Small Debye Length

1 = 0 for most of the capillary cross section in the limit of small Debye length, that is, for small A value. Therefore, it is necessary to solve for i/r only near the pore wall, where (r - a) A[). We can neglect the curvature effect, and the equation for y/ is exactly same as that for (f) in ID case  

Note For a thin diffuse layer, that is, a/Ap - oo, the velocity distribution equation reduces to 

This velocity expression is identical to Helmholtz-Smoluchowski equation for zero pressure gradient derived previously. 

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There will always be a charge transfer resistance (R i) associated with the ion exchange across the interface. Where there are very small Debye lengths in each phase (compared with the size of an ion) the exchange current I o can be evaluated from the relationship... 

When an+ 1 ions traverse the sheath in a short time compared to the field oscillations. Under this condition, an ion traversing the sheath experiences the sheath voltage prevailing at the time the ion entered the sheath. In the absence of collisions, the lED function will reflect precisely the variation of the sheath voltage with time. This quasi steady-state condition of cut+ 1 is satisfied for low RF frequencies or short ion transit times, i.e., thin sheaths (low sheath voltage or small Debye length). 

When treating the small Debye length case, we shall frequently use surface charge density and potential interchangeably, relating them through Eq. (7.1.12). 

The order of magnitude of the surface conductance may be estimated in the small Debye length limit by noting that approximately... 

Since for small Debye length the situation is completely complementary to the electroosmotic case, we may apply the result of Eq. (7.4.5), identifying Ap as the force per unit particle cross-sectional area exerted by the fluid on the particle. Assuming Stokes flow, we use STTixaU for the force on a single particle if there are n particles per unit volume, then the total force per unit volume is taken to be nF. With the potential drop measured over the suspension height H, it readily follows that... 

For small Debye length the particle fall speed is not affected by the induced electric field (Newman 1991). Therefore,... 

The thermal energy of a particle scales as kT, while with a — the van der Waals attractive energy scales as the Hamaker constant A (Eq. 8.1.20). Finally from Eq. (8.1.15), the energy of repulsion is seen to scale as ae for small zeta potentials, say f less than the Nernst potential at standard temperature (26 mV), and for small Debye length to radius ratio. 

For the limiting case of small Debye length ratio, shown schematically in Fig. 6.5.3A, (/t = 0 at the center and the solution is electrically neutral there, so the approximation = c = Cq employed in Eq. (6.5.12) is, in this case, exact. Moreover,... 

Rederive the small Debye length result for the electroosmotic velocity in a long circular capillary for a symmetrical dilute binary electrolyte (Eq. 6.5.23) with a constant surface charge density in place of constant surface (zeta) potential. Show that the appropriate boundary condition at the tube... 

This result reflects a noticeable increase in Wij stability ratios in the course of aggregation compared to the initial value W . The data demonstrate also that enhancement of the electrostatic repulsion, related to the increase in the Debye length Id or C potential, results in a retardation of the growth. Good correspondence between theory and experiment was observed at small Debye length, Aj, < 3-5 nm, in the so-called colloid regime. However, the effect of electrolyte on the PEC conformation became important at larger Xd and simulation failed to describe experiments [154]. 

The distribution of f/ which was obtained numerically in Figure 6.11 can be obtained analytically for small Debye length. It should be noted from the plot of the numerical solution that... 

Debye length in solid electrolytes. The Debye length Id describes the shielding of an electric field by the charge carriers. The electrostatic potential drops to 1/e of its value within Id- In contrast to diluted media with small concentrations of ions, such as liquid electrolytes, solid electrolytes have very small Debye lengths. 

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