# Combining phase field crystal methods with a Cahn-Hilliard model for binary alloys

## Introduction

Phase transitions in materials are typically accompanied by structural changes to the lattice symmetry [[1], [2], [3]]. These changes include individual lattice distortions [[1], [3]], grain rotations [[2]] and lattice defect formations [[4]]. All of these influence the microstructures that form in a material [[4], [5]]. In this paper, we introduce a modeling approach that couples the lattice symmetry with the composition field. Two continuum methods, namely a Cahn-Hilliard model and a phase field crystal model, are coupled and solved to describe a phase transition process.

The lattice symmetry and grain orientations in microstructures are known to affect physical properties of materials. For example, lithium diffusion in a battery electrode induces lattice deformation [[6], [7], [8], [9]], which affects lithium ion kinetics [[6]] and causes anisotropic expansion of electrodes [[7]]. Likewise, in a paraelectric to ferroelectric phase transitions, lattices transform from centrosymmetric to other point groups lacking an inversion centre. This transformation introduces stress-free spontaneous strains in the ferroelectric system [[1]]. At present, theoretical models like phase field methods describe complex microstructures in electrode/ferroelectric systems as a function of the composition field (lithium-ion concentration, temperature or polarization) [[10], [11], [12], [13]]. The Kobayashi-Warren-Carter phase field model [[9]] further accounts for crystallographic misorientation at grain boundaries during a phase transition process. While these modeling approaches provide insights on the position of phase and grain boundaries, they only account for grain orientations as an empirical parameter [[9]]. The current phase field approaches do not allow for lattices to distort independently. Consequently, the local strain fields arising from individual lattice distortions and the presence of defects in a material system are not explored.

Alternatively, a phase field crystal (PFC) method proposed by Elder and Grant [[14], [15]] describes atomistic details of material systems with periodic solutions. This modeling technique describes coarse-grained symmetry of a periodic system [[16], [17]], and is computed at faster time scales than the molecular dynamics simulations [[18]]. The PFC model has been applied to explore lattice defects in graphene [[19]] and nucleation problems in colloidal systems [[20]]. Binary alloy models [[21], [22], [23], [24]], an extension to the PFC formalism, was demonstrated to describe solidification [[23], [25]], crystallization [[26], [27]] and phase segregation processes [[28], [29]]. The PFC approach is a useful tool for multiscale modeling to describe the lattice symmetry of a material system.

In the current work, we combine the phase field crystal methods with a Cahn-Hilliard model in a 2D theoretical framework to model a phase transition process. The modeling approach couples two field parameters of a model system, namely the composition field and the coarse-grained lattice-symmetry distortions. The Cahn-Hilliard equation describes microstructures with a composition order-parameter field. The phase field crystal equation models a coarse-grained representation of lattice symmetry with peak density field as its order parameter. In the PFC equation, we introduce coordinate transformation coefficients to relate lattice symmetries in 2D point groups via affine transformations. These transformation coefficients are coupled with the composition field and influence the underlying lattice symmetry of a material system. As the composition field evolves following the Cahn-Hilliard equation, the transformation coefficients are updated in the PFC model. The PFC model computes the equilibrium lattice arrangements of the material system during composition evolution. Here, an assumption is that the dynamics of the PFC model is fast relative to the composition field dynamics. Using this coupled approach, we model the structural evolution of lattice distortions and defects during a phase transition process.

In this paper, we investigate the nature of the coupled CH-PFC methods by modeling three representative examples. First, we consider base cases to understand how transformation coefficients affect the coarse-grained lattice symmetries of the material system. Here, we stabilize hexagonal and square symmetries as representative lattice structures corresponding to two composition-field values. Second, we extend these base cases to investigate how the model interpolates the peak density field across a diffuse composition phase boundary. We model a representative binary alloy with hexagonal and square symmetry phases and explore the lattice distortions across diffuse interfaces. Finally, we model the composition field in the binary alloy to follow a Cahn-Hilliard type of diffusion and study the accompanying equilibrium lattice arrangements described by the peak density field. The simulations show lattice distortions at coherent interfaces and demonstrate structural evolution of lattice arrangements during a phase transition.

