An Analysis of Surface Relaxation in the Surface Cauchy--Born Model

TL;DR

This paper analyzes the error behavior of the Surface Cauchy-Born method in modeling surface relaxation of crystalline materials, revealing conditions for exponential accuracy improvements and proposing model enhancements.

Contribution

The paper provides an error analysis of the SCB method focusing on surface relaxation, identifying an alternative parameter for exponential error decay, and suggests an improved model with atomistic mesh enforcement.

Findings

Error in SCB is O(1) with mesh size in linearized 1D model.

Exponential error decay is achievable with respect to interaction potential stiffness.

Enforcing atomistic mesh spacing improves SCB accuracy at free boundaries.

Abstract

The Surface Cauchy-Born (SCB) method is a computational multi-scale method for the simulation of surface-dominated crystalline materials. We present an error analysis of the SCB method, focused on the role of surface relaxation. In a linearized 1D model we show that the error committed by the SCB method is O(1) in the mesh size; however, we are able to identify an alternative "approximation parameter" - the stiffness of the interaction potential - with respect to which the error in the mean strain is exponentially small. Our analysis naturally suggests an improvement of the SCB model by enforcing atomistic mesh spacing in the normal direction at the free boundary.