Diffuse double layer, equation

Yeung AT. (1992). Diffuse double-layer equations in SI units. ASCE Journal of Geotechnical Engineering 118(12) 2000-2005. 

Derive the general equation for the differential capacity of the diffuse double layer from the Gouy-Chapman equations. Make a plot of surface charge density tr versus this capacity. Show under what conditions your expressions reduce to the simple Helmholtz formula of Eq. V-17. 

The same system has been studied previously by Boguslavsky et al. [29], who also used the drop weight method. While qualitatively the same behavior was observed over the broad concentration range up to the solubility limit, the data were fitted to a Frumkin isotherm, i.e., the ions were supposed to be specifically adsorbed as the interfacial ion pair [29]. The equation of the Frumkin-type isotherm was derived by Krylov et al. [31], on assuming that the electrolyte concentration in each phase is high, so that the potential difference across the diffuse double layer can be neglected. 

Although a family of OgS - Jig8 values are allowed under Equation 7 the actual equilibrium state of the oxide/solution interface will be determined by the dissociation of the surface groups and the properties of the electrolyte or the diffuse double layer near the surface. For surfaces that develop surface charges by different mechanisms such as for semiconductor, there will be an equation of state or charge-potential relationship that is analogous to Equation 7 which characterizes the electrical response of the surface. 

We recall that the first integral in Equation 23a represents the change in electrical free energy in forming the diffuse double layer. This contribution to f, the free energy of formation of the charged interface, is positive and hence represents an unfavourable component which opposes the formation of the charged interface. 

As mentioned above, a substantial part of the electrical charge of the micelle surface has been shown to be neutralized by the association of the counter ions with the micelle. In the calculation based on Equation 12, however, the loss in entropy arising from this counter ion association is not taken into account. This is by no means insignificant in comparison to of Equation 12 (4). A major part of the counter ions are condensed on the ionic micelle surface and counteract the electrical energy assigned to the amphiphilic ions on the micellar surface. The minor part of the counter ions,in the diffuse double layer, are also restricted to the vicinity of the micellar surface. 

The main, currently used, surface complexation models (SCMs) are the constant capacitance, the diffuse double layer (DDL) or two layer, the triple layer, the four layer and the CD-MUSIC models. These models differ mainly in their descriptions of the electrical double layer at the oxide/solution interface and, in particular, in the locations of the various adsorbing species. As a result, the electrostatic equations which are used to relate surface potential to surface charge, i. e. the way the free energy of adsorption is divided into its chemical and electrostatic components, are different for each model. A further difference is the method by which the weakly bound (non specifically adsorbing see below) ions are treated. The CD-MUSIC model differs from all the others in that it attempts to take into account the nature and arrangement of the surface functional groups of the adsorbent. These models, which are fully described in a number of reviews (Westall and Hohl, 1980 Westall, 1986, 1987 James and Parks, 1982 Sparks, 1986 Schindler and Stumm, 1987 Davis and Kent, 1990 Hiemstra and Van Riemsdijk, 1996 Venema et al., 1996) are summarised here. 

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. 

So far, we have used the Maxwell equations of electrostatics to determine the distribution of ions in solution around an isolated, charged, flat surface. This distribution must be the equilibrium one. Hence, when a second snrface, also similarly charged, is brought close, the two surfaces will see each other as soon as their diffuse double-layers overlap. The ion densities aronnd each surface will then be altered from their equilibrinm valne and this will lead to an increase in energy and a repulsive force between the snrfaces. This situation is illustrated schematically in Fignre 6.12 for non-interacting and interacting flat snrfaces. 

One of the simplest equations is obtained using the Debye-Hiickel approximation (for low potentials) and the superposition principle. The latter assumes that the unperturbed potential near a charged surface can be simply added to that potential due to the other (unperturbed) surface. Thus, for the example shown in the Figure 6.12, it follows that /m = 2 /d/2- This is precisely valid for Coulomb-type interactions, where the potential at any point can be calculated from the potentials produced by each fixed charge, individually. However, the Poisson-Boltzmann equation is non-linear (this has to do with the fact that in the diffuse double-layer the ions are not fixed but move because of their kinetic energy) and so this is formally not correct although it still offers a useful approximation. 

This equation was derived above for the movement of a liquid through a stationary solid phase. Its application here to the movement of colloidal particles under experimental conditions that render the liquid medium immobile implies that the solid particle is large compared with the dimensions of the diffuse double layer k 1. It is customary to term this movement of the solid phase electrophoresis. The phenomenon is observed with particles suspended in a liquid (Fig. 6.139). 