## Coupled Cahn-Hilliard – phase-field crystal model

The aim is to couple the Cahn-Hilliard (CH) and the phase field crystal (PFC) methods, to explore structural changes to lattice symmetry during diffusion induced phase transition. In this section, we first introduce the two continuum models and explain how these methods are coupled in a 2D theoretical framework. Next, we describe the evolution of the two order parameters, namely the composition field and the peak density field. Finally, we discuss the numerical procedure followed to compute the coupled CH-PFC methods.

The first model is a Cahn-Hilliard method that describes the continuum composition field of a model system. This method utilizes a double-well free-energy function in terms of a composition field, , which is its order parameter. The second model is a phase field crystal (PFC) method that describes the coarse-grained lattice symmetry for the model system, and statistically illustrates lattice orientation, distortion and defect density. This approach describes a free energy functional that is minimized by a spatially-periodic order parameter, . In the current work, we couple the two models by using the composition field to influence the underlying lattice symmetry of the model system. The composition is not coupled to the peak density field via a homogeneous free energy, but rather as the coordinate transformation coefficients of the composition-dependent Laplacian , relative to a Cartesian basis. That is, each of the 5 Bravais lattices in the 2-dimensional space are stabilized by computing the Laplace operator in a transformed space on a coordinate plane. The transformation coefficients which control lattice deformations are described as functions of the composition field. These coefficients are updated during the evolution of the composition field.

The total free energy functional for the CH-PFC model is given by:

(1) |

Here, and describe the homogeneous energy contributions from the Cahn-Hilliard and PFC equations respectively. The composition gradient-energy coefficient is given by . The operator controls the coarse-grained lattice symmetry described by the particle density field. This operator is modeled as a function of the composition field and is discussed in detail later on in this section. The coefficients, and , correspond to the energy barrier height and to the local equilibrium states of respectively. The parameter , controls the second-order phase transition of the PFC model. In this paper, we model as a constant to always describe a crystalline-solid state. The parameters , relate the PFC equation to the first-order peak in an experimental structure factor. Further details on these coefficients are explained in the work by Elder and Grant [[15]].

Before proceeding with the model description,
we first normalize the free energy functional:

(2) |

The composition field, is normalized as , with local equilibrium states at and . The dimensionless peak density field, is given by . We set, , as a constant such that Eq. 2 always models a stable crystalline-solid phase for the peak density field [[15]]. With , the peak density field in Eq. 2 describes a hexagonal symmetry with a periodic spacing of at equilibrium. Note, is the length scale of the PFC model and . The gradient energy coefficient , is numerically calibrated such that the width of the diffuse composition interface spans over peaks described by the peak density field, . We introduce a constant, that relates the free energy normalizations of the Cahn-Hilliard and the PFC model. For simulations in this paper, we set .

The composition-dependent Laplacian in Eq. 1, introduces the composition-lattice symmetry coupling. Here, the composition terms enter the Laplacian via its coordinate transformation coefficients. The Laplace operator is written in terms of its second partial derivatives:

(3) |

where are the coordinate transformation coefficients. These coefficients are described as functions of the dimensionless composition field and correspond to the elements of a transformation matrix:

(4) |

The matrix , describes affine lattice transformations using hexagonal symmetry as the reference structure [[30]], [[31]]. With , the transformation matrix is an identity matrix and Eq. 2 describes a hexagonal symmetry in 2D [[15]]. In the current work, we choose the hexagonal and square symmetries to represent phases with compositions and respectively. These symmetries are chosen to illustrate exaggerated symmetry deformations during phase transition. The transformation coefficients in Eq. 4, , are modeled as linear functions of the dimensionless composition field, . We define to be the transformation coefficients corresponding to the hexagonal lattice , and is the deformation required to transform the lattice with a hexagonal symmetry to a square symmetry . Note, in both the hexagonal and square lattice symmetries, the transformation matrix encourages a periodic lattic- symmetry spacing of .

