Clausius-Mossotti/Lorentz-Lorenz

Continuum models have a long and honorable tradition in solvation modeling they ultimately have their roots in the classical formulas of Mossotti (1850), Clausius (1879), Lorentz (1880), and Lorenz (1881), based on the polarization fields in condensed media [32, 57], Chemical thermodynamics is based on free energies [58], and the modem theory of free energies in solution is traceable to Bom s derivation (1920) of the electrostatic free energy of insertion of a monatomic ion in a continuum dielectric [59], and Kirkwood and Onsager s... 

Clausius-Mossotti equation). In this expression, V designates the mole volume and Ae, Be, Cf,... are the first, second, third,... virial dielectric coefficients. A similar expansion exists for the refractive index, n, which is related to the (frequency dependent) dielectric constant as n2 = e (Lorentz-Lorenz equation, [87]). The second virial dielectric coefficient Be may be considered the sum of an orientational and a polarization term, Be = B0r + Bpo, arising from binary interactions, while the second virial refractive coefficient is given by just the polarization term, B = Bpo at high enough frequencies, the orientational component falls off to small values and the difference Be — B may be considered a measurement of the interaction-induced dipole moments [73],... 

This equation, called the Lorenz-Lorentz equation, was derived in 1880 by combining the Clausius-Mossotti expression for the local field with the idea of molecular polarizability. 

At low atomic densities N, the dielectric response of a medium can be written in the popular Clausius-Mossotti or Lorentz-Lorenz form as a function of N and a coefficient a that includes atomic or molecular polarizability ... 

The dilute limit emerges when a /z3 dilute limit, assume that the dielectric response of dense suspension follows the Lorentz-Lorenz or Clausius-Mossotti relation6 e = [(1 + 2Na/3)/(l - Na/3)] (This is the next approximate form when the number density N is too high to allow the linear relation e = 1 + Na.) Below what density N will this e be effectively linear in polarizability Expand... 

An increase in fractional free volume will reduce the number of polarisable groups per unit volume, and thereby reduce the relative permittivity of the polymer. Quantitatively, the effect may be estimated by means of the Clausius-Mossotti/Lorenz-Lorentz model for dielectric mixing (Bottcher, 1978) ... 

It is seen that the effect of the variation of the polarizability is of the same order of magnitude as the effect of statistical fluctuations in the dipole moments / described by the functions S2. For the highest frequencies which may be considered within the approximation introduced into the calculation, the correction to the Lorentz-Lorenz function is about 15 per cent larger than the corresponding correction for the static case, (i.e., to the Clausius-Mossotti function). A similar qualitative behavior may be expected for other noble gases under the same conditions. 

In 1906, J. C. Maxwell Garnett used the Maxwell Garnett theory, equation (12), for the first time to descibe the color of metal colloids glasses and of thin metal films. Equation (12) can be deviated from the Rayleigh scattering theory for spherical particles [21], or from the Lorentz-Lorenz assumption for the electrical field of a sphere and the Clausius-Mossotti Equation by using the polarizability of an metal particle if only dipole polarization is considered [22]. 

This relationship as such is not well obeyed for most compounds if the static or low-frequency relative permittivity is used, as can be judged from Table 11.1. The relationship can be correctly interpreted by using the relative permittivity due to electronic polarisation in the equation. With this in mind, substitution of the relationship given in Equation (11.10) into the Clausius-Mossotti equation yields the Lorentz-Lorenz equation ... 

Lorenz-Lorentz theory addressed the issue by extending the approach of Clausius-Mossotti to optical frequency fields (12,13). This extension relies on a spherical cavity (compare with the need for a needle-shaped cavity) and takes into account the effect of other charges. The only thorny issue is that a spherical cavity is not the best choice for anisotropic molecules. Nevertheless, the Lorenz-Lorentz approach has been widely used in studying optical properties of polymers (14). The expression of the local field is given by... 

Lorentz was the first to consider such problems for a reasonably defensible model of induced dipoles derive the local Lorentz field j and from this obtain the venerable Clausius Mossotti (or perhaps more properly Lorentz-Lorenz) formula. As shown schematically in Figure 1 (a) the molecules are assumed to be at sites on a cubic lattice with uniform macroscopic along the z axis. 

One of the presented structures is a monodispersion of subwavelength inclusions i (spheres) in dielectric host h. Fig. 2.22a. The other is polydispersion. Fig. 2.22b. The first situation can be described by the well-known MaxweU-Gamett model [171], the oldest effective medium model, obtained by the use of Clausius-Mossotti/ Lorenz-Lorentz equation. The other case is polydispersion, described by the implicit Bruggeman expression [172, 173]. 

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