In-line measurement of rheological properties of polymer melts

Shear viscosity (), first normal stress difference (N ₁), and extensional viscosity ( _E ) of polymer melts measured under processing conditions are important in process modeling, quality control, and process control. A slit rheometer that could simultaneously measure , N ₁, and the planar extensional viscosity ( _p ) was designed and tested by attaching it in-line to a laboratory model single-screw extruder. A tube (circular cross-section) rheometer to measure and the uniaxial extensional viscosity ( _u ) simultaneously was also designed and tested. Two commercial grades of LDPE (low density polyethylene) with melt index values of 6 and 12 were used as test materials for the study. Exit and hole pressure methods were used to estimate N ₁, and the entrance pressure drop method using the analyses of Cogswell, Binding, and Gibson (the last analysis used with the axisymmetric case only) was used to estimate _E .The hole pressure method was considered better than the exit pressure method to estimate N ₁ (due to the greater susceptibility of the latter to experimental errors). From the hole pressure method N ₁ was obtained from 100 kPa to 500 kPa over a range of shear rates from 40 s^–1 to 700 s^–1. Among the analyses used to estimate the extensional viscosity, Cogswell's is recommended due to its simpler equations without loss of much information compared to the other analyses. The range of extension rates achieved was 1 to 30 s^–1. The combination of the hole pressure and entrance pressure drop methods in a slit rheometer is a feasible design for a process rheometer, allowing the simultaneous measurement of the shear viscosity, first normal stress difference and planar extensional viscosity under processing conditions. Similarly, combining the entrance pressure drop measurements with a tube rheometer is also feasible and convenient.     

Viscoelastic plastic rheological model for particle filled polymer melts

A viscoelastic plastic model for suspension of small particles in polymer melts has been developed. In this model, the total stress is assumed to be the sum of stress in the polymer matrix and the filler network. A nonlinear viscoelastic model along with a yield criterion were used to represent the stresses in the polymer matrix and the filler network, respectively. The yield function is defined in terms of differential equations with an internal parameter. The internal parameter models the evolution of structure changes during floc rupture and restoration. The theoretical results were obtained for steady and oscillatory shear flow and compared with experimental data for particle filled thermoplastic melt. The experimental data included the steady state shear strress over a wide range of shear rates, the transient stress in a start up shear flow, stress relaxation after cessation of a steady state shear flow, the step shear and the oscillatory shear flow at various amplitudes.     

Dynamic and transient rheological properties of glass filled polymer melts

The dynamic and transient rheological properties of a low density polyethylene melt and a plasticized polyvinylchloride melt filled with glass beads were measured at 200 °C and 180 °C respectively in a modified Weissenberg Rheogoniometer R-17. Its main modification consisted of the use of a piezoelectric transducer instead of the conventional torsion bar, and of the interfacing of a microcomputer Apple II plus to the Rheogoniometer for data acquisition and analysis. The glass beads were pretreated with silane and titanate coupling agents to observe the effect of the chemical modification of the polymer filler interface on these properties. It was observed that both the dynamic viscosity and the storage modulus increased with the weight fraction, but this last parameter did appreciably affect the stress growth and stress relaxation curves of the polymeric matrices at low shear rates. The effect of coupling agents on these properties was varied.     

Utility of parallel-plate plastometer for rheological study of filled polymer melts

Summary In order to test the applicability of the parallel-plate plastometer to the rheological study of filled polymer melts, viscous properties of polyethylene and polystyrene filled with particles and short fibers were measured by this apparatus. Flow curves in the low range of shear rate were obtained by approximating the materials as Ostwald-de-Waele liquids. Up to about 0.5 of the volume fraction of the filler,, there was no problem and the utility of the apparatus was clearly shown. When increased more, the results were not necessarily self-consistent because the flow became unstable. Experiments on samples with high showed that the unstable flow resulted in an apparent yield stress which agreed with that estimated from the Casson's plots.

