Eur. Phys. J. E (2021) 44:119
https://doi.org/10.1140/epje/s10189-021-00113-x
THE EUROPEAN
PHYSICAL JOURNAL E
Regular Article - Living Systems
Hierarchical microphase separation in non-conserved
active mixtures
Yuting I. Lia and Michael E. Cates
DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd, Cambridge CB3 0WA, UK
Received 27 May 2021 / Accepted 18 August 2021 / Published online 27 September 2021
© The Author(s) 2021
Abstract Non-equilibrium phase separating systems with reactions, such as biomolecular condensates and
bacteria colonies, can break time-reversal symmetry (TRS) in two distinct ways. Firstly, the conservative
and non-conservative sectors of the dynamics can be governed by incompatible free energies; when both
sectors are present, this is the leading-order TRS violation, captured in its simplest form by ‘Model AB’.
Second, the diffusive dynamics can break TRS in its own right. This happens only at higher order in the
gradient expansion (but is the leading behaviour without reactions present) and is captured by ‘Active
Model B+’ (AMB+). Each of the two mechanisms can lead to microphase separation, by quite different
routes. Here we introduce Model AB+, for which both mechanisms are simultaneously present, and show
that for slow reaction rates the system can undergo a new type of hierarchical microphase separation,
which we call ‘bubbly microphase separation’. In this state, small droplets of one fluid are continuously
created and absorbed into large droplets, whose length-scales are controlled by the competing reactive and
diffusive dynamics.
1 Introduction
Recently, it has emerged that the non-membrane com-
partments of cells, also known as biomolecular con-
densates, can be fruitfully viewed as phase-separated
liquid–liquid mixtures undergoing chemical reactions
[1–7]. In contrast to passive binary fluids, cells con-
tinuously consume fuel, driving the system away from
equilibrium. Put differently, the liquid mixtures within
cells are examples of active matter. The character-
istics of phase separation in these biomolecular con-
densates are distinct from equilibrium counterparts,
in that microphase separation is often observed with-
out any energetic cause [4]. (In equilibrium, the latter
requires long-range interactions mediated by charged
species or block copolymers, for example. [8–10]) This
echoes studies of bacteria colonies whose phase separa-
tion (coarsening) is arrested by birth-and-death dynam-
ics to give patterns on a finite length scale [11]. This
suggests that, like the latter, the physics of biomolecular
condensates can be captured schematically by ‘active
field theories’ based on minimal, φ4-type ingredients
[12–17].
To construct top-down active field theories for par-
ticular systems, one can start with the desired symme-
try and conservation laws, and write down the lowest-
order theory. This follows lines similar to Hohenberg
and Halperin’s classification of equilibrium models [18].
a e-mail: yuting.li@damtp.cam.ac.uk (corresponding
author)
In the present work, we focus on theories for a sin-
gle scalar field φ, which could be (for bacteria or
swimming organisms) a rescaled density, or (for the
binary mixtures of interest here) a chemical compo-
sition variable. Within the resulting class of minimal
scalar models, several mechanisms have been found
to arrest phase separation. In our previous work, we
explored the lamellar phase and the droplet suspen-
sion phase in Model AB [16]. The latter is a canonical
representation of phase-separating conservative dynam-
ics coupled to non-conservative reactions, governed by
Model B and Model A, respectively, in such a way
that the underlying free energy structures in each sec-
tor are incompatible. This breaks detailed balance and
hence time reversal symmetry (TRS). The finite length
scale of the resulting microphase separation (between
regions rich in species 1 and 2, respectively) arises from
the balance between the reactions (which convert the
majority species into the minority) within the two sepa-
rated phases, and the conserved, diffusive currents that
transport species across the interfaces from high to low
chemical potential. In steady state, the interfaces can-
not be too far apart, otherwise the buildup of minority
species within each phase would cause a spinodal insta-
bility and create new interfaces. Microphase separation,
rather than bulk phase separation, occurs whenever the
stable fixed point composition of the homogeneous reac-
tion dynamics lies between the binodals of the conserva-
tive phase separation, so that the uniform bulk phases
are chemically unstable [11,16,19–22].
