Diffusion-controlled growth and transport in random composites
Roberts, Anthony P
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...[Show more] 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
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.
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