Phase Separation of Crystal Surfaces: A Lattice Gas Approach

TL;DR

This paper models the phase separation of crystal surfaces into facets using an Ising lattice gas, revealing temperature-dependent behaviors and growth dynamics relevant to alkali halides.

Contribution

It introduces a lattice gas model to study thermal faceting and surface decomposition, linking equilibrium shapes to phase diagrams and analyzing kinetic behaviors.

Findings

Low temperature separation involves logarithmic slow growth.

Higher temperature leads to power-law separation into multiple facets.

Out-of-equilibrium growth exhibits late-time power-law behavior.

Abstract

We consider both equilibrium and kinetic aspects of the phase separation (``thermal faceting") of thermodynamically unstable crystal surfaces into a hill--valley structure. The model we study is an Ising lattice gas for a simple cubic crystal with nearest--neighbor attractive interactions and weak next--nearest--neighbor repulsive interactions. It is likely applicable to alkali halides with the sodium chloride structure. Emphasis is placed on the fact that the equilibrium crystal shape can be interpreted as a phase diagram and that the details of its structure tell us into which surface orientations an unstable surface will decompose. We find that, depending on the temperature and growth conditions, a number of interesting behaviors are expected. For a crystal in equilibrium with its vapor, these include a low temperature regime with logarithmically--slow separation into three symmetrically--equivalent facets, and a higher temperature regime where separation proceeds as a power law in time into an entire one--parameter family of surface orientations. For a crystal slightly out of equilibrium with its vapor (slow crystal growth or etching), power--law growth should be the rule at late enough times. However, in the low temperature regime, the rate of separation rapidly decreases as the chemical potential difference between crystal and vapor phases goes to zero.