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

Diffusion-induced phase transitions typically change the lattice symmetry of the host material. In battery electrodes, for example, Li ions (diffusing species) are inserted between layers in a crystalline electrode material (host). This diffusion induces lattice distortions and defect formations in the electrode. The structural changes to the lattice symmetry affect the host material's properties. Here, we propose a 2D theoretical framework that couples a Cahn-Hilliard (CH) model, which describes the composition field of a diffusing species, with a phase-field crystal (PFC) model, which describes the host-material lattice symmetry. We couple the two continuum models via coordinate transformation coefficients. We introduce the transformation coefficients in the PFC method to describe affine lattice deformations. These transformation coefficients are modeled as functions of the composition field. Using this coupled approach, we explore the effects of coarse-grained lattice symmetry and distortions on a diffusion-induced phase transition process. In this paper, we demonstrate the working of the CH-PFC model through three representative examples: First, we describe base cases with hexagonal and square symmetries for two composition fields. Next, we illustrate how the CH-PFC method interpolates lattice symmetry across a diffuse phase boundary. Finally, we compute a Cahn-Hilliard type of diffusion and model the accompanying changes to lattice symmetry during a phase transition process.

Figure 1

Schematic representations of the Cahn-Hilliard–phase-field crystal (CH-PFC) method. (a) The reference blue-hexagonal symmetry deforms to a red-square symmetry under the transformation matrix described by Eq. (4), with composition field $c=1$. (b) A continuous change of the lattice symmetry from a square to a hexagon as a function of the composition field. The dashed quadrilaterals across the diffuse phase boundary illustrate the intermediate lattice symmetries.

Figure 2

Evolution of the peak density fields in representative systems forming (a) hexagonal and (b) square symmetries with homogeneous composition fields $c_{ij}=0$ and $c_{ij}=1$, respectively. Subfigures illustrate the peak density fields starting from a randomized initial state (far left), during evolution (center) and at the final equilibrium state (far right).

Figure 3

(a) The composition field of a representative binary alloy. (b) Variation of composition across AA in the simulation grid at $j=15$. (c) The equilibrium peak density field describing the underlying symmetry of the binary alloy corresponding to the composition field in (a). Square and hexagonal symmetries are described for phases with $c_{ij}=1$ and $c_{ij}=0$, respectively. The width of the composition phase boundary (illustrated by vertical dashed-lines) is numerically calibrated to span across $\sim 4$ peaks.

Figure 4

Phase transition showing structural evolution of lattices from a hexagonal (in blue with $c_{ij}=0$), to a square symmetry (in red with $c_{ij}=1$). Subfigures illustrate lattice transformations as a function of the composition field, $c_{ij}$. The pair of green arrows indicate the orientation of the square symmetry during phase transition. The dashed lines in (e) and (f) indicate a grain boundary in the square symmetry phase.

Figure 5

Schematic illustration of the atomic sites, peak positions, and coarse-grained lattice symmetry in a square phase system. The small-black dots represent atomic sites of the unit cell. The big-green dots schematically indicate peak positions in a CH-PFC simulation. The dashed-red lines connecting the peaks highlight an example of a coarse-grained lattice symmetry of the underlying unit cells. Note, in this figure the coarse-grained symmetry is four times the unit cell size.

Figure 6

The finite size effect of the computational domain on the square lattice symmetry. Computational domains of size (a) $L=100\delta x$, (b) $L=363\delta x$, and (c) $L=725\delta x$ are modeled with composition $c_{ij}=1$ to correspond with the square lattices. Inset figures illustrate lattice defects such as edge dislocations (small yellow circles) and grain boundaries (large green ellipses) in larger computational domains.

Figure 7

Edge-dislocations react with each other on further evolving the density field in Fig. 6. The subfigures (a–c) correspond to the inset box in Fig. 6.

Figure 8

Geometrical analysis of the (a) hexagonal and (b) square lattice symmetries formed at equilibrium in Figs. 2 and 2, respectively. The inset figures are sections of the equilibrium patterns with $\sim 50$ peaks. The lattice vectors $\mathit{a},\mathit{b}$ describe edges of a unit cell and enclose an interaxial angle $\mathbf{\gamma }$. The red-dashed lines in inset figures schematically illustrate $n$-fold symmetry of the patterns about $\mathit{c}$ direction (perpendicular to the paper).

Figure 9

Schematic illustration of 1D density fields described by the standard PFC model (in black with wavelength $a_0$) [21] and the CH-PFC model [in red with wavelength $\xi A\left(c\right)a_0$]. The black wave represents atomic sites separated by the lattice spacing $a_0$, while the red wave is a coarse-grained lattice unit $\xi A\left(c\right)a_0$ described by the CH-PFC model. In this illustration, the coarse-grained unit is five times the lattice spacing. In the CH-PFC simulations with $\xi \gg 1$, the peak separation is multiple times larger than a unit cell.