Relaxation Spectra Research Articles

Model for the Shape of the Relaxation Spectrum in Glass Formers.

Traditionally the broadness of the spectrum of the relaxation times observed in glass-forming materials has been rationalized by local heterogeneity, where a variety of atomistic environments leads to spectrum of single-exponential relaxation responses. However, the assumption of heterogeneity can break down when tested against the shape of the relaxation spectrum. An alternative homogeneous scenario assumes that the relaxation is inherently multiexponential. A recently developed switchback model [Medvedev, G. A. Phys. Rev. E 2023, 107 (3), 034122] naturally results in a multiexponential wedge-like spectrum that is consistent with the dielectric relaxation, light scattering, and the NMR data. As a particular case the switchback model allows for the spectrum to become single-exponential; under the heterogeneous scenario this would require the heterogeneities to completely vanish, which is hard to justify. Using data from photobleaching experiments and molecular dynamic simulations, it is shown that the relaxation spectrum may become single-exponential under large anisotropic deformation. This is interpreted as an argument in favor of the homogeneous scenario and specifically the switchback model for the relaxation of the glass formers.

A Comparative View of Becker, Lomnitz, and Lambert Linear Viscoelastic Models

We compare the classical viscoelastic models due to Becker and Lomnitz with respect to a recent viscoelastic model based on the Lambert W function. We take advantage of this comparison to derive new analytical expressions for the relaxation spectrum in the Becker and Lomnitz models, as well as novel integral representations for the retardation and relaxation spectra in the Lambert model. In order to derive these analytical expressions, we have used the analytical properties of the exponential integral and the Lambert W function, as well as the Titchmarsh’s inversion formula of the Stieltjes transform. In addition, we prove some interesting inequalities by comparing the different models considered, as well as the non-negativity of the retardation and relaxation spectral functions. This means that the complete monotonicity of the rate of creep and the relaxation functions is satisfied, as required by the classical theory of linear viscoelasticity.

Correlation of microstructure and magnetic softness of Si-microalloying FeNiBCuSi nanocrystalline alloy revealed by nanoindentation

Abstract Compared to the commercial soft-magnetic alloys, the high saturation magnetic flux density (B s) and low coercivity (H c) of post-developed novel nanocrystalline alloys tend to realize the miniaturization and lightweight of electronic products, thus attracting great attention. In this work, we designed a new FeNiBCuSi formulation with a novel atomic ratio, and the microstructure evolution and magnetic softness were investigated. Microstructure analysis revealed that the amount of Si prompted the differential chemical fluctuations of Cu element, favoring the different nucleation and growth processes of α-Fe nanocrystals. Furthermore, microstructural defects associated with chemical heterogeneities were unveiled using the Maxwell-Voigt model with two Kelvin units and one Maxwell unit based on creeping analysis by nanoindentation. The defect, with respect to a long relaxation time in relaxation spectra, was more likely to induce the formation of crystal nucleus that ultimately evolved into the α-Fe nanocrystals. As a result, Fe84Ni2B12.5Cu1Si0.5 alloy with refined uniform nanocrystalline microstructure exhibited excellent magnetic softness, including a high B s of 1.79 T and very low H c of 2.8 A/m. Our finding offers new insight into the influence of activated defects associated with chemical heterogeneities on microstructures of nanocrystalline alloy with excellent magnetic softness.

Direct Identification of the Continuous Relaxation Time and Frequency Spectra of Viscoelastic Materials

