-19*Y(6)-44*Y(7)
P(2)=20 13 .0341 +763.S167*q2)+45S02.S7*q2)**2-2Soo61.63 *C(2)* *3
&+ 3.S64 D+06 *q2)* *4- 2.4233 0+07 *q2)* *5+6 .95470+07* C(2)** 6
P(3)=2012.067 -1764.095*q3)+62S7 .52 *q3 )**2-21647 .96*q3 )**3-
&2.0119D+06*q3)**4+3.1716D+07*q3)**5-1.3SS50+OS*q3 )**6
P(4 )=2006.S017+2330.2424*q4)-54915.32*q4 )**2+ 1.62160+06*q4 )**3
&-2.496SD+07*C(4 )**4+ I.SS3S0+0S*q 4)**5-5.5531 0+OS*C(4)* *6
P(5)=2OO6.S34-2997 .314*C(5)-37906.61 *q5)**2+ 1.032S0+06*q5)**3
&-1.3306D+07*C(5)**4+S.411 0+07*C(5)**5-2.0S260+0S*C(5)**6
P( 6)=20 12.3 67 -9224.2655*C( 6)+ 33657. S*q 6)* *2-566S27 .S3 *C( 6)* *3
&+S.5676D+06*q 6)* *4-6.7 4S20+07 *q 6)** 5 +2 .OS370+0S *C( 6)* *6
P(7)=2011.9996-6247.911S*C(7)+5411.7566*q7)**2
&+250 11S.1 OS5*C(7)**3-4.16760+06*C(7)**4
DO lOS U=2,7
B3=B3+P(U)*Y(U)
B2=B2+Y(U)
10S CONTINUE
IF (B2 .EQ. 0.00+00) GOTO 455
W=(B3/B2)*4.1S7
GOT0456
455 W=S054.0
456 RETURN
FND
C
C----------------------------------------------------------------
C FUNCTION GIVING TIlE EQUILIBRIUM MOLE FRACTION OF CARBON IN FERRITE.
C
DOUBLE PRECISION FUNCTION XALPH(T)
DOUBLE PRECISION T,CTEMP
CTEMP=(T -273.00+(0)/900.00+00
XALPH=0.152S0-02-0.SS1 6D-02*CTEMP+O.24500-0 1*CTEMP*CTEMP
&-0.24170-01 *CTEMP*CTEMP*CTEMP+
&0.6966D-02*CTEMP*CTEMP*CTEMP*CTEMP
RETURN
END
C----- ---- ---- ---- ------ ---- -------- ----- ------ ------------ ------
C DATA FOR DIFFUSJVITIES OF TIlE THIRO ELEMENT IN AUSTENITE
C IN ORDER OF SI,MN,NI,CR,MO,CU
BLOCK DATA COEF
COMMON/DIFFUS/D22
DOUBLE PRECISION 022(7)
OAT A (022(1),1=2,7)/
& 6.40+00,2.41 D+OO,0.475D+00,4. 750+00,2. 73D+OO,I.00+00/
END
A-27
C------- -------------------- -- ---- --------------------------------
C FUNCTION TO CALCULATE DIFFUSIVITY OF THE TIURD ELEMENT
C CM**2/SEC
C
DOUBLE PRECISION FUNCTION DIFX(1)
IMPLICIT REAL*8(A-H,K-Z)
DOUBLE PRECISION CO(8),WO(8),D22(7)
COMMONffRANS/CO,WO
COMMONIDIFFUSlD22
Q=286000.0D+OO
K=8.314D+OO
00=0.70+00
1=1
10 1=1+1
IF(I .GT. 8) GOTO 20
IF(CO(l) .NE. 0) GOTO 20
GOT010
20 DlFX=D22(1)*OO*DEXP(-Q/Kff)
RETURN
END
C -- ---- ---- --------------- ------ ------ -------- ---------- ---------
C SUBROUTINE TO CALCULATE THE MAXIMUM FREE ENERGY CHANGE AVAILABLE
C FOR NUCLEA nON OF CEMENTITE FROM AUSTENlTE.
C KIRKALDY'S METHOD IS USED.
C SEE HASHIGUCHI ET AL., CALPHAD VOL.8 NO.2(l984) PP173-186
C
C M. TAKAHASHI, 20.11.1989
C
C
SUBROUTINE GCM(T,GMAX,xCEM)
IMPLICIT REAL*8 (A-H,K-Z)
COMMONffRANS/CO,WO
DOUBLE PRECISION X(8),Y(8),E(8,8),WW(8),x1(3),K(8)
&,FUN (3),G(8),CO(8), Y2(3)
COMMON/COEFF/G,E,WW,GFE3C
R=8.314D+OO
XGA=O.OlD+OO
C
C
DUMMY=l.OD-06
DO 1 1=1,7
X(I)=CO(I)
IF(X(I) .EQ. O.OD+OO)GOTO 1
IFLAG=I
1 CONTINUE
X(8)=C0(8)
YlNIT =4.0D+OO/3.0D+OO*X(IFLAG)
100 Y2(l)=YINIT-DUMMY
Y2(2)=YINIT +DUMMY
Y2(3)=YINIT
DO 101=1,3
FUN(I)=O.OO+OO
D02ll=2,7
1F(ll NE. IFLAG) THEN
Y(ll)=O.OD+OO
ELSE
Y(ll)= Y2(1)
ENDIF
2 CONTINUE
Y(8)= 1.00+00- Y(lFLAG)
C
C
MFE3C=R *T*DLOG(Y(8»+(1.0D+OO- Y(8»*WW(lFLAG)*Y(lFLAG)
MFE=R *T*DLOO(X(8»-R *T/2.0*(E(I,1 )*X(I )*X(I )+E(IFLAG,IFLAG)
&*X(lFLAG )*X(lFLAG»- R*T*X (1)*E(1,IFLAG )*X(lFLAG)
MC=R *T/3.0*DLOG(X(l »+R *T/3.0*(E(I,I)*X( I )+E(I ,IFLAG)
&*X(lFLAG»
MX=R *T*DLOG(X(lFLAG»+R *T*(E(I ,IFLAG)*X( I)
&+E(lFLAG,IFLAG)*X(IFLAG»
MM3C=R *T*DLOO(Y(lFLAG»+ Y(8)*WW(IFLAG)*( 1.00+00- Y(lFLAG»
C
FFC=GFE3C+MFE3C-MFE-MC
FMC=G(lFLAG)+MM3C-MX -MC
FUN(I)=FFC-FMC
A-28
CMUFEC=MFE3C*3.0D-tOO/4.0D-tOO-tGFE3C
C
ID CONTINUE
IF(ABS(FUN(3» .LT. I.OD-06) GOTO 20
YINIT =YINlT -2.0*DUMMY*FUN(3)/(FUN(2)-FUN( 1»
IF (YINlT .LT. O.OD-tOO)THEN
YINlT=1.0D-04
ENDIF
GOIDl00
20 GMAX=(FMC+FFC)/2.0D+OO*3.0D-tOO/4.0D-tOO
XCEM= YINIT*3.0/4.0
RElURN
END
C -- -------- --- ------------ ---- -------- ---------- -- ---- ----------
C SUBROUTINE ID CALCULATE PARAEQUlLIBRruM BAOUNDARY COMPOSmONS
C HILLERT-STAFFANSON SUB-REGULAR SOLUTION MODEL IS USED
C 1991.1.8 M.TAKAHASHI
C
SUBROUTINE FUNCP2(T XGA,IM)
IMPUCIT REAL*8 (A-H,K-Z)
DOUBLE PRECISION X(8),Y(8),G2(8),LFX(2,8),GM(8),AC(8)
&,C0(8),WO(8),X 1(3),FUN(3)
COMMON/TRANS/CO,WO
COMMON/HILI./G2,LFX,LCV ,GM,AC,DDG ,GFEC
R=8.314D-tOO
XGA=O.OO5D+OO
C
DUMMY =1.0D-08
100 Xl(l)=XGA-DUMMY
Xl(2)=XGA+DUMMY
Xl (3)=XGA
DO 101=1,3
K=CO(IM)/CO(8)
Y(IM)=K/(l.OD-tOO+K)
X(IM)=(l.OD+OO-Xl(1»*Y(IM)
X(l)=Xl(1)
DENO= 1.0D-tOO-X(l)
Y(8)=1.0D+OO-Y(IM)
X(8)=1.0D+OO-X(l)-X(IM)
X(I)=X(l)/DENO
X(IM)=X(IM)/DENO
X(8)=X(8)/DENO
EGO:- X(lM)*X( 1)*G2(1M)+ X(lM)*X(IM)*
& (LFX( 1,IM)+(3.0D+OO-4.0*X(lM»*LFX(2,IM»+X(l )*X( 1)*LCV
EG 1=-2.0*X( 1)*LCV +X(IM)*G2(IM)
EG2=X(8)*X(l )*G2(IM)+X(8)*X(8)*
& (LFX(l ,IM)+(1.0D+OO-4.0*X(lM»*LFX(2,IM»+X(I)*X(1 )*LCV
C
C
FUN(I)=X(8)*(
& GFEC-DDG/3 + R*T*DLOG(Y(8»+AC(IM)*Y(IM)*Y(IM)
& - R*T*DLOG(X(8»-R*T*DLOG(1.0D-tOO-X(I»
& - R*T/3*DLOG(X(I)/(1.0D+OO-X(l»)
& - EGO-EG 1/3 )
FUN(I)=FUN(I)+X(IM)*(
& GM(2)-DDG/3 + R*T*DLOG(Y(lM»+AC(IM)*Y(8)*Y(8)
& - R*T*DLOG(X(IM»-R*T*DLOG(I.0D+OO-X(l»
& - R*T/3*DLOG(X(l)/(1.0D+OO-X(l »)
& - EG2-EGl/3 )
C
10 CONTINUE
IF(ABS(FUN(3» .LT. 1.0D-06) GOTO 20
XGA=X( 1)-2.0*DUMMY*FUN(3)/(FUN(2)-FUN(l»
IF (XGA .LT. DUMMY) THEN
XGA=X( 1)/2.00+00
ENDIF
IF (XGA .GT. 1.00+(0) THEN
XGA=(1.0D-tOO+X( 1»(l.0
ENDIF
XGA=XGA*DENO
GOIDl00
20 XGA=Xl(3)
RETURN
A-29
END
C
C ------ -- ----- -- ---- -------- ------------ ------------------------
C SUBROUTINE TO CALCULATE EQUIUBRIUM BAOUNDARY COMPOS mONS
C IN FE-C-X TERNARY SYSTEM.
