Unlike in the bulk, at the nanoscale shape dictates properties. The imperative to understand and predict nanocrystal shape led to the development, over several decades, of a large number of mathematical models and, later, their software implementations. In this review, the various mathematical approaches used to model crystal shapes are first overviewed, from the century-old Wulff construction to the year-old (2020) approach to describe supported twinned nanocrystals, together with a discussion and disambiguation of the terminology. Then, the multitude of published software implementations of these Wulff-based shape models are described in detail, describing their technical aspects, advantages and limitations. Finally, a discussion of the scientific applications of shape models to either predict shape or use shape to deduce thermodynamic and/or kinetic parameters is offered, followed by a conclusion. This review provides a guide for scientists looking to model crystal shape in a field where ever-increasingly complex crystal shapes and compositions are required to fulfil the exciting promises of nanotechnology.

Christina Boukouvala and Joshua Daniel contributed equally

The online version contains supplementary material available at

Nanocrystals, defined as crystalline particles of size ranging from 1 to 1000 nm, have found a myriad of applications across science and engineering, for instance in optical devices [

Interest in nanocrystal shape and its control has, therefore, followed the trend of interest in nanostructures. Marks and Peng [

Yearly number of publications featuring the words “Wulff construction” (purple) or all of the words “nanoparticle”, “shape” and “thermodynamic” (orange) in their title or abstract. Data obtained from

The thermodynamic equilibrium shape of a nanocrystal, the simplest one to model, is governed by surface energy minimisation, as determined by Gibbs in 1873 [

This short review starts with a brief overview and disambiguation of the various mathematical models and terminologies used to model crystal shapes, from the century-old Wulff construction to the year-old (2020) approach to describe supported twinned nanocrystals. Next, we explore the multitude of published software implementations of Wulff-based shape models, describing for each their technical aspects, advantages and limitations. Finally, a discussion of the scientific applications of shape models to either predict shape or use shape to deduce thermodynamic and/or kinetic parameters is offered, followed by a conclusion.

In this section, the classic thermodynamic Wulff construction and its mathematical basis will first be introduced, followed by an overview of derivative but mathematically distinct Wulff-related constructions for (nano)crystals under various constraints such as twinned crystals, crystals governed by kinetic growth, alloys and supported crystals. We aim to keep this brief and the reader is directed to several more detailed publications for further details, if needed [

The “original” thermodynamic Wulff construction applies to the case of a single crystal in thermodynamic equilibrium. It states that the normal vector length to any external crystal facet will be proportional to the surface free energy of that facet, commonly expressed as:_{i} is the orientation-dependent surface free energy of facet _{i} is the normal distance from the centre of the particle (also referred to as the ‘Wulff point’ [_{w}) defining the crystal’s thermodynamic equilibrium shape:

Single crystal and modified (twinned) Wulff constructions. Gamma and v-plots viewed along < 110 > for the (

Wulff’s construction from 1901 was formally proven later by von Laue (1943) [

This thermodynamic approach, while limited to single crystals, has been abundantly used to understand and describe the shape of nanocrystals. In the common FCC system, adopted by Au, Ag, Cu, Al, and many others, thermodynamic crystal shapes lay somewhere in between a cube and an octahedron,

Nanocrystal shapes in the single crystal thermodynamic/kinetic, thermodynamic modified and kinetic modified Wulff constructions for FCC and HCP crystal structures. The FCC structures are twinned along the (111) plane, while the twinning plane varies for HCP as described in ref. [

The introduction of internal planar defects, namely twin boundaries formed at the nucleation and early stage of growth, leads to different underlying symmetry and potentially more complex crystal shapes. Particles with a single twin boundary or parallel sets of twin boundaries are referred to as singly-twinned or lamellar twinned particles (LTPs) [

For twins, the thermodynamic equilibrium shape is found by determining the thermodynamic Wulff shape for each crystal subunit, taking into account the twin boundary energy(ies), then “assembling” the final structure from these subunits. This approach, proposed by Marks in 1983 [_{m} of the individual single-crystal subunits:

A nanocrystal shape will only reach thermodynamic equilibrium given sufficient time and energy; many syntheses yield non-equilibrium shapes owing to kinetic effects. The kinetic Wulff construction aims to include these effects; it was first formally used by Frank et al

In the case of twinned crystals grown in kinetic conditions, a treatment similar to that of the thermodynamic approach can be applied by forming individual kinetically grown crystals (Eq.

