The saturation equation - an analytical expression for partial saturation during wicking flow in paper microfluidic channels

The design and fabrication of paper based microfluidic devices is critically dependent on modelling fluid flow through porous paper membranes. A commonly observed phenomenon is partial saturation, i.e., regions of the paper membrane not being filled completely due to pores of different sizes. The most comprehensive model till date of partial saturation during wicking flow in paper is the Richard’s equation. However, the nonlinear nature of this equation and the requirement for numerical solvers for its solution make it largely inaccessible to the paper microfluidics and lateral flow assay community. Moreover, the parameters used in Richard’s equation often need to be tweaked (or assumed) to match experimental data. This necessitates the need for a simple and appropriate model of partial saturation in paper membranes, easily usable by the wider research community. In the current work, we present an approach to model paper membranes as a bundle of parallel capillaries whose radii follow a two parameter log normal distribution. Application of the Washburn equation to the bundle provides a distribution of fluid fronts, which can be used to calculate saturation. Using this approach, we developed the first analytical expression for spatiotemporal variation of saturation in 1D wicking flow. Experimentally obtained data for spatiotemporal saturation for four different paper materials were fit to this analytical model to obtain parameters for each material. Results obtained from this analytical model match both experimental data as well as numerical results obtained from the Richards equation model well. The availability of an analytical expression for partial saturation in wicking flow promises to significantly increase access to such modelling among the wider research community.