Fig. b shows a schematic illustration of the Cahn-Hilliard – phase field crystal concept. In Fig. a, the transformation matrix describes lattice symmetry as a function of the composition field. For , the transformation matrix is an identity matrix, which describes the composition-dependent Laplacian (in Eq. 3) in an isotropic coordinate space. With the CH-PFC model stabilizes a hexagonal lattice symmetry at equilibrium, see blue hexagonal symmetry in Fig. a. However, for a system with , the transformation matrix introduces anisotropy in the transformation coefficients (in Eq. 3), which models the composition-dependent Laplacian in a transformed coordinate space. With the CH-PFC model results in a square symmetry at equilibrium, see the red square in Fig. a. Next, Fig. b schematically illustrates how the CH-PFC model interpolates the lattice symmetry across a diffuse phase boundary. Here, the transformation matrix is locally defined in space as a function of the composition field. For , the transformation matrix interpolates the peak density field to describe intermediate lattice symmetries between the square and the hexagonal, see the dashed quadrilaterals in Fig. b.

Next, we describe the evolution of the two order parameters during phase transition. Here, we assume that the elastic relaxation of the dimensionless peak density field, , is achieved instantaneously in comparison to the evolution of the composition field. Consequently, we model to be maintained throughout the phase transition process.

The composition field evolves using a generalized Cahn-Hilliard equation:

(5) |

Here, and is the dimensionless time variable . is the isotropic diffusion coefficient in Eq. 5 and is the size of the simulation grid. The variational derivative in Eq. 5, produces coupled terms connecting the peak density field and the composition field. In Eq. 5, it is of interest to note the two types of Laplace operators, , respectively. The Laplace operator is . This Laplacian computes the Cahn-Hilliard diffusion isotropically. The composition-dependent Laplacian describes its partial derivatives in a transformed-coordinate space, see Eq. 3. The transformation coefficients are influenced by the local composition field values and computes the derivatives of in a transformed-coordinate space. The propagation of the composition diffusion front given by Eq. 5 is affected by both the coarse-grained lattice arrangements and the local-composition of the model system. As the composition field evolves, the transformation coefficients in the composition-dependent Laplacian , are updated accordingly.

As the elastic relaxation is much faster than composition evolution, we introduce a time-like fictive variable to compute . The variable is treated as a rapidly changing parameter in comparison to the dimensionless time, . This variable is used as a relaxation parameter to approximate equilibrium of at each :

(6) |

Here, and are the sides of a rectangular simulation domain, and . Eq. 6 follows from the numerical scheme introduced by Melenthin et al. [[32]] that allows equilibrium states to be attained faster in comparison to the standard equation of motion of the PFC model [[15]]. Here, , is treated as a locally nonconserved order parameter, while the mass conservation, , is ensured globally. Other approaches to model faster dynamics for the peak density field can be found in the work by Heinonen et al. [[33]]. Note, the variational derivative in Eq. 6 introduces coupled composition-lattice symmetry terms. These coupled terms affect the symmetry of the periodic system.

Eqs. 5 and 6 are computed using an Euler discretization scheme in a 2D finite-difference framework. Simulation grids of size are modeled with periodic boundary conditions and with grid spacings of . At each grid point, the dimensionless composition and peak density fields are represented in their discrete forms as and respectively. The dimensionless composition time derivative in Eq. 4 is computed at regular time steps of , to track the evolving composition field. At each time step, , the transformation coefficients of the Laplace operator , are updated to correspond with the evolving composition field. Next, the equilibrium lattice symmetry at time , is identified by maintaining . This general numerical procedure is iterated. In other work, we apply the CH-PFC method to model Li-ion diffusion in electrode materials [[34]].

## CH-PFC simulations

In this section we investigate the nature of the CH-PFC methods by simulating a few representative examples. First, we explore how the transformation coefficients stabilize hexagonal and square symmetries as a function of the composition field. Using the hexagonal and square symmetries as base cases, we next model a representative binary alloy with diffuse interfaces. Here, we study how the model interpolates the peak density field across a diffuse phase boundary. Finally, we simulate a Cahn-Hilliard type of diffusion for the composition field and model the accompanying structural changes to the underlying lattice symmetry during a phase transition.

### Lattice symmetry

At first, we describe two representative systems (not necessarily a physical system) with homogeneous composition fields, and respectively. The composition fields are treated as fixed. Using the composition fields as input, we compute the peak density fields for the periodic systems. These representative systems will be generalized subsequently in the following subsections. Note, the peak density field is rapidly evolving with reference to the composition field dynamics, and is modeled with a fictive time in the subsequent computations, see Eq. 6.