C HJLLERT-ST AFFANSON SUB-REGULAR SOLUTION MODEL IS USED.
C I99I.1.8 M.TAKAHASHl
C
SUBROUTINE FUNCFC2(f ,XGA)
IMPLICIT REAL*8 (A-H,K-Z)
DOUBLE PRECISION X(8),Y(8),G2(8),LFX(2,8),GM(8),AC(8)
&,C0(8),WO(8),X 1(3),FUN(3)
COMMONffRANSlCO,WO
COMMONIHILl..IG2,LFX,LCV,GM,AC,DOO,GFEC
R=8.314D-+OO
XGA=O.OO5D-+OO
C
DUMMY=1.OD-OS
100 XI(l)=XGA-DUMMY
X1(2)=XGA+DUMMY
X1(3)=XGA
DO 101=1,3
X(l)=XI(l)
DENO= I .OD-+OO-X(l)
X(8)=1.0D+OO-X(l)
X(l)=X(l)/DENO
X(8)=X(8)/DENO
C
EGO=X(l )*X(l)*LCV
EGl =-2.0*X(l )*LCV
C
C
FUN(l)=GFEC-DOO/3
& - R*T*DLOG(1.0DtOO-X(l»
& - R*T/3*DLOG(X(l)/(l.ODtOO-X(l»)
& - EGO-EGl/3
C
10 CONTINUE
IF(ABS(FUN(3» .LT. 1.0D-06) GOTO 20
XGA=X(l )-2.0*DUMMY*FUN(3)/(FUN(2)-FUN( I»
IF (XGA .LT. DUMMY) THEN
XGA=X(1)/2.0D+OO
ENDIF
IF (XGA .GT. 1.0D-+OO)THEN
XGA=(I.OD+OO+X( 1»/2.0
ENDIF
XGA=XGA*DENO
GOTO 100
20 XGA=XI(3)
RE1URN
END
C
C ------------------------- ---- ------ -------- ------------ --------
C HILLERT-ST AFFANSON SUB-REGULAR SOLUTION MODEL IS USED
C 1991.1.8 M.TAKAHASHI
C IMF=I: F(X,Y) IMF=O: M(X,Y)
C
DOUBLE PRECISION FUNcrION FXY(f,C,X1,X2,Y2,IM,IMF)
IMPLICIT REAL*8 (A-H,K-Z)
DOUBLE PRECISION X(8),Y(8),G2(8),LFX(2,8),GM(8),AC(8)
&,CO(8),WO(8),FUN(3),C(8) .
COMMONffRANS/CO,WO
COMMONIHILl..IG2,LFX,LCV,GM,AC,DOO,GFEC
R=8.314D-+OO
C
C X( 1)=(C( 1)*(3.0*Y2-4.0*X2)-(C(IM)-X2»/(3 .0* Y2-4.0·C(lM»
X(1)=X1
X(lM)=X2
X(8)= 1.0D-+OO-X(I)-X(lM)
Y(lM)=Y2
Y(8)=1.0D-+OO-Y(IM)
DENO=1.0D+OO-X(l)
X(l)=X(I)/DENO
X(lM)=X(lM)/DENO
A-30
X(8)=X(8)/DENO
C
EGO=-X(lM)*X (1)*G2(lM)+ X(lM)*X (IM)*
& (LFX( I ,IM)+(3.0D+OO-4.0*X(lM»*LFX(2,IM»+X(l )*X( I )*LCV
EG I=-2.0*X(1 )*LCV +X(lM)*G2(lM)
EG2=X(8)*X(l )*G2(lM)+X(8)*X(8)*
& (LFX(l ,IM)+(I.OD+OO-4.0*X(lM»*LFX(2,IM»+X(l )*X( I )*LCV
C
IF (IMF .EQ. I) THEN
FXY =GFEC-DOO/3 + R*T*DLOG(Y(8»+AC(lM)*Y(lM)*Y(IM)
& -R*T*DLOG(X(8»-R*T*DLOO(I.OD+OO-X(I»
& - R*T/3*DLOG(X(l)/(I.OD+OO-X(l»)
& -EGO-EGl/3
ELSE
FXY =GM(2)-DOO/3 + R*T*DLOO(Y(lM»+AC(lM)*Y(8)*Y(8)
& - R*T*DLOG(X(lM»-R*T*DLOG(l.OD+OO-X(I»
& - R*T/3*DLOG(X(l)/(I.OD+OO-X(I»)
& - EG2-EGl/3
ENDIF
C
RElURN
END
C
C----- ------ ------ ---- ------ ---- ----------- -------- ---- ----------
A-31
C PROGRAM TO CALCULATE THE CARBON CONCENTRA nON IN AUSTENITE WHICH
C IS IN EITHER PARA- OR ORTHO- EQUlLffiRIUM WITH CEMENTITE.
C K1RKALDY'S METHOD IS USED.
C SEE HASHlGUCHI ET AL., CALPHAD VOL.8 NO.2(l984) PP173-186
C
C M. TAKAHASHI, 22.8.1989
C
IMPUOT REAL *8 (A-H,K-Z)
OOUBLE PREOSION C(8),Y(8),K(8),X(8)
&,C0(8),WO(8),CORTHO(8),WOR THO(8),CPARA(8),WP ARA(8)
COMMONffRANS/CO
READ(5,*) lALLOY
READ(5,*)(WO(l),I=l,7)
READ(5,*) CTEMPU,DTEMP,CTEMPL
CALL CONV(l,WO,CO)
WRITE(6,199) (W0(I),I=I,7)
WRITE(6,199) (CO(I),I=I,7)
199 FORMAT(4H C=,F7.4,4H SI=,F7.4,4H MN=,F7.4,
&4H NI=,F7.4,4H MO=,F7.4,4H CR=,F7.4,4H V=,F7.4)
WRlTE(6,*)
CC=O.ODtOO
00 11=1,7
CC=CC+CO(I)
CONTINUE
C0(8)=l.ODtOO-CC
KK=O.ODtOO
0021=2,7
K(1)=C0(I)/C0(8)
KK=KK+K(I)
2 CONTINUE
YY=O.OD-tOO
0031=2,7
Y(I)=K(I)/( l.ODtOOt-KK)
YY=YY+Y(I)
3 CONTINUE
Y(8)=l.OD-tOO-YY
C
C
WRlTE(6,*) T-ACM, C P-WT%C P-M.F.C O-WT%C'
&: O-M.F.C O-WT%X O-M.FX'
CTEMP=CTEMPU+DTEMP
200 CTEMP=CTEMP-DTEMP
IF (CTEMP .LT. CTEMPL) GOTO 300
T=CTEMP+273.0DtOO
CALL FUNCP(f,Y,K,CPARA)
CALL CONV(O,WPARA,CPARA)
CALL FUNCO(T,CORTHO)
CALL CONV(O,WORTHO,CORTHO)
WRITE(6,98) T -273.0D+OO,WP ARA(l ),CP ARA(l ),WORTHO(l ),CORTHO(l)
&,WORTHO(IALLOY),CORTHO(IAlLOY)
98 FORMAT(lH ,F8.2, 6012.4)
G0T0200
C
C
300 STOP
END
C
C----------------------------------------------------------------
C SUBROUTINE TO CALCULATE THE PARAEQU1L1BRlUM ACM TEMP.
C----- ---- ---- ---- ---- ------ ------ ------- ------ -------- ----------
C
SUBROUTINE FUNCP(T,Y,K,x)
IMPUCIT REAL *8 (A-H,K-Z)
OOUBLE PRECISION X(8),Y(8),E(7,7),W(7),Xl(3),K(8)
&,FUN (3),G(7),MM3C(8),MX (8)
R=8.314DtOO
XGA=O.005DtOO
C
C THERMODYNAMIC PARAMETERS
C IN ORDER OF SI,MN,NI,CR,MO,CU.
