Pressure dependent viscosity and dissipative heating in capillary rheometry of polymer melts

In high shear rate capillary rheometry the combined effect of pressure dependent viscosity and dissipative heating becomes significant. Analytical expressions are derived to treat curved Bagley plots and throttle experiments. End effects are taken into account by using an effective length over radius ratio. The non-adiabatic case is described using a lump heat transfer coefficient ? following Hay et al. (1999). The latter enters into the dissipative heating coefficient % MathType!MTEF!2!1!+- % feaaeaart1ev0aaatCvAUfKttLearuavTnhis1MBaeXatLxBI9gBam % XvP5wqSXMqHnxAJn0BKvguHDwzZbqegm0B1jxALjhiov2DaeHbuLwB % Lnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFf % euY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9 % q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqaba % WaaqaafaaakeaacqaH1oqzdaWgaaWcbaGaemiCaahabeaakiabg2da % 9iabeg8aYnaaCaaaleqabaGaeyOeI0IaeGymaedaaOWaaeWaaeaacq % WGJbWydaWgaaWcbaGaemiCaahabeaakiabgUcaRmaalyaabaGaeu4M % dWeabaGafmyBa0MbaiaaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacq % GHsislcqaIXaqmaaaaaa!4D6C! e_p = r ^{- 1} ( c_p + L \mathord

Capillary rheometry for polymer melts revisited

Capillary rheometry provides an efficient access to high shear rate flow properties relevant for processing. An automated gas driven capillary rheometer developed at BASF enables accurate measurements at imposed wall shear stress, thus supplementing instruments operating at imposed flow rate. A simplified treatment of dissipative heating based on the assumption of a radially flat temperature profile is outlined and justified by means of finite element simulations. The combined treatment of dissipation and pressure dependent viscosity yields relations to treat throttling experiments at imposed flow rate. Throttle pressure coefficients from a long die and an orifice agree for LDPE but significantly differ for PMSAN. The effect is explained on the basis of identical pressure coefficients for shear and elongational flows, with regard to a constant stress, however. The effect of melt compressibility is negligible in practical capillary rheometry if the temperature and pressure coefficients of the melt density are by an order of magnitude smaller than those of the viscosity. Gas pressure driven instruments allow an effective determination of wall slip velocities from Mooney plots. This is of advantage for the investigation of the mechanism of additives or processing aids. Furthermore, imposed pressure experiments are pertinent to investigate the spurt effect of HDPE and to demonstrate that two different slip processes contribute to the apparent flow curve above spurt.

A small scale capillary viscometer for polymer melts

Summary A small scale capillary viscometer that is capable of measuring the flow curves of 2–3 gms of molten polymer over the shear rate range 0.4 sec^–1 to 20,000 sec^–1 is described. A piezo electric crystal pressure transducer is used to measure the pressure directly above the capillary. Flow curves are given for anionic polystyrenes and polyethylene fractions, and it is shown that the viscosities measured agree with determinations in a cone and plate viscometer.

---

Zusammenfassung Es wird ein maßstäblich kleines Kapillarviskosimeter für 2–3 g geschmolzener Polymere beschrieben, das die Durchfiußkurve in Abhängigkeit vom Scherkoeffizienten im Bereich von 0,4 sec^–1 bis 20000 sec^–1 bestimmt. Verwendet wird ein piezoelektrischer Kristall-Druckaufnehmer, der den Druck genau über der Kapillare mißt. Angeführt werden Durchflußkurven für anionische Polystyrole und Polyäthylenanteile. Es wird gezeigt, daß die gemessenen Zähigkeiten mit denen von Kegel- und Plattendichtemessern übereinstimmen.

---

---

Paper presented at the Conference on Experimental Rheology, University of Bradford, April 17–19, 1968.     

Unstable capillary flow of reinforced polymer melts

The injection capillary flow of various unfilled and glass fibre or calcium carbonate filled polypropylene and nylon 6.6 melts is studied using either a single capillary of five capillaries in series, separated by small reservoirs. Only unfilled nylon 6.6 yields instability during flow through a single capillary due to mechanochemical degradation in the capillary at extremely high shear rates above 5 × 10⁵ s^?1. It is found that only short glass fibre reinforced polypropylene yields high frequency oscillations in the reservoir pressure and extrudate diameter and has discontinuity in the flow curve when the apparent shear rate is above 4 × 10⁵ s^?1 and the flow is through multiple capillaries. Further increase in the shear rate restores the stable flow. The intensity of the oscillations and the range of shear rate during which unstable flow occurs are increased with increasing melt temperature. The mechanism of this unstable flow is investigated by studying fibre orientation at the capillary entrance and exit using mouldings simulating capillary entry-exit flows.     

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.     

