Free energy charging formula

Other electrostatic processes studied include proton binding [43] and changing the molecular charge distribution [44], The free energy expansion formula (12.5) was used, including terms up to second order... 

The reorganization energy term derives from the solvent being unable to reorient on the same timescale as the electron transfer takes place. Thus, at the instant of transfer, the bulk dielectric portion of the solvent reaction field is oriented to solvate charge on species A, and not B, and over the course of the electron transfer only the optical part of the solvent reaction field can relax to the change in tire position of the charge (see Section 14.6). If the Bom formula (Eq. (11.12)) is used to compute the solvation free energies of the various equilibrium and non-equilibrium species involved, one finds that... 

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... 

Thanks to efficient recurrence formulae, multipole moments and multipole moment derivatives can be calculated at very high order with a low computational cost. The calculation of reaction field factors, however, may become computationally expensive at high order due to the increasing number of linear equations to be solved. Thus, in practice, the multipole moment expansion is cut off at a maximum value of f (/max), usually taken around 6. In order to get an order of magnitude of the error introduced by the truncation, let us consider Kirkwood s equations [5] for the free energy of a charge distribution of charges q, and r, in a spherical cavity of radius a ... 

A population of vacancies on one subset of atoms created by displacing some atoms into normally unoccupied interstitial sites constitute a second arrangement of paired point defects, termed Frenkel defects (Figure 2(b), (c)). Because one species of atom or ion is simply being redistributed in the crystal, charge balance is not an issue. A Frenkel defect in a crystal of formula MX consists of one interstitial cation plus one cation vacancy, or one interstitial anion plus one anion vacancy. Equally, a Frenkel defect in a crystal of formula MX2 can consist of one interstitial cation plus one cation vacancy, or one interstitial anion plus one anion vacancy. As with the other point defects, it is found that the free energy of a crystal is lowered by the presence of Frenkel defects and so a popnlation of these intrinsic defects is to be expected at temperatures above 0 K. The calculation of the number of Frenkel defects in a crystal can proceed along lines parallel to those for Schottky defects. The appropriate chemical equilibrium for cation defects is ... 

Similarly to the expressions found by Singer and Chandler [80] for the RISM/HNC equations, the KH approximation (4.f3) allows one to obtain the free energy functions in a closed analytical form avoiding the necessity of numerical coupHng parameter integration. The derivation is analogous for both RISM and 3D-RISM/KH equations [28], and is shown here in the context of the 3D approach. The excess part of the solvation chemical potential, in excess over the ideal translational term, can be related to the 3D site correlation functions by the Kirkwood s charging formula... 

If the multipole expansion is truncated at the charge level (i.e., the lowest one) and the center of the expansion is shifted to the different nuclei, a generalized Born approach is achieved. Thus, the Born formula for the free energy of an ion of charge q and radius a in a dielectric of permittivity e, namely. 

It follows from Deryagin s theory that the force of adhesion depends on the curvature [formula (1.62)] of the surfaces in contact. The influence of the properties of the surfaces on the adhesion is taken into account by the free energy /(O) the effect of capillary forces and the particle charges on the adhesive force, however, is not taken into consideration (see Chapter II, 12-14 on this point). 

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