Workshop II: Microfluidic Flows in Nature and Microfluidic Technologies
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Workshop II: Microfluidic Flows in Nature and Microfluidic Technologies IPAM UCLA April 18 - 22 2006. The mathematics of bio-separations: electroosmotic flow and band broadening in capillary electrophoresis (CE). Sandip Ghosal Mechanical Engineering Northwestern University.

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Sandip Ghosal Mechanical Engineering Northwestern University

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Workshop II: Microfluidic Flows in Nature and Microfluidic Technologies

IPAM UCLA April 18 - 22 2006

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

Sandip Ghosal

Mechanical Engineering

Northwestern University


Electrophoresis

Debye Layer of counter ions

+

+

+

+

- Ze

+

v

+

+

+

+

+

+

E

Electrophoretic mobility


Electroosmosis

v

Debye

Layer

~10 nm

E

Substrate

= electric potential here

Electroosmotic mobility


Thin Debye Layer (TDL) Limit

z

E

&

Debye Layer

(Helmholtz-Smoluchowski slip BC)


Application of TDL to Electroosmosis

E

100 micron

10 nm


Application of TDL to electrophoresis

z E

(Solution!)

Satisfies NS

Uniform flow in far field

Satisfies HS bc on particle

Force & Torque free

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


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

(for V

Ideal capillary


Sources of Band Broadening

  • Finite Debye Layers

  • Curved channels

  • Variations in channel properties ( , width etc.)

  • Joule heating

  • Electric conductivity changes

  • Etc.

(Opportunities for Applied Mathematics ….. )


Non uniform zeta-potentials

is reduced

Pressure Gradient

+

= Corrected Flow

Continuity requirement induces a pressure gradient which distorts the flow profile


What is “Taylor Dispersion” ?

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 on

Efficiency and Electroosmotically Driven

Transport in Capillary Electrophoresis”

Anal. Chem. 1992, 64, pg.2473-2478.


THE PROBLEM

Flow in a channel with variable zeta potential

Dispersion of a band in such a flow


Electroosmotic flow with variations in zeta


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)


Green Function

C

D


Green’s Function

1. Circular

2. Rectangular

3. Parallel Plates

4. Elliptical

5. Sector of Circle

6. Curvilinear Rectangle

7. Circular Annulus (concentric)

8. Circular Annulus (non-concentric)

9. Elliptical Annulus (concentric)

Trapezoidal = limiting case of 6


Effective Fluidic Resistance


Effective Radius & Zeta Potential

Q

Q


Application: Microfluidic Circuits

Loop i

Node i

(steady state only)


Application: Flow through porous media

E


Application: Elution Time Delays

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

Experiment 1

Protein

+

Mesityl Oxide

EOF

100 cm

Detector 3

(85 cm)

Detector 2

(50 cm)

Detector 1

(20 cm)


Application: Elution Time Delays

-

+


Best fit of theory to TR data

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


THE PROBLEM

Flow in a channel with variable zeta potential

Dispersion of a band in such a flow


Dispersion by EOF in a capillary

(on wall)

(in solution)


Formulation


O

O

The evolution of analyte concentration


Loss to wall

Advection

The evolution of analyte concentration

Solvability Condition


Asymptotic Solution

Dynamics controlled

by slow variables

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


RUN CZE MOVIE FILES


Experiments of Towns & Regnier

Anal. Chem. 64, 2473 (1992)

Experiment 2

300 V/cm

15 cm

M.O.

_

+

PEI 200

100 cm

Detector

remove


Theory vs. Experiment


Conclusion

The problem of EOF in a channel of general geometry and variable zeta-potential was

solved in the lubrication approx.

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

  • 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:

  • One dimensional reduced equations describing variations of analyte concentration.

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

Acknowledgement: supported by the NSF under grant CTS-0330604


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