Debye process

It can not be described by means of a single Debye process, but more complicated relaxation functions involving distributions of relaxation times (like the Cole-Cole function [117]) or distributions of energy barriers (like log-normal functions [118]) have to be used for its description. Usually a narrowing of the relaxation function with increasing temperature is observed. The Arrhenius temperature dependence of the associated characteristic time is ... 

In Refs. 80 and 81 it is shown that the Mittag-Leffier function is the exact relaxation function for an underlying fractal time random walk process, and that this function directly leads to the Cole-Cole behavior [82] for the complex susceptibility, which is broadly used to describe experimental results. Furthermore, the Mittag-Leffier function can be decomposed into single Debye processes, the relaxation time distribution of which is given by a mod-... 

The dielectric relaxation data for dimethylformamide (DMF) and dimethyl-acetamide (DMA) can be described by two Debye processes [9]. The low-frequency, high-amplitude process is attributed to rotational diffusion. For... 

Note that a pure Debye process is described as... 

The water spectra, calculated from Eq. (54), are depicted by dash-dotted lines in Figs. 6a-d for H20 and 6c-h for D20. The principal Debye process (55) is marked in Figs. 6a,d and 6g,h by open circles. Calculation for our mixed model is depicted by thick solid lines. To emphasize the contribution to s of transverse vibrations, we show by dashed lines the permittivity components generated by the a + b + c mechanisms. Therefore, the values of s, marked by dashed curves, do not account for the s L component. 

Figure 2. Frequency dependence of the real and imaginary permittivities in a simple Debye process. The half-height width of s (logy) can be shown to be 1.14.

For a non-Debye process, a good approximation is given by an HN-type expression where the frequency is changed to its reciprocal value [4,153] ... 

The interpolation of the experimental data was carried out by a least-squares fitting procedure of the DCF values. The most appropriate number of elementary Debye processes involved is determined by the minimum of the standard deviation x - The dielectric response obtained reflects some properties inherent in single particle dynamics. The best-fit curves of the experimental data are reported in Figure 19 Time dependence of the macroscopic dipole correlation... 

It is possible to show that the single-shell model equation can be presented as the sum of a single Debye process and... 

The Debye process 2 must be interpreted as an intracluster step which is coupled with the intercluster process 1. The immitance spectra show this Debye process as a sharp resonance, whereas the intercluster Cole-Cole process causes a... 

There seems to be a simple phase relation between the macroscopic frequency applied to the duster sample and the microscopic frequency of the two electrons within the cluster cores Vd macro = 21 jt Vd micro (D = Debye). Using the macroscopic frequency Vj, for the Debye process a microscopic frequency of 10 -10 ... 

All the intracellular events during a cycle are coupled in such a way to achieve strict logistics [1,2, 4]. For this reason intracellular relaxation is described by a single Debye process [8, 9], the relaxation time of which increases with the size of the cell. Relaxation of a cell ensemble should thus be controlled by directed fluctuations of the whole ensemble structure. If the kinetics is the same everywhere the influence of the colloid structure is explicitly accounted for by introducing the mode factor of a cell composed of y units, Toy> according to [13]... 

The real part n cOc) characterises the energy that can be stored in a growing cell ensemble at defined distances from a stationary state of reference. "(cOc) should describe dissipation during relaxation. Relaxation of the cell ensemble is then consequently considered as a superposition of many Debye processes, the fractions of which are given by the normalised universal mass fraction of the cells of different sizes (see Eq. 11). 

Processes during a cell cycle are evidenced to be closely controlled cooperative events, including synchronisation within the ensemble. This caused us to describe relaxation within each cell by a Debye process, the relaxation time of which should increase with the size of the cell involved ( finite-size effect ). In that way ensemble structure and relaxation processes of cell ensembles are strictly interrelated. The universal energy density distribution and the universal relaxation mode distribution turn out to be copies of each other. Consequently, the spectrum depends only on the universal properties of the ensemble structure, i.e. on the value of p. Since all the cell populations studied here belong to the / = 3 class, the linear relaxation behaviour should show the same features. 

