Mean-relaxation time

Systems having three or more relaxation times are very difficult to analyze, and the concept of a mean relaxation time has been developed to describe such reactions Bemasconi treats this problem. 

More complicated systems may feature three or even more relaxation times. These are difficult to analyze or interpret convincingly. The concept of a mean relaxation time can be used in these cases.14... 

Fig. 3.6.8 Comparison of estimated permeability for 35 sandstones. Left the log mean relaxation time model. Right the Coats-Timur model [33].

Fig. 4.7 Temperature dependence of the mean relaxation time (r) divided by the rheological shift factor for the dielectric normal mode (plus) the dielectric segmental mode (cross) and NSE at Qinax=l-44 A (empty circle) and Q=1.92 A (empty square) [7] (Reprinted with permission from [8]. Copyright 1992 Elsevier)...

As an example of potential clinical application of the proposed theoretical model, preliminary results of a phase I clinical trial are described below. We estimated the values of relaxation time and ratio Max/Max in adolescents with different results of endoscopy. We found that the mean relaxation time was significantly longer in subjects with a severe gastric and duodenal inflammation, namely, with ulcers and erosions compared to a healthy control group (p<0.05). The exhaled air of patients with milder forms of the disease and of the control group caused faster sensor relaxation after their interaction (Table 7.1). 

The peaks in s j dipolar (which are usually close to the peaks in e") can be used to determine the time or point in the cure process when the mean dipolar relaxation time has attained a specific value, i = 1/co, where to = 2nf is the frequency of measurement. The dipolar mobility as measured by the mean relaxation time t can be used as a molecular probe of the buildup in Tg. The time of occurrence of a given dipolar relaxation time as measured by a peak in a particular high frequency value of e"(co) can be quantitatively related to the attainment of a specific value of the resin s glass transition temperature. 

Chapter E is devoted to the mean-square dipole moment and mean rotational relaxation time derived from dielectric dispersion measurements. Typical data, both in helieogenic solvents and in the helix-coil transition region, are presented and interpreted in terms of existing theories. At thermodynamic equilibrium, helical and randomly coiled sequences in a polypeptide chain are fluctuating from moment to moment about certain averages. These fluctuations involve local interconversions of helix and random-coil residues. Recently, it has been shown that certain mean relaxation times of such local processes can be estimated by dielectric dispersion experiment. Chapter E also discusses the underlying theory of this possibility. 

At thermal equilibrium, the helical fraction and all other quantities characterizing the conformation of a helix-forming polypeptide are fluctuating from time to time about certain mean values which are uniquely determined by three basic parameters s, a, and N. The rates of these fluctuations depend on how fast helix units are created or disappear at various positions in the molecular chain. Recently, there has been great interest in estimating the mean relaxation times of these local helix-coil interconversion processes, and several methods have been proposed and tested. In what follows, we outline the theory underlying the dielectric method due to Schwarz (122, 123) as reformulated by Teramoto and Fujita (124). 

First, for M > 1WC the terminal relaxations are fairly well separated from the transition relaxations. In view of Eqs. (3.24) and (3.25), the values of rj0 and, /e° for M > Mc are properties of the terminal relaxations alone. The mean relaxation times of the terminal distribution are therefore given by Eqs. (3.26)—(3.28), combined with the experimental results in Section 5. 

Kline Stewart (1974) had previously found a mean relaxation time of 1.6 h for uptake of HTO by grass leaves in daylight. Stems were labelled much more slowly, with A21 about 50 h. Any diffusion downwards from leaves to stems would be against the transpiration flow, and the labelling was probably via uptake of HTO to soil. 

Here, Ae is the dielectric strength and x the mean relaxation time. The parameters a and /i describe the symmetric and asymmetric broadening of... 

A tentative conclusion from these considerations is, therefore, that the rate-determining factor for the second stage sorption is the rate at which polymer chains rearrange themselves in the presence of penetrant molecules. In this connection, we may remark that work is in progress concerning the correlation between and the mean relaxation time of a glassy polymer-diluent system (Odani, unpublished). 

The relaxation process may be accompanied by diffusion. Consequently, the mean relaxation time for such kinds of disordered systems is the time during which the relaxing microscopic structural unit would move a distance R. The Einstein-Smoluchowski theory [226,235] gives the relationship between x and R as... 

We first discuss the 2H NMR spin-lattice relaxation results of molecular glass formers at T< Tg. In Fig. 53, we present the mean relaxation time (7)), equal to the integral of the corresponding (nonexponential) relaxation function, for several glasses including a polymer (polybutadiene-d6). The temperature-... 

This equation also allows us to determine the mean relaxation time from q and /g. Actually, the mean relaxation time, (t), can be expressed as... 

This equation indicates that the mean relaxation time is the product of two terminal viscoelastic functions, the zero shear rate viscosity and the steady-state compliance. The mean relaxation time can also be expressed in terms of the relaxation modulus by means of the expression... 

Accordingly, the mean retardation time is always greater than the mean relaxation time. 

The a relaxation in the frequency domain can be interpreted in terms of a stretch mean relaxation time, as discussed below. According to Eq. (5.37), the response of a viscoelastic material to a shear stress a(t) is given by... 

According to Eq. (9.33), the mean relaxation time of the entangled network is... 

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