Computational and analytical studies of a new nonlocal phase-field crystal model in two dimensions

TL;DR

This paper introduces a nonlocal phase-field crystal (NPFC) model that accurately captures material structure factors, improves upon local models, and demonstrates robust numerical methods for simulating crystal structures in two dimensions.

Contribution

The paper develops a data-driven nonlocal PFC model that matches experimental structure factors more accurately than existing models and establishes its numerical robustness.

Findings

NPFC matches experimental structure factors up to the second peak.

Fourier spectral methods are convergent and asymptotically compatible.

Numerical experiments validate NPFC's effectiveness in simulating crystal structures.

Abstract

A nonlocal phase-field crystal (NPFC) model is presented as a nonlocal counterpart of the local phase-field crystal (LPFC) model and a special case of the structural PFC (XPFC) derived from classical field theory for crystal growth and phase transition. The NPFC incorporates a finite range of spatial nonlocal interactions that can account for both repulsive and attractive effects. The specific form is data-driven and determined by a fitting to the materials structure factor, which can be much more accurate than the LPFC and previously proposed fractional variant. In particular, it is able to match the experimental data of the structure factor up to the second peak, an achievement not possible with other PFC variants studied in the literature. Both LPFC and fractional PFC (FPFC) are also shown to be distinct scaling limits of the NPFC, which reflects the generality. The advantage of NPFC in retaining material properties suggests that it may be more suitable for characterizing liquid-solid transition systems. Moreover, we study numerical discretizations using Fourier spectral methods, which are shown to be convergent and asymptotically compatible, making them robust numerical discretizations across different parameter ranges. Numerical experiments are given in the two-dimensional case to demonstrate the effectiveness of the NPFC in simulating crystal structures and grain boundaries.