Pattern formation in precipitation processes. I. The theory of competitive coarsening

Abstract

The theory of competitive coarsening is formulated and analyzed for applications to the formation of macroscopic precipitation patterns. Previous work is extended to account for ‘‘local coarsening’’ which results if processes on smaller length scales are properly eliminated. Formally the theory includes the width of the local particle size distribution as a dynamical variable and avoids the assumption of monodispersivity. The linear analysis shows that long-wavelength instabilities of the spatially homogeneous system are suppressed in comparison with the results of previous work. Numerical simulations of the nonlinear model are applied to locally perturbed homogeneous systems and produce one precipitation band surrounded by a growing region essentially void of precipitate. Recurrent banding and satellite induction which result under the assumption of monodispersivity are not explained by this theory. The random patterns found experimentally in supercooled PbI2 solutions containing a gel are interpreted as dissolution patterns driven by exceptionally large particles which act as local perturbations. Using an analogy to percolation problems this point of view can rationalize part of the varying appearance of the patterns as well as the sensitivity to the initial salt concentration. The theory also qualitatively describes the ‘‘focusing effect’’ found in recent experiments on Liesegang band formation.