Onsager field

The internal field factors are now in general to be adjusted to take account of the differing environments of the molecules of the two species. In particular the reaction field introduced in the Onsager field will be different, even in very dilute solution, in the cavity surrounding the polar solute as compared to that of a non-polar solvent. 

In case (I), the theory that has been used to describe the EFISH analysis in earlier sections, the isolated molecule calculation provides the pz value and the internal field factors adjust the fields to allow for the polarization on the cavity surface. The elfect of reaction fields due to the additional polarization on the cavity walls induced by the permanent and induced dipoles on the central molecule is implicitly included in the low frequency Onsager field factor through the dielectric constant values. The choice of the high frequency dielectric constant ( o) in this formulation is rather ill defined and no account is taken of changes in the static or dynamic polarizabilities of the molecule as a result of the surrounding fields. 

Onsager resuir in this limit provided the lattice sums were evaluated for a continuum with Tra /3 s V/H but smaller reductions of the Lorenta result for discrete sums over more realistic cubic lattice points. Extensions to third order dipole interactions for rigid dipoles by Rosenberg and Lax (25) and with harmonic oscillator induced dipoles included by Cole (26) showed further differences from the Onsager result even for a continuum with the conclusion that for the model the true result lies somewhere between the Lorenta and Onsager field expressions. 

The Ising model has been solved exactly in one and two dimensions Onsager s solution of the model in two dimensions is only at zero field. Infomiation about the Ising model in tliree dunensions comes from high- and low-temperature expansions pioneered by Domb and Sykes [104] and others. We will discuss tire solution to the 1D Ising model in the presence of a magnetic field and the results of the solution to the 2D Ising model at zero field. 

Onsager s solution to the 2D Ising model in zero field (H= 0) is one of the most celebrated results in theoretical chemistry [105] it is the first example of critical exponents. Also, the solution for the Ising model can be mapped onto the lattice gas, binary alloy and a host of other systems that have Hamiltonians that are isomorphic to the Ising model Hamiltonian. 

That analyticity was the source of the problem should have been obvious from the work of Onsager (1944) [16] who obtained an exact solution for the two-dimensional Ising model in zero field and found that the heat capacity goes to infinity at the transition, a logarithmic singularity tiiat yields a = 0, but not the a = 0 of the analytic theory, which corresponds to a finite discontinuity. (Wliile diverging at the critical point, the heat capacity is synnnetrical without an actual discontinuity, so perhaps should be called third-order.)... 

Onsager s reaction field model in its original fonn offers a description of major aspects of equilibrium solvation effects on reaction rates in solution that includes the basic physical ideas, but the inlierent simplifications seriously limit its practical use for quantitative predictions. It smce has been extended along several lines, some of which are briefly sunnnarized in the next section. 

Onsager s original reaction field method imposes some serious lunitations the description of the solute as a point dipole located at the centre of a cavity, the spherical fonn of the cavity and the assumption that cavity size and solute dipole moment are independent of the solvent dielectric constant. 

Kirkwood generalized the Onsager reaction field method to arbitrary charge distributions and, for a spherical cavity, obtained the Gibbs free energy of solvation in tenns of a miiltipole expansion of the electrostatic field generated by the charge distribution [12, 1 3]... 

Fig. 7. The field-dependence of the charge-generation efficiency of a 2.0- lm thick (0) a l.l-).tm thick ( ), and 1.8-).tm thick (A) fuUerene/PMPS film obtained with positive charging and 340-nm irradiation (A). The soHd lines are calculated from the Onsager model. The best-fit curve is obtained with Tq = 2.7 nm and = 0.85. Also plotted is the charge-generation efficiency of a fuUerene/PVK film (+) obtained with positive charging and 340-nm irradiation (B). The soHd lines are calculated from the Onsager model. The best-fit curve is obtained with = 1.9 nm and = 0.9 (13).

A variety of methodologies have been implemented for the reaction field. The basic equation for the dielectric continuum model is the Poisson-Laplace equation, by which the electrostatic field in a cavity with an arbitrary shape and size is calculated, although some methods do not satisfy the equation. Because the solute s electronic strucmre and the reaction field depend on each other, a nonlinear equation (modified Schrddinger equation) has to be solved in an iterative manner. In practice this is achieved by modifying the electronic Hamiltonian or Fock operator, which is defined through the shape and size of the cavity and the description of the solute s electronic distribution. If one takes a dipole moment approximation for the solute s electronic distribution and a spherical cavity (Onsager s reaction field), the interaction can be derived rather easily and an analytical expression of theFock operator is obtained. However, such an expression is not feasible for an arbitrary electronic distribution in an arbitrary cavity fitted to the molecular shape. In this case the Fock operator is very complicated and has to be prepared by a numerical procedure. 

