Taylor dispersion in cyclic electric field-flow fractionation

Electric field-flow fractionation (EFFF), which combines a constant lateral electric field with an axial pressure-driven flow, can separate polyelectrolytes of different sizes in free solution. In large EFFF devices, fields are required to accomplish sharp resolution and such large fields can effectively immobilize the colloidal particles at the wall [Caldwell , Science 176, 269 (1972)]. Furthermore, particles with the same values of D/u(y)(e) cannot be separated by EFFF, where D is the molecular diffusivity and u(y)(e) is the electric field driven velocity on the lateral direction. It has been suggested that some of the difficulties associated with EFFF could be eliminated by using a cyclic transverse electric field [J. C. Giddings, Anal. Chem. 58, 2052 (1986)]. This technique in which a transverse cyclic electric field is combined with axial Poiseuille flow is called cyclic electric field-flow fractionation (CEFFF). In this paper, a multiple time scale analysis and regular expansions in the aspect ratio are used to determine the mean velocity and the dispersion coefficient of molecules in CEFFF. This problem was first studied for the case of large Peclet number and square wave electric fields by Shapiro and Brenner [M. Shapiro and H. Brenner, Phys. Fluids A 2, 1731 (1990)]. We extend the results of their study by developing results for both square wave and sinusoidal fields that are valid for all Peclet numbers. The dimensionless mean velocity (U-*) depends on the dimensionless frequency (Omega) and the product of the Peclet number (Pe) and the dimensionless amplitude of the lateral velocity driven by the applied field (R). The convective contribution to the dispersion coefficient is of the form Pe(2) f(Pe R,Omega). We also obtain the expressions for the mean velocity and the dispersion coefficient in the limit of small Omega. In this limit the results are essentially the time average of the unidirectional-EFFF results. Also, the mean velocities and dispersion are calculated for the case of square wave electric field and these results are compared with the large Peclet asymptotic results that were obtained by Shapiro and Brenner [M. Shapiro and H. Brenner, Phys. Fluids A 2, 1744 (1990)]. The results of mean velocity and dispersion coefficient for the square wave are also compared with those for sinusoidal fields.