C
DG=46150.0D+OO/3.0-19.205D+OO*T/3.0
GFE3C=I.332D+04-64.718*T +7.481 *T*DLOG(T)-DG
G(2)=28535.0DtOO-OO
A-32
CC
G(3)=-14263.0D-l-OO+ 1O.OD+OO*T-DG
C G(3)=-13532.0D+OO-00
G(4 )=20338.0D+OO-2.368D+OO*T- DC
G(5)=-244I 8.0D+OO+ 16.61 D+OO*T-2. 749D+OO*T*DLOG(T)-00
G(6)=-19644.0D+OO-0.628*T -00
G(7)=28535.0D+OO-DC
E(I.1)=891O.0D+OOff
E(I.2)=4.84DtOO-7370.0DtOOff
E( 1,3)=-4811.0D+OO{f
E( I ,4)=-2.2D+OO+ 7600.0D+OOff
E( 1,5)=24.4- 38400.0DtOOff
E( 1.6)=3.855DtOO-17870.0D+OOff
E(l.7)=4200.0D+OOff
C
E(2,2)=26048.0D+OOff
E(3,3)=2.406D+OO-175.6D+OOff
C E(3,3)=O.2D+OO
E(4,4)=-721.7D+OOff
E(5 ,5)= 7.655D+OO-3154.0D+OOff -0.661 D+OO*DLOG(T)
E(6.6)=-2330.0D+OOff
E(7.7)=-0.16lD+OO-7834.0D+OOff
W(2)=O.ODtOO
W(3 )=8351.0D+OO-15 .188D+OO*T
W(4)=O.0D+OO
W(5)= 1791.0D+OO
W(6)=O.0D+OO
W(7)=O.ODtOO
C
C
DUMMY=l.OD-06
lOO X1(l)=XGA-DUMMY
X I(2)=XGA+DUMMY
XI(3)=XGA
00 10 1=1,3
XX=O.OD+OO
FUN(I)=O.OD+OO
00 1 II=2.7
X(II)=(I.ODtOO-XI(I»*Y(II)
XX=XX+X(II)
CONTINUE
X(I)=X1(1)
X(8)=l.OD+OO-X(l )-XX
C
C
MFE3C=R*T*DLOG(Y(8»
MFE=R *T*DLOG(X(8»-R *T/2.0*E(l.1 )*X( I )*X(I)
MC=R *T[3.0*DLOG(X(I»+R *T[3.0*E(I.1 )*X( I)
00 2 Jl=2.7
IF (X(Jl) .EQ. O.OD+OO)GOTO 2
MFE3C=MFE3C+(I- Y(8»*W(Jl)*Y(Jl)
MFE=MFE-R *T/2.0*E(Jl,Jl )*X(J I )*X(JI )-R *T*X( 1)*E{l.Jl )*X(ll)
MC=MC+R *T[3.0*E(l) I)*X(Jl)
MX(Jl)= R*T*DLOG(X(Jl»
& +R*T*(E(I )1)*X(l )+E(Jl )1)*X(ll»
MM3C(Jl)= R*T*DLOG(Y(Jl»
& +Y(8)*W(Jl)
00312=2.7
MM3C(Jl )=MM3C(11)- Y(8)*W(J2)*Y(J2)
3 CONTINUE
2 CONTINUE
C
FUN(I)=FUN(I)+X(8)*(GFE3C+MFE3C-MFE-MC)
00 15 13=2,7
FUN (I)=FUN(I)+ X(13 )*(G(13 )+MM3C(13)- MX(13 )-MC)
15 CONTINUE
C
10 CONTINUE
IF(ABS(FUN(3» .LT. 1.0D-06) GOTO 20
XGA=X I(3)-2.0*DUMMY*FUN (3)/(FUN (2)-FUN (I»
GOTOlOO
20 XGA=X1(3)
RETURN
A-33
CC
END
C
C----- ----- -- ------ ---- ------ --- ---- -------~--------- ------------
C SUBROUTINE TO CALCULATE THE EQUILIBRIUM ACM TEMP.
C----- ---- ---- ------ ---- ------ ------ --------- ---------- ----------
C
SUBROUTINE FUNCO(T,x)
IMPLICIT REAL *8 (A-H,K-Z)
COMMON{fRANS{CO
DOUBLE PRECISION X(8),Y(8),E(7,7),W(7),xl (3),K(8)
&,FUN (3),G(7),MM3C(8),MX(8 ),CO(8)
R=8.314D+OO
XGA=O.OID+OO
C
C THERMODYNAMIC PARAMETERS
C IN ORDER OF SI,MN,NI,CR,MO,CU.
C
00=46150.0D+OO/3.0-19.205D+OO*T{3.0
GFE3C=I.332D+04-64.718*T +7.481*T*DLOG(T)-00
G(2)=28535.0D+OO-00
G(3)=-14263.0D+OO+ 1O.OD+OO*T-oo
C G(3)=-13532.0D+OO-00
G(4 )=20338.0D+OO-2.368D+OO*T-DG
G(5)=-24418.0D+OO+ 16.61D+oo*T -2.749D+OO*T*DWG(T)-DG
G(6)=-19644.0D+OO-0.628*T-00
G(7)=28535.0D+OO-DG
E(1,1)=8910.0D+OO{f
E(l.2)=4.84D+OO-7370.0D+OOrr
E(1,3)=-4811.0D+OO{f
E( 1,4)=-2.2D+OO+7600.0D+OO{f
E(l,5)=24.4-384oo.0D+OO{f
E(l,6)=3.855D+OO-17870.0D+OO/T
E(l,7)=42oo.0D+OO{f
C
E(2,2)=26048.0D+OO{f
E(3,3)=2.406D+OO-175.6D+OO{f
C E(3,3)=O.2D+OO
E(4,4)=-721. 7D+OO{f
E(5 ,5)=7 .655D+OO-3154.0D+OO{f -0.661 D+OO*DLOO(T)
E(6.6)=-2330.0D+OO{f
E(7. 7)=-0.161 D+00-7834.0D+OO{f
W(2)=O.0D+OO
W(3)=8351.0D+OO-15.188D+OO*T
W(4)=O.0D+OO
W(5)=1791.0D+OO
W(6)=O.0D+OO
W(7)=O.OD+OO
C
C
DUMMY = 1.0D-06
100 X1(l)=XGA-DUMMY
X1(2)=XGA+DUMMY
X1(3)=XGA
DO 10 1=1,3
XX=O.OD+OO
YY=O.OD+OO
FUN(I)=O.OD+OO
DO 1 ll=2,7
B=DEXP«GFE3C-G(ll)- W(ll»/Rff +E( 1,JI)*X 1(I»
X(ll)=CO(II)*( 4.0*X 1(I)-1.0D+OO){
& (3.0*B*(X 1(l)-CO(l»+4.0*CO(l)-1.0D+OO)
Y(II)=B*X(ll)
XX=XX+X(II)
YY=YY+Y(II)
CONTINUE
X(I)=X1(1)
X(8)=1.0D+OO-X(l )-XX
Y(8)=1.0D+OO-YY
C
C
MFE3C=R *T*DLOG(Y(8»
MFE=R *T*DLOO(X(8»-R *T/2.0*E(1, 1)*X( 1)*X(l)
A-34
MC=R*T/3.0*DLOO(X(I »+R *T/3.0*E(1 ,I)*X(I)
00211=2,7
IF (X(11) .EQ. 0.00+00) GOTO 2
MFE3C=MFE3C-+{I- Y(8»*W(11)*Y(J I)
MFE=MFE-R *T/2.0*E(JI,11 )*X(J 1)*X(11 )-R *T*X( 1)*E(I,11 )*X(J1)
MC=MC+R *T/3.0*E(I,J 1)*X(11)
MX(11)= R*T*DLOO(X(Jl)
& +R*T*(E(I,JI)*X(I)+E(11 ,JI)*X(J1»
MM3C(J1)= R*T*DLOO(Y(J1»
& +Y(8)*W(J1)
00312=2,7
MM3C(J1 )=MM3C(J1)- Y(8)*W(J2)*Y(J2)
3 CONTINUE
2 CONTINUE
C
FUN(I)=GFE3C+MFE3C-MFE-MC
C
10 CONTINUE
IF(ABS(FUN(3» .LT. 1.0D-06) GOTO 20
XGA=X 1(3)-2.0*DUMMY* FUN (3)/(FUN (2)- FUN (I»
GOTOlOO
20 XGA=XI(3)
RElURN
END
C
C-- -- ----- -- --- ------- -------- ------ ---- ------- --- ----- ---- --- ---
C SUBROUTINE TO CONVERT WT% TO MOLE FRACTION
C OR MOLE FRACTION TO WT%.
C ---- ----- ----- -- ----- --- ------- ----- --- ----- ----- -- --- ------- ---
C
SUBROUTINE CONV(N,W,C)
IMPUCIT REAL *8 (A-H,O-Z)
OOUBLE PRECISION AN(8), W(8), C(8), A(8)
AN(I)=12.0115D+OO
AN(2)=28.09D+OO
AN(3)=54.94D+OO
AN(4)=58.71D+OO
AN(5)=95.94D+OO
AN(6)=52.00D+OO
AN(7)=50.94D+OO
AN(8)=55.84D+OO
IF (N .EQ. 0) GOTO 1
W(8)= 100.00+00- W( 1)-W(2)- W(3)- W(4)- W(5)- W(6)- W(7)
AT=O.OD+OO
0021=1,8
A(I)=W (IYAN(I)
AT=AT+A(I)
2 CONTINUE
00 31=1,8
C(I)=W(IYAN(I)/ AT
3 CONTINUE
G0T04
C(8)= I.OD+OO-C( 1)-C(2)-C(3)-C( 4 )-C(5)-C(6)-C(1)
AT=O.OD+OO
0051=1,8
A(I)=C(I)* AN(I)
AT=AT+A(I)
5 CONTINUE
00 61=1,8
W (I)=C(I)* AN (I)/A1'*100.00+00
6 CONTINUE
4 REI1JRN
END
C--------- ---- -------- -- ------ ------- ------ -------- ---------- ----
A-35
CC PROORAM TO CALCULATE llIE GROWTI-I RATE OF PEARUTE ASSUMING 1lIE BUlK OR
C INTERFACE DIFFUSION CONTROL.
