Dynamics of Phase Separation of Crystal Surfaces

TL;DR

This paper studies the evolution of crystal surfaces into hill-and-valley patterns, revealing different coarsening dynamics in one and two dimensions influenced by anisotropy and phase boundary interactions.

Contribution

It establishes a connection between surface dynamics and the modified Cahn-Hilliard equation, and characterizes the coarsening behavior in different dimensions and mechanisms.

Findings

Logarithmic coarsening in quasi-1D patterns.

Power-law growth with exponents ~0.23 and ~0.13 in 2D.

Crystalline anisotropy significantly influences pattern evolution.

Abstract

We investigate the dynamical evolution of a thermodynamically unstable crystal surface into a hill-and-valley structure. We demonstrate that, for quasi one-dimensional ordering, the equation of motion maps exactly to the modified Cahn-Hilliard equation describing spinodal decomposition. Orderings in two dimensions follow the dynamics of continuum clock models. Our analysis emphasizes the importance of crystalline anisotropy and the interaction between phase boundaries in controlling the long time dynamics. We establish that the hill-and-valley pattern coarsens logarithmically in time for quasi one-dimensional orderings. For two-dimensional orderings, a power-law growth $L(t)\sim t^n$ of the typical pattern size is attained with exponent $n\approx 0.23$ and $n\approx 0.13$, for the two ordering mechanisms dominated by evaporation-condensation and by surface-diffusion respectively.