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119 Page 2 of 8 Eur. Phys. J. E (2021) 44 :119
Meanwhile, phase-separated states have also been
reported in fully conservative active systems described
by Active Model B+ (AMB+). This model is con-
structed by adding a complete set of lowest-order non-
equilibrium current terms to the standard (passive)
Model B [18] (with differing outcomes, in more than one
dimension, from an earlier, less complete, Active Model
B [12]) [14]. The active terms in AMB+ can send into
reverse the Ostwald process for droplets, which would
normally lead to full-phase separation by causing small
droplets to shrink and disappear while large ones grow
[17]. The reverse Ostwald process, which formally
emerges through a negative effective interfacial tension
in the expression for the Laplace pressure at a droplet
surface, instead equalizes the droplet sizes. (Note that
other definitions of interfacial tension do not become
negative, so that droplets remain stably spherical for
example.) In the absence of noise, this mechanism cre-
ates a set of uniformly sized droplets whose number is
governed by the initial condition. However, when noise
is present, this ultimately selects the length scale for
microphase separation by a subtle interplay of various
effects. (Those effects include noise-induced nucleation
of small droplets and coalescence of large ones, coupled
to a collective homogenization of droplet size caused by
reverse Ostwald [14].)
As just described, Model AB and AMB+ each involve
a single scalar field on its own. More elaborate active
field theories can be constructed by coupling a com-
positional scalar to a momentum-preserving fluid flow,
along lines established in passive systems as ‘Model H’
[18]. It is known that this coupling can give further dis-
tinctive routes to microphase separation, for example
when activity in effect reverses the sign of the inter-
facial tension that governs the stress exerted on the
fluid at the boundary of the droplet (but without doing
the same for the Ostwald currents). This happens when
the droplet is made of so-called contractile material
[13,15]. In this paper, we omit these further compli-
cations and consider only how to combine the types
of physics already captured by Model AB and AMB+,
in which there is no separate momentum conservation,
and the reaction-diffusion dynamics of φ give a com-
plete description of the pertinent physics.
In our previous work on Model AB [16], we broke
TRS only at the lowest possible order in (∇, φ). This
entails choosing chemical potentials for the Model B
and Model A sectors that could each give an equi-
librium system in the absence of the other, so that
μA,B = δFA,B/δφ, but also choosing FA = FB. This
leads to phase separation via the mechanism described
previously. However, in a given physical system, once
TRS is broken in this fashion it is likely to be bro-
ken elsewhere as well. In particular, we know that the
active currents represented by the φ-conserving AMB+
(which do not derive from any local chemical potential,
of the form δF/δφ or otherwise), can independently
cause phase separation via the reverse Ostwald pro-
cess [14]. Even if these are formally subdominant when
expanding in (∇, φ), such terms can in principle interact
in a strongly non-additive way with the microphase sep-
aration in Model AB, particularly if parameters are cho-
sen so that the two mechanisms act on well-separated
length- and time-scales.
Accordingly, in this paper we construct Model AB+
which includes both types of TRS breaking at once, and
use it to give a first account of the interplay between
the two mechanisms of microphase separation. Our aim
in this brief and exploratory study is not to compre-
hensively explore the parameter space, but rather to
show proof of principle that new physics can indeed
emerge from this combination of mechanisms. In par-
ticular, in the case where the chemical reactions are
slow, we will give numerical evidence for a hierarchical
phase separation in which the conservative and non-
conservative mechanisms act simultaneously at small
and large length-scales, respectively. This results in
a new, dynamical steady-state structure that we call
‘bubbly microphase separation’.
The paper is organized as follows. In Sect.2, we sum-
marize the original Model AB as presented in our previ-
ous work [16] and recall its various stationary patterns
before introducing Model AB+. Section 3 then explores
some steady states of Model AB+ with comparisons to
the corresponding Model AB pattern without the active
current terms from AMB+. These include droplet emul-
sions alongside the bubbly microphase-separated state.
We then present in Sect. 4 a mean-field Ostwald-type
calculation for growth of a φ > 0 (liquid) droplet in a
bath with φ < 0 (vapour). This explains a transition (or
crossover) between statistically different droplet emul-
sion states upon increasing the target density of the
chemical reactions above the dilute binodal of the con-
servative phase separation. Section 5 briefly summarizes
these findings and suggests directions for future work.
