Stretched exponential function

Abstract: The electrical conductivity of NaNO3–xAl2O3 composites has been studied over the wide range of temperature and frequency by means of impedance spectroscopy. The real part of the frequency dependent conductivity exhibits a simple power law feature and the dimensionless frequency exponent n has been determined. The conductivity spectra show scaling behaviour when the conductivity spectra are scaled by σdcT, where T is temperature in Kelvin. The real part of dielectric permittivity shows saturation at higher frequencies and a strong dispersion at lower frequencies. The imaginary part of permittivity varies inversely with frequency, due to the presence of dc conductivity. The frequency dependent plots of M″ and Z″ show that the conductivity relaxation is non-Debye in nature. The Kohlrausch-Williams-Watts stretched exponential function was used to describe the modulus spectra and the stretching exponent β is found to be temperature independent. The conductivity relaxation time has been estimated from the modulus spectra. The activation energy responsible for relaxation has been evaluated and it was found to be almost same as that of dc conductivity.

Abstract: The dynamics of aqueous solutions of a hydrophobically end-capped poly(oxyethylene)urethane is studied by means of pulsed field gradient NMR and dynamic light scattering (DLS). The results from the NMR measurements at higher concentrations reveal an anomalous diffusion behavior. However, when the observation time is increased the classical Fickian diffusion is recovered. These features are discussed in terms of effects of fractal structure and in the light of a coupling model. The concentration dependence of the self-diffusion coefficient is well described by a stretched exponential function. These results constitute the basis for the evaluation of the concentration-dependent coupling parameter

Abstract: The dynamic magnetic properties of powdered "x-ray amorphous" Fe 2 O 3 sample with mean particle size ~12.0 nm (σ=± 0.36) have been investigated using frequency dependent real part of ac susceptibility Xac' and time dependent dc magnetization studies. A frequency-dependent peak was observed in Xac' vs T measurements. The frequency dependence of the T p , the temperature at which the maximum in the ac susceptibility curve is observed, is described well by the Vogel-Fulcher law as well as the power law. The fitting with the Vogel-Fulcher law yields the relaxation time constant T 0 =5× 10 -8 s and the interaction parameter T 0 =34.0± 0.2 K. Whereas, the fitting with the conventional critical slowing-down law, the power law, yields T 0 =3.5× 10 -11 s, the freezing temperature τ g =35.1± 0.1 K and critical exponent zv=5.3± 0.4. The value of τ 0 =5× 10 -8 s obtained from the Vogel-Fulcher law fit gives an evidence of the presence of interacting spin clusters in our system and that obtained from the power law fit suggests the freezing of these spin clusters at ~35.1 K. Time dependence of the thermoremanent magnetization (TRM), represented well by the stretched exponential function, supports the presence of relaxing spin clusters with no unique relaxation time. The TRM is found to scale according to T log 10 (t/τ 0 ), with T 0 =3.5× 10 -11 s.