Permittivity frequency dependence

Dielectric measurements were used to evaluate the degrees of inter- and intramolecular hydrogen bonding in novolac resins.39 The frequency dependence of complex permittivity (s ) within a relaxation region can be described with a Havriliak and Negami function (HN function) ... 

In the equation s is the measured dielectric constant and e0 the permittivity of the vacuum, M is the molar mass and p the molecular density, while Aa and A (po2) are the isotope effects on the polarizability and the square of the permanent dipole moment respectively. Unfortunately, because the isotope effects under discussion are small, and high precision in measurements of bulk phase polarization is difficult to achieve, this approach has fallen into disfavor and now is only rarely used. Polarizability isotope effects, Aa, are better determined by measuring the frequency dependence of the refractive index (see below), and isotope effects on permanent dipole moments with spectroscopic experiments. 

Now let us examine what would happen to the response of the dielectric if we put an alternating voltage on the capacitor of frequency co. If CO is low (a few Hz) we would expect the material to respond in a similar manner to the fixed-voltage case, that is d (static) = e(co) = e(0). (It should be noted that eo, the permittivity of free space, is not frequency-dependent and that E(0)/eo = H, the static dielectric constant of the medium.) However, if we were to increase co to above microwave frequencies, the rotational dipole response of the medium would disappear and hence e(co) must fall. Similarly, as we increase co to above IR frequencies, the vibrational response to the field will be lost and e(co) will again fall. Once we are above far-UV frequencies, all dielectrics behave much like a plasma and eventually, at very high values, e(co)lto = 1. 

The temperature and frequency dependence of the complex dielectric permittivity a for both 2-chlorocydohexyi isobutyrate (CCHI) and poly(2-chlorocyclohexyl acrylate) (PCCHA) is reported. The analysis of the dielectric results in terms of the electric modulus suggests that whereas the conductive processes in CCHI are produced only by free charges, the conductivity observed in PCCHA involves both free charges and interfacial phenomena. The 4x4 RIS scheme presented which accounts for two rotational states about the CH-CO bonds of the side group reproduces the intramolecular correlation coefficient of the polymer. 

The dielectric measurements were carried out in a plate capacitor and frequency dependences of complex permittivity e = e — is (e and e" being its real and imaginary part, respectively) were determined [33] in the range /= 20 Hz- 200 kHz. 

If the electric field is reversed more rapidly, for example, at 1012 s-1, then the smallest molecules are no longer able to respond sufficiently rapidly before the electric field is reversed and therefore the permittivity falls, that is, the solution is no longer able to store the energy as a capacitor. For a polar liquid, the permittivity is frequency dependent in the range of 106 (radio frequencies) to 1012 s-1 (infrared frequencies). 

Though the original work is difficult to understand very good reviews about the van der Waals interaction between macroscopic bodies have appeared [114,120], In the macroscopic treatment the molecular polarizability and the ionization frequency are replaced by the static and frequency dependent dielectric permittivity. The Hamaker constant turns out to be the sum over many frequencies. The sum can be converted into an integral. For a material 1 interacting with material 2 across a medium 3, the non-retarded5 Hamaker constant is... 

We have introduced the effective complex susceptibility x ( ) = X,( )+ X ) stipulated by reorienting dipoles. This scalar quantity plays a fundamental role in subsequent description, since it connects the properties and parameters of our molecular models with the frequency dependences of the complex permittivity s (v) and the absorption coefficient ot (v) calculated for these models. 

Figure 28. Experimental frequency dependences of dielectric parameters recorded for liquid water (a) Real (curve 1) and imaginary (curve 2) parts of the complex permittivity at 27°C. The data are from Refs. 42 (solid lines) and 17 (circles), (b) Absorption coefficient. Solid line and crosses 1 refer to 1°C filled circles 2 refer to 27°C dashed line and squares 3 refer to 50°C. For lines the data from Ref. 17 were employed, for circles the data are from Ref. 42, for crosses and squares the data are from Ref. 53.

Figs. 32a-c illustrate the absorption spectra, calculated, respectively, for water H20 at 27°C, water H20 at 22.2°C, and water D20 at 22.2°C dotted lines show the contribution to the absorption coefficient due to vibrations of nonrigid dipoles. The latter contribution is found from the expression which follows from Eqs. (242) and (255). The experimental data [42, 51] are shown by squares. The dash-and-dotted line in Fig. 32b represents the result of calculations from the empirical formula by Liebe et al. [17] (given also in Section IV.G.2) for the complex permittivity of H20 at 27°C comprising double Debye-double Lorentz frequency dependences. 

Equations (281b) and (282) determine the frequency dependence of the reorienting complex permittivity e r(v). One can estimate the principal (Debye) relaxation time by using the relation... 

The frequency dependence of the complex permittivity pertinent to the vibration process is described by Eqs. (290a) and (293) ... 

