Williams-Landel-Ferry Equation Research Articles

Construction and examination of temperature master curve for asphalt with different aging extents

Master curve is widely applied to assess the rheological properties of asphalt. The traditional master curve describes the variation of asphalt rheological properties with frequency. However, compared to frequency, the influence of temperature on asphalt properties is of greater concern. This study attempted to establish a method to transform the traditional master curve into temperature master curve. To this end, asphalt binder with different aging extents were prepared using aging simulation tests, and traditional complex shear modulus master curves of asphalt binder were established based on frequency sweep test. These were then transformed into temperature master curves using the established temperature-frequency equation. Furthermore, the accuracy of the temperature master curve was assessed through temperature sweep tests. The results showed that the temperature-frequency equation obtained based on Williams Landel Ferry equation effectively described the relationship between temperature and reduced frequency. The temperature master curve demonstrated good applicability for asphalt at different aging levels, with the differences between complex shear modulus values calculated by temperature master curve and the experimental values generally within 20%. The temperature master curve accurately characterized the trend of complex shear modulus with temperature. However, this accuracy and applicability were primarily focused on the mid-temperature and high-temperature ranges. The temperature master curve had obvious deficiency in predicting the complex shear modulus at low temperature. The findings provide new methods and insights into the relationship between asphalt rheological properties and temperature.

Dynamic characteristics of bladder type attenuator for hydraulic systems

Flow ripples caused by the pump will lead to system failure or equipment damage. The bladder type attenuator is utilised to suppress flow ripples in piping systems, though its mathematic model has not been given out precisely. In this paper, the governing equations of the hollow cylindrical shell describing the bladder’s behaviour are studied. Considering the bladder’s viscoelastic materials, the constitutive function is obtained through DMA (Dynamic thermo-mechanical analysis) test and the WLF (Williams–Landel–Ferry) equation. Thus, the dynamic model of the attenuator can be written as an impedance expression, which consists of integral, fractional, constant, and differential components. Three experiments are carried out to validate the theoretical models. The static and dynamic models of the bladder are validated by experiments, indicating good agreement. The impedance model of the attenuator is validated by a hydraulic experiment assessed with the IL (insertion loss) index, showing high accuracy. All models proposed in this work are proven to be effective and accurate, and can be used to evaluate and optimise the bladder type attenuator theoretically.

Mapping positron annihilation lifetime spectroscopy data of a polymer to classical molecular dynamics simulations without shifting the glass transition temperature.

Positron annihilation lifetime spectroscopy (PALS) enables the nondestructive measurement of nanoscale cavities in materials. In this study, a strategy was proposed for mapping PALS measurement data of isotactic polypropylene to classical molecular dynamics (CMD) simulations. The discrepancy between simulated and experimental glass transition temperatures was resolved by shortening the polymer chains, rather than adjusting for the temperature, using the Williams-Landel-Ferry (WLF) equation. The effective probe radii of ortho-positronium (o-Ps), determined by comparing PALS data with CMD simulations, were ∼0.8nm, which was consistent with the o-Ps size given by the solution of the Schrödinger equation. The free-volume fraction corresponding to the effective probe radius was 12.3% at the glass transition temperature, close to the value estimated using Simha-Boyer theory. The cavity number density was proportional to the effective probe radius and decreased with temperature. The o-Ps effective probe radius was proportional to both the critical probe radius and the -1/3 power of the monomer number density, and increased with increasing temperature. These findings suggest that combining PALS measurements with CMD simulations may provide insight into cavities in polymeric materials without relying on the WLF equation.

Temperature and Frequency Dependence of the Dynamic Viscoelastic Properties of Silicone Rubber.

Temperature-frequency sweep tests were performed on silicone rubber to investigate the dynamic viscoelastic properties. The test results show that the viscoelasticity of silicone rubber presents significant temperature dependence and frequency dependence. The dynamic viscoelastic test curves at different temperatures can be shifted along the logarithmic frequency coordinate axis to construct smooth master curves at the reference temperature of 20 °C, covering a frequency range of 10 decades, which indicates thermorheological simplicity on a macro level and frequency temperature equivalence of the silicone rubber material in the experimental temperature range. The van Gurp-Palmen plot and Cole-Cole plot for the test data at various temperatures merge into a common curve, which further validates thermorheological simplicity. The temperature dependent shift factors of silicone rubber material were well characterized by the Williams-Landel-Ferry equation. Moreover, the fractional-order differential Kelvin (FDK) model, the fractional-order differential Zener (FDZ) model, and the improved fractional-order differential Zener (iFDZ) model were used to model the asymmetric loss factor master curve. The result shows that the iFDZ model is in good agreement with the test results, indicating that this model is suitable for describing the asymmetry of dynamic viscoelastic properties of silicone rubber.

