Unidad Multidiscliplinaria de Docencia e Investigación-Juriquilla, Facultad de Ciencias, Universidad Nacional Autónoma de México, CP 76230, Juriquilla, Querétaro, Mexico.
Phys Rev E. 2017 May;95(5-1):052804. doi: 10.1103/PhysRevE.95.052804. Epub 2017 May 22.
In this work, we provide a theoretical relationship between the spatial-dependent diffusion coefficient derived in the Fick-Jacobs (FJ) approximation and the macroscopic diffusion coefficient of a membrane that depends on the porosity, tortuosity, and the constriction factors. Based on simple mass conservation arguments under equilibrium as well as in nonequilibrium conditions, we generalize previous expressions for the effective diffusion coefficient of an irregular pore, originally obtained by Festa and d'Agliano for horizontal and periodic pores, and then extended by Bradley for tortuous periodic pores, to the case of pores with arbitrary geometry. Through a formal definition of the constrictivity factor in terms of the geometry of the pore, our results provide very clear physical interpretation of experimental measurements since they link the local properties of the flow with macroscopic quantities of experimental relevance in the design and optimization of porous materials. The macroscopic diffusion coefficient as well as the spatiotemporal evolution of the concentration profiles inside a pore have been recently measured by using pulse field gradient NMR techniques. The advantage of using the FJ approach is that the spatiotemporal concentration profile inside a pore of irregular geometry is directly related to the pore's shape and, therefore, that the macroscopic diffusion coefficient can be obtained by comparing the spatiotemporal concentration profiles from such experiments with those of the theoretical model. Hence, the present study is relevant for the understanding of the transport properties of porous materials where the shape and arrangement of pores can be controlled at will.
在这项工作中，我们提供了一个在菲克-雅可比（FJ）近似下导出的空间相关扩散系数与依赖于孔隙率、曲折度和收缩因子的膜的宏观扩散系数之间的理论关系。基于平衡和非平衡条件下的简单质量守恒论证，我们将最初由 Festa 和 d'Agliano 为水平和周期性孔隙获得的不规则孔隙有效扩散系数的表达式推广到具有任意几何形状的孔隙的情况，然后由 Bradley 对曲折周期性孔隙进行了推广。通过用孔隙几何形状定义收缩因子的正式定义，我们的结果提供了对实验测量的非常清晰的物理解释，因为它们将局部流动特性与多孔材料设计和优化中具有实验相关性的宏观量联系起来。最近，使用脉冲场梯度 NMR 技术测量了孔隙内的宏观扩散系数和浓度分布的时空演化。使用 FJ 方法的优点是，不规则几何形状的孔隙内的时空浓度分布直接与孔隙的形状有关，因此可以通过将这种实验的时空浓度分布与理论模型的时空浓度分布进行比较来获得宏观扩散系数。因此，本研究对于理解多孔材料的输运性质具有重要意义，因为可以根据需要控制孔隙的形状和排列。
