Diffuse double layer, thickness

The approximation in Eq. 6.72 requires that d be large compared to 1 Ik ( diffuse double layer thickness ) but small compared to the particle dimension. See Chaps. 4 and 6 in R. J. Hunter, op. cit.1 It should be noted in passing that V(r) is, speaking strictly, not a potential energy but is instead a potential of mean force, a statistical thermodynamic quantity (hence the dependence of statistical mechanics of the electrical double layer, Adv. Chem. Phys. 56 141 (1984). 

It is implied that the concentration c is that in the bulk of a solution at distances R significantly greater than the diffuse double layer thickness, 8=1/k, i.e. at R 5=1/k... 

Revil et al. [5] explain that the influence of temperature on the zeta potential is a result of changes in silanol equilibrium, adsorption equilibria, and diffuse double-layer thickness. Their analysis and data are geared towards the geophysical conununity, but the linear profiles they show describing change in zeta potential with temperature is consistent with Venditti et al. [3]. A linear zeta potential—temperature profile is again reported in a more recent paper by Revil et al. [6]. 

The value of the preexponential constant, const = (4k77ze) tanh (ze JAlLT), includes the dependence of a potential on these parameters and on the surface potential, ( )o. At high electrolyte concentrations, c 0.1-1 mol/dm, the double layer is compressed to a thickness on the order of fractions of a nm, while at low electrolyte concentrations, c 10 -10 mol/dm, the diffuse double layer thickness is on the order of tens of nanometers. 

As electrolyte is added to a colloidal suspension, the diffuse double-layer thickness around a particle is compressed so that the range of double-layer... 

The quantity 1 /k is thus the distance at which the potential has reached the 1 je fraction of its value at the surface and coincides with the center of action of the space charge. The plane at a = l//c is therefore taken as the effective thickness of the diffuse double layer. As an example, 1/x = 30 A in the case of 0.01 M uni-univalent electrolyte at 25°C. 

According to the Gouy-Chapman model, the thickness of the diffuse countercharge atmosphere in the medium (diffuse double layer) is characterised by the Debye length k 1, which depends on the electrostatic properties of the... 

In this situation, the tumor cells (or a group of cells viz. microtumor) are the immobile phase and the electroosmotic flow causes the water to move as a plug, the entire velocity gradient being concentrated at the cell surface in a layer of the same order of thickness as the diffuse double layer (Figure 5). In concentrated solutions, the thickness of the diffuse double is quite small (< 10 A) whereas in very dilute solutions (as are indeed... 

Figure 4.7. Variation of the potential with distance (in the diffuse double layer) for different concentrations and thicknesses of the double layer, d. ...

Equation 6.3 is identical to the equation that relates the charge density, voltage difference, and distance of separation of a parallel-plate capacitor. This result indicates that a diffuse double layer at low potentials behaves like a parallel capacitor in which the separation distance between the plates is given by k. This explains why k is called the double layer thickness. 

This thickness is about the same magnitude as the prediction based on the capacitor model (Equation (16)). The diffuse model is clearly superior, however, since it shows how the double layer thickness depends on the concentration and valence of the ions in the solution. 

From equation (7.9) it can be seen that, at low potentials, a diffuse double layer has the same capacity as a parallel plate condenser with a distance 1/k between the plates. It is customary to refer to 1/k (the distance over which the potential decreases by an exponential factor at low potentials) as the thickness of the diffuse double layer. 

When the Bom, double-layer, and van der Waals forces act over distances that are short compared to the diffusion boundary-layer thickness, and when the e forces form an energy hairier, the adsorption and desorption rates may be calculated by lumping the effect of the interactions into a boundary condition on the usual ccm-vective-diffusion equation. This condition takes the form of a first-order, reversible reaction on the collector s surface. The apparent rate constants and equilibrium collector capacity are explicitly related to the interaction profile and are shown to have the Arrhenius form. They do not depend on the collector geometry or flow pattern. 

Measurements of the rate of deposition of particles, suspended in a moving phase, onto a surface also change dramatically with ionic strength (Marshall and Kitchener, 1966 Hull and Kitchener, 1969 Fitzpatrick and Spiel-man, 1973 Clint et al., 1973). This indicates that repulsive double-layer forces are also of importance to the transport rates of particulate solutes. When the interactions act over distances that are small compared to the diffusion boundary-layer thickness, the rate of transport can be computed (Ruckenstein and Prieve, 1973 Spiel-man and Friedlander, 1974) by lumping the interactions into a boundary condition on the usual convective-diffusion equation. This takes die form of an irreversible, first-order reaction on tlie surface. A similar analysis has also been performed for the case of unsteady deposition from stagnant suspensions (Ruckenstein and Prieve, 1975). 

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