Electrical double layer Gouy-Chapman

The deviations from the Szyszkowski-Langmuir adsorption theory have led to the proposal of a munber of models for the equihbrium adsorption of surfactants at the gas-Uquid interface. The aim of this paper is to critically analyze the theories and assess their applicabihty to the adsorption of both ionic and nonionic surfactants at the gas-hquid interface. The thermodynamic approach of Butler [14] and the Lucassen-Reynders dividing surface [15] will be used to describe the adsorption layer state and adsorption isotherm as a function of partial molecular area for adsorbed nonionic surfactants. The traditional approach with the Gibbs dividing surface and Gibbs adsorption isotherm, and the Gouy-Chapman electrical double layer electrostatics will be used to describe the adsorption of ionic surfactants and ionic-nonionic surfactant mixtures. The fimdamental modeling of the adsorption processes and the molecular interactions in the adsorption layers will be developed to predict the parameters of the proposed models and improve the adsorption models for ionic surfactants. Finally, experimental data for surface tension will be used to validate the proposed adsorption models. 

The potential of each channel may be composed of two potentials. One is an oxidation-reduction potential generating at the boundary surface between the Ag electrode and the lipid membrane. The other is a Donnan potential at the boundary between the lipid membrane and the aqueous medium or more generally a Gouy-Chapman electrical double-layer potential formed in the aqueous medium [24]. Figure 7 shows a potential profile near the lipid membrane. The oxidation-reduction potential would not be affected by the outer solution in short time, because the lipid membrane had low permeability for water. Then the measured potential change by application of the taste solution is mainly due to the change in the surface electrical potential. 

To determine the spatial variation of a static electric field, one has to solve the Poisson equation for the appropriate charge distribution, subject to such boundary conditions as may pertain. The Poisson equation plays a central role in the Gouy-Chapman (- Gouy, - Chapman) electrical - double layer model and in the - Debye-Huckel theory of electrolyte solutions. In the first case the one-dimensional form of Eq. (2)... 

FIG. 1 The Gouy-Chapman electrical double layer. Net bound charge at a solid surface (v = 0) is countered by diffuse charge in solution. Coions in the diffuse double layer are omitted for clarity. 

The outer surface of the Stern layer is the shear surface of the micelle. The core and the Stern layer together constitute what is termed the kinetic micelle. Surrounding the Stern layer is a diffuse layer called the Gouy-Chapman electrical double layer, which contains the aN counterions required to neutralise the charge on the kinetic micelle. The thickness of the double layer is dependent on the ionic strength of the solution and is greatly compressed in the presence of electrolyte. 

Ion adsorption at day partides is a dynamic process, so that an exchange of ions can take place readily in response to the changing pH. These changes in pH influence the thickness of the Hehnholtz-Gouy-Chapman electrical double layer, and in turn the value of the so-called zeta potential (Q that behaves inversely to the viscosity (see Figure 2.20). 

The equation of state for ionized monolayers has been discussed by Hachisu [32]. This author has shown by independent derivations using three different approaches that the equation proposed by Davies [33] is applicable in the presence or absence of added electrolyte provided that the Gouy-Chapman electrical double-layer model applies. The Davies equation may be written... 

Poisson-Boltzmann equation — The Poisson-Boltz-mann equation is a nonlinear, elliptic, second-order, partial differential equation which plays a central role, e.g., in the Gouy-Chapman ( Gouy, Chapman) electrical double layer model and in the Debye-Hiickel theory of electrolyte solutions. It is derived from the classical Poisson equation for the electrostatic potential... 

Menestrina et al. discuss their interesting results in terms of the Gouy-Chapman electrical double-layer theory and suggest that molluscan hemo-cyanins are a class of channel-forming proteins. 

Some emphasis is given in the first two chapters to show that complex formation equilibria permit to predict quantitatively the extent of adsorption of H+, OH , of metal ions and ligands as a function of pH, solution variables and of surface characteristics. Although the surface chemistry of hydrous oxides is somewhat similar to that of reversible electrodes the charge development and sorption mechanism for oxides and other mineral surfaces are different. Charge development on hydrous oxides often results from coordinative interactions at the oxide surface. The surface coordinative model describes quantitatively how surface charge develops, and permits to incorporate the central features of the Electric Double Layer theory, above all the Gouy-Chapman diffuse double layer model. 

