Coarsening of three-dimensional droplets by two-dimensional diffusion: Part II. Theory

Abstract

Theoretical modeling of coarsening among a finite cluster of precipitates is implemented, using the multipole expansion method. This method requires the diffusion field to behave quasi-statically. Two approximate solutions were developed, one to monopolar order, and other to the dipolar order. The conventional Gibbs-Thomson equilibrium relationship was used as the boundary condition at the precipitate-matrix interface. Part I of this paper considers a liquid-liquid system in a mixed-dimensional geometrical configuration, wherein three-dimensional precipitates interact via a diffusion field constrained in two dimensions. This kind of geometric configuration is often encountered in island evolution dynamics and phase segregation in thin films. The initial experimental configuration of droplets provides the initial condition for the simulation. Both monopole and dipole approximations closely follow the experimentally observed scaling laws, characteristic for the mixed-dimensional coarsening (N−4/3 and\(\bar R\) 4, varied linearly with time, where N is the number of droplets in the experimental field of view, and\(\bar R\) is the average droplet radius). Good agreement is observed for time evolution of radii of some individual precipitates. Certain deviations appearing among the two approximate solutions and the experimental data are discussed.