Phase-field-crystal methodology for modeling of structural transformations

We introduce and characterize free-energy functionals for modeling of solids with different crystallographic symmetries within the phase-field-crystal methodology. The excess free energy responsible for the emergence of periodic phases is inspired by classical density-functional theory, but uses only a minimal description for the modes of the direct correlation function to preserve computational efficiency. We provide a detailed prescription for controlling the crystal structure and introduce parameters for changing temperature and surface energies, so that phase transformations between body-centered-cubic (bcc), face-centered-cubic (fcc), hexagonal-close-packed (hcp), and simple-cubic (sc) lattices can be studied. To illustrate the versatility of our free-energy functional, we compute the phase diagram for fcc-bcc-liquid coexistence in the temperature-density plane. We also demonstrate that our model can be extended to include hcp symmetry by dynamically simulating hcp-liquid coexistence from a seeded crystal nucleus. We further quantify the dependence of the elastic constants on the model control parameters in two and three dimensions, showing how the degree of elastic anisotropy can be tuned from the shape of the direct correlation functions.

Figure 1

Plots of $\Delta F_{\text{id}}$, where in all plots the ideal-gas energy is plotted with a solid line, and the expansion point for the polynomial form of the free energy is $n(\mathrm{r⃗})=0$. Equation (4) with (a) $\nu =0$ and $\xi =3$ (dash-dot) and $\nu =1$ and $\xi =1$ (dash); (b) $\xi =1$ and $\nu =1$ (dash-dot), $\nu =1.5$ (dash), and $\nu =2.45$ (dot); and (c) $\nu =1$ and $\xi =1$ (dash-dot), $\xi =0.5$ (dash), and $\xi =0.1675$ (dot).

Figure 2

Fourier space representation of 3D lattices, where the wave vector $k$ is given in units of the inverse lattice spacing $a=1$. Spectral peaks for (a) bcc, (b) fcc, and (c) sc lattices are plotted for ${\sigma }_v=0.05$.

Figure 3

(a) Two peak correlation functions ${\mathrm{Ĉ}}_2(k)$ for hcp (dot), fcc (dash), and bcc (dash-dot) lattices. In this illustration, the lattice spacing has been rescaled such that the first peak of the function overlaps for all three lattices. For simplicity, the case of zero effective temperature is shown here (i.e., $\sigma =0$). ${\alpha }_1$ and ${\alpha }_2$ are free parameters that set the width of each peak correspondingly. Planes illustrating the planar symmetry and the planar atomic density are shown in (b) for bcc $(110)$ (dominant) and $(200)$ planes and (c) for fcc $(111)$ (dominant) and $(200)$ planes.

Figure 4

(a) Phase diagram for an fcc lattice correlation function showing the coexistence between liquid, fcc, and bcc phases near a peritectic point. (b) Free-energy curves for the liquid (dot), fcc (solid), and bcc (dash) phases at a temperature close to the peritectic point of the phase diagram.

Figure 5

Effect of the $k=0$ mode illustrated using the 2D square correlation function. (a) Correlation functions with different $k=0$ mode amplitudes: ${\mathrm{Ĉ}}_2(0)=0$ (solid), ${\mathrm{Ĉ}}_2(0)=-1$ (dashed), and ${\mathrm{Ĉ}}_2(0)=-2$ (dotted). (b) Effect of the $k=0$ mode on the phase stability illustrated by free-energy curves for the liquid (black) and square (red/gray) phases for the three amplitudes shown in (a).

Figure 6

(a) Dynamic coexistence of hcp with a liquid. Three perpendicular planar cuts through the simulation volume are shown illustrating the hcp structure. (b) Free-energy curves for a quench using an hcp correlation function shown for the liquid (dot), bcc (dash), and hcp (solid). The energies clearly show the stability of the hcp lattice relative to a minimized bcc and liquid state at the reference density ($\mathrm{n̄}=0$).

Figure 7

(a) Elastic coefficient $m_b=(C_{11}+C_{12})$ for a square lattice from Eq. (14) as a function of correlation peak widths ${\alpha }_1={\alpha }_2$. Numerical (open circles) and analytical (filled circles) results are shown and fitted to Eq. (18) (dot). (b) Elastic coefficients $m_b$, $m_o$, and $m_s$ for the fcc lattice for bulk (filled circle), orthorhombic (open circle), and simple shear strain (filled diamonds) states as a function of the correlation function peak widths ${\alpha }_1={\alpha }_2$. Dotted lines shows fits to Eq. (18).

Figure 8

(a) Relative change in the elastic coefficient $m_b$ with temperature $\sigma$ for several correlation peak widths ${\alpha }_1={\alpha }_2=0.05$, $0.075$, $0.1$, $0.15$, and $0.175$ (bottom to top). (b) Dependence of the quadratic fit coefficient $b_{\alpha }$ in Eq. (24) on ${\alpha }_i$. Dotted line shows an exponential fit (see text).

Figure 9

Elastic coefficient prefactors $A_i$ as a function of ${\alpha }_2/{\alpha }_1$ for the orthorhombic strain $A_o$ (empty circle) and simple shear $A_s$ (filled circle).