The kinetic single crystal Wulff construction allows for shapes beyond the thermodynamic equilibrium by taking into account the growth-directing effects of surfactants, underpotential deposition, or other reaction additives and conditions [

The ability to add growth enhancements to re-entrant facets as well as disclinations and twin boundaries enabled the modified (twinned) Wulff construction to model a host of observed but previously unexplained nanocrystal shapes, i.e. shapes impossible to obtain by simply changing the surface energies/growth velocities (Fig.

The ability of homogeneous alloy particles to form more stable structures via surface segregation, and the resulting change in shape, has been incorporated in the alloy Wulff construction [

Nanocrystal applications may require or benefit from support on a substrate, for instance in catalysis where this can improve performance and reduce sintering [

Winterbottom and related constructions for supported crystals. Gamma and v-plots viewed along < 110 > for the (_{3} nanocube, adapted with permission from [_{3} nanoparticle, adapted with permission from [

This interfacial energy then replaces the surface energy term for that facet in an otherwise standard Wulff approach.

The thermodynamic Winterbottom construction has been used or invoked in multiple contexts, often related to supported catalysts, including the epitaxial growth and resulting orientation of La_{1.3}Sr_{1.7}Mn_{2}O_{7} on LaAlO_{3} [_{3} [

The _{j} and consequently the interfacial energy. Hence, _{j} is the proportion of the total interfacial area taken up by each individual subunit. The energy-minimizing shape can thus be determined, as for the single crystal Winterbottom construction by using

The case of a supported decahedral particle of FCC has been explored in detail [

The kinetic variants of both the single crystalline and modified Winterbottom constructions can be implemented by using growth velocities instead of surface energies. While no kinetic enhancement effects (concave surfaces, twin boundaries, etc

Additional variants of supported crystals have been developed for specialized contexts. The Summertop construction [

What has been called in the literature the atomistic Wulff construction is not in fact a distinct Wulff construction or arguably a mathematical approach at all, but we take the opportunity here to disambiguate its meaning and context. This is not a separate Wulff-related construction in the same sense as the models looked at until now—it is simply an application of the classic thermodynamic Wulff model.

The term atomistic Wulff construction came into use in 2005 with the first of a series of papers on modelling equilibrium-shaped Ru-based catalysts used in industrial ammonia synthesis [

Schematic of the steps required to generate an atomistic Wulff construction, here shown for a Au octahedron. The Wulff shape was rendered in Crystal Creator [

While filling a shape with model atoms is not by itself a Wulff construction, it is useful in a variety of contexts not covered by the continuum result. The resulting atomistic models of nanocrystals have been used for modelling surface adsorption [

While the notion of Wulff construction as a shape-predicting tool is generally well-known in the nanotechnology literature and its increased use is welcome, the existence and specific applicability of the multiple variants discussed above are not always acknowledged correctly. There is not, unfortunately, a single Wulff construction applicable to all cases, and much confusion could be avoided by unambiguously specifying which approach (thermodynamic vs kinetic, modified, supported, etc

An early source of confusion is Wulff’s 1901 paper [

Other names, not incorrect but also perhaps adding to the confusion, exist for the thermodynamic Wulff construction, such as “the equilibrium Wulff construction” [

Diagram of the terminology of the various shape constructions based on the considered surface property (energy for thermodynamic, growth velocity for kinetic), crystal structure (single or twinned crystal) and the growth environment (supported or unsupported)