Relaxation time and frequency spectra are not directly available by measurement. To determine them, an ill-posed inverse problem must be solved based on relaxation stress or oscillatory shear relaxation data. Therefore, the quality of spectra models has only been assessed indirectly by examining the fit of the experiment data to the relaxation modulus or dynamic moduli models. As the measures of data fitting, the mean sum of the moduli square errors were usually used, the minimization of which was an essential step of the identification algorithms. The aim of this paper was to determine a relaxation spectrum model that best approximates the real unknown spectrum in a direct manner. It was assumed that discrete-time noise-corrupted measurements of a relaxation modulus obtained in the stress relaxation experiment are available for identification. A modified relaxation frequency spectrum was defined as a quotient of the real relaxation spectrum and relaxation frequency and expanded into a series of linearly independent exponential functions that are known to constitute a basis of the space of square-integrable functions. The spectrum model, given by a finite series of these basis functions, was assumed. An integral-square error between the real unknown modified spectrum and the spectrum model was taken as a measure of the model quality. This index was proved to be expressed in terms of the measurable relaxation modulus at uniquely defined sampling instants. Next, an empirical identification index was introduced in which the values of the real relaxation modulus are replaced by their noisy measurements. The identification consists of determining the spectrum model that minimizes this empirical index. Tikhonov regularization was applied to guarantee model smoothness and noise robustness. A simple analytical formula was derived to calculate the optimal model parameters and expressed in terms of the singular value decomposition. A complete identification algorithm was developed. The analysis of the model smoothness and model accuracy for noisy measurements was carried out. The equivalence of the direct identification of the relaxation frequency and time spectra has been demonstrated when the time spectrum is modeled by a series of functions given by the product of the relaxation frequency and its exponential function. The direct identification concept can be applied to both viscoelastic fluids and solids; however, some limitations to its applicability have been pointed out. Numerical studies have shown that the proposed identification algorithm can be successfully used to identify Gaussian-like and Kohlrausch–Williams–Watt relaxation spectra. The applicability of this approach to determining other commonly used classes of relaxation spectra was also examined.

Evaluation of the water state and protein characteristics of Tibetan pork under the storage conditions of modified atmosphere packaging: Effect of oxygen concentration

To explore the changes in water status and protein characteristics of Tibetan pork (TP) under modified atmosphere packaging (MAP) with different oxygen concentrations compared to Duroc×Landrace×Yorkshire pork (DLY), the water holding capacity (WHC), water distribution, protein oxidation, and conformation of both types were determined. Results indicate that under MAP, TP pork and DLY pork exhibited higher water retention and lower protein oxidation compared to air packaging. However, with increased oxygen concentration in the MAP, protein oxidation intensified, leading to reduced WHC in the pork. Compared to DLY pork, TP pork in different packaging conditions maintained the integrity of protein secondary and tertiary structures, reducing protein cross-linking aggregation. The lower content of P3 in the two-dimensional relaxation spectra, shorter T1 and T2 relaxation times, and higher proton density suggest better water retention properties in Tibetan pork. These findings support the development of long-distance preservation and transportation technologies for TP pork.

Multi-parameter viscoelastic material model for denture adhesives based on time-temperature superposition and multiple linear regression analysis

BackgroundRestorative solutions designed for edentulous patients such as dentures and their accompanying denture adhesives operate in the complex and dynamic environment represented by human oral physiology. Developing material models accounting for the viscoelastic behavior of denture adhesives can facilitate their further optimization within that unique physiological environment. This study aims to statistically quantify the degree of significance of three physiological variables - namely: temperature, adhesive swelling, and pH - on denture adhesive mechanical behavior. Further, based on these statistical significance estimations, a previously-developed viscoelastic material modelling approach for such denture adhesives is further expanded and developed to capture these variables’ effects on mechanical behavior.MethodsIn this study a comparable version of Denture adhesive Corega Comfort was analysed rheologically using the steady state frequency sweep tests. The experimentally derived rheological storage and loss modulus values for the selected physiological variables were statistically analyzed using multi parameter linear regression analysis and the Pearson’s coefficient technique to understand the significance of each individual parameter on the relaxation spectrum of the denture adhesive. Subsequently, the parameters are incorporated into a viscoelastic material model based on Prony series discretization and time-temperature superposition, and the mathematical relationship for the loss modulus is deduced.ResultsThe results of this study clearly indicated that the variation in both the storage and loss modulus values can be accurately predicted using the oral cavity physiological parameters of temperature, swelling ratio, and pH with an adjusted R2 value of 0.85. The R2 value from the multi-parameter regression analysis indicated that the predictor variables can estimate the loss and storage modulus with a reasonable accuracy for at least 85% of the rheologically determined continuous relaxation spectrum with a confidence level of 98%. The Pearson’s coefficient for the independent variables indicated that temperature and swelling have a strong influence on the loss modulus, whereas pH had a weak influence. Based on statistical analysis, these mathematical relationships were further developed in this study.ConclusionsThis multi-parameter viscoelastic material model is intended to facilitate future detailed numerical investigations performed with implementation of denture adhesives using the finite element method.

