Multiscale Formulation for Assessment of Electroosmotic Flow in Paper-Based Microfluidics

In this work, multiscale techniques to model the pressure driven and electroosmotic flows in porous materials with paper-like microstructures are studied and applied. The multiscale technique is based on the definition of a representative volume element (RVE) of the material, where the microstructure is built from connected channels, where the fluid moves inside the void of the porous material. For fluid flow, the velocity is solved under incompressible flow conditions in the Stokes regime at the microscale level, while the homogeneous Darcy problem is solved at the macroscale level. Similarly, for electroosmotic flow, the velocity and pressure are also solved at the microscale under incompressible flow conditions in the Stokes regime. However, in this case a Helmholtz-Smoluchowsky term is considered at the surface of the solid microstructure. Such term is calculated by solving the electric field via the charge conservation equation. Consequently, the electroosmotic velocity is included in the fluid dynamic problem as a boundary condition, significantly reducing the computational demand. Afterward, once the homogenized velocity field of the microscale problem is obtained, an effective pressure-based permeability and an effective electroosmotic permeability are estimated at the macroscale. To validate the results, a comparison is made with experimental data and other numerical studies reported in the literature for common papers used in microfluidics, such as Whatman $\#$ 1 and Munktel 00A, but also through comparisons with direct numerical simulations. Finally, we propose a microcell structure for representing such papers for matching fluid flow and electrical properties. With such topology, electroosmotic and mixed fluid flow are solved in order to demonstrate the capabilities of the multiscale technique for representing different phenomena involved in paper-based microfluidics. With these microcells will be also possible to predict other physicochemical phenomena which are important for paper-based microfluidics such as capillary imbibition or scalar dispersion, among others.