Kohlrausch Williams Watts Relaxation Research Articles

Spatial and temporal heterogeneity of Kohlrausch–Williams–Watts stress relaxations in metallic glasses

We perform a molecular dynamics (MD) stress relaxation simulation for Zr50Cu40Al10 metallic glass to confirm that the time dependency of stress relaxation conforms with the Kohlrausch–Williams–Watts (KWW) equation, and to derive the temperature dependency of the Kohlrausch exponent βKWW. We also calculate local plastic deformation based on atomic strain, then discuss the morphology of relaxation and calculate the probability density of stress relaxation with respect to the characteristic time of relaxation from the number of deformed atoms. Afterward, we derive the time dependency of stress relaxation as a mode-averaged decay function, which expresses spatial and temporal heterogeneity. Both the results of simulation and calculation reproduce the KWW relaxation form and are in good agreement, confirming the spatially and temporally heterogeneous nature of KWW relaxation. The heterogeneity of the stress relaxation of metallic glass is determined by local stress changes caused by microscopic local plastic deformation.

The heterogeneous energy landscape expression of KWW relaxation.

Here we show a heterogeneous energy landscape approach to describing the Kohlrausch-Williams-Watts (KWW) relaxation function. For a homogeneous dynamic process, the distribution of free energy landscape is first proposed, revealing the significance of rugged fluctuations. In view of the heterogeneous relaxation given in two dynamic phases and the transmission coefficient in a rate process, we obtain a general characteristic relaxation time distribution equation for the KWW function in a closed, analytic form. Analyses of numerical computation show excellent accuracy, both in time and frequency domains, in the convergent performance of the heterogeneous energy landscape expression and shunning the catastrophic truncations reported in the previous work. The stretched exponential β, closely associated to temperature and apparent correlation with one dynamic phase, reveals a threshold value of 1/2 defining different behavior of the probability density functions. Our work may contribute, for example, to in-depth comprehension of the dynamic mechanism of glass transition, which cannot be provided by existing approaches.

Effect of change in the physical properties of water at its peculiar temperature points on the dielectric behavior of sodium polyacrylate

The behavior of sodium polyacrylate is studied via broadband dielectric spectroscopy. The non-Debye response in the frequency range 0.1 Hz–1.0 MHz, the non-Markovian dispersion in the infralow-frequency range, and the anomalies of dielectric permittivity and ac conductivity near “peculiar” temperatures of water (20, 35, 50, and 75°C) are ascertained experimentally. These facts make it possible to assign sodium polyacrylate to disordered media, in which the temperature behavior of dielectric parameters is described by the Kohlrausch–Williams–Watts relaxation function.

Simple method for the dielectric relaxation function investigation

A new simple method for determination of the numerical parameters of dielectric relaxation functions is presented. This analysis method uses a logarithmic scale for the time axis and the first derivative of relaxation function. Three reference points for the relaxation function are found; and new numerical parameters at these points are defined. The analysis of the Kohlrausch–Williams–Watts relaxation function is carried out. A number of useful formulas relating to the KWW exponent and the newly introduced numerical coefficients are found. The presented method gives the numerical parameters of a relaxation function and provides the new opportunities for dielectric studies.

High Dielectric Constant and Relaxation Mechanism of Water with Hydrated Copper(II) Ions in a Cucurbit[8]uril-Based Supramolecular Architecture

The dielectric properties of water confined in nanochannels of a cucurbit[8]uril-based supramolecular architecture containing water and hydrated copper(II) ions were investigated in the frequency range of 102 to 5 × 107 Hz at various temperatures around room temperature. Two relaxation processes were revealed and studied. The low-frequency dispersion described by the fractional power law is related to the relaxation of hopping charge carriers. The second orientational polarization contribution follows the Kohlrausch–Williams–Watts relaxation law with the characteristic relaxation time almost independent of temperature and gives rise to an exceptionally large static dielectric constant (∼500). The data suggest the existence of a third relaxation process with the characteristic frequency significantly larger than the measurement frequencies used in this work.

Structural relaxation of scintillating Ce-doped NaGd(PO3)4 glass

Structural relaxation of scintillating Ce-doped Na–Gd phosphate glass with a nominal composition of Ce:NaGd(PO3)4 was experimentally studied using non-isothermal thermo-mechanical analysis, and the relaxation process was described by the Tool–Narayanaswamy–Mazurin model. The distribution of relaxation times was expressed by the empirical Kohlrausch–Williams–Watts relaxation function with relaxation time directly proportional to dynamic viscosity. The model parameters and material constants were obtained by the nonlinear regression analysis of thermo-mechanical data. It has been concluded that the model used of structural relaxation correctly describes relaxation processes in studied Ce-doped NaGd(PO3)4 glass.

