Structural-relaxation time definition

Fig. 4.15 Momentum transfer (Q)-dependence of the characteristic time r(Q) of the a-relaxation obtained from the slow decay of the incoherent intermediate scattering function of the main chain protons in PI (O) (MD-simulations). The solid lines through the points show the Q-dependencies of z(Q) indicated. The estimated error bars are shown for two Q-values. The Q-dependence of the value of the non-Gaussian parameter at r(Q) is also included (filled triangle) as well as the static structure factor S(Q) on the linear scale in arbitrary units. The horizontal shadowed area marks the range of the characteristic times t mr- The values of the structural relaxation time and are indicated by the dashed-dotted and dotted lines, respectively (see the text for the definitions of the timescales). The temperature is 363 K in all cases. (Reprinted with permission from [105]. Copyright 2002 The American Physical Society)...

Let us demonstrate that the tendency to narrowing never manifests itself before the whole spectrum collapses, i.e. that the broadening of its central part is monotonic until Eq. (6.13) becomes valid. Let us consider quantity x j, denoting the orientational relaxation time at ( = 2. If rovibrational interaction is taken into account when calculating Kf(t) it is necessary to make the definition of xg/ given in Chapter 2 more precise. Collapse of the Q-branch rotational structure at T = I/ojqXj 1 shifts the centre of the whole spectrum to frequency cog. It must be eliminated by the definition... 

When a liquid supercools (i.e., does not crystallize when its temperature drops below the thermodynamic melting point), the liquidlike structure is frozen due to the high viscosity of the system. The supercooled liquid is in a so-called viscoelastic state. If the crystallization can be further avoided as the ten ierature continues to drop, a glass transition will happen at a certain temperature, where the frozen liquid turns into a brittle, rigid state known as a glassy state. A well-accepted definition for glass transition is that the relaxation time t of the system is 2 X10 s or the viscosity / isio Pas (an arbitrary standard, of course). 

Owing to their definite structures, most biomolecules have an appreciable permanent dipole moment which must lead to dielectric polarization via the rotational mechanism of preferential orientation. Thus pertinent experimental investigation permits a direct determination of the molecular dipole moments and rotational relaxation times (or rotational diffusion coefficients, respectively). These are characteristic factors for many macro-molecules and give valuable information regarding structural properties such as length, shape, and mass. 

A wide variety of chemical and spectroscopic techniques has been used to determine functionality in humic substances. Although nuclear magnetic resonance (NMR) spectroscopy has been used for a much shorter period of time than most other techniques for determining functional group concentrations, this technique has provided far more definitive information than all other methods combined. However, substantially more work must be done to obtain the quantitative data that are necessary for both structural elucidation and geochemical studies. In order to increase the accuracy of functional group concentration measurements, the effect of variations in nuclear Overhauser enhancement (NOE) and relaxation times must be evaluated. Preliminary results suggest that spectra of fractions isolated from humic substances should be better resolved and more readily interpreted than spectra of unfractionated samples. 

Compared to crystalline materials, the production and handling of amorphous substances are subject to serious complexities. Whereas the formation of crystalline materials can be described in terms of the phase rule, and solid-solid transformations (polymorphism) are well characterised in terms of pressure and temperature, this is not the case for glassy preparations that, in terms of phase behaviour, are classified as unstable . Their apparent stability derives from their very slow relaxations towards equilibrium states. Furthermore, where crystal structures are described by atomic or ionic coordinates in space, that which is not possible for amorphous materials, by definition, lack long-range order. Structurally, therefore, positions and orientations of molecules in a glass can only be described in terms of atomic or molecular distribution functions, which change over time the rates of such changes are defined by time correlation functions (relaxation times). 

Pioneering work of Tool [1946, 1948] on inorganic glasses using dilatometry indicated that volume relaxation after a temperature jump from an initial equilibrium state could not be described simply by a kinetic model in which the relaxation time T was solely dependent on the temperature. Tool therefore proposed that r was also a function of the structure of the glass, and this led to the definition of the Active temperature Tf. 

From this it can be seen that as the temperature of a supercooled liquid is being reduced, the time it takes for structural relaxations to occur increases very rapidly. If the cooling rate is kept constant, then at a certain temperature there will no longer be sufficient time for the liquid to return to equilibrium before the temperature is reduced. At this temperature the liquid undergoes a kinetic transition to the glassy phase and the structure is effectively fixed on an experimental time-scale. This is often defined [25, 44] as 100 s, which corresponds to a viscosity of -lO Pa s. It is also apparent from this definition of the glass transition that it is dependent on the rate at which the liquid is cooled a slower cooling rate will... 

The conventional, and very convenient, index to describe the random motion associated with thermal processes is the correlation time, r. This index measures the time scale over which noticeable motion occurs. In the limit of fast motion, i.e., short correlation times, such as occur in normal motionally averaged liquids, the well known theory of Bloembergen, Purcell and Pound (BPP) allows calculation of the correlation time when a minimum is observed in a plot of relaxation time (inverse) temperature. However, the motions relevant to the region of a glass-to-rubber transition are definitely not of the fast or motionally averaged variety, so that BPP-type theories are not applicable. Recently, Lee and Tang developed an analytical theory for the slow orientational dynamic behavior of anisotropic ESR hyperfine and fine-structure centers. The theory holds for slow correlation times and is therefore applicable to the onset of polymer chain motions. Lee s theory was generalized to enable calculation of slow motion orientational correlation times from resolved NMR quadrupole spectra, as reported by Lee and Shet and it has now been expressed in terms of resolved NMR chemical shift anisotropy. It is this latter formulation of Lee s theory that shall be used to analyze our experimental results in what follows. The results of the theory are summarized below for the case of axially symmetric chemical shift anisotropy. 

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