Statistical mechanics dielectric polarization

Almost all the formalism and the approximation schemes of Sections II and III have a natural extension to systems of polarizable dipolar particles, but the precise details of the extension depend on the way polarizability is introduced into the Hamiltonian. We refer to the two quite distinct Hamiltonian models that have been most thoroughly developed in this context as the constant-polarizability model and the fluctuating-polarizability model. The dielectric behavior of the former was first systematically investigated from a statistical mechanical viewpoint by Kirkwood and by Yvon, who considered the model almost exclusively in the absence of permanent dipole moments. (Kirkwood S subsequently pioneered an exact formulation of the statistical mechanics of polar molecules, but largely as a separate enterprise that did not attempt to treat the polarizability exactly.) The general case of polar-polarizable particles remained only very partially developed ... 

The Marcus treatment uses a classical statistical mechanical approach to calculate the activation energy required to surmount the barrier. It assumes a weakly adiabatic electron transfer process and non-equilibrium dielectric polarization of the solvent (continuum) as the source of activation. This model also considers the vibrational contributions of the inner solvation sphere. The Hush treatment considers ion-dipole and ligand field concepts in the treatment of inner coordination sphere contributions to the energy of activation [55, 56]. 

In polymer theory, the LDT result corresponds to the theory of large chain extensions P. J. Flory, Statistical Mechanics of Chain Molecules, Interscience, New York, 1969. Another mapping exists onto Debye s theory for dielectric properties of molecules with permanent dipoles P. Debye, Polar Molecules, reprinted, Dover, New York, 1958. 

The upper sign corresponds to a water-dielectric , and the lower one to a water-conductor type of interface. Equation (7) shows that a charge located next to a conductor will be attracted by its own image, and dielectrics in aqueous solutions will repel it. For a review of statistical-mechanical models of the double layer near a single interface we refer to [7], and here we would like only to illustrate how the image forces will alter the ion concentration and the electrostatic potential distribution next to a single wall. At a low electrolyte concentration the self-image forces will mostly dominate, and the ion-surface interaction will only be affected by the polarization due... 

Rigorous statistical mechanical analysis indicates that the dielectric relaxation function 0(f) of an isotropic system in the linear response regime is equivalent to an autocorrelation function of a microscopic polarization p(f) fluctuating through the molecular motion at equilibrium (Cole, 1967 Kubo, 1957) ... 

The problem of the interaction of dipoles was treated by Kirkwood (1939, 1940, 1946) and Frohlich (1958). Starting point is the statistical mechanics (Bbttcher 1973 Landau and Lifschitz 1979) where the contribution of the orientation polarization to the dielectric permittivity is expressed by... 

It can be concluded that remanent polarization and hence the piezoelectric response of a material are determined by Ae this makes it a practical criterion to use when designing piezoelectric amorphous polymers. The Dielectric relaxation strength Ae may be the result of either free or cooperative dipole motion. Dielectric theory yields a mathematical approach for examining the dielectric relaxation Ae due to free rotation of the dipoles. The equation incorporates Debye s work based on statistical mechanics, the Clausius-Mossotti equation, and the Onsager local field and neglects short-range interactions (43) ... 

There is an obvious choice for the simple case when only a single characteristic time is included. It goes back to Debye, who proposed it in a famous work on the dielectric properties of polar liquids, based on a statistical mechanical theory. We formulate the equation for the above mentioned simple mechanical relaxation process, associated with transitions between two conformational states only, and consider a creep experiment under shear stress. 

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