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Electrohydrodynamic instabilities in microfluidics Brian D. Storey Franklin W. Olin College of Engineering Needham MA EHD instability in microfluidics Lin, Storey, Oddy, Chen Santaigo PoF2004 ElMochtar, Aubry, Batton, LoC 2003 Chen, Lin, Lele, Santiago JFM 2005 Santos & Storey PRE 2008 Posner, Santiago, JFM 2006 Computation Experiment Lin, Storey, Santaigo JFM 2008 Hoburg and Melcher (JFM 1976) Web of science 1976-1985 8 citations by the author(s) 1982-1994 4 citations 2004-today 22 citations Electrohydrodynamics • Electrohydrodynamics is the interaction between electric fields and fluid motion. • Today we will be concerned with EHD of simple, miscible, electrolytes. What’s an electrolyte? A material in which the mobile species are ions and free movement of electrons is blocked. (Newman, Electrochemical Systems) Na + Cl - Na + Cl - Na + Cl - Na + Cl - Na + Cl - Na + Electrolytes and charged surfaces + + + + + + + + + + + - + + + + + - + + + - + + + + - + + + - 3 2.5 C 2 counter-ions 1.5 1 0.5 0 0 co-ions 1 2 3 X 4 5 Electric field + + + + + + + + + + + + + - + + + + + + - + + + - + + + - + + + + + + + + + + + + + + - + + + + + + - + + + - + + + + - - + - + - + + + - - + - Electroosmosis (200th anniversary) Electroosmosis in a channel (the simplest pump?) 1 -------------------------- 0.8 0.6 0.4 Y 0.2 Electric field Y0 -0.2 -0.4 -0.6 Electroneutral in bulk -0.8 -1 - -0 - - - -0.2- - - -0.4 - - - - -0.6- - - -0.8 - - - - -1 - Charge density Charge density Velocity 1 -0.98 0.8 -0.982 0.6 -0.984 0.4 -0.986 0.2 -0.988 0 -0.99 y y Double layers are typically thin -0.2 -0.992 -0.4 -0.994 -0.6 -0.996 -0.8 -0.998 -1 -1 0 0.2 0.4 0.6 Velocity 0.8 E U slip 1 1.2 0 0.2 0.4 Helmholtz-Smolochowski 0.6 Velocity 0.8 1 1.2 Electrohydrodynamic instability Experiments (Mike Oddy of J. Santiago’s group) Miscible interface High conductivity fluid 1 mm Low conductivity fluid V Model summary • Incompressible Navier-Stokes plus electric body force • Poisson-Nernst-Planck for ion transport binary, symmetric electrolyte; simplified by assuming fluid is nearly electro-neutral. • Helmholtz-Smolochowski electrokinetic slip boundary conditions Dv 2 Re P v e E Dt v 0 D 2 Ra ( is conductivity) Dt E 0 m a=F Mass is conserved Fluid conductivity goes with the flow Current is conserved, V=iR Lin, Storey, Oddy, Chen Santaigo PoF2004 Mechanism for charge generation E 0 E e E E 0 E E / E + + + + + + + High conductivity Electric field + + + + + + + + Low conductivity E e positive x E e positive x Mechanism for flow Dv 2 Re P v e E Dt v 0 D 2 Ra Dt E E / E 0 E Dimensionless parameters U H Rae ev D Re U ev H high low U Rv eo U ev E 2 H Electric Rayleigh number U ev Reynolds number Electrical conductivity ratio Ratio of electro-osmotic to electroviscous velocity Experiment vs. 2D Computation Lin, Storey, Oddy, Chen, Santiago, Phys Fluids 2004 Other configurations High conductivity center 2D Simulation (Storey, Phys D 2005) Experiment (Ponser & Santiago, JFM 2006) Low conductivity center 2D Simulation (Storey, Phys D 2005) Experiment (Ponser & Santiago, JFM 2006) Instability at T-junction 0.5, 0.75, 1, & 1.25 kV/cm Simulations with same basic model provided good agreement Chen, Lin, Lele, & Santiago, JFM 2005 Linear stability results 2D Linear Analysis with 1/ 2=10 3D Linear Analysis with 1/ 2=10 z Stable E (V/m) Ecrit x Rae Rae E (V/m) y Ecrit Stable Ecr,experiment ~ 35,000 V/m, Lin, Storey, Oddy, Chen, Santiago, Phys Fluids 2004 So 3D matters 3D DNS time Storey, Physica D, 2005 As does electroosmosis time Storey, Physica D, 2005 Thin channels • So aspect ratio matters, but can we model flow in thin channels with a 2D model? z y H x d 2 E 1 Thin Channel Approx. (Hele-Shaw) z y H x d 2 E 1 Solid- full 3D Dashed – this model Storey, Tilley, Lin, Santiago, Phys Fluids 2005 Hele-Shaw model works in linear regime, fails in non-linear regime 3D DNS Depth Ave Zeroth order 3D DNS Depth Ave Zeroth order Lin, Storey, Santiago JFM 2008 Higher order (includes EK dispersion) works better in NL regime 3D Simulation Full Depth Ave Zeroth order 3D Simulation Full Depth Ave Zeroth order Lin, Storey, Santiago JFM 2008 Depth-Averaged Model t = 0.0 s t = 0.5 s t = 1.0 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 Computation Experiment Lin, Storey, Santiago JFM 2008 Computational Results: depth-averaged model Experiment Simulation Lin, Storey, Santiago JFM 2008 So… • Depth averaged, 2D model for electrokinetic flow works. • Need to include electrokinetic dispersion in the model. • But what’s electrokinetic dispersion? Classic Taylor dispersion