Evolution of time constants during sol-gel transition

In developing a rheological constitutive equation for crosslinking polymers the determination and explanation of time constants are of great importance. For stoichiometrically balanced and imbalanced reactions of polydimethylsiloxane and polyurethane model polymers with appropriate cross-linkers, it is shown that the experimental data for the storage and loss moduli are well fitted within a large frequency range by a linear rheological constitutive equation with a truncated power law relaxation function.Special attention is given to the problem of estimating the different time constants. A method is proposed for the determination of the terminal relaxation time (in the case below the gel point) and a characteristic retardation time (in the case above the gel point) from the intersection points of the high and low frequency asymptotes of the storage modulus.     

Low frequency/shear rate measurements on polymer melts with a novel rheometer

To obtain the time dependent viscosity function over a wide experimental window, data from two different measuring instruments and methods were combined. The instruments involved were a conventional dynamic rheometer and the magnetoviscometer (MVM), a novel apparatus specially designed for measuring at low shear rates. For most of the investigated materials the MVM-data allow an expansion of the known time-range to higher values, thus giving additional information about the longest relaxation times in the systems, which in turn correspond to the high-molecular tail of the molar mass distribution. Especially in the long time range where it is difficult to get good dynamical data the experimental error of the MVM-data is very small.     

Comparison of stress-controlled and strain-controlled rheometers for large amplitude oscillatory shear

In linear viscoelastic region, it is well known that dynamic modulus and dynamic compliance can be converted to each other. However, it is questionable whether there exists an interconversion between large amplitude oscillatory shear (LAOS) data measured from different types of rheometers—stress-controlled and strain-controlled rheometers. Hence, we tried to prove the existence by use of polyethylene oxide (PEO) aqueous solutions with well-developed entanglements. From this experiment, we can conclude that a stress-controlled rheometer can simulate LAOS behavior measured from a strain-controlled rheometer under the conditions where inertia effect is not significant. Furthermore, it is investigated whether the LAOS data of the stress-controlled rheometer obey stress–frequency superposition as the strain–frequency superposition found by Cho et al. (J Rheol 54:27–63, 2010) from LAOS data measured by the strain-controlled rheometer. This scaling relation shows that the dimensionless stress amplitude is a function of zeta which is the product of the stress amplitude and linear viscoelastic function J′(ω). The plot shows that all of the data are superposed in a single curve without regard to frequency, molecular weight, and concentration of PEO aqueous solutions.     

The creep behavior of ideally atactic and commercial polymethylmethacrylate

Summary The shear creep behavior of polymethylmethacrylate, PMMA, samples has been studied in the neighborhood of and above their glass temperatures. One of the materials studied was ideally atactic with equal numbers of random isotactic and syndiotactic placements, while the other was a commercial or conventional PMMA which was about 76% syndiotactic. The glass temperatures,T _g , were found to be 106 and 117 °C respectively. Evacuation above the glass temperature for several weeks was necessary before reproducible creep compliance,J (t), curves could be obtained. It is believed that absorbed water plasticized the polar materials and its removal led to the shifting of theJ (t) curves to longer times. For both materials apparently successful temperature reduction was found to be possible within the temperature range of our investigations, i.e. up to 200 °C. Retardation spectra were calculated from the reduced curves and are compared. The temperature dependences, as described by the time scale shift factors,a _T , were similar when allowance is made for the different glass temperature. Botha _T curves could not be fitted to theWilliams, Landel, andFerry, WLF, free volume expression. These are the first examples of such a deviation for amorphous high polymers. It is proposed that the primary softening dispersion has two distinctly different groups of viscoelastic mechanisms contributing to it. On this basis the primary dispersion was decomposed into the two contributions. Both of the resulting temperature dependences were satisfactorily fitted to the WLF equation. Differences in the retardation spectra are noted. The glassy compliance of the commercial PMMA appears to be about twice that of the atactic PMMA.Data on commercial PMMA is incorporated in a thesis which has been submitted in partial fulfillment of the requirements for the degree of Master of Science in Materials Engineering, University of Pittsburgh, 1970.With 12 figures and 1 table     

