Self-similarity and coarsening of three dimensional particles on a one or two dimensional matrix

TL;DR

This study investigates self-similarity during coarsening of three-dimensional particles in systems with one or two-dimensional diffusion matrices, revealing power law growth and weak self-similarity breakdown.

Contribution

It predicts different scaling behaviors for 3D particles in 1D and 2D matrices, supported by numerical calculations confirming power laws and logarithmic factors.

Findings

Power law growth with exponent 1/7 in 3D/1D systems

Weak breakdown of self-similarity in 3D/2D systems

Numerical evidence of logarithmic factors in boundary motion laws

Abstract

We examine the validity of the hypothesis of self-similarity in systems coarsening under the driving force of interface energy reduction in which three dimensional particles are intersected by a one or two dimensional diffusion matrix. In both cases, solute fluxes onto the surface of the particles, assumed spherical, depend on both particle radius and inter-particle distance. We argue that overall mass conservation requires independent scalings for particle sizes and inter-particle distances under magnification of the structure, and predict power law growth for the average particle size in the case of a one dimensional matrix (3D/1D), and a weak breakdown of self-similarity in the two dimensional case (3D/2D). Numerical calculations confirm our predictions regarding self-similarity and power law growth of average particle size with an exponent 1/7 for the 3D/1D case, and provide evidence for the existence of logarithmic factors in the laws of boundary motion for the 3D/2D case. The latter indicate a weak breakdown of self-similarity.