The mathematics of bio-separations: electroosmotic flow and band broadening in capillary electrophoresis (CE)

Sandip Ghosal

Mechanical Engineering

Northwestern University

Workshop II: Microfluidic Flows in Nature and Microfluidic Technologies

IPAM UCLA April 18 - 22 2006

Electrophoresis

- Ze+

v

E

+++

+

++ +

+

+ +

Debye Layer of counter ions

€

v = μ epEElectrophoretic mobility

Electroosmosis

EDebye Layer~10 nm

Substrate = electric potential hereζ

€

v = μ eoE

v

Electroosmotic mobility

Thin Debye Layer (TDL) Limit

z

€

u(z)

€

φ(0) = ζ

Debye Layer

€

μ∂2u

∂z2+ ρ eE = 0

€

ε∂2φ

∂z2= −4πρ e

€

∂2

∂z2u−

εEφ

4πμ

⎛

⎝ ⎜

⎞

⎠ ⎟= 0

€

u(0) = 0

€

φ(0) = ζ&

€

u(z) =εE(φ −ζ )

4πμ

€

u(∞) = −εEζ

4πμ

(Helmholtz-Smoluchowski slip BC)

E

Application of TDL to Electroosmosis

E

€

u(wall) = −εζE

4πμ

10 nm

100 micron

€

μ∇2u−∇p = 0

€

u = u(wall) = −εζE

4πμ

Application of TDL to electrophoresis

z E

€

∇2φ = 0

€

∂φ∂n

= 0

€

φ ~ −Ez

€

u =εζ ∇φ4πμ

(Solution!)

Satisfies NS

Uniform flow in far field

Satisfies HS bc on particle

Force & Torque free

€

uep = −u(z → ∞) =εζE

4πμ

Morrison, F.A. J. Coll. Int. Sci. 34 (2) 1970

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 /[(μ eo + μ ep )E]

N ~ L2 /σ 2 = (μ eo + μ ep )V /(2D)

δμ ~ μ / N

Ideal capillary

€

N ~ 106

€

~ 30kV )(for V

Sources of Band Broadening

• Finite Debye Layers

• Curved channels

• Variations in channel properties ( , width etc.)

• Joule heating

• Electric conductivity changes

• Etc.

€

ζ

(Opportunities for Applied Mathematics ….. )

Non uniform zeta-potentials

€

ue ∝ ζ

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”

Eluted peaks in CE signals

Reproduced from:

Towns, J.K. & Regnier, F.E. “Impact of Polycation Adsorption onEfficiency and Electroosmotically DrivenTransport in Capillary Electrophoresis”Anal. Chem. 1992, 64, pg.2473-2478.

THE PROBLEM

1. Flow in a channel with variable zeta potential

2. Dispersion of a band in such a flow

Anderson & Idol Ajdari Lubrication Theory

(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

Formulation (Thin Debye Layer)

u(wall)=−εζ (x,y,z)E

4πμ

∇2φ=0E =−∇φ

L

ax

y

z

ρ0(∂tu+u⋅∇u)=−∇p+μ∇2u

∇ ⋅u=0

Slowly Varying Channels (Lubrication Limit)

L

aζ =ζ (x,y,z)

x

y

z

Asymptotic Expansion in

€

a /L <<1

u(x,y,z) =−′ p (x)μ

up +εE(x)4πμ

Ψ

∇H2up =−1 up(boundary)=0

∇H2Ψ =0 Ψ(boundary)=−ζ

Lubrication Solution

u(x,y,z) =−′ p (x)μ

up +εE(x)4πμ

Ψ

∇H2up =−1 up(boundary)=0

∇H2Ψ =0 Ψ(boundary)=−ζ

From solvability conditions on the next higher order equations:

E(x)A(x) =F F is a constant (Electric Flux)

−u pμ

A(x) ′ p (x)+εFΨ 4πμ

=Q Q is a constant (Volume Flux)

Green Function

∇H2up =−1 up(C)=0

∇H2Ψ =0 Ψ(C)=−ζ

D

C

up(x,y,z) =14π

G(y,z, ′ y , ′ z ;x)d ′ y d ′ z D∫

Ψ(x,y,z) =14π

ζ(x, ′ y , ′ z )ˆ n ⋅ ′ ∇ HC∫ Gd ′ s

ˆ n

Green’s Function

∇H2up =−1 up(C)=0 ∇H

2Ψ =0 Ψ(C)=−ζ

1. Circular 2. Rectangular3. Parallel Plates4. Elliptical5. Sector of Circle6. Curvilinear Rectangle7. Circular Annulus (concentric)8. Circular Annulus (non-concentric)9. Elliptical Annulus (concentric)