2 Model AB+
We will first briefly introduce both Model AB and
Active Model B+, before combining their two dynam-
ics to construct Model AB+. Model AB, constructed
as a combination of Model B and Model A, is a mini-
mal field theory for a scalar composition variable φ(x),
in reaction–diffusion systems in which the conservative
part of the dynamics, taken alone, would give phase
separation [16]:
∂tφ = −∇ · J − MAμA +
√
2εMAΛA
J = −MB∇μB +
√
2εMBΛB
μB = −αφ + βφ3 − κ∇2φ
μA = u(φ − φa)(φ − φt)
(1)
We assume that φa < φt. Hence, φa is the absorbing
state composition of the reaction diffusion dynamics
(this would be the zero density state for a birth-death
system), which marks the physical lower limit for φ,
whereas φt is the ‘target density’ of the chemical reac-
tions, such that there is a stable fixed point of the
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Eur. Phys. J. E (2021) 44 :119 Page 3 of 8 119
Fig. 1 Steady-state
patterns for
φt = −0.9, −0.5, 0, 0.4 for
two sets of (λ, ζ) values.
For λ = 0, ζ = 0 (as in
Model AB), the patterns
are stationary modulo
noise fluctuations and are
shown in the top panel. For
λ = −1, ζ = −4, the steady
states are dynamical with
droplets of the dense phase
(red) constantly spawn in
the dilute phase (blue),
snapshots of which are
shown in the bottom panel
dynamics at φ(x) = φt, if the diffusive tendency to
phase separate is switched off. In addition, MA,B are
mobility constants, ΛA,ΛB are spatiotemporal white
noises and α, β, κ, u are positive constants. The param-
eter 	 is temperature, or an equivalent noise parameter
in cases where the primary noise source is not thermal.
Given the form of chemical potential μB chosen here,
if we had chosen instead μA = μB, the conservative and
nonconservative dynamics would share a free energy
FB =
∫ (−αφ2/2 + βφ4/4 + κ(∇φ)2/2) dx so that the
model would be an equilibrium one. Much less obvi-
ously, but as discussed in our previous paper [16], if
μA is not equal to μB but differs from it only by terms
linear in φ, the system maps to an equilibrium model
with long-range attractions. Hence, by virtue of the
quadratic term in μA (alongside the absence of the cubic
term), Eq. (1) admits the lowest-order mismatch in μA
and μB that cannot be incorporated into an equilibrium
model by judicious matching of higher-order terms.
If the reactions are switched off (MA → 0), the diffu-
sive sector of the dynamics drives the system towards
bulk phase separation with the two coexisting phases
at the binodal densities, φ = ±φB = ±
√
α/β, as long
as the mean density (φ¯ = 1V
∫
dxφ(x) ) is between the
binodals. With reactions, broadly speaking, if the tar-
get density φt of the non-conservative sector is between
the binodals, the system exhibits micro-phase separa-
tion, where material is created in a dilute region, trans-
ported across the interface by the Model B current, and
eventually destroyed in a dense region. The morphology
depends on the value of φt, as shown in the upper panels
of Fig. (1). For φt close to zero, the system shows lamel-
lar patterns; otherwise, we either see a droplet phase or
its inverse, a bubble phase, depending on which bin-
odal φt is closer to. In this paragraph, and below, we
refer to the two bulk densities as though they were liq-
uid (φ > 0) and vapour (φ < 0) although of course
for biomolecular condensates φ is composition variable
describing whether a fluid mixture of species 1 and 2
is composed mainly of species 1 (φ > 0) or species 2
(φ < 0). In this context, a rightward current of ‘mate-
rial’ φ means a combination of a rightward current of
species 1 and a compensating negative current of species
2 at constant total mass density.
Model AB breaks time reversal symmetry solely by
having mismatched free energies in the nonconserved
(Model A) and conserved (Model B) sectors. But of
course, each sector can, in principle, break detailed bal-
ance on its own. In the Model A sector, this requires
gradient terms in μA that do not stem from a free
energy. At leading order in the gradient expansion, the
resulting ‘Model A+’ will not lead to microphase sepa-
ration under steady-state conditions because, without a
conservation law, there are no interfaces in the steady
state at which such terms could be large. Instead we
expect relaxation towards a uniform state at zero μA,
or φ = φt. These active ‘A+’ terms can of course mod-
ify the microphase separation caused by the competing
conserved and non-conserved dynamics in Model AB,
but we don’t expect them to introduce a second type
of microphase separation.
