Morphological Stability of a Particle Growing by Diffusion or Heat Flow

TLDR

The concentration field during diffusion‑controlled growth is assumed to satisfy Laplace’s equation, a condition commonly met in practice. The study aims to use spherical growth results to analyze the stability of nonspherical growth forms. The authors analyze shape stability by expanding small deviations from sphericity into spherical harmonics and tracking the time evolution of the expansion coefficients. They derive a dispersion law for harmonic amplitudes and show that the sphere is stable below a critical radius Rc (seven times the nucleation radius) and unstable above, with similar results for solidification.

Abstract

The stability of the shape of a spherical particle undergoing diffusion-controlled growth into an initially uniformly supersaturated matrix is studied by supposing an expansion, into spherical harmonics, of an infinitesimal deviation of the particle from sphericity and then calculating the time dependence of the coefficients of the expansion. It is assumed that the pertinent concentration field obeys Laplace's equation, an assumption whose conditions of validity are discussed in detail and are often satisfied in practice. A dispersion law is found for the rate of change of the amplitude of the various harmonics. It is shown that the sphere is stable below and unstable above a certain radius Rc, which is just seven times the critical radius of nucleation theory; analogous conclusions are obtained for the solidification problem. The results for the sphere are used to discuss the stability of nonspherical growth forms.