Theory of expansion and compression of polymeric materials: Implications for membrane solvent flow under compaction

We extend the classical Flory-Rehner theory for the expansion and compression of porous materials such as cross -linked polymer networks and membranes. The theory includes volume exclusion, affinity with the solvent, and finite stretching of the polymer chains. We also modify this equilibrium theory, which applies to equal expansion of a material in all directions, to the situation where a material can only expand in a single direction, as is the case when a thin polymer layer is tightly bound to a support structure. We further extend this equilibrium model to the case where a pressure is applied across a thin polymer layer, such as a membrane, and liquid flows across the membrane. The theory describes how in the direction of liquid flow the membrane is increasingly compacted and becomes less porous, and this effect increases when the applied pressure goes up. We present results of example calculations for a thick membrane showing significant changes in compaction across its thickness, and a thin membrane where compaction due to flow is minor. Lastly, we model the dynamics of the change in size of a porous material in time after a step change in the solventpolymer attraction parameter, which may be relevant, for instance, when the attraction parameter is highly temperature -dependent, and we change the temperature. The developed theory has direct implications for pressure -driven membrane separation processes ranging from reverse osmosis to organic solvent nanofiltration.