Effect of contact angle on the pressure needed for a liquid to permeate a cylindrical pore

Abstract

Effective separation of two immiscible liquids with filters requires a difference in the pressures to permeate between the two liquids, and setting the applied pressure between these two pressures. To help design such filters, we present an equation that enables the calculation of the pressure for a liquid phase to permeate through a smooth pore in the shape of a truncated cone as a function of (a) the contact angle of the liquid on the filters and (b) the angle of the pore wall. The equation was derived by considering the interfacial energy required to push a liquid meniscus from the top of the pore to the bottom, and then to exceed the maximum curvature required at the exit. This equation was tested experimentally by adding a hydrostatic head with water on the 3D-printed filters of acrylate polymer while systematically varying the pore radii and contact angle with water. Experimental results showed an increased pressure to permeate with higher contact angles while the equation predicted the opposite. We hypothesized that the reason for the disagreement was the assumption of a smooth pore. For a liquid on a rough pore wall, the curvature of the meniscus is not solely determined from the microscopic contact angle and the pore wall angle, but the liquid would adopt a lower curvature meniscus. Therefore, the developed equation was modified after reflecting the lower curvature, which showed much better agreement with the experimental results. The remaining discrepancy from the theory was attributed to the pressure fluctuation from the fluid flow occurring while adding water.