Diffusive tortuosity factor of solid and soft cake layers: A random walk simulation approach

Diffusive tortuosity factor of solid (colloidal) and soft (biofilm) cake layers formed during membrane filtration is investigated using the random walk simulation of solute tracers. Four different structures are investigated as possible candidate structures of the cake layers: simple cubic (SC), body-centered cubic (BCC), face-centered cubic (FCC), and random colloidal cake (RCC) structures. Low porosity of the deformable, compressible soft cake is mimicked by allowing overlaps among the adjacent particles and taking into account only the void spaces. When the volume fraction of each structure is less than its own maximum packing ratio of mono-dispersed solid spheres, Maxwell's theoretical prediction of the diffusive tortuosity factor is accurately superimposed on the simulation results, showing structural indifference of the tortuosity factor. However, when the soft cake is compressed so that the volume fraction becomes greater than the maximum packing ratio, the diffusive tortuosity factor surpasses Maxwell's theory and tends to diverge as the porosity reaches zero. The deviation of simulation results from the theory starts near the maximum packing ratio of each structure, and SC and FCC structures show higher diffusive tortuosity factors in comparison to the BCC structure. Most important, the RCC layer with a realistic irregular configuration has the highest diffusive tortuosity factor over almost the entire range of volume fraction. This implies that the solute diffusion within the soft cake layer of a random irregular structure is most hindered so that the concentration polarization and osmotic pressure of the solutes are accordingly enhanced on the membrane surface.