Fourier-spectral-method implementation of deformation in the phase-field crystal model: Further development

Abstract

The phase field crystal (PFC) method is a promising material modeling tool due to its atomistic resolution and diffusive time scale. In the PFC calculations, the Fourier spectral method (FSM) is often used due to its high accuracy and efficiency. In the previous work, I presented the scheme to implement FSM when the system is subjected to homogeneous deformation that can skew the computational domain. The scheme utilizes the coordinate transformation so that the deformation alters the Fourier wave vectors while the configuration of the discrete data is unaffected. As a result, the discretized data is always in the undeformed configuration, which allows for simple implementation of the discrete Fourier transform (DFT). Nevertheless, the previously-proposed scheme is complicated by the fact that the coordinate transformation is not performed on the real-space functions, which results in unnecessary complexity in interpreting the result from the DFT. Therefore, in this work, I present three developments to the previously-proposed FSM scheme. First, the FSM scheme is modified in such a way that the coordinate transformation to the undeformed configuration is applied to the real-space functions prior to the Fourier-transform step. In this manner, the configurations of the discrete data and its Fourier representation is unchanged by the deformation, resulting in clearer interpretation and simpler implementation of the scheme. Second, the FSM scheme is extended to solve the dynamic equation of the PFC model. Third, the real-space technique which arises from the coordinate-transformation step in the FSM scheme is discussed. The numerical calculations of the free energy are performed to validate the FSM and real space techniques, and the comparison of the accuracy between the two schemes is presented.