Kinetic Analytic Model for Time-Dependent Growth of Particles

Abstract

A kinetic model is used to describe the growth via nucleation process. The set of finite difference equations describing the growth are solved exactly for an equilibrium and a steady state. The equilibrium state is shown to be a special case of the more general steady state. For the time-dependent case the set of finite difference equations is reduced to a single partial differential equation (the Fokker-Planck equation). The partial differential equation is again solved analytically for the equilibrium, steady state, and time-dependent problem. In the latter case we have shown that the equation reduces to a diffusion equation and the cluster size distribution is described by a Gaussian distribution, with a time-dependent mean cluster size and an effective diffusion coefficient. The theoretical model is compared with the experimental data on the polymer-dispersed liquid crystals prepared by the polymerization-induced phase separation. The concept of critical radius is naturally introduced into the kinetic model and a very simple relation is obtained for the critical radius in terms of the measured quantities and a scaling parameter. A reasonably good agreement between experiment and theory is achieved, For microgravity experiments the diffusion coefficient is smaller than for 1-g experiments. The effect of gravity on critical radius is also indicated.