C
C GAMMNGAMMA+ALPHA AND GAMMA/GAMMA+ THETA INTERFACE COMPOsmONS
C ARE CALCULATED USING KlRKAillYS APPROXIMATE METHOD.
C
C 19915.12 M.TAKAHASHI
C
IMPUCIT REAL *8 (A-H,K-Z)
DOUBLE PRECISION X(8),Y(8),C0(8),WO(8),WW(8),K(8),GO(8)
&,G (8 ),G 2(8), G3(8),G4( 8),G M(8) ,LFX( 4,8) ,AC(8 ),E(8,8) ,EA( 8,8)
COMMONffRANS/CO,WO,K
COMMON/HILUG2,LFX,LCV,GM,AC,DDG,GFEC
& ,GFE3C,G3,G4,WW,E,EA,G,GO
READ(5,*) ISAMPLE
IS=O
10 IS=IS+1
IF (IS .GT. ISAMPLE) GOTO 100
READ(5,*) (WO(I),I=l,7)
READ(5,*) TI,DT,TF
CALL CONV(l,WO,CO)
C
WRITE(6,*) 'INITIAL COMPOSmON'
WRITE(6,l99)
WRITE(6,198) (WO(I),I=l,7)
WRITE(6,198) (CO(I),I=l,7)
199 FORMAT(lH,' C SI MN NI'
&,' CR MO CU ')
198 FORMAT(lH ,7F9.5)
WRITE(6,*)
C
CALL OMEGA(W)
C
CC=O.OD+OO
DO 11 1=1,7
CC=CC+CO(I)
11 CONTINUE
CO(8)= 1.0D+00-CC
KK=O.OD+OO
DO 121=2,7
K(I)=CO(I)/C0(8)
KK=KK+K(I)
IF (CO(I) .EQ. 0.00+99) GOTO 12
IM=I
12 CONTINUE
YY=O.OD+OO
DO 13 1=2,7
Y(I)=K(I)/(1.0D+OO+KK)
YY=YY+Y(I)
13 CONTINUE
Y(8)=l.OD+OO-YY
C
C
CfEMP=TI+DT
20 CfEMP=CTEMP-DT
T=CfEMP+273.0D+OO
IF (CfEMP .LT. TF) GOTO 100
IFLAGF=O
IFLAGC=O
C
C
C THERMODYNAMIC PARAMETERS IN ORDER OF SI,MN,NI,CR,MO,CU
C----------------------------------------------------------------
C THERMODYNAMIC PARAMETERS
C IN ORDER OF SI,MN,NI,CR,MO,CU.
C
DG=46150.0D+00/3.0-19.205D+OO*T/3.0
GFE3C=I.332D+04-64.718*T +7.481 *PDLOG(f)-DG
SFG=-11.906D-04*T +8.272D-06*T*T-15.079D-09*PT*T
& +12.857D-12*T*T*T*T
GO(l )=-65562.0D+OO+ 32.949*T
GO(3 )=-20520.0D+OO+4.086D+OO*T +1500.00+00* SFG
GO(5)=-1534.0D+00-19.472*T +2.749*T*DLOO(f)
A-36
G(2)=28535.0D+OO-00
G(3 )=-14263.0D+OO+ IO.OD+OO*T-oo
C G(3)=-13532.0D+OO-00
G(4 )=20338. OD+OO-2.368D+OO*T -00
G(5)=-24418.0D+OO+ 16.61 D+OO*T-2. 749D+OO*T'"DLCXJ(T)-DG
G(6)=-19644.0D+OO-0.628*T -00
G(7)=28535.0D+OO-00
C
C E(l,1 )=4.7859D+OO+5066.0D+OO[f
E(l,1 )=891O.0D+OO[f
C E(I,2)=4.84D+OO-7370.0D+OO[f
E(l,2)=14795.0D+oorr
E( I ,3)=-481 1.0D+oorr
E(I,4 )=-2.2D+OO+ 7600.0D+OO[f
E( 1.5)=24.4-38400.0D+oorr
E(l.6)=3.855D+OO-17870.0D+OOrr
E(l.7)=4200.0D+OO[f
C
E(2,2)=26048.0D+oorr
E(3.3)=2.406D+OO-175.6D+OO[f
C E(3.3)=O.2D+OO
E(4,4 )=-721. 7D+oorr
E(5.5)=7.655D+OO-3154.0D+OOrr -0.661 D+OO*DLCXJ(T)
E( 6.6)=- 2330.0D+OOrr
E(7 .7)=-0.161 D+OO-7834.0D+OO[f
C
EA(l.1 )=1.3D+OO
EA(2.2)=-13.31 D+00+44088.0rr
EA(3.3)=3 .082D+OO-4679.0[f + 1509.8*SFGrr
EA(4,4)=2.041 D+OO-2478.0[f +385.5*SFG[f
EA(5 ,5)=2.819D+OO-6039 .O[f
EA(6.6)=-0.219D+OO-4772.0[f +402.6*SFGrr
EA(7.7)=0.634D+OO-11270.0[f +IOO6.5*SFG[f
C
WW(2)=0.OD+OO
WW(3)=8351.0D+OO-15.188D+OO*T
WW(4)=0.OD+OO
WW(5)=1791.0D+OO
WW(6)=0.OD+OO
WW(7)=0.OD+OO
C----------------------------------------------------------------
G2(2)=123000D+OO
G2(3)=-48500D+OO
G2(4)=46000D+OO
G2(5)=-251160D+OO+118.0D+OO*T
G2(6)=-267200D+OO
G2(7)=-46000.0D+OO+ 55 .OD+OO*T
G3(2)=404180.OD+OO
G3(3)=-145500.0D+OO
G3(4)=138000.0D+OO
G3(5)=-153640.0D+OO+ 38.860*T
G3( 6)=- 267200.0D+OO
G3(7)=-46000.0D+OO+55.0*T
G4(l )=-65563.0D+OO+23.815*T
G4(2)=-64818.0D+OO+38.543*T
G4(3 )=-1800.0D+OO+ 1.276*T
G4( 4 )=-5650.0D+OO-3 .35 *T
G4(5)= I0460.0D+OO+O.628*T
G4(6)= I 0460.0D+OO+O.628*T
G4(7)=-627 6.OD+OO+3.34 7*T
G4(8)=( -I.OD+OO)*ENERGY(T ,n 0.T20)
C IF (T .LT. 1000.00+00) THEN
C G4(8)=8933.0D+OO-14.406*T +12.083D-03*T*T -11.51 D-06*T'"T'"T
C & +5.23D-09*T'"T'"T'"T
C ELSE
C G4(8)= 71659.0D+OO-216.84*T +24.773D-02*T'"T -12.661 D-05*T'"T'"T
C & +24.397D-09*T'"T'"T*T
C ENDIF
C ----- -- ---- -------- ---- ---- ------ --------- -------- ------ --------
C
C CALCULA nON OF PARAEQUlllBRlUM COMPOSmONS (GAMMNGAMMA+ALPHA)
C
C
CALL PEQGAO(f :X,Y,IM)
A-37
C CALL PEQGA(T :X,Y,IM)
C
PXIGA=X(1)
PX2GA=X(lM)
PXIAG=Y(I)
PX2AG= Y(lM)
C
C
C CALCULATION OF PARAEQUlllBRlUM COMPOS mONS (GAMMNGAMMA+ THETA)
C
C
C
CALL PEQGCO(T )(,Y ,IM)
PXIGC=X(l)
PX2GC=X(IM)
PXICG=O.25D+OO
PX2CG=Y(lM)
SC=5.9D-04/(1 OOO.OD+OO-T)/2
SC=I.OD-04/2.0D+OO/(131.12D+OO-0.1783*CTEMP
& +4.238D+OO*WO(IM»
SC=2.0*600.0* 1000.0/6.09D09/( 1OOO.OD+OO-CTEMP)
S=I.OD-04/(127.351 D+OO-0.17368D+OO*CTEMP-4.9195*W0(3)
& +1.7868*W0(5»
CCC=(PXI GA+PX 1GC)/2.0
D=DIFF(T,CCC,CCC* I.OOI,W)
C
C
C
C CALCULATION OF VOLUME DIFFUSION CONTROLLED GROwrn RATE OF PEARLITE
C
C
C
C
C
VELl= VD(PX 1GA,PX 1GC,PX 1AG,SC,D,S)
WRITE(6,*) ••
WRITE(6,*) 'Paraequilibrium carbon diffusion control'
WRITE(6,99) CTEMP,PXI GC,PX2GC,PX 1CG,PX2CG
WRITE(6,69) PXI GA,PX2GA,PXI AG,PX2AG
WRITE(6,68) VELl,D
99 FORMAT(1H ,'AT Temp=',F6.1,' CGC=',F8.6,' XGC=',
&F8.6,' CCG=',F8.6,' XCG=',F8.6)
69 FORMAT(lH,' CGA=',F8.6,' XGA=',
&F8.6,' CAG=',F8.6,' XAG=',F8.6)
68 FORMAT(lH,' V(bulk diff orc) 1 cm/sec=',D12.5
&,' Diff. Coef=',DlO.4)
C
C
C CALCULATION OF THE INTEFACE COMPOSmONS (GAMMMJAMMA+ALPHA)
C IDENTICAL ISOACTIVITY OF CARBON IN GAMMA IS ASSUMED
C
C CALCULATION OF ACTIVI1Y OF CARBON IN GAMMA
C
ACTIV =ACGO(T ,CO(1),CO(IM),E( 1,1),E( 1,IM»
write(6,*) 'ACTIV=',ACTIV
C
C
CALL NPGAO(T )(,Y,lM)
XI GANP=X(l)
X2GANP=X(lM)
XIAGNP=Y(l)
X2AGNP=Y(lM)
X2TRANS=X2ACO(T )( 1GANP ,ACTIV ,E(l, 1),E(l,IM»
write(6,*) 'X2GANP)(2TRANS = ')(2GANP)(2TRANS
IF (X2GANP .LT. X2TRANS) THEN
WRITE(6,*) 'ENTERING NPLE REGIME FOR FERRITE .... .'
