Gouy theory

I am disturbed that following Professor Hammes presentation the language in the discussion has changed. While discussing enzymes we looked at molecule properties closely related to the discussion of small molecules (e.g., atom position and motions). Now in the discussion of complex enzymes, especially in membranes, we have started to use bulk properties (e.g., we talk of phases, dielectric constants, Chapman-Gouy theory, etc.). Is it the view of the discussants that events in membrane-coupled-enzyme systems cannot be described by molecular events because of the complexity of the system resulting from extensive cooperativ-ity within the membrane (e.g., between lipids and proteins) ... 

Galculate the diffuse-layer charge according to the classical Gouy theory for a mercury electrode in contact with an aqueous electrolyte of 0.001 M NaF, the zeta potential being +8 mV and the temperature 28 K. The dielectric constant can be taken as that of water at this temperature. (Bockris)... 

The surface charge density at the slipping plane or electrokinetic surface charge density may be calculated from the potential by means of the Gouy theory (17) which for a plane surface gives ... 

Bolt, G. M. and M. Peech. 1953. The application of the Gouy theory to soil-water systems. Soil Sci. Soc. Amer. Proc. 17 2]0—213. 

It is not surprising that the Poisson-Boltzmann approach has been used frequently in computing interactions between charged entities. Mention may be made of the Gouy theory (Fig. 3.24) of the interaction between a charged electrode and the ions in a solution (see Chapter 6). Other examples are the distribution (Fig. 3.25) of electrons or holes inside a semiconductor in the vicinity of the semiconductor-electrolyte interface (see Chapter 6) and the distribution (Fig. 3.26) of charges near a polyelectrolyte molecule or a colloidal particle (see Chapter 6). 

Stem s Theory of the Double Layer.—The variations of capacity of the double layer with the conditions, the influence of electrolytes on the zeta-potential, and other considerations led Stern to propose a model for the double layer which combines the essential characteristics of the Helmholtz and the Gouy theories. According to Stern the double layer consists of two parts one, which is approximately of a molecular diame r in thickness, Is supposed to remain fixed to the surface, while ihe other is anlttfuse layer extending for me distance into tlie solution The fall of potential in the fixed layer is sharp while that in the diffuse layer is gradual, the decrease being exponential in nature, as required by equation (5). 

Two topical issues may be mentioned. The first is the definition of the potentials that are measured by different techniques, say by AFM, electrokinetically and externally imposed, and their relationships [11], The second is of a more theoretical nature and concerns modeling of the nondiffuse part of the double layer. The classical approach is through Stem theory [2], which in most cases is adequate, although it requires two additional parameters. A more recent development is in terms of ion correlations, essentially an advanced statistical theory whereby all coulombic ion-ion and ion-surface interaction pairs are counted and statistically summed [2]. This is a step forward over the smeared-out models of Gouy and Stern. The issue here is that cases must be found where deviations from Gouy theory cannot be interpreted on the basis of the Stem model... 

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. 

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. 

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. 

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