From Gradient Elasticity to Gradient Interatomic Potentials: The Case-Study of Gradient London Potential

Kostas Parisis,Bo Wang,Avraam Konstantinidis,Pu Hu,Andreas Giannakoudakis,Fei Shuang

doi:10.4236/jamp.2020.89137

Abstract

Motivated by the special theory of gradient elasticity (GradEla), a proposal is advanced for extending it to construct gradient models for interatomic potentials, commonly used in atomistic simulations. Our focus is on London’s quantum mechanical potential which is an analytical expression valid until a certain characteristic distance where “attractive” molecular interactions change character and become “repulsive” and cannot be described by the classical form of London’s potential. It turns out that the suggested internal length gradient (ILG) generalization of London’s potential generates both an “attractive” and a “repulsive” branch, and by adjusting the corresponding gradient parameters, the behavior of the empirical Lennard-Jones potentials is theoretically captured.

Highlights

A simplified version of gradient elasticity theory (GradEla) introducing an extra gradient term in the classical law of linear elasticity [1], has been shown to dispense with various difficulties encountered in the past
By adjusting the new phenomenological parameter characterizing the effect of the gradient (Laplacian) term, the behavior of various types of empirical interatomic potentials, such as the Lennard-Jones which we focus on, can be recovered
A brief review of the robust gradient elasticity (GradEla) model was given first with emphasis on its ability to remove the undesirable dislocation singularities predicted by the classical Hookean elasticity

Summary

Introduction

A simplified version of gradient elasticity theory (GradEla) introducing an extra gradient term (the Laplacian of Hookean stress) in the classical law of linear elasticity [1], has been shown to dispense with various difficulties encountered in the past. The main feature which made GradEla especially robust was the observation that, under certain conditions [4], solutions of GradEla can be obtained in terms of existing solutions of classical elasticity by solving a non-homogeneous Helmholtz equation This observation enabled to obtain explicit and easy-to-use nonsingular solutions for dislocations and disclinations, as well as for cracks. In a related presentation in a soft matter symposium at the University of Florida/Gainesville [10], the question was raised [11] whether or not such a gradient enhancement for the gravitational potential can be extended to modify interatomic potentials used for multiscale simulations in solid state and soft matter calculations This subject is currently under investigation by the Florida—Thessaloniki groups and a preliminary encouraging result is reported .

Review of GradEla

London’s Quantum Mechanical Potential

Gradient Modification of London’s Potential

Discussion—Future Directions