Integrating dynamic relaxation with inelastic deformation in metallic glasses: Theoretical insights and experimental validation

Metallic glasses (MGs), a metastable material far from equilibrium, exhibit intricate dynamic relaxation behaviors. The challenge lies in developing a model that accurately describes the dynamics and deformation mechanisms of MG. This paper introduces a model integrating dynamic relaxation with deformation behavior. Validation through dynamic mechanical analysis, stress relaxation, creep, and strain recovery tests confirm the existence of four deformation modes: elasticity, anelasticity from β relaxation, anelasticity from α relaxation at low temperatures, and viscoplasticity from α relaxation at high temperatures. The model captures all of these deformation modes. The dynamical mechanical spectrum and stress relaxation spectrum unveil dynamic features during glass to liquid transition, and a simple and effective experimental method was developed for identifying ultra-low-frequency dynamic relaxation. This work provides new perspectives on the study of relaxation dynamics in glassy states and establishes important connections between dynamic relaxation behavior and deformation mechanisms. These findings lay a theoretical and experimental foundation for the broad application of MGs.

Investigation of micromechanics and relaxation spectrum evolution in multiple recycled asphalt binders

Investigation of micromechanics and relaxation spectrum evolution in multiple recycled asphalt binders

Robust Recovery of Optimally Smoothed Polymer Relaxation Spectrum from Stress Relaxation Test Measurements.

The relaxation spectrum is a fundamental viscoelastic characteristic from which other material functions used to describe the rheological properties of polymers can be determined. The spectrum is recovered from relaxation stress or oscillatory shear data. Since the problem of the relaxation spectrum identification is ill-posed, in the known methods, different mechanisms are built-in to obtain a smooth enough and noise-robust relaxation spectrum model. The regularization of the original problem and/or limit of the set of admissible solutions are the most commonly used remedies. Here, the problem of determining an optimally smoothed continuous relaxation time spectrum is directly stated and solved for the first time, assuming that discrete-time noise-corrupted measurements of a relaxation modulus obtained in the stress relaxation experiment are available for identification. The relaxation time spectrum model that reproduces the relaxation modulus measurements and is the best smoothed in the class of continuous square-integrable functions is sought. Based on the Hilbert projection theorem, the best-smoothed relaxation spectrum model is found to be described by a finite sum of specific exponential-hyperbolic basis functions. For noise-corrupted measurements, a quadratic with respect to the Lagrange multipliers term is introduced into the Lagrangian functional of the model's best smoothing problem. As a result, a small model error of the relaxation modulus model is obtained, which increases the model's robustness. The necessary and sufficient optimality conditions are derived whose unique solution yields a direct analytical formula of the best-smoothed relaxation spectrum model. The related model of the relaxation modulus is given. A computational identification algorithm using the singular value decomposition is presented, which can be easily implemented in any computing environment. The approximation error, model smoothness, noise robustness, and identifiability of the polymer real spectrum are studied analytically. It is demonstrated by numerical studies that the algorithm proposed can be successfully applied for the identification of one- and two-mode Gaussian-like relaxation spectra. The applicability of this approach to determining the Baumgaertel, Schausberger, and Winter spectrum is also examined, and it is shown that it is well approximated for higher frequencies and, in particular, in the neighborhood of the local maximum. However, the comparison of the asymptotic properties of the best-smoothed spectrum model and the BSW model a priori excludes a good approximation of the spectrum in the close neighborhood of zero-relaxation time.

A theoretical perspective on segmental relaxation dynamics of model dendrimer

The internal segmental diffusion and the relaxation modes are investigated, and the relationship between the topology of model dendrimers is established, all within the bounds of analytical and simulation approaches. The precise theoretical and simulation approaches are thoroughly described after sufficient data is provided for model theories and their predictions. The analytical framework of the Generalised Langevin Equation is exactly replicated in the results. The primary focus of this review is on the NMR technique for forecasting the occurrence of the prominent processes, such as bond pulsation, segment reorientation, and total dendrimer rotation, utilizing the relaxation spectrum at various time scales. The utilization of two autocorrelation functions, namely time and orientational, is given significant emphasis. These functions illustrate the mobility of branches and their dependence on the generation number. The text also includes discussions of the approximations, assumptions, and limitations of each author’s chosen technique, when relevant.

