Relaxation Exponent Research Articles

Dielectric spectroscopy of polymer-metal composites across the percolation threshold

Polymer (polar/nonpolar)/metal composites (PMC) were prepared under different process conditions. In polar PMC, dipolar relaxation plays a predominant role below percolation threshold (fc) and anomalous low frequency dispersion (ALFD) becomes dominant above fc while ALFD is the only likely possibility for nonpolar PMC above fc. The magnitude of relaxation exponents "m", "p" and "n", evaluated from the experimental results using Jonscher's universal dielectric response (JUDR) laws, falls within the universal limit ~ [0, 1] with additional feature of strong dependence on volume fraction of conductor (f con ). The decrease in the relaxation exponent "m" with an increase of f con is directly linked with decrease in the number of dipoles of the polymer in the composite and is accompanied by a distribution of relaxation time due to increased heterogeneity of the system. The magnitude of the relaxation exponent "n" decreases at fc, due to the prevalence of Maxwell–Wagner–Sillar polarization contributed by uncorrelated electrons.

Percolation approach to glassy dynamics with continuously broken ergodicity.

We show that the relaxation dynamics near a glass transition with continuous ergodicity breaking can be endowed with a geometric interpretation based on percolation theory. At the mean-field level this approach is consistent with the mode-coupling theory (MCT) of type-A liquid-glass transitions and allows one to disentangle the universal and nonuniversal contributions to MCT relaxation exponents. Scaling predictions for the time correlation function are successfully tested in the F(12) schematic model and facilitated spin systems on a Bethe lattice. Our approach immediately suggests the extension of MCT scaling laws to finite spatial dimensions and yields predictions for dynamic relaxation exponents below an upper critical dimension of 6.

A generalized domain approach for describing the stress relaxation in a phase separated borosilicate glass

In this work we propose a generalized domain approach for modeling the stress relaxation behavior of phase õseparated borosilicate glasses. The generalization of the domain approach is done by taking two fundamental steps. First, we assume the boron groups to be the soft boundaries in the borosilicate glasses. Second, we introduce the time-dependent power law into the Kohlrausch function. Using the generalized domain approach, we explained why the value of the stretched exponent of the stress relaxation in borosilicate glasses anomalously deviates from the converged exponent value of 3/5. In addition, we introduce a pre-time, i.e., the time for glass to reach the structural equilibrium, into the generalized domain approach in order to compensate the variation of viscosity caused by phase separation. It is found that the pre-time is closely correlated with the time for borosilicate glasses to reach structural equilibrium before stress relaxation experiment.

Electrolyte effect on gelation behavior of oppositely charged nanocrystalline cellulose and polyelectrolyte

The electrolyte (NaCl) influences on the sol–gel transition of the complex solution composed of oppositely charged nanocrystalline cellulose (NCC) and polyelectrolyte (quaternized hydroxyethylcellulose ethoxylate, QHEC) were investigated by the rheological means in the present paper. Winter and Chambon theory was applicable to describe the sol–gel transition, and the critical gel points have been successfully determined. When increasing the NaCl concentration, more NCC were needed to form a critical gel due to the screening of the electrostatic interaction, and the larger loss tangent and relaxation exponent (n) values at the gel point demonstrated a less elastic nature of the complex solution with more NaCl. The results indicated the gel network was composed of entanglements and association of QHEC (as polymer network), as well as the electrostatic adsorption interaction between QHEC chains and NCC rods (as cross-linking). With the addition of NaCl, the screening effect led to the enhancement of the entanglements and weakening of the electrostatic adsorption, however, the gel strength decreased with increasing the NaCl amount, suggesting the electrostatic adsorption interaction played a more dominant role than the entanglements when the gel was formed. Moreover, the exponents of the scaling law η0∝ɛ−γ and Ge∝ɛz of the QHEC/NCC/NaCl solution revealed that the scaling law n=z/(z+γ) between n, γ, and z was only feasible at the highest NaCl concentration, as a result of that the intermolecular electrostatic interaction was completely screened, indicating the scaling law was only feasible when intermolecular interaction was small enough to be neglected.