in pressure driven flow “Physicochemical Hydrodynamics” Probstein Electrokinetic dispersion (looking in the thin direction) ueof, 1 ueof, 1 ueof, 2 High conductivity, E1 Low conductivity, E 2 High conductivity, E1 •Electroosmotic velocity depends upon the electric field •Electric field is high when conductivity is low •Low conductivity = high EO velocity Red; cond =10 Blue; cond =1 Ghosal, EP 2004 Baradawaj & Santiago JFM 2005 Ren & Li JCIS 2006 Sounart & Baygents JFM 2007 Dispersion acts as anisotropic diffusion 3D Simulation Full Depth Ave Zeroth order 3D Simulation Full Depth Ave Zeroth order Lin, Storey, Santiago JFM 2008 So… • Is flow stable in the shallow direction? • How does our shallow model break down? ueof, 1 High conductivity, E1 ueof, 1 ueof, 2 Low conductivity, E 2 High conductivity, E1 Example of axial conductivity gradients in EK Field Amplified Sample Stacking (FASS) t=0 - -- - - - - - + High Conductivity buffer Low Conductivity Sample High Conductivity buffer ES E t>0 - EB + UB - US --- -- - -- - EB - Stacked Analyte Burgi & Chein 1991, Analytical Chem. Unstable flow E=25,000 V/m, Conductivity ratio=10 Santos & Storey, PRE 2008 Flow in center similar to other observations High conductivity center 2D Simulation (Storey, Phys D 2005) Experiment (Ponser & Santiago, JFM 2006) Observations •“Shock” at the leading edge of the sample. •Vertical velocity at the channel walls pumps fluid toward the centerline. •Unstable flow only inside the sample region. Santos & Storey, PRE 2008 Stability measure Maximum vertical vel. along the centerline Santos & Storey, PRE 2008 Stability measure as function of applied field Unstable E field Santos & Storey, PRE 2008 A microfluidic EHD mixer E Field ElMochtar, Aubry, Batton, LoC 2003 Boy & Storey, PRE 2007 Time periodic forcing for enhanced mixing Boy & Storey, PRE 2007 Classic problem in electrochemistry Binary electrolyte (C+,C-) •Fixed potential •Fixed concentration of C+ •No flux of C- Current Solid surfaces are charge selective (electrode or ion exchange membrane). y x Steady state V=1 Double layer, Debye =0.01 4 3.5 E, flux of C+ 3 C+,C- 2.5 2 1.5 V=1 Double layer, Debye =0.01 1 0.5 0 0 Bulk is electro-neutral, linear conc. profile 0.2 0.4 x 0.6 0.8 1 Typical dimensionless Debye =0.0001 or less Current-voltage relationship 6 Resistor at low voltage 5 Observed Current 4 1D Solution 3 2 1 0 0 5 10 Voltage 15 20 Attributed to instability of double layers Zaltzman & Rubinstein, JFM 2007 Different views on bulk stability •Bulk instability. Grigin (1985, 1992) •Bulk instability, but not sufficient for mixing. Bruinsma & Alexander (1990) •Bulk instability. Rubinstein, Zaltzman, & Zaltzman (1995). •No bulk instability. Buchanan & Saville (1999) •No bulk instability. Highlighted problems with all earlier works reporting instability. Limited parameter space. Lerman, Zaltzman, Rubinstein (2005) Q: The model equations for bulk instability is the same as ours, why is there no bulk instability? Or is there? 0 Hoburg-Melcher limit Pe=∞, low V analysis c 1 2 v c c t Pe D 1 2 c c 0 D 1 Summary •D>1, Real, S2<0, Stable •D<1, Real, S2>0, Unstable •D=1, Imag, Oscillations D 1 ~ S2 S2 1 D D ; D D Storey, Zaltzman, & Rubinstein, PRE 2007 Bulk electroconvection, finite Pe low V analysis Storey, Zaltzman, & Rubinstein, PRE 2007 unstable L=-68 k=4.74 I D 1 D L Pe ; D 4 D 1 D 2 Current, Imax =4 Summary •D>1, Real, Stable •D<1, Real, Unstable (threshold) •D=1, Stable BE at finite voltage, D=0.1 Unstable Pe=9.9 Storey, Zaltzman, & Rubinstein, PRE 2007 Relationship between BE and microchannel EHD instability • Bulk instability can exist, in theory. • Threshold is different since conductivity gradient is driven • New bulk instability mechanism found when D+ < D-, that can occur at low V. • Many previous studies only considered D+=D. • An analysis looking for an application… Other example of flows driven by concentration polarization Device built for bio-molecule preconcentration From J. Han, MIT Instability observed From J. Han, MIT Stuff I didn’t show you.. Colloids, Posner Two phase, Zahn & Reddy Electrothermal, Ramos, Gonzalez, Castellanos, et al Two phase, Aubry et al Multi-species, Oddy & Santiago Acknowledgements • • • Collaborators: – Hao Lin, Rutgers – Juan Santiago, Stanford – Boris Zaltzman & Isaac Rubinstein, Ben Gurion University of Negev, Israel Undergraduate students – David Boy – Jobim Santos – Lee Edwards – Doug Ellwanger – Allison Schmidt – Mark Cavolowsky – Nina Cary – Angela Mao Funding – NSF – Olin College Depth averaged equations From the DA equations, we can reconstruct the full 3D fields.