Influence of pressure on end corrections in capillary rheometry

The research and modeling of the connection of polymer viscosity and the present pressure level have brought forth a variety of, often contradictory, results. While different machinery setups are used resulting in less comparison of the studies, a combined analytical and experimental approach to apply end corrections at correct pressure levels has not been presented yet. In this work, the effect of pressure on end corrections in die flow is assessed with a combination of a pressure chamber and an orifice die, and applied to capillary rheometry measurements at different pressure levels. The pressure dependence of the end corrections has been found to show good correlations with an applied modified Power-Law model. By comparing the results to regular end corrections, the error in computing the viscosity has been found to be largely independent of the shear-rate. Neglecting the influence of pressure on end corrections yields 0–12 % too high viscosities, depending on the average die pressure.     

A slit viscometer for polymer melts

Summary The construction of a new type of viscometer is described. The essential feature of this apparatus is the use of a capillary with rectangular cross-section, one side of the rectangle being at least ten times larger than the other. Membrane pressure gauges can be mounted into the capillary wall of such an apparatus without disturbing the flow pattern inside the capillary.Thus the pressure drop is measured between two points inside the capillary without a need for end corrections.The shear rate-shear stress relationship of a molten polyethylene was studied with this apparatus. The results are compared with those obtained with cylindrical capillaries.

---

Zusammenfassung Die Konstruktion eines neuen Viskosimetertyps wird beschrieben. Charakteristisch für dieses Viskosimeter ist die Verwendung einer Kapillare mit rechteckigem Querschnitt, dessen Rechteckseiten in einem Verhältnis von mindestens 101 gewählt werden. Membranmanometer können in die Kapillarwand eines derartigen Apparates eingebaut werden, ohne daß hierdurch das Strömungsprofil im Innern der Kapillare gestört wird.Auf diese Weise wird der Druckabfall zwischen zwei Stellen innerhalb der Kapillaren gemessen. Endkorrekturen werden überflüssig.Für ein geschmolzenes Polyäthylen wurde in diesem Apparat die Abhängigkeit des Geschwindigkeitsgefälles von der Schubspannung untersucht. Ein Vergleich mit den bei Verwendung zylindrischer Kapillaren erhaltenen Ergebnissen wurde angestellt.

     

A concentric-cylinder rheometer for polymer melts

Summary If one wishes to measure the viscosity of a polymer melt at high shear rates there are substantial fluid dynamical and heat transfer difficulties. Cone-plate instruments are limited because of secondary flows and because the fluid tends to leave the gap. In capillary-flow instruments, there are substantial radial temperature gradients and the possibility of flow irregularities.Similar difficulties are met in trying to study the response of melts to large-amplitude oscillatory shear, and fluid inertia must be added to the list. However, large-amplitude oscillatory shear is a test which is useful for studying non-linear viscoelasticity, and many flows of practical importance involve deformations outside the range of validity of the assumptions of linear viscoelasticity theory.A heavy duty rheometer has been designed in which shear viscosity can be measured at high shear rates, and which can also be used for large-amplitude oscillatory shear tests. The melt is sheared between concentric cylinders; the torque on the inner, stationary cylinder is monitored while the outer cylinder either rotates at steady speed or oscillates. The shear rate of frequency and amplitude are continuously variable over wide ranges.Careful consideration was given to the problems posed by hydrodynamic stability, fluid acceleration, heat generation and end effects, and the final design represents what the authors feel is a reasonable compromise between minimizing the influence of these factors and the basic practical requirement that the instrument have a reasonable cost and uncomplicated operating procedure.In large-amplitude oscillatory shear, the interpretation of the experimental results poses special problems. The stress response is periodic, but not sinusoidal so that it is not possible to apply the methods of linear viscoelasticity. A number of possibilities suggest themselves, but it has been concluded that the best method of representation involves the plotting of stress versus strain or rate-of-strain. This results in closed curves which have distinctive shapes depending on the basic nature of the fluid response.

