Desorption and Spatial Nonlocalities in Polymer-Penetrant Systems

DMS-9972013该奖项将支持两个主要问题的研究。第一个是聚合物解吸的数学模型的研究。 特别地，目标是模拟诸如结皮的异常行为，其中渗透物质可以被困在聚合物基质内。 第二部分的建议是一个研究的空间性质的聚合物扩散的非局部效应，这是由不同的长度尺度在聚合物网络。 在描述聚合物中扩散时，一个共同的问题是，在聚合物缠结网络中渗透剂分子的扩散不能用通常的模型（Fick定律）单独描述，而是必须考虑其他非局部效应，使用遗传积分核，这导致一个包含非线性偏积分微分方程的模型。 这个问题也许是最好的描述为一个移动的边界值问题，其中两个不同的运营商管理的现象，双方的边界。 在这项工作中，现有的模型将被完善，扩散和溶解的研究将进行，并与实验进行比较。 还将考虑不同的几何形状和溶胀对系统的影响。 由于它们不寻常但有用的特性，聚合物在工业生产中变得非常宝贵。 聚合物中的扩散不像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，这导致非局部效应。 一个数学模型，将这些影响制定和解决。 使用聚合物的一个领域是在药物应用中，其中聚合物通常用渗透剂（例如药物）饱和，然后允许缓慢释放渗透剂。 在各种配置，如圆柱形设备的数学研究，将进行分析这些现象。 因此，这项研究将对非数学领域产生直接影响，如材料科学和制药。

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.