Desorption and Spatial Nonlocalities in Polymer-Penetrant Systems

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DMS-9972013This award will support research on two main problems. The first is a study of mathematical models for polymer desorption. In particular, the goal is to model such anomalous behavior as skinning, where the penetrant substancecan become trapped inside the polymer matrix. The second part of the proposal is a study of the spatial nature of the nonlocal effects in polymer diffusion, which is caused by the disparate length scales in the polymer network. The common problem in describing diffusion in polymers is thatthe diffusion of the penetrant molecules in a polymer entanglement network cannot be described by the usual models (Fick's Law) alone.Rather, one must incorporate other nonlocal effects, using hereditary integral kernels.This results in a model that contains a nonlinear partial integrodifferential equation. This problem is perhaps best described as a moving boundary-value problem wheretwo distinct operators govern the phenomena on the two sides of the boundary. In this work, existing models will be refined, and studies of diffusion and dissolution will performed and compared with experiments. Different geometries and the effects of swelling on the system will also be considered. Because of their unusual, yet useful, properties, polymers have become invaluable in industrial production. Diffusion in polymers does not follow the standard physical laws as diffusion in other media, and hence a new mathematical approach is needed. In particular, the penetrant molecules diffuse through interconnected polymer "pockets", which leads to nonlocal effects. A mathematical model incorporating these effects will be formulated and solved. One area where polymers are used is in pharmaceutical applications, where the polymer is often saturated with a penetrant (such as a medicine), and then allowed to release the penetrant slowly. Several mathematical studies in various configurations, such as cylindrical devices, will be undertaken to analyze these phenomena. Thus the research will have direct impact on nonmathematical areas, such as materials science and pharmaceuticals.