Microrheology and Microstructure of Fmoc-Derivative Hydrogels

The viscoelasticity of hydrogel networks formed from the low-molecular-weight hydrogelator Fmoc-tyrosine (Fmoc-Y) is probed using particle-tracking microrheology. Gelation is initiated by adding glucono-δ-lactone (GdL), which gradually lowers the pH with time, allowing the dynamic properties of gelation to be examined. Consecutive plots of probe particle mean square displacement (MSD) versus lag time τ are shown to be superimposable, demonstrating the formation of a self-similar hydrogel network through a percolation transition. The analysis of this superposition yields a gel time t(gel) = 43.4 ± 0.05 min and a critical relaxation exponent n(c) = 0.782 ± 0.007, which is close to the predicted value of 3/4 for semiflexible polymer networks. The generalized Stokes-Einstein relation is applied to the master curves to find the viscoelastic moduli of the critical gel over a wide frequency range, showing that the critical gel is structurally and rheologically fragile. The scaling of G'/G″ as ω(0.795±0.099) ≈ ω(3/4) at high frequencies provides further evidence for semiflexible behavior. Cryogenic scanning electron micrographs depict a loosely connected network close to the gel point with a fibrillar persistence length that is longer than the network mesh size, further indications of semiflexible behavior. The system reported here is one of a number of synthetic systems shown to exhibit semiflexible behavior and indicates the opportunity for further rheological study of other Fmoc derivatives.

Investigation of the scaling law on gelation of oppositely charged nanocrystalline cellulose and polyelectrolyte

The sol–gel transition in the mixture system of oppositely charged polyelectrolyte (quaternized hydroxyethylcellulose ethoxylate, QHEC) and nanocrystalline cellulose (NCC) induced by electrostatic adsorption interaction was investigated by rheological means. Winter and Chambon theory was validated to be applicable for the critical gel point determination, and critical gel point have been successfully determined. With QHEC concentration increasing, more NCC were needed to form a critical gel, and smaller loss tangent and relaxation exponent (n) values at the gel point were observed, indicating the elastic nature of mixture was enhanced with QHEC increase. Gel strength behaved as a function of both QHEC and NCC concentrations, suggesting the gel network at the critical point was composed of entanglements and association of QHEC macromolecular chains, as well as the electrostatic adsorption interaction between QHEC chains and NCC rods. The calculated number of NCC rods per junction decreased from 0.30 to 0.01 when the QHEC concentration increased from 1.0wt% to 3.0wt%, indicating the electrostatic adsorption interaction between the NCC rods and QHEC chains was less significant to gel formation at higher QHEC concentrations. Therefore, the exponents of scaling law η0∝ϵ–γ and Ge∝ϵ z for the QHEC/NCC mixtures revealed that the scaling law n=z/(z+γ) between n, γ, and z was only feasible at highest QHEC concentration, since the intermolecular interaction (electrostatic adsorption interaction in this article) was so weak that can be neglected and the critical gel network was dominated by QHEC chain entanglements and association.

