Generalised diffusion equations for anomalous diffusion in polymer networks : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Physics, School of Natural Sciences, Massey University, Manawatu, New Zealand

Abstract

Fractional and generalised derivative equations have proven to be a powerful tool in the modelling of anomalous diffusion within complex fluids. The suitability of these equations arises from to their capacity to include memory effects prevalent in the fluid, as well as allowing for the inclusion of external forces and relevant boundary values. Fractional derivative equations may be derived from underlying continuous-time random-walk models, however, the fractional derivative equations (and their generalisations) are simpler to deal with. This thesis investigated the ability of fractional derivative equations (and their generalisations) to model long timescale anomalous diffusion phenomena observed in some visco-elastic polymer networks. Recent work has suggested that within certain physical polymer networks there is a tendency for internal stresses to continuously build and dissipate. This phenomenon manifests itself within recordings of probe-particle mean squared displacements at long time scales. The dynamic behaviour of these networks parallels behaviour observed in earthquakes, earning the phenomena the name gel or cyto quakes. This link suggests that statistical features involved in other stress driven events may provide insight into the modeling of gel quakes. Both temporal and spatial considerations relevant to these quaking systems will be outlined and modelled within this work.