Dispersion Frequency response

The stretched-exponential temporal response of Eq. (63), Section 2.1, a versatile and theoretically plausible correlation function, is one whose corresponding frequency behavior is now called Kohlrausch-Williams-Watts or just Kohlrausch [1854] model response, denoted here by Kk. It is also now customary to replace the a of the stretched-exponential equation by P or P, with A =D or 0. The k=D choice may be related to KD-model dispersive frequency response involving a distribution of dielectric relaxation (properly retardation ) times, and the A = 0 and 1 choices to two different distributions of resistivity relaxation times and thus to KO and K1-model responses, respectively. Note that the P parameter of the important K1 model is not directly related to stretched exponential temporal response, as are the other Kohlrausch models, but the DRTs of the KO and K1 models are closely related (Macdonald [1997a]). Further, although the KD and KO models are identical in form, they apply at different immittance levels and so represent distinct response behaviors. 

Barton [1966], Nakajima [1972], and Namikawa [1975] empirical relation, usually designated by BNN, has played a useful role for some time in the analysis of dispersed frequency response data (e.g. Dyre [1988], Macdonald [1996], Dyre and Schrpder [2000], Porto et al. [2000]). It involves a loosely defined parameter, p, expected to be of order 1, and Nakajima and Namikawa believed that it arose from correlation between electrical conduction and dielectric polarization, apparently because it involved both measured dc conductivity and a dielectric strength quantity Ae. 

Such efforts have met with limited success, and the reason usually advanced is our lack of understanding of the frequency dependence of molecular NLO properties. In classical electromagnetism, we refer to properties that depend on the frequency of radiation as dispersive and we say that (for example) dispersion is responsible for a rainbow. The blue colour of the sky is a dispersion effect, as is the red sky at night and morning. There is more to it than that, and you might like to read a more advanced text (Hinchliffe and Munn, 1985). 

The two systems discussed above demonstrate two mechanisms whereby the tensile strength of elastomers can be reinforced by the presence of rigid fillers. For the polymeric fillers dispersed within a vulcani-zate, the filler operates by raising the viscosity of the matrix, analogous to a decrease in temperature, but without affecting the dynamic, high frequency response (there is ample experimental evidence of the independence of Ty on presence of filler). There is also some indication that the rigidity of the filler affects the extent of reinforcement. 

The complex permittivity is obtained as follows For nondisperse materials (frequency-independent permittivity), the reflected signal follows the exponential response of the RC line-cell arrangement for disperse materials, the signal follows a convolution of the line-cell response with the frequency response of the sample. The actual sample response is found by writing the total voltage across the sample as follows ... 

If the dipoles are so tightly correlated that all move together, we have essentially one molecule with an enormous dipole moment, as in the low-frequency response of a single ferroelectric domain. The response is likely to be strongly non-linear at moderate field intensities. Another example is a polypeptide molecule coiled into a solid polar bar, and in this case it may be possible to uncoil the molecule and study it in a less corrdated motion. In the most interesting situation of biological material, polar long-chain molecules are dispersed in an ionic fluid, and the total moment associated with one molecule is a sum of more or less correlated permanent dipoles with a certainly-correlated ionic atmosphere. 

Note first that in this older picture, for both the attractive (van der Waals) forces and for the repulsive double-layer forces, the water separating two surfaces is treated as a continuum (theme (i) again). Extensions of the theory within that restricted assumption are these van der Waals forces were presumed to be due solely to electronic correlations in the ultra-violet frequency range (dispersion forces). The later theory of Lifshitz [3-10] includes all frequencies, microwave, infra-red, ultra and far ultra-violet correlations accessible through dielectric data for the interacting materials. All many-body effects are included, as is the contribution of temperature-dependent forces (cooperative permanent dipole-dipole interactions) which are important or dominant in oil-water and biological systems. Further, the inclusion of so-called retardation effects, shows that different frequency responses lock in at different distances, already a clue to the specificity of interactions. The effects of different geometries of the particles, or multiple layered structures can all be taken care of in the complete theory [3-10]. 