Glassy dynamics in a liquid of anisotropic molecules: Bifurcation of relaxation spectrum

In experimental and theoretical studies of glass transition phenomena, one often finds a sharp crossover in dynamical properties at a temperature Tcr. A bifurcation of a relaxation spectrum is also observed at a temperature TB≈Tcr; both lie significantly above the glass transition temperature. In order to better understand these phenomena, we introduce a new model of glass-forming liquids, a binary mixture of prolate and oblate ellipsoids. This model system exhibits sharp thermodynamic and dynamic anomalies, such as the specific heat jump during heating and a sharp variation in the thermal expansion coefficient around a temperature identified as the glass transition temperature, Tg. The same temperature is obtained from the fit of the calculated relaxation times to the Vogel–Fulcher–Tammann (VFT) form. As the temperature is lowered, the calculated single peak rotational relaxation spectrum splits into two peaks at TB above the estimated Tg. Similar bifurcation is also observed in the distribution of short-to-intermediate time translational diffusion. Interrogation of the two peaks reveals a lower extent of dynamic heterogeneity in the population of the faster mode. We observe an unexpected appearance of a sharp peak in the product of rotational relaxation time τ2 and diffusion constant D at a temperature Tcr, close to TB, but above the glass transition temperature. Additionally, we coarse-grain the system into cubic boxes, each containing, on average, ∼62 particles, to study the average dynamical properties. Clear evidence of large-scale sudden changes in the diffusion coefficient and rotational correlation time signals first-order transitions between low and high-mobility domains.

Combined Relaxation Spectra for the Prediction of Meat Quality: A Case Study on Broiler Breast Fillets with the Wooden Breast Condition.

This study evaluated the potential of using combined relaxation (CRelax) spectra within time-domain nuclear magnetic resonance (TD-NMR) measurements to predict meat quality. Broiler fillets affected by different severities of the wooden breast (WB) conditions were used as case-study samples because of the broader ranges of meat-quality variations. Partial least squares regression (PLSR) models were established to predict water-holding capacity (WHC) and meat texture, demonstrating superior CRelax capabilities for predicting meat quality. Additionally, a partial least squares discriminant analysis (PLS-DA) model was developed to predict WB severity based on CRelax spectra. The models exhibited high accuracy in distinguishing normal fillets from those affected by the WB condition and demonstrated competitive performance in classifying WB severity. This research contributes innovative insights into advanced spectroscopic techniques for comprehensive meat-quality evaluation, with implications for enhancing precision in meat applications.

Rheological Behavior Features of Feedstocks with a Two-Component Wax–Polyolefin Binder Compared to Analogs Based on Polyoxymethylene

Despite the large number of studies devoted to different compositions of polymer binders for PIM technology, the actual task is still a comparative analysis of the properties of different types of binders to determine their advantages and disadvantages and optimize the compositions used. In this regard, this study aims at the identification and comparative analysis of the rheological properties of the most demanded feedstocks with binders based on polyoxymethylene and a wax–polyolefin mixture under the condition of using identical steel powder filler. The rate of change in the volume fraction of the liquid phase of the binder in the compared feedstocks with temperature change was determined by the calculation–experimental method. As shown, the temperature dependence of the viscosity of feedstocks with a binder based on a polymer blend depends on factors with variable power, i.e., the viscosity change with temperature occurs by different mechanisms with their relaxation spectra. Thus, the principle of temperature–time superposition for feedstocks with multicomponent binders is not applicable, and the study of the viscosity of such materials should involve a wide range of shear rates and temperatures using experimental methods. Capillary rheometry was used to measure the flow curves of feedstocks based on polyoxymethylene and wax–polyolefin binders. The analysis of flow curves of feedstocks showed that feedstocks with a binder of solution–thermal type of debinding have significantly lower viscosity, which is an advantage for molding thin-walled products. However, their difference of 1.5 times sensitivity to the shear rate gradient leads to their lower resistance to “jets” and liquation of components because of shear rate gradients when molding products with elements of different cross-sectional areas.

