Theories Gouy-Chapman theory

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. 

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]. 

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. 

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... 

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. 

FIG. 5 Normalized concentration distribution in the pore of Figure 4 but charged with —0.05 C/m. The symbols are the same as in Figure 4, with the cations being the counterions. The anions (coions) of the RPM and SPM model are not distinguishable on the present scale. The dotted line is the prediction of the modified Gouy-Chapman theory and approximates the simulation results of the RPM. 

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. 

A full mathematical treatment of the Gouy-Chapman theory and the derivation of these equations is given in Appendix B.)... 

The charge of the diffuse EDL part (x > Xj) can be described by the equations of Gouy-Chapman theory, but with the value tj/j rather than /o ... 

APPENDIX B Derivation of the Main Equation of Gouy-Chapman Theory... 

APPENDIX B DERIVATION OE THE MAIN EQUATION OE GOUY-CHAPMAN THEORY... 

The Gouy-Chapman theory for metal-solution interfaces predicts interfacial capacities which are too high for more concentrated electrolyte solutions. It has therefore been amended by introducing an ion-free layer, the so-called Helmholtz layer, in contract with the metal surface. Although the resulting model has been somewhat discredited [30], it has been transferred to liquid-liquid interfaces [31] by postulating a double layer of solvent molecules into which the ions cannot penetrate (see Fig. 17) this is known as the modified Verwey-Niessen model. Since the interfacial capacity of liquid-liquid interfaces is... 

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]. 

According to the Gouy-Chapman theory, the surface charge densities on the aqueous and membrane sides, and o , respectively, can be expressed as... 

We recently synthesized several reasonably surface-active crown-ether-based ionophores. This type of ionophore in fact gave Nernstian slopes for corresponding primary ions with its ionophore of one order or less concentrations than the lowest allowable concentrations for Nernstian slopes with conventional counterpart ionophores. Furthermore, the detection limit was relatively improved with increased offset potentials due to the efficient and increased primary ion uptake into the vicinity of the membrane interface by surfactant ionophores selectively located there. These results were again well explained by the derived model essentially based on the Gouy-Chapman theory. Just like other interfacial phenomena, the surface and bulk phase of the ionophore incorporated liquid membrane may naturally be speculated to be more or less different. The SHG results presented here is one of strong evidence indicating that this is in fact true rather than speculation. 

When the ITIES is polarized with a potential difference 0, there is a separation of electrical charge across it. According to the Gouy-Chapman theory, the charges in the aqueous and organic diffuse layers are related to the potential drops and A0 in the respective layers by the equations... 

---