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Zusammenfassung Um die Anwendbarkeit des Parallel-Platten-Plastometers für die rheologische Untersuchung von Schmelzen gefüllter Polymerer zu prüfen, wurden mit einem solchen Gerät die viskosen Eigenschaften von Polyäthylen und Polystyrol-Proben gemessen, die mit Teilchen oder kurzen Fasern gefüllt waren. Im Bereich niedriger Schergeschwindigkeiten ließen sich die Fließkurven mit Hilfe der Ostwald-de Waele-Gleichung approximieren. Bis zu einer Füllstoff-Volumenkonzentration von etwa 50% traten keine Komplikationen auf, so daß die Eignung des Geräts für diese Fälle klar erwiesen werden konnte. Wenn jedoch noch weiter gesteigert wurde, waren die Ergebnisse durchweg nicht mehr widerspruchsfrei, da dann der Fließvorgang instabil wurde. Versuche mit hohen Volumenkonzentrationen zeigten, daß die Fließinstabilität eine scheinbare Fließgrenze zur Folge hat, deren Größe mit dem aus einer Casson-Darstellung abgeschätzten Wert in etwa übereinstimmt.

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With 8 figures and 1 table     

Peculiarities of rheological behavior of filled polymer melts in uniaxial stretching

The peculiarities of theological behavior of filled polymer melts in uniaxial extension in a wide range of strain rates (from 2× 10^–5 to 1 × 10^–1 s^–1) have been studied. Linear polyethylene and 1,4-polybutadiene containing up to 21.5 vol.% of carbon black, silica, calcium carbonate or glass fibers were used. It has been found that the transition from uniform to nonunion stretching due to the neck formation is typical of all specimen compositions, when they approach steady-state straining. Depending on the structure and rheological characteristics of the compositions general conditions for this transition have been established. The general regularities for varying the rheological characteristics of filled polymers in the course of their uniform stretching have been recognized. These regularities depend on the molecular characteristics of the polymer matrix and the presence in the compositions of the structural framework of high disperse filler or the network formed by the entangled fibers. Using polyethylene compositions it has been shown that the introduction of small amounts of disperse or fibrous fillers can give rise to acceleration of the relaxation process in filled polymers.     

Towards a rheological classification of flow induced crystallization experiments of polymer melts

Departing from molecular based rheology and rubber theory, four different flow regimes are identified associated to (1) the equilibrium configuration of the chains, (2) orientation of the contour path, (3) stretching of the contour path, and (4) rotational isomerization and a deviation from the Gaussian configuration of the polymer chain under strong stretching conditions. The influence of the ordering of the polymer chains on the enhanced point nucleation, from which spherulites grow, and on fibrous nucleation, from which the shish-kebab structure develops, is discussed in terms of kinetic and thermodynamic processes. The transitions between the different flow regimes, and the associated physical processes governing the flow induced crystallization process, are defined by Deborah numbers based on the reptation and stretching time of the chain, respectively, as well as a critical chain stretch. An evaluation of flow induced crystallization experiments reported in the literature performed in shear, uniaxial and planar elongational flows quantitatively illustrates that the transition from an enhanced nucleation rate of spherulites towards the development of the shish-kebab structure correlates with the transition from the orientation of the chain segments to the rotational isomerization of the high molecular weight chains in the melt. For one particular case this correlation is quantified by coupling the wide angle X-ray diffraction and birefringence measurements of the crystallization process to numerical simulations of the chain stretch of the high molecular weight chains using the extended Pom-Pom model in a cross-slot flow.     

Measurement of rheological and ultrasonic properties of food and synthetic polymer melts

A slit die viscometer has been used in conjunction with a co-rotating twin screw extruder to study the rheological behaviour of maize grits, potato powder and low density polyethylene, as a function of feed rate, screw speed and temperature. The shear viscosity of both maize and potato decreased with increasing feed rate. Increasing the temperature or screw speed at any given feed rate also reduced the viscosity. The ultrasonic velocity through the material has also been shown to be sensitive to the extruder operating conditions. Overall, the ultrasonic velocity decreased as screw speed and temperature increased and feed rate decreased.     