Effect of combining different factors on the rheological property of crude oil

ABSTRACT In this paper, a high-pressure rheometer (MARS 60) was used to conduct rheological experiments on crude oil under varying parameters, including temperature, static pressure, and methane (CH4) dissolution. Combining the Williams–Landel–Ferry equation (WLF equation), Barus equation, and Kishimoto – Fujita model (F-K model), a new model of combined shift factor was used to quantify the effects of different oil parameters on crude-oil viscosity. The results show that higher temperature, lower static pressure, and lower CH4 dissolution reduce crude-oil viscosity. Static pressure and CH4 dissolution have a more significant effect on viscosity in lower-temperature conditions when compared to higher-temperature conditions. Due to the combined effect of the saturation pressure, there is an optimal condition of CH4 dissolution for reducing the viscosity. The shift factor of CH4 concentration is defined as the ratio of the viscosity of crude oil under saturation pressure of CH4 and static pressure, and the results agree with the F-K model. A new model of the combined shift factor was developed with average deviations of the predicted shift factors for Shengli crude oil and Nanyang crude oil being 3.35% and 2.16%, respectively. Finally, the predicted and tested values are basically in agreement.

The roles of polymer relaxation phenomena, aqueous dye solubility and the physical properties of water in the mechanism of adsorption of a disperse dye on poly(ethylene terephthalate) fibres: part 5 analysis of polymer relaxation phenomena for different polymers

AbstractThe temperature dependent uptake of a commercial disperse dye on cotton and polyamide 66 fabrics at dyeing temperatures between 30°C and 130°C adhered to the Williams–Landel–Ferry equation, insofar as, very good correspondence was observed between plots of experimentally determined colour strength data points (log1/fk) and the respective structural relaxation times of the cellulose and nylon 66 polymers (logaT data points), as a function of the parameter (T − Tg). Adsorption of the dye on both types of fibre therefore concurs with the fundamental precept of the free volume model of dye diffusion. Comparison of the adherence of the uptake of the commercial dye on cotton and polyamide 66 fabrics with that secured on polyester fabric revealed that despite the major chemical and physical differences between the three types of fibre, the same dyeing mechanism likely applies to each fibre type. The marked temperature dependent uptake of the commercial grade disperse dye each of the three types of substrate is the consequence of two, different, but inherently interconnected, thermally activated phenomena, namely the relaxation times of the molecular rearrangements occurring within the respective cellulose, nylon 66 or poly(ethylene terephthalate) macromolecule, in which polymer glass transition assumes the principal role, and the aqueous solubility of the commercial grade disperse dye.

Experimental study of phase separation in dynamically asymmetric unentangled polymer blend.

We examined a phase separation process of a dynamically asymmetric blend of unentangled polyisoprene (PI) and poly(4-ethylstyrene) (PC2St) exhibiting the upper critical solution temperature. PI having the type-A dipole was the dielectrically active fast component, whereas PC2St was the dielectrically inert slow component whose dynamics can be detected by rheological measurements. To precisely model the phase separation process, it is important to estimate the composition dependence of the mobility, which is needed to describe the phase separation dynamics. For that purpose, we conducted dielectric and rheological measurements to determine the friction coefficient of each component in a homogeneous state sufficiently above the phase separation temperature. The temperature dependence of the friction coefficient of each component was reasonably expressed by the Williams-Landel-Ferry equation. Extrapolating this dependence obtained for blends of various compositions to the test temperature T* below the phase separation temperature, we were able to estimate the friction coefficient of the chain at T* as a function of the composition. This friction coefficient was then used to determine the mobility Λ defined for the material fluxes at T*. The time-dependent Ginzburg-Landau (TDGL) equation incorporating this Λ well described the experimentally observed phase separation dynamics. In particular, the 2D TDGL simulation with this Λ qualitatively captured the phase-separated structure observed with the optical microscope as well as broad dielectric mode distribution of the blend at T*.

