Dielectric constant complex

Principles in Processing Materials. In most practical apphcations of microwave power, the material to be processed is adequately specified in terms of its dielectric permittivity and conductivity. The permittivity is generally taken as complex to reflect loss mechanisms of the dielectric polarization process the conductivity may be specified separately to designate free carriers. Eor simplicity, it is common to lump ah. loss or absorption processes under one constitutive parameter (20) which can be alternatively labeled a conductivity, <7, or an imaginary part of the complex dielectric constant, S, as expressed in the foUowing equations for complex permittivity ... 

With the reader bearing in mind this framework, the Lifshitz theory of van der Waals interactions can readily be understood. According to the Lifshitz theory, van der Waals forces arise from the absorption of photons of frequency tu by a material with a complex dielectric constant... 

Refinements in the theory of interparticle long-range van der Waals forces (the Landau-Lifshitz theory) are within reach. New techniques are now available for measuring the complex dielectric constants of various media required for the implementation of that theory. 

There is a relation between I and the complex dielectric constant 8 (to) at an angular frequency 0), which can be written as follows (Fano, 1963) ... 

Fig. 4.2. Imaginary part e" of the complex dielectric constant versus real part with frequency as a parameter (Cole-Cole plot) at different temperatures. Arrows indicate the frequency of 10 Hz in each case. Insert shows thermal activation energy plot. (See Text)...

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. 

Fig. 11. Complex piezoelectric strain constant (20 Hz), complex Young s modulus (30 Hz), and complex dielectric constant (1kHz) of uniaxially drawn poly(D-propylene oxide) film plotted against temperature. Draw-ratio = 1.5. Degree of crystallinity=40%. Drawn after Furukawa and Fukada [Nature 221,1235 (1969)] by permission of Macmillan (Journals) Ltd.

Fig. 17. Complex dielectric constant, s — is", of the same film as in Fig. 16 plotted against frequency (Hayakawa and Wada (2), 1971)...

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. 

Anomalous dielectric dispetsion occurs when the frequency of the field is so high that the molecules do not have time to attain equilibrium with it. One may then use a complex dielectric constant... 

Combination of the two equations for Ihe complex dielectric constant and separation of real and imaginary parts gives... 

The symmelrical loss-frequency curve predicted by this simple theory is commonly observed for simple substances, but its maximum is usually lower and broader because of the existence of more than one relaxation time. Various functions have been proposed to represent the distribution of relaxation times. A convenient representation of dielectric behavior is obtained, according to the method of Cole and Cole, by writing the complex dielectric constant as... 

Mathematical equations, presented by Maxwell in 1864, are able to predict the behavior of microwave radiation s interaction with any type of food in any geometry. In order to do this, a single pair of parameters describing the electrical (or dielectric) properties of the food are required. This pair of parameters is known as the complex permittivity, or as is more commonly called in the United States, the complex dielectric constant. This parameter pair is defined as ... 

Here e represents the relative values of permittivity with respect to that of free space. The term permittivity is not in common usage in the United States in the field of food science. Instead, the term dielectric constant is used. Thus, in Equation 1, e is called the complex dielectric constant, e, the dielectric constant and e", the dielectric loss factor. 

For microwave radiation incident upon a slab from a direction perpendicular to its surface, a fraction of the energy will be reflected from the surface, Pr, depending upon its complex dielectric constant e. The main contribution to the magnitude of reflection however, is from the dielectric constant e. Errors due to neglecting e" are less than 5% for virtually all foods as is indicated by the 5% line in Figure 1. Neglecting the loss factor, an approximate equation for the fraction of microwave power reflected from an infinite slab food surface is given by ... 

The molecules form a thin absorbing layer of thickness d, with a complex dielectric constant n(w) = n](exact solution and various approximations.3. For thin films, Bell has obtained a reasonably simple approximation for the absorption of an SEW due to the overlying film. If the SEW are propagated a distance D on the metal substrate, with and without the thin film, the ratio of the SEW intensities is given by... 

When the perturbation is an electric field, the response is the material polarization and one can define a complex dielectric constant, which is the electrical equivalent of a compliance s = s — is" (dielectric spectroscopy Chapter 6). 

The Havriliak-Negami model was first established for the complex dielectric constant (equivalent to the complex compliance). Its mechanical translation can be written as... 

Water is highly polar. Its penetration in a polymer induces an increase of both components e and e" of the complex dielectric constant (Chapter 6), an increase of the conductivity and a decrease of the dielectric strength. Microdielectric sensors can be used to monitor water diffusion in thick samples, especially in composites (Kranbuehl, 2000). 

In optics, the complex refractive index is defined as N = n + ik = ( r(co))l/1, where the real index of refraction, n, and extinction coefficient, k, are related to the real and imaginary parts of the complex dielectric constant by ... 

Figure 3. Dielectric properties of barnacle muscle in the microwave frequency range are presented in the complex dielectric constant plane (%).

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

All sorts of biological particles of different effective complex dielectric constants behave similarly in an electrolyte medium. Figure 5 illustrates this fact. Neumann and Rosenheck s (AO) results on chromaffin vesicles are combined with E. coli data obtained by Sher and erythrocyte data obtained by Sher and silicon particles (full circles), also by Sher (A5). The total material fits convincingly the solid line of slope -1.5 which is demanded by the theoretical requirement that particle volume must be inversely related to the square of the threshold field strength mentioned above and discussed in greater detail elsewhere (Schwan and Sher (5A)). 

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. 

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