Unsteady rheometry: can we characterize weak gels with a controlled stress rheometer?

This paper is concerned with the influence of apparatus inertia effects in controlled stress rheometry. As evidenced on creep experiments, the coexistence of apparatus inertia and viscoelasticity leads to a coupling frequency. For weak gels, this coupling frequency is typically between 1 and 100 Hz. Therefore, frequency sweeps around and above this coupling frequency also corresponds to an effective shear stress sweep evolution due to a non-trivial resonant effect. In other words, frequency sweep experiments are not made at constant shear stress. The detailed modelling and analysis of this inertia effect on a typical weak gel shows a clear and fundamental limitation for its characterization using a controlled stress rheometer. Also, alternative approaches to standard rheometer software analysis are proposed to take this coupling effect into account. This paper was presented at the 3rd Annual Rheology Conference, AERC 2006, April 27-29, 2006, Crete, Greece.     

Using instrumental inertia in controlled stress rheometry

We describe a new method for characterizing the non-linear behavior of complex fluids at both small and large deformations. For creep measurements, we use the coupling between the instrumental inertia and the material‘s elasticity to follow the rheological behavior of a solution of iota carrageenan both above and below the yield stress. It is shown that this coupling selectively excites one particular frequency of the relaxation spectrum. An analytical calculation is used to quantify the non-linear behavior near the yield stress. The “free“ oscillations observed during the first few seconds allow us to choose the most appropriate mechanical model. Comparison with experiment shows that even above the yield stress, a linear model can still give independently reliable information about the changes in each element of the mechanical model. A comparison of free and forced oscillations in controlled stress rheometry shows both experimentally and theoretically the conditions under which the use of free oscillations is advantageous. Received: 4 September1997 Accepted: 13 January 1998     

Investigation of the linear flow regime of commercial polymers by numerical conversion of MVM creep measurements

A new experimental and numerical method has been developed to characterize the terminal flow behavior of polydisperse, commercial grade polymer melts over a wide dynamic range of time/frequency scales. Experimentally, an MVM rheometer specifically designed for long time scale (t 10⁴ s) creep measurements is used to measure the creep compliance of three commercial polymers: two high density polyethylenes and one polystyrene. The long time scale MVM creep data are complemented in the short time scale regime by creep data from an industrial plate-plate rheometer. The time-dependent creep data is combined and converted to a discrete retardation spectra using a nonlinear regularization algorithm to address the ill-posed nature of the interconversion. The retardation spectrum is analytically converted to dynamic moduli and compared with independently measured dynamic moduli. In the overlapping frequency region, calculations and measurements show excellent agreement and the combined data span a much larger dynamic range than either independent data set. The calculated and measured dynamic moduli data are combined and a retardation spectrum with a vastly expanded dynamic range is generated. Combining long time scale MVM creep compliance data and dynamic moduli data exploits the intrinsic sensitivities of controlled strain and controlled stress rheological experiments and is a powerful means to greatly expand the experimentally accessible dynamic range of time/frequency. This approach is particularly useful for commercial polymers with broad molecular weight distributions and commensurately large distributions of relaxation times.     

Gelation of a radiation crosslinked model polyethylene

A radiation crosslinked model linear low-density polyethylene (LLDPE) exhibits power-law relaxation,G(t) =St ^–n at its gel point (GP). The relaxation exponent has a value of about 0.46. The relaxation behavior is dominated by power laws, not only directly at GP, but in a very broad vicinity of GP and in a frequency window, which narrows with distance from the gel point. The power law exponent decreases with increasing radiation dose (increasing extent of crosslinking). Independent measurements of the gel fraction and the molecular-weight distribution of the radiated samples' soluble fraction support the rheological observations.Delivered as a Keynote Lecture at the Golden Jubilee Conference of the British Society of Rheology and Third European Rheology Conference, Edinburgh, 3–7 September, 1990.     

Viscoelastic properties of dibenzylidene sorbitol (DBS) physical gels at high frequencies

The rheological behavior of Dibenzylidene D-Sorbitol (DBS) gels formed in ethylene glycol, glycerol, mineral oil, ethanol, and chlorobenzene was studied using oscillatory squeezing flow viscometry. The storage (G ^′) and loss (G ^″) moduli were measured as a function of gellant concentration (0.5–2 w/w) and type of solvent. As expected greater values of gel strength (G ^′) were observed for gels containing higher concentrations of DBS. In addition, both storage and loss moduli of 2% systems were mostly frequency independent over the studied range, whereas 0.5% gels did exhibit some degree of dependence. We also found that the solvent plays an important role in the properties of the gels. Among the parameters that affect the viscoelastic properties of DBS gels, the solvent polarity and its ability to form hydrogen bonding may have significant effects on the gel rheology.     

