The fundamental relaxation time

In the Kirkwood approximation a single chain has a hydrodynamic radius R which scales like its geometric radius  

To this radius is associated a diffusion coefficient D = 7 /6mj, R. From D and R we can construct a characteristic time 

The scaling law for the inelastic scattering of light at a fixed wave vector q [eq. (VI.37)] is (within the Kiikwood approximation) 

Now we want to obtain a more physical feeling for this fundamental time T, which is the analog of the first relaxation time in the mode [Mcture. A very useful qualitative model for understanding the meaning of t is the dumbbell model introduced by Kuhn. Here one does not look at all the variables r. r 4-i giving the position of all beads on the chain, but one concentrates on a single variable—the total elongation r = r +i — r,. 

One then pictures the whole chain as a spring, with a certain elastic energy 

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This solution is characterized by two different relaxation times, whereas the fundamental relaxation time is two times longer than in the case of a single barrier process. 

As stated in section 5.7.2, measured mechanical and dielectric relaxation times are not expected to be equal to the fundamental relaxation times of the corresponding relaxing units in the polymer because of the effects of the surrounding medium. These effects are different for the two types of measurement, so it is unlikely that the measured relaxation times will be exactly the same. There are arguments that suggest that, in order to take this into account, the observed relaxation times for dielectric and mechanical relaxation should be multiplied by Soo/ s and respectively, before... 

We end this discussion of the fundamental relaxation time t by remarks on the structure of eq. (VI.42) ... 

In an entangled melt of chains the fundamental relaxation time T( scales like a strong power of the degree of polymerization N (t TV - ). The repta-tion model attempts to describe T( by a calculation of the wiggling motions of one chain inside of a tube fcxmed by its neighbors. It leads to a somewhat weaker exponent T( — N . The discrepancy is unexplained. 

For several samples of poly(7-benzyl-L-glutamate) with molecular weights between 6.4 X 10 and 57 X 10, Wada and collaborators showed by combining data from two laboratories that tq (or its equivalent in a somewhat different analysis) conformed to equation 14 with the correct, very strong dependence on molecular length so this relaxation time clearly represents end-over-end rotation. The others (tj, T2,...) might be attributed either flexural or accordionlike elon-gational modes. For the former, the fundamental relaxation time has been calculated as... 

The longest mode (p=l) should be identical to the motion of the chain. The fundamental correctness of the model for dilute solutions has been shown by Ferry [74], Ferry and co-workers [39,75] have shown that,in concentrated solutions, the formation of a polymeric network leads to a shift of the characteristic relaxation time A,0 (X0=l/ ycrit i.e. the critical shear rate where r becomes a function of y). It has been proposed that this time constant is related to the motion of the polymeric chain between two coupling points. 

There is a fundamental question concerning the nature of the self-motion of protons in glass-forming polymers. In Sect. 4.1 we have shown that the existing neutron scattering results on the self-correlation function at times close to the structural relaxation time r (Q-region 0.2t) with a KWW-like functional form and stretching exponents close to jSsO.5. 

As the average relaxation time becomes even longer than 104 s long times must be allowed before the sample itself achieves equilibrium. Eventually this becomes impractical and the sample becomes a glass. The longest volume relaxation times exceed the patience of the experimenter and the sample is allowed to remain in the nonequilibrium state. However, the sample does not remain in the same state as time increases because it will still relax toward the equilibrium state. The fundamental assumption of stationarity of the fluctuations is then violated and interpretation of the PCS becomes a problem. Such considerations have not stopped people from collecting data in this regime45, but they do preclude a clean interpretation. 

Si NMR comes close to detecting the fundamental step in viscous flow in silicates with good agreement between the time constant of the exchange process determined by NMR and the shear relaxation time of the K SiztOg liquid. 

Accordingly, since the dispersion and xa are obtained independently as separate and unrelated predictions, in such models the dispersion (or the time/frequency dependence) of the structural relaxation bears no relation to the structural relaxation time. This means it cannot govern the dynamic properties. As have been shown before [2], and will be further discussed in this chapter, several general properties of the dynamics are well known to be governed by or correlated with the dispersion. Therefore, neglect of the dispersion means a model of the glass transition cannot be consistent with the important and general properties of the phenomenon. The present situation makes clear the need to develop a theory that connects in a fundamental way the dispersion of relaxation times to xa and the various experimental properties. 

Although glass transition is conventionally defined by the thermodynamics and kinetic properties of the structural a-relaxation, a fundamental role is played by its precursor, the Johari-Goldstein (JG) secondary relaxation. The JG relaxation time, xjg, like the dispersion of the a-relaxation, is invariant to changes in the temperature and pressure combinations while keeping xa constant in the equilibrium liquid state of a glass-former. For any fixed xa, the ratio, T/G/Ta, is exclusively determined by the dispersion of the a-relaxation or by the fractional exponent, 1 — n, of the Kohlrausch function that fits the dispersion. There is remarkable similarity in properties between the JG relaxation time and the a-relaxation time. Conventional theories and models of glass transition do not account for these nontrivial connections between the JG relaxation and the a-relaxation. For completeness, these theories and models have to be extended to address the JG relaxation and its remarkable properties. 

In the formulation of the transport equations, several characteristic time scales are defined. In this framework these time scales are considered fundamental in the classification and the understanding of the dominant mechanisms in the suspension flow. The particle relaxation time Tgp was already defined in (10.98). The particle-particle collision time t, is defined by ... 

In the case of vibrational responses the population relaxation times may be dominating the coherence decays. In addition, it can be essential to incorporate the multilevel nature of molecular vibrators into the response. The rate of repopulation of the ground state is seldom equal to the decay of the fundamental V = 1 state, so there can be bottlenecks in the ground state recovery. Following... 

The physical and mechanical properties of materials are the primary concern for their use in applications. However, these properties reflect the motion of the constituent molecules, which makes study of the latter essential to the fundamental understanding necessary for developing new technologies. The structural dynamics is quantified by a time constant, t, which is a measure of the time scale for reorientation of a small molecule or the correlated conformational transitions of a few backbone bonds in a polymer. For both liquids and polymer melts the structural relaxation time (and viscosity, /, which is roughly proportional to t) varies with temperature, with Arrhenius behavior... 

Relaxation parameters of interest for the study of polymers include 1) 13C and H spin-lattice relaxation times (T1C and T1H), 2) the spin-spin relaxation time T2, 3) the nuclear Overhauser enhancement (NOE), 4) the proton and carbon rotating-frame relaxation times (T p and T p), 5) the C-H cross-relaxation time TCH, and 6) the proton relaxation time in the dipolar state, T1D (2). Not all of these parameters provide information in a direct manner nonetheless, the inferred information is important in characterizing motional frequencies and amplitudes in solids. The measurement of data over a range of temperatures is fundamental to this characterization. 

The fundamental measurements of dielectric constant and resistivity in multiphase systems follow directly from methods used for solid systems (Curtis, 1915). The material resistivity (or electrical conductivity) together with the permittivity are useful parameters for calculating the charge relaxation time of the material. 

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