Onsager factor

Of course, the response of a conductor to a solute charge distribution is complete , while that of a dielectric medium is not. So, in COSMO models, the more simply evaluated conductor-polarization free energy is scaled by a factor of 2(s — l)/(2e -f-1) after its computation (i.e., by the Onsager factor in the case of the SM5C model, however, the scaling factor is (e — l)/e - see Section 11.3.3). 

The last exponential factor takes into account a possibility of the free-positron annihilation occuring during the Ps formation time, ips (on the order of some picoseconds) with the annihilation rate 1/t2 (< 2 ns-1). Obviously, the contribution from this factor is negligible. In nonpolar molecular media at room temperature rc is 300 A. Typical thermalization lengths b of electrons are < 100 A. Thus, the Onsager factor, 1 — err, is also very close to unity. Therefore, to explain observable values of Ps yields, which never exceed 0.7, we must conclude according to Eq. (11) that the terminal positron spur contains on average 2 to 3 ion-electron pairs. 

If the term in the derivative of the field factor were negligible the expression on the left of this equation would be defined completely in terms of macroscopic measurable quantities. The specifics of the chosen cavity model enter the field factor derivative where Lorentz-Lorenz and Onsager factors may be mixed. The most commonly used procedure is to employ Onsager for the static field and Lorentz factors for the optical fields. For Fj (i = 0,1),... 

In general the Onsager factor F is an unspecific correction. A more detailed discussion can be found in reference Bottcher (1973). 

In difference to the Onsager factor F, the interaction of dipoles plays an important role in condensed systems. This is especially true also for polymeric materials. Specific interactions between molecules and segments in the case of polymers can be caused, for instance, by hydrogen bonding, steric interactions, etc., and can lead to associations of molecules or segments. 

Table 7.5 focuses on the cavity field term only and compares PCM and the classical Onsager factor. The latter is a fairly good approximation of PCM when the molecular cavity is almost spherical [i.e., in the case of HCHO and (CH3)3CCHO]. In the other cases the difference is larger, the maximum deviation occurring for HC = CCHO, whose molecular cavity, more or less of a cylindrical shape, is badly approximated by a sphere. 

Finally, to simplify the electrostatic algorithm some methods start with a dielectric permittivity e = oo corresponding to a conductor like solvent and then correct the result either by the ratio 2(e-l)/(2e-i-l)of the Onsager factor of the actual dielectric permittivity of the solvent to the limiting value when e = oo, or by the ratio of the Born factors ( -l)/e. Itis noteworthy that the difference may not be negligible for low dielectric constants. This approximation has first been introduced in the COSMO model (Klamt and Schuurmann 1993). A version of the PCM model, known as C-PCM (Barone and Cossi 1998) uses the same approximation to evaluate the electrostatic solvation term. 

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]. 

The course of the recombination processes in a particular system depends on several factors. One of the most important ones is the polarity of the system. Both geminate and bulk recombination processes are strongly influenced by the Coulomb attraction between electrons and cations, and the range of this interaction in condensed matter is determined by the dielectric constant e. The range of the Coulomb interaction in a particular system is usually represented by the Onsager radius, r, which is defined as the distance at which the electrostatic energy of a pair of elementary charges falls down to the thermal level kj,T. 

The exact treatment yields expressions which have the same form as the expressions given above only the numerical factors are different. The more detailed theory for the diffusion-convection problem between plane walls was developed by Furry, Jones, and Onsager (F10) and that for the column constructed from two concentric cylinders by Furry and Jones (Fll). Recently more attention has been given to the r61e of the temperature dependence of the transport coefficients in column operation (B9, S15). 

Exercise. Formulate the Onsager relations for nonlinear systems of diffusion type. Exercise. Sometimes Pe(x) contains an additional phase-space factor w(x), so that its dependence on 6 is displayed by... 

The formal description of thermodiffusion in the critical region has been discussed in detail by Luettmer-Strathmann [79], The diffusion coefficient of a critical mixture in the long wavelength limit contains a mobility factor, the Onsager coefficient a = ab + Aa, and a thermodynamic contribution, the static structure factor S(0) [7, 79] ... 

The asymptotic value of s, so = 0.54, falls in between the predictions of Forster model, sa = 1/n2 = 0.5, which assumes infinitely thin point dipoles, and the Onsager value, sa = 3/( hr1 + 1) = 0.6, which considers point dipoles contained in spherical cavities. It is reasonable to think that real molecules fall in between these two limits. The solvent screening factors obtained for the data set, along with the fitted screening function, Eq. 14, and the Forster and Onsager values are plotted in Fig. 8. 

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