Desorption and Spatial Nonlocalities in Polymer-Penetrant Systems

DMS-9972013该奖项将支持对两个主要问题的研究。第一部分是聚合物解吸的数学模型研究。特别是，目标是模拟这种异常行为，如结皮，即渗透性物质可能被困在聚合物基质中。该建议的第二部分是对聚合物扩散中非局部效应的空间性质的研究，该效应是由聚合物网络中不同的长度尺度引起的。在描述聚合物中的扩散时，常见的问题是渗透剂分子在聚合物纠缠网络中的扩散不能仅用通常的模型(Fick定律)来描述，而必须考虑其他非局域效应，使用遗传积分核，这导致了一个包含非线性偏积分微分方程的模型。这个问题也许最好被描述为一个移动边值问题，其中两个不同的算子控制着边界两侧的现象。在这项工作中，将改进现有的模型，并进行扩散和溶解的研究，并与实验进行比较。不同的几何形状和膨胀对系统的影响也将被考虑。由于其不同寻常但有用的特性，聚合物在工业生产中已变得无价。聚合物中的扩散不像在其他介质中的扩散那样遵循标准的物理定律，因此需要一种新的数学方法。特别是，渗透剂分子通过相互连接的聚合物“口袋”扩散，从而导致非局部效应。一个包含这些影响的数学模型将被表述和求解。聚合物的一个应用领域是在制药应用中，聚合物通常被渗透剂(如药物)饱和，然后被允许缓慢释放渗透剂。将进行几项不同配置的数学研究，如圆柱形装置，以分析这些现象。因此，这项研究将对材料科学和制药等非数学领域产生直接影响。

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.