Solution to the multi-particle diffusion problem with applications to Ostwald ripening-I. Theory

The multi-particle diffusion problem (MDP), which is a general problem concerning n-domains interacting through their diffusion fields is treated by an embedding technique, wherein growing and shrinking domains are represented by point sources or sinks. The diffusion solution exterior to the phase domains is constructed using potential theoretic techniques by an adaptation of Ewald's method for calculating lattice sums. Here a periodic representation is applied to a random particle basis and used to circumvent the semiconvergent behavior of the monopole sums normally encountered in such embedding methods. The phenomenon of Ostwald ripening is then discussed in terms of the MDP, and a novel formulation is developed based on the concept of an interaction matrix, the elements of which are Ewald sums. The formal relationship of these interaction matrix elements to Madelung's constant is discussed, with emphasis on an analytical description of how local diffusional interactions influence the coarsening rate. A comparison is presented of MDP results with the statistical behavior predicted by Lifshitz, Slyozov, and Wagner.