Calculation of discrete retardation spectra from creep data — II. Analysis of measured creep curves

An algorithm for the calculation of discrete, logarithmic equidistant retardation spectra from creep or recovery is applied to experimental data in different regions of consistency. Spectra in the glass-rubber transition region are given for technical poly(styrene), poly(methylmethacrylate), and poly(carbonate) as well as the course of all characteristic compliance type functions. The spectrum in the terminal region of a poly(styrene) of narrow molecular weight distribution is calculated both from creep and recovery data. The course of the dynamic moduli calculated from the spectra and by direct conversion is found in excellent agreement to measurements by means of a dynamic viscometer.Dedicated to Prof. Dr. H. Janeschitz-Kriegl at the occassion of his 70th birthday.     

A class of linear viscoelastic models based on Bessel functions

In this paper we investigate a general class of linear viscoelastic models whose creep and relaxation memory functions are expressed in Laplace domain by suitable ratios of modified Bessel functions of contiguous order. In time domain these functions are shown to be expressed by Dirichlet series (that is infinite Prony series). It follows that the corresponding creep compliance and relaxation modulus turn out to be characterized by infinite discrete spectra of retardation and relaxation time respectively. As a matter of fact, we get a class of viscoelastic models depending on a real parameter \(\nu > -1\). Such models exhibit rheological properties akin to those of a fractional Maxwell model (of order 1/2) for short times and of a standard Maxwell model for long times.     

A melt rheometer with integrated small angle light scattering

A stress-controlled commercial rheometer has been equipped with small-angle light scattering (SALS) instrument working in a range of wave-vectors between 0.5 and 4.2 m^–1 By a specially designed optical system the scattered light is focused directly on the square chip of a CCD-detector with high dynamical range allowing quantitative two-dimensional (2 D) intensity measurements of polarized and depolarized scattering. The rheo SALS system is suited for simultaneous fast measurements of 2D SALS and rheological material functions of polymer melts and liquid crystals. As a first application results on the influence of shear rate on the late stage of spinodal decomposition of a polybutadiene/polyisoprene blend are reported. A second application concerns a creep experiment in the liquid crystalline phase of a lyotropic non-ionic surfactant with simultaneous measurement of the time-dependent compliance and associated changes of the 2D intensity patterns of the polarized and depolarized scattering.     

Viscoelastic measurement of complex fluids using forced oscillating torsion resonator with continuously varying frequency capability

Traditional torsional resonators, often obtaining the viscoelastic moduli of complex fluids only at one or several given discrete frequencies, lack the continuously varying frequency capability. This is an obvious disadvantage of the traditional torsional resonator technique. This paper presents an improved strategy, based on our previous discrete-frequency-measuring method (Wang et al., J Rheol 52:999–1011, 2008), to overcome such restriction and thus accomplish the continuously varying frequency capability of the traditional torsional resonator for measuring the viscoelastic properties of complex fluids. The feasibility of this strategy is demonstrated with the Newtonian fluids (several water–glycerol solutions) of viscosities varying from 10 to 1,400 cp by using our homemade torsion resonator apparatus in the 10 ~ 2,500 rad/s frequency range (continuous frequencies). Some results for typical viscoelastic polymers (two polyethylene oxide (PEO) aqueous solutions) are also given. Additionally, a comparison of the PEO results is made with the common rheometer technique. It is demonstrated that this improved strategy could enable the traditional torsional resonators, with one oscillating resonance mode, to work as the microrheological technique and the common rheometer technique in the continuous frequency range.     

Presentation of the creep curves of solid biological materials by a simplified mathematical version of the generalized Kelvin-Voigt model

Published creep equations, derived from the Kelvin-Voigt model with a discrete retardation time spectrum were used to generate data for linear presentation of the creep function. It is shown that 2–3 term creep functions, containing 4–6 constants, could be reduced to an algebraic expression having two constants only. Consequently, entire creep curves of a variety of biological and food materials could be described by a single type of equation in the form of:P (t) = k ₀ + k₁ t + t/(k₂ + k₃t) whereP (t) is the creep parameter (compliance, strain or deformation),k ₀ a constant representing the instantaneous compliance,k ₁ a constant representing the steady-state flow, andk ₂ andk ₃ the characteristic constants of the creep function (t) obtained by regression from the linear relationship oft/ (t) vst.     