Constrained inversion of rheological data to molecular weight distribution for polymer melts

In this paper an inverse method for converting linear viscoelastic properties of polymer melts into molecular weight distribution (MWD) is presented. It relies on the use of the double reptation mixing rule and the Generalized Exponential function (GEX) describing the MWD; i.e. an a priori knowledge of the nature of the solution is postulated following previous attempts (Nobile et al. 1996b; Nobile and Cocchini 1999, 2001) where the Tuminello kernel was adopted to get analytical results. Here a MWD dependent kernel, recently proposed in the literature (Thimm et al. 1999), is applied, still obtaining analytical results but avoiding the consequences of the crude step-like approximation of the Tuminello kernel. In particular, the steady-state compliance and the elastic modulus at small frequencies do not suffer from the underestimation discussed in the cited papers. At high frequencies, the Rouse modes contribution has also been considered in the model. The present approach turns out to be simpler, more accurate and robust than the previous one. Moreover, the results can be more clearly stated in the frame of the theory. Careful and easy-to-use formulas for the zero-shear-rate viscosity, the steady-state compliance, the modulus and frequency at the G'-G' crossover, have been obtained as a function of the MW averages. As expected, the comparison among the results obtained through the two kernels explicitly confirms the minor effects of the kernel on the rheology of polydisperse polymers. Concerning the molecular weight averages values, the predictions obtained from the dynamic moduli measurements typically differ less than 10% from those obtained from the reference measurements obtained with other methods.     

Representation of the rheological properties of polymer melts in terms of complex fluidity

In dynamic rheological experiments melt behavior is usually expressed in terms of complex viscosity ^* () or complex modulusG ^* (). In contrast, we attempted to use the complex fluidity ^* () = 1/µ ^* () to represent this behavior. The main interest is to simplify the complex-plane diagram and to simplify the determination of fundamental parameters such as the Newtonian viscosity or the parameter of relaxation-time distribution when a Cole-Cole type distribution can be applied. ^* () complex shear viscosity - () real part of the complex viscosity - () imaginary part of the complex viscosity - G ^* () complex shear modulus - G() storage modulus in shear - G() loss modulus in shear - J ^* () complex shear compliance - J() storage compliance in shear - J() loss compliance in shear - shear strain - rate of strain - angular frequency (rad/s) - shear stress - loss angle - ^* () complex shear fluidity - () real part of the complex fluidity - () imaginary part of the complex fluidity - ₀ zero-viscosity - ₀ average relaxation time - h parameter of relaxation-time distribution     

Numerical modeling of the forming of fiber bundles from polymer melts

A mathematical model for the process of forming of synthetic fibers moving as a bundle is formulated. Three main versions are considered: forming of exposed bundles, forming in shafts with blowing, and stretching of fibers by means of an ejector. Low and high-speed forming regimes are also considered within the framework of the Maxwell model of a viscoelastic fluid. The calculations performed showed that the parameters of the fiber bundle produced depend on the method of forming used and on the local conditions in high-speed stretching, accompanied by oriented crystallization. Dnepropetrovsk State University, Dnepropetrovsk 320625. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 1, pp. 184–192, January–February, 1999.     

Flow birefringence of polymer melts: Application to the investigation of time dependent rheological properties

Summary Transient stresses including normal stresses, which are developed in a polymer melt by a suddenly imposed constant rate of shear, are investigated by mechanical measurement and, indirectly, with the aid of the flow birefringence technique. For the latter purpose use is made of the so-called stress-optical law, which is carefully checked.It appears that the essentially linear model of the rubberlike liquid, as proposed byLodge, is capable of describing the behaviour of polymer melts rather well, if the applied total shear does not exceed unity. In order to describe also steady state values of the stresses successfully, one should extend measurements to extremely low shear rates.These statements are verified with the aid of a method which was originally designed bySchwarzl andStruik for the practical calculation of interrelations between linear viscoelastic functions. In the present paper dynamic shear moduli are used as reference functions.

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Zusammenfassung Mit der Zeit anwachsende Spannungen, darunter auch Normalspannungen, wie sie sich nach dem plötzlichen Anlegen einer konstanten Schergeschwindigkeit in einer Polymerschmelze entwickeln, werden mit Hilfe mechanischer Messungen und indirekt mit Hilfe der Strömungsdoppelbrechung untersucht. Für den letzteren Zweck wird das sogenannte spannungsoptische Gesetz herangezogen, dessen Gültigkeit sorgfältig überprüft wird.Es ergibt sich, daß das im Wesen lineare Modell der gummiartigen Flüssigkeit, wie es vonLodge vorgeschlagen wurde, sich recht gut zur Beschreibung des Verhaltens von Polymerschmelzen eignet, solange der im ganzen angelegte Schub den Wert Eins nicht überschreitet. Um auch stationäre Werte der Spannungen in die Beschreibung erfolgreich einzubeziehen, sollte man die Messungen bis zu extrem niedrigen Schergeschwindigkeiten ausdehnen.Die gemachten Feststellungen werden mit Hilfe einer Methode verifiziert, die vonSchwarzl undStruik ursprünglich für die praktische Berechnung von Beziehungen zwischen Zustandsfunktionen entwickelt wurde, die dem linear viskoelastischen Verhalten entsprechen. In der vorliegenden Veröffentlichung dienen die dynamischen Schubmoduln als Bezugsfunktionen.

