Diffuse electrical double-layer theory

Double integration with respect to EA yields the surface excess rB+ however, the calculation requires that the value of this excess be known, along with the value of the first differential 3TB+/3EA for a definite potential. This value can be found, for example, by measuring the interfacial tension, especially at the potential of the electrocapillary maximum. The surface excess is often found for solutions of the alkali metals on the basis of the assumption that, at potentials sufficiently more negative than the zero-charge potential, the electrode double layer has a diffuse character without specific adsorption of any component of the electrolyte. The theory of diffuse electrical double layer is then used to determine TB+ and dTB+/3EA (see Section 4.3.1). 

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. 

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

The generation of colloidal charges in water.The theory of the diffuse electrical double-layer. The zeta potential. The flocculation of charged colloids. The interaction between two charged surfaces in water. Laboratory project on the use of microelectrophoresis to measure the zeta potential of a colloid. 

Several SCM s have been described in the literature. The more commonly used models include the Constant Capacitance Model (Schindler and Stumm, 1987), the Diffuse Double Layer Model (Stumm et al., 1970) and the Triple Layer Model (Davis et al., 1978 Yates et al, 1974). All are based on electric double layer theory but differ in their geometric description of the oxide-water interface and the treatment of the electrostatic interactions. 

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. 

Equations (l)-(4) are the foundations of electrical double layer theory and are often used in modeling the adsorption of metal ions at interfaces of charged solid and electrolyte solutions. In a typieal TLM, the outer layer capacitance is often fixed at a lower value (i.e., C2 = 0.2 F/m ), whereas iimer layer capacitance (Ci) can be adjusted to between 1.0 and 1.4 F/m [25]. It should be noted that the three-plane model (TPM) is a variation of the classical triple-layer model, in which the outer layer eapaeitanee is not fixed. Although the physical presentations of the TLM and TPM are identical as shown in Fig. 2, i.e., both involve a surface layer (0), an inner Helmholtz plane (p), and an outer Helmholtz plane d) where the diffuse double layer starts, a one-step protonation process (i.e., 1 piC approach) is, in general, assumed in the TPM, in eontrast to a two-step protonation process (i.e., 2 p/C approach) in the TLM. Another distinct difference is that pair-forming ions are assumed to be on the outer Helmholtz plane in the TPM but on the inner Helmholtz plane in the TLM. In our study, the outer layer capacitance is allowed to vary while the pair-forming ions are placed on the iimer Helmholtz plane with a complete set of surface eomplexation reactions being considered. Therefore, our approach represents a hybrid of the TPM and TLM. 

It may be appreciated that electrokinetic phenomena are determined by electric properties at the plane of shear rather than at the real surface. In the following sections of this chapter, the relation between the measured property and is further analyzed. This is done for electroosmosis, electrophoresis, streaming current, and streaming potential. The sedimentation potential will not be discussed any further, because in practice this phenomenon does not play an important role. The electrokinetic charge density may then be derived from using the theory for the diffuse electrical double layer. 

The pioneering Gouy-Chapman theory can be used to quantitatively describe the diffuse electrical double layer. The electrical potential at a distance X from the colloidal particle surface (i f(x)) is described by the onedimensional Poisson s equation that relates the number of charges per unit volume (or space charge density, p) to /(x). 

Appendix 102 Gouy-Chapman theory of the diffuse electrical double-layer... 

IHP) (the Helmholtz condenser formula is used in connection with it), located at the surface of the layer of Stem adsorbed ions, and an outer Helmholtz plane (OHP), located on the plane of centers of the next layer of ions marking the beginning of the diffuse layer. These planes, marked IHP and OHP in Fig. V-3 are merely planes of average electrical property the actual local potentials, if they could be measured, must vary wildly between locations where there is an adsorbed ion and places where only water resides on the surface. For liquid surfaces, discussed in Section V-7C, the interface will not be smooth due to thermal waves (Section IV-3). Sweeney and co-workers applied gradient theory (see Chapter III) to model the electric double layer and interfacial tension of a hydrocarbon-aqueous electrolyte interface [27]. 

Two kinds of barriers are important for two-phase emulsions the electric double layer and steric repulsion from adsorbed polymers. An ionic surfactant adsorbed at the interface of an oil droplet in water orients the polar group toward the water. The counterions of the surfactant form a diffuse cloud reaching out into the continuous phase, the electric double layer. When the counterions start overlapping at the approach of two droplets, a repulsion force is experienced. The repulsion from the electric double layer is famous because it played a decisive role in the theory for colloidal stabiUty that is called DLVO, after its originators Derjaguin, Landau, Vervey, and Overbeek (14,15). The theory provided substantial progress in the understanding of colloidal stabihty, and its treatment dominated the colloid science Hterature for several decades. 

At present it is impossible to formulate an exact theory of the structure of the electrical double layer, even in the simple case where no specific adsorption occurs. This is partly because of the lack of experimental data (e.g. on the permittivity in electric fields of up to 109 V m"1) and partly because even the largest computers are incapable of carrying out such a task. The analysis of a system where an electrically charged metal in which the positions of the ions in the lattice are known (the situation is more complicated with liquid metals) is in contact with an electrolyte solution should include the effect of the electrical field on the permittivity of the solvent, its structure and electrolyte ion concentrations in the vicinity of the interface, and, at the same time, the effect of varying ion concentrations on the structure and the permittivity of the solvent. Because of the unsolved difficulties in the solution of this problem, simplifying models must be employed the electrical double layer is divided into three regions that interact only electrostatically, i.e. the electrode itself, the compact layer and the diffuse layer. 

It is assumed that the quantity Cc is not a function of the electrolyte concentration c, and changes only with the charge cr, while Cd depends both on o and on c, according to the diffuse layer theory (see below). The validity of this relationship is a necessary condition for the case where the adsorption of ions in the double layer is purely electrostatic in nature. Experiments have demonstrated that the concept of the electrical double layer without specific adsorption is applicable to a very limited number of systems. Specific adsorption apparently does not occur in LiF, NaF and KF solutions (except at high concentrations, where anomalous phenomena occur). At potentials that are appropriately more negative than Epzc, where adsorption of anions is absent, no specific adsorption occurs for the salts of... 

The surface sites and complexes lie in a layer on the mineral surface which, because of the charged complexes, has a net electrical charge that can be either positive or negative. A second layer, the diffuse layer, separates the surface layer from the bulk fluid. The role of the diffuse layer is to achieve local charge balance with the surface hence, its net charge is opposite that of the sorbing surface. Double layer theory, applied to a mixed ionic solution, does not specify which ions make up the diffuse layer. 

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 ... 

So far we have talked only in terms of electrostatic potentials. Can we use this theory to find the charge density on the surface (Oo) In order for the electrical double-layer to be neutral overall, it follows that the total summed charge in the diffuse layer must equal the surface charge. Thus, it follows that... 

What is the diffuse layer, and what is its relation to the Gouy-Chapman theory of electrical double layers ... 

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