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 \(\alpha \) 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.

众所周知，弹性效应会引起表面不稳定性。在本文中，我们分析了一维离散系统，该系统可以揭示类似于在邻近表面上外延生长的“阶梯成束”现象的图案形成机理。表面台阶经受长距离成对相互作用，呈一般的Lennard-Jones（LJ）型势能形式。它的特征是两个指数m和n描述了在小距离和大距离处相互作用势的奇异和衰减行为，因此被称为广义LJ（m，  n） 潜在的。我们对阶跃配置的渐近性质和最小能量的值（尤其是它们对m和n的依赖以及附加参数\（\ alpha \））进行系统分析指示相互作用范围。我们的结果表明，在聚束和非聚束状态之间存在相变。此外，我们的某些陈述适用于能源的任何临界点，不一定适用于最小化。这项工作扩展了Luo等人的技术和结果。（SIAM多尺度模型Simul 14（2）：737–771，2016年）着重于LJ（0,2）电势的情况（起源于各步之间的弹性单极和偶极相互作用）。作为副产品，我们的结果还导致一个众所周知的事实，即经典LJ（6,12）电位未显示出阶跃聚束型现象。