Free energy of the bcc-liquid interface and the Wulff shape as predicted by the Phase-Field Crystal model

TL;DR

This study uses the phase-field crystal model to calculate the free energy of bcc crystal-liquid interfaces across orientations and temperatures, revealing anisotropic behaviors and resulting in Wulff shapes that evolve from spherical to polyhedral forms.

Contribution

It provides a detailed computation of interface free energies for bcc crystals at various orientations and temperatures using the PFC model, and derives equilibrium Wulff shapes.

Findings

Maximum free energy at {100} orientation for all temperatures.

Minimum free energy shifts from {111} to {211} as temperature increases.

Wulff shapes transition from spherical to polyhedral with increasing temperature.

Abstract

The Euler-Lagrange equation of the phase-field crystal (PFC) model has been solved under appropriate boundary conditions to obtain the equilibrium free energy of the body centered cubic crystal-liquid interface for 18 orientations at various reduced temperatures in the range $\epsilon\in\left[0,0.5\right]$. While the maximum free energy corresponds to the $\left\{ 100\right\} $ orientation for all $\epsilon$ values, the minimum is realized by the $\left\{ 111\right\} $ direction for small $\epsilon\,(<0.13)$, and by the $\left\{ 211\right\} $ orientation for higher $\epsilon$. The predicted dependence on the reduced temperature is consistent with the respective mean field critical exponent. The results are fitted with an eight-term Kubic harmonic series, and are used to create stereographic plots displaying the anisotropy of the interface free energy. We have also derived the corresponding Wulff shapes that vary with increasing $\epsilon$ from sphere to a polyhedral form that differs from the rhombo-dodecahedron obtained previously by growing a bcc seed until reaching equilibrium with the remaining liquid.