Electrical double layer free energy

Note that Eqs. (5.17) and (5.18) hold among Fj, F, and y, that is, F = F = yA and that in this case the electrical double-layer free energy F is equal to the electric work Fei, namely,... 

What is the Gibbs free energy of an electric double layer The energy of an electric double layer plays a central role in colloid science, for instance to describe the properties of charged polymers (polyelectrolytes) or the interaction between colloidal particles. Here, we only give results for diffuse layers because it is simpler and in most applications only the diffuse layer is relevant. The formalism is, however, applicable to other double layers as well. 

If the dissociation of ionizable groups on the particle surface can be regarded as complete, then N = and Sc can be dropped so that the surface free energy increase Ft is just equal to the electrical part of the double-layer free energy F (Eq. (5.4)), namely. 

The free energy of interaction of two interacting bodies may be expressed as a sum of electrical double layer interaction energy (in general, repulsive) and the van der Waals attraction energy (in general, attractive) ... 

Loeb, AL Overbeek, JTG Wiersema, PH, The Electrical Double Layer Around a Spherical Colloid Particle, Computation of the Potential, Charge Density, and Free Energy of the Electrical Double Layer Around a sperical Colloid Particle M.I.T. Press Cambridge, MA, 1961. Lorentz, HA, Wied, Ann. 11, 70, 1880. 

Free Energies of Electrical Double Layers at the Oxide-Solution Interface... 

The aim of this paper is not to add to the current debate but to present a simple graphical method of analysing the free energy of formation of the electrical double layer at the oxide/solution interface ( 1). This will provide a simple way of visualizing the complementary roles of chemical reactions or surface properties of... 

Detailed numerical examples of the behaviour of the surface charge and surface potential when the electrical double layer of two identical amphoteric surfaces overlap and interact are available in the literature (8). Examples of the differences between the form of the interaction free energy under constant... 

The main, currently used, surface complexation models (SCMs) are the constant capacitance, the diffuse double layer (DDL) or two layer, the triple layer, the four layer and the CD-MUSIC models. These models differ mainly in their descriptions of the electrical double layer at the oxide/solution interface and, in particular, in the locations of the various adsorbing species. As a result, the electrostatic equations which are used to relate surface potential to surface charge, i. e. the way the free energy of adsorption is divided into its chemical and electrostatic components, are different for each model. A further difference is the method by which the weakly bound (non specifically adsorbing see below) ions are treated. The CD-MUSIC model differs from all the others in that it attempts to take into account the nature and arrangement of the surface functional groups of the adsorbent. These models, which are fully described in a number of reviews (Westall and Hohl, 1980 Westall, 1986, 1987 James and Parks, 1982 Sparks, 1986 Schindler and Stumm, 1987 Davis and Kent, 1990 Hiemstra and Van Riemsdijk, 1996 Venema et al., 1996) are summarised here. 

AF4 is the change in the free energy of the electrical double layer accompanying the adsorption of charged trains on the charged surface, and if the Debye-Hiichel approximation is applied, it is given by... 

The Gibbs free energy of the electric double layer... 

The Gibbs free energy of an electric double layer is negative because it forms spontaneously. Roughly, it increases in proportion to the square of the surface potential. 

We start with the first approach and calculate the change in Gibbs free energy for two approaching double layers. The Gibbs free energy of one isolated electric double layer per unit area is (Eq. 4.35)... 

M. Manciu, E. Ruckenstein On the Chemical Free Energy of the. Electrical Double Layer, LANGMUIR 19(2003)1114-1120. 

Chemical Free Energy of the Electrical Double Layer... 

To get the interaction potential we must first evaluate the free energy of formation of the electrical double layer between two charged bodies. This is defined as the work done in charging up the surfaces. The process by which uniformly charged surfaces are charged up from a neutral reference state has been discussed by Yerwey and Overbeek [4], who have shown that the electrostatic work of charging a surface is given by the simple formula... 

In this chapter, mathematical procedures for the estimation of the electrical interactions between particles covered by an ion-penetrable membrane immersed in a general electrolyte solution is introduced. The treatment is similar to that for rigid particles, except that fixed charges are distributed over a finite volume in space, rather than over a rigid surface. This introduces some complexities. Several approximate methods for the resolution of the Poisson-Boltzmann equation are discussed. The basic thermodynamic properties of an electrical double layer, including Helmholtz free energy, amount of ion adsorption, and entropy are then estimated on the basis of the results obtained, followed by the evaluation of the critical coagulation concentration of counterions and the stability ratio of the system under consideration. 

The Helmholtz free energy, Fe, comprised the free energy due to the presence of electrical double layer, Fe d, and that due to the presence of membrane, Feis, that is,... 

In order to determine a system thermodynamically, one has to specify some independent parameters (e.g. N, T, P or V) besides the composition of the system. The most common choice in MC simulation is to specify N, V and T resulting in the canonical ensemble, where the Helmholtz free energy A is the natural thermodynamical potential. However, MC calculations can be performed in any ensemble, where the suitable choice depends on the application. It is straightforward to apply the Metropolis MC algorithm to a simple electric double layer in the iVFT ensemble. It is however, not so efficient for polymers composed of more than a few tens of monomers. For long polymers other algorithms should be considered and the Pivot algorithm [21] offers an efficient alternative. MC simulations provide thermodynamic and structural information, but time-dependent properties are not accessible. If kinetic or time-dependent properties are of interest one has to use molecular dynamic or brownian dynamic simulations. 

The ideas underlying elemental structures models are to establish microstructures experimentally, to compute free energies and chemical potentials from models based on these structures, and to use the chemical potentials to construct phase diagrams. Jonsson and Wennerstrom have used this approach to predict the phase diagrams of water, hydrocarbon, and ionic surfactant mixtures [18]. In their model, they assume the surfactant resides in sheetlike structures with heads on one side and tails on the other side of the sheet. They consider five structures spheres, inverted (reversed) spheres, cylinders, inverted cylinders, and layers (lamellar). These structures are indicated in Fig. 12. Nonpolar regions (tails and oil) are cross-hatched. For these elemental structures, Jonsson and Wennerstrom include in the free energy contributions from the electrical double layer on the water... 

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