Two simulation grids of size are modeled with periodic boundary conditions. The transformation matrices at each grid point, for the two representative systems with and are computed following Eq. 4:

(7) |

Matrices, and describe the transformation coefficients to model the hexagonal and square lattice symmetries respectively. Note, the determinant of the matrices in Eq. 7 are and respectively. The difference in the determinants , indicates an area change between the square and hexagonal lattices. This is because, in the current work we model hexagonal and square symmetries to assume equal lattice spacing of . Therefore the number density of peaks changes with lattice symmetry.

Using the transformation matrices in Eq. 7 we next compute the peak density fields of the periodic systems. The simulation grids are initialized with random peak-density field values, – a condition that we will refer to as the “random initial seed”. Starting from this random state and average density, , the evolution of the peak density field, Eq. 6, is iterated until equilibrium is reached.

Fig. b shows the evolution of density fields from randomized initial states, for the two homogeneous composition fields, and , respectively. During evolution, individual grains with hexagonal and square lattice symmetries nucleate in Fig. a and Fig. b respectively. Note, grains of different sizes and lattice orientations form during a CH-PFC simulation, see ’During evolution’ in Fig. b. At the grain boundaries, lattice symmetries distort to form coherent interfaces. At equilibrium, individual grains arrange to minimize lattice misfits at the grain boundaries. A coarse-grained representation of hexagonal and square lattice symmetries are formed in Fig. a and Fig. b respectively.

Fig. b shows the formation of multiple grains in homogeneous composition fields and identifies the position/orientation of the grain boundaries in the model system. In Fig. a, the density peaks that model the hexagonal symmetry are of circular shape. However, for the square symmetry in Fig. b, the density peaks are ellipsoidal in shape. This difference in the density peak shapes is explained from the use of transformation matrices and in Eq. 7. The transformation matrix for hexagonal symmetry, describes an isotropic composition-dependent Laplacian, . This computes the density peaks to be of circular shape. While, the transformation matrix for a square symmetry, introduces transformation coefficients in the composition-dependent Laplace operator, see Eq. 7 and Eq. 3. These transformation coefficients shear the density peaks to an ellipsoidal shape. Similar ellipsoidal density peaks are observed in the anisotropic PFC simulations [[23], [35]]. Furthermore, the density peaks near grain boundaries in both Fig. a and Fig. b, appear smeared and deviate from the regular ellipsoidal/circular shapes. Here, an interpretation is that the smeared appearance indicates lattice distortion at the interfaces to maintain coherency between neighboring grains.

### Diffuse interface

Next, we investigate the model behaviour to interpolate the peak density field across a diffuse interface in a representative binary alloy. Here, the hexagonal and square lattice symmetries at compositions and are used as base cases, and correspond to the two phases of the binary alloy. A representative binary alloy with diffuse phase boundaries is modeled and its composition field is treated to be fixed. The equilibrium lattice symmetry for this system with heterogeneous composition field is computed.

A periodic simulation grid of size is modeled. Here, two phases with and separated by a sharp interface is assumed in the initial state:

(8) |

Next, the composition field is evolved following Eq. 5, without any influence from the peak density field. That is, . The composition time derivative is iterated until the phase boundary begins to smooth and is then held fixed. This is to explore the coupling of the fast kinetics of for a single interation of . Fig. 3(a) illustrates the composition of a binary alloy with diffuse phase boundaries. Fig. 3(b) shows the composition variation across the simulation grid at .

Following Eq. 3-4, the transformation matrix, , is next computed with describing the discrete composition field shown in Fig. 3(a):

(9) |

Here, defines the transformed space for the composition-dependent Laplace operator at each grid point. Using this transformation matrix as an input, the equilibrium peak density field is next computed, Eq. 6.

