Characterization of Transverse Channel Concentration Profiles Obtainable with a Class of Microfluidic Networks

Abstract

We analyze mathematically a previously reported class of passive microfluidic mixing networks. The networks produce nonhomogeneous concentrations in the output channel, resulting in diverse concentration profiles. We formally prove that all profiles obtainable with this class of networks can be described as polynomials of degree no higher than the number of input channels less one. We derive explicit formulas for the calculation of resultant output concentration profiles and conversely for the calculation of input concentrations needed to obtain set output profiles.

Characterization of transport in microfluidic gradient generators

“…The corresponding minimized RMS deviation at that position decreases as well. These results provide theoretical support to the insight of Sager et al (2006) that the N streams may be thought of as sampling a desired function, where the granularity of the sampling decreases with N.…”

“…It was observed originally by Dertinger et al (2001) that pyramidal schemes generate a profile that is polynomial in nature, and the degree of the resulting polynomial profile is one less than the number of concentration inlets. This was later proved mathematically by Sager et al (2006), who further demonstrated that the discrete streams generated by the network can be thought of as sampling a specific function, termed the ''impulse response polynomial.'' They developed a general method for calculating the concentration of each sampling stream and the coefficients of the impulse response polynomial, given the concentrations of the inlets and the properties of the network.…”

“…(6), to be valid, it is required that each input stream have the same viscosity , a reasonable assumption for streams varying only by the concentration of a dilute species. We denote C i * as the concentration of the ith input stream, which can be calculated generally using methods described elsewhere [e.g., Sager et al (2006)]. …”

“…The flow is in the positive y direction. c Examples of staircase input boundary conditions (solid) and associated impulse response functions (dashed), after Sager et al (2006) property that each sampling stream entering the gradient chamber has equal flow rates.…”

Controlling Mass Transport in Microfluidic Devices

“…Early works on gradient generation focused on the mass-conservation relationship solely between the resulting concentration profile and the number of inlet streams (124, 137). The transport process—namely a detailed hydrodynamic profile of flow in rectangular channels—was not considered; the length of each mixing channel was deliberately kept long to assure that complete diffusive mixing was achieved (106).…”