Relaxation spectra - single mode

Vertical scales have been adjusted so that Rg(0) values agree and so that the vertical axes both span a factor of two in Rg. 

This section considers reports of the frequency-dependent dielectric functions, primarily for polymers in nondilute solutions. By analogy with the treatment of the storage and loss moduli in Chapter 13, the two-parameter temporal scaling approach in that chapter leads to expectations for the dynamic dielectric and dielectric loss functions and their frequency dependences, including for the dynamic dielectric function 

Dielectric relaxation spectra of cfs-polyisoprene in benzene, at concentrations from the dilute up almost to the melt, were obtained by Adachi, et a/. (4). The polymer s and were 86 and 102 kDa, respectively. Relaxation spectra are shown in Eigure 7.9 lower-frequency stretched exponentials and higher-frequency power laws describe each spectrum well, though both frequency regimes were not reached with every solution. The exponentials are clearly stretched, with 5 1 the power-law exponents x e (1.2,1.38) are seen to be close to those of other polyisoprene systems. For the most concentrated solutions at very high frequency, an additive constant reflects the first visibility of the higher-frequency segmental diffusive modes, as explored by Adachi, era/. (31). 

In a study of highly concentrated cw-polyisoprenes in toluene, Adachi, et al. determinede (co) with4.5 53kDaforconcentrations41-100wt% (31-33). 

V 10 Hz, a second stretched exponential and then a power law in co appear as functions to be added to the first power law to generate the displayed lines. The second power-law s exponent is in the range 1.05-1.10. 

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The storage and loss moduli, G and G", are obtained from the relaxation spectrum in the usual way—that is, using G = Gi[co rl/(l + co zl)] G — G,[mT /(l -P The longest relaxation mode of the relaxation modulus in Eq. (3-67) is the dominant one it accounts for 96% of the zero-shear viscosity. Thus, the reptation model predicts that for a nearly monodisperse melt, the relaxation spectrum is dominated by a single relaxation time, T = Ta. This is in reasonable accord with experimental data at low and moderate frequencies (see the dashed line in Fig. 3-29). As the frequency increases, however, there... 

It might be argued that this agreement reflects the method of data treatment rather than a fundamental confirmation of Eq. (2.1) and certain other analogous relations for dynamic behavior. That is, both experiments are treated with relations deduced from a single normal mode calculation used to calculate both r] and the relaxation spectrum H, and both yield the product Thus, rj may be obtained from H (t) as... 

Smalley and co-workers have probed intramolecular vibrational relaxation by viewing the yields and the time-dependence of the fluorescence from Sj in alkylated benzenes. They focus attention on those ring modes whose vibrational frequencies are unshifted by alkylation these are vibrations with nodes at the alkylated ring carbon atom. The absorption lines are sharp, but as the alkyl chain is lengthened, the emission spectrum develops a broad relaxed component, while the intensity of the sharp unrelaxed resonance fluorescence diminishes in intensity as the intensity of the relaxed spectrum increases. The time-dependence of the relaxed and unrelaxed emission is found to be a single exponential decay, so unfortunately, the rapid intramolecular dephasing decay has not yet been followed. 

Relaxation of Single and of Entangled Macromolecules. In the absence of hydrodynamic interactions (HI) the normal modes of a polymer are Rouse modes, which act as overdamped harmonic oscillators. With HI the Rouse modes are still nearly normal modes, but the relaxation spectrum is modified. The HI are screened in semidilute solutions. At higher concentrations and in bulk disentanglement by reptation and tube renewal dominate slow viscoelastic processes. 

If this is now squared, using the double reptation formula, Eq. 6.3 5, we obtain many relaxation terms that correspond to the cross terms for each pair of terms in the above summations. This will broaden the spectrum of relaxation times compared to the single-relaxation-time approximation. Nevertheless, because the Doi-Edwards relaxation spectrum is so narrow (i.e., the modes higher than the first mode have very little weight), inclusion of these extra modes does not improve the predictions of the double reptation theory very much. The major reason the basic double reptation model does poorly in describing the shape of the peaks in is... 

It should be noted that the decomposition shown in Eq. 3.7.2 is not necessarily a subdivision of separate sets of spins, as all spins in general are subject to both relaxation and diffusion. Rather, it is a classification of different components of the overall decay according to their time constant. In particular cases, the spectrum of amplitudes an represents the populations of a set of pore types, each encoded with a modulation determined by its internal gradient. However, in the case of stronger encoding, the initial magnetization distribution within a single pore type may contain multiple modes (j)n. In this case the interpretation could become more complex [49]. 

In a typical situation we are interested in the absorption mode of a dynamic spectrum,/abs (co), which equals the real part of the complex function f(co) given by equation (145). In most cases of unsaturated spectra the relaxation matrix which describes single-quantum transitions can be replaced by a constant — E/T2(effective) which is characteristic of the experimental conditions involved and reflects the inhomogeneity of the external magnetic field B0. The absorption mode spectrum is given by ... 

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