The effect of entanglements on the rheological behavior of polybutadiene critical gels

We investigated the stress relaxation behavior of critical gels originating from six nearly monodisperse, highly entangled polybutadiene melts of different molecular weight from 18000 to 97 000 g/mole. The polymers were vulcanized by a hydrosilation reaction which takes place nearly exclusively at the pendant 1,2-vinyl sites distributed randomly along the polybutadiene chain. The BSW spectrum represents the relaxation of the initial uncrosslinked precursor. A characteristic parameter is the longest relaxation time of the precursor. Crosslinking increases this longest time even further. Surprisingly, the relaxation spectrum of the precursor is not altered much by the crosslinking except for an additional long time behavior. At the gel point (critical gel), this long time behavior is self-similar. It follows the typical power law as described by the Chambon-Winter gel equation, G(t) = St ^–n , in the terminal zone. The critical relaxation exponent was found to be close to n = 0.5 over a range of stoichiometric ratios and for all precursor molecular weights analyzed. A new scaling relationship was found between the gel stiffness, S, and the precursor molecular weight of the form: S M _w ^zn , where exponent z from the zero shear viscosity-molecular weight relationship, ₀ M _w ^z , is commonly found to be z = 3.3 – 3.6.     

Stress relaxation and creep of polymer gels in solvent under uniaxial and biaxial deformations

Stress relaxation and creep of polymer gels in solvent under various deformation modes such as uniaxial, strip-biaxial, and equibiaxial were theoretically investigated. The magnitudes of relaxed stress and the creep at equilibrium under each deformation mode were derived by a thermodynamic consideration of gel system. Combining a constitutive equation of gel with the equation of motion of polymer network, the stress and strain under each deformation mode have been formulated. The theory proposed here was applied to the rectangular gels under various deformations to calculate the stress relaxation and creep behavior of polymer networks in solvent. Two methods different in treatment of swelling under the constant strain or stress were employed for the calculation: one is based on the assumption that the swelling proceeds isotropically, and the other considers the anisotropic swelling process. The results obtained by the two methods mainly differ in the diffusion mode determining the swelling behavior. The possibility of undershoot of relative strain in load-free direction in the creep is also expected.     

The evolution of linear viscoelasticity during the vulcanization of polyethylene

The evolution of linear viscoelasticity during the vulcanization of polyethylene is studied through the gel point. The material in the vicinity of the gel point is described by two scaling laws: one characterizes the viscoelasticity at the critical point and a second characterizes the evolution of viscoelasticity near the gel point. Time Resolved Mechanical Spectroscopy is used to observe both scaling phenomena. The material at the gel point displays power law relaxation over five decades of time with a power-law relaxation exponent equal to 0.32. This study conforms with previous findings that this exponent is composition-dependent. The longest relaxation time diverges in the vicinity of the gel point as _max |p _c - p| ^–1/, and we find = 0.2. This result conforms with previous reports that this exponent may be independent of composition. The Arrhenius flow activation energy for this material undergoes three-fold changes during crosslinking up to the gel point. A single-adjustable-parameter stretched-exponential-power law relaxation function is an inadequate representation of crosslinked materials over any significant range of extent of the reaction up to the gel point.     

A simple extension apparatus for dilute gelatin gels

Summary An apparatus has been designed and constructed which allows the rupture properties of dilute gelatin gels to be studied. Breaking stresses in the range 1.7×10⁴ to 1×10⁶ dyne cm^–2 have been determined; extension at break varied between 46 and 155%.The initial slope of the load-extension graph is linear and can be used to calculateYoung's modulus and can be related to Bloom jelly strength.

---

Zusammenfassung Es ist ein Apparat konstruiert und gebaut worden, welcher die Untersuchung der Brucheigenschaften von verdünnten Gelatine-Gelen gestattet. Die Bruchspannungen im Bereich von 1,7×10⁴ bis 1×10⁶ Dyn cm^–2 sind bestimmt worden. Die Dehnung am Bruchpunkt variierte zwischen 46 und 155%.Das Anfangsstück (ansteigend) der Belastungs/Dehnungskurve ist linear und kann zur Berechnung desYoungschen Moduls verwendet und mit der Gallertfestigkeit nachBloom in Beziehung gesetzt werden.

