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

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
Electrophoresis Technologies

Debye Layer of counter ions

+

+

+

+

- Ze

+

v

+

+

+

+

+

+

E

Electrophoretic mobility


Electroosmosis
Electroosmosis Technologies

v

Debye

Layer

~10 nm

E

Substrate

= electric potential here

Electroosmotic mobility


Thin debye layer tdl limit
Thin Debye Layer (TDL) Limit Technologies

z

E

&

Debye Layer

(Helmholtz-Smoluchowski slip BC)


Application of tdl to electroosmosis
Application of TDL to Electroosmosis Technologies

E

100 micron

10 nm


Application of tdl to electrophoresis
Application of TDL to electrophoresis Technologies

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



Sandip ghosal mechanical engineering northwestern university

Light from UV source Technologies

Sample Injection Port

Sample (Analyte)

UV detector

Buffer (fixed pH)

+

--

CAPILLARY ZONE ELECTROPHORESIS


Capillary zone electrophoresis cze fundamentals
Capillary Zone Electrophoresis (CZE) Fundamentals Technologies

(for V

Ideal capillary


Sources of band broadening
Sources of Band Broadening Technologies

  • 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
Non uniform zeta-potentials Technologies

is reduced

Pressure Gradient

+

= Corrected Flow

Continuity requirement induces a pressure gradient which distorts the flow profile


What is taylor dispersion
What is “Taylor Dispersion” ? Technologies

G.I. Taylor, 1953, Proc. Royal Soc. A, 219, 186

Aka “Taylor-Aris dispersion” or “Shear-induced dispersion”


Eluted peaks in ce signals
Eluted peaks in CE signals Technologies

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.


Sandip ghosal mechanical engineering northwestern university

THE PROBLEM Technologies

Flow in a channel with variable zeta potential

Dispersion of a band in such a flow



Formulation thin debye layer
Formulation (Thin Debye Layer) Technologies

y

a

x

z

L


Slowly varying channels lubrication limit
Slowly Varying Channels (Lubrication Limit) Technologies

y

x

a

z

L

Asymptotic Expansion in


Lubrication solution
Lubrication Solution Technologies

From solvability conditions on the next higher order equations:

F is a constant (Electric Flux)

Q is a constant (Volume Flux)


Green function
Green Function Technologies

C

D


Green s function
Green’s Function Technologies

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




Application microfluidic circuits
Application: Microfluidic Circuits Technologies

Loop i

Node i

(steady state only)



Application elution time delays
Application: Elution Time Delays Technologies

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)



Sandip ghosal mechanical engineering northwestern university

Best fit of theory to TR data Technologies

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


Sandip ghosal mechanical engineering northwestern university

THE PROBLEM Technologies

Flow in a channel with variable zeta potential

Dispersion of a band in such a flow


Dispersion by eof in a capillary
Dispersion by EOF in a capillary Technologies

(on wall)

(in solution)


Formulation
Formulation Technologies


The evolution of analyte concentration

O Technologies

O

The evolution of analyte concentration


The evolution of analyte concentration1

Loss to wall Technologies

Advection

The evolution of analyte concentration

Solvability Condition


Asymptotic solution
Asymptotic Solution Technologies

Dynamics controlled

by slow variables

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


Sandip ghosal mechanical engineering northwestern university

RUN CZE MOVIE FILES Technologies


Experiments of towns regnier
Experiments of Towns & Regnier Technologies

Anal. Chem. 64, 2473 (1992)

Experiment 2

300 V/cm

15 cm

M.O.

_

+

PEI 200

100 cm

Detector

remove


Theory vs experiment
Theory vs. Experiment Technologies


Conclusion
Conclusion Technologies

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