Dielectric constant dependence

The fact that the dielectric constant depends on the frequency gives SPFM an interesting spectroscopic character. Local dielectric spectroscopy, i.e., the study of s(w), can be performed by varying the frequency of the applied bias. Application of this capability in the RF range has been pursued by Xiang et al. in the smdy of metal and superconductor films [39,40] and dielectric materials [41]. In these applications a metallic tip in contact with the surface was used. 

The dielectric constant of a polymer (K) (which we also refer to as relative electric permittivity or electric inductive capacity) is a measure of its interaction with an electrical field in which it is placed. It is inversely related to volume resistivity. The dielectric constant depends strongly on the polarizability of molecules tvithin the polymer. In polymers with negligible dipole moments, the dielectric constant is low and it is essentially independent of temperature and the frequency of an alternating electric field. Polymers with polar constituents have higher dielectric constants. When we place such polymers in an electrical field, their dipoles attempt... 

The dielectric constant depends on the electric field strength of the environment (this fact was discussed in the treatment of hydration in Chapter 2). If the molecules in the environment contain charges (e.g., dipoles), the degree of ordering of these molecules would affect the strength of the electric field. Water molecules are dipoles, and as such have the capability to affect the strength of the electric field, and consequently, the dielectric constant... 

Continuum solvation models consider the solvent as a homogeneous, isotropic, linear dielectric medium [104], The solute is considered to occupy a cavity in this medium. The ability of a bulk dielectric medium to be polarized and hence to exert an electric field back on the solute (this field is called the reaction field) is determined by the dielectric constant. The dielectric constant depends on the frequency of the applied field, and for equilibrium solvation we use the static dielectric constant that corresponds to a slowly changing field. In order to obtain accurate results, the solute charge distribution should be optimized in the presence of the field (the reaction field) exerted back on the solute by the dielectric medium. This is usually done by a quantum mechanical molecular orbital calculation called a self-consistent reaction field (SCRF) calculation, which is iterative since the reaction field depends on the distortion of the solute wave function and vice versa. While the assumption of linear homogeneous response is adequate for the solvent molecules at distant positions, it is a poor representation for the solute-solvent interaction in the first solvation shell. In this case, the solute sees the atomic-scale charge distribution of the solvent molecules and polarizes nonlinearly and system specifically on an atomic scale (see Figure 3.9). More generally, one could say that the breakdown of the linear response approximation is connected with the fact that the liquid medium is structured [105],... 

Static dielectric constant depends on the displacement of ions from their regular positions in an applied electric field. 

Approaches have been developed which modify the traditional double-layer theory by assuming that the dielectric constant depends upon position. This dependence is considered to occur because the dielectric constant is affected by the electric field [9-12]. However, the dielectric constant can also be changed by the presence of hairs on the surface. Indeed, hairy chains on the surface constrain the orientation of the water molecules nearby, and, as a result, in the region near the surface the dielectric constant becomes lower... 

As with normal electrolytes (Section 2.12.1), the dielectric constant depends linearly on concentration ... 

Methods for Determining the Dielectric Constant. The method used by Drude1 for determining the dielectric constant depends upon the fact that, unless dispersion (to be discussed below) takes place, the dielectric constant may be determined from the relationship... 

If the film cannot be freely deformed in its plane, the piezoelectric current is called t/33 or dj. If the variation in the electric field is measured per unit of stress, g coefficients are obtained that are connected by the correlation of g = d/e where e is the dielectric constant depending on the film thickness. Constants g and tf are most widely used in the design of electromechanical transducers. The yield from the conversion of mechanical energy into electrical energy is represented by the electromechanical coupling coefficient ATjby Eq. (3.3). 

The selection rules governing photon absorption in solids determine the oscillator strength of the optical transition and its energy dependence. The expressions obtained for the imaginary component of the optical dielectric constant depend on whether the transition is allowed in the dipole approximation and on whether the simultaneous absorption or emission of a phonon is involved. In pure single-crystal materials, the absorption coefficient can be described conveniently by relationships that take the general form [4]... 

