Interfacial phenomena play a considerable role in many physical, chemical and biological systems. Some of these systems exhibit interaction of several interfaces. This thesis contains a study of two types of systems characterized by interacting interfaces. First, a theory of the formation of nanoscale porous structures in oxides of metals grown by anodization is developed. It is shown that a growing oxide layer can become unstable which yields the formation of a spatially irregular array of pores. The conditions for the instability of the oxide layer are found. A weakly nonlinear analysis is performed and it is shown that the system evolution near the instability threshold is described by the Kuramoto-Sivashinsky equation. Farther from threshold, in the long-wave approximation, a system of strongly nonlinear evolution equations is derived that describes the formation of deep irregular pores. The other part of this thesis includes the study of a biological double-membrane consisting of two coupled lipid bilayers, typical of some intracellular organelles and bacteria. We first consider a multi-component double-membrane in which the curvatures of the two membranes and the distance between them are coupled to the lipid chemical compositions. Secondly, we examine the effect of non-equilibrium chemical fluxes across a double-membrane. We consider the dependence of the fluxes on the concentration of the transported chemical and on the membrane curvature, as well as the coupling of the intermembrane distance with the intermembrane concentration. For both systems, a phenomenological free-energy functional is formulated and the derived nonlinear evolution equations are studied. In the first case, we focus on phase separation in the double-membrane system. In the second case, we concentrate on the effect of the intermembrane chemical transport on the membrane dynamics. Linear stability analysis is performed and the domains of parameters are found in which the double-membrane is stable. For parameter values corresponding to an unstable membrane, numerical simulations reveal various types of complex dynamics, including the formation of stationary, spatially-periodic patterns in the first system, and in the second system, oscillations in the chemical concentration and membrane shape.

Last modified

• 09/19/2018

Creator

• Christine Sample

DOI

• https://doi.org/10.21985/N2X154

Subject

• Applied Mathematics

Keyword

• Mathematics

Date created

• 2008-09-03

Resource type

• Dissertation

Rights statement

• In Copyright