Kinetics of surface growth with coupled diffusion and the emergence of a universal growth path

Abstract

Surface growth, by association or dissociation of material on the boundaries 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 complexity in describing the kinetics of such processes arises due to their inherent coupled interaction with the diffusing constituents that are needed to sustain the growth, and the influence of local stresses on the growth rates. Moreover, changes in concentration of solvent within the bulk of the body, generated by diffusion, can affect volumetric changes, thus leading to an additional interacting growth mechanism. In this paper we present a general theoretical framework that captures these complexities to describe the kinetics of surface growth while accounting for coupled diffusion. Then, by combination of analytical and numerical tools, applied to a simple growth geometry, we show that the evolution of such growth processes rapidly tends towards a universal path that is independent of initial conditions. This path, on which surface growth mechanisms and diffusion act harmoniously, can be extended to analytically portray the evolution of a body from inception up to a treadmilling state, in which addition and removal of material are balanced.