Linear viscoelastic shear and bulk relaxation moduli in poly(tetramethylene oxide) (PTMO) using united-atom molecular dynamics

A quantitative understanding and prediction of the viscoelastic relaxation behavior of polymer melts is required to build up predictive multiscale models, which can be utilized for the practical design of viscoelastic materials. We report the application of the Green–Kubo method to compute the common shear relaxation modulus and the rarely reported bulk relaxation modulus for a realistic united atom model of poly(tetramethylene oxide) (PTMO), which is one key components of the segmented polyurethane (PU) elastomers. The temperature and molecular weight dependent viscoelastic behavior is investigated in detail by computing the bulk relaxation modulus K(t) along with shear relaxation modulus G(t). Our results provide new rheological data for PTMO melt based on the united atom model, which incorporates the chemical details of the polymer backbone. The predicted stress relaxation are mapped onto master curves using the time–temperature superposition principle with horizontal and vertical shift factors. The computed shift factors agree with the Williams–Landel–Ferry equation and appear to be similar for the shear and bulk relaxation modulus. The underlying dynamics for the shear and bulk relaxation modulus appear to stem from the same mechanistic origin as both processes seem to sample similar kinetic signature of the reference time scale of the underlying thermally activated processes in the liquid. Our results demonstrate furthermore, that MD simulations with the chemical details beyond coarse-grained models can sample the Rouse dynamics and the transitions towards entanglement, which is evident by the emergence of a plateau in shear relaxation modulus and scaling relations in the diffusion behavior of the monomer at larger molecular weights.

Microscopic Dynamics and Viscoelasticity of Vitrimers

We implement a hybrid molecular dynamics/Monte Carlo simulation to study the microscopic dynamics and the macroscopic rheology of vitrimers with a fast bond exchange rate. We show that the linear viscoelastic properties and mean squared displacement of the vitrimers collapse onto master curves by applying the same shift factors that follow the Williams–Landel–Ferry equation at low temperatures and Arrhenius-like behavior at high temperatures. The linkage between the microscopic dynamics and the linear rheology of vitrimers is established using the generalized Stokes–Einstein relationship, which efficiently extends the timescale of simulations and predicts the viscoelasticity. The values of the shift factors are related to the characteristic decay time of the intermediate scattering function, which is accessible in scattering experiments. The same results hold in the case of an all-atom model of an ionic liquid. Our methodology provides a microscopic basis for the time-superposition principle and predicts the macroscopic rheology of thermo-rheologically simple vitrimers.

Reliability of Calculation of Dynamic Modulus for Asphalt Mixtures Using Different Master Curve Models and Shift Factor Equations.

To develop a mechanistic-empirical pavement design system for Norwegian conditions, this paper evaluates the influence of the adoption of different models and shifting techniques on the determination of dynamic modulus master curves of asphalt mixtures. Two asphalt mixture types commonly used in Norway, namely Asphalt Concrete (AC) and Stone Mastic Asphalt (SMA) containing neat bitumen and polymer-modified bitumen, were prepared by the roller compactor, and their dynamic moduli were determined by the cyclic indirect tensile test. The dynamic modulus master curves were constructed using the standard logistic sigmoidal model, a generalized logistic sigmoidal model and the Christensen–Anderson–Marasteanu model. The shifting techniques consisted of log-linear, quadratic polynomial function, Arrhenius, William–Landel–Ferry and Kaelble methods. The absolute error, normalised square error and goodness-of-fit statistics encompassing standard error ratio and coefficient of determination were used to appraise the models and shifting methods. The results showed that the standard logistic sigmoidal model and the Williams–Landel–Ferry equation had the most suitable fits for the specimens tested. The asphalt mixtures containing neat bitumen had a better fit than the ones containing polymer-modified bitumen. The Kaelble equation and log-linear equation led to similar results. These findings provide a relevant recommendation for the mechanistic-empirical pavement design system.

