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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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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
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
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
Application: Microfluidic Circuits Loop i Node i (steady state only)
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)
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)
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)
Experiments of Towns & Regnier Anal. Chem. 64, 2473 (1992) Experiment 2 300 V/cm 15 cm M.O. _ + PEI 200 100 cm Detector remove
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