Distributions of Relaxation Times

The relaxation times, t, and corresponding moduli, G, constitute what is called the distribution or spectrum of relaxation times. The relaxation spectrum given in Eq. (3-40) is a distinctive feature of the Rouse model that can be tested experimentally. A simple type of rheological experiment from which this spectrum can be obtained is small-amplitude oscillatory deformation, discussed in Section 1.3.1.4. In this test, at low frequencies, (o /x, the Rouse model predicts the usual terminal relaxation behavior G — Gco rf, and G = Gcnxi. More significantly, at higher frequencies, where co is in the range 1/t, oj 1/tat, the Rouse model predicts a power-law frequency dependence of G and G  

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In principle, the relaxation spectrum H(r) describes the distribution of relaxation times which characterizes a sample. If such a distribution function can be determined from one type of deformation experiment, it can be used to evaluate the modulus or compliance in experiments involving other modes of deformation. In this sense it embodies the key features of the viscoelastic response of a spectrum. Methods for finding a function H(r) which is compatible with experimental results are discussed in Ferry s Viscoelastic Properties of Polymers. In Sec. 3.12 we shall see how a molecular model for viscoelasticity can be used as a source of information concerning the relaxation spectrum. 

The dynamic mechanical properties of PTFE have been measured at frequencies from 0.033 to 90 Uz. Abmpt changes in the distribution of relaxation times are associated with the crystalline transitions at 19 and 30°C (75). The activation energies are 102.5 kj/mol (24.5 kcal/mol) below 19°C, 510.4 kJ/mol (122 kcal/mol) between the transitions, and 31.4 kJ/mol (7.5 kcal/mol) above 30°C. 

The notion that 1// noise results from a distribution of relaxation times, operating simultaneously and independently, was first proposed by A. van der Ziel, Physica 16 (1950) 359. 

In studies of superparamagnetic relaxation the blocking temperature is defined as the temperature at which the relaxation time equals the time scale of the experimental technique. Thus, the blocking temperature is not uniquely defined, but depends on the experimental technique that is used for the study of superparamagnetic relaxation. In Mossbauer spectroscopy studies of samples with a broad distribution of relaxation times, the average blocking temperature is commonly defined as the temperature where half of the spectral area is in a sextet and half of it is in a singlet or a doublet form. 

In summary, the NFS investigation of FC/DBP reveals three temperature ranges in which the detector molecule FC exhibits different relaxation behavior. Up to 150 K, it follows harmonic Debye relaxation ( exp(—t/x) ). Such a distribution of relaxation times is characteristic of the glassy state. The broader the distribution of relaxation times x, the smaller will be. In the present case, takes values close to 0.5 [31] which is typical of polymers and many molecular glasses. Above the glass-to-liquid transition at = 202 K, the msd of iron becomes so large that the/factor drops practically to zero. 

Natural rocks seldom have a single pore size but rather a distribution of pore sizes. If all pores are in the fast-diffusion limit, have the same surface relaxivity and have no diffirsional coupling, then the pores will relax in parallel with a distribution of relaxation times that corresponds to the distribution of the pore sizes. The magnetization will decay as a sum of the exponentials as described by Eq. (3.6.4). 

The oil and water response in Figure 3.6.7 could easily be distinguished in this example with kerosene as the oil. If the oil was a crude oil with a broad distribution of relaxation times, the oil may have non-zero response at relaxation times shorter than the Tx cut-off. This could result in mistaking a part of the oil response as BVI. The correct approach in this case is to use diffusion measurements to distinguish between water and oil. This will be discussed under fluid identification (Section 3.6.9). 

R. M. Kroeker, R. M. Henkelman 1986, (Analysis of biological NMR relaxation data with continuous distributions of relaxation-times),/. Mag. Reson. 69, 218. 

The inset shows a unimodal distribution of relaxation times r = I 1 obtained by a CONTIN analysis. Besides CONTIN there is a number of alternative techniques [51] for the determination of the distribution from the correlation function. Detailed discussions of this topic have been given by Stock and Ray [52] and by Stepanek [50]. 

The distribution of relaxation times H(r) can be estimated from a stress relaxation or Er(() curve plotted on a log t scale by... 

The methods described above give continuous distributions of relaxation times. However, the molecular theories of Viscoelasticity of polymers as... 

Master curves are important since they give directly the response to be expected at other times at that temperature. In addition, such curves are required to calculate the distribution of relaxation times as discussed earlier. Master curves can be made from stress relaxation data, dynamic mechanical data, or creep data (and, though less straightforwardly, from constant-strain-rate data and from dielectric response data). Figure 9 shows master curves for the compliance of poly(n. v-isoprene) of different molecular weights. The master curves were constructed from creep curves such as those shown in Figure 10 (32). The reference temperature 7, for the... 

The distribution of relaxation times f/(ln t) is a constant over several decades of time. What is the shape of the stress-re taxation curve over this time interval ... 