Trapezoidal = limiting case of 6

Effective Fluidic Resistance

−u pμ

A(x) ′ p (x)+εFΨ 4πμ

=Q

Q =pa −pb

8μLπa*

4 −εζ*

4πμπa*

2 φa −φb

L

a* =8

π <u p−1A−1 >

⎡

⎣ ⎢ ⎤

⎦ ⎥

1/4

ς* =−18π

<Ψ u p−1A−1 >

<A−1 ><u p−1A−1 >1/ 2

φa −φb = E(x)dxa

b

∫ =F

A(x)a

b

∫ dx=FL <A−1 >

Effective Radius & Zeta Potential

ζ =ζ (x,y,z)pa

pb

φaφb

Q

pa

φa

pbφbQ

€

2a*

€

ζ =ζ*

Application: Microfluidic Circuits

Node i

Loop i

(Δφi −IiR)ii

∑ =0

Iii

∑ =0

Qii

∑ =0

(Qii

∑ −α iΔpi −βiΔφi) =0

(steady state only)

Application: Flow through porous media

E

Q =αΔp+βΔφ

Application: Elution Time Delays

100 cm

EOF

Detector 3 (85 cm)

Detector 2 (50 cm)

Detector 1 (20 cm)

Protein +Mesityl Oxide

Experiment 1

Towns & Regnier [Anal. Chem. Vol. 64, 2473 1992]

Application: Elution Time Delays

+ -

ζ0

ζ1

€

dX

dt= u = −

ε < ζ > E

4πμ= F(X)

€

x = X(t)

€

ζ =ζ0 + e−αx (ζ 1 −ζ 0 )

€

ζ =ζ0

Best fit of theory to TR data

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

THE PROBLEM

1. Flow in a channel with variable zeta potential

2. Dispersion of a band in such a flow

Dispersion by EOF in a capillary

€

2a0

€

ζ =ζ0

€

c(r, x, t)(in solution)

€

s(x, t)(on wall)

€

ue = −εζE

4πμ

€

Length = a0

Time = a0 /ue

T =εt X =εx

r€

Re = uea0 /ν

Pe = uea0 /D

Formulation

∂tc+(u+uep̂ x )⋅∇c =Pe−1∇2c

−Pe-1(∂rc)r =1 =∂ts(x,t)

Re(∂tu+u⋅∇u) =−∇p+∇2u

∇ ⋅u=0ur=1 =ˆ x ζ

∂ts= f(s,cw,L ) =Kacw(sm −s) −Kds

ζ =g(s,L ) =1−βs

cw ≡c(1,x,t)

The evolution of analyte concentration

−Pe-1 ∂c∂r

⎛ ⎝

⎞ ⎠ r=1

=ε∂Ts

1r

∂∂r

r∂c∂r

⎛ ⎝

⎞ ⎠

=εPe∂Tc+u∂Xc+uep∂Xc+V∂rc( )−ε2∂XXcO

O

⇒ c=c(X,T)

The evolution of analyte concentration

−Pe-1 ∂c1

∂r⎛ ⎝

⎞ ⎠ r=1

=∂Ts0

1r

∂∂r

r∂c1

∂r⎛ ⎝

⎞ ⎠

=Pe∂Tc0 +u0∂Xc0 +uep∂Xc0( )

Solvability Condition

LHS=1π

2πr (LHS0

1

∫ ) dr=−2 Pe ∂Ts0 =RHS

∂Tc0 + uep+ ζ( )∂Xc0 =−2∂Ts0

Advection Loss to wall

c =c+Pe4

(1−2r2)∂ts+Pe24

(2−6r2 +3r4)(ζ −u)∂xc

∂tc+(u+uep)∂xc =∂x(Deff∂xc) +g

∂ts= f(s,cw,....)

cw =c−Pe4

∂ts−Pe24

(ζ −u)∂xc

Deff =Pe-1 +Pe48

ζ −u( )2

g=−2∂ts+Pe12

∂x ζ −u( )∂ts[ ]

u=Δp8L

+ ζ −18

Red ζ

dt−

148

Reddt

ΔpL

⎛ ⎝

⎞ ⎠

Asymptotic Solution

Dynamics controlledby slow variables

Ghosal, J. Fluid Mech. 491, 285 (2003)

RUN CZE MOVIE FILES

Experiments of Towns & Regnier

+

remove100 cm

15 cm300 V/cm

PEI 200

_

Detector

Experiment 2Anal. Chem. 64, 2473 (1992)

M.O.

Theory vs. Experiment

N =X2 /σ 2

Conclusion

The problem of EOF in a channel of general geometry and variable zeta-potential was solved in the lubrication approx.

1. Full analytical solution requires only a knowledge of the Green’s function for the cross-sectional shape.

2. Volume flux of fluid through any such channel can be described completely in terms of the effective radius and zeta potential.

The problem of band broadening in CZE due to wall interactions was considered. By exploiting the multiscale nature of the problem an asymptotic theory was developed that provides:

1. One dimensional reduced equations describing variations of analyte concentration.

2. The predictions are consistent with numerical calculations and existing experimental results.

Acknowledgement: supported by the NSF under grant CTS-0330604