Mathematical modeling and stochastic simulation of soft materials

Date
2014
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University of Delaware
Abstract
Soft materials are all around us; they may appear as consumer products, foods, or biological materials. The interest in studying the properties of soft materials both experimentally and theoretically has steadily increased due to their wide range of industrial applications. One example of a soft material is wormlike micellar solutions. Depending on the temperature and composition, these solvent-surfactant-salt mixtures may exhibit close to mono-exponential or, alternatively, power-law or stretched-exponential stress decay. Of particular interest to this thesis is the development of stochastic models that can capture the stress relaxation behavior of such materials in the small strain limit, which is non-exponential in time as opposed to exponential. Continuous time random walk (CTRW) or subordinated Langevin processes are utilized to model systems exhibiting non-exponential relaxation behavior or anomalous diffusion. Stochastic simulations using the CTRW approach or the subordination method are carried out in this thesis for one-dimensional systems in which the probability density distribution of particle positions is described by a fractional Fokker-Planck equation (FFPE). The equivalence of the CTRW simulation and the subordination simulation with that of the FFPE is analyzed through the simulation of an ensemble of particle trajectories. The simulated particle dynamics suggest that CTRW processes or subordinated Langevin dynamics can be included in soft material mesoscale dynamics to capture the anomalous transport. To model the non-exponential stress relaxation dynamics of soft gel systems (three-dimensional fluids), stochastic models are simulated using transient network theory as developed and combined with the CTRW and subordinated Langevin processes. This approach enables us to connect the microstructural dynamics of certain soft gel-like materials with macroscale experimental observations by examining the material properties under homogeneous shear flow conditions. The study shows that transient network models combined with CTRW or subordination processes can successfully predict the non-exponential stress relaxation dynamics of soft materials. Future work should include the understanding of this class of models for other types of flows, e.g. inhomogeneous flows, as well as inclusion of the CTRW approach in a transient network model with the network topology tracked.
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