S TANFORD M ICROFLUIDICS L ABORATORY A D EPTH -A VERAGED M ODEL F OR E LECTROKINETIC F LOWS I N A T HIN M ICROCHANNEL G EOMETRY Hao Lin, 1 Brian D. Storey 2 and Juan G. Santiago 1 1. Mechanical Engineering Department, Stanford University 2. Franklin W. Olin College of Engineering IMECE, November 15 th, 2004, Anaheim, CA

S TANFORD M ICROFLUIDICS L ABORATORY Motivation: Generalized EK flow with conductivity gradients Field amplified sample stacking (FASS) Electrokinetic instability (EKI) Rajiv Bharadwaj Michael H. Oddy

S TANFORD M ICROFLUIDICS L ABORATORY Previous Work Lin, Storey, Oddy, Chen & Santiago 2004, Phys. Fluids. 16(6): – Instability mechanism: induced by bulk charge accumulation; stabilized by diffusion (Taylor-Melcher- Baygents) – 2D and 3D linear analyses – 2D nonlinear computations Storey, Tilley, Lin & Santiago 2004 Phys. Fluids, in press. – Depth-averaged Hele-Shaw analysis (zeroth-order) Chen, Lin, Lele & Santiago 2004 J. Fluid Mech., in press – Instability mechanism: induced by bulk charge accumulation; stabilized by diffusion (Taylor-Melcher- Baygents) – Depth-averaged linear analyses – Convective and absolute instability Experiment 2D Computation

S TANFORD M ICROFLUIDICS L ABORATORY Thin-Channel Model Practicality Consideration – 2D depth-averaged model significantly reduces the cost of 3D computation – Model well captures the full 3D physics Develop flow model for generalized electrokinetic channel flows – Eletrokinetic instability and mixing – Sample stacking – Other EK flows which involves conductivity gradients x y z d H E 11 22

S TANFORD M ICROFLUIDICS L ABORATORY Full 3D Formulation (Lin et al.) H. Lin, Storey, B., M. Oddy, Chen, C.-H., and J.G. Santiago, “Instability of Electrokinetic Microchannel Flows with Conductivity Gradients,” Phys. Fluids 16(6), , C.-H. Chen, H. Lin, S.K. Lele, and J.G. Santiago, “Convective and Absolute Electrokinetic Instabilities with Conductivity Gradients,” J. Fluid Mech., in press, 2004.

S TANFORD M ICROFLUIDICS L ABORATORY Depth Averaged Model Asymptotic Expansion based on the aspect ratio  = d/H which is similar to lubrication/Hele-Shaw theory Equations are depth- averaged to obtain in-plane (x,y) governing equations Flows in the z-direction are integrated and modeled u z x

S TANFORD M ICROFLUIDICS L ABORATORY Depth Averaged Equations Convective dispersion: Taylor-Aris type Momentum: Darcy-Brinkman- Forchheimer H. Lin, Storey, B., and J.G. Santiago, “A depth-averaged model for electrokinetic flows in a thin microchannel geometry,” to be submitted, 2004.

S TANFORD M ICROFLUIDICS L ABORATORY Field Amplified Sample Stacking (FASS) + t > Stacked Analyte - t = 0 High Conductivity buffer Low Conductivity SampleHigh Conductivity buffer UBUB USUS ESES EBEB E EBEB Rajiv Bharadwaj

S TANFORD M ICROFLUIDICS L ABORATORY 1D Simplification (y-invariant) y E x Dispersion effects include: EOF variation in x Vertical circulation in z uu eo, 1eo, 2 w z x High Conductivity Low Conductivity

S TANFORD M ICROFLUIDICS L ABORATORY FASS: Model vs DNS DNS Model DNS Model Model w/o Dispersion DNS Model Model w/o Dispersion

S TANFORD M ICROFLUIDICS L ABORATORY FASS: Model vs DNS DNS Model Model w/o Dispersion Model  RMS Time (s) DNS Model w/o Dispersion

S TANFORD M ICROFLUIDICS L ABORATORY Motivation: Electrokinetic Instability (EKI) No gradient  =  m 1 mm (Michael H. Oddy) (C.-H. Chen) (Rajiv Bharadwaj)

S TANFORD M ICROFLUIDICS L ABORATORY Linear Analysis: 2D vs 3D 3D Linear Analysis Stable E cr,experiment ~ 0.3 kv/cm, E cr,2D ~ 0.04 kv/cm, E cr,3D ~ 0.18 kv/cm H. Lin, Storey, B., M. Oddy, Chen, C.-H., and J.G. Santiago, “Instability of Electrokinetic Microchannel Flows with Conductivity Gradients,” Phys. Fluids 16(6), , D Linear Analysis Stable

S TANFORD M ICROFLUIDICS L ABORATORY EKI: Linear Analysis Model 3D Linear zeroth-order momentum

S TANFORD M ICROFLUIDICS L ABORATORY EKI: Nonlinear Simulation Experiment Model t = 0.0 s t = 0.5 s t = 1.5 s t = 2.0 s t = 2.5 s t = 3.0 s t = 4.0 s t = 5.0 s t = 1.0 s

S TANFORD M ICROFLUIDICS L ABORATORY Conclusions and Future Work Developed depth-averaged model for general EK flows in microchannels Model validated with DNS and experiments Future work: – Modeling and optimization of realistic FASS applications – Modeling and optimization of EKI mixing