Macroscopic relaxation functions

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... 

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... 

An alternative approach to DS study is to examine the dynamic molecular properties of a substance directly in the time domain. In the linear response approximation, the fluctuations of polarization caused by thermal motion are the same as for the macroscopic rearrangements induced by the electric field [27,28], Thus, one can equate the relaxation function < )(t) and the macroscopic dipole correlation function (DCF) V(t) as follows ... 

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. 

The macroscopic correlation function can be expressed as a product of the relaxation functions g z/zj) at all stages of self-similarity of the fractal system considered [47,154] ... 

Figure 31. Semilog plot of the macroscopic correlation function of the 20-pm sample ( ) and the 30-pm sample (A) at the temperature corresponding to percolation. The solid lines correspond to the htting of the experimental data by the KWW relaxation function. (Reproduced with permission from Ref. 2. Copyright 2002, Elsevier Science B.V.)...

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. ... 

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). 

The most studied relaxation processes from the point of view of molecular theories are those governing relaxation function, G,(t), in equation [7.2.4]. According to the Rouse theory, a macromolecule is modeled by a bead-spring chain. The beads are the centers of hydrodynamic interaction of a molecule with a solvent while the springs model elastic linkage between the beads. The polymer macromolecule is subdivided into a number of equal segments (submolecules or subchains) within which the equilibrium is supposed to be achieved thus the model does not permit to describe small-scale motions that are smaller in size than the statistical segment. Maximal relaxation time in a spectrum is expressed in terms of macroscopic parameters of the system, which can be easily measured ... 

The molecular dynamics simulations by Brot et al. of dipoles interacting with a Stockmayer or off center Stockmayer potential in a two dimensional disk and three dimensional sphere as described in 2.4 provide and 2 rotational correlation functions as well as equilibrium properties. The more extensive 2D calculations (at less cost) produced single molecule inner disk and macroscopic disk in vacuo correlations (as defined in 1.4 and 3.2) together with the experimental macrosopic relaxation function 90 (t) for (U>). A distinction needs to be made here between the single particle function denoted byf( )(t)of the form the inner disk or sphere function (t)... 

Before we come to these models, we will first introduce a basic law of statistical thermodynamics which we require for the subsequent treatments and this is the fluctuation-dissipation theorem . We learned in the previous chapter that the relaxation times showing up in time- or frequency dependent response functions equal certain characteristic times of the molecular dynamics in thermal equilibrium. This is true in the range of linear responses, where interactions with applied fields are always weak compared to the internal interaction potentials and therefore leave the times of motion unchanged. The fluctuation-dissipation theorem concerns this situation and describes explicitly the relation between the microscopic dynamics in thermal equilibration and macroscopic response functions. 

Tbe dklcctric pennittivity of such subsystem can be expressed in terms of a macroscopic response function t) representing the respoose of tbe subsystem to a step-function external electric 6eld. The subsystem is regarded as being in thermodynamical equilibrium before application of tbe field step, which upseu the equilibrium until a new one is reached. Upon removing the external field the subsystem usually relaxes to equilibrium again, unless this relaxation is hindered by some structural means. 

Importantly for the structural recovery of glasses, the model predicts an equilibrium decay function which is of KWW form (jS= 1 —n), see equation (81), for even a single primitive species. Thus the requirement of a non-exponential decay function is fulfilled by the model. Although the other models use a broad relaxation function to describe behavior, they neither make the prediction of equation (91), nor can the general equation (89) result from them. In general, n and t can be functions of Tand d however, we treat only the case where t is a function of 8, i.e. t = T (r, 5(t )). Then, rewriting equation (86) in terms of t and identifying the macroscopic variable 0 with the departure from equilibrium 5, we find that, for isothermal volume recovery, equation (89) becomes... 

Chapter 4 deals with the local dynamics of polymer melts and the glass transition. NSE results on the self- and the pair correlation function relating to the primary and secondary relaxation will be discussed. We will show that the macroscopic flow manifests itself on the nearest neighbour scale and relate the secondary relaxations to intrachain dynamics. The question of the spatial heterogeneity of the a-process will be another important issue. NSE observations demonstrate a subhnear diffusion regime underlying the atomic motions during the structural a-relaxation. 

Where p defines the shape of the hole energy spectrum. The relaxation time x in Equation 3 is treated as a function of temperature, nonequilibrium glassy state (5), crosslink density and applied stresses instead of as an experimental constant in the Kohlrausch-Williams-Watts function. The macroscopic (global) relaxation time x is related to that of the local state (A) by x = x = i a which results in (11)... 

The second term in Eq. (3) is, as stated at the end of Section IV, the relaxation mean of the integral I[St ] for the information-minimizing (generalized canonical) S, having the basic macroscopic functions p + t dpjdt)0, wa + t duJ0t)o, E + t(SE/dt)0. This ", may be calculated directly, but with the involvement of Lagrange multiplier functions (see below), and the same will follow for l[8F Then, to the first order,... 

The Kirkendall effect in metals shows that during interdiffusion, the relaxation time for local defect equilibration is often sufficiently short (compared to the characteristic time of macroscopic component transport) to justify the assumption of local point defect equilibrium. In those cases, the (isothermal, isobaric) transport coefficients (e.g., Dh bj) are functions only of composition. Those quantitative methods introduced in Section 4.3.3 which have been worked out for multicomponent diffusion can then be applied. 

In elastomer samples with macroscopic segmental orientation, the residual dipolar couplings are oriented as well, so that also the transverse relaxation decay depends on orientation. Therefore, the relaxation rate 1/T2 of a strained rubber band exhibits an orientation dependence, which is characteristic of the orientational distribution function of the residual dipolar interactions in the network. For perfect order the orientation dependence is determined by the square of the second Legendre polynomial [14]. Nearly perfect molecular order has been observed in porcine tendon by the orientation dependence of 1/T2 [77]. It can be concluded, that the NMR-MOUSE appears suitable to discriminate effects of macroscopic molecular order from effects of temperature and cross-link density by the orientation dependence of T2. 

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