Finding direct correlation functions for desired two-dimensional lattices with a phase-field crystal

Abstract

The phase-field crystal model is one of the most successful models with which to describe crystallization at a near-atomic scale with a larger timescale compared with other atomistic methods when direct correlation functions (DCFs) for the desired lattice are specified. However, the DCFs are, in general, not known and are hard to obtain from essential material information, such as primitive lattice vectors and atom positions. In this paper, we propose a method of obtaining two-point DCFs for desired two-dimensional lattices. The proposed optimization scheme is simple in that it minimizes the temporal change in free energy with respect to the target lattice using a gradient descent. In numerical experiments, we successfully obtained DCFs not only for well-known two-dimensional lattices (i.e., triangular, square, rectangular, honeycomb, and kagome lattices) but also for five nontrivial lattices (i.e., maple leaf, ladybug, trellis, Lieb, and CaVO lattices). We also show that these five lattices can be simulated using a phase-field crystal with at least five modes.