Rheological complexity in simple chain models

Dynamical properties of short freely jointed and freely rotating chains are studied using molecular dynamics simulations. These results are combined with those of previous studies, and the degree of rheological complexity of the two models is assessed. New results are based on an improved analysis procedure of the rotational relaxation of the second Legendre polynomials of the end-to-end vector in terms of the Kohlrausch-Williams-Watts (KWW) function. Increased accuracy permits the variation of the KWW stretching exponent beta to be tracked over a wide range of state points. The smoothness of beta as a function of packing fraction eta is a testimony both to the accuracy of the analytical methods and the appropriateness of (eta(0)-eta) as a measure of the distance to the ideal glass transition at eta(0). Relatively direct comparison is made with experiment by viewing beta as a function of the KWW relaxation time tau(KWW). The simulation results are found to be typical of small molecular glass formers. Several manifestations of rheological complexity are considered. First, the proportionality of alpha-relaxation times is explored by the comparison of translational to rotational motion (i.e., the Debye-Stokes-Einstein relation), of motion on different length scales (i.e., the Stokes-Einstein relation), and of rotational motion at intermediate times to that at long time. Second, the range of time-temperature superposition master curve behavior is assessed. Third, the variation of beta across state points is tracked. Although no particulate model of a liquid is rigorously rheologically simple, we find freely jointed chains closely approximated this idealization, while freely rotating chains display distinctly complex dynamical features.

Impossibility of pressure-induced crossover from ferroelectric to nonergodic relaxor state in aPb(Mg1∕3Nb2∕3)0.7Ti0.3O3crystal: Dielectric spectroscopic study

Relaxor behavior induced by hydrostatic pressure up to $0.95\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$ in the $\mathrm{Pb}({\mathrm{Mg}}_{1∕3}{\mathrm{Nb}}_{2∕3}{)}_{0.7}{\mathrm{Ti}}_{0.3}{\mathrm{O}}_{3}$ (PMN-30PT) ferroelectric crystal was studied using dielectric spectroscopy. With increasing pressure we observed the decrease of the ferroelectric phase transition temperature $({T}_{C})$, the suppression and smearing of the dielectric anomaly at ${T}_{C}$, and the appearance of strong relaxorlike dielectric dispersion below the temperature of the permittivity maximum $({T}_{m})$. Such kinds of pressure-induced alteration are inherent in compositionally disordered perovskite ferroelectrics. It is usually believed to signify a crossover from the ferroelectric ground state to the nonergodic relaxor ground state in which the dipole moments of polar nanoregions (PNRs) are frozen in a way characteristic of dipole glasses. Surprisingly, our analysis of the dielectric spectra in PMN-30PT at high pressure did not reveal any glassy freezing of dipole dynamics. This means that the nature of the high-pressure-induced ground state is different from the nonergodic relaxor state observed in canonical relaxors at ambient pressure. At $T>{T}_{C}$ the dielectric spectra measured in PMN-30PT under different pressures are qualitatively similar. They are composed of two contributions that follow the Kohlrausch-Williams-Watts (KWW) and the Curie--von Schweidler (CS) relaxation patterns, respectively. The dielectric susceptibility related to the KWW relaxation provides the major contribution to the total dielectric constant. The shapes of the frequency and temperature dependences of this susceptibility remain practically unaffected by pressure. Contrary to the canonical relaxors the KWW relaxation time does not obey the Vogel-Fulcher law. On the other hand the CS-related susceptibility, which is significant only at low frequencies, considerably increases with increasing pressure and the shapes of its frequency and temperature dependences change radically. At $T<{T}_{C}$ the KWW and CS relaxation processes are not observed at ambient pressure, but persist at $0.8\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$. The KWW characteristic relaxation time varies with temperature according to the Arrhenius law. We propose that the observed variation of properties results from the pressure-induced crossover from the sharp order-disorder-type ferroelectric phase transition, which is triggered by the cooperative interactions among dynamic (in the high-temperature phase) PNRs to the diffuse displacive-type ferroelectric transition, which is related to the growth of PNR dimensions.

