UC Riverside

In this work, we study the problem of fluid mixing in microfluidic chips. The motivation for studying this problem comes from the process of sample preparation for chemical, biological, medical and environmental experiments, which often require preparation of fluid mixtures with desired concentrations.

We assume that fluids are manipulated in discrete units called droplets. The input set of droplets consist of two distinct fluids: the reactant, which is the fluid of interest, and the buffer fluid that is used to dilute it. The goal is to produce a target set of droplets with prespecified reactant concentrations. In the model we study, the mixing process in a microfluidic chip can be abstractly represented as a mixing graph.

A mixing graph is a collection of micro-mixers (nodes) connected by micro-channels (edges) that converts an input set of droplets I into a set of output droplets T, by applying a sequence of 1:1 mixing operations. This graph may also produce some waste, which are superfluous droplets of fluid not used in the target set. Computational complexity of most natural questions regarding such mixing graphs remain open. For example, it is not even known whether it is decidable for a given target set to be produced without waste. Current work in the literature contains only heuristic approaches that compute mixing graphs while attempting to optimize certain objectives, including minimizing waste, reactant usage, the depth of the graphs, and more.

Our first contribution is an efficient algorithm for computing mixing graphs for single-droplet targets. Our algorithm produces significantly less waste than state-of-the-art algorithms in an experimental comparison. We also provide a bound on its worst-case performance that is significantly better than those for earlier algorithms. Our second result concerns the variant of the problem where the objective is to design a mixing graph that perfectly mixes a collection of input droplets with arbitrary concentrations. We provide a complete characterization of input sets for which such graphs exist, and an efficient algorithm to construct these graphs. In addition, we provide several other results about properties of mixing graphs and the computational complexity of computing mixing graphs of fixed depth.