Dielectrics, frequency-dependent response

The expression immediately gives an estimate of the enthalpy of adsorption in taking an atom from the gaseous (vacuum) state to a liquid, or to a composite medium like a zeolite, characterised by its measured dielectric frequency dependent response (o>). It is, exactly as for the electrostatic Bom self-energy in taking an ion from vacuum to water ... 

The coefficient AHam itself varies with separation /. It takes the form of a sum over all frequencies at which fluctuations can occur wherein each term depends on the frequency-dependent responses of materials A, B, and m to electromagnetic fields. These responses are written in terms of "dielectric" functions sA, b, and em that are extracted from absorption spectra. It is the differences in these dielectric responses that create interactions. To first approximation,... 

Since the frequency-dependent response of the solvent is included in the kernel of the integrals collected into X , the resulting hyperpolarizabilities will also depend on the frequency spectrum of the dielectric function e(u ) of the solvent. When e(w) is described by the Debye formula (i.e. in terms of a single relaxation mode), i.e. 

Frequency-Dependent Response of Dielectrics Basic Aspects... 

The dielectric medium is normally taken to have a constant value of e, but may for some purposes also be taken to depend for example on the distance from M. For dynamical phenomena it can also be allowed to be frequency dependent i.e. the response of the solvent is different for a fast reaction, such as an electronic transition, and a slow reaction, such as a molecular reorientation. 

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. 

From frequency dependent dielectric loss measurements, the transitions associated with solvent dipole reorientations occur on a timescale of 10-n -10-13 s. By contrast, the time response of the electronic contribution to the solvent polarization is much more rapid since it involves a readjustment in electron clouds . The difference in timescales for the two types of polarization is of paramount importance in deciding what properties of the solvent play a role in electron transfer. The electronic component of the polarization adjusts rapidly and remains in equilibrium with the charge distribution while electron transfer occurs. The orientational component arising from solvent dipoles must adopt a non-equilibrium distribution before electron... 

Let us summarize by modeling the velocity autocorrelation function using Debye-Huckel type interactions between charged point defects in ionic crystals, one can evaluate the frequency-dependent conductivity and give an interpretation of the universal dielectric response. 

A detailed analysis of the dielectric response of PMMA has been performed [75]. The frequency dependence of the dielectric loss, e"y at a few temperatures is shown in Fig. 108. An increase in temperature is associated with a shift towards higher frequency, as expected, but also with quite a significant increase in the height of the peak maximum. 

Figure 15.6 shows the influence of pressure on the dielectric response of pmn [14], The broad frequency-dependent peaks and the frequency dispersion in both s and e" are the characteristic signatures of the rl state. It is seen that pressure shifts the peaks to lower temperatures and suppresses e and e" in the high temperature phase. The suppression of e"(T) is mostly due to the suppression of e T) as e"(T) = e (T) tan<5(T), where tan <5 represents the dielectric loss. These pressure effects are characteristic of ABO3 rls and are well understood [14],... 

The molecule is often represented as a polarizable point dipole. A few attempts have been performed with finite size models, such as dielectric spheres [64], To the best of our knowledge, the first model that joined a quantum mechanical description of the molecule with a continuum description of the metal was that by Hilton and Oxtoby [72], They considered an hydrogen atom in front of a perfect conductor plate, and they calculated the static polarizability aeff to demonstrate that the effect of the image potential on aeff could not justify SERS enhancement. In recent years, PCM has been extended to systems composed of a molecule, a metal specimen and possibly a solvent or a matrix embedding the metal-molecule system in a molecularly shaped cavity [62,73-78], In particular, the molecule was treated at the Hartree-Fock, DFT or ZINDO level, while for the metal different models have been explored for SERS and luminescence calculations, metal aggregates composed of several spherical particles, characterized by the experimental frequency-dependent dielectric constant. For luminescence, the effects of the surface roughness and the nonlocal response of the metal (at the Lindhard level) for planar metal surfaces have been also explored. The calculation of static and dynamic electrostatic interactions between the molecule, the complex shaped metal body and the solvent or matrix was done by using a BEM coupled, in some versions of the model, with an IEF approach. 

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],... 