Comparison of different shear rheometers with regard to creep and creep recovery measurements

The performance of two shear rheometers with regard to creep and creep recovery measurements is investigated. The first one is a commercially available stress-controlled rheometer that uses an air bearing, and the second one is a magnetic bearing torsional creep apparatus that was built at the Institute of Polymer Materials in Erlangen. The creep and creep recovery measurements were performed in the linear-viscoelastic regime of two polyethylene melts at a temperature of 150°C. The creep compliances of the polyethylenes measured by both rheometers are in excellent agreement. The recoverable compliances of both polyethylene melts, however, have lower values in the case of the commercial rheometer than in the case of the magnetic bearing rheometer. The experimental parameters of creep recovery experiments and the features of the two bearings that are responsible for the different results of the two rheometers are discussed. The influence of the level of the applied shear stress on the short time behavior is investigated as well as the residual torque of the bearings which influences the long time region of the recoverable compliances. Also addressed are the influence of the momentum of inertia of the rotor and the bearing friction which is different for the two rheometers.     

Dynamic study of gelatin gels by creep measurements

We have investigated the dynamical properties of gelatin gels using creep measurements. A commercial apparatus (Carrimed CSL500) was modified in order to increase the deformation of the gel and to take advantage of the inertia of the system. When a step stress is applied, the very first response of these materials is an oscillating strain owing to a coupling of the high elasticity of the gelatin gels and the inertia of the apparatus. From these damped oscillations, we have extracted the elastic and loss moduli as a function of frequency, which allows us to widen the frequency range (toward high frequencies) of measurement. After subtraction of the oscillations, we have obtained the compliance funtion from which, using Ferry's formalism, we can calculate the relaxation time distribution function over a very large time range (10^–3–10⁴ s). We show that the dynamics of gelatin gels is governed by two very different characteristic times. We interpret the faster relaxation time as relaxation at the scale of the gel network mesh-size, while the slower time we assign to relaxations involving the lifetime of the gelatin gel cross-links. It is now possible to use creep measurements as an alternative to the forced oscillatory function determination, as the same data can be obtained but, more quickly, and over a large frequency range. This gives us more indication of the gel's structure (gel network behaviour, kinetics of ageing) than all the laborious methods previously necessary.     

The determination of a general time creep compliance relation of linear viscoelastic materials under constant load and its extension to nonlinear viscoelastic behavior for the Burger model

Linear viscoelastic materials yield a creep function which only depends on time if creep experiments are performed under constant stress ₀. In practice, this condition is very difficult to realize, and as a consequence, the experiments are performed under constant force. For small strains the difference between the conditions of constant stress and constant force is negligible. Otherwise, the decrease in cross-section has to be taken into account and leads to increasing stress in the course of time for creep experiments under constant load. The Boltzmann superposition principle is solved under the condition of constant load and for strains . The creep complicance C(t; ₀) defined by the ratio becomes, in principle, dependent on the initial stress ₀. As a consequence, a set of creep compliance curves cannot be approximated with a simple parameter fit. Already the application of the solution on the Burger model yields a creep compliance curve with all three creep ranges. Furthermore, the mathematical structure of the time creep compliance relation of the Burger model allows nonlinear viscoelastic extension via the introduction of the yield strength _max and a nonlinearity parameter n _l . The creep behavior of PBT and PC can be described in the range of long times up to initial stresses ₀, being 75% for PBT and 60% for PC of the yield stress _max with only two or one free fit parameter, respectively.     

