The Peierls—Nabarro Model of Dislocations: A Venerable Theory and its Current Development

Abstract

Dislocations are central to the understanding of mechanical properties of crystalline solids. While continuum elasticity theory describes well the long-range elastic strain of a dislocation for length scales beyond a few lattice spacings, it breaks down near the singularity in the region surrounding the dis-location center, known as the dislocation core. There has been a great deal of interest in describing accurately the dislocation core structure on an atomic scale because of its important role in many phenomena of crystal plasticity [1–3]. The core properties control, for instance, the mobility of dislocations, which accounts for the intrinsic ductility or brittleness of solids. The core is also responsible for the interaction of dislocations at close distances, which is rel-evant to plastic deformation. Two types of theoretical approaches have been employed to study dislo-cation core properties. The first is based on direct atomistic simulations usi-ng either empirical interatomic potentials or ab initio calculations. Empirical potentials involve the fitting of parameters to a predetermined database and hence may not be reliable in predicting the core properties, where severe dis-tortions like bond breaking, bond formation and switching necessitate a quan-tum mechanical description of the electronic degrees of freedom. On the other hand, ab initio total energy calculations, though considerably more accurate, are computationally expensive for the studies of dislocation properties. The second approach is based on the framework of the Peierls–Nabarro (P–N) model which holds the promise of becoming a plausible alternative to direct atomistic sim-ulations. For this reason, there has been a resurgence of interest in the simple and tractable P–N model for studying the dislocation core structure and mobil-ity. In particular, the P–N model permits easy estimation of the key dislocation 793 S. Yip (ed.), Handbook of Materials Modeling, 793–811. c 2005 Springer. Printed in the Netherlands. 794 G. Lu characteristics of nucleation and mobility, directly from quantities (GSF ene-rgy, see later) accessible through standard quantum mechanical or empirical atomistic computations.