Free Energy Functionals for Efficient Phase Field Crystal Modeling of Structural Phase Transformations

The phase field crystal (PFC) method is a promising technique for modeling materials with atomic resolution on mesoscopic time scales. While numerically more efficient than classical density functional theory (CDFT), its single mode free energy limits the complexity of structural transformations that can be simulated. We introduce a new PFC model inspired by CDFT, which uses a systematic construction of two-particle correlation functions that allows for a broad class of structural transformations. Our approach considers planar spacings, lattice symmetries, planar atomic densities, and atomic vibrational amplitudes in the unit cell, and parameterizes temperature and anisotropic surface energies. The power of our approach is demonstrated by two examples of structural phase transformations.

Figure 1

(a) Direct pair correlation function for a square lattice at two effective temperatures ($\sigma =0.82$, solid and $\sigma =0.5$, dotted, see text). The two peaks in the correlation function each represent one of two planes in the unit cell with interplanar spacings ${\lambda }_1$ and ${\lambda }_2$ (see inset). (b) Correlation function for a simple cubic (sc) lattice at two different temperatures. In 3D, a cubic lattice is represented by three planes in the unit cell.

Figure 2

Effect of peak width ${\alpha }_i$ in the correlation functions on planar interface width $W_i$. Left: Three values of ${\alpha }_i$ and their resulting interface widths. Top right: Peak shapes corresponding the interface widths to the left. Bottom right: Dependence of simulated interface width on the correlation peak width (see text).

Figure 3

(a) Phase diagram for a square lattice correlation function showing coexistence between liquid, solid, and triangular phases. The inset contains stripes of square and triangular lattices in a liquid phase quenched to a temperature $\sigma =0.79$. The square phase density was initialized to $n_{\mathrm{sq}}=0.067$ and the triangular phase density to $n_{\mathrm{tri}}=0.076$. The mean density of the system is set to $\overline{n}=0.07$ and the surface energy parameters are ${\alpha }_1=\sqrt{2}$ and ${\alpha }_2=1$. (b) Phase diagram for a fcc correlation function. The insets show the peritectic point and the energy curves for the liquid, fcc and bcc states at $\sigma =0.06$.

Figure 4

2D nucleation from triangles to squares with noise amplitude of $A_n=0.01$ (a) A liquid with density $\overline{n}=0.09$ is quenched into the triangular region of the phase diagram ($\sigma =0.82$), marked ($A$) in Fig. 3a, produces grains with triangular symmetry. (b)–(d) A further quench into the square portion of the phase diagram ($\sigma =0.6$), marked ($B$) in Fig. 3a, illustrating the nucleation and subsequent growth of square grains at the vertices of the triangular grains.