Long-term behavior of mineral-filled polymer composites modeled by discrete spectra

A discrete spectra transformation technique is used for the processing and analysis of long-term stress relaxation and creep compliance data of mineral-filled polymer composites. A non-linear regression simultaneously adjusts the parameters of N discrete relaxation or retardation spectra. For small N the solution is insensitive to the choice of regression starting value sets. From the relaxation time spectrum a corresponding discrete retardation spectrum and creep compliance can be calculated using the Laplace transform and vice versa. The analysis of long-term (more than 1200 days) both relaxation and retardation experimental data demonstrates the applicability of the transformation technique. Comparisons of the experimental and calculated spectra are given. The influence of the filler amount is demonstrated.     

Creep performance of PVC aged at temperature relatively close to glass transition temperature

In order to predict the mechanical performance of the polyvinyl chloride(PVC) at a high operating temperature,a series of short-term tensile creep tests(onetenth of the physical aging time) of the PVC are carried out at 63 C with a small constant stress by a dynamic mechanical analyzer(DMA).The Struik-Kohlrausch(SK) formula and Struik shifting methods are used to describe these creep data for various physical aging time.A new phenomenological model based on the multiple relaxation mechanisms of an amorphous polymer is developed to quantitatively characterize the SK parameters(the initial creep compliance,the characteristic retardation time,and the shape factor) determined by the aging time.It is shown that the momentary creep compliance curve of the PVC at 63 C can be very well fitted by the SK formula for each aging time.However,the SK parameters for the creep curves are not constant during the aging process at the elevated temperatures,and the evolution of these parameters and the creep rate versus aging time curves at the double logarithmic coordinates have shown a nonlinear phenomenon.Moreover,the creep master curves obtained by the superposition with the Struik shifting methods are unsatisfactory in such a case.Finally,the predicted results calculated from the present model incorporating with the SK formula are in excellent agreement with the creep experimental data for the PVC isothermally aged at the temperature relatively close to the glass transition temperature.     

A newly designed coaxial cylinder-type dynamic rheometer

A newly designed dynamic rheometer is described. This apparatus is designed for the purpose of the investigation of the physical ageing process of gels and for the determination of the viscoelastic properties of polymer melts. With this instrument the dynamic moduli can be determined very accurately in a range of about 8 decades (– 2 < logG < 5.3 and – 3 < logG < 5.4;G andG in N/m²) and in a frequency range of more than 6 decades (5 10^–5 – 100 Hz) with an accuracy and stability of 0.025%. Results are shown of measurements on a silicon oil (with loss angles deviating only little from 90°), on a gel of polyvinyl chloride in Reomol (with loss angles deviating only little from 0°) and on a polystyrene melt for which, at low frequencies, the dependencies of the storage and loss moduli on the frequency are quadratic and linear, respectively.Dedicated to the Sixtieth birthday of Prof. Dr. H. R. K. N. Janeschitz-Kriegl, Johannes Kepler University, Linz, Austria.     

A New Technique for Characterization of Low Impedance Materials at Acoustic Frequencies

Measuring the mechanical properties of low impedance rubbery polymers at acoustic frequencies is a challenging problem due to the small signal amplitudes, relatively high loss, and the long wavelength of stress waves. One such material is solid polyurea (PU), an elastomeric copolymer, which has excellent chemical, thermal, and mechanical properties and is widely used as a coating (e.g. in truck bed lining) or blast protection (advanced helmet designs and concrete structures) material. Moreover, due to its heterogeneous structure, PU has a wide transition of thermo-mechanical behavior from rubber-like to glassy compared to most engineering polymers, which translates to a broader loss spectrum in frequency domain. In this study, we have developed a new test technique by modifying the split Hopkinson pressure bar and using ball impact to measure Young’s storage and loss moduli of polyurea at kHz frequencies. This will therefore fill the frequency gap between the dynamic mechanical analysis (DMA) and ultrasonic (US) wave measurement. The measured Young’s storage and loss moduli from this technique are compared with the master curves of the moduli developed using experimental data of dynamic mechanical analysis and ultrasonic wave measurements. This technique is a direct measurement which provides more reliable data in the kHz frequency range and can be used to evaluate the reliability of other indirect estimations including master curves. The utility of this technique is not limited to polyurea and it can be used to characterize other low impedance materials at kHz frequencies.     