Chain dynamics of alternating polymers P(C n EG4)

Alternating or sequence defined polymers attract the attention of an increasing number of researchers recently. Due to their different blocks, they are very customizable and material properties can be tuned. In this publication, we present dynamical studies with focus on polymer dynamics, investigated by rheology and fast field cycling (FFC) NMR. The molecular weight dependencies of the relaxation time and zero-shear viscosity could hint to entanglement effects; however, the spectral shape of the viscoelastic data resembles the polymer dynamics of unentangled melts. Taking both techniques into account, i.e. rheology and FFC NMR leads to the conclusion that the relaxation spectra of the alternating polymer are similar to those of linear polymers.

Research on Material Viscoelasticity and Its Influence on Indentation Rolling Resistance

Viscoelastic materials are applied in several fields, and their relaxation characteristics are intricately related to the failure mechanism of sealing components and the generation of indentation rolling resistance in belt conveyors. Therefore, it is imperative to explore the relaxation characteristics of viscoelastic materials using characterization models. This article focuses on exploring these characterization models and the indentation rolling resistance of viscoelastic materials. The research comprises the following aspects: (1) A 2N + 2 element generalized Maxwell constitutive model is proposed for the relaxation behavior of viscoelastic materials to address the limitations of conventional relaxation models. (2) We conducted numerical calculations based on the relaxation modulus to solve the relaxation spectrum using several relaxation spectrum models. The findings showed that the model parameters were dependent on the testing time range. (3) The relationship between the indentation rolling resistance and relaxation model parameters was evaluated based on the theoretical foundation of the indentation rolling resistance calculation.

Change in Superparamagnetic State Induced by Swift Heavy Ion Irradiation in Nano-Maghemite

The effect of swift heavy ion irradiation on sol–gel-prepared maghemite nanoparticles was studied by 57Fe transmission Mössbauer spectroscopy and X-ray diffractometry (XRD). The room temperature Mössbauer spectra of the non-irradiated nano-maghemite showed poorly resolved magnetically split, typical relaxation spectra due to the superparamagnetic state of the nanoparticles. Significant changes in the line shape, indicating changes in the superparamagnetic state, were found in the Mössbauer spectra upon irradiation by 160 MeV and 155 MeV 132Xe26+ ions with fluences of 5 × 1013 ion cm−2 and 1 × 1014 ion cm−2. XRD of the irradiated maghemite nanoparticles showed a significant broadening of the corresponding lines, indicating a decrease in the crystallite size, compared to those of the non-irradiated ones. The results are discussed in terms of the defects induced by irradiation and the corresponding changes related to the change in particle size and consequently in the superparamagnetic state caused by irradiation.

Viscoelastic characterization of the mucus from the skin of loach (II) using linear viscoelastic property

ABSTRACT A structuralized Rivlin-Sawyers model with inhomogeneous shear-rate effect on the component of relaxation spectrum was proposed here to express the viscoelastic property of loach mucus. The linear viscoelastic property of the mucus was obtained using the experimental frequency sweep data at small strain, and the viscosity in the shear-rate decreasing region with 0.0631 s−1 as the maximum shear rate was fitted by the model with inhomogeneous shear-rate effect. Both the predictions of the steady viscosity curves with four maximum shear rates, i.e., 0.004 s−1, 0.01 s−1, 0.1 s−1, and 1 s−1, and that of the transient shear viscosity in the step rate experiment at 0.1 s−1 show reasonable characterization of the model on the viscoelasticity of the loach skin mucus.

Sampling Points-Independent Identification of the Fractional Maxwell Model of Viscoelastic Materials Based on Stress Relaxation Experiment Data.

