Amplitude Expansion Phase Field Crystal (APFC) Modeling based Efficient Dislocation Simulations using Fourier Pseudospectral Method
Xinyi Wei, Yangshuai Wang, Kai Jiang, Lei Zhang
TL;DR
The paper tackles efficient mesoscale simulation of crystalline defects, particularly dislocations, by leveraging the amplitude expansion of the phase-field crystal (APFC) model to capture lattice deformations without full atomistic resolution. A Fourier pseudospectral solver is developed to solve the APFC equations, exploiting periodicity and a semi-implicit time discretization; amplitudes $\eta_j$ are evolved via $\frac{\partial \eta_j}{\partial t} = -|\mathbf{k}_j|^2 \frac{\delta F}{\delta \eta_j^*}$ with an explicit energy functional $F$. The authors establish local analyticity results for the APFC system and demonstrate spectral convergence in 2D triangular and 3D BCC lattices by comparing reconstructed strain fields against continuum elasticity theory, observing correct far-field decay. With $O(D \log D)$ cost per step from FFTs, the method enables efficient large-scale defect dynamics and lays groundwork for extensions to grain boundaries and atomistic-to-continuum coupling.
Abstract
Crystalline defects critically influence material properties, necessitating accurate simulation methods. Existing approaches, from atomic-scale configurations to continuum elasticity, face inherent limitations in modeling dislocation-induced lattice deformation. The amplitude expansion of the phase field crystal (APFC) model bridges this gap with a mesoscopic description. This paper introduces a computationally efficient Fourier pseudospectral method for solving the APFC equations. The method exploits system periodicity and solution analyticity--the latter's rigorous proof remaining an open question, as discussed herein--to enable precise implementation of periodic boundary conditions. Numerical experiments on 2D triangular and 3D body-centered cubic lattices demonstrate that the method accurately reproduces the strain fields of edge dislocations, matching continuum theory predictions. These results confirm the APFC model's potential for capturing complex defect structures at the mesoscale, paving the way for simulating more intricate defect dynamics.