In what follows, we therefore focus on enhanc-
ing Model AB by adding active gradient terms to
the conservative sector where such terms can lead to
microphase separation in their own right. The terms in
question are those of Active Model B+ [14],
∂tφ = −∇ · J
J = MB
[
−∇
(
μB + λ |∇φ|2
)
+ ζ(∇2φ)∇φ
]
+
√
2εMBΛB (2)
where μB is the same as before. The λ and ζ terms are
the lowest-order Landau–Ginzburg terms that break
TRS in systems with φ conservation. (One other term
at this order can be absorbed into a φ-dependent
square-gradient coefficient κ(φ) without breaking the
free energy structure.) It has been shown that the λ
and ζ terms not only shift the binodals, but that the
latter can also lead to effectively negative surface ten-
sions for either liquid droplets (φ > 0) or vapour bub-
bles (φ < 0), albeit not both at once, depending on the
sign of ζ [14]. The tension in question is the one gov-
erning Laplace pressures and hence Ostwald ripening,
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119 Page 4 of 8 Eur. Phys. J. E (2021) 44 :119
not other physics (such as fluctuations at the interface
between phases which remains stable). The resulting
reverse Ostwald process causes microphase separation
into an emulsion of finite droplets that do not coarsen,
which is of course not seen in equilibrium Model B. In
renormalization-group terms, this microphase separa-
tion appears to be connected with a strong coupling
regime at large enough values of (λ, ζ). As such, these
variables remain important even though, being higher
order in the Landau–Ginzburg expansion, they are for-
mally irrelevant in the neighbourhood of the Wilson–
Fisher fixed point which controls the critical onset of
bulk phase separation [23].
This suggests that equally important physics could
also be lost by ignoring these non-integrable gradient
terms in the conservative sector for systems with both
conserved and non-conserved dynamics, whereas we did
ignore them when constructing Model AB as a canon-
ical model for that case [16]. In light of this, we now
construct Model AB+ by adding the λ and ζ terms
to the conservative sector of Model AB, so that (1) is
replaced by:
∂tφ = −∇ · J − MAμA +
√
2εMAΛA
J = MB
[
−∇
(
μB + λ |∇φ|2
)
+ ζ(∇2φ)∇φ
]
+
√
2εMBΛB
μB = −αφ + βφ3 − κ∇2φ
μA = u(φ − φa)(φ − φt)
(3)
The rest of this paper addresses Model AB+ as
written in the form (3). All numerical simulations are
performed in d=2 with periodic boundary conditions.
Spatial derivatives are computed in real space follow-
ing the methods detailed in “Appendix” A of [14].
Time integration is performed using the explicit Euler–
Maruyama method [24]. Throughout the paper, unless
otherwise stated, we set MB = 1, α = β = 0.25, κ =
1, u = (−φa + φt/2), φa = −10. Note that we make
no attempt at a systematic exploration of parameter
space, but focus on selected regimes where new physics
can be expected by adding nonzero λ, ζ terms to param-
eter sets that we used previously to study Model AB.
(In particular, the somewhat arbitrary choice for u was
adopted in our previous paper [16] and is kept here for
consistency. In principle, one can simply set u = 1 and
the results will be qualitatively the same.)
3 Steady states
Model AB+ has at most one stationary state of uniform
density, which lies at the target density φt. (Any other
uniform state has φ˙ = 0 from the reactions, with no
diffusive currents that could balance this.) Linear sta-
bility analysis finds this uniform state to be unstable to
spatial perturbations if
MB
α˜2
4κ
> MAu˜ (4)
where α˜ = 3βφ2t −α, u˜ = u(−φa +φt). The two sides of
the inequality represent characteristic relaxation rates
via diffusion and reactions, respectively, on length scales
set by κ. In many biological situations, relatively rapid
thermal diffusion of chemical species is accompanied
by a relatively slow reaction-driven turnover time, so
that this ‘slow reaction’ condition is easily obeyed.
Hence, in this paper we focus on the unstable regime
far away from the threshold of linear instability. We fur-
ther assume that φa ≤ −φB so the reaction dynamics
is approximately linear in φ(x) for values lying between
the binodals. This means that we have an approximate
symmetry (λ, ζ, φ) → (−λ,−ζ,−φ) in our system, sim-
ilar to Active Model B+ where this symmetry is exact
[14].