IFLAGF=1
XIGA=XIGANP
X2GA=X2GANP
XIAG=XIAGNP
X2AG=X2AGNP
ELSE
CALL EQGAO(T )(,Y,ACTIV,lM)
XIGA=X(l)
X2GA=X(lM)
XIAG=Y(1)
X2AG=Y(lM)
A-38
ENDIF
C
C CALCULATION OF THE INfER FACE COMPOSITIONS (GAMMA/GAMMA+ THETA)
C IDENTICAL ISOACTIVIlY OF CARBON IN GAMMAA T THE INTERFACES IS ASSUMED
C
C
CALL NPGCO(T X,Y ,IM)
XIGCNP=X(I)
X2GCNP=X(lM)
XICGNP=O.25D-tOO
X2CGNP=Y(lM)
X2TRANS=X2ACO(T Xl GCNP ,ACTI V,E(l,1 ),E(l,IM))
IF(X2GCNP .GT. X2TRANS) THEN
WRITE(6,*) 'ENTERING NPLE REGIME FOR CEMENTITE .....:
IFLAGC=I
XIGC=XIGCNP
X2GC=X2GCNP
XICG=XICGNP
X2CG=X2CGNP
ELSE
CALL EQGCO(T X,Y,ACTIV,IM)
XIGC=X(l)
X2GC=X(lM)
X ICG=0.25D+00
X2CG=Y(lM)
ENDIF
C
C CALCULATION OF BOUNDARY DIFFUSION ffiNTROll..ED GROWTH RATE OF PEARLITE
C
VEL2= VBD(T X2GAX2GC,SC,CO(IM),S,IM)
C
50 WRITE(6,*) 'Boundary diffusion controlled growth'
WRITE(6,98) CTEMP X IGCX2GCX ICG,X2CG
WRITE(6,67) XIGAX2GAXIAGX2AG
WRITE(6,66) VEL2
98 FORMAT(lH ,'AT Temp=',F6.1,' CGC=',F8.6: XGC=',
&F8.6,' CCG=',F8.6,' XCG=',F8.6)
67 FORMAT(lH,' CGA=',FS.6,' XGA=',
&F8.6,' CAG=',F8.6,' XAG=',F8.6)
66 FORMAT(lH,' V(boundary diff of X) /cm/sec=',DI2.5)
C
C
G0T020
C
lOO END
C
C----------------------------------------------------------------
C SUBROtJIlNE TO CALCULATE INTERFACE ffiMPOSITIONS (GAMMNGAMMA+ALPHA)
C Kirkaldy's approximate method is used to obtain the flrst
C guess value of NPLE calculation
C
SUBROUTINE NPGAO(T X,Y ,IM)
IMPUCIT REAL*8 (A-H,K-Z)
DOUBLE PRECISION X(8),Y(8),CO(8),WO(8),WW(8),K(8)
&XX(3 ),G(8) ,G2(8),G 3(8 ),G4(8), GM (8), GO(8)
&,LFX (4,8),AC (8),E(8,8) ,EA(8 ,8),FUN (3)
COMMON/TRANS/CO,WO,K
COMMON/HILl/G2,LFX,LCV,GM,AC,DDG,GFEC,
& GFE3C,G3 ,G4,WW,E,EA,G ,GO
C
C write(6,*) 1M=',IM
R=8.314D-tOO
XGA=CO(l )+{).OlD+OO
ITER=O
ACC=1.0D-05
DXI=l.OD-lO
200 XX(I)=XGA-DXI
XX(2)=XGA+DX 1
XX(3)=XGA
ITER=ITER+I
IF(ITER .GT. lO) THEN
ACC=ACC*10
ITER=O
ENDIF
A-39
CDO 220 1=1,3
Xl=XX(l)
C
Al=DEXP(GO(l)/Rff +E(l ,l)*Xl)/
& (l.OD+OO+EA(l,l )*Xl*DEXP(GO(l)/Rff»
A2=DEXP(GO(IM)/Rff +E(l ,IM)*X 1)/
& (l.OD+OO+E(l,IM)*Xl*DEXP(GO(l)/Rff»
C write(6,*) 'Al,A2=',Al,A2
Yl=Al*Xl
C
Y2=CO(IM)
X2=Y7/A2
C write(6,*) 'X2AC="X2
XO=l.OD+OO-Xl-X2
YO=l.OD+OO-YI-Y2
C
FFG=DLOG(X0)-1.0/2.0*E(l,l )*X l*X 1-1.0/2.0*E(IM,IM)*X2*X2
& -E(l ,IM)*X 1*X2
FFA=DLOG(YO)-1.0/2.0*EA(l,l )*Yl*Yl-l.0/2.0*EA(lM ,lM)*Y2*Y2
& -E(1,IM)*Yl*Y2
C
FUN(I)=G4(8)/Rff +FFG-FFA
C FUN(I)= YO-XO*DEXP(G4(8)/Rff
C & +EA(l,l )/2.0*Yl*Yl +E(l,IM)*Yl*Y2
C & +EA(IM,IM)/2.0*Y2*Y2
C & -E(l,l)/2.0*Xl*Xl-E(l,IM)*Xl*X2
C & -E(IM,IM)/2.0*X2*X2)
220 CONTINUE
IF (ABS(FUN(3» .LT. ACC) GOTO 100
DUMMY =XGA-2.0*DX 1*FUN(3)/(FUN(2)-FUN(1»
C write(6,*) 'DUMMY=',DUMMY
C write(6,*) 'FUNl ,2,3=',(FUN(Ill).rn= 1,3)
IF (DUMMY .LT. 0.00+00) THEN
DUMMY =XGA/2.00+00
ENDIF
IF (DUMMY .GT. 0.25D+OO) THEN
DUMMY =(0.25D+OO+XGA)/2.0
ENDIF
XGA=DUMMY
C write(6,*) 'FUN3,xl=',FUN(3),XGA
GOT02oo
100 X(l)=Xl
X(IM)=X2
Y(l)=Yl
Y(IM)=Y2
C
RETURN
END
C
C----------------------------------------------------------------
C SUBROUTINE TO CALCULATE INTERFACE COMPOsmONS (GAMMNGAMMA+Al.PHA)
C Kirkaldy's approximate method is used to obtain the fIrst
C guess value.
C
SUBROUTINE EQGAO(T ,x,Y,ACTIV ,IM)
IMPUCIT REAL*8 (A-H,K-Z)
DOUBLE PRECISION X(8), Y(8),CO(8),WO(8),WW(8),K(8)
& ,xX(3 ),G(8) ,G2(8), 03(8) ,G4(8), GM(8), GO(8)
&,LFX (4,8 ),AC(8 ),E(8 ,8),EA(8,8) ,FUN (3)
COMMON/TRANS/CO,WO,K
COMMONIHILLlG2,LFX,LCV,GM,AC,DDG,GFEC,
& GFE3C,03,G4,WW,E,EA,G,GO
C
R=8.314D+OO
XGA=CO(l )*1.10+00
DX1=1.0D-06
200 XX(l)=XGA-DXl
XX(2)=XGA+DXl
XX(3)=XGA
DO 2201=1,3
Xl=XX(l)
Al=DEXP(GO(l)/Rff +E(l,l)*Xl)/
& (l.OD+OO+EA(l,l)*Xl*DEXP(GO(l)/Rff)
A-40
CA2=DEXP(GO(lM)/Rff +E(I,IM)"'Xl)/
& (1.0D+OO+E(l,IM)"'Xl"'DEXP(GO(l)/Rff»
Yl=Al"'Xl
C
X2=X2ACO(T,Xl ,ACTIY,E(l ,1),E(l,lM»
C X2=X2AC(T ,xl,ACTIY,LCY,G2(lM»
C X2=CO(lM)"'(1.0D-+OO-Al )"'XII
C & «1.0D-+OO-A2)"'CO(I)+(A2-Al)"'Xl)
Y2=A2*X2
XO=1.0D+OO-XI-X2
YO=1.0D-+OO-Y 1-Y2
C
FFG=DLOG(X0)-1.0/2.0"'E(l,l)"'XI"'XI-I.0/2.0"'E(lM,IM)"'X2"'X2
& -E(l,IM)*Xl*X2
FFA=DLOG(YO)-1.0/2.0*EA(l,l )"'YI"'YI-I.0/2.0*EA(lM ,IM)*Y2*Y2
& -E( I ,IM)*Y I*Y2
C
FUN(1)=G4(8)fRff +FFG-FFA
C FUN(1)= YO-XO*DEXP(G4(S)fRff
C & +EA(l,l)/2.0*Yl*Yl+E(l,lM)*Yl*Y2
C & +EA(lM,IM)/2.0*Y2*Y2
C & -E(I,1 )/2.0'" X 1*XI-E(l,lM)*Xl *X2
C & -E(lM,lM)/2.0*X2"'X2)
