Polarizable spheres

To answer this question, let us first consider a neutral molecule that is usually said to be polar if it possesses a dipole moment (the term dipolar would be more appropriate)1 . In solution, the solute-solvent interactions result not only from the permanent dipole moments of solute or solvent molecules, but also from their polarizabilities. Let us recall that the polarizability a of a spherical molecule is defined by means of the dipole m = E induced by an external electric field E in its own direction. Figure 7.1 shows the four major dielectric interactions (dipole-dipole, solute dipole-solvent polarizability, solute polarizability-solvent dipole, polarizability-polarizability). Analytical expressions of the corresponding energy terms can be derived within the simple model of spherical-centered dipoles in isotropically polarizable spheres (Suppan, 1990). These four non-specific dielectric in-... 

If solvent (or environment) relaxation is complete, equations for the dipole-dipole interaction solvatochromic shifts can be derived within the simple model of spherical-centered dipoles in isotropically polarizable spheres and within the assumption of equal dipole moments in Franck-Condon and relaxed states. The solvatochromic shifts (expressed in wavenumbers) are then given by Eqs (7.3) and (7.4) for absorption and emission, respectively ... 

Such specifics of interaction between dipoles of H O, cations and anions mostly determine the structure of water solution. The simplest idea of it is provided by the statistical theory of diluted solutions of strong electrolytes proposed by Peter Joseph Debye (1884-1966) and Erich Armand Hiickel (1896-1980) in 1923. Under this theory ions are treated as rigid non-polarizable spheres separated by a uniform medium with high value of the dielectric constant. At that, structure of the solution is function of distances dipoles H O and ions. Depending on it, it is customary to distinguish molecular and supramolecular structure. Molecular structure is determined by a direct effect of ions on the orientation and mobility of water dipoles and is manifested first of all by the formation of hydrates. Supramolecular structure is caused by undisturbed interaction of H O molecules between each other (Figure 1.2). 

The need to calculate activities coefficients occurred in the 1920 s. The search for methods of their determination was conducted in two major directions from the position of remote electrostatic and close Coulomb interactions between ions. Remote electrostatic interactions in diluted solutions of strong electrolytes were studied by Peter Debye and E. Hiickel. In 1923 they proposed a theory of ion interaction in diluted solutions. According to this theory, electrolytes in a solution are totally dissociated and their ions behave as rigid non-polarizable spheres with charges in the center, which are distributed in a uniform medium with high values of... 

For almost every case, o is a symmetric matrix ( xy=Oyx, etc.). Now suppose that the scattering body is not just a polarizable sphere but has vibrational modes of its own - normal modes, Q, described by... 

In the case of pure induced polarization the solute molecules Bj may be considered as polarizable spheres associated with a molecular permittivity (s )j and Py = 0. When these molecules are immersed in a large quantity of nonpolar solvent the total permittivity of the solution s is practically that of the solvent. 

In the case of isotropic polarizabilities and uniaxial permanent dipoles the electric fields which actually work on the molecules are given by Eqs. (3.18) and (3.34). The internal field of polar polarizable spheres is given by... 

In the 1970s, S. S. Dukhin s group was perhaps the first to recognize that the electrophoretic mobility of polarizable particles must generally depend on the electric field [9]. In a series of Russian papers, which have yet to gain widespread attention, they predicted perturbations of the mobility as AZ oc and thus nonlinear electrokinetic motion At/ oc, which they have come to call the Stotz-Wien effect. For the case of a steady weak field applied to an ideally polarizable sphere of radius a, A. S. Dukhin derived an expansion for the mobility ... 

In the 1970s, Shilov and Estrella-Lopis first recognized that electrohydrodynamics (what we now call ICEO) can contribute to the motion of particles in low-frequency, nonuniform electric fields [17], in addition to DEP, although the effect has not been studied much in theory or experiment. Shilov and Simonova analyzed the problem of an ideally polarizable sphere in a uniform field gradient and made the remarkable prediction that the particle does not move. Due to equal and opposite motions by DEP and ICEP, the sphere levitates in the field while driving a steady ICEO flow, but this is a unique case. 

The value of the induced potential depends on not only the externally applied electric field but also the electric properties of both the solid surface and the liquid electrolyte and geometry of the solid surface. To date, more attention has been given to the induced potential on ideally polarizable surfaces. To obtain the induced potential of an ideally polarizable sphere, two assumptions are applied after the polarization, the electric field lines near the conducting surface are distorted and go around the conducting sphere (expressed as ,. = 0 as shown in Fig. 3) and the potential at the ideally polarized surface is identical (expressed as = 0 at r < a). Utilizing the assumptions above, Dykhin and Bazant solved the Laplace s equation [2, 3] and got the analytical solution of the induced potential on a polarized cylinder and sphere. The analytical solutions to the Laplace s equation are only limited to cylinder and sphere geometries, because the boundary conditions for the Laplace s equation are much easier in these cases. The obtained induced potential on an ideally polarizable sphere is... 

The approximate calculation of the solvent reorganization energy for the charge transfer between spherical reactants was carried out by Marcus [2]. Under the assumption that the distance between the centres of reactants R is much larger than their radii a and b and that the reactants can be described by non-polarizable spheres with the charges strictly and uniformly distributed over their surfaces, the expression for Eg obtained in Ref. [2] has the form... 

Stotz-Wien effect. For the case of a steady weak field applied to an ideally polarizable sphere of radius a, A.S. Dukhin derived an expansion for the mobility. 

Moreover, it is known that molten alkali fiuoride mixtures (LiF, NaF, KF) behave as a bath of polarizable spheres containing free cations (Li, Na" ", K ) and free F anions [15]. In that case, the fluorine chemical shift for a given composition is given by ... 

In the work referred to, the reference system adopted is that of polar-polarizable spheres in a mean field. 

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