Figure 35. Frequency dependence in the submillimeter wavelength region of the real (a, b) and imaginary (c, d) parts of the complex permittivity. Solid lines Calculation for the composite HC-HO model. Dashed lines Experimental data [51]. Dashed-and-dotted lines show the contributions to the calculated quantities due to stretching vibrations of an effective non-rigid dipole. The vertical lines are pertinent to the estimated frequency v b of the second stochastic process. Parts (a) and (c) refer to ordinary water, and parts (b) and (d) refer to heavy water. Temperature 22.2°C.

For the closed description of the electron transfer in polar medium, it is necessary to express the reorganization energy in the formula (27) via the characteristics of polar media (it is assumed that the high-temperature approach can be always applied to the outer-sphere degrees of freedom). It was done in the works [12, 19, 23], and most consistently in the work of Ovchinnikov and Ovchinnikova [24] where the frequency dependence of the medium dielectric permittivity e(co) is taken into account exactly, but the spatial dispersion was neglected. 

These are obtained by introducing an explicit time dependence of the permittivity. This dependence, which is specific to each solvent is of a complex nature, cannot in general be represented through an analytic function. What we can do is to derive semiempirical formulae either by applying theoretical models based on measurements of relaxation times (such as that formulated by Debye) or by determining through experiments the behaviour of the permittivity with respect to the frequency of an external applied field. 

The solution of the time-dependent HF or KS Equation (2.184) can be obtained within a time-dependent coupled HF or KS approaches (TDHF or TDDFT) by expanding all the involved matrices (F, R, C and e) in powers of the field components. It has to be noted that the solvent-induced matrices present in F(,(R) depend on the frequency-dependent nature of the field as they depend on the density matrix R and as they are determined by the value of the solvent dielectric permittivity at the resulting frequency. 

The dispersion interaction between an atom and a metal surface was first calculated by Lennard-Jones in 1932, who considered the metal as a perfect conductor for static and time-dependent fields, using a point dipole for the molecule [44], Although these results overestimate the dispersion energy, the correct l/d3 dependence was recovered (d is the metal-molecule distance). Later studies [45 17] extended the work of Lennard-Jones to dielectrics with a frequency-dependent dielectric constant [48] (real metals may be approximated in this way) and took into account electromagnetic retardation effects. Limiting ourselves to small molecule-metal distances, the dispersion interaction of a molecule characterized by a frequency-dependent isotropic polarizability a embedded in a dielectric medium with permittivity esol (note that no cavity is built around the molecule) reads ... 

The chemical nature of the metal appears in Equation (2.333) via the frequency-dependent permittivity emet(w), evaluated at the imaginary frequency iw. For simple metals, the Drude form is often reasonable ... 

A frequency dependence of complex dielectric permittivity of polar polymer reveals two sets or two branches of relaxation processes (Adachi and Kotaka 1993), which correspond to the two branches of conformational relaxation, described in Section 4.2.4. The available empirical data on the molecular-weight dependencies are consistent with formulae (4.41) and (4.42). It was revealed for undiluted polyisoprene and poly(d, /-lactic acid) that the terminal (slow) dielectric relaxation time depends strongly on molecular weight of polymers (Adachi and Kotaka 1993 Ren et al. 2003). Two relaxation branches were discovered for i.s-polyisoprene melts in experiments by Imanishi et al. (1988) and Fodor and Hill (1994). The fast relaxation times do not depend on the length of the macromolecule, while the slow relaxation times do. For the latter, Imanishi et al. (1988) have found... 

Note that a frequency-dependent stress-optical coefficient C w) can be introduced by comparing the stress tensor and the relative permittivity tensor... 

Eq. (4), frequency-dependent, such that the limit for a(w) in Eq. (8) becomes physically acceptable. Under conditions appropriate to the correct limit, the normalized real and imaginary parts of the complex permittivity and the normalized dielectric conductivity take on the form depicted in Fig. (1). Here, is the relaxation time in the limit of zero frequency (diabatic limit). Irrespective of the details of the model employed, both a(w) and cs(u>) must tend toward zero as 11 + , in contrast to Eq. (8), for any relaxation process. In the case of a resonant process, not expected below the extreme far-infrared region, a(u>) is given by an expression consistent with a resonant dispersion for k (w) in Eq. (6), not the relaxation dispersion for K (m) implicit in Eq. 

In ferroelectrics the major contributor to tan 3 is domain wall movement which diminishes as the amplitude of the applied field diminishes the value applicable to pyroelectric detectors will be that for very small fields. The permittivity is also very sensitive to bias field strength, as is its temperature coefficient. The properties of some ferroelectrics - the relaxors - are also frequency dependent. It is important, therefore, to ensure that when assessing the suitability of a ferroelectric for a particular application on the basis of measured properties that the measurements have been made using values of the parameters (frequency, field strength etc.) appropriate to the application. This is not always done. 

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