220 CONTINUE .
IF (ABS(FUN(3» .LT. 1.0D-06) GOTO lOO
DUMMY =XGA-2.0"'DX 1*FUN(3)/(FUN(2)-FUN(I»
IF (DUMMY .LT. 0.00-+00) THEN
DUMMY =XGA/2.00-+00
ENDIF
IF (DUMMY .GT. 1.0D-+OO)THEN
DUMMY =( I.OD+OO+XGA)/2.0
ENDIF
XGA=DUMMY
GOT02oo
lOO X(l)=Xl
X(lM)=X2
Y(I)=Yl
Y(lM)=Y2
C
RElURN
END
C
C--- -- ------ ---- -- ------ ---- -- ------ --------- --------------------
C SUBROUTINE TO CALCULATE INTERFACE COMPOSmONS (GAMMNGAMMA+ALPHA)
C Kirkaldy's approximate method is used to obtain the flISt
C guess value. -paraequilibrium-
C
SUBROUTINE PEQGAO(T ,x,Y,IM)
lMPUCITREAL*S (A-H,K-Z)
DOUBLE PRECISION X(S),Y(S),CO(S),WO(S),WW(S)
&,K(S),XXX(3 ),FUN 1(3) ,G(S) ,G2(S), G3(S),G4(S) ,GM(S)
&,LFX( 4,S) ,AC(S),E(S,S),EA(S,S) ,GO(S)
COMMONITRANS/CO,WO,K
COMMON/HILUG2,LFX,LCY,GM,AC,DDG,GFEC
& ,GFE3C,G3,G4,WW,E,EA,G,GO
C
R=S.31 4D-+OO
C
Xl=O.OlD-+OO
DX1=1.0D-06
200 XXX(I)=XI-DXl
XXX(2)=X1 +DX1
XXX(3)=Xl
DO 2201=1,3
Xl=XXX(1)
Al=DEXP(G4(1)/Rff +E(l,I)"'Xl)/
& (1.0D+OO+EA(I,I)*Xl *DEXP(G4(l)/Rff»
Yl=Al"'Xl
X2=( 1.0D-+OO-Xl)*K(lM)/(l.OD+oo+K(1M»
Y2=( 1.0D-+OO-Yl )*K(lM)/(I.OD+oo+K(IM»
XO=1.0D-+OO-XI-X2
YO=1.0D+OO-YI-Y2
C
A-41
FFG=R *T*DLOG(XO)-R *T/2.0*E( 1,1)*X I *XI
& -R*T/2.0*E(IM,IM)*X2*X2
& -R*T*E(I,IM)*XI *X2
FFA=R*T*DLOG(YO)-R*T/2.0*EA(l,I)*YI *YI
& -R*T/2.0*EA(lM,IM)*Y2*.Y2
& -R*T*E(l,IM)*YI*Y2
F2G=R *T*DLOG(X2)+R *T*(E( I,IM)*XI +E(lM,IM)*X2)
F2A=R *T*DLOG(Y2)+R *T*(E(I,IM)*YI +EA(IM,IM)*Y2)
FUN I (I)=XO* (G4(8 )+FFG-FF A)+X2 *(G4(IM)+F2G- F2A)
C
220 CONTINUE
IF (FUNI(3) .LT. 1.0D-06) GOTO 300
DUMMYI=XI-2.0*DX I*FUNI (3)/(FUNI (2)-FUN I(I»
IF (DUMMYI .LT. O.OD+OO)THEN
DUMMY I=X 1/2.00+00
ENDIF
XI=DUMMYI
GOT0200
300 X(l)=XI
X(IM)=X2
Y(l)=YI
Y(IM)=Y2
C
RETURN
END
C
C--------. ---------------------------------------.- .. ---... -. ----
DOUBLE PRECISION FUNCTION ACGO(T ,X1,X2,E I1 ,E12)
IMPUCIT REAL*8 (A-H,K-Z)
C
ACGO=DEXP(DLOG(XI)+EII *XI+EI2*X2)
C
RETURN
END
C
C-----------.- ----.--.-- -----------------------------------------
DOUBLE PRECISION FUNCTION X2ACO(T ,X I,A,E II ,E12)
IMPUCIT REAL*8 (A-H,K-Z)
C
X2ACO=(DLOG(A)-DLOG(XI)-EII *XI)/E12
C
RETURN
END
C
C--- -- ---- ---- ------ ---- ---- ---- --------- ------- ----- ------------
DOUBLE PRECISION FUNCTION ENERGY(T,TlO,T20)
DOUBLE PRECISION T,TlO,T20,F,T7
T7=T-1.00+02*T20
IF (T7 .LT. 3.00+(2) GOTO I
IF (T7 .LT. 7.0D+(2) GOTO 2
IF (T7 .LT. 9.210+(2) GOTO 3
IF (T7 .LT. 1.0D+(3) GOTO 5
F=(3.381 D+02-3.31 D+OO*(T7 -1.OD+03 )+9 .83 D-03* (T7 -9 .9999999D+02
&)** 1.96D+OO-7 .11 D+OO*DSIN(0.034*(T7 -1.OD+(3»)/( -4.1 87D+OO)
GOT04
F= 1.38D+OO*T7 -1.4990+03
G0T04
2 F=1.65786D+OO*T7-1.58ID+03
G0T04
3 F= 1.30089D+OO*T7 -1.33ID+03
G0T04
5 F=-I.OD+OO*(l.20D+02-(T7-9 AD+02)*(0.73333333333D+OO»
4 ENERGY=(1.41D+02*TlO + F)*4.187D+OO
RETURN
END
C
C----- ---- ------ ---- -- ------ ---- ---- ----- -- ------------ ----------
C FUNCTION TO CALCULATE PARAEQun..mRIUM CARBON DIFFUSION
C CONTROLLED GROWTII RATE OF PEARUTE
C
DOUBLE PRECISION FUNCTION VD(XGA,XGC,XAG,SC,O,S)
IMPUCIT REAL*8 (A-H,K-Z)
A=O.72D+OO
SA=S*7.0D+OO/8.0D+OO
A-42
SCEM=S/8.0D+OO
CCEM=O.250+00
VD=D/ A*S**21SNSCEM*(XGA-XGC)/
& (CCEM-XAG)/S*(l.OD+OO-SC/S)
REruRN
END
C
C----------- ------ ---- ------ ------ ------------- -------- ------ ----
C FUNCllON TO CALCULATE BOUNDARY DIFRJSION CONTROLLED GROWTH RATE
COFPEARUTE
C
ooUBLE PRECISION FUNCllON VBD(T ,XGA,XGC,SC,XBAR,SJM)
IMPUCIT REAL*S (A-H,K-Z)
ooUBLE PRECISION CK(S),Q(S)
CK(3)=1.l26D-OS
CK(5)=2.455D-09
Q(3)=14117S.0D+OO
Q(5)=13S516.0D+OO
R=S.314D+OO
SA=S*7.0D+OO/S.OD+OO
SCEM=S/8.0D+OO
KDD=CK(IM)*DEXP( -Q(lM)/RfI)
VBD= 12.0D+OO*KDD/SNSCEM*(XGA-XGC)/
& XBAR*(1.0D+OO-SC/S)
RETIJRN
END
C
C----------------------------------------------------------------
C SUBROUTINE TO CONVERT Wf% TO MOLE FRACTION
C OR MOLE FRACTION TO Wf%.
C
SUBROUTINE CONV(N,Wf,C)
IMPUCIT REAL*S (A-H,O-Z)
ooUBLE PRECISION AN(S), Wf(S), C(S), A(S)
AN(l)=12.0115D+OO
AN(2)=2S.09D+OO
AN(3)=54.94D+00
AN(4)=5S.71D+00
AN(6)=95.94D+00
AN(5)=52.ooD+00
AN(7)=63.55D+00
AN(S)=55.84D+OO
IF (N .EQ. 0) GOTO 1
WT(S)= 100.00+00- WT(l)- WT(2)- Wf(3)- WT(4)- WT(5)- WT(6)- Wf(7)
AT=O.OD+OO
oo21=I,S
A(I)=WT(I)/AN(I)
AT=AT+A(I)
2 CONTINUE
00 3 1=I,S
C(I)=WT(lYAN(I)/AT
3 CONTINUE
GOT04
C(S)= 1.0D+OO-C( 1)-C(2)-C(3)-C( 4)-C(5)-C(6)-C(7)
AT=O.OD+OO
00 5 1=I,S
A(I)=C(I)* AN(I)
AT=AT+A(I)
5 CONTINUE
00 6 1=I,S
WT(I)=C(I)* AN(I)/AT*I00.0D+OO
6 CONTINUE
4 RETIJRN
END
C
C----------------------------------------------------------------
C FUNCllON GIVING LFG LN(ACTIVITY) OF CARBON IN AUSTENITE.
C
ooUBLE PRECISION FUNCTION CG(X,T,W,R)
ooUBLE PRECISION J,OO,DUMMY,T J{,W,X
J=I-DEXP(-W/(R*n)
OO=DSQRT( 1-2*(1 +2* J)*X+(1 +S* J)*X*X)
DUMMY =5*DLOG« 1-2*X)/X)+6*W/(R *T)+«3S575.0)-(
&13.4S)*T)/(R*T)
A-43
CG=DUMMY +DLOG«(oo-1 +3*X)/(00+ 1-3*X»**6)
RElURN
END
C
C----- -- ---- ---- ------ ------ ---- --------- -------- ---- ------------
C RJNCTION GIVING DIFFERENTIAL OF LN(ACITVITY) OF CARBON IN
C AUSTENITE. DIFFERENTIAL IS wrrn RESPECf TO X.