Effect of Cross-Linking Density on Horizontal and Vertical Shift Factors in Linear Viscoelastic Functions of Epoxy Resins

Epoxy resins are an important class of thermosetting resins, and their network structure, obtained by the curing reaction of epoxy and amine compounds, plays an important role in the material properties. We here revisited a time–temperature superposition (TTS) principle applied to the dynamic viscoelastic functions of epoxy resins, in which the network was well defined and systematically varied on the basis of the length of n-alkyl diamine. The superimposition of isothermal curves in the frequency domain required not only a horizontal shift but also a vertical shift, regardless of the length of n-alkyl diamine. The temperature dependence of the horizontal shift factor, aT, could be well expressed by the Williams–Landel–Ferry equation. The fitting parameter, called C2, increased with decreasing alkyl chain length of the diamine, meaning that the thermal expansion of the free volume was suppressed due to greater cross-linking density. This was qualitatively confirmed by a full-atomistic molecular dynamics (MD) simulation. Meanwhile, the vertical shift factor, bT, increased with increasing temperature, and the extent was smaller with increasing cross-linking density. This can be explained in terms of the entropic contribution to the modulus in the temperature region above the glass transition temperature. The entropy change estimated using the isobaric molar heat capacity from the MD simulation strongly supported this hypothesis. The knowledge here obtained should be useful for a better design of thermosetting polymers as well as for the prediction of long-term durability.

Numerical and Experimental Investigations of Polymer Viscoelastic Materials Obtained by 3D Printing.

The aim of this research is to determine the relaxation and creep modulus of 3D printed materials, and the numerical research is based on the finite volume method. The basic material for determining these characteristics is ABS (acrylonitrile butadiene styrene) plastic as one of the most widely used polymeric materials in 3D printing. The experimental method for determining the relaxation functions involved the use of a creep test, in which a constant increase of the stress of the material was performed over time to a certain predetermined value. In addition to this test, DMA (dynamic mechanical analysis) analysis was used. Determination of unknown parameters of relaxation functions in analytical form was performed on the basis of the expression for the storage modulus in the frequency domain. The influence of temperature on the values of the relaxation modulus is considered through the determination of the shift factor. Shift factor is determined on the basis of a series of tests of the relaxation function at different constant temperatures. The shift factor is presented in the form of the WLF (Williams-Landel-Ferry) equation. After obtaining such experimentally determined viscoelastic characteristics with analytical expressions for relaxation modulus and shift factors, numerical analysis can be performed. For this numerical analysis, a mathematical model with an incremental approach was used, as developed in earlier works although with a certain modification. In the experimental analysis, the analytical expression for relaxation modulus in the form of the Prony series is used, and since it is the sum of exponential functions, this enables the derivation of a recursive algorithm for stress calculation. Numerical analysis was performed on several test cases and the results were compared with the results of the experiment and available analytical solutions. A good agreement was obtained between the results of the numerical simulation and the results of the experiment and analytical solutions.

Evaluation for Low Temperature Performance of SBS Modified Asphalt by Dynamic Shear Rheometer Method

Finding an alternative or supplementary test method to evaluating the low temperature performance of asphalt is an area of considerable interest. This paper tries to explore the possibility of using the dynamic shear rheometer (DSR) method for assessing the low temperature properties of styrenebutadienestyrene (SBS) modified asphalt. In the study, 60/80 and 80/100 pen grade asphalt binders, named binder A-70, binder B-70 and binder C-90, are used to produce the SBS modified asphalt samples. After that, the low temperature performance of the asphalt binders is characterized by using bending beam rheometer (BBR) test. The results indicate that the low temperature performance of the different binders is related to the source of the binder. The low temperature performance of asphalt could be improved with the addition of the SBS. The DSR test is used to develop the complex modulus master curves for binders. Based on the principle of time–temperature conversion, the glass transition temperature of asphalt is calculated by the Williams–Landel–Ferry (WLF) equation. The glass transition temperatures (Tg) of base asphalt and the SBS modified asphalt are determined by the viscoelastic parameters of the master curve and the WLF equation coefficients based on the time–temperature superposition principle. By establishing the relationship between the critical temperature and the Tg of the asphalt binder, the effectiveness of the method established in this paper is verified. The advantage of this method is the ability to use the DSR test for the rapid evaluation of the low temperature performance of asphalt, which is able to reduce testing materials and save testing time as well. The glass transition temperature of the SBS modified asphalt is closely associated with aging degree, asphalt source and the SBS content.

