Continuous Spectrum Of Relaxation Times Research Articles

Material Grain Size Determines Relaxation-Time Distributions in Slow-Dynamics Experiments

By measuring the elastic slow dynamics behavior of various consolidated granular materials and metallic alloys, we show that the evolution of the wave velocity as a function of time can be reproduced by using a continuous relaxation-time spectrum, whose main features can be extracted from the experiments. Here we propose to adopt a Weibull distribution of the relaxation times, that we find to display a maximum, which is related to the material grain size and independent of the conditioning amplitude. These results show that the role of the grain-size distribution is fundamental to quantify slow dynamics effects, as well as to clarify its mesoscopic physical origin. Moreover, they can pave the way for the implementation of advanced characterization methods or nondestructive testing techniques capable of retrieving information on the different spatial scales or defects in the materials under investigation.

A comparison of crack propagation theories in viscoelastic materials

Crack propagation in viscoelastic materials cannot be understood with the use of classical fracture mechanics, which predicts no dependence on the speed of propagation, unless cohesive models like Barenblatt or Dugdale are introduced, as done by Knauss & Schapery first in the 1970s. However, there is another approach, suggested qualitatively by deGennes in 1996, and quantitatively by Persson and Brener in 2005, which attempts an energy (power) balance by considering viscoelastic dissipation in the bulk of the material. Here, we revisit the main results of the two theories and show that they lead to approximately the same scaling laws not just for the standard material, but also for power law materials (which have a continuous spectrum of relaxation times). Recent findings by Schapery have concluded that the shape of the cohesive law results essentially in a shift in velocity which depends both on cohesive law shape and viscoelastic properties. Therefore, the Persson-Brener cutoff radius in the integral of dissipation can be chosen to fit approximately the cohesive model results to match the shift of the reference velocity.

Phenomenological model of viscoelasticity for systems undergoing sol–gel transition

A material undergoing sol–gel transition evolves from the pre-gel (sol) state to the post-gel state through the critical gel state. It is well-known that critical gels exhibit power-law rheology. The faster decay of the relaxation modulus in the pre-gel state can be empirically described by modifying this power-law decay with a stretched exponential factor. A phenomenological analytical expression for the relaxation modulus in the post-gel state is proposed by invoking the symmetry associated with the evolution of the relaxation time on either side of the critical gel state and by accounting for natural constraints. This expression, which depends on the extent of cross-linking, can be suitably transformed to obtain analytical expressions for the dynamic moduli and the continuous relaxation time spectrum. Thus, the proposed model facilitates a comprehensive description of viscoelastic evolution from the pre-gel to the post-gel states. It is validated by carrying out experiments on a model colloidal gel-forming system and by considering other diverse gel-forming systems studied in the literature. After calibrating the parameters of the phenomenological model, it is found to be in excellent agreement with experimental data. Such a well-calibrated phenomenological model can be used to determine any linear viscoelastic response over a wide range of frequencies and extents of cross-linking encompassing the entire sol–gel transition.

Effect of Microwave Irradiation on the Microinhomogeneity of α-Relaxation Processes in Polyvinyl Alcohol

The internal friction spectra λ = f(T) and the temperature dependence ν = f(T) of the frequency of free damping in the temperature range from –150°C to 120°C of microwave-irradiated and nonirradiated samples of polyvinyl alcohol were obtained. A theoretical analysis of nature of the peak located on the ascending branch of the α-peak of relaxation losses on the internal friction spectrum is carried out, and its disappearance after the microwave irradiation is explained. A comparison of changes in the continuous spectrum of relaxation times in the region of glass-transition temperature is performed. It is established that microwave irradiation expands the spectrum of relaxation time, and, consequently, increases the relaxation microinhomogeneity of the α-relaxation process.

Relaxation time spectra of basaltic lavas between 500–1150 °C reveal patterns of Kramers–Kronig inconsistency of the complex viscoelastic shear modulus

Abstract An important, yet until now neglected, aspect of the viscoelastic behaviour of lavas is the Kramers–Kronig consistency of their complex viscoelastic shear modulus. The most general linear viscoelastic model – the generalized Maxwell body with continuous relaxation time spectrum – produces a consistent storage and loss modulus, as can be verified by Kramers–Kronig formulae. We reprocessed the original datasets of the high-precision laboratory data by James et al. (2004) supplied as pairs of magnitude of the complex viscoelastic shear modulus and the loss angle. We introduce the magnitude-borne and loss-angle-borne logarithmic relaxation time spectra and their ratio as a suitable indicator of the linear viscoelastic inconsistency. The basaltic lavas from Etna, Hawai’i and Vesuvius have shown a general convergence to the ideal consistency with increasing temperature, although each sample with an individual inconsistency pattern. The biggest surprise is the inconsistency ratio rising to ~20 in Etna 1992 top sample at 786 °C. Such a high inconsistency level still waits for an explanation and for the discoveries of its class-mates either in laboratory or field experiments.