C
DOUBLE PRECISION RJNCTION DCG(X,T,W,R)
DOUBLE PRECISION J,OO,Doo,X,T,W,R
J=I-DEXP(-W/(R*n)
00=DSQRT(l-2*(l +2*J)*X+(1 +8*1)*X*X)
DDG=(0.5/DG)*( -2-4 *J+2*X + 16*J*X)
OCG=-«1 0/(1-2*X»+(5/X»+6*«OOO+ 3)/(00-1 +3*X
&)-(000-3)/(00+ 1-3*X»
RE1URN
END
C
C----- -- ------ ------ ------ ------ --------- -------- ---------- ------
SUBROUTINE OMEGA(W)
C SUBROUTINE TO CALCULATE THE CARBON CARBON INTERACTION ENERGY IN
C AUSTENITE, AS A RJNCTION OF ALLOY COMPOSmON. BASED ON .MUCG 18
C THE ANSWER IS IN JOULES PER MOL. **7 OCTOBER 1981**
COMMON/TRANS/CO,WO,K
OOUBLE PRECISION C(8),W,P(8),B 1,B2,Y(8),T10,TIO,B3,XONE
&,CO(8),WO(8),K(8)
INTEGERI,U
DO 1 1=1,8
C(I)=WO(I)
OONT1NUE
B3=0.00+00
C(8)=C(I)+C(2)+C(3)+C(4)+C(5)+C(6)+C(7)
C(8)= 100.0D+OO-C(8)
C(8)=C(8)/55.84D+OO
C(1)=C(I)/12.0115D+OO
C(2)=C(2)/28.09D+OO
C(3)=C(3)/54.94D+OO
C(4)=C(4)/58.71D+OO
C(5)=C(5)/95.94D+OO
C(6)=C(6)/52.0D+OO
C(7)=C(7)/50.94D+OO
B 1=C( 1)+C(2)+C(3 )+C( 4)+C(5)+C(6)+C(7)+C(8)
DO 107 U=2,7
Y(U)=C(U)/C(8)
107 CONTINUE
DO 106 U=I,8
C(U)=C(U)/B 1
106 CONTINUE
XONE--C(l)
C XONE=DINT(lOOOO.OD+OO*XONE)
C XONE=XONE/l0000
B2=O.OD+OO
T1 0=Y(2)*( -3)+ Y(3)*2+ Y(4 )*12+ Y(5)*( -9)+ Y(6)*( -1)+ Y(7)*( -12)
TIO=-3* Y(2)-37 .5*Y(3 )-6*Y( 4 )-26*Y(5)-19*Y( 6)-44 *Y(7)
P(2)=20 13.0341 +763.8167*C(2)+45802.87*C(2)**2-280061.63*C(2)**3
&+3.864 D+06 *C(2) **4- 2.42~3 D+07*C (2)* *5+6.9547 D+07 *C(2) **6
P(3)=2012.067 -1764.095*C(3)+6287 .52*C(3)**2-21647 .96*C(3 )**3-
&2.0119D+06*C(3)**4+3.1716D+07*C(3)**5-1.3885D+08*C(3)**6
P(4 )=2006.80 17+2330.2424*C( 4)-54915 .32*C(4 )**2+ 1.6216D+06*C(4 )**3
&-2.4968D+07*C(4)**4+ 1.8838D+08*C( 4)**5-5 .5531D+08*C(4)* *6
P(5)=2006.834-2997 .314 *C(5)-37906.61*C(5)**2+ 1.0328D+06*C(5)* *3
&-1.3306D+07*C(5)**4+8.411 D+07*C(5)**5-2.0826D+08*C(5)**6
P(6)=2012.367 -9224.2655*C(6 )+33657.8*C(6)**2-566827 .83*C(6 )**3
&+8.5 676D+06*C( 6)**4-6. 7482D+07 *C( 6)** 5 +2.083 7D+08*C( 6)* *6
P(7)=20 11.9996-6247 .9118*C(7)+5411. 7566*C(7)**2
&+250118.1085*C(7)**3-4.1676D+06*C(7)**4
00 108 U=2,7
B3=B3+P(U)*Y(U)
B2=B2+Y(U)
108 CONTINUE
IF (B2 .EQ. O.OD+OO)GOTO 455
W=(B3/B2)*4.187
GOT0456
455 W=8054.0
AM
456 RETURN
END
C
C----------------------------------------------------------------
C FUNCTION GIVING THE EQUILIBRIUM MOLE FRACTION OF CARBON IN
C FERRITE.
C
OOUBIE PRECISION FUNCTION XALPH(I)
OOUBLE PRECISION T,CTEMP
CTEMP=(T -273 .OD+OO)/900.0D+00
XALPH=0.1528D-02-0.8816D-02*CTEMP+O.2450D-0 I *CTEMP*CTEMP
&-0.241 7D-01 *CTEMP*CTEMP*CTEMP+
&0.6966D-02*CTEMP*CTEMP*CTEMP*CTEMP
RETURN
END
C
C----- ---- ------ ---- ---- ------ ------ --------- -------- ------------
C SUBROUTINE TO CALCULATE THE PARAEQUILIBRIUM INTERFACE CHEMISTRY
C
SUBROUTINE PEQGCO(T X,Y ,lM)
IMPUCIT REAL*8 (A-H,K-Z)
DOUBLE PRECISION X(8),Y(8),C0(8),WO(8),WW(8)
&,K(8),X I (3 ),FUN(3 ),G(8),G2(8),G 3(8),G4(8),GM(8)
&,LFX( 4,8) ,AC(8),E(8,8), EA(8,8) ,GO(8)
COMMONffRANS/CO,WO,K
COMMON/HILL/G2,LFX,LCV,GM,AC,DDG,GFEC
& ,GFE3C,G3,G4,WW,E,EA,G,GO
R=8.314D+OO
XGA=O.005D+OO
C
C
DUMMY=1.0D-08
100 XI(I)=XGA-DUMMY
Xl(2)=XGA+DUMMY
Xl(3)=XGA
00 101=1,3
XX=O.OD+OO
FUN(I)=O.OD+OO
KK=K(IM)
Y(IM)=KK/(1 +KK)
X(IM)=(l-X 1(I»*Y(IM)
X(I)=X 1(I)
X(8)= 1.0D+OO-X(I )-X(lM)
Y(8)=1.0D+OO-Y(IM)
C
DENO= 1.0D+OO
C
MFE3C=R *PDLOG(Y(8»
MFE=R*PDLOG(X(8YDENO)
& -R*T/2.0*E(I,I)*X(I)*X(I)IDENOIDENO
MC=R *T/3.0*DLOG(X(I )IDENO)
& +R*T/3.0*E(I,I)*X(l)IDENO
MFE3C=MFE3C+( 1-Y(8»*WW(lM)*Y(IM)
MFE=MFE-R *T/2.0*E(lM,lM)*X(IM)*X(IM)IDENOIDENO
MFE=MFE-R *PX(l )*E(1 ,IM) *X(IM)IDENOIDENO
MC=MC+R *T/3 .O*E( I ,IM)*X(IM)IDENO
MX= R*PDLOG(X(lMYDENO)
& +R *T*(E( 1,IM)*X(l )+E(IM,IM)*X(IM»)/DENO
MM3C= R*PDLOG(Y(IM»
& +Y(8)*WW(IM)- Y(8)*WW(IM)*Y(IM)
C
FUN(I)=X(8)*(GFE3C+MFE3C-MFE-MC)
FUN(I)=FUN(I)+X(IM)*(G(IM)+MM3C-MX -MC)
C
C WRITE(6,*) IM,E(l,I),E(l,IM),E(lM,IM)
C
10 CONTINUE
IF(ABS(FUN(3» .LT. I.OD-6) GOTO 20
XGA=X 1(3)-2.0*DUMMY*FUN(3)/(FUN(2)-FUN( 1»
GOTO 100
20 XGA=Xl(3)
RETURN
END
C
A-45
C----------------------------------------------- -----------------
C SUBROUTINE TO CALCULATE THE EQUILIBRIUM INTERFACE CHEMISTR Y
C
SUBROUTINE NPGCO(T ,x,Y ,IM)
IMPUCIT REAL·S (A-H,K-Z)
OOUBLE PRECISION X(S),Y(S),CO(S),WO(S),WW(S)
&,K(S) ,xX(3), FUN (3) ,GO(S)
&,G(S), G2(S ),G3(S),G4(S), GM (S ),M X(S) ,MM3C(S)
&,LFX (4 ,S),AC(S),E(S ,S),EA(S,S)
COMMONffRANS/CO,WO,K
COMMON/HILlIG2,LFX,LCV,GM,AC,DDG,GFEC
& ,GFE3C,G3,G4,WW,E,EA,G,GO
C
R=S.314D+OO
C
XGC=CO(IM)·O.990+00
DUMMY=1.0D-6
100 XX(l)=XGC-DUMMY
XX(2)=XGC+DUMMY
XX(3)=XGC
00 101=1,3
X1=XX(l)
FUN(I)=O,OD+OO
B=DEXP«GFE3C-G(IM)-WW(IM»/R{f +E(1 ,IM)·X1)
Y2=C0(IM)·4.0/3.0
X2=Y2/B
XS=1.0D+OO-X1-X2
YS= 1.00+00- Y2