Vortex creep down to 0.3 K in superconducting Fe(Te,Se) single crystals

We report on a study of the vortex creep in Fe${}_{1+\ensuremath{\delta}}($Te${}_{x}$Se${}_{1\ensuremath{-}x}$) single crystals ($x=0.5$ and $0.4$) down to 0.28 K ($\ensuremath{\sim}{T}_{c}/50$) and up to ${\ensuremath{\mu}}_{0}{H}_{a}=2$T. The relaxation of the current density [$J(t)$] has been measured during 20 hours and the decay of $J(t)$ can be well described by a $J(t)\ensuremath{\propto}{[ln(t/{t}_{0})]}^{\ensuremath{-}1/\ensuremath{\mu}}$ law. We show that the relaxation exponent $\ensuremath{\mu}$ tends towards 0 for $T<2$K and ${\ensuremath{\mu}}_{0}{H}_{a}<0.1$T [i.e. $J(t)\ensuremath{\rightarrow}{({t}_{0}/t)}^{\ensuremath{\alpha}}$] and increases for increasing $T$ and/or ${H}_{a}$. Our measurements strongly suggest that the logarithmic creep rate $R=\ensuremath{-}dln(J)/dln(t)$ remains finite at zero temperature (${R}_{|T\ensuremath{\rightarrow}0}\ensuremath{\rightarrow}2$%) and hence that quantum creep plays a dominant role in the relaxation process at low temperature. A maximum is observed in both the temperature and field dependence of $R(t=100\text{s},T,{H}_{a}),$ which can be associated to a crossover from a single vortex (one-dimensional) to a bundle (three-dimensional) creep regime.

Short-term market reaction after trading halts in Chinese stock market

In this paper, we study the dynamics of absolute return, trading volume and bid–ask spread after the trading halts using high-frequency data from the Shanghai Stock Exchange. We deal with all three types of trading halts, namely intraday halts, one-day halts and inter-day halts, of 203 stocks in Shanghai Stock Exchange from August 2009 to 2011. We find that absolute return, trading volume, and in case of bid–ask spread around intraday halts share the same pattern with a sharp peak and a power law relaxation after that. While for different types of trading halts, the peaks’ height and the relaxation exponents are different. From the perspective of halt reasons or halt durations, the relaxation exponents of absolute return after inter-day halts are larger than those after intraday halts and one-day halts, which implies that inter-day halts are most effective. From the perspective of price trends, the relaxation exponents of excess absolute return and excess volume for positive events are larger than those for negative events in case of intraday halts and one-day halts, implying that positive events are more effective than negative events for intraday halts and one-day halts. In contrast, negative events are more effective than positive events for inter-day halts.

Gelation and Cross-Linking in Multifunctional Thiol and Multifunctional Acrylate Systems Involving an in Situ Comonomer Catalyst

Dynamic rheology in combination with Fourier transform infrared spectroscopy (FTIR) is used to examine the gelation kinetics, mechanism, and gel point of novel thiol–acrylate systems containing varying concentrations of an in situ catalyst. Gelation, as evidenced from the gel time determined using the Winter–Chambon criterion, is found to occur more quickly with increasing catalyst concentration up until a critical catalyst concentration of 22 mol %, whereupon the gel time lengthens. Such a minimum in gel time may be attributed to changes in the number of available reaction sites and percentage conversion required for gelation. Chemical conversions at the gel point measured for representative samples are consistent with theoretical values calculated using Flory–Stockmayer’s statistical approach, confirming our hypothesis. Relaxation exponents of 0.97 and fractal dimensions of 1.3 are calculated for all samples, consistent with coarse-grained discontinuous molecular dynamics (DMD) simulations. The elevated...

Measurement of the true transverse nuclear magnetic resonance relaxation in the presence of field gradients

A measure of the nuclear spin transverse relaxation time T2, as determined using the nuclear magnetic resonance Carr-Purcell Meiboom-Gill (CPMG) experiment, provides unique information characterizing the microstructure of porous media which are themselves ubiquitous across fields of petrophysics, biophysics, and chemical engineering. However, the CPMG measurement is sensitive to diffusion in large magnetic field gradients. Under such conditions an effective relaxation time T2,eff is observed instead, described by a combination of relaxation and diffusion exponents. The relaxation exponent always varies as nte (where n is the number, and te is the temporal separation, of spin echoes). The diffusion exponent varies as nte (k), where 1 < k ≤ 3, although the exact analytic form is often unknown. Here we present a general approach to separating the influence of relaxation and diffusion by utilizing a composite diffusion exponent. Any T2,eff component with a power of k > 1 is removed to provide a measure of the true T2 relaxation time distribution from CPMG data acquired in the presence of a strong background gradient. We apply the technique to discriminate between the effects of relaxation and diffusion in porous media using catalysts and rocks as examples. The method is generally applicable to any CPMG measurements conducted in the presence of a static magnetic field gradient.