Heavy-tail properties of relaxation time distributions underlying the Havriliak–Negami and the Kohlrausch–Williams–Watts relaxation patterns

A detailed discussion of asymptotic properties of the Havriliak–Negami and the Kohlrausch–Williams–Watts relaxation time distributions is presented. The heavy-tail property of the Havriliak–Negami relaxation time distribution, leading to the infinite mean relaxation time, is discussed. In contrast, the existence of the finite mean relaxation time for the Kohlrausch–Williams–Watts response is shown. The discussion of the Cole–Davidson and the Cole–Cole cases is also included. Using the Tauberian theorems we show that these properties are determined directly by the asymptotic behavior of the considered empirical functions.

Dispersion and Absorption in Media with Stretch Exponential and Fractional Power Relaxation

Dispersion and absorption in a media with stretch exponential or Kohlrausch–Williams– Watts relaxation are considered. The frequency-dependent dispersion and absorption coefficients, phase and group velocities are obtained and compared to Debye–Mandelshtam– Leontovich expressions.

Dielectric relaxation behavior of poly(acrylonitrile- co-methacrylonitrile) microcapsules dispersed in a silicone matrix

The dielectric relaxation behavior of poly(acrylonitrile- co-methacrylonitrile) dispersed in a cured polydimethyl siloxane (PDMS) matrix as microcapsules was investigated over multiple thermal cycles and at varying concentrations. The copolymer microcapsules contained an isopentane core. In the PDMS matrix this copolymer displayed a pronounced relaxation signal at temperatures above the glass transition of the copolymers due to Maxwell–Wagner–Sillars (MWS) relaxation. The mechanism of MWS relaxation interpreted by the Havriliak–Negami and Kohlrausch–Williams–Watts relaxation functions was found to be very similar to previous studies of neat polyacrylonitrile and its copolymer. The activation energy of the relaxation decreased over successive thermal cycling coincident with a decreasing strength of the relaxation. These observations were attributed to the decreasing concentration of nitrile groups due to intramolecular cyclizations.

On the Rajagopal relaxation-time distribution and its relationship to the Kohlrausch–Williams–Watts relaxation function

A detailed analysis of the relationship of the Kohlrausch–Williams–Watts relaxation function to its recently proposed approximation resulting from the Rajagopal log-relaxation-time distribution is presented. Our considerations are based on the interpretation of the relaxation function as a survival probability of the initial state of a relaxing system expressed by means of the weighted average of an exponential decay with respect to the distribution of the effective relaxation time. Such an interpretation of the relaxation function enforces certain framework of analysis in which the basic probabilistic rules have to be fulfilled. In our approach, to compare the properties of the relaxation function resulting from the Rajagopal relaxation-time distribution and the Kohlrausch–Williams–Watts relaxation function, we use the strict relationship of the latter to the one-sided stable relaxation-rate distribution.

Ionic conduction and dielectric relaxation in Ag +/Na + ion-exchanged aluminosilicate glasses: mixed mobile ion effect and KWW relaxation

Ag +/Na + ion-exchanged aluminosilicate glasses with uniform concentration profiles were prepared, and their electrical conductivities were investigated as functions of the ion-exchange ratio and the initial glass compositions. In the case of the ion-exchanged glasses of x20Ag 2O–(1− x)20Na 2O–10Al 2O 3–70SiO 2 in mol%, the conductivity, σ, and its activation energy, E σ , showed a minimum and a maximum at the same ion-exchange ratio x=0.3, respectively, and the mixed mobile ion effect (MMIE) was observed. The fully ion-exchanged sample attained σ=3.5×10 −5 S/cm at 200 °C, which was 1.5 orders of magnitude larger than that of initial glass. In the case of x25Ag 2O–(1− x)25Na 2O–25Al 2O 3–50SiO 2, the mixed mobile ion effect was also observed at x=0.5. The maximum conductivity of 2×10 −4 S/cm at 200 °C was obtained in the fully ion-exchanged glass sample. The electric relaxation analysis was also conducted on both systems, and Kohlrausch–Williams–Watts (KWW) fractional exponent β was obtained as a function of x. The decrease of β was observed near x≈0.3 in the former system, while that of the later system was independent of the ion-exchange ratio. Based on the structural analysis results, the observed behaviors were investigated from the point of view of the occupation of Ag + ions on the non-bridging oxygen-site (NBO-site) and the charge compensation-site (CC-site) of AlO 4 tetrahedral unit.

