Williams Landel Ferry Research Articles

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.

Viscoelastic characterization of high‐density polyethylene membranes under the combined effect of the temperature and the gravity for thermoforming applications

AbstractNumerical simulation of the heating stage of thermoplastics in thermoforming requires a good knowledge of the behavior of the materials used. To this end, a study is being conducted on the characterization of the viscoelastic behavior of a circular membrane, made of high‐density polyethylene (HDPE), under the combined effect of temperature and the force of gravity. The experimental tests were carried out in a convection oven for five temperatures (100, 110, 120, 130 and 140°C). For the numerical characterization of the viscoelastic behavior, two viscoelastic models were considered: the classic Kelvin‐Voigt model and the new three‐parameter modified Burger's model (Jeffrey model) that we propose. The mechanical parameters of both models were identified using the Levenberg‐Marquardt algorithm. The thermal‐dependency of the viscosity was characterized by two thermal models: the Arrhenius law and the William‐Landel‐Ferry (WLF) equation.

Development of master curves for nonlinear viscoelastic behaviour of nanofiller-reinforced composites

Multi-walled carbon nanotube-reinforced polypropylene (MWCNT/PP) nanocomposites were prepared through solution mixing and characterized for their nonlinear viscoelastic behaviour. The effect of applied stress, time and temperature was studied to obtain compliance responses. Time–temperature superposition principle was applied on experimentally obtained isothermal compliance curves using horizontal shift factors from William–Landel–Ferry and Arrhenius models. Horizontal shifting was found to be helpful in determining temperature-dependent short- and long-term behaviour of the developed nanocomposite material. Neat PP material was found to be more prone to deformation at considered temperatures. 1% MWCNT PP nanocomposite was observed with lower compliance as compared to other samples.

Mechanical Behavior of Melt-Mixed 3D Hierarchical Graphene/Polypropylene Nanocomposites.

The mechanical properties of novel low percolation melt-mixed 3D hierarchical graphene/polypropylene nanocomposites are analyzed in this study. The analysis spans a broad range of techniques and time scales, from impact to tensile, dynamic mechanical behavior, and creep. The applicability of the time–temperature superposition principle and its limitations in the construction of the master curve for the isotactic polypropylene (iPP)-based graphene nanocomposites has been verified and presented. The Williams–Landel–Ferry method has been used to evaluate the dynamics and also Cole–Cole curves were presented to verify the thermorheological character of the nanocomposites. Short term (quasi-static) tensile tests, creep, and impact strength measurements were used to evaluate the load transfer efficiency. A significant increase of Young’s modulus with increasing filler content indicates reasonably good dispersion and adhesion between the iPP and the filler. The Young’s modulus results were compared with predicted modulus values using Halpin–Tsai model. An increase in brittleness resulting in lower impact strength values has also been recorded.

Verification and Validation of openInjMoldSim, an Open-Source Solver to Model the Filling Stage of Thermoplastic Injection Molding

In the present study, the simulation of the three-dimensional (3D) non-isothermal, non-Newtonian fluid flow of polymer melts is investigated. In particular, the filling stage of thermoplastic injection molding is numerically studied with a solver implemented in the open-source computational library O p e n F O A M ® . The numerical method is based on a compressible two-phase flow model, developed following a cell-centered unstructured finite volume discretization scheme, combined with a volume-of-fluid (VOF) technique for the interface capturing. Additionally, the Cross-WLF (Williams–Landel–Ferry) model is used to characterize the rheological behavior of the polymer melts, and the modified Tait equation is used as the equation of state. To verify the numerical implementation, the code predictions are first compared with analytical solutions, for a Newtonian fluid flowing through a cylindrical channel. Subsequently, the melt filling process of a non-Newtonian fluid (Cross-WLF) in a rectangular cavity with a cylindrical insert and in a tensile test specimen are studied. The predicted melt flow front interface and fields (pressure, velocity, and temperature) contours are found to be in good agreement with the reference solutions, obtained with the proprietary software M o l d e x 3 D ® . Additionally, the computational effort, measured by the elapsed wall-time of the simulations, is analyzed for both the open-source and proprietary software, and both are found to be similar for the same level of accuracy, when the parallelization capabilities of O p e n F O A M ® are employed.