Extending out into solution from the electrical double layer (or the compact double layer, as it is sometimes known) is a continuous repetition of the layering effect, but with diminishing magnitude. This extension of the compact double layer toward the bulk solution is known as the Gouy-Chapman diffuse double layer. Its effect on electrode kinetics and the concentration of electroactive species at the electrode surface is manifest when supporting electrolyte concentrations are low or zero. 

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]. 

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... 

In 1910, Georges Gouy (1854-1926) and independently, in 1913, David L. Chapman (1869-1958) introduced the notion of a diffuse electrical double layer at the surface of electrodes resulting from a thermal motion of ions and their electrostatic interactions with the surface. 

The second term in equation (9) is the usual electrostatic term. Here vA is the valency of the unit and e is the elementary charge, and ip(z) is the electrostatic potential. This second term is the well-known contribution accounted for in the classical Poisson-Boltzmann (Gouy -Chapman) equation that describes the electric double layer. The electrostatic potential can be computed from the charge distribution, as explained below. 

In the electrochemical literature one finds the Gouy-Chapman (GC) and Gouy-Chapman-Stern (GCS) approaches as standard models for the electric double layer [9,10]. 

As we have seen, the electric state of a surface depends on the spatial distribution of free (electronic or ionic) charges in its neighborhood. The distribution is usually idealized as an electric double layer one layer is envisaged as a fixed charge or surface charge attached to the particle or solid surface while the other is distributed more or less diffusively in the liquid in contact (Gouy-Chapman diffuse model, Fig. 3.2). A balance between electrostatic and thermal forces is attained. 

Gouy-Chapman, Stern, and triple layer). Methods which have been used for determining thermodynamic constants from experimental data for surface hydrolysis reactions are examined critically. One method of linear extrapolation of the logarithm of the activity quotient to zero surface charge is shown to bias the values which are obtained for the intrinsic acidity constants of the diprotic surface groups. The advantages of a simple model based on monoprotic surface groups and a Stern model of the electric double layer are discussed. The model is physically plausible, and mathematically consistent with adsorption and surface potential data. 

For present purposes, the electrical double-layer is represented in terms of Stem s model (Figure 5.8) wherein the double-layer is divided into two parts separated by a plane (Stem plane) located at a distance of about one hydrated-ion radius from the surface. The potential changes from xj/o (surface) to x/s8 (Stem potential) in the Stem layer and decays to zero in the diffuse double-layer quantitative treatment of the diffuse double-layer follows the Gouy-Chapman theory(16,17 ... 

According to the Gouy-Chapman theory of the diffuse electric double layer (chapter 3 of [18]), for a uni-univalent electrolyte,... 

Ionic surfactants are electrolytes dissociated in water, forming an electrical double layer consisting of counterions and co-ions at the interface. The Gouy-Chapman theory is used to model the double layer. In conjunction with the Gibbs adsorption equation and the equations of state, the theory allows the surfactant adsorption and the related interfacial properties to be determined [9,10] (The Gibbs adsorption model is certainly simpler than the Butler-Lucassen-Reynders model for this case.). 

Gouy and Chapman (1910-13) independently used the Poisson-Boltzmann equations to describe the diffnse electrical double-layer formed at the interface between a charged snrface and an aqueous solution. 

This is the important Poisson-Boltzmann (PB) equation and the model used to derive it is usually called the Gouy-Chapman (GC) theory. It is the basic equation for calculating all electrical double-layer problems, for flat surfaces. In deriving it we have, however, assumed that all ions are point charges and that the potentials at each plane x are uniformly smeared out along that plane. These are usually reasonable assumptions. 

Figure 7.4. Schematic model of the Electrical Double Layer (EDL) at the metal oxide-aqueous solution interface showing elements of the Gouy-Chapman-Stern-Grahame model, including specifically adsorbed cations and non-specifically adsorbed solvated anions. The zero-plane is defined by the location of surface sites, which may be protonated or deprotonated. The inner Helmholtz plane, or [i-planc, is defined by the centers of specifically adsorbed anions and cations. The outer Helmholtz plane, d-plane, or Stern plane corresponds to the beginning of the diffuse layer of counter-ions and co-ions. Cation size has been exaggerated. Estimates of the dielectric constant of water, e, are indicated for the first and second water layers nearest the interface and for bulk water (modified after [6]).

Before we proceed to the Gouy-Chapman theory of electrical double layers, it is worthwhile to note that relations similar to Equations (45) and (47) can also be derived for double layers surrounding spherical particles. The equation for surface charge density takes the form... 

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