To model the lattice symmetry of the binary alloy shown in Fig. 3(a), the simulation grid is initialized with random peak density field values, . Using from Eq. 9, the evolution of the peak density field, Eq. 6, is iterated to find the equilibrium lattice-symmetry for the model system. Fig. 3(c) shows the equilibrium lattice-arrangements described for the heterogeneous composition field (shown in Fig. 3(a)). Lattices with square symmetry are stabilized in the phase with , and hexagonal symmetry is observed in the phase with . At the phase boundaries, , the coupled CH-PFC model describes a coarse-grained representation of deformed lattices. Here, the density peaks are smeared to illustrate the lattice distortion at the phase boundaries, see Fig. 3(c). Note, the composition phase boundary is numerically calibrated to span over density peaks (about 25 grid spacings). Fig. 3 provides an atomistic insight into the coarse-grained lattice arrangements across a diffuse phase boundary.

### Phase transition

Up to this point, we only modeled the microscopic configurations at fixed compositions. However, to model phase transition with microscopic insights on the coarse-grained lattice symmetry, we need to simulate the evolution of the composition field. The binary alloy in Fig. 3 is considered as the initial state, and we next extend the simulation to describe the propagation of the diffusion front. A representative Cahn-Hilliard type of diffusion for the composition field is modeled. During the phase transition, the equilibrium lattice arrangements of the underlying system is computed. An assumption made in this simulation is that the dynamics of elastic relaxation (equilibrating the peak density field) is several times faster than the diffusion of the composition field. Using this CH-PFC approach we investigate how composition field influences the lattice arrangements in a model system during phase transitions.

Taking as an initial state, the lattice arrangements described for a binary alloy from Fig. 3, the phase transition is modeled by allowing composition to diffuse into the simulation domain. The composition field is held fixed at , for and throughout the simulation. This boundary condition is a proxy for having a consistent composition reservoir. The composition field on the remaining part of the simulation grid, , is allowed to evolve with time. The composition time derivative, Eq. 5, is iterated from to , in dimensionless time intervals of . Note, the composition evolution at , receives input from the equilibrium peak density field calculated for the time step. The composition field is tracked as , until a homogenous phase is obtained.

The composition field at each evolution step, , is used as an input to compute the transformation matrix in Eq. 9. At a given time step, , the transformation matrix , is used to calculate the equilibrium peak-density-field following Eq. 6. The composition and peak density fields are iterated until the phase transition is complete.

Fig. f shows the structural evolution of the coarse-grained lattice arrangements during the phase transition. At the intial state , the coarse-grained lattice symmetry for the heterogeneous composition, is described, see Fig. a. Here, two coherent phases with square and hexagonal symmetries are formed in domains with and respectively. Note in Fig. a, the edges of the square lattices are mostly aligned with the axes of the simulation grid. A pair of green arrows in Fig. a illustrates the orientation of square lattices in the simulation grid. Across the diffuse phase boundary, hexagonal and square lattices are distorted to maintain coherency, see Fig. a. Next, in Fig. f(b-e), as the composition field diffuses into the simulation domain, the hexagonal lattice symmetry is transformed to a square symmetry.

In Fig. b, the phase with square symmetry occupies of the simulation grid. Here, it is interesting to note that square lattices begin to rotate uniformly as the diffusion front propagates through the simulation grid. In Fig. f(c-d), the square lattice symmetry is observed to rotate further (e.g., orientation of the green arrows in Fig. f(c-d)). We interpret that the square lattices rotate to maintain coherency with the neighboring hexagonal phase. Note, the periodic boundary conditions on the simulation domain further enforce an additional strain on the peak density field. This is discussed in detail in the next section of this paper. In Fig. e, a grain boundary (as indicated by the dashed line) is formed in the square symmetry phase. This grain boundary migrates in the square symmetry phase and remains in the homogeneous phase, see Fig. f. At , the phase transition is complete with a homogenous composition field and a phase with square symmetry is described at equilibrium, see Fig. f.

## Discussion of the CH-PFC model

The coupled Cahn-Hilliard – phase field crystal model provides a theoretical framework to describe continuum phase transition with microscopic insights. There are several issues we feel remain to be clarified in interpreting the simulations. Among these issues are three questions: Do the peaks in the CH-PFC simulations represent atomic sites or illustrate the underlying lattice symmetry? Are the total number of peaks in a simulation grid conserved? In Eq. 1 why was the composition field coupled with the peak density field only via the Laplace operator? In this section, we discuss these key details of the coupled CH-PFC model and explore potential further work.