Resilient modulus prediction of RAP using the Contact Dynamics Method

Abstract The Optimal Recycling of Reclaimed Asphalts in low-traffic Pavements (ORRAP) project concerns a cold recycling of 100% Reclaimed Asphalt Pavement (RAP) without binder addition, in base and subbase layers of low-traffic pavements. The mastic coating of aggregates induces a viscoelastic behaviour and changes the resilient modulus with frequency and temperature. This modulus is obtained with repeated load triaxial tests and is a key element for pavement design. In this context, this paper presents a numerical alternative to predict this influence with a discrete approach. Contact Dynamics (CD) simulations are used to reproduce the resilient modulus test on a set of rigid spherical particles with a viscoelastic contact, based on the Burgers’ model. The tests were carried out with cylindrical samples at several frequencies for two temperatures (20 °C and 40 °C). The proposed model is calibrated in a given range of temperatures and frequencies regarding the experimental data, and then used to predict resilient modulus values at unreachable frequencies in the laboratory, with one set of parameters for each temperature. Finally, using the Williams-Landel-Ferry (WLF) equation, a master curve of resilient modulus at 20 °C as a function of the frequency was established. The Time-Temperature Superposition Principle (TTSP) reveals that the values at high frequencies and 40 °C correspond to the data at low frequencies and 20 °C.

Viscosity characterization of confined bitumen considering microaggregate‐bitumen interactions

AbstractMicroaggregate‐bitumen interactions in asphalt mixtures cause nonuniform rheological properties of bitumen adhering to aggregates. It is necessary to quantify microaggregate‐bitumen interactions so that the properties of the confined bitumen can be precisely characterized. This study developed a viscosity model of confined bitumen film considering microsubstrate–bitumen interactions. An exponential function was introduced to characterize the spatial distribution of viscosity under the effects of these microinteractions. The apparent viscosity of the bitumen film was formulated as an expression of thickness comprising bulk and interfacial viscosities and a decay index. The bulk viscosity showed a temperature dependence that corresponded with the Williams–Landel–Ferry model. The interfacial viscosity was higher than the bulk viscosity at high temperatures but lower at low temperatures, which indicates that the effect of substrate–bitumen interactions is bidirectional and depends on temperature. This phenomenon could be attributed to the microstructural transition of interfacial bitumen from an oriented layer to a disentangled one. Molecular dynamics simulations demonstrated the microstructural changes of interfacial bitumen under the effect of microaggregate‐bitumen interactions.

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.

Degradation Model for the Tensile Strength of PVC and Rubber Gasket Materials Exposed to Benzene and PCE-Saturated Aqueous Solutions

In this paper, a tensile strength degradation model is developed for polyvinyl chloride (PVC) pipe and three rubber gasket materials commonly used in stormwater drains and wet utilities; the degradation model considers exposure to a single contaminant at its saturation level. The contaminant considered included benzene and tetrachloroethylene (PCE) because of their prevalence. The materials considered are unplasticized Polyvinyl chloride (u-PVC) utilized as the pipe material, and three types of rubber gaskets: Polychloroprene (Neoprene-CR), Acrylonitrile (nitrile) butadiene rubber (Buna-N), and fluoroelastomer rubber (FKM—Grade A VitonTM). First, the degradation rates of these materials are experimentally quantified using accelerated testing; then using the experimental results and Williams–Landel–Ferry (WLF) method a degradation model is developed. Finally, a simplified method is introduced to relate the tensile strength degradation of PVC to the penetration rate of the contaminants.

Generalised superposition models for rheologically complex starch-nanohybrid films and integrational construction of master-curves

We propose an integrational method to construct master-curves, using synthetic simultaneous multifrequency scans to avoid intrinsic temperature control variances and better predict Tg and Tm. Proposed generalised time–temperature superposition models for rheologically complex materials are presented. The methodologies and models were applied to thermoplastic starch films modified with polyhedral oligomeric silsesquioxanes (POS). We confirmed their thermorheologically complex behaviour, which in principle cannot be used to construct master curves. However, based on properties predicted by the models, we were able to establish temperature constraints where the time-temperature principle was valid. Our models have higher coefficient of determination than the Williams–Landel–Ferry (WLF) or Arrhenius models currently applied in rheologically simple materials. The addition of POS increased the internal ordering of the films, leading to more compact molecular arrangement, made the films more resilient to deformation and their master-curves begin at higher frequencies than the control.

