Transport of Deformable Particles in Porous Media

Filtration is critical to the purification in food and residual water in fossil fuels industries. In the pharmaceutical industry, new drug delivery techniques to transport and deliver drugs in a more efficient way use a filtration process in the fabrication and purification of liposomes—small, soft vesicles formed by a lipid bilayer that encapsulate a water solution where the drugs are stored. And in a similar vein, new designs of microfiltration devices are used to capture circulating tumor cells, and then be able to design better treatments. In all these cases, the transport of soft colloids, soft vesicles and other small deformable particles through the tortuous network is a key figure-of-merit. To further the ability to design membranes for fractionation of deformable particles, we have developed a (2-d) computational multiscale approach. We use a combination of the extended finite element method to describe the creeping fluid flow (Re ~ 0) inside a portion of a filtration membrane with an embedded interface, coupled with a particle method that interpolates the particle's membrane (interfacial) position and corresponding fields using least square fitting.

We have now modeled (in 2-d) how the combination of porous network geometry (porosity, pore size, and connectivity) and soft particle deformation-related properties (elasticity and surface tension) influence the particles' sieving coefficients. The focus in this paper is on the relationship between the elasticity of the particle and the porosity of the system, and its effects on the overall permeability of the system (and the sieving of the particles). First we present the base case of two immiscible fluids (such as an oil droplet in water), and show that the transport of such particles consistently scale with the concepts of a critical pressure gradient across the pore being determined by the ratio of the particle and pore diameter and the interfacial tension, as has bee analytically derived for the case of an axisymmetric spherical droplet entering a capillary. The base case of a single pore and droplet is then extended to include arrays of obstacles (creating a porous network) and multiple droplets. In this case, the applied trans-network prerssure drop is not the same presssure gradient each droplet experiences when it needs to deform to pass between obstacles. These cases provide a separate set of scaling rules that can guide membrane design for fractionation purposes.

Finally, and most importantly we examine the difference between the simple immiscible droplet and an elastic particle. In the former case, the "penalty" for deformation is in creating curvature, while in the latter case the deformation requires creation of surface area. Thus, we find that 

-        elastically deformable particles have different modes of deformations, and

-        the pressure landscapes on their permeation are drastically different than surface tension based particles.