Figure 6.1. Time-dependent adjustment of the dielectric induction D, following application (at f = 0) of an electric field Es to the sample, in the case of a Debye process, where (f) = exp(-f/x), or processes described by the Kdhlrausch-Williams-Watts (KWW) stretched exponential function, where (f) = exp[-(f/x) ] with 0 < Pkww 1-...

The important point of this discussion is that once the Debye process is modified to include a time-dependent local viscosity, realistic dielectric results are obtained for polymers. To repeat the D-B contention, Eq. 

Loss peaks obtained for solid polymers are generally much broader than a Debye process, and indeed broader than is the case for many relaxations in solution. In many cases they are grossly unsymmetrical and even show structural features. For broad and even unsymmetrical loss peaks their shapes are often analysed in terms of a standard distribution function as described on p. 106. It should be stressed, however, that anomalous peak shapes can result from an artifact in data analysis when relaxations occur at low frequencies (10 to 10 Hz). The conventional technique for this frequency range is to measure the decay current (or less commonly, the... 

Fig. 5.5. Real part left) and imaginary part right) of the dynamic compliance associated with a mechanical Debye-process...

The dynamic compliance of the single-time relaxation process, in the literature also addressed as Debye-process , thus has a simple form, being a function of the product cur and A J only. Separation into the real and the imaginary part yields... 

In fact, these properties are not specific to the Debye-process, but have a deeper basis which extends their validity. According to the Kramers-Kronig relations, J and J" are mutually dependent and closer inspection of the equations reveals that it is impossible, in principle, to have a loss without a simultaneous change in J. Both effects are coupled, the reason being, as mentioned above, the validity of the causality principle. 

The loss curve has a characteristic width, the total width at half height amounting to 1.2 decades. Compared to the loss at the resonance frequency of an oscillating system, the loss curve of the Debye-process is much broader. A halfwidth of 1.2 decades in fact presents the lower limit for all loss curves found in relaxing systems. Loss curves which are narrower are therefore indicative of the presence of oscillatory contributions, or more generally speaking, indicate effects of moments of inertia. 

A simple check, if a measured dynamic compliance or a dielectric function agrees with a Debye-process, is provided by the Cole-Cole plot . Let us illustrate it with a dielectric single-time relaxation process. If we choose for the dipolar polarization an expression analogous to Eq. (5.66) and take also into account the instantaneous electronic polarization with a dielectric constant u, the dielectric function e uj) shows the form... 

Having established the properties of the single-time relaxation process, we have now also a means to represent a more complex behavior. This can be accomplished by applying the superposition principle, which must always hold in systems controlled by linear equations. Considering shear properties again, we write for a dynamic compliance J (uj) with general shape a sum of Debye-processes with relaxation times ti and relaxation strengths AJi... 

The identical function can be used in order to describe the result of a creep experiment on the system. One has just to substitute the dynamic compliance of the Debye-process by the associated elementary creep function, as given by Eq. (5.61). This leads to... 

Alternative to the shear compliances, J t) and J (o ), one can also use for the description of the properties under shear the shear moduli, G t) and As we shall find, this can drastically change the values of the relaxation times. Let us first consider a single Debye-process, now in combination with a superposed perfectly elastic part, and calculate the associated dynamic modulus. We have... 

Rather than representing the viscoelastic properties of a given sample in the form of Eq. (5.73), i.e. by a superposition of Debye-processes which are specified by AJ/ and r, one can perform an analogous procedure based on single-time relaxation processes specified by AG/ and f/. We then write in the integral form... 

In order to show that this is true, we have to prove that the time dependent modulus for the Debye-process does indeed equal the exponential function exp—(t/f). For the proof, we calculate first the primary response function, /x(t), by use of Eq. (5.31) ... 

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