Diffusion-controlled growth and transport in random composites

Abstract

In the first part of this thesis we consider three different models of diffusioncontrolled growth; phase-field models, diffusion-limited aggregation, and stochastic non-Laplacian growth models. Singular perturbation theory is employed to derive solutions to a phase-field model of solidification for a variety of far-field boundary conditions. Where possible, computational techniques are used to verify the accuracy of the analytical methods. We consider the ramifications of interpreting the computational parameter of the model as a physical quantity. Our preliminary calculations suggest a modification of the boundary condition at the solid-liquid interface. Order of magnitude estimates indicate that the correction is of the order of the standard Gibbs-Thompson under-cooling correction. The fractal properties of diffusion-limited aggregation (DLA) and dielectric breakdown (DBM) clusters are compared. It is found that significant differences occur between the models. These differences indicate that the the morphological properties of discrete diffusion-controlled growth models are sensitive to variations in local growth rules. This is contrary to a long standing conjecture in the theory of pattern formation in discrete models. To model a variety of discrete growth processes where Laplacian assumptions are not satisfied we develop a formal method for modeling non-Laplacian growth. The method is based on assigning rules to lattice-based walkers which are consistent with continuum equations. The rules are derived from a finite difference scheme. The method is used to develop a model for the study of diffusive growth in finiteconcentration fields. Agreement with analytic results shows the method is wellfounded and able to reproduce morphologies realized in many physical systems. We apply the method to electro-chemical deposition and study the interplay between the electrostatic and diffusion fields. We also examine the effect of a local (nonuniform) flow field on deposition in a hydrodynamic flow. In the second part of this thesis we investigate the effective conductivity and elastic moduli of two classes of random media. The first is defined by the level-cut(s) of a Gaussian random field while the second is comprised of overlapping hollow spheres. The three point solid-solid correlation function is derived for each of the models and utilized in the evaluation of rigorous variational bounds on the effective properties. Simulations are used to calculate the effective conductivity for a variety of different realizations of the models at several different conductivity contrasts. The simulation data lie between the bounds in all cases. It is found that the effective properties are relatively insensitive to microstructure within suitably defined morphological classes. Between these classes very large differences are observed. The features of the microstructure which most strongly effect composite properties are described. Finally we demonstrate the ability of the theoretical random media to successfully model a range of composites including porous rocks and foamed solids.