Understanding complex dynamics of interfacial reconstruction in polyampholyte hydrogels undergoing mechano-chemo-electrotaxis coupling

Polyampholyte (PA) hydrogels have attracted significant attention for their superior mechanical strength and toughness compared with other conventional hydrogels. In this study, we present a novel thermodynamic approach to understanding the mechano-chemo-electrotaxis coupling and interfacial dynamics in PA hydrogels. Flory–Huggins theory, carried out through an interfacial free-energy model, is the foundation for the quantitative study of the mechanically constitutive relationship of the PA gels. The proposed free-energy model is further extended to describe the mechano-chemo-electrotaxis switching and interfacial dynamics by co-relating the Williams–Landel–Ferry equation and scaling laws. It was concluded that the interfacial bonding strength is the key factor influencing the mechanical strength and reconstruction reversibility of the PA macromolecular gel system. The resulting analytical outcomes showed good agreement with the reported experimental data. We opine that the proposed model will guide the future application of PA hydrogels.

Rheology of carbohydrate blends close to the glass transition: Temperature and water content dependence of the viscosity in relation to fragility and strength

Several modifications of the Williams-Landel-Ferry (WLF) equation that incorporate the water-content dependence of the viscosity are introduced and applied to the fitting the zero-shear viscosity of a systematic series of maltopolymer-maltose blends for water contents w between 4% and 70% (M. Dupas-Langlet et al., Carbohydrate Polymers 213 (2019) 147–158). These models include a previously published model that addresses the water-content dependence of the viscosity via a Gordon-Taylor-type modification of the C2 coefficient of the WLF equation. New models that are based on two simple assumptions are introduced: 1. The viscosity at the glass transition temperature Tg decreases exponentially with the water content and 2. The WLF coefficient C2 depends linearly on the water content. The modified WLF models allow to extract the so-called isoviscosity lines, that connect points of varying temperature and water content that are characterized by the same viscosity. Based on data obtained between T = −15 °C and 70 °C using shear rheology (w = 30–70% w/w) and dynamic mechanical thermal analysis (w = 4–9% w/w), we conclude that the models provide a good fit of the experimental data, and that additional data, specifically very close to the glass transition line, is needed, to assess the hypotheses underlying the various modified WLF models. It is established that the viscosity at Tg is dependent on the composition and decreases with the content of maltose and water. The modified WLF models are used to determine Angell’s fragility parameter m and Roos’ strength parameter S. m and S are observed to increase, respectively decrease with increasing water and maltose content, signifying an increasing temperature dependence of the viscosity close to Tg with decreasing diluent content. The application of the isoviscosity concept to unit operations in the food and pharmaceutical industry is discussed. Specifically, we show how to analyze atomization, agglomeration, sintering and compaction using the isoviscosity concept.

On the Temperature Dependence of the Glass Transition Activation Energy

The temperature dependence of the glass transition activation energy for amorphous organic polymers and inorganic glasses is calculated using the Williams–Landel–Ferry equation for the relaxation time (viscosity). The results obtained are in satisfactory agreement with the experiment. The nature of the dependence of the glass transition activation energy on the temperature is discussed.

Rheological characterization of vegetable oil blends: Effect of shear rate, temperature, and short‐term heating

AbstractRheological characterization of canola–olive and canola–sesame blends was carried out. The canola–olive and canola–sesame blends were prepared in the vol/vol ratios of 90:10, 70:30, and 50:50. The rheology was studied at shear rates 1–132 s−1 and temperatures 15–65°C (288.15–338.15 K) before and after heating for 5, 10, and 15 min. Viscosity varied linearly with percentage composition of blend. Power law, Sisko, Cross, and Carreau equations for viscosity‐shear relation, and Arrhenius and Williams–Landel–Ferry (WLF) equations for viscosity‐temperature relation were evaluated. The reduced χ2 values for the 70:30 canola–olive blend at 25°C were 7.84 × 10−5, 2.28 × 10−5, 1.26 × 10−6, and 1.23 × 10−6 for Power law, Sisko, Cross, and Carreau equations, respectively. Best fits were produced by Cross and Carreau for viscosity‐shear relation and by WLF equation for viscosity‐temperature relation. Short‐term heating was had no significant effect on the rheological profiles of the pure oils and their blends.Practical ApplicationsRheological characterization of vegetable oil blends could be helpful to engineers and scientists in formulation of vegetable oil blends for edible purposes and in designing the required process equipment. Curve fitting studies in rheology are especially helpful because the fitted equations can be used to solve complex flow problems encountered in engineering. The study could also be of interest to researchers and formulators in the field of biolubricants as well as medicinal oil formulations.

