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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.

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Sandip Ghosal Mechanical Engineering Northwestern University

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  1. 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

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

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

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

  5. Application of TDL to Electroosmosis E 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.2473-2478.

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

  15. Electroosmotic flow with variations in zeta

  16. Formulation (Thin Debye Layer) y a x z L

  17. Slowly Varying Channels (Lubrication Limit) y x a 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 Potential Q Q

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

  24. Application: Flow through porous media E

  25. 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)

  26. Application: Elution Time Delays - +

  27. Best fit of theory to TR data Ghosal, Anal. Chem., 2002, 74, 771-775

  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. O O The evolution of analyte concentration

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

  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. _ + PEI 200 100 cm 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-0330604

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