Connection Between Intrapore Free Energy, Molecule Permeation, and Selectivity of Nanofiltration Membranes

Purification and separation by nanofiltration is a fascinating interdisciplinary topic that is being actively studied both in polymer science and nanoscience/nanotechnology. However, a comprehensive understanding of the separation mechanism based on a sound physical foundation for the nanofiltration process of aqueous solutions remains very challenging and not well established. To confront this issue, in this study we developed a physical model that is motivated by classic polymer physics for the permeation of polymer chains in confining channels and based on the physics of Brownian motion in the pressure field, thus generalizing the extensively employed Arrhenius-type equation. Our study showed that the intrapore free energy plays a critical role to separate different species in the process of nanofiltration. One main result of our study is that flux behaviors of molecules are controlled by a characteristic pressure defined as critical pressure. Furthermore, we found that this critical pressure directly connects with a free energy barrier, which has been investigated in depth by the potential of mean force for molecules confined in nanopores in extensive previous studies. Another main result of our research is that we clarified in dilute solutions whether or not the relation between solute retention and pressure drop is a linearly decaying function in the process of nanofiltration. Our study showed that the observed decaying relation depends on the regime of pressure drop: if the pressure drop is on the order of the critical pressure, it is decaying in a nonlinear manner; otherwise, it approaches a linear dependence. Our research demonstrated that these two regimes can be wrapped up into an exponentially decaying relation between solute retention and the critical pressures of solute and water, which in turn are determined by their intrapore free energies. These results are universal for the permeation of both polymer chains and hydrated small molecules and mostly independent of molecular and nanopore details, and they can be applied to determine the confinement free energy barrier of molecule permeation particularly for single polymer permeation in experiments, which is still not fully understood. We expect that these findings will provide a priori insight into the process of designing separation membranes with tailored properties.