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

2. Electrophoresis + + + + - Ze + v + + + + + + E
Debye Layer of counter ions + + + +
- Ze + v + + + + + + E
Electrophoretic mobility

3. Electroosmosis v Electroosmotic mobility E Substrate
Debye
Layer
~10 nm E
Substrate
= electric potential here
Electroosmotic mobility

4. Thin Debye Layer (TDL) Limitz E
&
Debye Layer
(Helmholtz-Smoluchowski slip BC)

5. Application of TDL to Electroosmosis
100 micron
10 nm

6. 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

7. Slab Gel Electrophoresis (SGE)

8. Light from UV source
Sample Injection Port
Sample (Analyte)
UV detector
Buffer (fixed pH) + --
CAPILLARY ZONE ELECTROPHORESIS

9. Capillary Zone Electrophoresis (CZE) Fundamentals
(for V
Ideal capillary

10. 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 ….. )

11. Non uniform zeta-potentials
is reduced
Pressure Gradient +
= Corrected Flow
Continuity requirement induces a pressure gradient which distorts the flow profile

12. What is “Taylor Dispersion” ?
G.I. Taylor, 1953, Proc. Royal Soc. A, 219, 186
Aka “Taylor-Aris dispersion” or “Shear-induced dispersion”

13. 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

14. THE PROBLEM
Flow in a channel with variable zeta potential
Dispersion of a band in such a flow

15. Geometry Cylindrical symmetry Plane Parallel Amplitude Small
Electroosmotic flow with variations in zeta
Anderson & Idol
Ajdari
Lubrication Theory
(Ghosal)
Geometry
Cylindrical symmetry
Plane Parallel
Amplitude
Small
Wavelength
Long
Variable
zeta
zeta,gap
Reference
Chem. Eng. Comm. Vol
Phys. Rev. Lett. Vol
Phys. Rev. E Vol
J. Fluid Mech. Vol

16. Formulation (Thin Debye Layer)x z L

17. Slowly Varying Channels (Lubrication Limit)x z L
Asymptotic Expansion in

18. Lubrication Solution
From solvability conditions on the next higher order equations:
F is a constant (Electric Flux)
Q is a constant (Volume Flux)

19. Green Function C D

20. 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

21. Effective Fluidic Resistance

22. Effective Radius & Zeta PotentialQ Q

23. Application: Microfluidic Circuits
Loop i
Node i
(steady state only)

24. Application: Flow through porous media

25. Application: Elution Time Delays
Towns & Regnier [Anal. Chem. Vol. 64, ]
Experiment 1
Protein +
Mesityl Oxide
EOF
100 cm
Detector 3
(85 cm)
Detector 2
(50 cm)
Detector 1
(20 cm)

26. Application: Elution Time Delays- +

27. Best fit of theory to TR data
Ghosal, Anal. Chem., 2002, 74,

28. THE PROBLEM
Flow in a channel with variable zeta potential
Dispersion of a band in such a flow

29. Dispersion by EOF in a capillary
(on wall)
(in solution)

30. Formulation

31. The evolution of analyte concentration

32. The evolution of analyte concentration
Solvability Condition
Advection
Loss to wall

33. Asymptotic Solution Dynamics controlled by slow variables
Ghosal, J. Fluid Mech. 491, 285 (2003)

34. RUN CZE MOVIE FILES

35. Experiments of Towns & Regnier
Anal. Chem. 64, 2473 (1992)
Experiment 2
300 V/cm
15 cm
M.O. _ +
100 cm
PEI 200
Detector
remove

36. Theory vs. Experiment

37. 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