Electroosmotic velocity profile

Figure 2 Electroosmotic velocity profiles in narrow channels with r/8 values of 1, 2, 5, 10, and 50, respectively (lower to upper curves).

Fig. 1 Electroosmotic velocity profiles for various values keeping a fixed value of kR = 10. The velocity profiles...

A pbotobleacbed marker in a microcbaimel is shown inset of a measured electroosmotic velocity profile. The contrast of the inset image has been adjusted for clarity (Reprinted frinn [4])... 

Sinton D, Li D (2003) Electroosmotic velocity profiles in microchannels. Coll Surf A 222 273-283... 

Figure 23-16 (a) Electric double layer is created by negative silica surface and excess cations in the diffuse part of the double layer in solution near the wall. The wall is negative and the diffuse part of the double layer is positive, (b) Predominance of cations in diffuse part of the double layer produces net electroosmotic flow toward the cathode when an external field is applied, (c) Electroosmotic velocity profile is uniform over more than 99.9% of the cross section of the capillary. A capillary is required to maintain constant temperature in the liquid. Temperature variation in larger-diameter tubes causes bands to broaden. 

The flow profiles of electrodriven and pressure driven separations are illustrated in Figure 9.2. Electroosmotic flow, since it originates near the capillary walls, is characterized by a flat flow profile. A laminar profile is observed in pressure-driven systems. In pressure-driven flow systems, the highest velocities are reached in the center of the flow channels, while the lowest velocities are attained near the column walls. Since a zone of analyte-distributing events across the flow conduit has different velocities across a laminar profile, band broadening results as the analyte zone is transferred through the conduit. The flat electroosmotic flow profile created in electrodriven separations is a principal advantage of capillary electrophoretic techniques and results in extremely efficient separations. 

Fig. 9.4. Flow velocity profiles vs. channel diameter for pressure driven flow (top) and electroosmotically driven flow (bottom). Reproduced from [33], with permission.

The external electric field is in the direction of the pore axis. The particle is driven to move by the imposed electric field, the electroosmotic flow, and the Brownian force due to thermal fluctuation of the solvent molecules. Unlike the usual electroosmotic flow in an open slit, the fluid velocity profile is no longer uniform because a pressure gradient is built up due to the presence of the closed end. The probability of the particle position is obtained by solving the Fokker-Planck equation. The penetration depth is found to be dependent upon the Peclet number, which is a measure of significance of the convective electroosmotic flow relative to the Brownian diffusion, and the Damkohler number, which is a ratio of the characteristic diffusion-to-deposition times. 

Luo and Andrade have reexamined [66] the potential of CEC by comparing the effect of the conclusions of the Rice-Whitehead theory [35] of doublelayer overlap on the determination of minimum dp with those which result from more recent treatments of the velocity profile in electroosmotic flow. They concluded that, for ionic strength <10 mM, the particle size can again be less than 1 pm, and that plate numbers up to 1 x 106 should be theoretically possible. An obstacle to the realization of such efficiencies in CEC is, however, the consequence of the recognition [6] by Giddings that there is no satisfactory mathemat-... 

Representation of electroosmotic flow of a fluid (e.g., water) in a glass capillary. Only ions near the glass walls are shown. Note that the velocity profile here is flatter (so-called plug flow ) than the parabolic profile seen with an external pressure drop along a tube. 

Speed of plane kinematic wave Speed of kinematic shock moving up from container bottom True electrophoretic velocity in electrophoresis cell Speed with which a point with ionic fraction Xg in solution moves Electroosmotic velocity in electrophoresis cell Speed of ion exchange zone front Liquid velocity in electrophoresis cell Maximum fluid velocity at center of circular or straight channel with fully developed velocity profile, Eq. (4.2.14) Velocity at free surface of... 

A last variant we mention is capillary zone electrophoresis (Gordon et al. 1988). It employs an electroosmotically driven flow in a capillary, arising from an electric field applied parallel to the capillary, which is charged when in contact with an aqueous solution (Section 6.5). The flow has a nearly flat velocity profile (Fig. 6.5.1), thereby minimizing broadening due to Taylor dispersion of the electrophoretically separated solute bands. 

This formula for the electroosmotic velocity past a plane charged surface is known as the Helmholtz-Smoluchowski equation. Note that within this picture, where the double layer thickness is very small compared with the characteristic length, say alX t> 100, the fluid moves as in plug flow. Thus the velocity slips at the wall that is, it goes from U to zero discontinuously. For a finite-thickness diffuse layer the actual velocity profile has a behavior similar to that shown in Fig. 6.5.1, where the velocity drops continuously across the layer to zero at the wall. The constant electroosmotic velocity therefore represents the velocity at the edge of the diffuse layer. A typical zeta potential is about 0.1 V. Thus for = 10 V m" with viscosity that of water, the electroosmotic velocity U 10 " ms, a very small value. 

Velocity profiles across the capillary have a Poisseuille shaped flow and the expression predicts that the electroosmotic coefficient of permeability should vary with the square of the radius. In practice, it is found generally that this law is not as satisfactory as the Helmholtz-Smoluchowski approach for predicting electroosmotic behavior in soils. The failure of small pore theory may be because most clays have an aggregate structure with the flow determined by the larger pores [6], Another theoretical approach is referred to as the Spiegler Friction theory [25,6]. Its assumption, that the medium for electroosmosis is a perfect permselective membrane, is obviously not valid for soils, where the pore fluid comprises dilute electrol d e. An expression is derived for the net electroosmotic flow, Q, in moles/Faraday,... 

Note that in the small double layer thickness approximation, the character of motion of liquid in the capillary is that of plug flow with the velocity U. If the thickness of the double layer is small, but finite, the velocity profile looks like the one shown in Fig. 7.9. For the characteristic values C = 0.1 V, = 10 m, we have for water U = m/s. Thus, electroosmotic motion has a very low velocity. 

In microfluidic systems, due to the very thin EDL compared to the microchannel dimension, the electroosmotic velocity distribution and EDL potential profile inside the EDL region become insignificant. Thus we do not need to solve the Poisson equation together with the Boltzmann distribution. Instead the electroosmotic flow velocity at the edge of the EDL (i.e., from the diffuse layer to the bulk phase) is given by the Smoluchowski equation [1], expressed as... 

---