Weighting factor, relaxation

Relaxation processes are probably the most important of the interactions between electric fields and matter. Debye [6] extended the Langevin theory of dipole orientation in a constant field to the case of a varying field. He showed that the Boltzmann factor of the Langevin theory becomes a time-dependent weighting factor. When a steady electric field is applied to a dielectric the distortion polarization, PDisior, will be established very quickly - we can say instantaneously compared with time intervals of interest. But the remaining dipolar part of the polarization (orientation polarization, Porient) takes time to reach its equilibrium value. When the polarization becomes complex, the permittivity must also become complex, as shown by Eq. (5) ... 

In expression 5, W is a weighting factor as before, aoab is the measured cross relaxation rate and (Tab is the calculated cross relaxation rate at any stage in the calculation as given by expression 3. Since a is directly proportional to 1/r, this expression reduces to expression 2 in the limit of a single rigid conformer, if W is adjusted to absorb the proportionality constants. 

For these first experiments, a temperature relatively close to Tg, T=123°C, was chosen with the intention of minimizing the relaxation of stress and chain orientation during the quenching the weight-average relaxation time of sample SI at 123°C is calculated from that at 140°C and the thermal shift factor between 123°C and 140°C Xw(123°C) 380s. On the other hand the cooling time of the stretched specimens can be estimated to a few seconds [19], which is very small compared to the polymer relaxation time at the temperature of the experiments. 

It is also possible to derive a set of relaxation times and weighting factors numerically by optimization or approximation methods from the experimental data. In that case, the relaxation times have no real physical meaning and are simply numerical/empirical pareimeters which allows one to represent the viscoelastic behaviour as a sum of decaying exponentials which are handy to use for n imerical analysis. 

In a macroscopically disordered system such as a microcrystalline powder or a glassy frozen solution, all possible orientations p occur with weighting factors sin p. The EPR spectrum of such a disordered system depends on whether reorientation by rotational diffusion is very slow, moderate, or very fast on the EPR timescale. In the following we assume isotropic Brownian rotational diffusion with an isotropic value Riso of the diffusion tensor and a transverse relaxation time of 150 ns. 

Specific models for internal motions can be used to interpret heteronuclear relaxation, such as restricted diffusion and site-jump models. However, model-free formal methods are preferable, at least for the initial analysis, since available experimental data generally are insufficient to completely characterize complex internal motions or to uniquely determine a specific motional model. The model-free approach of Lipari and Szabo for the analysis of relaxation data has been used for proteins and even for peptides. It attempts to reproduce relaxation rates by a weighted product of spectral density functions with different correlation times The weighting factors are identified as order parameters for the molecular rotational correlation time and optional further local correlation times r. The term (1-S ) would then be proportional to the amplitude of the corresponding internal motion. However, the Lipari-Szabo approach is based on the assumption that molecular and local correlation times are not coupled, i.e. they should be distinct enough (e.g. differing by at least a factor of 10 in time) to allow for this separation. However, in small molecules the rates of these different processes are of the same order of magnitude, and the requirements of the Lipari-Szabo approach may not be fulfilled. Molecular dynamics simulation provide a complementary approach for the interpretation of relaxation measurements. 

The solidity of gel electrolytes results from chain entanglements. At high temperatures they flow like liquids, but on cooling they show a small increase in the shear modulus at temperatures well above T. This is the liquid-to-rubber transition. The values of shear modulus and viscosity for rubbery solids are considerably lower than those for glass forming liquids at an equivalent structural relaxation time. The local or microscopic viscosity relaxation time of the rubbery material, which is reflected in the 7], obeys a VTF equation with a pre-exponential factor equivalent to that for small-molecule liquids. Above the liquid-to-rubber transition, the VTF equation is also obeyed but the pre-exponential term for viscosity is much larger than is typical for small-molecule liquids and is dependent on the polymer molecular weight. 

Under these conditions, Eq. (32) indicates the maximum extent to which a particular mode p can reduce S(Q,t) as a function of the momentum transfer Q. Figure 10 presents the Q-dependence of the mode contributions for PE of molecular weights Mw = 2000 and Mw = 4800 used in the experiments to be described later. Vertical lines mark the experimentally examined momentum transfers. Let us begin with the short chain. For the smaller Q the internal modes do not influence the dynamic structure factor. There, only the translational diffusion is observed. With increasing Q, the first mode begins to play a role. If Q is further increased, higher relaxation modes also begin to influence the... 

---