Temperature dependence of relaxation

Increasing temperature invariably reduces the magnitude of the relaxation time, by as much as a factor of 10 . Despite this enormous effect on the time of relaxation, for a thermorheologically simple material, temperature has no effect on the rate at which the stress decays (i.e., the shape of the relaxation function). At least for linear polymers, the terminal relaxation function and height of the rubbery plateau are minimally affected by temperature. Segmental 

The functional form of the temperature dependence can vary. For a thermally activated process, wherein the relaxation mechanism is governed by the microscopic success rate in transitioning an energy barrier, Arrhenius behavior obtains 

Very often relaxation data is interpreted in terms of free volume, for example, using the theories of Bueche [19] and Fujita [20]. The idea that free volume governs molecular mobility gives rise to the two most common forms for the temperature dependence of polymer viscoelasticity. If the free volume goes to zero at absolute zero temperature, the equation of Williams-Landel-Ferry (WLF) can be derived [4,17] 

As an alternative, the assumption can be made that the unoccupied volume is not all free , or accessible for the relaxing species. This approach leads to the prediction that the free volume equals zero at some temperature T, which is above zero Kelvin. The resulting expression for the temperature dependence of the relaxation times is the Vogel-Fulcher-Teymann-Hess (VFTH] equation [4,17] 

Since relaxation times must be measured at temperatures wherein their magnitudes fall within the experimental window of the available instmmentation, they are rarely measured for different materials at the same temperature. This introduces uncertainty into any attempt to compare the temperature dependence of different materials. Recently a normalization scheme has been widely adopted for the comparison of data obtained on different polymers, and even on small molecule, glass-forming liquids. In this method, shift factors are plotted as a function of inverse temperature normalized by Tg, or by a dynamic glass transition temperature, defined as one at which the relaxation time assumes some arbitrary value (e.g., 100 s). 

---

M.L. Williams, R.E. Landel, and J.D. Ferry, The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming Uquids, J. Am. Chem. Soc., 77, 3701-3707, 1955. 

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. 

M. L. Williams, R.R Landel, and J.D. Ferry The Temperature Dependence of Relaxation Mechanisms in Amorphous Polymers and Other Glass-Forming Liquids. ... 

As the temperature is further lowered, the natural processes that maintain the Boltzmann distribution (relaxation processes) may be no longer able to keep up with the rate of transitions induced by the microwave radiation. Power saturation leads to a decrease in signal at low temperatures and high levels of microwave power. Because the rate and temperature dependence of relaxation processes is very different in different systems, different paramagnetic species saturate at different levels of power and are best observed at different temperatures. Organic radicals are best observed at relatively high temperature and low levels of power transition metals, especially in systems in which S > 7, are usually observed at cryogenic temperatures because of their rapid relaxation rates. 

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) ... 

From the biological area, iron-sulfur clusters in biomolecules such as rubredoxin mononuclear Fe-S clusters (Rao et at., 1972), plant-type ferredoxin 2Fe-2S clusters (Johnson, 1975) and bacterial-type ferredoxin 4Fe-2S clusters (Thompson et at., 1974) are readily distinguished from one another by their Mossbauer spectra. The temperature dependence of relaxation effects can provide information about the types of internuclear interaction and can even lead to estimates of the distance between paramagnetic sites, for example, the two 4Fe-4S clusters in ferredoxin in Peptococcus aerogenes (Adman etal., 1973). 

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]... 

The temperature dependence of relaxation in glass-forming liquids is often described by the Vogel-Tammann-Fulcher (VTF) equation... 

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 relaxation kinetics of the Arrhenius and Eyring types were found for an extremely wide class of systems in different aggregative states [7,52-54]. Nevertheless, in many cases, these laws cannot explain the experimentally observed temperature dependences of relaxation rates. Thus, to describe the relaxation kinetics, especially for amorphous and glass-forming substances [55-59], many authors have used the Vogel-Fulcher-Tammann (VFT) law ... 

The WLF equation applies to amorphous polymers in the temperature range of Tg to about Tg + lOO C. In this equation J is the reference temperature, these days taken to be the T, while and C2 are constants, initially thought to be universal (with Cx = 17.44 and C2 = 51.6), but now known to vary somewhat from polymer to polymer. These experimental observations bring up a number of interesting questions. What is the molecular basis of the time-temperature superposition principle What is the significance of the log scale and what does the superposition principle tell us about the temperature dependence of relaxation behavior And what about the temperature dependence of a7 at temperatures well below 2 ... 

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... 

---