Electroosmotic flow and dispersion in microfluidics Sandip Ghosal - - PowerPoint PPT Presentation

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

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 ~ R3 interfacial forces ~ R2 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 = m

eoE

v

Electroosmotic mobility

Electrophoresis

• Ze

+ v E + + + + + + + + + +

Debye Layer of counter ions

v = m

epE

Electrophoretic mobility

Equilibrium Debye Layers

+ + + + + + +

Counter-ion (-) Co-ion (+)

is the mean field

φ(z)

ci = ci0 exp(- zief /kBT ) ε

2f

z2 = - r e

(Poisson)

ρe = cizi

i

￥

e

Gouy-Chapman Model

λ

(l << L)

(Neutral)

+ + + + + + +

Counter-ion (-) Co-ion (+)

φ

z

ζ

λD

x = zief /kBT <<1

If in GC model,

exp(x) 1+ x

2f

z2 + f l D

2 = 0

f = z exp(- z/l D) l D

2 = ekBT

4p ( ci0

i

￥ zi

2e2)- 1

Debye-Huckel Model (zeta potential) then For 1M KCl

λD ~ 0.3nm

Thin Debye Layer (TDL) Limit

z

u(z)

φ(0) =z

Debye Layer

µ

2u

z2 + r eE = 0

ε

2f

z2 = - 4pr e

2

z2 u - eEf 4pm

• = 0

u(0) = 0

φ(0) =z

&

u(z) = eE(f - z ) 4pm u( ) = - eEz 4pm

(Helmholtz-Smoluchowski slip BC)

Electroosmotic Speed

E 10 nm 100 micron 10 nm

ue

µ

2u

z2 = 0

u(z = η/ 2) = υε = − εζΕ 4πµ

u(z) = ue

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

c(x,t)

x

ueo

u = ueo + uep

c t + u c x = D

2c

x 2

σ 2 ~ 2Dt = 2DL/u = 2DL/[(m

eo + m ep)E]

N ~ L2 /s 2 = (m

eo + m ep)V /(2D)

dm~ m/ N

Ideal capillary

N ~ 106

~ 30kV)

(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

ue z

Continuity requirement induces a pressure gradient which distorts the flow profile

is reduced Pressure Gradient + = Corrected Flow

What is “Taylor Dispersion” ?

Deff = D + a2umax

2

192D

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 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

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(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, 103-128 Anderson, J.L. & Idol, W.K. Chem. Eng. Commun., 1985, 38, 93-106

distance: velocity: ue = − ες0Ε0

4πµ

a0

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

dx dt = u = ￡ z = (L - x).1+0.x L =1- x L x = L(1- e- t / L ) t +L

(at small times)

x

ζ = 0 ζ =1

L

x

L

t

Application: Elution Time Delays +

• dX

dt = u = - e <z > E 4pm = F(X)

x = X(t)

ζ =ζ0 +e- ax(z 1 - z 0) ζ =ζ0

Best fit of theory to TR data

Ghosal, Anal. Chem., 2002, 74, 771-775

Anderson & Idol Ajdari Ghosal

Geometry Cylindrical symmetry Plane Parallel Amplitude Small Wavelength Long Variable zeta zeta,gap zeta,gap Reference

• Chem. Eng. Comm.
• Vol. 38 1985
• Phys. Rev. Lett. Vol.

75 1995

• Phys. Rev. E Vol. 53

1996

• J. Fluid Mech. Vol. 459 2002

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

ζ1 z 0 = - 0.326

Diffusivity of Mesityl Oxide

WILKE-CHANG FORMULA

Theory vs. Experiment

Ghosal, S., Anal. Chem., 2002, 74, 4198-4203

36

(II) The Coupled Problem

CZE with wall interactions in round capillary

2a0

ζ =ζ0

c(r,x,t)

(in solution)

s(x,t)

(on wall)

ue = - ez 0E 4pm

Length = a0 Time = a0 /ue

Re = uea0 /n Pe = uea0 / D

(less than 1)

(greater than 10)

Flow+Transport Equations

Method of strained co-ordinates

T =et, X =ex

φ = φ(r, X,T )

Asymptotic Solution

Dynamics controlled by slow variables

S.Ghosal JFM 2003 491 285 S.Datta & S.Ghosal Phy. of Fluids (2008) 20 012103

(a0

x <<1)

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.

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