Energy Scaling and Asymptotic Properties of One-Dimensional Discrete System with Generalized Lennard-Jones (m, n) Interaction

It is well known that elastic effects can cause surface instability. In this paper, we analyze a one-dimensional discrete system which can reveal pattern formation mechanism resembling the “step-bunching” phenomenon for epitaxial growth on vicinal surfaces. The surface steps are subject to long-range pairwise interactions taking the form of a general Lennard-Jones (LJ)-type potential. It is characterized by two exponents m and n describing the singular and decaying behaviors of the interacting potential at small and large distances, and henceforth are called generalized LJ (m, n) potential. We provide a systematic analysis of the asymptotic properties of the step configurations and the value of the minimum energy, in particular their dependence on m and n and an additional parameter α indicating the interaction range. Our results show that there is a phase transition between the bunching and non-bunching regimes. Moreover, some of our statements are applicable for any critical points of the energy, not necessarily minimizers. This work extends the technique and results of Luo et al. (SIAM Multiscale Model Simul 14(2):737–771, 2016) which concentrates on the case of LJ (0,2) potential (originated from the elastic force monopole and dipole interactions between the steps). As a by-product, our result also leads to the well-known fact that the classical LJ (6,12) potential does not demonstrate the step-bunching-type phenomenon.