Some typical steady states are shown in Fig. (1) for
λ = −1, ζ = −4, where the Ostwald tension for droplets
of the dense phase (φ > 0) are negative, as shown in
Tjhung et. al. [14]. These are directly compared with
the pure Model AB case (λ = ζ = 0). For low values
of φt, where pure AB is in a uniform state we see an
emulsion of these dense (‘liquid’) droplets in the dilute
phase (‘vapour’, φ < 0), stabilized by reverse Ostwald.
At high φt, chosen so that the Model AB mechanism
leads to a microphase-separated state with liquid phase
in the majority (hence an emulsion of vapour bubbles
in liquid), the active current terms cause phase inver-
sion of this state so that the system is liquid-in-vapour.
This is inevitable for sufficiently negative λ, ζ where the
conservative dynamics causes microphase separation on
a relatively short length scale since, importantly, these
terms only stabilize liquid in vapour droplets and not
vice versa (unless their signs are reversed, in which case
so is the whole phase diagram).
Because even at these parameter values the binodals
φ2,1 are only modestly shifted from the equilibrium val-
ues ±φB, the global phase volumes of the liquid and
vapour phases remain primarily under the control of
the reaction dynamics and are hence set by φt. At large
enough values of this quantity, the majority liquid phase
unavoidably percolates, but the minority phase of dis-
connected large vapour bubbles contain within them
small liquid droplets that are stabilized by reverse Ost-
wald. (The interior of such a bubble thus resembles a
piece of the liquid-in-vapour emulsion seen at smaller
φt.) These small liquid droplets are continuously pro-
duced within the vapour bubbles but then grow, dif-
fuse and merge into the surrounding liquid phase. This
behaviour closely resembles the ‘bubbly phase separa-
tion’ reported previously for AMB+, except that in the
latter case, there is only a single vapour domain in the
system which can effectively be viewed as a bulk phase
separation between the liquid-in-vapour emulsion and
excess liquid [14]. (Note also that [14] mainly addresses
the case with λ, ζ > 0 for which the identities of the
‘liquid’ and ‘vapour’ phases are, trivially, interchanged
from those in the present discussion.) In the presence of
the Model AB mechanism, this type of emulsion/liquid
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Eur. Phys. J. E (2021) 44 :119 Page 5 of 8 119
bulk phase separation is itself unstable since the bulk
liquid phase is not at the target density. Accordingly,
for slow reaction dynamics the system must homogenize
once again at some larger scale so that the chemical
conversion of the majority into the minority species in
each neighbourhood can be balanced by diffusive mass
transport. Accordingly we see a finite density of the
large liquid-in-vapour emulsion bubbles whose size can-
not grow further. Echoing the language of [14], we refer
to this state as ‘bubbly microphase separation’.
A snapshot of the life-cycle of droplets is shown
in Fig. (2) showing the growth of a small nucleated
droplet, its growth, and coalescence into the surround-
ing liquid phase. As stated previously, this echoes the
findings of [14] for the life-cycle of liquid droplets within
a bulk phase separation between an emulsion of such
droplets and excess liquid.
4 Ostwald dynamics in model AB+
In the mean field limit, the binodals of AMB+, defined
as the steady-state coexisting densities for a flat inter-
face, can be computed analytically [13]. We will not
show the calculation here but merely quote the results
for λ = −1, ζ = −4: the dense binodal shifts to
φ1 = 1.10 and the dilute binodal shifts to φ2 = −0.86.
These replace the values ±φB = ±1 for the passive
model (with α = −β as chosen above).
In addition, around circular two-dimensional droplets,
the density φ close to the interface is further modified
by an offset depending on the radius of the droplet R.
In equilibrium Model B, this curvature-induced shift
around a liquid droplet is proportional to σ/R, where
σ is the surface tension. As shown in [14] for AMB+,
the surface tension defined this way can be negative for
sufficiently negative of λ and ζ, thus reversing the Ost-
wald ripening for droplets, arresting their growth at a
finite size. If we now switch on the chemical reactions
to give Model AB+, then so long as these reactions
are slow (which is the regime addressed in this paper)
the dynamics on the length-scale of the interface width
ξ0 
√
κ/2α is dominated by the conservative current.
This means that we can carry across from AMB+ the
results for the curvature-induced offset to the binodals.