Since interest in crystal shape started gaining momentum, initially to model minerals and later nanoparticles, an increasing variety of computational tools have been developed to implement Wulff-like constructions. The list of available tools is quite large and includes both web-based and desktop applications such as Wulffman [

One approach to solving the Wulff construction problem, implemented in Wulffman [

Wulffman is one of the first user-friendly applications developed to calculate the Wulff shape for crystals of any crystallographic symmetry. The user needs to define, at minimum, the shape’s point group symmetry, selecting either one of the 32 crystal classes, one of the 2 icosahedral point groups or a custom-made point group, the crystallographic vectors and angles, and the desired crystal facets along with their surface energies. Additional input can be provided to simulate dynamic Wulff constructions, to construct partially faceted shapes with smooth connecting regions, or to introduce non-symmetry related facets, a feature that can be exploited to simulate cleavage. Finally, multiple shapes can be created independently in the same window allowing for simulation of shape intersections. The user can define the colour and visibility of the facets and vertices, obtain geometric information such as selected distances, point coordinates and facet areas, while statistics on the number of facets, edges and vertices, as well as total volume and surface area are readily provided. Shapes can be exported as still or animated GIF files. Wulffman is a very versatile tool, however it features quite old graphics and has not been updated since 2002. It is available to download as a binary or C++ source code [

WulffPack [

Another popular approach to render crystal shapes is that of vertex elimination. It is based on the principle that the projection of any point belonging in the Wulff shape on any facet plane normal must be shorter than the normal vector’s length. Hence, after the equivalent facets are populated based on the (user-defined) crystal symmetry, a gamma plot-like vector normal to the plane and with length equal to the corresponding surface energy is defined for each facet. Possible shape vertices are then calculated by the intersection of sets of three planes. If the projection of a vertex vector onto one of the gamma vectors is longer than the length of the gamma vector then the vertex is out of the shape and is eliminated. The remaining vertices are arranged in order, such that they circumscribe the facets. This approach is typically slower than the dual Wulff shape but additional techniques can be implemented to make the algorithm faster: in Wulffmaker [

The Wulffmaker [

WinXMorph [

WinXMorph excels at rendering and allows control over the colour, transparency, lighting, and texture of the crystal, independent of any symmetry relations, achieving artistic to realistic visualisations. Calculated shapes can be exported as .stl or. gif files and as virtual reality files in the VRMLV2.0 format; some information is given on the total volume and area of the constructed crystal.

SOWOS [

NanoCrystal [

Crystal Creator is a standalone user-friendly GUI capable of both kinetic and thermodynamic, single crystal and twin Wulff constructions [

The user input required in Crystal Creator includes selecting one of the three supported crystal structures, setting the surface energies/growth velocities of low index facets, choosing a twinning type and twin plane (or none), setting an optional shell thickness and adjusting some computational parameters such as the grid size. The three dimensional shape is then displayed and can be saved as an image (.png, .tif, .jpeg, etc.) or Matlab figure (.fig) while the coordinates of the grid points in the shape are exported in a text file. The format of this file matches the required DDSCAT input file format; the second file needed for DDSCAT can be interactively produced within Crystal Creator.

The above list of Wulff construction tools is not exhaustive and one can find additional tools. BCN-M [

In addition to the tools developed explicitly to model crystal shape, a few large computational packages also include some Wulff-related features, such as VESTA 3, [

It is worth noting that the Winterbottom construction, explicitly mentioned only in Wulffmaker and Wulffpack, can essentially be modelled by any tool that allows for the addition of a single non-symmetry related facet such as in Wulffman, WinXMorph and SOWOS. The surface energy attributed to the extra facet is the interface energy. Similarly, it is in principle possible to model twinned crystals without the dedicated functions available in Crystal Creator or WinXMoprh: one needs to edit the available codes, as has been reported for BCC twin shapes [