For the SCC of type II an example of a RTD modelled is shown in Figure 7, The model used is the dispersion model (sec Esq. 6). The values of the model parameters determined arc a Bodenstein number of 8.8 and a mean residence time of 0.6 s. It clearly shows that the model for the RTD explains the frequency response measurement up to a frequency of 2 Hz, At the frequency of 2 Hz the signal-to-noise ratio of 100 is reached. Any mixing processes which affect the transfer function above this frequency cannot be identified. 

Gangwall et al. [47] were the first to apply Fourier analysis for the evaluation of the transport parameters of the Kubin-Kucera model. Gunn et al. applied the frequency response [80] and the pulse response method [83] in order to determine the coefficients of axial dispersion and internal diffusion in packed beds from experiments performed at various Reynolds numbers. Bashi and Gunn [83] compared the methods based on the analytical properties of the Fourier and the Laplace transforms for the calculation of transport coefficients. MacDonnald et al. [84] discussed the applications of the method of moments to the analysis of the profiles of skewed chromatographic peaks. When more than two parameters have to be determined from one single run, the moment analysis method is less suitable, because only the first and second moments are reliable (see Figure 6.9). Therefore, only two parameters can be determined accurately. 

The frequency dispersion of porous electrodes can be described based on the finding that a transmission line equivalent circuit can simulate the frequency response in a pore. The assumptions of de Levi s model (transmission line model) include cylindrical pore shape, equal radius and length for all pores, electrolyte conductivity, and interfacial impedance, which are not the function of the location in a pore, and no curvature of the equipotential surface in a pore is considered to exist. The latter assumption is not applicable to a rough surface with shallow pores. It has been shown that the impedance of a porous electrode in the absence of faradaic reactions follows the linear line with the phase angle of 45° at high frequency and then... 

In the frequency response method, first applied to the study of zeolitic diffusion by Yasuda [29] and further developed by Rees and coworkers [2,30-33], the volume of a system containing a widely dispersed sample of adsorbent, under a known pressure of sorbate, is subjected to a periodic (usually sinusoidal) perturbation. If there is no mass transfer or if mass transfer is infinitely rapid so that gas-solid mass-transfer equilibrium is always maintained, the pressure in the system should follow the volume perturbation with no phase difference. The effect of a finite resistance to mass transfer is to cause a phase shift so that the pressure response lags behind the volume perturbation. Measuring the in-phase and out-of-phase responses over a range of frequencies yields the characteristic frequency response spectrum, which may be matched to the spectrum derived from the theoretical model in order to determine the time constant of the mass-transfer process. As with other methods the response may be influenced by heat-transfer resistance, so to obtain reliable results, it is essential to carry out sufficient experimental checks to eliminate such effects or to allow for them in the theoretical model. The form of the frequency response spectrum depends on the nature of the dominant mass-transfer resistance and can therefore be helpful in distinguishing between diffusion-controlled and surface-resistance-controlled processes. 

Solvents with vanishing molecular dipole moments but finite higher order multipoles, such as benzene, toluene, or dioxane, can exhibit much higher polarity, as reflected by its influence on the ET energetics, than predicted by the local dielectric theory [228], Full spatially dispersive solvent response formulation is required in this case [27-29, 104, 229-233], There are different approaches to the problem of spatial dispersion. The original formulation by Kornyshev and co-workers [27c, 28] introduces the frequency-dependent screening effect on the basis of heuristic arguments. More recent approaches are based on the density-function theory [104,197],... 

Absorption and dispersion spectra steady-state frequency-response... 