Selection of a Relaxation Function to Describe the Dissipative α-Relaxation Process in Latex Polymers

Discrete relaxation times D(τ) in latex polymers are calculated using the phenomenological model of a standard linear body and continuous relaxation time spectra H(τ) involving the Kohlrausch function in the temperature range from –150 to +250°C. The higher relaxation inhomogeneity of the modified polymers confirms the previously obtained experimental data on its estimates across the width of the continuous spectrum of relaxation times with the involvement of dependence λ/λmax = f {(Т – Тmax)/Тmax}.

Cooperativity in Molecular Dynamics Structural Models and the Dielectric Spectra of 1,2-Ethanediol

Linear relationships are established between the experimental equilibrium correlation factor and the molecular dynamics (MD) mean 〈cos ϕ〉 value of the О–Н···О bond angle and the longitudinal component of the unit vector of the mean statistical dipole moment of the cluster in liquid 1,2-ethanediol (12ED). The achievements of modern MD models in describing the experimental dispersion of the permittivity of 12ED by both continuous and discrete relaxation time spectra are analyzed. The advantage computer MD experiments have over dielectric spectroscopy for calculating relaxation time and determining the molecular diffusion mechanisms of the rearrangement of the network 12ED structure, which is more complex than water, is demonstrated.

Development of Prony series models based on continuous relaxation spectrums for relaxation moduli determined using creep tests

Prony series models have been widely utilized to represent relaxation moduli. However, negative model parameters and wavy fitting curves are common observations of Prony series models of relaxation moduli constructed based on creep tests without establishing a continuous relaxation spectrum. This study developed a method of constructing relaxation modulus models in the form of Prony series based on creep test data via establishing continuous spectrums of relaxation times. Uniaxial compressive creep tests were performed on two types of asphalt mixtures at five test temperatures. The measured creep compliances at every test temperature were modeled using the generalized Kelvin model with two Kelvin elements, which provided excellent model fitting with R2 values larger than 0.99. The modeled creep compliances at every temperature were converted into the relaxation moduli according to the constitutive relations of the generalized Kelvin model. Master curves of the relaxation moduli were constructed using the modified power law model. The continuous spectrum of relaxation times was therefore established for each mixture type. The relaxation modulus model in terms of the continuous relaxation spectrum was approximated using the form of Prony series. Prony series models with various terms were constructed for each mixture type; the parameters of these Prony series models were all positive. It was demonstrated that, when specifying one relaxation time per decade equidistantly on a base-10 logarithmic axis, the constructed Prony series model for each mixture type had a reasonable number of terms and generated a fitting curve of the relaxation modulus with negligible waviness and an R2 value larger than 0.99.The method developed in this study provided an effective solution to build a Prony series model of the relaxation modulus with a limited number of terms, guaranteed positive model parameters, an excellent goodness of fit, and a fitting curve with negligible waviness.

Regularization methods for finding the relaxation time spectra of linear polydisperse polymer melts

The calculation of discrete or continuous relaxation time spectra from rheometric measurables of polydisperse polymers is an ill-posed problem. In this paper, a curve fitting method for solving this problem is presented and compared to selected models from the literature. It is shown that the new method is capable of correctly predicting the molecular mass distributions of linear polydisperse polymer melts as well as their relaxation time spectra.