The Debye-Hiickel approximation to the diffuse double-layer problem produces a number of relatively simple equations that introduce a variety of double-layer topics as well as a number of qualitative generalizations. In order to extend the range of the quantitative relationships, however, it is necessary to return to the Poisson-Boltzmann equation and the unrestricted Gouy-Chapman theory, which we do in Section 11.6. 

Equation (62) describes the variation in potential with distance from the surface for a diffuse double layer without the simplifying assumption of low potentials. It is obviously far less easy to gain a feeling for this relationship than for the low-potential case. Anticipation of this fact is why so much attention was devoted to the Debye-Hiickel approximation in the first place. Note that Equation (62) may be written... 

Outside the Stern surface the double layer continues to be described by Equation (63) or one of its approximations. The only modifications of the analysis of the diffuse double layer required by the introduction of the Stern surface are that x be measured from 6 rather than from the wall and that 06 be used instead of 0O as the potential at the inner boundary of the diffuse layer. 

The possible usefulness of this relationship — which is known as the Hiickel equation — should not be overlooked. Throughout Chapter 11 we were concerned with the potential surrounding a charged particle. Equation (11.1) provides a way of evaluating the potential at the surface, i/ o, in terms of the concentration of potential-determining ions. Owing to ion adsorption in the Stern layer, this may not be the appropriate value to use for the potential at the inner limit of the diffuse double layer. Although f is not necessarily identical to i/ o, it is nevertheless a quantity of considerable interest. 

Equation (69) may also be integrated analytically. Although we do not consider the actual solutions, which are rather complex, Figure 12.8 shows graphically the results of these integrations for water at 25°C, assuming / = 10-l5 V-2 m2. The abscissa shows values of j/0, the potential at the inner limit of the diffuse double layer, with rj /e plotted on the ordinate. It must be remembered that this last quantity equals f according to Equation (39) —which we... 

It is postulated that one of the ions of the adsorbed 1 1 electrolyte is surface active and that it forms an ionized monolayer at the solid/liquid interface. All counterions are assumed located in the diffuse double layer (no specific adsorption). Similions are negatively adsorbed in the diffuse double layer. Since the surface-containing region must be electrically neutral, the total moles of electrolyte adsorbed, n2a, equals the total moles of counterions in the diffuse double layer which must be equal to the sum of the moles of similions in the diffuse double layer and the charged surface, A[Pg.158]

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. 

Keywords Navier-Stokes equation, diffuse double-layer, Darcy s law... 

When Di > i>2, the effective Debye—Hiickel length X (which now depends on ip(x)) is larger than that obtained for the Poisson—Boltzmann equation. Consequently, the diffuse double layer is larger in the vicinity of a charged surface, as predicted earlier.4 7 9 However, when V2 > Vi (small counterions), X < X and the diffuse double layer is compressed. The effect is proportional to the ionic strength and is, in general, small for typical electrolyte concentrations, since n(v — v[Pg.337]

The Poisson-Boltzman (P-B) equation commonly serves as the basis from which electrostatic interactions between suspended clay particles in solution are described ([23], see Sec.II. A. 2). In aqueous environments, both inner and outer-sphere complexes may form, and these complexes along with the intrinsic surface charge density are included in the net particle surface charge density (crp, 4). When clay mineral particles are suspended in water, a diffuse double layer (DDL) of ion charge is structured with an associated volumetric charge density (p ) if av 0. Given that the entire system must remain electrically neutral, ap then must equal — f p dx. In its simplest form, the DDL may be described, with the help of the P-B equation, by the traditional Gouy-Chapman [23-27] model, which describes the inner potential variation as a function of distance from the particle surface [23]. 

There is a range of equations used describing the experimental data for the interactions of a substance as liquid and solid phases. They extend from simple empirical equations (sorption isotherms) to complicated mechanistic models based on surface complexation for the determination of electric potentials, e.g. constant-capacitance, diffuse-double layer and triple-layer model. 

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. 

Equations 3.12 and 3.13 reveal that when I increases, k increases but /0 decreases, leading to a constant a. The term a is related to the experimental CEC. Since the concentration and valence of the charged solution constituents dictate the distance that the diffuse layer could extend into the bulk solution, when / increases (either by increasing ion concentration or by increasing ion valence), the drop in potential occurs a short distance from the solid surface, thereby creating a thinner diffuse double layer (Fig. 3.23). 

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