Towards a Theory of Mixing Graphs: A Characterization of Perfect Mixability

TL;DR

This paper characterizes which sets of fluid droplets can be perfectly mixed in microfluidic devices, providing an efficient algorithm to test perfect mixability and construct mixing graphs with polynomial complexity.

Contribution

It offers a complete characterization of perfectly mixable droplet sets and an efficient polynomial-time algorithm to determine and construct such mixing graphs.

Findings

Complete characterization of perfect mixability.

Polynomial-time algorithm for testing perfect mixability.

Construction of polynomial-size mixing graphs.

Abstract

Some microfluidic lab-on-chip devices contain modules whose function is to mix two fluids, called reactant and buffer, in desired proportions. In one of the technologies for fluid mixing the process can be represented by a directed acyclic graph whose nodes represent micro-mixers and edges represent micro-channels. A micro-mixer has two input channels and two output channels; it receives two fluid droplets, one from each input, mixes them perfectly, and produces two droplets of the mixed fluid on its output channels. Such a mixing graph converts a set I of input droplets into a set T of output droplets, where the droplets are specified by their reactant concentrations. The most fundamental algorithmic question related to mixing graphs is to determine, given an input set I and a target set T, whether there is a mixing graph that converts I into T. We refer to this decision problem as mix-reachability. While the complexity of this problem remains open, we provide a solution to its natural sub-problem, called perfect mixability, in which we ask whether, given a collection C of droplets, there is a mixing graph that mixes C perfectly, producing only droplets whose concentration is the average concentration of C. We provide a complete characterization of such perfectly mixable sets and an efficient algorithm for testing perfect mixability. Further, we prove that any perfectly mixable set has a perfect-mixing graph of polynomial size, and that this graph can be computed in polynomial time.