The utilization of cross-stream diffusion under laminar flow for precise analyte handling plays a critical role in microfluidic biochemical assays such as sample preparation, concentration gradient generation, and molecular interactions. The non-uniform velocity profile along the cross-section of a rectangular microchannel with arbitrary aspect ratio under pressure-driven flow results in unique, heterogeneous species transport including Taylor dispersion and position-dependent diffusion scaling law. Although numerical methods such as finite difference method, finite element method, the method of lines and lattice Boltzmann (LB) method have been used for quantitative study of the phenomena, they inherently suffer from several limitations, such as difficulty to provide direct, physical insight into the underlying transport mechanism and prohibitive computational cost to suppress the artificial numerical diffusion (ND). To address these issues, several analytical models have been proposed, which share several common assumptions such as large aspect ratio and neglecting depth-wise diffusion due to the non-uniform axial velocity in the 3D convection-diffusion equation, markedly limiting their utility. In this paper, we present a three dimensional (3D) analytical model to investigate the diffusion of analyte between two cross streams in rectangular microchannels with arbitrary aspect ratios under pressure-driven flow. The 3D convection-diffusion equation is solved in a Fourier series form using a double integral transformation method and associated eigensystem calculation. Therefore, the model for the first time is capable of capturing the non-uniform transport rate (i.e., the ‘butterfly effect’) and the position-dependent scaling-law of diffusion (1/3-power at the channel wall and 1/2-pwer at the half-depth plane) through an analytical solution. Our analytical model was extensively validated against both experimental and numerical data in terms of the concentration distribution, diffusion scaling law and the mixing efficiency with excellent agreement (the relative error is much less than 0.5% in various benchmark test cases.) Quantitative comparison between our analytical model and other prior analytical models in extensive parameter space was also performed, which convincingly demonstrates that our model accommodates much broader transport regimes and more practical microfluidic applications.

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