Diffuse double layer theory

The increased cation concentration in (he DDL develops a countertendency for cation diffusion away from the surface. The diffusion tends to equalize cation concentrations throughout the solution phase. Combining the equations for cation attraction and diffusion yields the Boltzmann equation  

The treatment of double-layer phenomena is straightforward when the change of electric potential with distance from the colloid surface can be adequately estimated. This distribution can be considered to arise from the termination of individual lines of 

Hydroxyoxide surfaces often appear to have concentration-dependent anion exchange capacities, but such behavior can also be explained by the collapse of the double layer at high salt concentrations. Under these conditions, positively charged sites are no longer masked by the DDLs of the predominantly negatively charged soil matrix. 

FIGURE 8.8. Electrical potential and ion concentration between interacting negatively charged platelets. 

In theory, one can calculate the distribution of electric potential within the double layer for any combination of colloid charge, salt concentration, counter-ion valence and interparticle distance. The Boltzmann equation (8.15) can then be used to calculate cation and anion distributions. From such distributions, cation exchange, colloid swelling, and anion repulsion can be inferred but the calculations are complex, tedious, and often only approximate. 

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The surface potential is not accessible by direct experimental measurement it can be calculated from the experimentally determined surface charge (Eqs. 3.1 - 3.3) by Eqs. (3.3a) and (3.3b). The zeta potential, calculated from electrophoretic measurements is typically lower than the surface potential, y, calculated from diffuse double layer theory. The zeta potential reflects the potential difference between the plane of shear and the bulk phase. The distance between the surface and the shear plane cannot be defined rigorously. 

If qM is evaluated experimentally, e.g. from the integration of differential capacity curves, A02 can be calculated using eqn. (44) of the diffuse double layer theory. Figure 3 shows the variation of A02 with... 

A consequence of eqn. (108) is that the true transfer coefficient, at, can be calculated from the apparent transfer coefficient with the double layer correction calculated from the diffuse double layer theory [6]... 

D.C. Grahame, Diffuse double layer theory for electrolytes of unsymmetrical valence types, J. Chem. Phys. 21 (1953) 1054-1060. 

This model is based on the Gouy-Chapman theory (diffuse double-layer theory). The theory states that in the area of the boundary layer between solid and aqueous phase, independently of the surface charge, increased concentrations of cations and anions within a diffuse layer exists because of electrostatic forces. In contrast to the constant-capacitance model, the electrical potential does not change up to a certain distance from the phase boundaries and is not immediately declining in a linear manner (Fig. 14 a). Diffusion counteracts these forces, leading to dilution with increasing distance from the boundary. This relation can be described physically by the Poisson-Boltzmann equation. 

The Diffuse-Double-Layer Theory of Gouy and Chapman... 

It is evident now why the Helmholtz and Gouy-Chapman models were retained. While each alone fails completely when compared with experiment, a simple combination of the two yields good agreement. There is room for improvement and refinement of the theory, but we shall not deal with that here. The model of Stem brings theory and experiment close enough for us to believe that it does describe the real situation at the interface. Moreover, the work of Grahame shows that the diffuse-double-layer theory, used in the proper context (i.e., assuming that the two capacitors are effectively connected in series), yields consistent results and can be considered to be correct, within the limits of the approximations used to derive it. 

In spite of its inherent complexity, it is possible to obtain (j) as a function of potential, by first integrating numerically the relationship between C, and E (cf. Eq. 20G) to obtain q versus E, and then using diffuse-double-layer theory (cf. Eq. 14G) to calculate (j) as a... 

Many methods have been used to determine the value of the PZC on solid electrodes. The one that seems to be most reliable, and relatively easy to perform, is based on diffuse-double-layer theory. Measurement of the capacitance in dilute solutions (C < 0.01 M) should show a minimum at , as seen in Eq. 15G and Fig. 4G. Lowering the concentration yields better defined minima. Modem instrumentation... 

Use diffuse-double-layer theory (Eq. 14G), with ([) replaced by < )... 

Fig. 5G Experimental test of the diffuse-double-layer theory. Concentration of NaF (a) O.OIM (b) O.IOM. Solid lines experimental values of C. Dashed lines C calculated from Eq. 19G. Reprinted with permission from Grahame, J. Am. Chem Soc. 76, 4819. Copyright 1954, the American Chemical Society.

Application of Diffuse-Double-Layer Theory in Plating... 

It is interesting to see how diffuse-double-layer theory can be put to practical use in a field in which progress over the years has been achieved by trial and error, rather than through fundamental research. 

Though a bit artificial, the subdivision into a Stern and a diffuse part has proven its value. One reason Is that the diffuse part can be described with relatively simple analytical equations that become exact at sufficiently large distance from the surface. These two parts play central roles in electrokinetics, colloid stability and many other phenomena where diffuse double layer theory is found to apply well. From the more theoretical side, the diffuse part is characterized by relatively low potentials, so that deviations from ideality are... 

Hyperbolic functions often occur in diffuse double layer theory. Some of their properties are summarized in appendix 2. 

Although the diffuse ions do not contribute to AG , it is nevertheless possible to use diffuse double layer theory to compute this quantity, because in this model the screening Is described, telling us by how much y/° rises with increasing [Pg.267]

Hyperbolic functions are combinations of positive and negative exponentials. They resemble gonlometrlc functions and derive their names from the fact that they describe the coordinates of points on rectangular hyperbolas. They are often encountered In diffuse double layer theory. ... 

We calculate 0 for 7 = 0.1 according to the diffuse double-layer theory (equation 40c in Chapter 9). ffp is calculated from [=A10H ] as follows ... 

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