The Kohlrausch-Williams-Watts (KWW) and the Havriliak-Negami (HN) relaxation functions have been widely used to describe the relaxation behavior of glass-forming liquids and complex systems. While the HN relaxation function is a frequency function, the natural domain of the KWW relaxation function is time (although it has also been used with frequency-domain spectroscopies). A relationship among the parameters of the two models is suggested by the fact that both models yield an accurate description of real data. Nevertheless, this relationship cannot be an analytical one, since it is known that the HN and the KWW relaxation functions are not exactly Fourier transforms of each other. In order to find out the nature of this relationship, a method which makes use of a distribution of relaxation times is proposed here. Numerical simulations following the KWW model have been assumed to describe the relaxation behavior in time; likewise, the HN description was assumed to be valid for the frequency domain. From this work, a connection among the parameters of both models is obtained, which is expected to be valid for those experimental data that can be described by either the KWW or the HN model. This is the case for most, if not all, measurements on the dynamics in complex systems and glass-forming liquids that frequently appear in the literature. The proposed procedure has been tested by using dielectric-spectroscopy measurements, both in frequency and time domains to study the α relaxation in a glass-forming polymeric system, poly(hydroxy ether of bisphenol-A).

The Kohlrausch-Williams-Watts (KWW) and the Havriliak-Negami (HN) relaxation functions have been widely used to describe the relaxation behavior of glass-forming liquids and complex systems over the last several years. The HN relaxation function is a frequency-domain function while the KWW function applies for the time domain. In a previous paper we discussed the interconnections between these two functions by presenting a method where we found that the best HN description in the frequency domain corresponds to a given KWW function in the time domain. From that work we proposed several empirical relationships that allow us to determine the HN parameters corresponding to a given set of KWW ones. It was also outlined how to proceed in the opposite way, i.e., to obtain the best KWW time-domain description corresponding to a given HN relaxation function in the frequency domain. This is what we develop in this work by varying the HN parameters in search of the values of the best KWW fits. Likewise we can put a limit to the region where the HN and the KWW functions are compatible in the sense that they can be equally used by just choosing the right parameter change. A confident confirmation of this procedure is that when the HN parameters reported in the literature for the α relaxation of glass-forming liquids are considered, their values fall directly upon this region, where we find that the HN and KWW functions deviate less. However, for other dynamical processes like secondary relaxations of polymers or the α relaxation of polymer blends the HN parameters reported indicate that a single KWW relaxation function is not able to describe the time-decay behavior.

Using broad-band dielectric spectroscopy ${\text{(10}}^{\text{−2}}{\text{–10}}^9\hspace{0.17em}\mathrm{Hz})$ the merging of the α- and β-relaxations has been investigated for a polymer, syndiotactic poly(methyl methacrylate) (PMMA), in which the dielectric losses are dominated by a strong β-relaxation. The asymmetrically shaped β-relaxation cannot be described by a Cole–Cole function, not even at low temperatures where the α- and β-relaxations are well separated in frequency. At higher temperatures close to $T_g$ (=404 K), the weak α-relaxation enters our dynamic window and rapidly merges with the β-relaxation. To investigate this merging process we first used a simple addition of two Havriliak–Negami (HN) equations to fit the spectra. The obtained relaxation time for the β-relaxation then displays a kink in its temperature dependence close to $T_g.$ To gain further understanding of the merging, the data were analyzed by means of a regularization method in order to calculate the corresponding distributions of relaxation times directly from the actual measurements without the need of invoking any parametrical fitting functions. From these distributions we obtained the relaxation functions in time domain which were analyzed by adopting an ansatz proposed by Williams which implies that the α- and β-relaxations are statistically independent processes. The dynamics in the merging region of PMMA can then be described using extrapolations of the β-relaxation from low temperatures and the kink in the temperature dependence of the relaxation time of the β-relaxation obtained in the simple addition approach, is explained in terms of overlapping effects between the α- and β-processes, i.e., without invoking any change of relaxation mechanism.