The van der Waals interaction depends on the dielectric properties of the materials that interact and that of the medium that separates them. ("Dielectric" designates the response of material to an electric field across it Greek Si- or Si a- means "across.") The dielectric function e can be measured experimentally by use of the reflection and transmission properties of light as functions of frequency. At low frequencies, the dielectric function e for nonconducting materials approaches a limit that is the familiar dielectric constant. The dielectric function actually has two parts, one that measures the polarization properties and the other that measures the absorption properties of the material. 

P. J. W. Debye, Polar Molecules (Dover, New York, reprint of 1929 edition) presents the fundamental theory with stunning clarity. See also, e.g., H. Frohlich, "Theory of dielectrics Dielectric constant and dielectric loss," in Monographs on the Physics and Chemistry of Materials Series, 2nd ed. (Clarendon, Oxford University Press, Oxford, June 1987). Here I have taken the zero-frequency response and multiplied it by the frequency dependence of the simplest dipolar relaxation. I have also put a> = if and taken the sign to follow the convention for poles consistent with the form of derivation of the general Lifshitz formula. This last detail is of no practical importance because in the summation Jf over frequencies fn only the first, n = 0, term counts. The relaxation time r is such that permanent-dipole response is dead by fi anyway. The permanent-dipole response is derived in many standard texts. 

Dielectric relaxation — Dielectric materials have the ability to store energy when an external electric field is applied (see -> dielectric constant, dielectric - permittivity). Dielectric relaxation is the delayed response of a dielectric medium to an external field, e.g., AC sinusoidal voltage, usually at high frequencies. The resulting current is made up of a charging current and a loss current. The relaxation can be described as a frequency-dependent permittivity. The real part of the complex permittivity (e1) is a measure of how much energy from an external electric field is stored in a material, the imaginary part (e") is called the loss factor. The latter is the measure of how dissipative a material is to an exter-... 

The response of a pure, homogeneous medium to the applied fields may be characterized quite generally by a complex frequency-dependent dielectric function e(o>), which can be written in terms of its real and imaginary parts as... 

It is now well understood that the static dielectric constant of liquid water is highly correlated with the mean dipole moment in the liquid, and that a dipole moment near 2.6 D is necessary to reproduce water s dielectric constant of s = 78 T5,i85,i96 holds for both polarizable and nonpolarizable models. Polarizable models, however, do a better job of modeling the frequency-dependent dielectric constant than do nonpolarizable models. Certain features of the dielectric spectrum are inaccessible to nonpolarizable models, including a peak that depends on translation-induced polarization response, and an optical dielectric constant that differs from unity. The dipole moment of 2.6 D should be considered as an optimal value for typical (i.e.. 

There are several complications in using this technique for a-Si H, first of which is the frequency dependence. The capacitance is measured by the response to a small alternating applied electric field. The depletion layer capacitance is obtained only when the free carriers within the bulk of the semiconductor can respond at the frequency of the applied field, dielectric relaxation time. 

We have presented response methods that provide procedures for calculating frequency-dependent molecular properties for a molecular subsystem coupled to a stmctured environment. We have shown that the molecular subsystem is treated on a quantum mechanical level and the stmctured environment as a classical subsystem. We have presented the stmctured environment, classical subsystem, as a heterogeneous dielectric media or a molecular mechanics force field. We have demonstrated that the interactions between the quantum mechanical and classical subsystems are part of the energy functional used for optimizing the MCSCF electronic wave function. 

The conducting properties of a liquid in a porous medium can provide information on the pore geometry and the pore surface area [17]. Indeed, both the motion of free carriers and the polarization of the pore interfaces contribute to the total conductivity. Polymer foams are three-dimensional solids with an ultramacropore network, through which ionic species can migrate depending on the network structure. Based on previous works on water-saturated rocks and glasses, we have extracted information about the three-dimensional structure of the freeze-dried foams from the dielectric response. Let be d and the dielectric constant and the conductivity, respectively. Dielectric properties are usually expressed by the frequency-dependent real and imaginary components of the complex dielectric permittivity ... 

Some Theoretical CfMuaderatifms.— Since the first application of fast-response time-domain methods to dielectric measurement, the value of a direct time domain convra sion of the data has been universally recognized. It is after all a strange paradox that, in order to obtmn information about time-dependent molecular motions fi om the time-dq>endent reflection coefficient, P(t), one must go through the stage of the frequency-dependent reflection codficient Yet so far, no mathematically rigorous treat-... 

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