A new insight into the dependence of relaxation time on frequency in viscoelastic surfactant solutions: From experimental to modeling study

HypothesisThe relaxation time in viscoelastic surfactant solutions is a function of temperature, salt/surfactant concentrations, resting conditions, as well as shear frequency. The simplistic assumption of a single and constant relaxation time is not representative of all relaxation modes in these solutions especially at high frequencies. ExperimentsSteady-state and oscillatory measurements are carried out to study the effects of high temperature, concentration and resting condition on the rheology of surfactants/salt mixtures including a non-ionic and a zwitterionic/anionic surfactant system. Furthermore, a novel semi-empirical rheological model is deducted based on Cates theory.This model introduces, for the first time, a frequency-dependence for the continuous relaxation time spectrum. FindingsAt high temperatures, the non-ionic surfactant become more viscoelastic and the zwitterionic/anionic system loses its viscoelasticity. The addition of surfactant/salt improves the viscoelasticity of both systems, and, for the zwitterionic/anionic mixture, increasing the resting temperature improves its viscoelasticity. In addition, the proposed model significantly improves predictions of traditional Maxwell model for different viscoelastic surfactant solutions (using data from this study and the literature) for a considerable range of surfactant and salt combinations at a wide range of temperature.

Extraction of viscoelastic functions from creep data with ringing

Sometimes the creep ringing phenomena are observed as results of the coupling between the instrumental inertia and the viscoelasticity of the materials. Especially, biopolymer solutions are apt to be the victim of creep ringing because creep test is usually conducted to enlarge the frequency range of dynamic data. To overcome the creep ringing problems, we developed a general method which extracts pure material response functions from creep data with ringing by the use of Laplace transform and nonlinear regression method. The method is not based on spring-dashpot models and gives continuous relaxation time spectrum.

Rheological signatures of gelation and effect of shear melting on aging colloidal suspension

Colloidal suspensions that are out of thermodynamic equilibrium undergo physical aging wherein their structure evolves to lower the free energy. In aqueous suspension of Laponite, physical aging accompanies increases of elastic and viscous moduli as a function of time. In this work we study temporal evolution of elastic and viscous moduli at different frequencies and observe that freshly prepared aqueous suspension of Laponite demonstrates identical rheological behavior reported for the crosslinking polymeric materials undergoing chemical gelation. Consequently at a certain time tan{\delta} is observed to be independent of frequency. However, for samples preserved under rest condition for longer duration before applying the shear melting, the liquid to solid transition subsequent to shear melting shows greater deviation from classical gelation. We also obtain continuous relaxation time spectra from the frequency dependence of viscous modulus. We observe that, with increase in the rest time, continuous relaxation time spectrum shows gradual variation from negative slope, describing dominance of fast relaxation modes to positive slope representing dominance of slow relaxation modes. We propose that the deviation from gelation behavior for the shear melted suspensions originates from inability of shear melting to completely break the percolated structure thereby creating unbroken aggregates. The volume fraction of such unbroken aggregates increases with the rest time. For small rest times presence of fewer number of unbroken aggregates cause deviation from the classical gelation. On the other hand, at high rest times presence of greater fraction of unbroken aggregates subsequent to shear melting demonstrate dynamic arrest leading to inversion of relaxation time spectra.

Relaxation inhomogeneity of the structure of latex polymers modified with water-soluble phthalocyanine

The possibility to quantitatively analyze change in the relaxation microinhomogeneity of latex polymers during variation in the method of their preparation and modification with ionic phthalocyanine has been shown. The probability of estimating the width of the continuous relaxation-time spectra via comparison of the temperature dependences of the given internal-friction spectra and the temperature dependence of variation in the discrete relaxation time of the α process has been demonstrated. The correlation between the temperature-frequency variations of the polymeric relaxation modulus of the systems under the free-oscillation regime and the degree of their relaxation microinhomogeneity has been considered.

Free energy and states of fractional-order hereditariness

Complex materials, often encountered in recent engineering and material sciences applications, show no complete separations between solid and fluid phases. This aspect is reflected in the continuous relaxation time spectra recorded in cyclic load tests. As a consequence the material free energy cannot be defined in a unique manner yielding a significative lack of knowledge of the maximum recoverable work that can extracted from the material. The non-uniqueness of the free energy function is removed in the paper for power-laws relaxation/creep function by using a recently proposed mechanical analogue to fractional-order hereditariness.