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a _T shift factor - B _ij Finger deformation tensor - C stress-optical coefficient, (m²/N) - f (p _jl ) undetermined scalar function - G shear modulus, (N/m²) - G(t) time dependent shear modulus, (N/m²) - G() shear storage modulus, (N/m²) - G() shear loss modulus, (N/m²) - G _r reduced shear storage modulus, (N/m²) - G _r reduced shear loss modulus, (N/m²) - H() shear relaxation time spectrum, (N/m²) - k Boltzmann constant, (Nm/°K) - n _ik refractive index tensor - p undetermined hydrostatic pressure, (N/m²) - p _ij ,p _ik stress tensor, (N/m²) - p ₂₁ shear stress, (N/m²) - p ₁₁ –p ₂₂ first normal stress difference, (N/m²) - p ₂₂ –p ₃₃ second normal stress difference, (N/m²) - q shear rate, (s^–1) - t, t time, (s) - T absolute temperature, (°K) - T ₀ reference temperature, (°K) - x the ratiot/ - x position vector of a material point after deformation, (m) - x position vector of a material point before deformation, (m) - ₀, ₁ constants in eq. [37] - ₀, ₁ constants in eq. [37] - shear deformation - (t, t) time dependent shear deformation - _ij unity tensor - n flow birefringence in the 1–2 plane - (q) non-Newtonian shear viscosity, (N s/m²) - ^* () complex dynamic viscosity, (N s/m²) - | ^* ()| absolute value of complex dynamic viscosity, (N s/m²) - () real part of complex dynamic viscosity, (N s/m²) - () imaginary part of complex dynamic viscosity, (N s/m²) - (t — t) memory function, (N/m² · s) - v number of effective chains per unit of volume, (m^–3) - temperature dependent density, (kg/m³) - ₀ density at reference temperatureT ₀, (kg/m³) - relaxation time, (s) - integration variable, (s) - (x) approximate intensity function - ₁ (x) error function - extinction angle - _m orientation angle of the stress ellipsoid - circular frequency, (s^–1) - 1 direction of flow - 2 direction of the velocity gradient - 3 indifferent direction - t time dependence The present investigation has been carried out under the auspices of the Netherlands Organization for the Advancement of Pure Research (Z. W. O.).North Atlantic Treaty Organization Science Post Doctoral Fellow.Research Fellow, Delft University of Technology.With 11 figures and 2 tables     

Quasi-linear rheological behaviour of polymer melts: Comparison between mechanical and improved flow birefringence measurements

Summary The latest improvement of the rotor unit (cone- and plate-type) for the measurement of the flow-birefringence of polymer melts is described. As a consequence of this improvement, the influence of parasitic birefringences, as produced in the corners near the windows, is minimized. It is shown that the limitation of the range of available rates of shear, as experienced with previous versions of the apparatus, was due to those parasitic birefringences which predominated at rates of shear where the main birefringence caused an optical path difference of one, two or more whole wavelengths (fringes) and was, as a consequence, virtually zero. This explains why great troubles were experienced in particular with polymers possessing a high back-bone anisotropy. The validity of the linear stress-optical rule up to shear rates, where melt fracture occurs in capillary flow, is well understood in terms of the wriggling motion of chain molecules. Comparison with direct mechanical measurements was very satisfactory.