The powerful three dimensional visualisations of the various Wulff construction tools can provide an insight into experimentally observed shapes, construct models based on theoretical data, supply the required shape model to further analyse crystal properties, or serve as an aid to understand how crystal symmetry is reflected in the crystal shape. Ultimately, the choice of tool depends on the required construction type (single crystal, twinned, Winterbottom, double Winterbottom), the material’s crystal structure, the preferred tool interface (application or source code) and the purpose of the construction. For instance, if twinning is required one would use Crystal Creator [

Summary of the main Wulff-modelling software and their shape construction capabilities

Computational approach | Tool | Wulff | Winterbottom | Modified Wulff | Continuum model | Atomistic model | Stoichiometry for binary systems | Programme language | |
---|---|---|---|---|---|---|---|---|---|

Twin | Enhancement | ||||||||

Dual shape | Wulffman | √ | √ | × | × | √ | × | – | C+ + (1998) |

WulffPack | √ | √ | √ only Dh & Ih | × | √ | √ | √some systems | Python (2020) | |

Closest vertex Technique | Wulffmaker | √ | × | × | × | √ | × | – | Wolfram Mathematica (2012) |

WinXMorph | √ | × | √ | × | √ | √ | × | Delphi 6 (2005) | |

SOWOS | √ | √ | × | × | √ | √ | √some systems | Fortran90 (2013) | |

NanoCrystal | √ | × | × | × | × | √ | √some systems | C+ + , Matlab, PHP (2018) | |

Growth front Isosurface | Crystal Creator | √ | × | √ not Ih | √ | √ | × | – | Matlab (2019) |

The Wulff construction and its various implementations are most commonly used to predict nanoparticle shape based on available knowledge of the exposed facets and the corresponding surface energies as depicted schematically in Fig.

Flow chart illustrating the forward and inverse Wulff construction modelling

Surface energies can be obtained numerically from first principles calculations (e.g. DFT, Møller–Plesset perturbation theory [

Experimental approaches on surface energy measurements are scarcely reported and are restricted to very few systems. Examples involve cleavage methods [

Finally, morphing from one shape to another can help instruct the necessary changes in a synthesis to obtain the desired shape. For example, Wulff shape models show that a {100}-faceted FCC cube will shift into a cuboctahedron, expressing both {100} and {111} facets, and eventually into a {111}-bound octahedron, as the {100} to {111} growth velocity ratio increases. Therefore, if starting with a synthesis of cubes one needs to inhibit {111} growth to generate octahedral nanocrystals.

The phrase inverse Wulff construction [

Mathematically, the thermodynamic Wulff construction involves the determination of facet surface areas

In practice, measuring surface areas can be challenging, and often determining _{2}O, [

Given one surface energy value, often determined by DFT, others can be deduced using the ratios. Such quantitative results can be used as inputs for other thermodynamic material models [

As interest in synthesising (nano)crystals increased over the past decades, a plethora of mathematically distinct Wulff-based approaches have been developed to describe and predict their shapes. These models were reviewed and include thermodynamic and kinetic approaches, single crystal and twin shapes, as well as suspended and supported systems, as summarized in Fig.

All except the most recent construction, the modified Winterbottom, have been implemented in one or more openly or commercially available software packages, making crystal shape modelling quick and user-friendly. We described such approaches alongside their capabilities and limitations, thus providing a guide for their use in a variety of contexts.

Looking to the future, the increasingly complex crystal shapes and compositions developed to fulfil the exciting promises of nanotechnology will require increasingly agile descriptive and predictive tools. Wulff-related constructions are capable of supporting such scientific advances owing to their many different modifications accumulated over a century. At present, all of the required mathematical foundations are available, and only further, more streamlined and complete implementations are to be developed and hopefully widely adopted.

Not applicable.

JD and ER outlined the initial structure of the manuscript, and JD performed the initial literature review and wrote Sect.

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 804523). C.B. is thankful for funding from the Engineering and Physical Sciences Research Council (EPSRC, Standard Research Studentship (DTP), EP/R513180/1).

Data available upon request.

The authors declare that they have no competing interests.

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