Steady Advection-Diffusion in Complex Microfluidic Mixers

Lipid nanoparticles are a crucial ingredient of mRNA vaccines. However, their synthesis is still throttled by a poor understanding of their self-assembly. These particles are non-equilibrium structures, and their exact properties depends on the history of their microenvironment during their formation. The turbulent impinging jet mixers which are the gold standard for the synthesis of lipid nanoparticles do not allow such a granular determination of the microenvironment. A proper understanding of these particle’s assembly cannot proceed without precise understanding of the concentration fields in the mixers used to make them. Microfluidic systems, due to the regularity of their flow and concentration profiles, allow such a precise characterization.

In this paper, we combine theoretical work on advection-diffusion in the Hele-Shaw cell with techniques for the computation of Schwarz-Christoffel maps to obtain flow and concentration profiles everywhere in arbitrary polygonal microfluidic mixers. Using a sequence of conformal maps, we reduce the advection-diffusion equation in a complex mixer to a simple streamline coordinate formulation, which we can then solve analytically. The solutions thus obtained can then be combined with numerical population balance models to model lipid nanoparticle growth in micromixers. Analytical knowledge of the conditions in the micromixer thus enables us to build a comprehensive framework for the synthesis of self-assembled particles in microfluidic systems.