Temperature dependence of relaxation times

The answer to our question at the beginning of this summary therefore has to be as follows. When you want to locate the glass transition of a polymer melt, find the temperature at which a change in dynamics occurs. You will be able to observe a developing time-scale separation between short-time, vibrational dynamics and structural relaxation in the vicinity of this temperature. Below this crossover temperature, one will find that the temperature dependence of relaxation times assumes an Arrhenius law. Whether MCT is the final answer to describe this process in complex liquids like polymers may be a point of debate, but this crossover temperature is the temperature at which the glass transition occurs. 

Publications on the temperature dependence of 13C relaxation [190-199] are concerned with simple molecules such as carbon disulfide, iodomethane, and acetonitrile [190-194], Any interpretation of the temperature dependence of relaxation times requires a knowledge of the relative contributions of dipole-dipole and spin-rotation relaxation, as the former becomes progressively slower and the latter steadily faster with increasing temperature. In special cases, such as that of cyclopropane [200], the effects of both contributions can almost cancel each other. 

One theory that describes the temperature dependence of relaxation time and structural recovery is the Tool-Narayanaswamy-Moynihan (TNM) model developed to describe the often nonlinear relationship between heating rate and Tg. In this model, the structural relaxation time, x, is referenced as a function of temperature (T), activation enthalpy (Ah ), universal gas constant (R), hctive temperature (7)), and nonlinearity factor (x) (Tool, 1946 Narayanaswamy, 1971 Moynihan et al., 1976) ... 

To proceed further, we make use of the Adam-Gibbs model for the temperature dependence of relaxation time x(T) of cooperative rearranging regions in glassforming liquids [41]... 

During annealing, the temperature dependence of relaxation time of macroscopic quantities is often described by the Narayanaswamy-Moynihan equation... 

Williams, Landel and Ferry introduce their famous WLF-equation for describing the temperature dependence of relaxation times as a universal function of T and Tg... 

The dynamic characteristics of adsorbed molecules can be determined in terms of temperature dependences of relaxation times [14-16] and by measurements of self-diffusion coefficients applying the pulsed-gradient spin-echo method [ 17-20]. Both methods enable one to estimate the mobility of molecules in adsorbent pores and the rotational mobility of separate molecular groups. The methods are based on the fact that the nuclear spin relaxation time of a molecule depends on the feasibility for adsorbed molecules to move in adsorbent pores. The lower the molecule s mobility, the more effective is the interaction between nuclear magnetic dipoles of adsorbed molecules and the shorter is the nuclear spin relaxation time. The results of measuring relaxation times at various temperatures may form the basis for calculations of activation characteristics of molecular motions of adsorbed molecules in an adsorption layer. These characteristics are of utmost importance for application of adsorbents as catalyst carriers. They determine the diffusion of reagent molecules towards the active sites of a catalyst and the rate of removal of reaction products. Sometimes the data on the temperature dependence of a diffusion coefficient allow one to ascertain subtle mechanisms of filling of micropores in activated carbons [17]. 

Figure 1. Temperature dependence of relaxation times derived from phosphorescence depolarization data.

The temperature dependences of relaxation times of the p and y processes of lactose and the secondary mode of octa-O-acetyl-lactose are presented in Fig. 6. In order to determine relaxation times of P- and y- modes of lactose and octa-O-acetyl-lactose the Cole-Cole and Havriliak-Negami functions were used respectively. Activation energies of all secondary relaxations were estimated from the Arrhenius fits... 

Figure 5.26 (a) Zeeman diagram for [Fe802(0H)i2(tacn)6]Br8 calculated with D = -0.2, E/D — 0.19 cm" and with applied field along the easy z) axis, (b) Temperature dependence of relaxation time measured in different applied fields. Reprinted with permission from Sangregorio et al., 1997 [52]. Copyright (1997) American Physical Society... 

Figure 15 shows the 3D plot of s relative to neutralized chitosan, evidencing the a process in the high temperature-low frequency range whose Arrhenius-type temperature dependence of relaxation times is shown in the inset. The secondary process observed in the low-temperature region is due to local main chain motion via the glycosidic bond, influenced by the amino side group (see [149] for discussion). 

The modulus formalism is being applied more frequently to fit data of systems largely influenced by conductivity, such as composites [154], semi-crystalline polymers [143], ionic liquids [63] and biological systems [138]. This allows a better resolution of relaxation processes and leads to similar shape parameters and temperature dependence of relaxation times to those achieved by using complex permittivity [143]. 

Figure 25 shows the temperature dependence of relaxation time for the relaxation processes in the internal and interfacial regions of the ultrathin PS1.46M film sandwiched between the SiO layers. Since it was hard to distinguish the temperature-Ta relations between the vacuum deposited and laminated films, each data point was averaged over six independent measurements including both vacuum deposited and laminated films. The average thickness was about 40 nm. For comparison, the dashed curve in Fig. 25 denotes the bulk data obtained by the Vogel-Fulcher equation [72, 73] ... 

The horizontal shift factor reflects the temperature dependence of relaxation time, and the vertical shift factor reflects the... 

Electron spin relaxation times reflect both intramolecular and intermolecular dynamic processes. In favorable cases relaxation times can be estimated from progressive power-saturation measurements, but more accurate values can be obtained by pulsed time domain techniques. For metal ions with low-lying excited states the temperature dependence of relaxation times can be used to determine the energy of the excited state. ... 

Now, if we assume that a dipolar molecule has a spherical form of volume (Afiyna and rotates in continuous medium with viscosity q [units g.cm s (Poise)], then the friction force may be written as = 87iqa and Xo = /ksT. This model is very simple, however, it predicts a correct magnitude and temperature dependence of relaxation times for dipoles in an isotropic hquid. 

Pouliquen et al. (1997) used NMR technique to study water and lipid reserves in seeds. The temperature dependence of relaxation time was used to identify differences in the thermodynamic properties of water between dry seeds and during germination (Figure 8.3). 

In brief, the VTF equation is commonly used to describe the temperature dependence of relaxation time of a fluid ... 

Fig. 27. Temperature dependence of relaxation time for the conformational transitions evaluated using the jump diffusion model with damped vibrations. ( ) PB, ( ) PIB, ( ) PCP, (x) PE. (Reprinted with permission from [ 130]. Copyright 1999 American Chemical Society, Washington)...

The relaxation and electron dynamics properties of this polymer are changed sufficiently as a hydrogen atom is replaced by a sulfur-based alkyl group. Such substitution leads to the stronger temperature dependence of relaxation times in laser-modified PATAC as compared with trans-VA ... 

C. THE WLF EQUATION AND THE RELATION OF TEMPERATURE DEPENDENCE OF RELAXATION TIMES TO FREE VOLUME... 

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