Stretched exponential relaxation time

In contrast to the stretched exponential relaxation time, the Debye relaxation time T2 does not vary with hydration and stays around t2 = 4.7 0.4 ps and 2.5 0.6 ps for Fi and F2, respectively (Fig. 120). These values are noticeably larger than the bulk values for the same water model (2.5 and 0.9 ps [648]). The parameter a in equation (31) reflects the fraction of these weakly bound water molecules with Debye rotational dynamics. The amplitude a increases with hydration level, as it is shown in Fig. 121. At low hydrations, a is negligibly small and therefore cannot be estimated from the fits with a reasonable accuracy. At the surface of a rigid lysozyme, we have detected the appearance of the water molecules with Debye-like rotational dynamics only when Ny, > 300. On... 

Fig. 5. The characteristic frequencies QR and time exponents (3 in the stretched exponential relaxation function obtained for the randomly labelled PDMS melt at 100 °C. (Reprinted with permission from [44]. Copyright 1989 Steinkopff Verlag, Darmstadt)...

Stretched exponential relaxation is a fascinating phenomenon, because it describes the equilibration of a very wide class of disordered materials. The form was first observed by Kohlrausch in 1847, in the time-dependent decay of the electric charge stored on a glass surface, which is caused by the dielectric relaxation of the glass. The same decay is observed below the glass transition temperature of many oxide and polymeric glasses, as well as spin glasses and other disordered systems. 

The long-pathway rearrangement processes expected for fragile materials at low temperatures are expected to be rare, to involve a local disruption of the otherwise well-structured amorphous medium, and to be relatively long-lived on the usual molecular time scale. These features all contribute to a substantial lengthening of the mean relaxation time /rci(7 ), Eq. (36), with declining temperature. Furthermore, the landscape diversity of deep traps and of the configuration space pathways that connect them should produce a broad spectrum of relaxation times, just as required by stretched-exponential relaxation functions, Eq. (34). 

The exponent value of 0.6 in Jonscher regime is considered to arise by the ion-ion interactions, usually of the coulombic type. During the process of the hopping of the ions, even separate hopping events may have a broad distribution of relaxation times, and this effect can manifest as stretching of the relaxation times. Ngai s coupling model accounts for stretched exponential relaxation and considers it as a consequence of... 

Actually, up to the present time, many-body relaxation is still an unsolved problem in condensed matter physics. In his magical year of 1905, Einstein solved the problem of diffusion of pollen particles in water discovered in 1827 by the botanist, Robert Brown. In this Brownian diffusion problem, the diffusing particles are far apart and do not interact with each other and the correlation function is the linear exponential function, exp(-t/r). It is by far simpler a problem than the interacting many-body relaxation/diffusion problem involved in glass transition. It is a pity that Einstein in 1905 was unaware of the experimental work of R. Kohlrausch and his intriguing stretch exponential relaxation function, exp[-(t/r) ], published in 1847 and followed by other publications by his son, F. Kohlrausch. 

The rate dependencies of the ferroelectric material properties are also reflected in the dynamics after fatigue. Initially, most of the domain system will be switched almost instantaneously [235], and only a small amount of polarization will creep for longer time periods [194]. A highly retarded stretched exponential relaxation was observed after bipolar fatigue treatment [235], and these observations correlated well with the thermally activated domain dynamics. If the overall materials response was represented in a rate-dependent constitutive material law 236], however, then a growing defect cluster size would retard the domain dynamics considerably. Hard and soft material behaviors were also representable as different barrier heights to a thermally activated domain wall motion, as demonstrated by the theoretical studies of Belov and Kreher [236]. 

At short times, the orientation of the central water molecule is fixed by the H bonds to its neighbors. It performs oscillations around the HB direction that are nearly harmonic. This dynamic behavior is described by Cj (t). At longer times, the bonds break, the cage begins to relax, and the particle can reorient itself, losing its memory of its initial orientation. Thus, the first-order rotational correlation function eventually decays to zero by a stretched exponential relaxation. The RCM model demonstrates that the higher order correlation functions are thus determined from Ci(r)[98] and that in the decoupling approximation Fh(<2, t) = FiiQ, t)FR Q, t). The Fh(<2, ) can be written... 

With increasing polymer concentrations, S(q,t) of Faraone, et al. s PMMA samples shows an increasingly prominent slow relaxation. The relaxation first appears at concentrations they identify as being below the nominal overlap concentration. Their bimodal spectra are represented well as a sum of two stretched exponentials in time, the fast relaxation having the larger amplitude and being close-to-exponential (0.9 < 5 1). The mean decay rate of the fast exponential scales as q the relax-... 

In real systems, a distribution in the characteristic time may lead to a stretched exponential decay. In the thermally activated regime where the relaxation of the magnetization is due to the Orbach mechanism, the temperature dependence of the relaxation time may be described by an Arrhenius law of the form ... 

Fig. 10. Arhennius plot of relaxation time obtained by stretched exponential fits of decay of nBx for n-type and p-type a-Si H (Street et al., 1988).

The slow relaxation of the occupied band tail density and of the conductivity a are accurately described by the stretched exponential time dependence... 

Thus, even when the elongation rate, as defined by equation 3.76, is constant, the separation of two material points increases exponentially with time. As stress relaxation occurs exponentially, it is clear that at high elongation rates the stress will increase very rapidly. In a purely viscous liquid the stress relaxes instantaneously and consequently this high resistance to stretching does not occur. 

Fig. 4.22 Arrhenius representation of the relaxation rates obtained from fitting stretched exponentials to the spectra of PB at Q=1.88 A" at different temperatures. The three symbols represent three different sets of experiments carried out in separate experimental runs. The solid line displays the viscosity time scale. The dashed line indicates the Arrhenius behaviour of the low-temperature branch. (Reprinted with permission from [188]. Copyright 1992 The American Physical Society)...

Fig. 4.24 Temperature dependence of the characteristic times obtained from the fits of Spair(Q,t) to stretched exponentials with =0.41 at Qmax=l-48 A (filled circle) and 2.71 A (empty circle). Dashed-dotted line corresponds to the Vogel-Fulcher-like temperature dependence of the viscosity and the solid line to the Arrhenius-like temperature dependence of the dielectric -relaxation. (Reprinted with permission from [189]. Copyright 1996 The American Physical Society)...

The data were fitted to a stretched exponential function (Eq. 4.9) setting the stretching parameter to its dielectric value. The solid lines included in Fig. 6.3 display the resulting curves. These fits lead to the Q-dependent characteristic relaxation times TKww(Q)> hich are converted to average relaxation times by Eq. 5.25 (see Fig. 6.4). 

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