Erratum: Nonsteady relaxation and critical exponents at the depinning transition [Phys. Rev. E87, 032122 (2013)

Fil: Ferrero, Ezequiel E.. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Patagonia Norte; Argentina. Comision Nacional de Energia Atomica. Centro Atomico Bariloche; Argentina

Nonsteady relaxation and critical exponents at the depinning transition

We study the non-steady relaxation of a driven one-dimensional elastic interface at the depinning transition by extensive numerical simulations concurrently implemented on graphics processing units (GPUs). We compute the time-dependent velocity and roughness as the interface relaxes from a flat initial configuration at the thermodynamic random-manifold critical force. Above a first, non-universal microscopic time-regime, we find a non-trivial long crossover towards the non-steady macroscopic critical regime. This "mesoscopic" time-regime is robust under changes of the microscopic disorder including its random-bond or random-field character, and can be fairly described as power-law corrections to the asymptotic scaling forms yielding the true critical exponents. In order to avoid fitting effective exponents with a systematic bias we implement a practical criterion of consistency and perform large-scale (L~2^{25}) simulations for the non-steady dynamics of the continuum displacement quenched Edwards Wilkinson equation, getting accurate and consistent depinning exponents for this class: \beta = 0.245 \pm 0.006, z = 1.433 \pm 0.007, \zeta=1.250 \pm 0.005 and \nu=1.333 \pm 0.007. Our study may explain numerical discrepancies (as large as 30% for the velocity exponent \beta) found in the literature. It might also be relevant for the analysis of experimental protocols with driven interfaces keeping a long-term memory of the initial condition.

Solvents effects in the formation and viscoelasticity of DBS organogels

The gel behavior of dibenzylidene sorbitol (DBS) in different solvents was studied by rheology, optical microscopy and transmission electron microscopy. It was found that DBS molecules would self-assemble into fibrils with a helical structure, and the fibrillar diameter decreased with increasing solvent polarity. The gel dissolution temperature was extremely sensitive to the DBS concentration and solvent polarity, which could be explained by the Gibbs–Thomson equation through interfacial stress. The phase diagram, which classified the sol, the gel and the cluster regime in different solvents was determined by rheology and optical microscopy. The critical gel concentration was found to increase as the difference in the polar and hydrogen-bonding components of the solubility parameter (Δδph) between the gelator and solvents decreased. Meanwhile, the difference in the critical gel concentration in different solvents could explain the difference in the relaxation exponent and the gel strength at the gel point well. In the stable gel state, the plateau modulus depended on the gelator concentration according to a power-law scaling, GoN ∝ c2, which was consistent with entanglement theory and independent of the type of solvent. However, the gelator–solvent interaction and its temperature dependence were found to affect the stability of the gel substantially under large amplitude oscillatory shear. The critical strain would increase as Δδph decreased, and showed more evident temperature dependence on solvents with a smaller Δδph.

Monitoring degradation of matrix metalloproteinases-cleavable PEG hydrogels via multiple particle tracking microrheology

