Sandip Ghosal Associate Professor Mechanical Engineering Department Northwestern University, Evanston, IL, USA Electroosmotic flow and dispersion in microfluidics IMA Tutorial: Mathematics of Microfluidic Transport Phenomena December 5-6, 2009

Courtesy: Prof. J. Santiago’s kitchen A kitchen sink (literally!) experiment that shows the effect of electrostatic forces on hydrodynamics

3 On small scales things are different! 2 R body forces ~ R 3 interfacial forces ~ R 2 interfacial charge dominates at small R

Electroosmosis through porous media E FLOW Charged Debye Layers Reuss, F.F. (1809) Proc. Imperial Soc. Naturalists of Moscow

Electroosmosis E Debye Layer ~10 nm Substrate = electric potential here v Electroosmotic mobility

Electrophoresis - Ze + v E Debye Layer of counter ions Electrophoretic mobility

Equilibrium Debye Layers Counter-ion (-) Co-ion (+) is the mean field (Poisson) Gouy-Chapman Model (Neutral)

Counter-ion (-) Co-ion (+) z If in GC model, Debye-Huckel Model (zeta potential) then For 1M KCl

Thin Debye Layer (TDL) Limit z Debye Layer & (Helmholtz-Smoluchowski slip BC)

Electroosmotic Speed E 10 nm 100 micron 10 nm

Slab Gel Electrophoresis (SGE)

Sample Injection Port Sample (Analyte) Buffer (fixed pH) + -- UV detector Light from UV source CAPILLARY ZONE ELECTROPHORESIS

Capillary Zone Electrophoresis (CZE) Fundamentals Ideal capillary (for V

“Anomalous dispersion” mechanisms In practice, N is always LESS than this “ideal” (diffusion limited) value. Why? Joule heating Curved channels Wall adsorption of analytes Sample over loading ……….

Non uniform zeta-potentials Continuity requirement induces a pressure gradient which distorts the flow profile is reducedPressure Gradient + = Corrected Flow

What is “Taylor Dispersion” ? G.I. Taylor, 1953, Proc. Royal Soc. A, 219, 186 Aka “Taylor-Aris dispersion” or “Shear-induced dispersion”

Zone Broadening by Taylor Dispersion AB Resolution Degraded Signal Weakened Clean CE “Dirty” CE Time Delay

Parabolic profile due to induced pressure Experiment using Caged Fluorescence Technique - Sandia Labs EOF suppressed E Laser sheet (activation) Caged Dye Detection

19 (I) The Flow Problem: what does the flow profile look like in a micro capillary with non-uniformly charged walls? (II) The Transport Problem: what is the time evolution of a sample zone in such a non-uniform but steady EOF? (III) The Coupled Problem: same as (II) but the EOF is unsteady; it is altered continuously as the sample coats the capillary. Mathematical Modeling

20 (I) The Flow Problem

Formulation (Thin Debye Layer) L a x y z

Slowly Varying Channels (Lubrication Limit) L a x y z Asymptotic Expansion in

Lubrication Solution From solvability conditions on the next higher order equations: F is a constant (Electric Flux) Q is a constant (Volume Flux)

Lubrication Theory in cylindrical capillary Boundary conditions Solution Ghosal, S., J. Fluid Mech., 2002, 459, Anderson, J.L. & Idol, W.K. Chem. Eng. Commun., 1985, 38, distance: velocity:

The Experiments of Towns & Regnier 100 cm EOF Detector 3 (85 cm) Detector 2 (50 cm) Detector 1 (20 cm) Protein + Mesityl Oxide Experiment 1 Towns J. & Regnier F. Anal. Chem. 64, 2473 (1992)

Understanding elution time delays (at small times)

Application: Elution Time Delays + -

Best fit of theory to TR data Ghosal, Anal. Chem., 2002, 74,

Anderson & Idol Ajdari Ghosal GeometryCylindrical symmetry Plane Parallel AmplitudeSmall WavelengthLong Variablezetazeta,gap Reference Chem. Eng. Comm. Vol Phys. Rev. Lett. Vol Phys. Rev. E Vol J. Fluid Mech. Vol Electroosmotic flow with variations in zeta ( Lubrication Theory )

30 (II) The Transport Problem

The Experiments of Towns & Regnier + remove 100 cm 15 cm 300 V/cm (fixed) PEI 200 _ Detector Experiment 2 M.O. Towns J. & Regnier F. Anal. Chem. 64, 2473 (1992) zeta potential

Taylor Dispersion in Experiment 2 X EOF

Experiment 2: determining the parameters

Diffusivity of Mesityl Oxide WILKE-CHANG FORMULA

Theory vs. Experiment Ghosal, S., Anal. Chem., 2002, 74,

36 (II) The Coupled Problem

CZE with wall interactions in round capillary (in solution)(on wall) (less than 1) (greater than 10)

Flow+Transport Equations

Method of strained co-ordinates

Asymptotic Solution Dynamics controlled by slow variables S.Ghosal JFM S.Datta & S.Ghosal Phy. of Fluids (2008)

DNS vs. Theory

Shariff, K. & Ghosal S. (2004) Analytica Chimica Acta, 507, 87-93

Eluted peaks in CE signals Reproduced from: Towns, J.K. & Regnier, F.E. “Impact of Polycation Adsorption on Efficiency and Electroosmotically Driven Transport in Capillary Electrophoresis” Anal. Chem. 1992, 64, pg

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Summary Problem of EOF in a channel of general geometry was discussed in the lubrication approximation. Full analytical solution requires only a knowledge of the Green’s function for the cross-sectional shape. In the case of circular capillaries, the lubrication theory approach can explain experimental data on dispersion in CE. The coupled “hydro-chemical” equations were solved using asymptotic methods for an analyte that adsorbs to channel. walls and alters its zeta potential.