The Effects of Asphalt Migration on the Dynamic Modulus of Asphalt Mixture

Asphalt migration is one of the significant detrimental effects on asphalt pavement performance. In order to simulate the state after the occurrence of asphalt migration amid asphalt pavement layers and further investigate the effects of asphalt migration on the dynamic modulus of asphalt mixture, samples with different asphalt contents layers were firstly separated into the upper and lower half portions and then compacted together. By conducting the dynamic modulus test with the Superpave Simple Performance Tester (SPT), the variation laws of the dynamic modulus (|E*|) and the phase angle (δ) at different testing temperatures and loading frequencies were analyzed in this paper. Further, the dynamic modulus and the stiffness parameter (|E*|/sinδ) at the loading frequency of 10 Hz and testing temperature of 50 °C were illustrated. Simultaneously, the master curves of the dynamic modulus and phase angle of asphalt mixtures under different testing conditions were constructed to better investigate the effects of asphalt migration on the dynamic modulus by means of Williams–Landel–Ferry (WLF) equation and Sigmoidal function. Results show that, after the asphalt migration, the dynamic modulus of asphalt mixtures increase with the increasing loading frequency while they decrease with the increasing testing temperature; the dynamic modulus and the stiffness parameter are the highest when asphalt mixtures have the optimum asphalt content layers, and then decrease with the incremental difference of asphalt content in the upper and lower half portions. Besides this, different from the master curves of dynamic modulus, the master curves of phase angle firstly increase with the increase of loading frequency to the highest point and then decrease with the further increase of loading frequency and are not as smooth as that of dynamic modulus. It can be concluded that the asphalt migration has compromised the mixture’s mechanical structure, and the more asphalt migrates, the weaker the mechanical properties of asphalt mixture will be. Additionally, based on the shift factors and master curves in the time–temperature superposition principle (TTSP), the effects of asphalt migration on the dynamic modulus and the variation laws of the dynamic modulus of asphalt mixture after the occurrence of asphalt migration can be better construed at the quantitative level.

Temperature dependence of relaxation time of hydrolyzed polyacrylamide solution for enhanced oil recovery

Among enhanced oil recovery (EOR) polymers that are now commercial, partially hydrolyzed polyacrylamide (HPAM) is the most commonly used polymer. The relaxation time, which is a key parameter describing the shear-thickening behavior of HPAM polymer solutions, was experimentally measured and quantitatively analyzed. From dynamic frequency sweep tests, the elastic modulus and viscous modulus of various polymer solutions (polymer concentration 500–3000ppm, NaCl salinity 1000–10,000ppm) were measured as a function of temperature (25–70℃), where the cross-over method was carried out to obtain the relaxation time. The two parameters in the Williams–Landel–Ferry (WLF) equation for polymer melts, which describe the temperature effect on relaxation time, were calculated by fitting the measured relaxation time; the first parameter is universally constant, the second parameter depends on conditions of the polymer solution (polymer concentration and salinity). The estimated relaxation times using this proposed method matched well with the measured relaxation time with a root mean square error of 0.1261s.

Analysis of Non‐Isothermal Viscous Flow Coalescence at Micro Scale

The non‐isothermal coalescence of two spherical bodies caused by capillary‐induced viscous flow was analyzed. Based on this analysis, a new dimensionless number (K number) was introduced for defining thermal coalescence regimes. Based on the value of this number, coalescence may or may not be affected by thermal effects in different cases. To make this clearer, the conventional coalescence models of Frenkel‐Eshelby and Pokluda were modified by assuming viscosity as a temperature dependent variable. This was conducted by considering the effects of temperature on the viscosity of the involved material through evaluating different expressions including linear and Reynolds and Williams‐Landel‐Ferry (WLF) equations. The results of the modified models for the bridge growth rate show that temperature changes significantly affect the kinetics of coalescence, particularly when the characteristic times of coalescence and heat conduction are in the same order, i.e., moderate K numbers. This analysis is applicable for diverse situations since viscous flow coalescence occurs in various physical and industrial applications of particles or droplets.

Comparison between thermal, rheological and failure properties for the performance grading of asphalt cements

Thermal, rheological and failure properties of nine extracted and recovered asphalt cements were investigated to predict cracking susceptibility. Glass transition temperatures, non-isothermal Ozawa crystallization constants and binder fragilities were determined but found to lack a strong correlation with established performance indicators. All binders exhibited thermorheologically simple behavior and master curves for complex modulus and phase angle were readily constructed by applying the time-temperature superposition (TTS) principle. Failure properties of the binders were determined using ductility and double-edge-notched tension (DENT) tests. It was found that the ductility and approximate critical crack tip opening displacement (CTOD) are correlated, although the latter provides more reasonable (lower) values of strain tolerance. The Williams-Landel-Ferry (WLF) equation described the shift factors in dynamic shear, and the WLF constants C1 and C2 correlate reasonably well with ductility and CTOD. It was found that the phase angle was sensitive to asphalt cement quality, with poor performing binders showing a limiting phase angle temperature, T(δ = 45°), nearly 25 °C higher than those of superior performing materials. This investigation demonstrates that recovered binders can be prematurely hardened, likely due to overheating during production or the incorrect use of reclaimed asphalt. As this problem is not recognized when testing material from the asphalt cement supply tank, it is recommended to test extracted and recovered binder from the loose hot mix asphalt. This will assure that the material as placed actually meets performance requirements for the local climate.

