Complex dielectric constant dispersion

The most familiar method of evaluating is by dielectric dispersion experiments, in which the real and imaginary parts of the complex dielectric constant over those of the solvent are determined as functions of frequency. It is the value of referring to the state of vacuum that can be correlated with the molecular structure of the solute. Polymers cannot be dispersed in the gaseous state. Furthermore, solvents effective for polypeptides are usually polar, and only approximate theories are presently available for the estimate of vacuum < 2> from dielectric measurements with polar solvents. Therefore the dipolar information about polypeptides is always beset with ambiguity in absolute magnitude as well as in interpretation. 

The Onsager cavity description of solvation treats the solvent as a dielectric continuum. The dielectric dynamics of the solvent is typically characterized by the frequency-dependent complex dielectric constant s(co). The measurement of (co) for a neat solvent is conventionally called a dielectric dispersion measurement. 

Figure 4. Equivalent circuit for the f3-dispersion of a cell suspension and corresponding plot in the complex dielectric constant plane (9)...

Styrene and 1-hexene have been selectively hydrogenated as well as substituted acetylenes, alkyne diols, stilbene and other unsaturated hydrocarbons with these palladium montmorillonites. A size selectivity was invoked to explain the enhanced hydrogenation activity of certain clay catalysts presumably due to the differences in interlamellar spacings of the clay which will depend on degree of hydration, concentration of Pd(II) complex, dielectric constant of the solvent used to disperse the reactants and other factors. 

Equation (11.25) is called the Debye dispersion relation or the Debye equation. The complex dielectric constant is defined to be... 

Dispersion of the measured complex dielectric constant is known from dielectric relaxation experiments. The complex dielectric constant e may be represented as... 

Fig. 9.3. The real part e and imaginary part e" of the complex dielectric constant near a region of Debye dispersion.

Other empirical distributed elements have been described, which can be expressed as a combination of a CPE and one or more ideal circuit elements. Cole and Cole found that frequency dispersion in dielectrics results in an arc in the complex e plane (an alternative form of presentation) with its center below the real axis (Fig. 10a) [16]. They suggested the equivalent circuit shown in Fig. 10(b), which includes a CPE and two capacitors. For ft) —> 0, the model yields capacitance Co and for ft) —> oo the model yields capacitance Coo- The model can be expressed with the following empirical formula for the complex dielectric constant... 

In general, bulk dielectric dispersion in solids and liquids is well known and described in the literature [24, 297, 298]. In this chapter, the dispersion of capacitances at electrode surfaces in solutions will be discussed. The complex dielectric constant is described as... 

Now, as already mentioned in Section 1.3, the h of that section s Eq. (6) is of just the same form as the well-known Cole-Cole dielectric dispersion response function (Cole and Cole [1941]). In its normalized form, the same / function can thus apply at either the impedance or the complex dielectric constant level. We may generalize this result (J. R Macdonald [1985a,c,d]) by asserting that any IS response... 

Conductive-system dispersive response may be associated with a distribution of relaxation times (DRT) at the complex resistivity level, as in the work of Moynihan, Boesch, and Laberge [1973] based on the assumption of stretched-exponential response in the time domain (Eq. (118), Section 2.1.2.7), work that led to the widely used original modulus formalism (OMF) for data fitting and analysis, hi contrast, dielectric dispersive response may be characterized by a distribution of dielectric relaxation times defined at the complex dielectric constant or permittivity level (Macdonald [1995]). Its history, summarized in the monograph of Bbttcher and Bordewijk [1978], began more than a hundred years ago. Until relatively recently, however, these two types of dispersive response were not usually distinguished, and conductive-system dispersive response was often analyzed as if it were of dielectric character, even when this was not the case. In this section, material parameters will be expressed in specific form appropriate to the level concerned. 

It is known that the complex dielectric constant of the resonator materials follows the dielectric dispersion equation, which is expressed as the superposition of electronic and ionic polarization ... 

Equation (B3.3.7) is the fundamental equation of classical dispersion theory [4]. Because Xe is related to the high-frequency dielectric constant by Eq. (3.30), and the high-frequency dielectric constant is the square of the refractive index (Eq. 3.30), it appears that the dielectric constant and refractive index also should be treated as complex numbers. To indicate this, we ll rewrite Eqs. (3.19) and (3.30) using and to distinguish the complex dielectric constant and refractive index from, e and n, the more familiar, real quantities that apply to non-absorbing media ... 