From (9.32), which shows that is determined by the least effective conductivity mechanism, it is evident that this static conductivity may be quite critically dependent upon the presence of proton-donor or proton-acceptor impurities and, in this sense, the behaviour of ice is quite analogous to that of electronic semiconductors like germanium or silicon. Mole fractions of impurity which are significant are of the order of io . The high-frequency conductivity and relaxation time t are, from (9.30) and (9.35), rather less sensitive to impurity content than is doping levels. The static dielectric constant depends on purity in a rather complex way, as we shall discuss presently. 

However, we can still follow the steps from the previous section if we assume that the dielectric constant depends on R through the local volume fraction occupied by the solvent (R), the number of monomers in that volume element Ai(R), and the local temperature r(R), that is. 

If a polymer contains polar groups whose moments do not cancel out, the actual value of the dielectric constant depends very strongly on the molecular conformation, which in turn may depend on the dipoles, because strong repulsion between parallel dipoles will cause the conformation to... 

In general, polarizability is difficult to determine experimentally. However, the ratio of the capacity of a condenser in a vacuum to that in the medium under consideration, i.e., the dielectric constant of the medium, can be measured. At low frequencies, the dielectric constant of electrical nonconductors is almost independent of the frequency. At high frequencies, the dielectric constant depends on the frequency, since the permanent dipoles are no longer able to establish a preferred orientation, because of rapid alteration of the field. 

It is desirable to have direct evidence from other experimental methods such as neutron diffraction and scattering. The dielectric constant of the hydrogen sulphide clathrate increased as the temperature decreased down to 10 - 15 K [13]. It decreased at the lower temperature. The temperature of the maximum of the dielectric constant depended on the occupancy x. However, it was always higher than the temperature of the heat capacity peak. The mechanism of the phase transition at 7.6 K will be discussed below. 

The dielectric properties of water and other polar substances are measured by the relative dielectric constant, e, valid in a static electric field. For a propagating EM field, the dielectric constant depends on the frequency of the radiation, for the reasons just given. At light frequencies, the oscillation of dipoles or the vibrations of nuclei are unimportant. Then, as follows from Maxwell s equations, the dielectric constant for a substance is equal to the square of the refractive index (not proven) ... 

Figure 3. Dielectric constant dependence on frequency for Epiclon-derived Pis. The e values at 1 kHz, 10 kHz and 100 kHz were experimentally obtained on a LCR METER, while those at 1 MHz were estimated by Maxwell s identity using ellipsometrical measurements...

Substitute the Born electrostatic free energy Gei from Equation (22.63) into Equation (22.64), use Dvapor = 1 and recognize that water s dielectric constant depends on temperature,... 

In this equation the quantity F(T/T )d ln(T/rQ) is the distribution of relaxation times, while the quantity in brackets is the Debye relaxation function, that is, see Eq. (5). A particularly useful form for representing the complex dielectric constant dependence on frequency is given by the following equation ... 

Apart from an occasional reference to polymers, the equations developed in Sections IV and V are general and not necessarily limited to long-chain molecules. However, their application to small molecules is handicapped by the lack of information on Dg, though y can usually be estimated reasonably well because of the preponderance of x-ray data on small molecules. Smyth has reviewed, quite extensively, the dielectric properties of polar solids. In his work he attributed the low values of e to solidification, which usually fixes the molecule with such rigidity in the lattice that little or no orientation of the dipoles in an externally applied field is possible. Therefore the orientation polarization is zero, and the dielectric constant depends on the same factors as those in the nonpolar molecular solid. The dielectric constant temperature curves of these polar molecules show curves of great discontinuity at the melting point, for in... 

Since LTCCs are basically composite structures of glass and crystals, controlling their dielectric constant depends largely on the combination of constituent materials of the composite and its material composition (volume fraction of the constituent materials). In addition, the dielectric constant of... 

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