Stretched exponential relaxation

The relaxation data of Fig. 6.20 are not described by a single time constant, but instead extend over more than four orders of magnitude in time. The time dependence follows a stretched exponential 

The solid lines in the data of Fig. 6.20 are fitted to this expression, with values of P ranging from 0.45-0.70 as indicated. 

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. 

It is apparently a general characteristic of glassy disorder, although there has been considerable debate over the relation between the stretched exponential decay and the microscopic relaxation mechanisms. 

The bonding disorder of a glass suggests that a decay with a single time constant is not expected, but instead an average over the structural configurations. One possibility is a local variation in decay rates described by a distribution of time constants i t), so that 

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Figure 6. Shown is the correlation between the liquid s fragility and the exponent p of the stretched exponential relaxations, as predicted by the RFOT theory, superimposed on the measured values in many liquids taken from the compilation of Bohmer et al. [50]. The dashed line assumed a simple gaussian distribution with the width mentioned in the text. The solid line takes into account the existence of the highest barrier by replacing the barrier distribution to the right of the most probable value by a narrow peak of the same area the peak is located at that most probable value. Taken from Ref. [45] with permission.

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)...

A particular characteristic feature of dynamic processes in the vicinity of the glass transition is the ubiquity of the Kohlrausch-Williams-Watts (KWW) stretched exponential relaxation 1,7-9... 

Thus, the non-Debye dielectric behavior in silica glasses and PS is similar. These systems exhibit an intermediate temperature percolation process associated with the transfer of the electric excitations through the random structures of fractal paths. It was shown that at the mesoscale range the fractal dimension of the complex material morphology (Dr for porous glasses and porous silicon) coincides with the fractal dimension Dp of the path structure. This value can be obtained by fitting the experimental DCF to the stretched-exponential relaxation law (64). 

Kohlrausch (1847) used this analysis to explain his original observation of stretched exponential relaxation. 

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. 

Whether this is more than a convenient parameterization is debatable. The stretched exponential in eq. (35) has no direct relation to stretched exponential relaxation of bulk magnetization (see Campbell et al. 1994). As will be discussed in sect. 8 the only clear-cut case is the highly dilute spin glass. It was shown by Uemura et al. (1984) that above the glass transition temperature root-exponential relaxation occurs, that is p = 0.5. That... 

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]. 

Lupascu, D.C., Eedosov, S., Verdier, C., von Seggern, H., and Rddel, J. (2004) Stretched exponential relaxation in fatigued lead-zirconate-titanate. J. Appl. Phys., 95, 1386-1390. 

J. R. Macdonald and J. C. PhUhps [2005] Topological Derivation of Shape Exponents for Stretched Exponential Relaxation, J. Chem. Phys., to he published. 

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... 

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