The design of hydrogel matrices for cell encapsulation and tissue regeneration has become increasingly complex. Oftentimes, researchers seek to recapitulate specific biophysical and biochemical cues critical for the resident cell population and an in depth understanding of changes in the local microstructure and rheological properties of the synthetic matrix during enzymatic degradation would be extremely beneficial. Multiple particle tracking microrheology (MPT) enables simultaneous characterization of rheological properties and visualization of the microstructure in an evolving hydrogel scaffold. MPT measures the Brownian motion of fluorescently labeled probe particles embedded in the material, which is directly related to rheological properties using the Generalized Stokes–Einstein Relation (GSER). Here, we study a hydrogel scaffold consisting of a four-arm poly(ethylene glycol) (PEG) end functionalized with norbornene that is cross-linked with both a nondegradable PEG–dithiol and a matrix metalloproteinase (MMP) degradable peptide sequence (KCGPQG↓IWGQCK) using thiol–ene chemistry. The material degradation is measured as a function of time and extent of degradability, focusing on measuring the gel–sol transition. Using time–cure superposition, we determine the critical degradation time and critical extent of degradability for this specific gel formulation as tc = 1.85 h and pc = 0.589, respectively, and the critical relaxation exponent, n = 0.16. Finally, spatial information gained by MPT measurements quantifies the heterogeneity within the scaffold showing that these hydrogels degrade homogeneously when collagenase is introduced in solution at a concentration of 0.1–0.3 mg mL−1. Understanding the microstructural and rheological properties of a material near the gel–sol transition enables researchers to improve their insight as to how cells remodel their microenvironment when encapsulated in gels, and more precisely design and manipulate this parameter to improve three-dimensional culture systems.

Influence of alkali and temperature on glucomannan gels at high concentration

The effect of type and concentration of alkali on 5 g/100 mL of glucomannan gels was examined in depth by mechanical and thermo-rheological analysis. Four KGM gels were made from aqueous glucomannan solutions deacetylated with 0.6 mol/L and 1 mol/L of KOH (A0 and A1) and NaOH (B0 and B1), respectively. At 25 °C all physical networks were too tight and dense due to the large number of extended junction zones, as indicated by the high breaking force (BF), complex modulus (G∗) and low strain amplitude (γmax). The influence of alkali concentration was noticeable only in KOH gels, of which A0 was significantly more elastic and time stable than A1 over both short and longer-time scales. 0.6 mol/L KOH was the most suitable alkali, producing the most stable and homogeneous gel, with the highest level of connectivity, the lowest relaxation exponent (n) and the highest gel strength (S).Increasing temperature resulted in loss of elasticity in A0, where γmax, G∗ and S decreased, and the loss factor (tan δ), creep and recovery compliances (J) and n increased. Thus, the networks become less flexible and more unstable at high temperatures, especially between 25 and 50 °C.

Gelation of attractive particles in polymer melt

The physical gelation of the attractive carbon black particles in the isotactic polypropylene melt is studied. Following the method proposed by Winter and Chambon, oscillatory rheology is used to study the kinetic gelation process. The gel time, the relaxation exponent of the critical gel, the gel stiffness and the activation energy for the gelation are determined. The influences of the filler concentration and the experimental temperature on the gelation process are also investigated. The results suggest that the gelation of the filler particles in the polymer melt mainly follows second-order kinetics, and the critical gel should be formed by the rigid particles and the inter-particle polymer layers.

Rheological study of gel phenomena during epoxide network formation in the presence of sepiolite

AbstractThe dynamic behaviour during the crosslinking of an epoxy polymer near the gel point was monitored using rheological multiple frequency experiments. The influence of a needle‐shaped inorganic nanofiller, sepiolite, either non‐modified or organically surface modified during the cure process in the presence of an aliphatic and an aromatic hardener was investigated. The validity of various criteria for determining the gel point was examined for the crosslinking of these filled thermosets. The Winter–Chambon criterion at the gel point is obeyed by the unfilled and by the non‐modified sepiolite‐filled epoxy matrix with either of the two hardeners. However, physical gels are formed in the presence of the organically modified sepiolite and the Winter–Chambon criterion is not valid. For all the systems investigated, the nanofiller reduces the time to reach gelation. Critical relaxation exponents and gel strength have been determined indicating a more elastic and stronger gel in the presence of the aliphatic hardener. © 2012 Society of Chemical Industry

Unified physics of stretched exponential relaxation and Weibull fracture statistics