For porous electrodes, an additional frequency dispersion appears. First, it can be induced by a non-local effect when a dimension of a system (for example, pore length) is shorter than a characteristic length (for example, diffusion length), i.e. for diffusion in finite space. Second, the distribution characteristic may refer to various heterogeneities such as roughness, distribution of pores, surface disorder and anisotropic surface structures. De Levie used a transmission-line-equivalent circuit to simulate the frequency response in a pore where cylindrical pore shape, equal radius and length for all pores were assumed [14]. 

Dispersion (frequency dependence according to the laws of relaxation) is the correspondent frequency domain concept of relaxation permittivity as a function of frequency. Even if the concept of relaxation is linked with step functions, it can of course be studied also with sine waves. An ideal step function contains all frequencies, and dispersion can be analyzed with a step function followed by a frequency (Fourier) analysis of the response signal or with a sinusoidal signal of varying frequency. 

These data can also be presented as conductance and susceptance versus frequency, as in Figure 4.17. The very broad nature of the dispersion can easily be seen in this figure. The conductance levels out at low frequencies, indicating the DC conductance level of the skin. The susceptance seems to reach a maximum at approximately 1 MHz, which should then correspond to the characteristic frequency of the dispersion. This frequency response... 

The best model takes all recent knowledge about relaxation processes, frequency dispersion, diffusion, fractals, and so on into account. It is an electrical equivalent to the skin (i.e., it has the same frequency response). Furthermore it is simple and uses symbols in a way that makes it easy to understand the outlines of the electrical properties of the different substructures of the skin. The model takes care of all requirements of an electrical model of the skin—but unfortunately it does not exist The most correct of the existing models is therefore the one best adapted for the target group. 

The frequency response of si for PAN-CSA prepared from chloroform and subsequently briefly exposed to m-cresol vapor (crpc 20S/cm) [193] (Fig. 46.26) is characteristic of localized electrons. si is positive at all optical frequencies the scattering due to disorder in these materials has broadened and washed out the dielectric zero crossings. Lorentzian dispersion due to a localized polaron [146] is evident in bi around 12,000 cm (1.5 eV) and for this material increases positively with decreasing wavenumber in the far IR, characteristic of a material with a small residual band gap or localized carriers. Lower conductivity PAN-HCl [193] (ctdc 10 S/cm) materials show even less dispersion with wavenumber. si for these materials is also positive over the whole range and shows only a modest... 

Since conductive-system dispersive response may be transformed and shown graphically at the complex dielectric level, and dielectric dispersion may be presented at the complex resistivity level, frequency-response data alone may be insufficient to allow positive identification of which type of process is present, since there may be great similarity between the peaked dispersion curves that appear in plots of p"(co) and of e"(co) or of e"(cd) = e"((o) - (otjcoev). Here, e is the permittivity of vacuum. This quantity has usually been designated as b, as in other parts of this book. Its designation here as f avoids ambiguity and allows clear distinction between it and e(0) = e (0) = o, the usage in the present section. 

An important conclusion from the paper by Brug et al. [1984] is that involvement of a CPE at solid electrodes used for studies of the impedance of Faradaic reactions can severely influence the frequency dispersion of interfacial admittance, leading to large errors in the determined Faradaic rate parameters. However, those authors note that it is feasible to account for the CPE effect correctly and to check the results of impedance analysis with respect to their internal consistency. The latter can be checked by a Kramers-Kronig analysis (cf. Lasia [1999]) which requires, however, detailed frequency-response data. Their approach was supported by experimental impedance studies on proton reduction at single and polycrystalline Au electrodes and on reduction of tris-oxalato-Fe(lII) (Brug et al. [1984]). 

Analysis of Dispersed, Conducting-System Frequency-Response Data, J. Non-Cry St. Solids 197, 83-110. 

A fundamental problem of binaural hstening is known as in-head localization (IHL) and is apparently due to the lack of a dispersed wavefront striking the aural cavity. Other problems such as bone conduction (BC), the frequency response and resonance challenges of small aperture assemblies, variable leakage... 

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