Linear viscoelastic behavior of asphalt mixes from dynamic frequency response functions

SummaryIn the small strain domain, asphalt mixes (AM) have a linear viscoelastic (LVE) behavior that is strongly dependent on frequency and temperature. The maximum ratio of modulus values can be up to 1000, and traditional elastic analyses are not pertinent. The possibility to characterize AM from frequency response functions (FRFs) was studied. A new optimization process using the finite element method (FEM) has been developed to back‐calculate the LVE properties of AM from FRFs. The numerical optimization process was applied to a reference material with averaged LVE properties determined from tension‐compression tests performed on a wide variety of AM types. The LVE properties were modeled considering the 3‐Dim version of the model 2S2P1D (2 Springs, 2 Parabolic elements, and 1 Dashpot). Reference FRFs for the considered reference material were obtained from FEM simulations. Three different configurations that may be of interest for practical tests were studied at five different temperatures. The proposed numerical optimization method consists in performing separate optimizations at each temperature to obtain the LVE properties for the considered temperature. Then values obtained at each temperature are considered to optimize 2S2P1D and Williams Landel Ferry (WLF) Equation constants to simulate the global LVE behavior of the material. The accuracy of the process was assessed regarding both the calculated FRFs and the complex modulus evaluation. Results indicate that the proposed optimization process converges almost perfectly towards the reference FRFs. The simulated complex modulus values are also in very good agreement with the values of the reference material.

Frequency-dependent material properties of copper and aluminum alloys

In this paper, the dynamic behavior of copper and aluminum alloys as used in electric drive units is investigated. Isothermal frequency sweeps are performed from 0.1 up to 50 Hz at temperatures of up to 400 °C. An evaluation of the test results at a constant frequency of 1 Hz shows a decrease in the storage modulus and an increase in the material damping. Considering all frequencies, a supplementary frequency dependency related to the material composition is detected. The lower the volume fraction of the alloying elements, the higher the impact of temperature and frequency on the material properties. The variations of the material parameters allow applying the time temperature superposition principle to estimate the dynamic behavior beyond the limited tested frequency range and temperatures by fitting the Williams–Landel–Ferry equation. Additional thermal aging of CU-ETP specimens does not affect the storage modulus, but diminishes the material damping. The findings show that each material has to be tested with regard to its respective application, material composition and manufacturing process. Furthermore, they demonstrate the relevance of considering the frequency-dependent material properties of low-alloyed copper and aluminum, especially in case of temperatures above 100 °C.

The Connection Between the Parameters of WLF Equation and of Arrhenius Equation

AbstractThe description of the change in characteristic temperatures of thermomechanical and viscoelastic properties of polymers and elastomers with deformation frequency or of temperature dependence of polymer properties is widely achieved by two equations: (1) the Williams‐Landel‐Ferry (WLF) equation and (2) the Arrhenius equation. Mostly the WLF equation is used. Often the distinction between the two descriptions is based on the argument: if volume processes play the key role, then WLF equation is the right one, if thermally activated processes play the key role, then Arrhenius equation is the right one. Both equations are based on the activation of processes, and always the temperature is the variable, which activate the processes. Both descriptions are methods to parameterize the temperature dependence of properties or the change of characteristic temperatures, as glass‐rubber transition temperature, with deformation rate. Also, the so‐called ‘volume processes’ are controlled by temperature, but the thermal activation can be small in energy to initiate the change in spatial position from one site to another for a molecule. This means both descriptions should be congruent. In this article, the congruence is shown and the relation between WLF parameters and Arrhenius parameters will be established. For this, a slight modification of the usual Arrhenius equation is necessary. Also, other descriptions are discussed in short: Doolittle equation and Vogel‐Fulcher equation, they were or are used to describe the change of viscosity with temperature in melts or solutions.