The simplest SCRF model is the Onsager reaction field model. In this method, the solute occupies a fixed spherical cavity of radius Oq within the solvent field. A dipole in the molecule will induce a dipole in the medium, and the electric field applied by the solvent dipole will in turn interact with the molecular dipole, leading to net stabilization. 

Molecules do not consist of rigid arrays of point charges, and on application of an external electrostatic field the electrons and protons will rearrange themselves until the interaction energy is a minimum. In classical electrostatics, where we deal with macroscopic samples, the phenomenon is referred to as the induced polarization. I dealt with this in Chapter 15, when we discussed the Onsager model of solvation. The nuclei and the electrons will tend to move in opposite directions when a field is applied, and so the electric dipole moment will change. Again, in classical electrostatics we study the induced dipole moment per unit volume. 

Incomplete Dissociation into Free Ions. As is well known, there are many substances which behave as a strong electrolyte when dissolved in one solvent, but as a weak electrolyte when dissolved in another solvent. In any solvent the Debye-IIiickel-Onsager theory predicts how the ions of a solute should behave in an applied electric field, if the solute is completely dissociated into free ions. When we wish to survey the electrical conductivity of those solutes which (in certain solvents) behave as weak electrolytes, we have to ask, in each case, the question posed in Sec. 20 in this solution is it true that, at any moment, every ion responds to the applied electric field in the way predicted by the Debye-Hiickel theory, or does a certain fraction of the solute fail to respond to the field in this way In cases where it is true that, at any moment, a certain fraction of the solute fails to contribute to the conductivity, we have to ask the further question is this failure due to the presence of short-range forces of attraction, or can it be due merely to the presence of strong electrostatic forces ... 

The reason for the success of this type of data fitting is that for moderately large barriers it becomes unimportant whether escape for the image potential is treated within the framework of the Onsager or the RS model. An indication that the Onsager description is, nevertheless, more appropriate is the intersection of j(/ ) curves for variable temperature at high electric fields. This is a characteristic feature of Onsager processes 24J. 

It should be kept in mind that all transport processes in electrolytes and electrodes have to be described in general by irreversible thermodynamics. The equations given above hold only in the case that asymmetric Onsager coefficients are negligible and the fluxes of different species are independent of each other. This should not be confused with chemical diffusion processes in which the interaction is caused by the formation of internal electric fields. Enhancements of the diffusion of ions in electrode materials by a factor of up to 70000 were observed in the case of LiiSb [15]. 

Perhaps the best starting point in a review of the nonequilibrium field, and certainly the work that most directly influenced the present theory, is Onsager s celebrated 1931 paper on the reciprocal relations [10]. This showed that the symmetry of the linear hydrodynamic transport matrix was a consequence of the time reversibility of Hamilton s equations of motion. This is an early example of the overlap between macroscopic thermodynamics and microscopic statistical mechanics. The consequences of time reversibility play an essential role in the present nonequilibrium theory, and in various fluctuation and work theorems to be discussed shortly. 

The first approximate calculation was carried out by Debye and Hiickel and later by Onsager, who obtained the following relationship for the relative strength of the relaxation field AE/E in a very dilute solution of a single uni-univalent electrolyte... 

Pimblott (1993) has used MC and ME methods for the external field (E) dependence of the escape probability (Pesc) for multiple ion-pair spurs. At low fields, Pesc increases linearly with E with a slope-to-intercept ratio (S/I) very similar to the isolated ion-pair case as given by Onsager (1938). Therefore, from the agreement of the experimental S/I with the Onsager value, one cannot conclude that only isolated ion-pairs are involved. However, the near equality of S/I is contingent on small Pesc, which is not expected at high fields. 

With r = 28.45 nm, r = 3.0 nm, and r = 8.39 nm, Bartczak and Hummel (1986) compute the escape probability Pesc = 0.0336, 0.0261, and 0.0230 respectively for N = 1, 2, and 3. While the first is comparable to the Onsager value, the latter are new results. The kinetics of recombination for the isolated pair, found by Bartczak and Hummel (1987) using MC, is very similar to that obtained by Abell e al. (1972). For N > 1, these authors found the recombination kinetics to be faster than that for the isolated pair. For two pairs, the calculated escape probability increased with the external field, but not as strongly as for N = 1. 

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