Electrostatic interactions free energy

The next-order correction terms to Derjaguin s formula and HHF formula can be derived as follows [13] Consider two spherical particles 1 and 2 in an electrolyte solution, having radii oi and 02 and surface potentials i/ oi and 1/ 02, respectively, at a closest distance, H, between their surfaces (Fig. 12.2). We assume that i/ oi and i//q2 are constant, independent of H, and are small enough to apply the linear Debye-Hiickel linearization approximation. The electrostatic interaction free energy (H) of two spheres at constant surface potential in the Debye-Hlickel approximation is given by... 

The first theory of polyelectrolyte complex formation was proposed by Voom and Overbeek [14, 15]. This mean field model was used to describe the binodal compositions, the water content and the critical salt concentration as a function of the polymer chain length. This theoretical description uses the Debye-Hiickel approximation the approximations within the derivation of the electrostatic interaction free energy are therefore only valid at low charge densities. The correlation effects at high concentrations of salt and monomeric units are neglected, and ion pairing effects such as counterion condensation are not taken into account. Despite these limitations, the experimental results could be described reasonably well [13]. 

Perhaps the first attempt to interpret the proton-proton correlation, based on electrostatic interactions, was made by Bjerrum (1923). If e is the proton charge and Rfjfj the proton-proton distance in the diacid, then the interaction free energy was assumed to be given by... 

We developed the Analytical Generalized Born plus Non-Polar (AGBNP) model, an implicit solvent model based on the Generalized Born model [37-40,44, 66] for the electrostatic component and on the decomposition of the nonpolar hydration-free energy into a cavity component based on the solute surface area and a solute-solvent van der Waals interaction free energy component modeled using an estimator based on the Born radius of each atom. 

In Fig. 6.15 the interaction free energy distance curves are shown for several layer thicknesses adsorbed polymer in a good solvent. The weak minimum due to the van der Waals force decreases with increasing layer thickness. In Fig. 6.16, a typical interaction free energy curve is shown in the presence of van der Waals attraction, electrostatic repulsion and steric repulsion due to adsorbed polymer. Note the absence of a primary minimum. 

Glendinning, A.B. Russel, W.B. The electrostatic repulsion between charged spheres from exact solutions to the linearized Poisson-Boltzmann equation. J. Colloid Interface Sci. 1983, 93, 95-111 Carnie, S.L. Chan, D.Y.C. Interaction free energy between identical spherical colloidal... 

The basic idea of the DLVO theory is that the stability of lyophobic colloids in aqueous systems is determined by the combination of van der Waals attraction and electrostatic repulsion and that the two are exactly additive. In other words, the total interaction free energy VT would at any value of h... 

Figure 3.12 Typical forms of the total-interaction free energy for (a) electrostatically stabilised systems [curves (i), (ii), and (iii) refer to increasing electrolyte concentration], (b) sterically stabilised systems [curves (i), (ii), and (iii) refer to constant density of polymer chains, but decreasing <5, arising from decreasing values of a].

When the solute molecule dissolves in the solvent two interactions are possible which produce a negative free energy change, these being Van der Waals and electrostatic interactions. The energy associated with the Van der Waals interactions is approximately proportional to the molecular surface area of the solute, while the electrostatic forces... 

The situation is still more complex in the presence of surfactants. Recently, a self-consistent electrostatic theory has been presented to predict disjoining pressure isotherms of aqueous thin-liquid films, surface tension, and potentials of air bubbles immersed in electrolyte solutions with nonionic surfactants [53], The proposed model combines specific adsorption of hydroxide ions at the interface with image charge and dispersion forces on ions in the diffuse double layer. These two additional ion interaction free energies are incorporated into the Boltzmann equation, and a simple model for the specific adsorption of the hydroxide ions is used for achieving the description of the ion distribution. Then, by combining this distribution with the Poisson equation for the electrostatic potential, an MPB nonlinear differential equation appears. 

P(h) is the normal pressure of a film of thickness h, whereas Po is the normal pressure of a sufficiently thick film such that the interaction free energy is zero. Notably, n(h) is the net force per unit area acting across the film, i.e. normal to the interfaces. Thus n(h) is simply equal to —dVr/dh, where Vj is the net force that results from three main contributions, van der Waals, electrostatic and steric forces, i.e. 

The electrostatic calculations result in elements of the interaction free energy matrix llG,yll. It is written in the following format by our program and contains all the energies required to solve for the pH-dependent properties of interest. The form and content of the data file is as follows ... 

A enp is the change in the internal electronic kinetic and electronic and nuclear coulombic energy of the solute upon relaxation in solution, which is driven by the favorable electric polarization interaction with the solvent, while Cp is the electrostatic polarization free energy expressed in terms of the generalized Bom approximation (equation 40). 

Further, the solute is gradually coupled to the solvent bath through solvent-solute dispersion-repulsion (van der Waals) and electrostatic interactions by means of a coupling parameter The free energy terms associated with this step are evaluated as an integral over Hence, solvent-solute van der Waals, Cvdw and electrostatic, Cei interaction free energies are given as ... 

As with SCRF-PCM only macroscopic electrostatic contribntions to the Gibbs free energy of solvation are taken into account, short-range effects which are limited predominantly to the first solvation shell have to be considered by adding additional tenns. These correct for the neglect of effects caused by solnte-solvent electron correlation inclnding dispersion forces, hydrophobic interactions, dielectric saturation in the case of... 

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