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

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.