Single-relaxation-time model

The general linear solid leads to the single relaxation time model the solution of (2.4) for the case of oscillating strain leads to... 

Fig. 2. Temperature dependence of the storage modulus G, loss modulus G", relaxation time t and ratio (n2/nj) of the equilibrium numbers of conformers for a single relaxation time model...

Fig. 11a and b. Debye single relaxation time model for dipole orientation showing a) permittivity and b) loss factor as a function of the product of the angular frequency w and the dipole realxation time rd. The relaxed permittivity is e, and the unrelaxed permittivity is e ... 

Fig. 14. Cole-Cole diagrams illustrating dipole relaxation behavior. a) Debye single relaxation time model, b) Williams-Watts expression with p = 0.5. c) Cole-Cote expression with...

As with mechanical relaxation, the single-relaxation-time model is inadequate for polymers and the Cole-Cole plots are not semicircles. Figure 7-... 

DC Transient-Current Method. In this method a step voltage is applied to the sample and the current response is measured by a fast-response electrometer. For the single- relaxation-time model, the current response would be given by equation (7-9). In recent years this method has been of renewed interest because with the advent of modem computing methods, it is possible to Fourier-transform the response in the time domain to obtain the frequency response. Several Fourier-transform dielectric spectrometers have been designed. We may note the one of historical significance due to Johnson et al.15, as well as modem commercial instruments.16 The method has the great... 

Alternative analytical methods are the JCAHR isothermal single relaxation time model [Kovacs et al., 1979] and the peak shifting technique, [Hutchinson, 1992]. None of these attempt to use the data to predict long-term aging effects for which a different approach is necessary. 

The validity of the viscoelastic model (5.32) has been tested against experimental and molecular dynamics simulation results [26, 27, 28]. The detailed comparison has established that the viscoelastic model works remarkably well for wavenumbers k km, where km denotes the first peak position of the static structure factor S k). However, it has also been found that the situation is not so satisfactory for smaller wavenumbers, where the viscoelastic model is shown in some circumstances to yield even qualitatively incorrect results. This failure was attributed to the fact that the single relaxation time model (5.31) cannot describe both the short-time behavior of the memory function, dominated by the so-called binary collisions, and in particular the intermediate and long-time behavior where in the liquid range additional slow processes play an important role (see the next subsection). It is obvious that these conclusions demand a more rigorous consideration of the memory function, which lead to the development of the modern version of the kinetic theory. Nevertheless, the viscoelastic model provides a rather satisfactory account of the main features of microscopic collective density fluctuations in simple liquids at relatively large wavenumbers, and its value should not be undervalued. 

Alternative analytical methods are the KAHR isothermal single relaxation time model (Kovacs et al. 1979) and the peak shifting technique (Hutchinson 1992). 

A dielectric experiment is analogous to a DMA experiment where the charge is mathematically comparable to the strain and the voltage is comparable to the stress. The mathematical analysis is similar and, assuming a single relaxation time model for simplicity, it can be shown that the complex dielectric constant has the form e = s - is" with... 

Fig. 34. Single relaxation time model curves for dielectric parameters s, and tan S. The curves were calculated using eo = 10, co = 2 and a relaxation time of 10 sec. (discussed in McCrum and co-workers, Ref 69).

Dielectric experiments that involve studies of the relaxation function (r) are denoted as time-domain experiments, while those related to the complex permittivity function e (co) are considered dynamic experiments. The latter have the advantage of introducing an experimental timescale ( l/(o), which, when compared to the different intrinsic timescales of the system (the relaxation time x), provides useful information on the molecular level. In terms of the single-relaxation-time model of Debye (1921,1929), the complex permittivity for a dipolar mechanism can be written as follows ... 

It is very uncommon that the relaxation processes in polymers can be described by a single relaxation time model, and eq. (10.17) has been modified empirically. The Cole-Cole equation includes a symmetric broadening factor P which takes values between 0 and 1 (single relaxation time) ... 

If we apply to the creep experiment the single relaxation time model, the simple Maxwell model, eq. 3.2.9, or eq. 3.2.16, we obtain... 

The discussion so far has been limited to the single relaxation time model. In polymers, however, the measured absorption and sound speed vs frequency curves are much broader than predicted by the single relaxation time model. This observation is interpreted to mean that there is a continuous distribution of... 

An Evaluation of the Debye-Onsager Model. The simplest treatment for solvation dynamics is the Debye-Onsager model which we reviewed in Section II.A. It assumes that the solvent (i) is well modeled as a uniform dielectric continuum and (ii) has a single relaxation time (i.e., the solvent is a Debye solvent ) td (Eq. (18)). The model predicts that C(t) should be a single... 

The single relaxation time approximation corresponds to a stochastic model in which the fluctuating force on a molecule has a Lorentzian spectrum. Thus if the fluctuating force is a Gaussian-Markov process, it follows that the memory function must have this simple form.64 Of course it would be naive to assume that this exponential memory will accurately account for the dynamical behavior on liquids. It should be regarded as a simple model which has certain qualitative features that we expect real memory functions to have. It decays to zero and, moreover, is of a sufficiently simple mathematical form that the velocity autocorrelation function,... 

In a quantitative way, these simple models are not powerful enough to account for the behaviour of a real polymer. One of their prime shortcomings is, that they describe processes with a single relaxation time only. To illustrate this, we consider the stress relaxation of a Maxwell model. In a stress relaxation curves for three different values of the relaxation time have been indicated with this way of plotting the three curves have the same shape and are shifted with respect to the log t axis. Also a more realistic relaxation curve is drawn, which extends over a much broader region of the log t scale than a single element. 

Deviations from predicted relaxation behavior have been observed for large proteins (3 -7 ), polymers (8y9j and highly associated small molecules (10). Particularly prominent are observations of Ti field dependences and low NOE s within the so-called "extreme spectral narrowing region," where single correlation time models predict field independence of Tp and full NOE s. 

The results of the fit are shown in Table I where it is seen that at most temperatures the 95% confidence limit for (X overlaps zero and hence there is little deviation from single relaxation time behaviour according to this analysis. Another inference is that the value of deduced from this procedure probably decreases with increasing temperature but the most important conclusion is that the relaxation time T is not particularly sensitive to the model chosen, the values obtained agreeing to within experimental error irrespective of whether the value of was allowed to be fitted as an unknown parameter, or whether it was clamped at zero (Debye case). 

The simplest model of hindered dipole orientation is due to Debye28), and assumes a single relaxation time for all molecular species. The Debye model, plus a term to... 

Materials that exhibit a single relaxation time constant can be modeled by the Debye relation which appears as a characteristic response in the permittivity as a function of frequency. The complex permittivity diagram is called Cole-Cole diagram constructed by plotting e" vs. e with frequency as independent parameter. 

In modeling the interaction of a liquid with plate modes, the high frequency of operation necessitates the consideration of viscoelastic response by the liquid. For the simple liquids examined, good agreement was obtained by modeling the liquid as a Maxwellian fluid with a single relaxation time r. When the Maxwellian fluid is driven in oscillatory flow with cot < 1, it responds as a Newtonian fluid characterized by the shear viscosity, rj. For wt > 1, the oscillation rate approaches the rate of molecular motion in the liquid and energy ceases to be dissipated in... 

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