Lattice imperfections studied by use of lattice Green’s functions

Abstract

This paper explores the use of lattice Green’s functions for calculating the static structure of defects in lattices, in that the atoms of the lattice interact with their neighbors with an arbitrary nonlinear (short-range) potential. The method is hierarchical, in which Green’s functions are calculated for the perfect lattice, for increasingly complicated defect lattices, and finally the nonlinear structure problem is iterated until a converged solution is found. For the case where the defect must be embedded within a very large linear system, and the slip plane, cleavage plane, nonlinear zone, etc., can be made small compared to the system size, Green’s functions are a very powerful method for studying the physics of defects and their interactions. As an illustration of the method, we report numerical calculations for an interfacial crack emitting dislocations from an interface between two joined two-dimensional hexagonal lattices. The supercell size was 4×${10}^6$, and the crack length was 101 lattice spacings. After the Green’s functions were obtained for the defective lattice, the dislocation and crack structures were obtained in a minute or less, making possible detailed studies of the defects with various external loads, force laws, defect relative positions, etc., with negligible computer time. With practical supercomputer times, supercell and defect sizes one or two orders larger are feasible, thus making possible a realistic calculation of three-dimensional nucleation events on cracks, etc.