Macroscopic relaxation

The transport properties on this model system have been presented by mean of the calculation of the chemical diffusion coefficient representing the macroscopic relaxation of density fluctuations. 

Relation (14) gives equivalent information on dielectric relaxation properties of the sample being tested both in frequency and in time domain. Therefore the dielectric response might be measured experimentally as a function of either frequency or time, providing data in the form of a dielectric spectrum s (co) or the macroscopic relaxation function [Pg.8]

For example, when a macroscopic relaxation function obeys the simple exponential law... 

Figure 13 plots the relaxation times ratio x, / x j and the amplitude A corresponding to the macroscopic relaxation time of the decay function determined by (25). Near the percolation threshold, x, /xi exhibits a maximum and exhibits the well-known critical slowing down effect [152], The description of the mechanism of the cooperative relaxation in the percolation region will be presented in Section V.B. 

It is well known [54,270] that the macroscopic dielectric relaxation time of bulk water (8.27 ps at 25°C) is about 10 times greater than the microscopic relaxation time of a single water molecule, which is about one hydrogen bond lifetime [206,272-274] (about 0.7 ps). This fact follows from the associative structure of bulk water where the macroscopic relaxation time reflects the cooperative relaxation process in a cluster of water molecules. 

In the context of the model presented above, the microscopic relaxation time of a water molecule is equal to the cutoff time of the scaling in time domain To-For the most hydrophilic polymer, PVA, the strong interaction between the polymer and the water molecule results in the greatest value of microscopic relaxation time To, only 10% less than the macroscopic relaxation time of the bulk water. The most hydrophobic polymer, PVP, has the smallest value of a single water molecule microscopic relaxation time, which is almost equal to the microscopic relaxation time of bulk water (see Table III). Therefore, weakening the hydrophilic properties (or intensifying the hydrophobic properties) results in a decreasing of interaction between the water and the polymer and consequently in the decrease of To-... 

The values of Tj, plotted on Fig. 12 leads to an apparent activation energy of 40 + 5 KJ/mole. To compare these results with macroscopic relaxation, we used the well-known WLF time-temperatiu e superposition equation. According to this equation, the value of the principal or glass transition relaxation time t, at a temperature Tj can be deduced from its value Tb at Tb following ... 

The local reorientation processes observed in FAD and the macroscopic relaxation have the same temperature behavior. Similar observations have already been made using NMR or Fluorescence polarization under continuous excitation However,... 

The application of luminescence techniques to the study of macro-molecular behaviour has enjoyed an enormous growth rate in the last decade. The attraction of such methods lies in the degrees of both specificity and selectivity afforded to the investigator. Consequently the polymer may be doped or labelled at sufficiently low concentration levels of luminophore as to induce minimal perturbation of the system. Polarized photoselection techniques offer particular attraction in the study of relaxation phenomena both in solution and solid states. In principle, astute labelling can allow elucidation of the molecular mechanisms responsible for the macroscopic relaxations exhibited by the raacromolecular system. In addition, luminescent probes can address the microviscosity of their environment. 

Many attempts have been made in order to interpret the exponential decay function on a. microscopic basis and to relate the macroscopic relaxation times to those of a single polar unit t. ... 

It is often assumed, although without actual proof, that there exists a definite proportionality between the molecular and the macroscopic relaxation times. A relation between r and r could only be established if the actual field acting on the molecule were known. Equation 27 is obtained if this field is identified either with the external or with the Lorentz internal field. In the first case the macroscopic relaxation time is identical with the molecular one. In the second case r is proportional to t. If Onsager s model is used,88 it may be shown that in first approximation, the cavity field G is itself subject to dispersion ... 

The procedure above has not in any sense derived the macroscopic relaxation equations only some formal conditions have been stated under which the structures of the microscopic and macroscopic equations become the same. One crucial point, which certainly deserves further comment, is the physical basis of the Markov approximation. This approximation removes the memory effects from (5.5) so that the structures of the microscopic and macroscopic equations become similar. For this approximation to be useful, the memory kernel must decay much more rapidly than the density fields. The projected time evolution will guarantee that this is the case, provided these fields decay much more slowly than other variables in the system. 

In Section 10.2 we saw that the macroscopic relaxation equations can be used to determine correlation functions. In this section we summarize the traditional methods for deducing the macroscopic relaxation equations of fluid mechanics. In subsequent sections these equations are used to determine the Rayleigh-Brillouin spectrum. The first step in the derivation of the relaxation equation involves a discussion of conservation laws. 

The typical decay behavior of the dipole correlation function of the microemulsion in the percolation region is presented at Fig. 24. Figure 25 shows the temperature dependence of the effective relaxation time, defined within the fractal parameters, and corresponding to the macroscopic relaxation time Tjjj of the KWW model. In the percolation threshold T, the exhibits a maximum and reflects the well-known critical slowing down effect (131). 

In more complicated cases, the perturbations of equilibrium distributions over vibrational, rotational, and sometimes translational degrees of freedom must be taken into account. It must be borne in mind that the relation between macroscopic relaxation and the reaction times is far from always defining the extent of non-equilibrium. The true criterion for the reaction-induced perturbation of equilibrium distribution is formulated in terms of the microscopic relaxation and of the reaction rate constants which define the relation between the rates of changes in the population of the given quantum states of reactants caused by these two processes. For this reason, the study of elementary rates of relaxation processes is of essential kinetic interest. 

This equation is widely used to extract the microscopic probabilities from the macroscopic relaxation times. 

One can hope that these will not greatly affect comparisons of macroscopic relaxation functions rather than microscopic functions and that better treatments as from Fulton s methods for example will clarify these questions. Even so, It seems fair to claim that a better basis now exists for extracting useful information from Kerr effect measurements and to explore questions of whether rotational reorientations in time are diffusion or Brownian motion like at one extreme infrequently by large jumps at the other or something in between. With developments in instrumentation of the sort suggested above there appear to be real possibilities for studies of dynamics of simpler molecules to complement those by other methods. 

Besides frequency, time is another critical parameter for the description of dielectric phenomena in polymers. The mathematical analysis of the time-dependent response is based largely on the (macroscopic) relaxation function 0(r), which describes the change of the system after the removal of an applied stimulus (in the present case, the electric field, in the case of DMA, the stress). Dipole orientation, which follows the application (at time r = 0) of a static... 

Berg et al. (17) in frog nodes, is likely affected by larger errors because it was based on voltage fluctuation measurements (8) and it was calculated using "standard Hodgkin-Huxley (HH) parameters (18) rather than macroscopic relaxation data from the same nodes. 

As we have seen, the time dependence of a macroscopic relaxation process always reflects the underlying microscopic dynamics. We may now proceed and look for kinetical equations which correctly describe the time dependence of the observed retarded responses. 

RIVAIL - You mentioned the Powles Glarum equation as the only one which links the macroscopic relaxation time to the microscopic one. Is there a special reason, specific / f electrolyte solutions, to prefer this equation to other ones such as KKVR, except simplicity ... 

The local reorientation processes observed in FAD and the macroscopic relaxation have the same temperature behavior. Similar observa-tinuous excitation (40). However, in this latter technique, the slopes of the curves depend on the choice of the model, so that the confidence one could put in the agreement (or discrepancy) between spectroscopic and mechanical results relies directly on the confidence one put in the model arbitrarily chosen to treat the fluorescence polarization data. ... 

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