We present fully atomistic molecular dynamics simulations for a realistic model of a glass-forming polymer: polyisoprene. The simulations are carried out at 363 K and extend until 20 ns. We calculate the self-part of the Van Hove correlation function $G_s\mathrm{}(r,t),$ the mean-squared displacement $〈r^2\mathrm{}\left(t\right)\mathrm{}〉,$ the second-order non-Gaussian parameter ${\alpha }_2\mathrm{}\left(t\right),$ and the incoherent intermediate scattering function $F_s\mathrm{}(Q,t)\mathrm{}$ for the main chain protons. In addition, we also calculate the density-density correlation function $F(Q,t)/F(Q,0\mathrm{})\mathrm{}$ and the second-order autocorrelation function $M_2\mathrm{}\left(t\right)\mathrm{}$ for different C-H bonds of the main chain. ${\alpha }_2\mathrm{}\left(t\right)\mathrm{}$ shows a broad maximum centered at a time $t^{*}\approx 4\mathrm{ps},$ which corresponds to the intermediate region of $〈r^2\mathrm{}\left(t\right)\mathrm{}〉$ between microscopic dynamics and sublinear diffusion. The analysis of $F_s\mathrm{}(Q,t),$ $F(Q,t)/F(Q,0\mathrm{}),$ and $M_2\mathrm{}\left(t\right)\mathrm{}$ focuses on the second slow step which is associated to the α relaxation. Following the usual experimental procedure this decay is described in terms of a Kohlrausch-Williams-Watts (KWW) function: $A\exp \left[-\right(t/\tau {)}^{\beta }].$ In the Q range below $Q_{\max },$ where $Q_{\max }$ is the value at which the static structure factor shows its first maximum, the Q dependence of the KWW relaxation time of $F_s\mathrm{}(Q,t)\mathrm{}$ follows a law $\tau \sim Q^{-2/\beta }.$ This kind of Q dependence corresponds to a Gaussian behavior of $G_s\mathrm{}(r,t)\mathrm{}$ and $F_s\mathrm{}(Q,t).$ This law has been experimentally found in this Q range for different polymers. In the higher Q range—not easily accessible experimentally—strong deviations from the Gaussian behavior manifest. This crossover from Gaussian to non-Gaussian behavior can be understood in the framework of the mode coupling theory as well as in terms of a crossover from homogeneous to heterogeneous dynamics. This last interpretation opens a possible way of rationalizing the apparent contradiction between the neutron scattering and relaxation techniques results concerning dynamical heterogeneity of the α relaxation.

We have applied fluorescence lifetime imaging (FLIM) to the autofluorescence of different kinds of biological tissue in vitro, including animal tissue sections and knee joints as well as human teeth, obtaining two-dimensional maps with functional contrast. We find that fluorescence decay profiles of biological tissue are well described by the stretched exponential function (StrEF), which can represent the complex nature of tissue. The StrEF yields a continuous distribution of fluorescence lifetimes, which can be extracted with an inverse Laplace transformation, and additional information is provided by the width of the distribution. Our experimental results from FLIM microscopy in combination with the StrEF analysis indicate that this technique is ready for clinical deployment, including portability that is through the use of a compact picosecond diode laser as the excitation source. The results obtained with our FLIM endoscope successfully demonstrated the viability of this modality, though they need further optimization. We expect a custom-designed endoscope with optimized illumination and detection efficiencies to provide significantly improved performance.

We report quasielastic neutron scattering experiments exploring the $\alpha$ relaxation in polyisoprene over an unprecedented range in momentum transfer. Corroborating and validating earlier molecular dynamics simulations, the measurements reveal a crossover from a Gaussian regime of sublinear diffusion to a strongly non-Gaussian regime at short distances. We show that a consistent interpretation in terms of a distribution of finite jumps underlying the $\alpha$ process is possible. This model leads to a time-dependent non-Gaussian parameter exhibiting all features revealed so far from various simulations.