Epicycle method for elasticity limit calculations

Abstract

The task of finding the smallest energy needed to bring a solid to its onset of mechanical instability arises in many problems in materials science, from the determination of the elasticity limit to the consistent assignment of free energies to mechanically unstable phases. However, unless the space of possible deformations is low-dimensional and a priori known, this problem is numerically difficult, as it involves minimizing a function under a constraint on its Hessian, which is computationally prohibitive to obtain in low-symmetry systems, especially if electronic structure calculations are used. We propose a method that is inspired by the well-known dimer method for saddle point searches but that adds the necessary ingredients to solve for the lowest onset of mechanical instability. The method consists of two nested optimization problems. The inner one involves a dimer-like construction to find the direction of smallest curvature as well as the gradient of this curvature function. The outer optimization then minimizes energy using the result of the inner optimization problem to constrain the search to the hypersurface enclosing all points of zero minimum curvature. We use the method to investigate potential failure mechanisms of graphene and to determine the free energy of mechanically unstable Pt-W and Cu-W alloys.

Identifying rhenium substitute candidate multiprincipal-element alloys from electronic structure and thermodynamic criteria

“…Of these, 32 involved mechanically and thermodynamically unstable structures. To assign them a plausible free energy (as required by the Calphad framework), we used the recently developed inflection detection approach, combined with the epicycle method [41,42], using the robustrelax_vasp command of Alloy Theoretic Automated Toolkit (ATAT) [43,44] with its defaults settings. In addition, to ensure that free energy differences between different crystal structures are accurately reproduced, 36 lattice dynamics calculations were performed using the supercell method [45] with the fitfc code [43] within the harmonic approximation.…”