Mechanical Relaxation Time Scales in a Zr–Ti–Ni–Cu–Be Bulk Metallic Glass

The relaxation time scales in a commercial-grade Zr41.25Ti13.75Ni10Cu12.5Be22.5 (at.%) bulk metallic glass were examined using transient and dynamic mechanical experiments. The viscoelastic and sub-Tg relaxations were well described by the Kohlrausch–Williams–Watts relaxation function. A large activation energy (4.0 eV) and small nonexponentiality parameter (approximately 0.5) were observed for viscoelastic relaxation above Tg consistent with the cooperative nature of atomic movements leading ultimately to viscous flow. Conversely, a small activation energy (0.1 eV) and large nonexponentiality parameter (approximately 0.9) were observed for the sub-Tg relaxation suggesting localized atomic adjustments which may involve different structural units or mechanisms. The glass transition was manifested as a decoupling of the sub-Tg and viscoelastic relaxation. The resulting transition temperature determined at a selected time scale was in agreement with the value obtained from calorimetric studies.

ISOTHERMAL RELAXATION OF RUBBED POLYSTYRENE THIN FILMS PROBED WITH OPTICAL BIREFRINGENCE MEASUREMENTS

Optical birefringence measurements were used to observe the isothermal relaxation of rubbed polystyrene films on silica substrates below the glass transition temperature. The relaxation dynamics were described by the Kohlrausch–Williams–Watts relaxation function with Arrhenius temperature dependence on relaxation time. The isothermal results were in good agreement with results obtained using a 1 K min -1 heating rate.

Self-motion and the alpha relaxation in a simulated glass-forming polymer: crossover from Gaussian to non-Gaussian dynamic behavior.

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)(r,t), the mean-squared displacement <r(2)(t)>, the second-order non-Gaussian parameter alpha(2)(t), and the incoherent intermediate scattering function F(s)(Q,t) for the main chain protons. In addition, we also calculate the density-density correlation function F(Q,t)/F(Q,0) and the second-order autocorrelation function M2(t) for different C-H bonds of the main chain. alpha(2)(t) shows a broad maximum centered at a time t(*) approximately 4 ps, which corresponds to the intermediate region of <r(2)(t)> between microscopic dynamics and sublinear diffusion. The analysis of F(s)(Q,t), F(Q,t)/F(Q,0), and M2(t) focuses on the second slow step which is associated to the alpha relaxation. Following the usual experimental procedure this decay is described in terms of a Kohlrausch-Williams-Watts (KWW) function: A exp[-(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)(Q,t) follows a law tau approximately Q(-2/beta). This kind of Q dependence corresponds to a Gaussian behavior of G(s)(r,t) and F(s)(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 alpha relaxation.

Glassy dynamics of simulated polymer melts: Coherent scattering and van Hove correlation functions

Whereas the first part of this paper dealt with the relaxation in the β-regime, this part investigates the final relaxation (α-relaxation) of a simulated polymer melt consisting of short non-entangled chains in the supercooled state above the critical temperature of ideal mode-coupling theory (MCT). The temperature range covers the onset of a two-step relaxation behaviour down to a temperature merely 2% above . We monitor the incoherent intermediate scattering function as well as the coherent intermediate scattering function of both a single chain and the melt over a wide range of wave numbers q. Upon approaching the coherent α-relaxation time of the melt increases strongly close to the maximum q max of the collective static structure factor Sq and roughly follows the shape of Sq for q q max. For smaller q-values corresponding to the radius of gyration the relaxation time exhibits another maximum. The temperature dependence of the relaxation times is well described by a power law with a q-dependent exponent in an intermediate temperature range. Deviations are found very close to and far above , the onset of which depends on q. The time-temperature superposition principle of MCT is clearly borne out in the whole range of reciprocal vectors. An analysis of the α-decay by the Kohlrausch-Williams-Watts (KWW) function reveals that the collective KWW stretching exponent and KWW relaxation time show a modulation with Sq. Furthermore, both incoherent and coherent KWW times approach the large-q prediction of MCT already for q > q max. At small q, a q-3 power law is found for the coherent chain KWW times similar to that of recent experiments.

