Abstract
Nanoporous membranes are heterogeneous structures, with heterogeneity manifesting at the microscale. In examining particle transport through such media, it has been observed that this transport deviates from classical diffusion, as described by Fick's second law. Moreover, the classical model is physically unsustainable, as it is non-causal and predicts an infinite speed of concentration perturbation propagation through a substantial medium. In this work, we have derived two causal models as extensions of Fick's second law, where causality is linked to the effects of inertial memory in the nanoporous membrane. The results of the derived models have been compared with each other and with those obtained from the classical model. It has been demonstrated that both causal models, one with exponentially fading inertial memory and the other with power-law fading memory, predict that the concentration perturbation propagates as a damped wave, leading to an increased time required for the cumulative amount of molecules passing through the membrane to reach a steady state compared to the classical model. The power-law fading memory model predicts a longer time required to achieve a stationary state. These findings have significant implications for understanding cell physiology, developing drug delivery systems, and designing nanoporous membranes for various applications.