Percolation of Deformable Particles through Porous Networks

The transport of micron-size particles such as bacteria, cells, or synthetic lipid vesicles through porous spaces is a process relevant to membrane-based processes including purification and preparation bio-pharma products. Often, the motion of these particles depends on their ability to squeeze through small constrictions, making their capacity to deform an important factor for their permeation. However, it is still unclear how the mechanical behavior of these particles affects collective transport through porous networks. To address this issue, we present a method to reconcile the pore-scale mechanics of the particles with the Darcy scale to understand the motion of a deformable particle through a porous network. We first show that particle transport is governed by a mechanical instability occurring at the pore scale which leads to a binary permeation response on each pore. Then, using the principles of directed bond percolation, we are able to link this microscopic behavior to the probability of permeating through a random porous network. We show that this instability, together with network uniformity, are key to understanding the nonlinear permeation of particles at a given pressure gradient. The results are then summarized by a phase diagram that predicts three distinct permeation regimes based on particle properties and the randomness of the pore network. Finally, we consider how this analysis is affected by making the particles visco-elastic versus simply viscous.