Gouy Chapman

Several features of the behavior of the Gouy-Chapman equations are illus-... 

The Gouy-Chapman treatment of the double layer runs into difficulties at small Kx values when is large. For example, if is 300 mV, yo is 12 and if Co is, say, 10" mol/1, then the local concentration of negative ions near the surface, given by Eq. V-1, would be C = = 160 mol/1 The trouble... 

Assume is -25 mV for a certain silica surface in contact with O.OOlAf aqueous NaCl at 25°C. Calculate, assuming simple Gouy-Chapman theory (a) at 200 A from the surface, (b) the concentrations of Na and of Cr ions 10 A from the surface, and (c) the surface charge density in electronic charges per unit area. 

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. 

Stahlberg has presented models for ion-exchange chromatography combining the Gouy-Chapman theory for the electrical double layer (see Section V-2) with the Langmuir isotherm (. XI-4) [193] and with a specific adsorption model [194]. 

Chemical properties of deposited monolayers have been studied in various ways. The degree of ionization of a substituted coumarin film deposited on quartz was determined as a function of the pH of a solution in contact with the film, from which comparison with Gouy-Chapman theory (see Section V-2) could be made [151]. Several studies have been made of the UV-induced polymerization of monolayers (as well as of multilayers) of diacetylene amphiphiles (see Refs. 168, 169). Excitation energy transfer has been observed in a mixed monolayer of donor and acceptor molecules in stearic acid [170]. Electrical properties have been of interest, particularly the possibility that a suitably asymmetric film might be a unidirectional conductor, that is, a rectifier (see Refs. 171, 172). Optical properties of interest include the ability to make planar optical waveguides of thick LB films [173, 174]. 

Here a few core equations are presented from tire simplest tlieory for tire electric double layer tire Gouy-Chapman tlieory [41]. We consider a solution of ions of valency and z in a medium witli dielectric constant t. The ions... 

How can Equation (11.79) be solved Before computers were available only simple ihapes could be considered. For example, proteins were modelled as spheres or ellipses Tanford-Kirkwood theory) DNA as a uniformly charged cylinder and membranes as planes (Gouy-Chapman theory). With computers, numerical approaches can be used to solve the Poisson-Boltzmann equation. A variety of numerical methods can be employed, including finite element and boundary element methods, but we will restrict our discussion to the finite difference method first introduced for proteins by Warwicker and Watson [Warwicker and Watson 1982]. Several groups have implemented this method here we concentrate on the work of Honig s group, whose DelPhi program has been widely used. 

Fig. 2. Schematic diagram of a suspended colloidal particle, showing relative locations of the Stem layer (thickness, 5) that consists of adsorbed ions and the Gouy-Chapman layer (1 /k) which dissipates the excess charge, not screened by the Stem layer, to 2ero ia the bulk solution (108). In the absence of a...

Stem layer, the Gouy-Chapman layer dissipates the surface charge. 

Since the interface behaves like a capacitor, Helmholtz described it as two rigid charged planes of opposite sign [2]. For a more quantitative description Gouy and Chapman introduced a model for the electrolyte at a microscopic level [2]. In the Gouy-Chapman approach the interfacial properties are related to ionic distributions at the interface, the solvent is a dielectric medium of dielectric constant e filling the solution half-space up to the perfect charged plane—the wall. The ionic solution is considered as formed... 

These results reduce to the linear Gouy-Chapman theory if all the dq... 

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. 

To our knowledge this is quite a new formula for the differential capacitance. It is vahd whenever charging is equivalent to a shift in space of the position of the wall. We can verify that it is fulfilled for the Gouy-Chapman theory. One physical content of this formula is to show that for a positive charge on the wall we must have g (o-) > (o-) in order to have a positive... 

FIG. 7 Parsons-Zobel plot of 1/Q as a function of the inverse Gouy-Chapman capacitance 1 /Cqc- The plot is calculated analytically from Eqs. (54) and (85) at zero charge density. The straight line represents the case = a = For the upper... 

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... 

It is natural to consider the case when the surface affinity h to adsorb or desorb ions remains unchanged when charging the wall but other cases could be considered as well. In Fig. 13 the differential capacitance C is plotted as a function of a for several values of h. The curves display a maximum for non-positive values of h and a flat minimum for positive values of h. At the pzc the value of the Gouy-Chapman theory and that for h = 0 coincide and the same symmetry argument as in the previous section for the totally symmetric local interaction can be used to rationalize this result. 

Gouy-Chapman and Stem Models of the Double Layer... 

Fig. 20.8 Gouy-Chapman diffuse layer model of the double layer...

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 )...

The Stern model (1924) may be regarded as a synthesis of the Helmholz model of a layer of ions in contact with the electrode (Fig. 20.2) and the Gouy-Chapman diffuse model (Fig. 20.10), and it follows that the net charge density on the solution side of the interphase is now given by... 

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. 

---