Secondary dielectric β-relaxation in amorphous poly(ethylene terephthalate): combined thermally stimulated and isothermal depolarization current investigations

The complex β dielectric relaxation process in amorphous poly(ethylene terephthalate) (PET) is investigated by thermally stimulated depolarization currents (TSDC) and isothermal depolarization currents (IDC) techniques. TSDC and IDC data are analyzed in the framework of the Ngai coupling model in the [−150; −40°C] temperature range using the Marchal method and involving a Kohlrausch–Williams–Watts (KWW) functional form of the polarization time decay. The temperature dependence of the characteristic τ KWW relaxation time and of the β KWW stretching parameter of the KWW relaxation function derived from both depolarization current methods for the β relaxation of PET are determined and then compared. The very good agreement observed between the two sets of experimental stretching parameters determined by TSDC and IDC confirms the high potentiality of the TSDC technique to study complex relaxation processes as such as the β relaxation of PET. The two different overlapped relaxation processes involved in the β relaxation have been confirmed throughout the determination of two different temperature behaviors of the β KWW parameter. Each process of complex β relaxation is characterized by a low and nearly temperature-independent KWW stretching parameter. We show that the β KWW value of the high-temperature component, involving phenyl group motions, is slightly lower than the one corresponding to the low-temperature process assigned to the motions of carbonyl groups.

Time and temperature invariances in the evolution of properties through the glass transition

In this paper we analyze relaxation phenomena of amorphous materials near or below the glass transition. A phenomenological theory is suggested that maintains the main ingredients of the widely accepted models, i.e., those quasi-universal properties of structural relaxation which are well established, while a new approach is adopted for constructing the overall relaxation under a given temperature history. The evolution of the relaxational part of a property p under time–temperature changes is described by a first order relaxational equation that states that the instantaneous advance of the relaxation is proportional to the amount of deviation from equilibrium. The model consistently combines three different principles: (a) Linearity of response, (b) time–temperature re-scaling, and (c) power law relaxation at short times. This is achieved by imposing the following requirements on the relaxational equation: that the equation be expressed as a unique function of the reduced time; and that it provides the Kohlrausch–Williams–Watts relaxation law in the particular case of a temperature jump experiment. In addition, the relaxation time is not a function of fictive temperature. This approach provides as an outcome a new type of superposition over past perturbations. The analysis of rate heating/cooling experiments shows that the model reproduces the hysteresis of the fictive temperature and the peaks in heat capacity curves frequently observed in experiment. The physical meaning of the shift relationship between cooling rate and fictive temperature is critically examined on the basis of scaling properties and relaxational properties and some limitations of the standard result are identified. A more general and physically reasonable relationship is obtained by rigorous derivation in the framework of the new model. It is therefore demonstrated that that relationship is not related to nonlinearity, contrary to what is widely believed. In addition, it is shown that the more general relationship involves the parameter β describing the slowing down of the relaxation. This provides the basis for new relations to be inferred between apparently different phenomenological properties. An explanation is advanced for the observed correlations between measured parameters in the Tool–Narayanaswamy–Moynihan phenomenology. © 2000 American Institute of Physics.

Interpretation of the TSDC fractional polarization experiments on the α‐relaxation of polymers

We report on the interpretation of the thermally stimulated depolarization current (TSDC) experiments, with partial polarization methods, on the dielectric α-relaxation. The results obtained on polyvinyl acetate are rationalized on the basis of the Boltzmann superposition principle in combination with a Kohlrausch–Williams–Watts (KWW) time decay of the polarization (with the β exponent essentially temperature independent and equal to the value determined by conventional dielectric methods at Tg). From this analysis of the global TSDC spectrum we found a complex temperature dependence of the KWW relaxation time, which is Arrhenius-like at the lowest temperatures but crosses over to the Vogel–Fulcher behavior observed above Tg in the temperature range of the TSDC peak. On the basis of these results, we found the way of predicting the TSDC spectra measured after partial polarization procedures. We found that, the distribution of activation energies and compensation behavior deduced by following the standard way of analysis are associated to the assumption of an Arrhenius-like temperature dependence of the α-relaxation time in the temperature range explored by TSDC. Therefore we conclude that both the distribution of activation energies and compensation behavior obtained by following the standard way of analysis do not give a proper physical picture of the α-relaxation of glassy polymers around the glass-transition temperature. Our results also show that the partial polarization TSDC methods are not able to give insight about the actual existence or not of a distribution of relaxation times at the origin of the nonexponentiality of the α-relaxation of polymers. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 2105–2113, 2000