Gouy Potential

Of these three contributions to the free energy, the most important in terms of its effect on the pH-dependence of adsorption is the coulombic term AG9° 1. This is a measure of the interaction of the charged ion witft the Gouy potential plane of adsorption. This surface potential is assumed by... 

The surface, interfacial, and diffuse charge densities (respectively, Og, and a j) and the electrostatic (Gouy) potentials at the surface and adsorption planes (respectively i( g and j)... 

In the VSC-VSP model the charge density Og of adsorbed cations exerts an effect on the surface and Gouy potentials which can be quite profound where relatively large amounts of cations are adsorbed. This effect, not considered in the classical double-layer theory as presented by James and Healy ( ), can help us.to explain, for example, a rise in the increase in adsorption with increased pH that is different from that otherwise predicted. 

It therefore seems reasonable to say that as a first approximation, the interfacial dielectric constant within the first layer of adsorbed water molecules may be taken as the lower value of 6.0. It is this lower value which we used in the Levine ( ) expression for the solvation free-energy term, and in the surface potential-Gouy potential relation in the VSC-VSP model in order to obtain the predicted adsorption shown in Figure 7 and tabulated in Table I. Our value for of 2.17 k lies well within the diameter (2.76 A) of the first adsorbed layer of water within which the approximation of a 6.0 dielectric constant should be valid. 

While previous work has often been conducted under conditions where only trace quantities of lead or other heavy metals have been placed in contact with an adsorbent, very few of these approaches have dealt with the problems faced as the adsorbent sites begin to be filled. The usefulness of the VSC-VSP model in taking this into account is illustrated here by demonstration of the effect of charged adsorbed species on the electrostatic potential which acts on the adsorbing ions. When a given number of equivalents of adsorbent are placed in contact with a comparatively large number of moles of cations, some of which will attach to the adsorbent, adsorption will be further opposed in two ways. First, of course, the process of adsorption will reduce the number of sites available for further adsorption. Second, the Gouy potential is said by Bowden ad. (7) to decrease from the... 

The region of the gradual potential drop from the Helmholtz layer into the bulk of the solution is called the Gouy or diffuse layer (29,30). The Gouy layer has similar characteristics to the ion atmosphere from electrolyte theory. This layer has an almost exponential decay of potential with increasing distance. The thickness of the diffuse layer may be approximated by the Debye length of the electrolyte. 

Instead of an exact calculation, Gouy and Chapman have assumed that (4) can be approximated by combining the Poisson equation with a Boltzmann factor which contains the mean electrical potential existing in the interface. (This approximation will be rederived below). From this approach the distribution of the potential across the interface can be calculated as the function of a and from (2) we get a differential capacitance Cqc- It has been shown by Grahame that Cqc fits very well the measurements in the case of low ionic concentrations [11]. For higher concentrations another capacitance in series, Q, had to be introduced. It is called the inner layer capacitance and it was first considered by Stern [1,2]. Then the experimental capacitance Cexp is analyzed according to ... 

Since the potential verifies the Poisson equation the nonlinear Gouy-Chapman theory is recovered. In what follows we summarize some results of the nonlinear Gouy-Chapman (NLGC) theory that are useful for the subsequent part of this work. 

In Fig. 8 density profiles are presented for several values of charge density a on the wall and for the wall potential h = — and h= Fig. 9 contains the corresponding ionic charge density profiles. For the adsorptive wall potential h < 0) the profiles q z) in Fig. 9(a) and j (z) in Fig. 8(a) are monotonic, as in the Gouy-Chapman theory. For a wall which is neutral relative to the adsorption A = 0 the density profiles are monotonic with a maximum at the wall position. This maximum also appears on the charge... 

Fig. 20.9 Experimental capacitance-potential curve for O-OOI m KCl and calculated curve using the Gouy-Chapman model. The experimental curve and the theoretical curve agree at potentials (us R.H.E.) near the p.z.c. Note the constant capacitance of 17 x 10 F m at negative potentials (after Bockris and Drazic )...

Nakagaki1U) has given a theoretical treatment of the electrostatic interactions by using the Gouy-Chapman equation for the relation between the surface charge density oe and surface potential /. The experimental data for (Lys)n agrees very well with the theoretical curve obtained. 

The physical meaning of the g (ion) potential depends on the accepted model of an ionic double layer. The proposed models correspond to the Gouy-Chapman diffuse layer, with or without allowance for the Stem modification and/or the penetration of small counter-ions above the plane of the ionic heads of the adsorbed large ions. " The experimental data obtained for the adsorption of dodecyl trimethylammonium bromide and sodium dodecyl sulfate strongly support the Haydon and Taylor mode According to this model, there is a considerable space between the ionic heads and the surface boundary between, for instance, water and heptane. The presence in this space of small inorganic ions forms an additional diffuse layer that partly compensates for the diffuse layer potential between the ionic heads and the bulk solution. Thus, the Eq. (31) may be considered as a linear combination of two linear functions, one of which [A% - g (dip)] crosses the zero point of the coordinates (A% and 1/A are equal to zero), and the other has an intercept on the potential axis. This, of course, implies that the orientation of the apparent dipole moments of the long-chain ions is independent of A. 

FIGURE 10.2 Potential dependence of differential capacitance calculated from Gouy-Chapman theory for z+ = = 1 and various concentrations (1) 10 , (2) 10 , (3) 10 M. 

The physical meaning of the g" (ion) potential depends on the accepted model of ionic double layer. The proposed models correspond to the Gouy Chapman diffuse layer, with or without allowance for the Stern modification and/or the penetration of small counterions above the plane of the ionic heads of the adsorbed large ions [17,18]. The presence of adsorbed Langmuir monolayers may induce very high changes of the surface potential of water. For example. A/" shifts attaining ca. —0.9 (hexadecylamine hydrochloride), and ca. -bl.OV (perfluorodecanoic acid) have been observed [68]. 

The non-steady-state optical analysis introduced by Ding et al. also featured deviations from the Butler-Volmer behavior under identical conditions [43]. In this case, the large potential range accessible with these techniques allows measurements of the rate constant in the vicinity of the potential of zero charge (k j). The potential dependence of the ET rate constant normalized by as obtained from the optical analysis of the TCNQ reduction by ferrocyanide is displayed in Fig. 10(a) [43]. This dependence was analyzed in terms of the preencounter equilibrium model associated with a mixed-solvent layer type of interfacial structure [see Eqs. (14) and (16)]. The experimental results were compared to the theoretical curve obtained from Eq. (14) assuming that the potential drop between the reaction planes (A 0) is zero. The potential drop in the aqueous side was estimated by the Gouy-Chapman model. The theoretical curve underestimates the experimental trend, and the difference can be associated with the third term in Eq. (14). 

To evaluate the contribution of the SHG active oriented cation complexes to the ISE potential, the SHG responses were analyzed on the basis of a space-charge model [30,31]. This model, which was proposed to explain the permselectivity behavior of electrically neutral ionophore-based liquid membranes, assumes that a space charge region exists at the membrane boundary the primary function of lipophilic ionophores is to solubilize cations in the boundary region of the membrane, whereas hydrophilic counteranions are excluded from the membrane phase. Theoretical treatments of this model reported so far were essentially based on the assumption of a double-diffuse layer at the organic-aqueous solution interface and used a description of the diffuse double layer based on the classical Gouy-Chapman theory [31,34]. 

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