Consider N liquid droplets of radius Rj  ξ0 for
j ∈ (1, 2, . . . , N) in a dilute bath in a domain of vol-
ume V with periodic boundary conditions. We now sin-
gle out one droplet of radius Ri and treat the rest as
homogeneous background. Let φ±(x) be the composi-
tion field inside and outside the singled-out droplet,
respectively, and define variables ψ±(x) as the devia-
tion from the modified binodal densities:
φ−(x) = φ2 + ψ−(x) |x| ≥ Ri
φ+(x) = φ1 + ψ+(x) 0 ≤ |x| < Ri (5)
For reasons given in [16], provided that φt is close to
the modified dilute binodal φ2 (so that there is a large
excess of vapour around the droplet), and so long as the
timescale separation between conservative and noncon-
servative dynamics holds (slow reaction regime), we can
linearize the deterministic part of Eq. (3) to obtain,
∂tψ− = D−∇2ψ− + (1 − Δ)g0− + g1−ψ−
+ V −1
N∑
j=1
2πRjJ
j
−
∂tψ+ = D+∇2ψ+ + g0+ + g1+ψ+
(6)
where D± = MB(−α + 3β(φ1,2)2) and we have defined
g(φ) = −MAμA(φ), g0± = g(φ1,2), g1± = g′(φ1,2),Δ =
V −1
∑
j πR
2
j for conciseness. The (1 − Δ) prefactor
accounts for the change in the chemical turnover as a
result of the N droplets. In the last term in the equa-
tion for ψ−, J
j
− denotes the current at the surface of the
jth droplet and thus this term represents the injection
of mass by all the droplets. We refer to Sect. 3.3 of our
previous paper [16] for a more detailed account of the
calculation. Note that we neglect the κ term here as ψ
varies on a length-scale much larger than the interfacial
width. The linearized equations need to be solved with
the appropriate boundary conditions: at |x| → 0 and
|x| → ∞, we only need ψ± to be finite; at the bound-
ary of the droplet, we require ψ± = δ±(R), where δ±(R)
is the aforementioned offset for the dense and dilute
phase, respectively.
We will now sketch the perturbative Ostwald cal-
culation with results presented only at key steps and
refer to Sec. 3.2 of [16] for more detailed explanations
of the scheme. This is inspired by the standard Ostwald
calculation for both passive and active systems [14,17]
and its adaptation to multiple droplets [16,19], but we
implemented it here, for the first time, to include both
the nonconservative reaction terms in the bulk phases
(following [16]) and the effects of active currents on the
matching conditions (following [14]) which cause rever-
sal of the Ostwald dynamics.
For fixed droplet radii {Rj}, we solve for the station-
ary state of the linearized equations with ψ±(Ri) =
δ±(Ri), then compute the current J i± at the inter-
face with the quasi-static solutions as a function of the
droplet radii {Rj},
J i+ = D+k+(c+ − δ+(Ri))
I1(k+Ri)
I0(k+Ri)
J i− = −D−k−(ci− − δ−(Ri))
K1(k−Ri)
K0(k−Ri)
ci− = c−(1 − λ) − (g1−V )−1
N∑
j=1
2πRjJ
j
−
(7)
where K0,1, I0,1 are the modified Bessel functions, k± =√
−g1±/D± and c± = −g0±/g1± ≈ φt − φ2,1. At the sur-
face of the singled-out ith droplet (treating the rest
as a homogeneous bath), the mismatch of the diffu-
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119 Page 6 of 8 Eur. Phys. J. E (2021) 44 :119
Fig. 2 Three snapshots of
the life-cycle of a dense
droplet created inside the
dilute phase, as indicated
by the arrow. The
simulation parameters are
uMA = 10
−6, φt = 0.6
sive currents on the two sides of the interface J i± shifts
the droplet boundary, thus changing the radius of the
droplet,
∂tRi =
J i+(Ri) − J i−({Rj})
φ1 − φ2 (8)
So far, we have reduced the multiple-droplet dynam-
ics to N coupled equations for {Rj} by treating each
droplet in isolation. One can then immediately spot a
solution where all the droplets have the same radius.