     

Incipient behaviour of gelatin gels

The present work describes isothermal (25°C) gelation measurements for gelatin gels over a range of concentrations. Methods for the estimation of the gel time are discussed, and data compared with two recent models. The isothermal time growth of modulus is also investigated, and the superposition of such data discussed. For concentrations close to the critical gel concentration C ₀ there are significant deviations in the latter case. These may be related to the approach to the biphasic region of the phase diagram suggested by other workers.     

Rheological properties of skim milk gels at various temperatures; interrelation between the dynamic moduli and the relaxation modulus

The rheological properties of rennet-induced skim milk gels were determined by two methods, i.e., via stress relaxation and dynamic tests. The stress relaxation modulusG _c (t) was calculated from the dynamic moduliG andG by using a simple approximation formula and by means of a more complex procedure, via calculation of the relaxation spectrum. Either calculation method gave the same results forG _c (t). The magnitude of the relaxation modulus obtained from the stress relaxation experiments was 10% to 20% lower than that calculated from the dynamic tests.Rennet-induced skim milk gels did not show an equilibrium modulus. An increase in temperature in the range from 20° to 35 °C resulted in lower moduli at a given time scale and faster relaxation. Dynamic measurements were also performed on acid-induced skim milk gels at various temperatures andG _c (t) was calculated. The moduli of the acid-induced gels were higher than those of the rennet-induced gels and a kind of permanent network seemed to exist, also at higher temperatures. G storage shear modulus,N·m^–2; - G loss shear modulus,N·m^–2; - G _c calculated storage shear modulus,N·m^–2; - G _c calculated loss shear modulus,N·m^–2; - G _e equilibrium shear modulus,N·m^–2; - G _ec calculated equilibrium shear modulus,N·m^–2; - G(t) relaxation shear modulus,N·m^–2; - G _c (t) calculated relaxation shear modulus,N·m^–2; - G ^*(t) pseudo relaxation shear modulus,N·m^–2; - H relaxation spectrum,N·m^–2; - t time,s; - relaxation time,s; - angular frequency, rad·s^–1. Partly presented at the Conference on Rheology of Food, Pharmaceutical and Biological Materials, Warwick, UK, September 13–15, 1989 [33].     

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 torsional creep apparatus

Summary This paper describes a torsional creep apparatus for the determination of the creep compliance in shear as a function of time. The instrument is suitable for the measurement of compliances below 5·10^–8 m²/N, in the time range between 1 and 10⁵ seconds. Within five minutes and with an accuracy of 1 C, the temperature of the specimen can be adjusted to any value between –175 and –200 C. From the torsional creep measurements can be calculated the dynamic shear modulus and the corresponding damping. The lowest damping, determined in this way, is several times 10^–3.

---

Zusammenfassung Ein Apparat zur Messung des Torsions-Kriechverhaltens wird beschrieben. Das Instrument ist zur Bestimmung von Nachgiebigkeiten kleiner als 5·10^–5 m²/N im Zeitbereich zwischen 1 und 10⁵ Sekunden geeignet. Die Temperatur der Probe läßt sich innerhalb 5 Minuten und mit einer Genauigkeit von 1 C auf jeden Wert zwischen –175 und –200 C einstellen. Aus der gemessenen Kriechfunktion können der dynamische Schubmodul und die dynamische Dämpfung berechnet werden. Eine untere Grenze für die Dämpfung, die man auf diese Weise noch bestimmen kann, ist einige Male 10^–3.

---

---

The work was partly sponsored by the Office of Naval Research, Washington, D. C., under contract number N 62558-3884.     

On stability of the Hill equation with damping

We consider the Hill equation with damping describing the parametric oscillations of a torsional pendulum excited by varying the moment of inertia of the rotating body. Using the method of a small parameter, we analytically calculate a fundamental system of solutions of this equation in the form of power series in the excitation amplitude with accuracy O(²) and verify conditions for its stability. In the first-order approximation in , we prove that the resonance domain exists only if the excitation frequency is sufficiently close to the double natural frequency of the pendulum; the corresponding equation of the stability boundary is obtained.Published in Neliniini Kolyvannya, Vol. 7, No. 2, pp. 169–179, April–June, 2004.     