The complicated nature of materials often necessitates a statistical approach to understanding and predicting their underlying physics. One such example is the empirical Weibull distribution used to describe the fracture statistics of brittle materials such as glass and ceramics. The Weibull distribution adopts the same mathematical form as proposed by Kohlrausch for stretched exponential relaxation. Although it was also originally proposed as a strictly empirical expression, stretched exponential decay has more recently been derived from the Phillips diffusion-trap model, which links the dimensionless stretching exponent to the topology of excitations in a glassy network. In this paper we propose an analogous explanation as a physical basis for the Weibull distribution, with an ensemble of flaws in the brittle material serving as a substitute for the traps in the Phillips model. One key difference between stretched exponential relaxation and Weibull fracture statistics is the effective dimensionality of the system. We argue that the stochastic description of the flaw space in the Weibull distribution results in a negative dimensionality, which explains the difference in magnitude of the dimensionless Weibull modulus compared to the stretching relaxation exponent.

Omori law for eruption foreshocks and aftershocks

Using the 1973–2009 worldwide catalogs for M ≥ 4.8 seismicity and VEI ≥ 0 volcano eruptions, we compare the properties of seismic damage patterns contemporary with eruption with the properties of foreshocks and aftershocks of classic tectonic earthquakes. Using superposed epoch analysis, we demonstrated that the seismicity rate after eruption decreases as a power law similar to the Omori law of earthquake aftershocks. We further show that a complete mapping of Omori law of earthquake aftershocks onto eruption aftershocks does exist as Rerup(t) = (K0.10β(VEI))/[(t+c)p], volcanic explosivity index (VEI), being an empirical measure that exponentially scales with eruption size. β close to 0.4 is the value reported for M = 5–6.5 earthquakes from the same catalog. The p values are in the 0.7 range, i.e., robustly smaller than the 0.9–1.0 range for earthquake aftershocks we estimated in the volcanic area. K value for eruptions is 2–10 times smaller than for earthquakes, and it scales with VEI values. All those parameters characterize a slower damage relaxation after eruptions than after earthquakes. When earthquakes' foreshock rates are proposed to be independent of the main shock magnitude, we resolved a strong increase in foreshock rates including an increase of the p′ value of the inverse Omori law prior eruptions with eruption size. These patterns, all emerging from mean field analysis, are evidence of the volcanic eruptions being contemporary with a stochastic brittle damage in the Earth crust. These results suggest a generic damage relaxation within the Earth crust as power law distributed after or before events. The loading and relaxation exponents and the damage rate emerge as being controlled by the loading rate, as reported during lab‐scale experiments. The more impulsive the loading, i.e., km/s for the slip velocity during earthquakes against km/h for dyke propagation, the faster the relaxation (0.9–1.0 p values for earthquakes' aftershocks against 0.7 for eruptions' aftershocks). Before eruptions, the larger the impending events, the higher the p values. All the observations converge toward the amplitude and frequency of the stress step to drive the Omori law parameters as qualitatively reproduced by the rate and state friction law response of brittle crust faults to loading.

A utilization of internal/external quasi-static magnetic field gradients: transport phenomenon and magnetic resonance imaging of solid polymers

Two topics concerning the utilization of internal and external magnetic field gradients are discussed. The first topic concerns the transport of conformons in the quasi-ordered phase of a regioregular, π-conjugated polymer, poly(4-methylthiazole-2,5-diyl), using a combination of longitudinal and transverse relaxation dispersion measurements. The effects of conformon diffusion on the relaxation exponent were successfully estimated via a numerical approach that used the Fourier spectrum solution of the one-dimensional Bloch-Torrey equation. The second topic demonstrates a low-cost magnetic resonance imaging method using a magnetic field gradient from a neodymium ferromagnet. Spin echo experiments are shown for three-layered thin films of PDMS/PTFE/PDMS under a large magnetic field.