First, the peak density field in CH-PFC simulation describes the coarse-grained lattice symmetry of the underlying atomic arrangements. Individual peaks do not represent atomic sites, however the arrangement of peaks indicates the unit cell symmetry of the model system. Similarly, a grain boundary in a CH-PFC simulation is a coarse-grained approximation of the underlying lattice orientations, distortions and defects. Fig. 5 provides a schematic illustration of the difference between atomic sites, peak positions and coarse-grained lattice symmetry. In Fig. 5, the small-black dots indicate atomic sites, which correspond to the deterministic positions of atoms in the unit cell. The big-green dots highlight representative peak positions modeled by a CH-PFC method. The dashed-red lines connecting the peaks in Fig. 5, indicate an example of a coarse-grained lattice symmetry. In Fig. 5, the side of the coarse-grained lattice is four times that of the unit cell. However, in our CH-PFC simulations, the coarse-grained lattice is several times larger than a unit cell.

Second, the number density of peaks in the CH-PFC simulations are not necessarily conserved. Let us consider Fig. b, where hexagonal and square symmetries are described on identitical computational grids of size . The total number of peaks in both these symmetry systems are not necessarily the same. This can be explained from two reasons: First, the transformation matrices in Eq. 7, and , describe lattice symmetries with an area difference . Second, the periodic boundary conditions on the computational grid enforces a strain on the peak density field. That is, on an infinitely large grid size, the peak density field would assume the fundamental length scale of specified by the transformation coefficients in Eq. 3. However, by modeling this density field on a periodic grid with finite dimensions, we strain the lattice symmetry spacing and force the peak density field to satisfy periodic boundary conditions. To minimize these imposed strains and to simultaneously maintain periodicity, the CH-PFC model introduces (or removes) peaks to (or from) the simulation grid. For future applications of the CH-PFC model, the computational grid size is to be calibrated to correspond to the closest fundamental length scale of the peak density field. Alternatively, numerical correction terms to Eq. 6 to conserve the number of peaks can be used [[36]].

Finally, in Eq. (1), the composition and the peak density fields are coupled only via the Laplace operator. That is, the coarse-grained lattice symmetry described by Eq. 1 is solely determined by the coordinate transformation coefficients of the Laplace operator. These transformation coefficients (which are functions of the composition field) describe lattice transformations with hexagonal symmetry as the reference structure. In this paper, we assumed the ideal free energy contribution from other non-linear terms in Eq. 1, to be independent of the composition field for a couple of reasons: First, this assumption allows the CH-PFC model to stabilize a reference hexagonal lattice symmetry for composition field, . Second, Eq. 1 will always describe a crystalline/ordered state for the model system. This is because the driving force for the peak density field towards the disordered state (controlled by term ) is not a function of the composition field.

## Summary

We introduced a 2D theoretical framework, which combined a Cahn-Hilliard (CH) model and a phase field crystal (PFC) model, to describe a phase transition process. In this CH-PFC method, the composition field was coupled to the coarse-grained lattice symmetry (peak density field) of the periodic system. The CH-PFC modeling approach captured the effects of microscopic configurations, such as lattice orientations, distortions and presence of defects, on the phase-transition process. Furthermore, the model described the structural evolution of the coarse-grained lattice symmetry during a phase change.

Using the CH-PFC approach, we stabilized representative lattice symmetries (hexagonal and square) as a function of the composition field. Here, we found that multiple grains formed in a single phase, and identified the position and orientation of grain boundaries. Next, in a binary alloy, we described the coarse-grained distortion of lattice symmetry across a diffuse phase boundary. Finally, we modelled a representative phase transition process – here, the CH-PFC simulations modeled grain rotations and grain boundary migrations during phase change.

## Acknowledgements

A.R.B acknowledges the support of the Lindemann
trust fellowship. The authors would like to acknowledge the support
by the grant DE-SC0002633 funded by the U.S. Department of Energy,
Office of Science, in carrying out this work. Further, the authors
wish to thank Dr. Rachel Zucker for useful discussions on phase field
crystal modeling methods.

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