From dynamic modulus via different relaxation spectra to relaxation and creep functions

The main goal of the paper is to compare predictive power of relaxation spectra found by different methods of calculations. The experimental data were obtained for a new family of propylene random copolymers with 1-pentene as a comonomer. The results of measurements include flow curves, viscoelastic properties, creep curves and rubbery elasticity of copolymer melts. Different relaxation spectra were calculated using independent methods based on different ideas. It lead to various distributions of relaxation times and their “weights”. However, all of them correctly describe the frequency dependencies of dynamic modulus. Besides, calculated spectra were used for finding integral characteristics of viscoelastic behaviour of a material (Newtonian viscosity, the normal stress coefficient, steady-state compliance). In this sense all approaches are equivalent, though it appears impossible to estimate instantaneous modulus. The most crucial arguments in estimating the results of different approaches is calculating the other viscoelastic function and predicting behaviour of a material in various deformation modes. It is the relaxation and creep functions. The results of relaxation curve calculations show that all methods used give rather similar results in the central part of the curves, but the relaxation curves begin to diverge when approaching the high-time (low-frequency) boundary of the relaxation curves. The distributions of retardation times calculated through different approaches also appear very different. Meanwhile, predictions of the creep curves based on these different retardation spectra are rather close to each other and coincide with the experimental points in the wide time range. Relatively slight divergences are observed close to the upper boundary of the experimental window. All these results support the conclusion about a rather free choice of the relaxation time spectrum in fitting experimental data and predicting viscoelastic behaviour of a material in different deformation modes. Received: 15 March 2000 Accepted: 18 September 2000     

Dynamic viscoelastic behaviour of wheat flour doughs

Summary The general dynamic mechanical properties of wheat flour doughs have been investigated for a range of strains over a frequency spectrum of 0.032–32 Hz. An approximately linear stress-strain relationship was found to hold at the lower range of strains imposed. Departures from the linear response of the dough are explained in terms of a protein chain-starch granule interaction and the accompanying time dependence in terms of a stress-biased diffusion process. The influence of water and temperature on the dynamic moduli are shown, indicating that a frequency-temperature and a water absorption-frequency superposition principle is applicable. The transient stress relaxation modulus is calculated from the dynamic moduli to demonstrate the ability of dynamic methods to investigate the time scale inaccessible to transient techniques.     

Compliance of actin filament networks measured by particle-tracking microrheology and diffusing wave spectroscopy

We monitor the time-dependent shear compliance of a solution of semi-flexible polymers, using diffusing wave spectroscopy (DWS) and video-enhanced single-particle-tracking (SPT) microrheology. These two techniques use the small thermally excited motion of probing microspheres to interrogate the local properties of polymer solutions. The solutions consist of networks of actin filaments which are long semi-flexible polymers. We establish a relationship between the mean square displacement (MSD) of microspheres imbedded in the solution and the time-dependent creep compliance of the solution, <Δr ²(t)>=(k _B T/πa)J(t). Here, J(t) is the creep compliance, <Δr ²(t)> is the mean-square displacement, and a is the radius of the microsphere chosen to be larger than the mesh size of the polymer network. DWS allows us to measure mean square displacements with microsecond temporal resolution and Ångström spatial resolution. At short times, the mean square displacement of a 0.96μm diameter sphere in a concentrated actin solution displays sub-diffusion. <Δr ²(t)>∝t ^{, with a characteristic exponent =0.78±0.05, which reflects the finite rigidity of actin. At long times, the MSD reaches a plateau, with a magnitude that decreases with concentration. The creep compliance is shown to be a weak function of polymer concentration and scales as J _p ∝c –1.2±0.3. This exponent is correctly described by a recent model describing the viscoelasticity of semi-flexible polymer solutions. The DWS and video-enhanced SPT measurements of the compliance plateau agree quantitatively with compliance measured independently using classical mechanical rheometry for a viscous oil and for a solution of flexible polymers. This paper extends the use of DWS and single-particle-tracking to probe the local mechanical properties of polymer networks, shows for the first time the proportionality between mean square displacement and local creep compliance, and therefore presents a new, direct way to extract the viscoelastic properties of polymer systems and complex fluids.}