Stretched-exponential mode

One recalls that the spectral modes here are stretched exponentials with very different stretching exponents. Unlike pairs of pure-exponential modes, one cannot simply identify a mode as "fast" or "slow" because it decays sooner or later than another mode stretched-exponential modes decay to some extent on a multiplicity of time scales. However, the observations that (i) the slow modes have a common, diffusive, q-dependence, the same for large and small probes, and (ii) the fast modes have a different, non-diffusive (b bj 0) but also common q-dependence, nearly the same for large and small probes, supports our grouping of modes as "fast" or "slow". It would have been equally possible to refer to the modes as sharp, with p 1, and broad, with Pf < p. 

According to the Rouse model the mode correlators (Eq. 3.14) should decay in a single exponential fashion. A direct evaluation from the atomic trajectories shows that the three major contributing Rouse modes decay with stretched exponentials displaying stretching exponents jSof (1 13=0.96 and 2,3 jS=0.86). We note, however, that there is no evidence for the extreme stretching of in-... 

Figure 2 Pump-probe data for the asymmetric stretching CO mode of W(CO)6 ( 1990 cm-1) in supercritical ethane at the critical density (6.88 mol/L) and 343 K. The heavy line is a fit to a single exponential. The lifetime, Ti, equals 278 ps. Data scans in other solvents, temperatures, and densities were of similar quality.

The model exhibits the expected Rouse relaxation modes at high temperature. At lower temperature, the chain motions can still be decomposed into Rouse normal modes, but the normal modes no longer relax via single exponential decays. Instead, the decay of each mode is described by a stretched exponential, and the stretching increases decreases) as the mode number increases. In addition, the temperature-dependence of the relaxation rates is described by the VFTH expression. The emergence of these features of real glasses in such a simple model suggests that such features are insensitive to molecular details. 

Figure 11. Relaxation, 4>(r), of the center of energy is plotted for wave packets propagated by the normal modes of cytochrome c hydrated by 400 water molecules (circles) and myoglobin (squares). Curve is a stretched exponential, Eq. (33), with p = 2v = 0.52, the value fit to the computed energy diffusion data for cytochrome c plotted in Fig. 10, and time constant, t — 11 ps.

Here Qa is the mean value of property Q averaged over basin a (at energy ), and (X) is the spectral weight in the continuum limit of the modes with exponential decay constant X. If 2(0 in fact has the stretched exponential form, then (X) will be proportional to the Laplace transform F(X), for which both numerical (Lindsey and Patterson, 1980) and analytical (Helfand, 1983) studies are available. In the simple exponential decay limit= 1, F(X) reduces to an infinitely narrow Dirac delta function but it broadens as p decreases toward the lower limit to involve a wide range of simple exponential relaxation rates. 

The 20.4 kDa dextran had concentrations up to 166 g/L. QELSS spectra were bimodal, the slower mode corresponding to Dp of the 864 kDa dextrans. Data followed a simple exponential. Daivis, et al. [103] used QELSS and PFGNMR to measure Dp of 110 kDa polystyrene, M /M = 1.06 in 110 kDa polyvinylmethylether, M /Mn 1.3, in toluene. From Fig. 24a, the two physical techniques agree, except perhaps at the largest c. Stretched exponentials describe well each data set. 

Multiple trains of evidence, including (i) the transition in Ds(c) from an exponential to a stretched-exponential, and (ii) observations of Floudas, et cd. (40) that the relative intensities of the fast and slow solvent rotation modes change rapidly in a narrow band of concentrations, suggest that there is a fundamental change in solvent behavior near cp 0.4. 

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). 

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