Let this radius be R, the currents J i− can then be
obtained explicitly by substituting the ci− equation into
the J i− equation:
J i−(R) = −D−k−
K1
K0
(c−(1 − λ)
−δ−(R))
(
1 − 2λD−k−
g1−R
K1
K0
)
(9)
where the arguments of the modified Bessel functions
K1,0 have been omitted to ease notation. Substituting
Eq. (9) into Eq. (8) yields the growth rate R˙ as a func-
tion of R. The resulting R˙(R) curve is plotted in Fig.
(3a) for one droplet in a box of size 2562 for φt < φ1 and
φt > φ1. Both plots show a stable fixed point: smaller
droplets grow towards the stable radius, and larger
droplets shrink towards it. These facts, respectively,
explain the spontaneous emergence of liquid droplets
within the vapour phase (given the presence of noise at
short scales) and also the arrested growth of droplets
shown in Fig. (1). There are, however, some important
differences between the two plots in Fig. (3a). Observe
that the curve for the currents inside the droplet J+(R)
is similar in the two panels, whereas the current in the
vapour phase J−(R) is qualitatively different. In the
case where φt < φ1, ψ−(x) → c− ≈ (φt − φ1) < 0
as |x| → ∞ whereas ψ−(R) → 0 for large R, leading
to diffusive currents away from the droplet on the out-
side. In contrast, for φt > φ1, we have positive ψ− at
infinity, thus the current outside the droplet is always
toward the droplet. This has profound consequences:
for φt < φ1, the zero of the R˙ has an upper bound
at δ−(R) = c−, which is independent of the reaction
rate MA; but for φt > φ1, there is no upper bound
for the stationary fixed-point radius (at least at the
mean field level). Comparisons of the above analysis
against numerical simulations with low noise are shown
in Fig. (3b): although the perturbative Ostwald calcu-
lation does not quantitatively capture the exact time
evolution of R, due to the quasi-static assumption, the
stable radii between theory and simulations agree rea-
sonably well.
As one increases the number of droplets in the Ost-
wald calculation (by changing λ = NπR2/V in Eq. (9)),
the stable fixed-point radius decreases, also observed in
[16]; though qualitatively, the fixed-point radius is still
substantially larger for φt > φ1 than φt < φ1. This
explains why, at fixed noise level, the emulsion of small
and rather monodisperse liquid droplets seen in the bot-
tom left panel of Fig. 1 takes on a quite different char-
acter in the two bottom centre panels, with relatively
large and quite polydisperse droplets. Clearly in this
high phase-volume emulsion state the droplets are nei-
ther uniform nor well separated, so that any further
attempt to rationalize their statistics requires a much
better understanding of the role of noise in the steady-
state kinetics. This lies beyond our scope; as noted pre-
viously, the issue is a surprisingly complicated one even
in AMB+ where chemical reactions are absent [14].
For even larger values of φt, the Ostwald calculation
above continues to qualitatively capture the growth of
initially small liquid droplets within vapour regions, but
as the droplets grow larger, they come into contact and
either form or merge into the continuous phase of liquid
which is present in the resulting regime of bubbly or
hierarchical microphase separation.
5 Conclusion
In this paper, we have furthered our investigation into
scalar field theories for non-equilibrium phase separat-
ing systems. Specifically, we have presented an initial
study of the effect of having diffusive dynamics that
breaks detailed balance on its own, on top of a mis-
match of chemical potentials between the conservative
dynamics and the non-conservative dynamics. This mis-
match is separately a hallmark of broken time rever-
sal symmetry (broken detailed balance) in active phase
separation whenever both types of dynamics are present
at once. The resulting new model, Model AB+, incor-
porates the λ and ζ terms, describing active currents
in Active Model B+, into Model AB from our previous
paper [16] which focuses on the chemical potential mis-
match in isolation. The latter causes microphase sepa-
ration whenever the chemical reactions drive the system
123
Eur. Phys. J. E (2021) 44 :119 Page 7 of 8 119
(a) (b)
Fig. 3 a Plots of the growth rate R˙ and the currents inside
J+ and outside J− for a dense (liquid) droplet of radius R
in a dilute (vapour) bath for φt < φ1 and φt > φ1 (recall
φ1 = −0.86). In both cases, the R˙ curve crosses the X-
axis from above, indicating a stable fixed point for R. The
key difference is that in the left panel the J− curve is pos-
itive as R → ∞ whereas J− remains negative on the right
panel. b Comparisons between theoretical predictions (dot-
ted line) against simulations (solid line) for φt < φ1 and
φt > φ1. All plots are produced with the same reaction rate
uMA = 1 × 10−6. The noise level used in the simulations is
ε = 0.001
towards a target density that lies between the binodals
of a conserved, diffusive phase separation.
