Force and heat current formulas for many-body potentials in molecular dynamics simulation with applications to thermal conductivity calculations

TL;DR

This paper derives unified force and heat current formulas for many-body potentials in molecular dynamics, enabling accurate thermal conductivity calculations for covalently bonded materials.

Contribution

It introduces a pairwise force expression for many-body potentials that obeys Newton's third law, facilitating unambiguous stress and heat current calculations.

Findings

Derived force and heat current formulas applicable to many-body potentials.

Validated formulas on silicon, diamond, graphene, and carbon nanotubes.

Analyzed effects of cell size and simulation time on results.

Abstract

We derive expressions of interatomic force and heat current for many-body potentials such as the Tersoff, the Brenner, and the Stillinger-Weber potential used extensively in molecular dynamics simulations of covalently bonded materials. Although these potentials have a many-body nature, a pairwise force expression that follows Newton's third law can be found without referring to any partition of the potential. Based on this force formula, a stress applicable for periodic systems can be unambiguously defined. The force formula can then be used to derive the heat current formulas using a natural potential partitioning. Our heat current formulation is found to be equivalent to most of the seemingly different heat current formulas used in the literature, but to deviate from the stress-based formula derived from two-body potential. We validate our formulation numerically on various systems descried by the Tersoff potential, namely three-dimensional silicon and diamond, two-dimensional graphene, and quasi-one-dimensional carbon nanotube. The effects of cell size and time used in the simulation are examined.