Using Low-Field NMR To Infer the Physical Properties of Glassy Oligosaccharide/Water Mixtures

Low-field NMR (LF-NMR) is usually used as an analytical technique, for instance, to determine water and oil contents. For this application, no attempt is made to understand the physical origin of the data. Here we build a physical model to explain the five fit parameters of the conventional free induction decay (FID) for glassy oligosaccharide/water mixtures. The amplitudes of the signals from low-mobility and high-mobility protons correspond to the density of oligosaccharide protons and water protons, respectively. The relaxation time of the high-mobility protons is described using a statistical model for the probability that oligosaccharide hydroxyl groups form multiple hydrogen bonds. The variation of energy of the hydrogen bond is calculated from the average bond distance and the average angle contribution. Applying the model to experimental data shows that hydrogen atoms screen the water oxygen atoms when two water molecules solvate a single hydroxyl group. Furthermore, the relaxation time of the oligosaccharide protons is independent of its molecular weight and the water content. Finally, inversion of the FID using the inverse Laplace transform gives the continuous spectrum of relaxation times, which is a fingerprint of the oligosaccharide.

Resonance sensors for measuring dielectric spectra of liquid crystals in a wide frequency range

Designs of resonance sensors covering a wide frequency range of 1–104 MHz and intended for measuring the dispersion of the permittivity of liquid crystals are proposed. Using an example of a 4-methoxybenzyliden-4′-buthylanilin (MMBA) liquid crystal, it is shown that these sensors are characterized by a high sensitivity and allow measurements of temperature and field dependences of dielectric spectra of samples with volumes of ∼1 mm3. The behavior of dispersion of the MMBA permittivity observed experimentally is described by the Debye equation with a continuous spectrum of relaxation times within a certain range. A numerical method is proposed that allows reconstruction of the relaxation-time distribution function from the measured spectra of the real component of the permittivity. This distribution function represents the features of both the motion of molecules and the processes of intramolecular vibrations of movable fragments.

Indirect detection of structural changes in micellar solutions by rheological measurements

At higher concentration levels the inner structure of micellar solutions cannot be detected directly by optical means. Nevertheless the flow behavior of the micellar solutions reflect their micellar structures. Hence, in this study the material behavior of micellar surfactant solutions was investigated by rheometrical means in steady and oscillatory shear flows with and without thermal preload. The rheological behavior of the solutions was found to be strongly dependent on the concentration. The complex material structure is modeled according to the flow behavior by discrete and continuous relaxation time spectra depending on the concentration. It is found that the material possesses besides the memory for mechanical preload also a strong memory for thermal preload which depends on the strength of preload.

An experimental study on the flow behavior of micellar solutions

At higher concentration levels, the inner structure of micellar solutions cannot be detected directly by optical means. Nevertheless, the flow behavior of the micellar solutions reflects their micellar structures. Hence, in this study the material behavior of micellar surfactant solutions was investigated by rheometrical means in steady and oscillatory shear flows. The flow behavior of the solutions was found to be strongly dependent on the concentration of the surfactants. At very low concentrations, the surfactant solution shows Newtonian behavior. With increasing concentration, a transition to shear thinning behavior and increasing viscoelasticity was found. The complex material structure is modeled according to the flow behavior by discrete and continuous relaxation time spectra, depending on the concentration.

Relaxation Time Spectrum of Hydrogels by CONTIN Analysis

ABSTRACT:CONTIN is a general‐purpose program for inverting noisy linear algebraic and integral equations by means of inverse Laplace transform. This study explored the application of CONTIN analysis to determine the relaxation time distribution spectra for food gels, including gellan, carrageenan, whey protein, and gelatin gels, based on stress‐relaxation data. CONTIN results represent the continuous relaxation time spectra when the number of the terms in the discrete Maxwell stress‐relaxation model approached infinity. The CONTIN results for gellan gels were correlated to the texture properties of gels from compression tests with respect to the effects of calcium concentrations. CONTIN analysis may be a very effective tool in elucidating the microstructural properties of a hydrogel from mechanical testing.

Determination of the relaxation time spectrum from dynamic moduli using an edge preserving regularization method

For the numerical determination of a continuous relaxation time spectrum from rheological data so-called regularization methods are necessary. The regularization method based on Tikhonov regularization and commonly used in rheology assumes the relaxation time spectrum to be smooth. Accordingly, this method is not able to properly resolve spectra containing edges or at least large curvatures as proposed in the literature. In order to reconstruct such spectra a novel edge preserving regularization (EPR) method was developed which extends the properties of the commonly used method. In order to demonstrate the efficiency of EPR it is compared with the commonly used regularization method by means of simulated and experimental data. This reveals that EPR is able to prove whether or not the theoretically predicted edge containing spectra for high-molecular-weight, monodisperse polymers are compatible with experimental data.