Generalized Taylor-Aris dispersion in discrete spatially periodic networks: microfluidic applications

A theory is presented for the lumped parameter, convective-diffusive transport of individual, noninteracting Brownian solute particles ("macromolecules") moving within spatially periodic, solvent-filled networks--the latter representing models of chip-based microfluidic chromatographic separation devices, as well as porous media. Using graph-theoretical techniques, the composite medium is conceptually decomposed into a network of channels (the edges) through which the solute is transported by a combination of molecular diffusion and either "piggyback" entrainment within a flowing solvent or an externally applied force field acting upon the solute molecules. A probabilistic choice of egress channel for a solute particle exiting the intersection (vertex) of the channels is furnished by an imperfect mixing model. A spatially periodic, Taylor-Aris-like "method-of-moments" scheme is applied to this transport model, leading to discrete matrix equations for computing the network-scale particle velocity vector U(*) and dispersivity dyadic D(*) in terms of the prescribed microscale transport parameters and network geometry characterizing the basic unit cell of which the spatially periodic device is comprised. The ensuing algebraic equations governing the vertex-based, discrete unit-cell "fields" P(0)(infinity)(i) and B(i) (i=1,2,...,n), whose paradigmatic summations yield U(*) and D(*), constitute discrete analogs of classical continuous macrotransport phenomenological parameters, P(0)(infinity)(r) and B(r), with r a continuous position vector defined within the unit cell. The ease with which these discrete calculations can be performed for complex networks renders feasible parametric studies of potential microfluidic chip designs, particularly those pertinent to biomolecular separation schemes. Application of this discrete theory to the dispersion analysis of pressure-driven flow in spatially periodic serpentine microchannels is shown to accord with existing results previously derived using classical continuous macrotransport theory.