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 talk, we analyze a one-dimensional discrete system which can reveal pattern formation mechanism resembling the "step-bunching" phenomena in epitaxial growth on vicinal surfaces. The surface steps are subject to pairwise long range interactions which takes 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 potential for $x\\ll 1$ and $x\\gg 1$. (We henceforth call these potentials generalized LJ $(m,n)$ potentials.) We provide a systematic study of the asymptotic properties of the step configurations and the value of the minimum energy for different regimes of $m$ and $n$.

Both linear stability and nonlinear minimization problems are considered. Our results show that there is a phase transition between the bunching and non-bunching regimes. Some of our statements also hold for critical points of the energy, not just minimizers. This work extends the technique and results of Luo-Xiang-Yip 2016 which concentrates on the case of LJ (0,2) potential (originated from the elastic force monopole and dipole interactions between the steps). We emphasize on discrete models but continuum description will also be discussed. This is joint work with Tao Luo and Yang Xiang.