Kinetics of surface growth and coupled diffusion in a continuum framework

Abstract

Surface growth by association or dissociation of material on the boundary of a body is ubiquitous in both natural and engineering systems. It is the fundamental mechanism by which biological materials grow, starting from the level of a single cell, and is increasingly applied in engineering processes for fabrication and self-assembly. A significant challenge in modeling such processes arises due to the inherent coupled interaction between the growth kinetics, the local stresses, and the diffusing constituents needed to sustain the growth. Moreover, the volume of the body changes not only due to surface growth but also by variation in solvent concentration within the bulk of the body. In this thesis we present a general theoretical framework that captures these phenomena and describes the kinetics of surface growth while accounting for coupled diffusion.We then use a combination of analytical and numerical tools to study growth in simple geometries.

In the particular problems of growth on flat and on spherical surfaces, we show that the growth process typically involves two stages. Initially, the body deforms, primarily due to diffusion with minimal growth. This is followed by a second stage during which surface growth and diffusion act harmoniously. It is shown that during this latter stage the body evolves along a 'universal path' that is independent of initial conditions. We make use of this path to analytically describe the evolution of a body from inception up to treadmilling, the latter being a steady state in which the addition and removal of material are balanced. Among the wide spectrum of possible applications for our general continuum framework, we chose to examine in this thesis a specific problem of high interest in microbiology. In this context, we explore the kinetics of degradation of biomatter in the natural environment.

This process plays a key role in the global carbon cycle; it hasn't so far been thoroughly elucidated due to the complexity that arises from the coupling of mechanisms that are at the interface of biology, chemistry, mechanics and physics. Micro-scale experiments performed in a highly controlled fashion reveal highly nonlinear degradation kinetics, that classical scaling arguments fail to explain. It is shown that our model captures the complexity of the experimental observation. It can reveal distinct signatures of different bacterial strains and can thus predict degradation processes that are beyond the reach of the laboratory setting.