Thin-domain modeling of mass transport in microchannels, with application to diffusive mixing

Abstract

Because of the significance of diffusion on the microfluidic length scale, diffusion-based mixers are a valuable and popular component in lab-on-a-chip designs, with many variations emerging. The design of such mixers, however, is limited by existing simulation tools, which are often slow and poorly suited to the special geometry of microchannels. In this article, we derive the necessary conditions for a reduced-dimension model of flow and diffusion in a thin (i.e., height much less than width) channel. We subsequently show that the resulting mass transport equation can be solved in a streamfunction-pressure coordinate system, providing a simpler, more accurate, and more stable solution than is possible using standard methods in the x–y coordinate system. Simulations by this method were fast, requiring less than 30 min even for complex domains; the bulk of the time was spent on noncomputational tasks such as mesh generation. The approach was applied to three problems: a simple mixer based on the T sensor of Yager and co-workers, and then two variations on the base case in which obstructions were introduced to increase the available diffusion time. Although the obstructions changed the flow pattern significantly, there was little effect on the outlet concentration profile. It was observed from the model equations that if the flow rate is unchanged, the improvement in mixing can be expressed in terms of an upper bound based on the increase in pressure drop. In the case studied, the pressure drop change was small, so the mixing was not improved very much.