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.

Effect of intermolecular interactions on the viscoelastic behavior of unentangled polyamide melts

We have studied the effect of ionic interactions on the dynamics of polyamides in the molten state, both in the terminal relaxation regime and in the vicinity of the glass transition, using amorphous and end-blocked PA 6I polyamides. The density of interactions was varied by substituting different amounts of 5-sulfoisophthalic acid lithium salt. First, the effect of molecular weight distribution for PA 6I was studied in detail. As the molecular weight increased, a clear rubbery plateau appeared and the longest relaxation time was shifted to lower frequencies. The reference temperature used to construct the master curves was chosen to be Tg so that all master curves overlap in the glassy region. The main effect of introducing ionic groups was the increase in Tg by 10–40 °C. For comparable molecular weights, master curves of PA 6I and substituted polyamides overlapped in the complete range of frequencies when using an appropriate reference temperature. The reference temperature, at least up to 15 mol. %, seems to be correlated to the width of the glass transition that increases with increasing ionic content. The Rouse model was used to fit the viscoelastic response of all unentangled polyamides, showing that hydrogen bonds and/or ionic interactions did not play a role in the viscoelasticity of these polyamides in the molten state. The dynamic fragility indices determined from William–Landel–Ferry parameters decrease with increasing ionic content. Small-angle x-ray scattering measurements show that no segregation of ionic domains occurs in unentangled ionic copolyamides, while entangled ionic copolyamides show only weak segregation.We have studied the effect of ionic interactions on the dynamics of polyamides in the molten state, both in the terminal relaxation regime and in the vicinity of the glass transition, using amorphous and end-blocked PA 6I polyamides. The density of interactions was varied by substituting different amounts of 5-sulfoisophthalic acid lithium salt. First, the effect of molecular weight distribution for PA 6I was studied in detail. As the molecular weight increased, a clear rubbery plateau appeared and the longest relaxation time was shifted to lower frequencies. The reference temperature used to construct the master curves was chosen to be Tg so that all master curves overlap in the glassy region. The main effect of introducing ionic groups was the increase in Tg by 10–40 °C. For comparable molecular weights, master curves of PA 6I and substituted polyamides overlapped in the complete range of frequencies when using an appropriate reference temperature. The reference temperature, at least up to 15 mol. %, se...

Frequency-dependent viscoelastic properties of Chinese fir (Cunninghamia lanceolata) under hygrothermal conditions. Part 1: moisture adsorption

Abstract To elucidate the frequency-dependent viscoelasticity of wood under a moisture non-equilibrium state, changes in stiffness and damping as a function of frequency were investigated during the moisture adsorption process. The moisture adsorption processes were carried out at six temperatures (30–80°C) and three relative humidity levels (30, 60 and 90% RH). During the moisture adsorption process, the wood stiffness decreased, and damping increased with the increment of moisture content (MC). Regardless of the moisture adsorption time, the wood stiffness increased, and damping decreased with the increasing testing frequency. Based on the re-organized Williams-Landel-Ferry (WLF) model, the time-moisture superposition (TMS) relation was assumed to be applicable for developing a master curve of wood stiffness during the moisture adsorption process. The frequency ranges of the stiffness master curves spanned from 16 to 23 orders of magnitude at temperatures ranging from 30 to 80°C. However, the TMS relation was not able to predict the wood damping properties during the moisture adsorption process due to the multi-relaxation system of the wood and the non-proportional relationship between free volume and MC at transient moisture conditions.

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.

Influence of blend ratio and water content on the rheology and fragility of maltopolymer/maltose blends

The rheological behavior of blends of a fractionated maltopolymer (Mw = 1.4⋅104 Da) and the disaccharide maltose is investigated as a function of water content and temperature, with emphasis on the viscosity and molecular relaxations in the approach to the glass transition. Shear rheology is combined with dynamic mechanical thermal analysis to probe viscosities between 1 mPa s and 1012 Pa s. Differential scanning calorimetry is used to determine glass transition and enthalpy relaxation of the carbohydrate blends. The rheology data are fitted with a modified version of Williams–Landel–Ferry equation (Williams et al., 1955). The fragility of the blends is quantified using Angell’s fragility parameters m and F1/2 (Angell, 1991) and Roos’ strength parameter S (Roos, 1995b). The increase in fragility of the maltopolymer systems with increasing water and maltose contents is interpreted as a reduction of the entanglement density and an interference of water molecules with the hydrogen bonding between the carbohydrate chains.