Scaling for microfluidic mixing

Mixing in microfluidic channels or chambers is typically dominated by molecular diffusion. A common method used to evaluate mixing involves the examination of a time series of instantaneous concentration maps of fluid tracers (colorimetric or fluorescent) that enable visualization of fluid layering and simultaneous diffusive mixing. A scale often used to characterize micromixer performance is the global deviation of these concentration maps. While useful, this measurement scale does not provide a sensitive metric for evaluating fluid layering in the mixing process. This paper proposes an analytical approach that examines spatial concentration gradients and a global gradient-based scale, a normalized L2 norm of the gradient map, for micromixer performance evaluation. This gradient-based scale is complementary to deviation-based scales and is especially useful for the class of micromixers that enhance mixing by stretching and folding of fluids, whether the dominant mode of mixing is diffusion or chaotic advection. The algorithm is easy for micromixer designers to implement and will reveal performance metric information that remains implicitly hidden when deviation-based scales are used. The use of gradient-based mixing performance evaluation is illustrated with baker's transform, a series of discrete mappings similar to kneading dough. The changes in both the deviation-based and gradient-based scale created by discrete fluid stretching and folding are discussed. The results from the one-dimensional discrete mixing problem are extended to a realistic mixing problem that simulates continuous stretching and folding.