Electrohydrodynamic instabilities in microfluidics Brian D. Storey Franklin W. Olin College of Engineering Needham MA

EHD instability in microfluidics Posner, Santiago, JFM 2006 Chen, Lin, Lele, Santiago JFM 2005 ElMochtar, Aubry, Batton, LoC 2003 Lin, Storey, Oddy, Chen Santaigo PoF2004 Lin, Storey, Santaigo JFM 2008 Computation Experiment Santos & Storey PRE 2008

Hoburg and Melcher (JFM 1976) Web of science citations by the author(s) 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 +

Electrolytes and charged surfaces counter-ions co-ions

Electroosmosis (200 th anniversary) Electric field

Electroosmosis in a channel (the simplest pump?) Charge density Velocity Y Y Electric field Electroneutral in bulk

Double layers are typically thin Helmholtz-Smolochowski

Electrohydrodynamic instability Experiments (Mike Oddy of J. Santiago’s group) 1 mm V High conductivity fluid Low conductivity fluid Miscible interface

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 Lin, Storey, Oddy, Chen Santaigo PoF2004 m a=F Mass is conserved Fluid conductivity goes with the flow Current is conserved, V=iR

Mechanism for charge generation High conductivityLow conductivity Electric field E

Mechanism for flow E

Dimensionless parameters Electric Rayleigh number Reynolds number Ratio of electro-osmotic to electroviscous velocity Electrical conductivity ratio

Experiment vs. 2D Computation Lin, Storey, Oddy, Chen, Santiago, Phys Fluids 2004

Other configurations High conductivity center Low conductivity center 2D Simulation (Storey, Phys D 2005) Experiment (Ponser & Santiago, JFM 2006) 2D Simulation (Storey, Phys D 2005)

Instability at T-junction 0.5, 0.75, 1, & 1.25 kV/cm Chen, Lin, Lele, & Santiago, JFM 2005 Simulations with same basic model provided good agreement

Linear stability results E cr,experiment ~ 35,000 V/m, x y z 2D Linear Analysis with  1 /  2 =10 Stable Ra e E (V/m) 3D Linear Analysis with  1 /  2 =10 Stable Ra e E (V/m) Ecrit Lin, Storey, Oddy, Chen, Santiago, Phys Fluids 2004

So 3D matters 3D DNS Storey, Physica D, 2005 time

As does electroosmosis Storey, Physica D, 2005 time

Thin channels So aspect ratio matters, but can we model flow in thin channels with a 2D model? x y z d H E 11 22

Thin Channel Approx. (Hele-Shaw) Solid- full 3D Dashed – this model x y z d H E 11 22 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 Experiment Computation 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 Lin, Storey, Santiago JFM 2008

Computational Results: depth-averaged model Lin, Storey, Santiago JFM 2008 Experiment Simulation

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) Electroosmotic velocity depends upon the electric field Electric field is high when conductivity is low Low conductivity = high EO velocity High conductivity, E 1 u eof, 1 u eof, 2 High conductivity, E Low conductivity, E 2 u eof, 1 u eof, 2 1 u eof, 1 High conductivity, E Red; cond =10Blue; 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? High conductivity, E 1 u eof, 1 u eof, 2 High conductivity, E Low conductivity, E 2 u eof, 1 u eof, 2 1 u eof, 1 High conductivity, E

Example of axial conductivity gradients in EK Field Amplified Sample Stacking (FASS) + t > Stacked Analyte - t = 0 High Conductivity buffer Low Conductivity SampleHigh Conductivity buffer UBUB USUS ESES EBEB E EBEB 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 x y Binary electrolyte (C+,C-) Fixed potential Fixed concentration of C+ No flux of C- Solid surfaces are charge selective (electrode or ion exchange membrane). Current

Steady state V=1 E, flux of C+ Bulk is electro-neutral, linear conc. profile Double layer, Debye =0.01 Typical dimensionless Debye = or less

Current-voltage relationship Resistor at low voltage 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?

Hoburg-Melcher limit Pe=∞, low V analysis 0 Summary D>1, Real, S 2 <0, Stable D 0, Unstable D=1, Imag, Oscillations Storey, Zaltzman, & Rubinstein, PRE 2007

Bulk electroconvection, finite Pe low V analysis Current, I max =4 unstable L=-68 k=4.74 Summary D>1, Real, Stable D<1, Real, Unstable (threshold) D=1, Stable Storey, Zaltzman, & Rubinstein, PRE 2007

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 From J. Han, MIT Device built for bio-molecule preconcentration

Instability observed From J. Han, MIT

Stuff I didn’t show you.. Colloids, Posner Two phase, Zahn & Reddy Two phase, Aubry et al Electrothermal, Ramos, Gonzalez, Castellanos, 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.