毛细管电泳中的电渗流和区带展宽课件

The mathematics of bio-separations: electroosmotic flow and band broadening in capillary electrophoresis (CE) Sandip Ghosal Mechanical Engineering Northwestern University Workshop II: Microfluidic Flows in Nature and Microfluidic Technologies IPAM UCLA April 18 - 22 2006 Electrophoresis - Ze + v E + + + + + + + + + + Debye Layer of counter ions Electrophoretic mobility Electroosmosis E Debye Layer 10 nm Substrate = electric potential here v Electroosmotic mobility Thin Debye Layer (TDL) Limit z Debye Layer & (Helmholtz-Smoluchowski slip BC) E Application of TDL to Electroosmosis E 10 nm 100 micron 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) Sample Injection Port Sample (Analyte) Buffer (fixed pH) + - UV detector Light from UV source CAPILLARY ZONE ELECTROPHORESIS Capillary Zone Electrophoresis (CZE) Fundamentals Ideal capillary (for V 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 Continuity requirement induces a pressure gradient which distorts the flow profile is reduced Pressure Gradient + = Corrected Flow 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 1. Flow in a channel with variable zeta potential 2. Dispersion of a band in such a flow Anderson & Idol Ajdari Lubrication Theory (Ghosal) GeometryCylindrical symmetry Plane Parallel AmplitudeSmall WavelengthLong Variablezetazeta,gapzeta,gap Reference Chem. Eng. Comm. Vol. 38 1985 Phys. Rev. Lett. Vol. 75 1995 Phys. Rev. E Vol. 53 1996 J. Fluid Mech. Vol. 459 2002 Electroosmotic flow with variations in zeta Formulation (Thin Debye Layer) L a x y z Slowly Varying Channels (Lubrication Limit) L ax y z 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 D C Greens 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 Node i Loop i (steady state only) Application: Flow through porous media E Application: Elution Time Delays 100 cm EOF Detector 3 (85 cm) Detector 2 (50 cm) Detector 1 (20 cm) Protein + Mesityl Oxide Experiment 1 Towns & Regnier Anal. Chem. Vol. 64, 2473 1992 Application: Elution Time Delays +- Best fit of theory to TR data Ghosal, Anal. Chem., 2002, 74, 771-775 THE PROBLEM 1. Flow in a channel with variable zeta potential 2. Dispersion of a band in such a flow Dispersion by EOF in a capillary (in solution) (on wall) Formulation The evolution of analyte concentration O O The evolution of analyte concentration Solvability Condition Advection Loss to wall Asymptotic Solution Dynamics controlled by slow variables Ghosal, J. Fluid Mech. 491, 285 (2003) RUN CZE MOVIE FILES Experiments of Towns & Regnier + remove 100 cm 15 cm 300 V/cm PEI 200 _ Detector Experiment 2 Anal. Chem. 64, 2473 (1992) M.O. 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 Greens 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 dimensi