It should be noted that a number of experimental observations do not agree with the Bakhshiev-Mazurenko model (1) the time-dependent range of relaxational shifts of spectra is considerably wider than that described by Eq. (2.9), which may be associated with the existence of a distribution of relaxation times 89 94, (2) the bandwidth of the fluorescence spectrum varies significantly during relaxation(93) (3) substantial deviations from exponential... 

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 ... 

This picture is usually known as the heterogeneous scenario. The distribution of relaxation times g (In r) can be obtained from < (t) by means of inverse Laplace transformation methods (see, e.g. [158] and references therein) and for P=0.5 it has an exact analytical form. It is noteworthy that if this scenario is not correct, i.e. if the integral kernel, exp(-t/r), is conceptually inappropriate, g(ln r) becomes physically meaningless. The other extreme picture, the homogeneous scenario, considers that all the particles in the system relax identically but by an intrinsically non-exponential process. 

Thus the SINs presumably have a fine-scale, microheterogeneous character, while the PU Itself has a relatively broad distribution of relaxation times relative to that of the PMMA. This overall... 

By using the normalization condition for the distribution of relaxation times,... 

One of the widely used methods of analysis of kinetic data is based on extraction of the distribution of relaxation times or, equivalently, enthalpic barrier heights. In this section, we show that this may be done easily by using the distribution function introduced by Raicu (1999 see Equation [1.16] above). To this end, we use the data reported by Walther and coworkers (Walther et al. 2005) from pump-probe as well as the transient phase grating measurements on trehalose-embedded MbCO. Their pump-probe data have been used without modification herein, while the phase grating data (also reproduced in Figure 1.12) have been corrected for thermal diffusion of the grating using the relaxation time reported above, r,, and Equation (1.25). 

In Figure 1.13 two recombination steps are clearly distinguishable, especially in the pump-probe data. These have been fitted previously by two separate stretched exponentials. Here, we used a single distribution of relaxation times, which accounted for both recombination steps. 

Once the best-fit parameters are obtained, the distribution of relaxation times or, equivalently, barrier heights may be easily computed from Equation (1.16). The distribution function corresponding to the data in Figure 1.13 is plotted for the two types of measurements in Figure 1.14. 

FIGURE 1.14 Distribution of relaxation times for the recombination of photodissociated Mb-CO. 

Gordon Atkinson I think there is, at least in principle, a way to examine a system for distribution of very similar paths. This is by relaxation technique, particularly ultrasonic absorption. A distribution of similar, though not identical paths, implies a distribution of relaxation times centered at a certain frequency. But there is a subtle experimental problem involved in distinguishing between a single relaxation time and a rather closely spaced distribution of relaxation times. 

The Tte of the 3Fe-4S centre in succinate ubiquinone reductase between 4 and 8 K is decreased by interaction with paramagnetic cytochrome b.98 To mitigate the impact of spectral diffusion the relaxation times were measured by a picket-fence sequence with 100 pulses. Analysis of the powder pattern distribution of relaxation times indicated that the anisotropic dipolar interaction dominated over isotropic scalar interaction and a lower limit of 10 A was estimated for the distance between the iron-sulfur cluster and the heme. 

Computation, using the dynamic RIS model, of the relaxation times for POM helices. The bimodal distribution of relaxation times is rationalized with a simple model. 

Note that h is proportional to n1/2 in 0-solvents, and thus to N112. For 0 = 0 the flow disturbance is zero, the chain is said to be free draining, and the original Rouse model is recovered. For hP, flow in the coil interior is presumed to be substantially reduced, the chain is frequently said to behave as an impenetrable coil, and the Zimm model is obtained. Equations (4.10-4.12) continue to apply for all values of h, although the distribution of relaxation times depends on h. Some results for the two limiting cases and large N are ... 

The shear modulus G(l) of a relaxing viscoleastic substance is a more sensitive probe of the overall distribution of relaxation times, as it does not depend so completely on either end of the relaxation spectrum. Although the present one-dimensional model cannot comprehend shear, it may be useful to study the analogous relaxation function. The relaxation function //(In x), is defined10 by... 

In general, the dielectric relaxation in a polar polymer exhibits a wide distribution of relaxation time. In this case, the real and imaginary parts of the dielectric constant are written in the form ... 

Results of the next section indicate that the three-dimensional lattice acts much like an Einstein solid with a single phonon frequency vE. Viscoelastic experiments (5) on rosin in the glassy state are consistent with the sharp distribution of relaxation times thus predicted. In the dielectric case where each oscillator contains a dipole,... 

Cubic Array. In the first section we demonstrated that, in appropriate limits, the relaxation time associated with a particular normal mode is related to the characteristic frequency as the inverse square [Eq. (1.6)]. This basic relation is independent of dimensionality of the array and provides a connection between lattice vibrational spectra and the distribution of relaxation times function, as (6)... 

A series of curves drawn for the same molecular weights of Fig. 5 is presented in Fig. 6. Notice that the modulus function falls off much more rapidly in the three-dimensional case, reflecting the sharper distribution of relaxation times. (The broken line indicates the slope-1/2 associated with the linear chain behavior.)... 

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