Considerable development has been observed in the area of applying fractional-order rheological models to describe the viscoelastic properties of miscellaneous materials in the last few decades together with the increasingly stronger adoption of fractional calculus. The fractional Maxwell model is the best-known non-integer-order rheological model. A weighted least-square approximation problem of the relaxation modulus by the fractional Maxwell model is considered when only the time measurements of the relaxation modulus corrupted by additive noises are accessible for identification. This study was dedicated to the determination of the model, optimal in the sense of the integral square weighted model quality index, which does not depend on the particular sampling points applied in the stress relaxation experiment. It is proved that even when the real description of the material relaxation modulus is entirely unknown, the optimal fractional Maxwell model parameters can be recovered from the relaxation modulus measurements recorded for sampling time points selected randomly according to respective randomization. The identified model is a strongly consistent estimate of the desired optimal model. The exponential convergence rate is demonstrated both by the stochastic convergence analysis and by the numerical studies. A simple scheme for the optimal model identification is given. Numerical studies are presented for the materials described by the short relaxation times of the unimodal Gauss-like relaxation spectrum and the long relaxation times of the Baumgaertel, Schausberger and Winter spectrum. These studies have shown that the appropriate randomization introduced in the selection of sampling points guarantees that the sequence of the optimal fractional Maxwell model parameters asymptotically converge to parameters independent of these sampling points. The robustness of the identified model to the measurement disturbances was demonstrated by analytical analysis and numerical studies.

Novel hyper-viscoelastic approach to modelling elastomer mechanic behaviour with relaxation spectrum

Elastomers, such as polyurethane, exhibit hyperelastic behaviour and distinct rate dependence under a wide range of strain rates. Herein, a methodology for developing a hyper-viscoelastic constitutive model for succinctly characterising the mechanical behaviour of polyurethane under large deformations and a wide range of strain rates is proposed. The model is constructed using a long-term hyperelastic formulation combined with a rate-dependent viscoelastic model. This model is based on internal state variables and a finite strain formulation. A polyurethane elastomer was used as a sample to develop this model, by way of testing wide strain-rate mechanical behaviour (0.001–5000s−1) and viscoelastic behaviour. The viscoelastic behaviour was obtained using the equivalent time–temperature superposition relation. A parameter-fitting algorithm was adopted to separate the long-term behaviour from the viscoelastic behaviour. The long-term behaviour was modelled using a Mooney–Rivlin hyperelastic model, and its parameters were identified using the least-squares method. The time-dependent viscous effect calculated using the Prony series-based model was approximately determined based on the continuous relaxation spectrum. The characterisation capacity was confirmed through a thorough comparison with experimental curves. This model can be extended to other elastomers, even under tension, without any specific limitations of the internal components.

PolyWeight: A free and open-source program for determination of molecular weight distribution of linear polymers

This paper introduces PolyWeight, a Python software featuring a user-friendly graphical user interface (GUI), which offers two distinct approaches for MWD determination: an analytical relation-based method and a parametric model-based method. By utilizing dynamic moduli, users can calculate MWD as well as molecular weight averages such as Mn, Mw, and Mz. The functionality of PolyWeight is validated using synthetic data and real data obtained from the literature, exhibiting good agreement. Log files and datasets used in this work are available on GitHub. Program summaryProgram Title: PolyWeightCPC Library link to program files:https://doi.org/10.17632/rn8t4mmvyd.1Developer's repository link:https://github.com/a-minotto/PolyWeightLicensing provisions: GPLv3Programming language: Python 3.xNature of problem: Analytical rheology enables the determination of molecular weight distribution (MWD) by solving inverse problems. The reptation theory establishes a liaison between the viscoelastic behavior and MWD, permitting the application of mathematical resolution methods like regularization and parameterization. PolyWeight offers two solutions: a parameterization-based method that assumes a generalized exponential distribution (GEX) for the MWD, and an analytical relationship-based method that utilizes the relaxation spectrum.Solution method: The analytical relationship-based solution incorporates a mathematical formulation that considers the relaxation spectrum obtained from the dynamic moduli using the NLREG software. The numerical integration of the relaxation spectrum is performed utilizing the integrate.quad function from the SciPy library. An optimization process is employed in the parametric solution to perform a multiobjective fit of the viscoelastic models to the dynamic moduli. The integrals in this case are also calculated using the integrate.quad package from the SciPy library. The optimization procedure utilizes the lmfit library, where a Minimizer object is created based on the optimization function and subsequently utilized with the minimize function, employing the default Levenberg-Marquardt algorithm for the fit.