Onsager continuum limit

Jepsen and Friedman found, however, that for microscopic impurities, (2.34a) and (2.34b)—in contrast to (2.34c)—no longer appeared to be satisfied beyond the lowest order iny in the low-density approximation they were considering, which left open the asymptotic form the microscopic results would have. Equation 2.33 reveals that only if the Onsager approximation (2.30d) were satisfied in the molecular solvent would (2.33) and (2.34) be the same. The reason for this will become clear in our discussion of the y->0 limit below, where we show that only in the Onsager continuum limit, in which (2.30d) becomes exact, is the dielectric response to each solvent dipole that of a vacuum in a macroscopic sphere surrounding the solvent dipole. Thus only in the Onsager continuum limit are the assumptions satisfied under which one can identify each solvent particle as a macrosphere within which 6= 1, and so assure the identity of the full set of ratios in (2.33) to (2.34). 

In the quantum mechanical continuum model, the solute is embedded in a cavity while the solvent, treated as a continuous medium having the same dielectric constant as the bulk liquid, is incorporated in the solute Hamiltonian as a perturbation. In this reaction field approach, which has its origin in Onsager s work, the bulk medium is polarized by the solute molecules and subsequently back-polarizes the solute, etc. The continuum approach has been criticized for its neglect of the molecular structure of the solvent. Also, the higher-order moments of the charge distribution, which in general are not included in the calculations, may have important effects on the results. Another important limitation of the early implementations of this method was the lack of a realistic representation of the cavity form and size in relation to the shape of the solute. 

Naturally Onsager s derivation cannot be exact for a molecular fluid, since on a microscopic scale the surrounding medium cannot be treated as a continuum. On the other hand we have recovered the same expression as an exact result in the limit y->0. This is done by choosing a dipole interaction as given by (2.44), where 0 fulfills (2.74). As we have just shown, this... 

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 decreases in permittivity when ions are added to a polar solvent were traditionally interpreted in terms of saturation or solvation of local ionic environments (76) until Hubbard and Onsager (77) (78) worked out a continuum theory of the kinetic depolarization effect. This arises from the fact that part of the electric field solvent dipoles near an ion is from the moving ion and similarly for the ion in the field of reorienting dipoles with the consequences that both responses are delayed in proportion to the relaxation time of the solvent polarization. The remarkably simple Hubbard-Onsager expression for the resulting decrement of static (or better limiting low frequency) permittivity can be written... 

The earlier attempt to approach the electrostatic contribution to the free energy of solvation is due to Kirkwood (1934). This model is based on a multipole expansion of the charge distribution of the solute at the center of a spherical cavity surroimded by a continuum represented by the dielectric permittivity of the solvent. When this expansion is limited to rank 1 which corresponds to a pure dipole fi, one finds the Onsager model (Onsager 1936) in which the electrostatic contribution to the free energy of solvation by a solvent of dielectric constant e of a molecule having a dipole moment in a cavity of radius a takes the expression ... 

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