Towards a Theory of Mixing Graphs: A Characterization of Perfect Mixability (Extended Abstract)

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.One of the most fundamental functions of LoC devices is mixing of different fluids. In particular, in applications related to sample preparation, the objective is to produce desired volumes of pre-specified mixtures of fluids. In typical applications only two fluids are involved, in which case the process of mixing is often referred to as dilution. The fluid to be diluted is called reactant and the diluting fluid is called buffer. For example, in clinical diagnostics common reactants include blood, serum, plasma and urine, while phosphate buffered saline is often used as buffer [16].There is a variety of different technologies that can be used to manufacture microfluidic devices for fluid mixing. In our work, we consider microfluidic chips that involve a collection of tiny components called micro-mixers connected by micro-channels. In such chips, input fluids are injected into the chip using fluid dispensers, then they travel, following appropriate micro-channels, through a sequence of micro-mixers in which they are subjected to mixing operations, and are eventually discharged into output reservoirs. We focus on droplet-based chips, where fluids are manipulated in discrete units called droplets. In such chips each micro-mixer has exactly two input and two output channels. It receives one droplet of fluid from each input, mixes them perfectly, producing two identical droplets on its outputs. Specifically, if the input droplets have (reactant) concentrations a, b, then the produced droplets will have concentration 1 2 (a + b). It foll...