Nonlinear dielectric constant equation

This equation shows that the alternating capacitance of different frequencies corresponds to each order of the nonlinear dielectric constant. Signals corresponding to 333, 3333 and 33333 were obtained by setting the reference signal of the lock-in amplifier in Figure 16.1 to frequency cop, 2 ujp and 3 uip of the applied electric field, respectively. 

The linear polarizability, a, describes the first-order response of the dipole moment with respect to external electric fields. The polarizability of a solute can be related to the dielectric constant of the solution through Debye s equation and molar refractivity through the Clausius-Mosotti equation [1], Together with the dipole moment, a dominates the intermolecular forces such as the van der Waals interactions, while its variations upon vibration determine the Raman activities. Although a corresponds to the linear response of the dipole moment, it is the first quantity of interest in nonlinear optics (NLO) and particularly for the deduction of stracture-property relationships and for the design of new... 

Solvation Effects. Many previous accounts of the activity coefficients have considered the connections between the solvation of ions and deviations from the DH limiting-laws in a semi-empirical manner, e.g., the Robinson and Stokes equation (3). In the interpretation of results according to our model, the parameter a also relates to the physical reality of a solvated ion, and the effects of polarization on the interionic forces are closely related to the nature of this entity from an electrostatic viewpoint. Without recourse to specific numerical results, we briefly illustrate the usefulness of the model by defining a polarizable cosphere (or primary solvation shell) as that small region within which the solvent responds to the ionic field in nonlinear manner the solvent outside responds linearly through mild Born-type interactions, described adequately with the use of the dielectric constant of the pure solvent. (Our comments here refer largely to activity coefficients in aqueous solution, and we assume complete dissociation of the solute. The polarizability of cations in some solvents, e.g., DMF and acetonitrile, follows a different sequence, and there is probably some ion-association.)... 

Rocchia W, Alexov E, Honig B (2001) Extending the applicability of the nonlinear Poisson-Boltzmann equation Multiple dielectric constants and multivalent ions. J Phys Chem B 105 6507-6514... 

A consequence of the complex interplay of the dielectric and thermal properties with the imposed microwave field is that both Maxwell s equations and the Fourier heat equation are mathematically nonlinear (i.e., they are in general nonlinear partial differential equations). Although analytical solutions have been proposed under particular assumptions, most often microwave heating is modeled numerically via methods such as finite difference time domain (FDTD) techniques. Both the analytical and the numerical solutions presume that the numerical values of the dielectric constants and the thermal conductivity are known over the temperature, microstructural, and chemical composition range of interest, but it is rare in practice to have such complete databases on the pertinent material properties. 

PB models36 37 solve Equation 7.8 or its nonlinear extension for stronger fields by using a finite difference grid and treating the shape of the protein in detail, while continuing to use macroscopic dielectric constants for both the protein and the solvent. 

Eigure 2 compares the LPM results for the nonlinear Poisson-Boltzmann equation (Eq. 13) and the analytical solutions of the linearized equation (Eq. 35), together with a numerical solution using the multigrid method. The parameters are the bulk ionic molar concentration c o = — 4 M, = c oNa where Na is Avogadro s number, z = 1 is the dielectric constant of the... 

Application of Born s equation to photoconductivity data implies that only electronic polarization processes occur. On a longer time scale, electrostriction leads to an increase of density and subsequently to an increase of the dielectric constant around the ion. Furthermore, for the first few shells of atoms or molecules around the charge, nonlinear dielectric effects must be considered (see Section 1.6). The Bom equation is derived under the assumption of a continuum model of the dielectric. A more adequate approach would take into account the positions of the atoms or molecules around the ion. Such attempts have been made for organic solids (Sato et al., 1981) and rare gas solids (Lyons and Sceats, 1970). 

Equation (8.18) is vahd for linear dielectric materials (i.e., polarization is directly proportional to electric field). Ferroelectric materials used in high-X capacitor dielectric materials are nonlinear. Figure 8.34 illustrates the relationships between polarization, electric field, and dielectric constant for a ferroelectric material as given by the following equation ... 

W. Rocchia, E. Alexov, and B. Honig,/. Phys. Chem. B, 105, 6507 (2002). Extending the Applicability of the Nonlinear Poisson-Boltzmann Equation Multiple Dielectric Constants... 

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