Electrical double layer Gouy-Chapman equation

The most widely used theory of the stability of electrostatically stabilized spherical colloids was developed by Deryaguin, Landau, Verwey, and Overbeek (DLVO), based on the Poisson-Boltzmann equation, the model of the diffuse electrical double layer (Gouy-Chapman theory), and the van der Waals attraction [60,61]. One of the key features of this theory is the effective range of the electrical potential around the particles, as shown in Figure 25.7. Charges at the latex particles surface can be either covalently bound or adsorbed, while ionic initiator end groups and ionic comonomers serve as the main sources of covalently attached permanent charges. 

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... 

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. 

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. 

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... 

This simple equation is, however, only valid for R Xp- If the radius is not much larger than the Debye length we can no longer treat the particle surface as an almost planar surface. In fact, we can no longer use the Gouy-Chapman theory but have to apply the theory of Debye and Hiickel. Debye and Hiickel explicitly considered the electric double layer of a sphere. A result of their theory is that the total surface charge and surface potential are related by... 

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)... 

Conversely, according to the description of the electrical double layer based on the Stern-Gouy-Chapman (S-G-C) version of the theory [24], counter ions cannot get closer to the surface than a certain distance (plane of closest approach of counter ions). Chemically adsorbed ions are located at the inner Helmholtz plane (IHP), while non-chemically adsorbed ions are located in the outer Helmholtz plane (OHP) at a distance x from the surface. The potential difference between this plane and the bulk solution is 1 ohp- In this version of the theory, Pqhp replaces P in all equations. Two regions are discernible in the double layer the compact area between the charged surface and the OHP in which the potential decays linearly and the diffuse layer in which the potential decay is almost exponential due to screening effects. 

S. L. Carnie, G.M. Torrie, The Statistical Mechanics of the Electrical Double Layer, Advan. Chem. Phys. 56 (1984) 141 253. (Gouy-Chapman and more advanced models, including integral equation theories, discrete charges, simulations.)... 

FIGURE 3.15 Dimensionless mean electrostatic potential (a) and surface-ion distribution function (b) as predicted by the Gouy-Chapman-Stern (GCS) and modified Poisson-Boltzmann (MPB) theories for a 1 1 electrolyte with a = 0.425 nm and c = 0.197 M. (Outhwaite, Bhuiyan, and Levine, 1980, Theory of the electric double layer using a modified Poisson-Boltzmann equation. Journal of the Chemical Society, Faraday Transactions 2 Molecular and Chemical Physics, 76, 1388-1408. Reproduced by permission of The Royal Society of Chemistry.)... 

Figure 4.11 Comparison of the predictions of the Debye-Huckel and Gouy-Chapman equations for the potential 4> in the electrical double layer. The Debye-Hiickel equation can be used with insignificant error up to —50 mV. (From Ref. 3.)...

The electrostatic double-layer force can be calculated using the continuum theory, which is based on the theory of Gouy, Chapman, Debye, and Hiickel for an electrical double layer. The Debye length relates the surface charge density of a surface to the electrostatic surface potential /o via the Grahame equation, which for 1 1 electrolytes can be expressed as... 

Oil/water interfaces are classified into the ideal-polarized interface and the nonpolarized interface. The interface between a nitrobenzene solution of tetrabutylam-monium tetraphenylborate and an aqueous solution of lithium chloride behaves as an ideal-polarized interface in a certain potential range. Electrocapillary curves of the interface were measured. The results are analyzed using the electrocapillary equation of the ideal-polarized interface and the Gouy-Chapman theory of diffuse double layers. The electric double layer structure consisting of the inner layer and the two diffuse double layers on each side of the interface is discussed. Electrocapillary curves of the nonpolarized oil/water interface are discussed for two cases of a nonpolarized nitrobenzene/water interface. 

In the following sections, different surface complexation models will be introduced. General aspects and specific models will be discussed. The components of surface complexation theory will be presented, as well as some recent developments covering, for example, the use of equations for the diffuse part of the electrical double layer for electrolyte concentrations, for which the traditional Gouy-Chapman equation is not recommended or a generalization of Smit s compartment model [6] for situations in which the traditional models are at a loss. 

Two potential improvements compared to the common practice are introduced. Both refer to the description of the diffuse layer. The commonly applied surface complexation models involve the Poisson-Boltzmann approximation for diffuse-layer potential of the electric double layer (resulting in the Gouy-Chapman equation for flat plates in most apphcations). 

The diffuse double layer can be described by the Gouy-Chapman equation, which is a solution of the Poisson-Boltzmann equation for a planar diffuse double layer. The Poisson-Boltzmann equation relates the electrical potential to the distribution and concentration of charged species. The distribution of charges in the electrolyte solution is described by Boltzmann distributions. 

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... 

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