IF (X(S) .LT. O.OD+OO)THEN
WRITE(6,·) 'XS,x1=',xS,X1
ENDIF
C
C
MFE3C=R·PDLOG(YS)
MFE=R·T·DLOO(XS)-R ·T/2.0·E(l,l )·XI·X 1
MC=R ·T/3.0·DLOG(X l)+R ·T/3.0·E(l, WX1
MFE3C=MFE3C+(1- YS)·WW(IM)·Y2
MFE=MFE-R ·T/2.0·E(IM,IM)·X2·X2-R ·PX 1·E(l,IM)·X2
MC=MC+R ·T/3.0·E(I,IM)·X2
MX(IM)= R·PDLOG(X2)
& +R·P(E(l,IM)·XI+E(IM,IM)·X2)
MM3C(IM)= R·PDLOG(Y2)
& +YS·WW(IM)
MM3C(IM)=MM3C(IM)- YS·WW(IM)·Y2
C
FUN(I)=GFE3C+MFE3C-MFE-MC
C
10 CONTINUE
IF(ABS(FUN(3» .LT. 1.0D-02) GOTO 20
DUMMYl=XGC-2.0·DUMMY·FUN(3 )/(FUN(2)-FUN(1»
IF (DUMMY1 .LT. O.OD+OO)THEN
DUMMY1=XGC/2.0D+OO
ENDIF
XGC=DUMMY1
GOTO 100
20 X(l)=X1
X(IM)=X2
Y(l)=O.250+00
Y(IM)=3.0D+OO/4.0D+OO·Y2
RETURN
END
C
C----- ---- ---- -------- -- ------ ------ ------- -------- ---------- ----
C SUBROUTINE TO CALCULATE THE EQUIUBRIUM INTERFACE CHEMISTRY
C
SUBROUTINE EQGCO(T ,x,Y,ACTIV,IM)
IMPUCIT REAUS (A-H,K-Z)
OOUBLE PRECISION X(S),Y(S),CO(S),WO(S),WW(S)
&,K(S),xx(3 ),FUN (3 ),GO(S)
&,G(S),G2(S),G3(S) ,G4(S ),GM (S),MX(S),M M3C(S)
& ,LFX( 4,S ),AC(S),E(S ,S),EA(S ,S)
COMMONffRANS/CO,WO,K
COMMON/HILlIG2,LFX,LCV,GM,AC,DDG,GFEC
& ,GFE3C,G3,G4,WW,E,EA,G,GO
A-46
C
R=S.314D+OO
C
XGC=CO(IM)*O.99D+OO
7 X2=X2ACO(T.XGC.ACTIV,E( 1,1),E(l ,IM»
C7 X2=X2AC(T ,xGC,ACTIV ,LCV ,G2(IM»
IF (X2 .LT. O.OD+OO)THEN
XGC=(CO(l)+XGC)I2.0
G0T07
ENDIF
DUMMY = 1.0D-6
100 XX(l)=XGC-DUMMY
XX(2)=XGC+DUMMY
XX(3)=XGC
DO 10 1=1,3
X1=XX(l)
FUN(I)=O.OD+OO
B=DEXP«GFE3C-G(IM)- WW(IM»)/R/f +E( 1,IM)*X 1)
C X2=C0(IM)*(4.0*X1-1.0D+OO)/
C & (3.0*B*(X1-CO(1»+4.0*CO(1)-1.0D+OO)
C X2=X2AC(T,X1.ACTIV,LCV,G2(IM»
X2=X2ACO(T ,x1.ACTIV,E(l,1),E(l,IM»
Y2=B*X2
XS=1.0D+OO-X1-X2
YS= 1.0D+OO-Y2
IF (X(S) .LT. O.OD+OO)THEN
WRITE(6,*) 'XS,x1=',xS,X1
ENDIF
C
C
MFE3C=R *T*DLOO(YS)
MFE=R *T*DLOG(XS)-R *T/2.0*E(l ,1)*X1*X 1
MC=R *T/3.0*DLOG(X l)+R *T/3.0*E(l, 1)*X1
MFE3C=MFE3C+{1- YS)*WW(IM)*Y2
MFE=MFE-R *T/2.0*E(IM,IM)*X2*X2-R *T*X l*E( 1,1M)*X2
MC=MC+R *T/3.0*E(l,IM)*X2
MX(IM)= R*T*DLOO(X2)
& +R*T*(E(1.1M)*Xl+E(IM,IM)*X2)
MM3C(IM)= R*T*DLOG(Y2)
& +YS*WW(IM)
MM3C(1M)=MM3C(1M)- YS*WW(IM)*Y2
C
FUN(l)=GFE3C+MFE3C-MFE-MC
C
10 CONTINUE
IF(ABS(FUN(3» .LT. 1.0D-02) GOTO 20
DUMMY1=XGC-2.0*DUMMY*FUN(3)/(FUN(2)-FUN(1 »
IF (DUMMY 1 .LT. O.OD+OO)THEN
DUMMY1=XGC/2.0D+OO
ENDIF
XGC=DUMMY1
ooTO 100
20 X(l)=X1
X(IM)=X2
Y(l)=O.25D+OO
Y(IM)=3.0D+OO/4.0D+OO*Y2
RETURN
END
C
C----------------------------------------------------------------
C FUNCTION TO CALCULATE EQUILIBRIUM FUNCTION F AND M
C IMF=l: F(X,Y) IMF=O: M(X,Y)
C
DOUBLE PRECISION FUNCTION FXYO(T,C,x1,x2,Y2,IM,IMF)
IMPUCIT REAL*S (A-H,K-Z)
DOUBLE PRECISION C(8),CO(8),WO(8),WW(8),K(8),G0(8)
&,G (8 ),G 2(8), G3(S) ,G4(8 ),GM (8) ,LFX (4,8) ,AC(8) ,E(8 ,8) ,EA(8 ,8)
COMMON/TRANS/CO,WO,K
COMMONIHILUG2,LFX,LCV,GM,AC,DDG,GFEC
& ,GFE3C,G3.G4,WW.E,EA,G,GO
R=S.314D+OO
C
C
CG=X1
A-47
XI =(C( 1)* (3 .0*Y2-4 .0*X2 )-(C(IM)- X2) )/(3 .0* Y2-4.0*C (1M»
X8=1.0D+OO-XI-X2
Y8=l.OD+OO- Y2
C
MFE3C=R*T*DLOG(YS)
MFE=R *T*DLOG(XS)-R *T/2.0*E(l,1 )*X I*X 1
MC=R *T/3.0*DLOG(X I)+R *T/3.0*E(l,1 )*XI
MFE3C=MFE3C+(I- YS)*WW(IM)*Y2
MFE=MFE-R *T/2.0*E(IM,IM)*X2*X2-R *T*X I*E( I,IM)*X2
MC=MC+R *T/3 .O*E( 1,IM)*X2
MX= R*T*DLOG(X2)
& +R *T*(E( 1,IM)*X 1+E(lM ,IM)*X2)
MM3C= R*T*DLOG(Y2)
& +YS*WW(lM)
MM3C=MM3C- YS*WW(lM)*Y2
C
IF (IMF .EQ. I) THEN
FXYO=GFE3C+MFE3C-MFE-MC
ELSE
FXYO=G(lM}+MM3C-MX-MC
ENDIF
C
RETURN
FND
C
C------- --------------------------------- ------------------------
C FUNcnON GIVING mE CARBON DIFFUSIVITY IN AUSTENITE
C
DOUBLE PREOSION FUNCTION DIFF(T ,X I ,X2,W)
IMPUCIT REAL*S(A-H,K-Y), INTEGER(1)'z)
DOUBLE PRECISION D(3OO),CARB(3OO)
C HH=PLANCK CONSTJ/S, KK=BOLTZMAN CONST. JIK
HH=6.6262D-34
KK=1.3S062D-23
Z=12
A5=1.0D+OO
R=S.31432D+OO
C
C DIFF=DIFFUSIVITY OF CARBON IN AUSTENITE, CM**/SEC
C Z=COORDINA nON OF INTERSITAL SITE
C PSI=COMPOsmON DEPENDENCE OF DIFFUSION COEFFICIENT
C mETA=NO. C ATOMS/ NO. FE ATOMS
C ACTIV=AcnVITY OF CARBON IN AUSTENITE
C R=GAS CONSTANT
C X=MOLE FRAcnON OF CARBON
C T=ABSOLUTE TEMPERATURE
C SIGMA=SITE EXCLUSION PROBABUTY
C W=CARBON CARBON INTERAcnON ENERGY IN AUSTENITE
C
DASH=(KK*T/HH)*DEXP( -(21230.0D+OO/T})*DEXP( -31.84D+OO)
XINCR=(X I-X2)/3OO.0D+OO
DO 11=1,300
CARB(1)=X2+(l-I)*XINCR
X=CARB(I)
THET A=XI(AS-X)
ACITV=CG(X,T,W,R)
ACITV=DEXP(ACTIV)
DAcnV=DCG(X,T,W,R)
DACTIV=DACTIV* Acnv
DAcnV=DACITV* AS/«AS+ mET A)**2)
SIGMA=A5-DEXP« -(W»)/(R *n)
PSI=ACTIV*(AS+Z*«A5+ THET A)/(A5-(AS+Z/2)*mET A+(Z/2)*(AS +Zj2)*
&(AS-SIGMA)*mET A*mET A»)+(AS+ mET A)*DACTIV
D(I)=DASH*PSI
CONTINUE
113=0
CALL DQSES(XlNCR,D,300,ANS,ERROR)
DIFF=ANS/(XI-X2)
RETURN
FND
C- -------- -- ---- ---- ------ ------ ----- ------- --------- ------------
A-48