Crystal Interfaces. Part I. Semi-Infinite Crystals

Abstract

This paper presents a calculation of the stresses and energies for interfaces between different crystals. Only simple cases corresponding to interfacial dislocations of pure screw or edge type are treated. The difference between the two crystals is expressed in terms of different elastic constants and variable interfacial bondings. Two kinds of interfacial forces appear: tangential forces with a periodic character and normal forces. The latter are induced by different normal displacements of the two contact surfaces of the crystals due to the equal and opposite tangential interfacial forces and are accounted for by assuming a linear relation between force and relative displacement. By using a periodic parabolic model to represent the periodic potential associated with the tangential forces, it is shown that the contribution of the normal force to the interfacial energies is negligible for the approximation used. When these normal forces are neglected, the Peierls-Nabarro representation of the interfacial forces yields simple results in terms of an interfacial rigidity modulus μ and an effective elastic constant λ+ defined by 1/λ+=1/λa+1/λb=(1−σa)μa+(1−σb)/μb, where σ is Poisson's ratio and the two crystals are designated by a and b. It is seen that ½λa≤λ+≤λa, where λa is the smaller of λa and λb. Apart from being proportional to μc, the interfacial energy E depends only on a parameter β=2π(c/p)(λ+/μ), where p is the dislocation spacing and the lattice constant c of the reference lattice is given by 2/c=1/a+1/b. The dependence on β is such that below a poorly defined value of the order of unity the variation of E is very rapid while beyond this it remains practically constant. It is further shown that the elastic energies stored in the two crystals are in the ratio λa/λb and that less than 2% of these energies is stored at distances from the interface greater than half the distance between dislocations. Similar results are shown to hold for simple twist and tilt boundaries.