Debye length ratio

Equation (6.5.12), subject to the boundary conditions indicated, can be solved analytically only under limiting conditions on the Debye length described below. In general, it must be solved numerically. Figure 6.5.2 gives numerical solutions obtained by Gross Osterle (1968) for a constant surface potential 2-79 for various values of the Debye length ratio A It can be seen that for A " <0.1 the potential is zero over most of the capillary cross section, whereas for A " > 10 the potential is nearly constant over the cross section. 

Figure 6.5.2 Dimensionless potential distribution across a cylindrical capillary for different values of the Debye length ratio A " and a constant surface potential = C = 2.79 (after Gross 8c Osterle 1968).

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,... 

Calculate the Debye length and the Debye length ratio X /a. What is the pressure gradient developed across the porous plug at any instant of time ... 

For a general case where the particle diameter-to-Debye length ratio is large (a the mobility equation becomes ... 

Thermal diffusivity Temperature sensitivity Temperature difference Thickness of tube Aspect ratio, relation of Cp/Cy Fluid dielectric constant Wall zeta potential Dimensionless temperature Friction factor, Debye length Mean free path Dynamic viscosity Kinematic viscosity Bejan number Density... 

The appropriate model of the interface for e.g. Pt/LiCFaSOa-PEO depends on the concentration of charged species in the PEO. When the salt PEO ratio is less than 1 10 the Debye length will also be less than the size of a mobile charge. In this case again the Helmholtz model of the double layer will be appropriate. 

Here the same scaling as in (4.1.1), (4.1.2) has been employed. Once again, e is the square of the ratio of the Debye length to the dimensional thickness of the compartment (L). 

A satisfactory explanation of the experimental results in terms of the existing theories cannot be provided. An interpretation using a scaling of the activity coefficient vs. the ratio of the Debye length 1D to the contour length L has been discussed in [38] for narrow distributed NaPSS standards. This interpretation,... 

Figure 40. Numerically calculated impedance for ion blockage. The variation of the defect concentration leads to the transition from Warburg to a pure semicircular behavior. The impedances are normalized such that the points of highest frequency coincide. (For the mobility ratio, ratio of thickness to Debye length and charge numbers the following values are assumed = lOultJi, UA = 101, = l = -z )m...

The ratio of plate half-width yl to Debye length X is ... 

Fig. 11 shows the plot of Nusselt number, eq.(lOO), as function of heat generation due to resistance S and the ratio of tube radius to Debye length Z. 

The evidence of electrokinetic salt rejection by a microporous inorganic material was given by Jacazio et al. [63] based on the model of Osterle [26-28]. Experiments were carried out on the salt rejecting properties of compacted clay through which saline solutions were forced under high pressures. In accordance with the model the performance of the porous material was shown to depend on three main parameters the ratio of the Debye length to effective pore... 

Electrically charged particles in aqueous media are surroimded by ions of opposite charge (counterions) and electrolyte ions, namely, the electrical double layer. The quantity He represents the energy of repulsion caused by the interaction of the electrical double layers. The expression for He depends on the ratio between the particle radius and the thickness of the electrical double layer, k, called the Debye length. For K.a > 5 (Quemada and Berli, 2002) ... 

In liquids of low dielectric constant, dispersants tend not to form ionic species in solution, but can form ions in adsorbed films on particle surfaces where acid-base interactions and proton transfer occurs between the particle surface and the dispersant. Particle potentials develop when adsorbed dispersant ions desorb into the organic medium where they become the counter-ions. Zeta-potentials well over a hundred millivolts result from the stronger acid-base interactions. Debye lengths in concentrated dispersions are typically 5 to 20 nm, and the DLVO energy barriers, of ten exceed 25 kT with stability ratios of 10° or more. 

Notice that the ratio between Debye length and rod separation dtoi can be written as... 

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. 

We next calculate how the electroosmotic flow and potential change in a long capillary for different ratios of Debye length to radius. We shall allow for pressure gradients, but the mean velocity and tube radius are assumed sufficiently small that inertia effects can be neglected. The dilute electrolyte solution in the capillary is taken to be binary. 

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