Diffusion is a transport process of great interest as it pertains to many problems in biology, medicine and chemistry. In particular, studies of diffusion taking place in porous systems could help to model processes such as toxins spreading in soils or groundwater, transport of materials through permeable membranes or polyelectrolyte migration in gel electrophoresis. A more thorough understanding of the details of diffusion at the microscopic level could be useful in designing optimized gels for electrophoresis as well as more precise sieving methods. In numerical studies, a common approach is to model diffusion by using a Monte Carlo (MC) algorithm of a random walk on a lattice. In this work, we use methodologies to solve these algorithms exactly and thus efficiently obtain precise results.
This thesis consists of two studies, in the form of publishable manuscripts, of the biased and unbiased random walk of small particles on a lattice. First, we model the diffusion of point-like particles in networks of identical periodic square and cubic cavities interconnected by holes. The geometry of these cavities is parameterized in order to study these systems for a variety of different cavity structures. The diffusion inside these systems is modeled by a random walk on a lattice with an algorithm previously developed by the Slater group. In this approach, the MC algorithm can be solved exactly, thereby reducing the time required to obtain precise results which would otherwise require lengthy simulations. We test theoretical predictions and validate analytical results for various asymptotic cases. Additionally, we develop interpolation functions to fit our exact numerical data.
In the second study, we develop a new matrix method of exact calculation by expanding and improving a previous model to overcome its limitations. Our new model is capable of treating random walks biased by periodic time-dependent fields. We chose to model the case of particles diffusing in a 2-D network of trap-shaped obstacles in the presence of an external force (of finite frequency and arbitrary strength). In these studies, we examine the effect that varying the frequency of the applied force has on the mean velocities of particles of different sizes. Using this approach, we are able to qualitatively reproduce known experimental results. As in the first project, this new algorithm can be solved exactly and thus provides the same advantages, including a significant reduction in the time required to obtain precise results.
| Identifer | oai:union.ndltd.org:uottawa.ca/oai:ruor.uottawa.ca:10393/28028 |
| Date | January 2008 |
| Creators | Torres, Francis |
| Publisher | University of Ottawa (Canada) |
| Source Sets | Université d’Ottawa |
| Language | French |
| Detected Language | English |
| Type | Thesis |
| Format | 91 p. |
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