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Zusammenfassung Eine Beschreibung der jüngsten Verbesserung an der Rotorzelle (Kegel-Platte-Typ) für die Messung der Strömungsdoppelbrechung polymerer Schmelzen wird gegeben. Als Folge dieser Verbesserung wird der Einfluß der parasitären Doppelbrechung, die in den Ecken neben den Fenstern auftritt, minimalisiert. Es wird gezeigt, daß die Begrenzung des Bereiches der verfügbaren Schergeschwindigkeiten, wie sie in früheren Versionen des Apparates gefunden wurde, auf das Vorherrschen der parasitären Doppelbrechungen bei Schergeschwindigkeiten zurückzuführen ist, bei denen die Hauptdoppelbrechung optische Weglängen von einer, zwei oder mehreren ganzen Wellenlängen verursacht und daher effektiv null wird. Dies erklärt, warum große Schwierigkeiten gerade bei solchen Polymeren auftraten, die eine stärkere Kettenanisotropie aufweisen. Die Gültigkeit der linearen spannungsoptischen Regel bis zu Schergeschwindigkeiten, bei denen in der Kapillaren Schmelzbruch auftritt, kann man aufgrund der Ringelbewegung der Makromoleküle gut verstehen. Ein Vergleich mit direkten mechanischen Messungen ergab sehr befriedigende Ergebnisse.

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With 8 figures and 2 tables     

Assessing the practical utility of the hole-pressure method for the in-line rheological characterization of polymer melts

The most promising method capable of providing accurate measurements of the first and second normal-stress differences in shear flows at shear rates typical of polymer processing is the so-called hole-pressure method, but its use has not been as widespread as would be expected, namely due to the experimental difficulties associated with performing such experiments accurately. In this work, we use a small-scale modular slit die to assess the practical utility of the method for in-line monitoring of polymer melt flow. We provide a quantitative analysis of intrinsic error sources and use state-of-the-art data acquisition tools to minimize errors associated with pressure transducers. Our results demonstrate that the method can be used to accurately measure the viscosity and first normal-stress difference in melts but probably not the second normal-stress difference because the intrinsic errors are too high, even when the influence of all the potential error sources is minimized or eliminated.     

An extension of the Curtiss-Bird theory for the rheological properties of concentrated polymer solutions and melts

The constitutive equation for the elastic reptating rope model is considered in the limit at which the rope becomes inextensible. It is shown that in this limit a constitutive equation is obtained which differs essentially from the Curtiss-Bird theory if the correlation length is much smaller than the contour length of the rope. This constitutive equation contains an extra term related to correlations between segments. The differences between this result and the Curtiss-Bird theory are considered, and consequences are indicated.     

Finite rise time step strain modeling of nearly monodisperse polymer melts and solutions

The step shear strain experiment is one of the fundamental transient tests used to characterize the rheology of viscoelastic polymer melts and solutions. Many melts and solutions exhibit homogeneous deformation and stress relaxation; in these cases the transient dynamics can be modeled by completely ignoring momentum effects and imposing singular kinematics. Recently, however, it has been observed that there are certain classes of nearly monodisperse melts and solutions that exhibit anomalous nonhomogeneous deformation and stress relaxation (Morrison and Larson (1990), Larson, Khan, and Raju (1988), Vrentas and Graessley (1982), and Osaki and Kurata (1980)). We demonstrate that, for these classes, a finite rise time must be incorporated, some source of inhomogeneity must be present, and a small amount of added Newtonian viscosity is necessary. We examine five nonlinear and quasilinear models; the Johnson-Segalman, Phan Thien Tanner, Giesekus, White-Metzner, and Larson models. We determine which mathematical features of the models are necessary and/or sufficient to describe the observed experimental behavior.     

A rheological measurement of three polyethylene melts

Summary Three polyethylene melts were measured on aWeissenberg Rheogoniometer and a capillary rheometer. A method of extrapolating zero shear values of viscosity from the capillary data, described bySabia, is used and the results compared with those measured by the Rheogoniometer.Paper to be given to a Meeting of the British Society of Rheology on September 28th, 1964.     

Nonisothermal flow of polymer melts

The Curtiss-Bird theory for polymer melts is modified to allow the temperature to be a function of time. The resulting constitutive equation involves integrals over the temperature history of the polymer. The predictions of this constitutive equation for the inception of uniaxial elongation with simultaneous cooling at a constant rate are calculated and compared with the experimental data of Matsumoto and Bogue. Qualitative agreement between the theory and the data is obtained when the link tension coefficient in the Curtiss-Bird theory is not equal to zero, but such agreement cannot be obtained when this coefficient is equal to zero. The extension of the theory to include spatially varying temperature is also discussed.