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     

Free oscillatory shear measurements —an interesting application of constant stress rheometers in the creep mode

A constant stress rheometer in the creep mode was used to perform free oscillatory shear measurements on soft solids. The results obtained are in good to excellent agreement compared to forced oscillatory shear measurement data. Depending on the rheologic properties of the sample and the moment of inertia of the rotating device of the measurement system, free oscillations are suited to confirm or supplemented forced oscillatory measurement results.Dedicated to Professor W. Nitsch on the occasion of his 60^th birthday.     

A Measuring System for Bulk and Shear Characterization of Polymers

This paper describes a novel measuring system for investigating the influence of pressure and temperature on the mechanical properties of time-dependent polymer materials. The system can measure the volume and the shear relaxation moduli of solid polymer specimens simultaneously subjected to temperatures from −50 to +120°C with a precision of ±0.01°C, and pressures from atmospheric to 500 MPa with a precision of ±0.1 MPa. The paper demonstrates the measuring capabilities of the apparatus. For poly(vinyl) acetate (PVAc) are presented sample measurements of the shear relaxation modulus as function of time, pressure and temperature; specific volume; the bulk creep compliance; the coefficient of thermal expansion; the bulk modulus; and the pressure drop experiments which simulate conditions to which a material is exposed during the injection molding process. The shear moduli may be measured in the range from 1 to 4,000 MPa with the relative error of 3%.The error of volumetric measurements is 0.05%, which corresponds to 0.00005 cm³/g. In all cases results are shown as measured, no additional smoothing or filtering was employed. This paper is dedicated to Professor Nicholas W. Tschoegl on the occasion of his 87th birthday, for his contributions to the field of time-dependent bulk properties of polymeric materials.     

Thermal contraction and volume relaxation of amorphous polymers

Two experimental techniques are described for the determination of the change of specific volume of polymers with temperature and aging time, which allow measurements between – 160 °C and + 200 °C. Four technical amorphous polymers, PS, PVC, PMMA and PC have been investigated. Volume-temperature curves under constant rate of cooling are presented and interpreted with respect to relaxation processes known from other physical investigations. The rate dependence of dilatometric glass transition temperatures is compared with the time dependence of rheometric glass transition temperatures from shear creep data. Volume relaxation data at constant aging temperature are presented. Aging is found to proceed until very low temperatures in the glassy state for e.g. PMMA.For polystyrene, a comparison is made between the predictions of a very simple theory of volume relaxation due to Kovacs with experimental data, using additional information from volume temperature curves and the time temperature shift of the shear creep transition. The theory predicts a rate of volume relaxation which is much lower than that found by experiment.     

The development of rheometry for strain-sensitive gelling systems and its application in a study of fibrin–thrombin gel formation

This paper reports simulated sequential frequency sweep data which have been reconstructed from time resolved viscoelastic data obtained by Fourier transform mechanical spectroscopy. Comparisons of the results show that the recording of anomalous values of the stress relaxation power law exponent α at the Gel Point under ‘rapid’ gelling conditions may be due to inappropriate rheological techniques. An appropriate rheometrical criterion is established for the application of sequential frequency sweeps in order to obtain accurate values of α in the formation of strain-sensitive, rapidly formed gels. Furthermore, using appropriate rheometry, we report values of α for fibrin–thrombin gels formed by the addition of thrombin to a physiologically relevant level of human fibrinogen, and relate these values to the microstructure of the fibrin gel network in terms of a fractal dimension. The present study is the first to report a modification of the fractal characteristics of incipient clots in fibrin–thrombin gels due to the availability of thrombin. This work confirms the hypothesis that the self-similar (fractal) stress relaxation behaviour recorded at the Gel Point of samples of coagulating blood (Evans et al. 2010a, b) is associated with the microstructural characteristics of the incipient blood clot’s fibrin network.     

Drilling fluid shear stress overshoot behavior

Shear stress overshoot behavior was studied in four drilling fluid systems and ten bentonite dispersions. These overshoot properties, also described by the American Petroleum Institute as gel strengths, were measured after gelation times of 10 s to 24 h at temperatures of 20–80 °C. Two different rheometers were used to measure overshoot behavior. Gel strength development with time followed a first-order model. Gel development rates at 20 °C varied from 0.005 to 0.01 min^–1 for drilling fluid systems and from 0.0004 to 0.02 min^–1 for bentonite dispersions. Increasing the gelation temperature for each drilling fluid system caused an increase in the gelling rate constant. Comparison of gel strengths in bentonite dispersions were made using a Fann 35 A viscometer and a Weissenberg Rheogoniometer. Higher gel strength values observed using the Rheogoniometer were believed to be due to differences in instrument spring stiffness and fixture inertia.