Gouy-Chapman theory electrostatic potential

Figure 17.6 Distribution of the electrostatic potential for an ensemble of hard sphere ions and dipoles in contact with a hard wall the straight line is the prediction of the Gouy-Chapman theory. Data taken from Ref. 8.

The diffuse layer is described by the Gouy—Chapman theory of 1913 [21, 22], which is based on the same equations as the Debye—Hiickel theory of 1923 for electrolytes, which describes the electrostatic potential around an ion in a given ionic atmosphere [23]. 

This model is based on the Gouy-Chapman theory (diffuse double-layer theory). The theory states that in the area of the boundary layer between solid and aqueous phase, independently of the surface charge, increased concentrations of cations and anions within a diffuse layer exists because of electrostatic forces. In contrast to the constant-capacitance model, the electrical potential does not change up to a certain distance from the phase boundaries and is not immediately declining in a linear manner (Fig. 14 a). Diffusion counteracts these forces, leading to dilution with increasing distance from the boundary. This relation can be described physically by the Poisson-Boltzmann equation. 

We use the Gouy-Chapman theory for the diffuse layer which is based on the Poisson-Boltzmann (P.B.) equation for the potential distribution. Although the different corrections to the P.B. equation in double-layer theory have been investigated (20, 21, 22, 23), it is difficult to state precisely the range of validity of this equation. In the present problem the P.B. equation seems a reasonable approximation at 0.1M of a 1-1 electrolyte to 50mV for the mean electrostatic potential pd at the ohp (24) this upper limit for pd increases with a decrease in electrolyte concentration. All the values for pd calculated in Tables I-IV are less than 50 mV— most of them are well below. If n is the volume density of each ion type of the 1-1 electrolyte in the substrate, c the dielectric constant of the electrolyte medium, and... 

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. 

In most of the earlier works, the average electrostatic potential in the plane where the reacting ions are located was chosen for il/. This plane was often identified with the outer Helmholtz layer. In this case, the potential calculated according to the Gouy-Chapman theory was used for... 

Through the Gouy-Chapman theory, we can arrive to expressions for the electrostatic potential and ion concentration as a function of distance from a charged surface and finally - what is most important for applications - the expressions for the potential energy interaction between two double layers as a function of distance between the particles or surfaces. 

At higher salt concentration (c>0.5 M) the screening exceeds to an even greater extent that predicted by Gouy-Chapman theory, and the transition temperature flattens off to a plateau value at salt concentrations of c —1.5 - 2 M. Independent measurements using charged partitioning spin labels indicate that the surface potential of the bilayers is completely screened at this salt concentration. Thus from Fig. 2.7 it appears that the electrostatic phase transition temperature shift is effectively screened in >2 M... 

The next step is to determine the electrical charge and potential distribution in this diffuse region. This is done by using relevant electrostatic and statistical mechanical theories. For a charged planar surface, this problem was solved by Gouy (in 1910) and Chapman (in 1913) by solving the Poisson-Boltzmann equation, the so called Gouy-Chapman (G-C) model. 

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-Huckel theory of electrolyte solutions. It is derived from the classical -> Poisson equation for the electrostatic potential... 

Few outer-sphere electrode reactions have precursor-state concentrations that are measurable [21] so that it is usual to estimate wp and ws from double-layer models. The simplest, and by far the most commonly used, treatment is the Frumkin model embodied in eqns. (8) and (8a) whereby, as noted in Sect. 2.2, the sole contributor to wp and ws is presumed to be electrostatic work associated with transporting the reactant from the bulk solution to the o.H.p. at an average potential Gouy-Chapman (GC) theory [58],... 

There are two versions of the physical description of this system. According to the Gouy-Chapman (G-C) theory, counter ion thermal energy runs counter to the electrostatic attraction and a secondary diffuse layer in which the potential decays almost exponentially because of screening effects is generated. Both layers are under dynamic equilibrium. The electrical potential difference, between the stationary phase and the bulk eluent can be theoretically estimated. Figure 3.2 depicts the G-C model. 

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