This study was motivated by the ability of the active
current terms to reverse the classical Ostwald process
[14], which is also associated with a distinct strong cou-
pling regime renormalization group calculations [23]. As
a result of this, Model AB+ has two distinct mechanis-
tic channels for microphase separation. Our work has
shown that these can interact in nontrivial ways, giving
two distinct types of emulsion state and also a ‘bubbly’
microphase separation in which the two processes are
operative hierarchically on different scales. To be more
specific, we focused our studies on parameter regimes
where the surface tension for droplets of the dense
‘liquid’ phase is negative. For φt close to the dilute
‘vapour’ binodal φ1, suspensions of stable droplets of
finite size are observed. Ostwald calculations of a single
droplet reveal that if φt < φ1 there is an upper limit
on the droplet size as uMA → 0 determined entirely
by the diffusive dynamics, whereas for φt > φ1 such an
upper limit does not exist. The dynamics become more
complicated for multiple droplets, though the qualita-
tive conclusion remains that there are two structurally
distinct liquid-in-vapour emulsions, as reflected in the
phase diagram. On the other hand, for larger values of
φt, the system settles into the bubbly microphase sep-
arated state, characterized by large but finite domains
with densities at the two binodals, but with droplets of
the liquid phase continuously created within the vapour
domains before merging with the surrounding fluid.
Thus far, we explored some interesting behaviour of
Model AB+, but made no attempt to systematically
scan its multidimensional parameter space. There is,
accordingly, plenty of room to find new physics beyond
that presented here. For example, it would be interest-
ing to see what is the interplay between the reverse Ost-
wald dynamics and the limit cycles observed in Model
AB [16]. In these cycles, there is no stationary phase
separation but a cycle where phase separation leads
to a slow reduction in global density until the system
rehomogenizes, whereupon the global density reverses
until the system phase separates again. (This is pos-
sible when the local reaction rate is sufficiently non-
linear in density.) It is not clear how this oscillation
might interact with a conserved dynamics that favours
microphase separation, especially when this is itself
a highly dynamical process governed by a nontrivial
life-cycle for droplets. Finally, as reported separately
for Model AB and AMB+, the noise strength 	 plays
an important role in selecting the number and size of
droplets in the emulsion state [14,16] and this aspect
deserved further study.
Finally, we have not made any attempt to connect
the parameters or mechanisms of Model AB+ directly
with any specific microscopic examples of active phase
separation, whether within living cells [1–4], bacterial
colonies [11], or elsewhere. There have been successes in
applying similar Swift–Hohenberg–Halperin-type mod-
els to ab-initio simulations and real experiments [25–
27], but this is generally not easy because the model
is constructed top-down to have a minimal structure.
While the results are generic, only a comparison with
more microscopic, bottom-up treatments can determine
whether individual parameters are large or small in
any particular case. For example, in purely conser-
vative phase separation with ABPs (Active Brownian
Particles), it was found that hard-core repulsion leads
directly to the λ term in AMB+ but that to recover the
ζ term, additional (e.g., soft-core) interactions are also
required [14]. Despite these difficulties, we believe that
the exploration of a generic model such as Model AB+
can suggest mechanistic explanations for structure for-
mation in binary active systems that might otherwise
be quite puzzling. A possible example is when there is
emergent structure on more than one length scale, as
arises in bubbly microphase separation (bottom right
panel of Fig. 1).
123
119 Page 8 of 8 Eur. Phys. J. E (2021) 44 :119
Acknowledgements We thank Cesare Nardini, Rajesh
Singh and Elsen Tjhung for valuable discussions and code
donations. YIL thanks the Cambridge Trust and the Jardine
Foundation for a PhD studentship. This work was funded
in part by by the European Research Council under the
Horizon 2020 Programme, ERC grant agreement number
740269. MEC is funded by the Royal Society.
Author contribution statement
All authors contributed equally to the paper.
Open Access This article is licensed under a Creative Com-
mons Attribution 4.0 International License, which permits
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to the original author(s) and the source, provide a link to
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To view a copy of this licence, visit http://creativecomm
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