IMA Tutorial: Mathematics of Microfluidic Transport Phenomena December 5-6, 2009 Electroosmotic flow and dispersion in microfluidics Sandip Ghosal Associate Professor Mechanical Engineering Department Northwestern University, Evanston, IL, USA E-mail: s-ghosal@northwestern.edu http://www.mech.northwestern.edu/fac/ghosal
A kitchen sink (literally!) experiment that shows the effect of electrostatic forces on hydrodynamics Courtesy: Prof. J. Santiago’s kitchen
On small scales things are different! 2 R body forces ~ R 3 interfacial forces ~ R 2 interfacial charge dominates at small R 3
Electroosmosis through porous media FLOW Charged Debye Layers E Reuss, F.F. (1809) Proc. Imperial Soc. Naturalists of Moscow
Electroosmosis v Debye E Layer ~10 nm Substrate = electric potential here v = m eo E Electroosmotic mobility
Electrophoresis Debye Layer of counter ions + + + + + v - Ze + + + + + + E v = m ep E Electrophoretic mobility
Equilibrium Debye Layers (Neutral) φ ( z ) is the mean field (l << L ) c i = c i 0 exp( - z i e f / k B T ) λ 2 f ε z 2 = - r e (Poisson) ￥ ρ e = c i z i e i + + + + + + + Gouy-Chapman Model Co-ion (+) Counter-ion (-)
λ D ~ 0.3 nm z If in GC model, x = z i e f / k B T << 1 exp( x ) 1 + x then Debye-Huckel Model 2 f z 2 + f 2 = 0 l D f = z exp( - z / l D ) λ D ￥ z i 2 = e k B T 2 e 2 ) - 1 l D 4 p ( c i 0 φ + + + + + + + ζ i Co-ion (+) (zeta potential) For 1M KCl Counter-ion (-)
Thin Debye Layer (TDL) Limit 2 u 2 f ε µ z z 2 = - 4 pr e z 2 + r e E = 0 � � 2 z 2 u - e E f � � = 0 u ( z ) � � 4 pm φ (0) = z & u (0) = 0 u ( z ) = e E ( f - z ) 4 pm Debye Layer u ( ) = - e E z φ (0) = z 4 pm (Helmholtz-Smoluchowski slip BC)
Electroosmotic Speed E 100 micron 10 nm 2 u µ z 2 = 0 u e η / 2) = υ ε = − εζ Ε u ( z = 4 πµ 10 nm u ( z ) = u e
Slab Gel Electrophoresis (SGE)
Light from UV source Sample Injection Port Sample (Analyte) UV detector Buffer (fixed pH) + -- CAPILLARY ZONE ELECTROPHORESIS
Capillary Zone Electrophoresis (CZE) Fundamentals c ( x , t ) u eo x u = u eo + u ep 2 c c t + u c x = D x 2 σ 2 ~ 2 Dt = 2 DL / u = 2 DL /[( m eo + m ep ) E ] N ~ L 2 / s 2 = ( m eo + m ep ) V /(2 D ) dm ~ m / N ~ 30 kV ) N ~ 10 6 (for V Ideal capillary
“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 u e z is reduced Pressure Gradient + = Corrected Flow Continuity requirement induces a pressure gradient which distorts the flow profile
What is “Taylor Dispersion” ? D eff = D + a 2 u max 2 192 D G.I. Taylor, 1953, Proc. Royal Soc. A, 219 , 186 Aka “Taylor-Aris dispersion” or “Shear-induced dispersion”
Zone Broadening by Taylor Dispersion A B Clean CE Time Delay “Dirty” CE Resolution Degraded Signal Weakened
Parabolic profile due to induced pressure Experiment using Caged Fluorescence Technique - Sandia Labs E Detection Caged Dye EOF suppressed Laser sheet (activation)
Mathematical Modeling (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. 19
(I) The Flow Problem 20
Formulation (Thin Debye Layer) y a x z L
Slowly Varying Channels (Lubrication Limit) y x a z L 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 distance: a 0 velocity: u e = − ες 0 Ε 0 4 πµ Ghosal, S., J. Fluid Mech., 2002, 459 , 103-128 Anderson, J.L. & Idol, W.K. Chem. Eng. Commun., 1985, 38 , 93-106
The Experiments of Towns & Regnier Towns J. & Regnier F. Anal. Chem. 64 , 2473 (1992) Experiment 1 Protein + EOF Mesityl Oxide 100 cm Detector 3 Detector 2 Detector 1 (85 cm) (50 cm) (20 cm)
Understanding elution time delays t ζ = 0 ζ = 1 x L x L z = ( L - x ).1 + 0. x dx = 1 - x dt = u = ￡ L L x = L (1 - e - t / L ) t + L (at small times)
Application: Elution Time Delays - + x = X ( t ) ζ = ζ 0 + e - a x ( z 1 - z 0 ) ζ = ζ 0 dX dt = u = - e < z > E = F ( X ) 4 pm
Best fit of theory to TR data Ghosal, Anal. Chem., 2002, 74, 771-775
Electroosmotic flow with variations in zeta Anderson & Idol Ajdari Ghosal ( Lubrication Theory ) Geometry Cylindrical Plane symmetry Parallel Amplitude Small Wavelength Long Variable zeta zeta,gap zeta,gap Chem. Eng. Comm. Phys. Rev. Lett. Vol. J. Fluid Mech. Vol. 459 2002 Reference Vol. 38 1985 75 1995 Phys. Rev. E Vol. 53 1996
(II) The Transport Problem 30
The Experiments of Towns & Regnier Towns J. & Regnier F. Anal. Chem. 64 , 2473 (1992) Experiment 2 300 V/cm (fixed) 15 cm M.O. _ + PEI 200 100 cm Detector remove zeta potential
Taylor Dispersion in Experiment 2 EOF X
Experiment 2: determining the parameters ζ 1 = - 0.326 z 0
Diffusivity of Mesityl Oxide WILKE-CHANG FORMULA
Theory vs. Experiment Ghosal, S., Anal. Chem., 2002, 74 , 4198-4203
(II) The Coupled Problem 36
CZE with wall interactions in round capillary 2 a 0 ζ = ζ 0 s ( x , t ) c ( r , x , t ) (on wall) (in solution) u e = - ez 0 E 4 pm Length = a 0 Re = u e a 0 / n (less than 1) Time = a 0 / u e (greater than 10) Pe = u e a 0 / D
Flow+Transport Equations
Method of strained co-ordinates T = e t , X = e x φ = φ ( r , X , T )
( a 0 x << 1) Asymptotic Solution Dynamics controlled by slow variables S.Ghosal JFM 2003 491 285 S.Datta & S.Ghosal Phy. of Fluids (2008) 20 012103
DNS vs. Theory
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.2473-2478.
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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. http://www.mendeley.com/profiles/sandip-ghosal/
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