The complex dielectric constant of a suspension e of orientated ellipsoidal particles with the dielectric constant Cp at the particle volume fraction < ) dispersed in a continuous medium with a complex dielectric constant , can be calculated from the Maxwell-Wagner-Sillars equation [77] ... 

The second procedure belongs to a reaction-field-like approach. The most used model is that of Linder it is based on the extension of the concept of reaction field of a dipole subject to fluctuations exclusively electric in origin, and relates the dispersion free energy of a molecule immersed in a solvent to the molecular polarizability and the complex dielectric constant. More realistic versions of Linder s model have been elaborated by Rinaldi et al. and by Aguilar and Olivares del... 

Rather, correlations are presented with the dielectric constant or with solvent polarity . It is true that the magnitude of the effects observed is frequently in the range of 0.2 -0.5 Hz as predicted by Raynes. Some evidence for specific interaction is suggested by the collision complex model 50> and by the deviations observed in chloroform solutions, acids and gases. A few investigators have noted that no correlation was found with the refractive index of the solvents suggesting that dispersion forces are not important for 2/H H, at least in those compounds studied. 

The electrical properties of polyelectrolyte complexes are more closely related to those of biologically produced solids. The extremely high relative dielectric constants at low frequencies and the dispersion properties of salt-containing polyelectrolyte complexes have not been reported for other synthetic polymers. Neutral polyelectrolyte complexes immersed in dilute salt solution undergo marked changes in alternating current capacitance and resistance upon small variations in the electrolyte concentration. In addition, their frequency-dependence is governed by the nature of the microions. As shown in... 

Macroscopic solvent effects can be described by the dielectric constant of a medium, whereas the effects of polarization, induced dipoles, and specific solvation are examples of microscopic solvent effects. Carbenium ions are very strong electrophiles that interact reversibly with several components of the reaction mixture in addition to undergoing initiation, propagation, transfer, and termination. These interactions may be relatively weak as in dispersive interactions, which last less than it takes for a bond vibration (<10 14 sec), and are thus considered to involve "sticky collisions. Stronger interactions lead to long-lived intermediates and/or complex formation, often with a change of hybridization. For example, onium ions are formed with -donors. Even stable trityl ions react very rapidly with amines to form ammonium ions [41], and with water, alcohol, ethers, and esters to form oxonium ions. Onium ion formation is reversible, with the equilibrium constant depending on the nucleophile, cation, solvent, and temperature (cf., Section IV.C.3). 

The nature of the analyte interactions with liophilic ions could be electrostatic attraction, ion association, or dispersive-type interactions. Most probably all mentioned types are present. Ion association is essentially the same as an ion-pairing used in a general form of time-dependent interionic formation with the average lifetime on the level of 10 sec in water-organic solution with dielectric constant between 30 and 40. With increase of the water content in the mobile phase, the dielectric constant increases and approaches 80 (water) this decrease the lifetime of ion-associated complexes to approximately 10 sec, which is still about four orders of magnitude longer than average molecular vibration time. 

The dielectric constant is a natural choice of order parameter to study freezing of dipolar liquids, because of the large change in the orientational polarizability between the liquid and solid phases. The dielectric relaxation time was calculated by fitting the dispersion spectrum of the complex permittivity near resonance to the Debye model of orientational relaxation. In the Debye dispersion relation (equation (3)), ij is the frequency of the applied potential and t is the orientational (rotational) relaxation time of a dipolar molecule. The subscript s refers to static permittivity (low frequency limit, when the dipoles have sufficient time to be in phase with the applied field). The subscript oo refers to the optical permittivity (high frequency limit) and is a measure of the induced component of the permittivity. 

A major difference between the two methods of initiation is that the solvent in y-ray studies is almost inevitably the monomer itself, and these generally have lower dielectric constants than the chlorocarbon solvents most often used in the chemically initiated systems. As a result, it is not possible to compare the values of kp +) obtained from each technique without accounting for this difference in solvation. Classically, propagation involves charge dispersion in forming the transition-state complex and hence a reduction in the polarity of the system. Thus media of lower solvation power should favourably influence the process. (See reference 114 for more detailed discussion.) Experimentally the values of kp(+) from radiation-induced polymerizations are consistently higher than those obtained using stable salts as initiators, and this simplistic picture therefore seems to be confirmed. Dunn has recently carried out a detailed compilation of the available data on / p(+) and readers will find this an excellent distillation of the current position. 

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