---

Zusammenfassung Bei der Viskositätsmessung an Polymerschmelzen unter hohen Schergeschwindigkeiten treten strömungsdynamische und Wärmeübergangsprobleme in Erscheinung. Kegel-Platte-Rheometer sind nur bei relativ geringen Deformationen brauchbar, weil bei höherer Scherbeanspruchung die Meßsubstanz infolge Sekundärströmungserscheinungen bzw. Trägheitskräften aus dem Meßspalt austritt. Bei Kapillarrheometern ergeben sich unter hohen Scherbedingungen gravierende radiale Temperaturgradienten und Fließinstabilitäten.Die gleichen Probleme finden sich, wenn das Verhalten von Schmelzen bei oszillatorischer Beanspruchung mit großer Scheramplitude untersucht werden soll. Allerdings ist diese Beanspruchungsart besonders geeignet, wenn die nicht-lineare Viskoelastizität studiert werden soll — und viele praktische Strömungsverhältnisse lassen sich durchaus nicht im Rahmen der linearen Viskoelastizitätstheorie behandeln.Es wurde ein robustes Rheometer vom Couettetyp entworfen, mit dem die Scherviskosität bei hohen Schergeschwindigkeiten gemessen werden kann und das ebenso für Oszillationsversuche mit großer Scheramplitude geeignet ist. Der stetige oder oszillatorische Antrieb erfolgt auf den äußeren Zylinder, während am inneren, ruhenden Zylinder das übertragene Drehmoment gemessen wird.Besonderer Wert wurde bei dem Entwurf des Rheometers den Problemen beigemessen, die sich durch hydrodynamische Stabilität, Beschleunigungsvorgänge, Wärmeentwicklung sowie Endeffekte ergeben. Die endgültige Form des Rheometers ist in den Augen der Autoren ein vernünftiger Kompromiß zwischen der Minimierung der genannten Einflüsse und den praktischen Voraussetzungen, daß ein Rheometer nicht zu teuer ist, aber einfach zu bedienen sein muß.Die Interpretation der experimentellen Ergebnisse solcher Oszillationsversuche ist einigermaßen problematisch. Der resultierende Spannungsverlauf ist zwar periodisch, aber nicht sinusförmig, so daß die Methoden der linearen Viskoelastizität nicht anwendbar sind. Obwohl sich mehrere Möglichkeiten anbieten, dürfte die beste Darstellungsmethode der Ergebnisse durch die Auftragung der Spannung in Abhängigkeit von der Scherung bzw. der Schergeschwindigkeit gegeben sein. Dieses ergibt geschlossene Kurven, deren besonderes Aussehen durch die Eigenschaften des Testmaterials gekennzeichnet ist.

---

C Cauchy-Green strain tensor - C ^–1 Finger strain tensor - h gap spacing - k thermal conductivity - T difference between maximum and minimum temperatures in sheared fluid - II second invariant of rate-of-strain tensor;: - material constant of second-order fluid - material constant of second-order fluid - ₀ shear strain amplitude - shear rate in simple shear - ₀ maximum shear rate; equal to ₀ - rate-of-deformation tensor; grad velocity plus its transpose - material constant in eq. [8] - viscosity - ^* complex viscosity - dynamic viscosity - characteristic time of fluid - µ material constant with units of viscosity - fluid density - viscous or extra stress tensor - frequency (radians/sec.) With 3 figures     

The rheological characterisation of bread dough using capillary rheometry

An Australian hard wheat flour–water dough has been characterised using parallel plate and capillary rheometers over an extensive range of apparent shear rates (10^− 3–10³ s^− 1) relevant to process conditions. Torsional measurements showed that the shear viscosity of the dough increased with strain to a maximum value and then decreased, suggesting a breakdown of the dough structure. Both torsional and capillary experiments revealed the shear-thinning behaviour of the dough. The wall slip phenomenon in capillary rheometry was investigated and found to be diameter dependent and occurred at a critical shear stress of approximately 5–10 kPa. A two-regime power law behaviour was observed, with the power law index approximately 0.3 in the low shear rate range increasing to 0.67 in the high shear rate range. Pressure fluctuation was observed in the capillary data and increased with shear rate, in particular, at shear rates approaching 10⁴ s^− 1. The results demonstrate that capillary rheometry is a viable means of rheologically testing dough at high shear rates provided pressure fluctuation is carefully monitored and capillary rheometry corrections, including wall slip, are accounted for.     

A non-affine network model for polymer melts

In this paper, a network model of polymer melts is proposed in which network junction points move non-affinely. In this non-affine motion, junction points follow particle paths as seen by an observer rotating at the fluid element's net rigid-rotation rate. The speed at which junction points move is reduced as the network segments near their maximum extensions. In order to maintain a frame invariant model, it is necessary that the vorticity be decomposed into two portions, such that, = _R + _D . The deformational vorticity, _D , arises from shear deformation and is frame invariant while the rigid vorticity, _R , is frame dependent. A constitutive equation based on this finitely extensible network strand (FENS) motion is developed. The model illustrates how rotations that cause changes in the relative orientation of a fluid element with its surroundings can be incorporated into a constitutive equation using the deformational vorticity. The FENS model predicts a shear-thinning viscosity, and the Trouton viscosity predicted by the model is finite for all elongation rates. Finally, stochastic simulation results are presented to justify a mathematical approximation used in deriving the constitutive equation.     

A differential constitutive equation for polymer melts

A modification of the Giesekus constitutive equation is derived by incorporating (approximately, via the Peterlin approximation) the finite extensibility of polymer molecules into dumbbell kinetic theory along with the anisotropic hydrodynamic drag suggested by Giesekus. The constitutive equation that is obtained retains much of the simplicity of Giesekus' constitutive equation, but it involves terms that are cubic in the stress as well as those that are quadratic. It is shown that the constitutive equation quantitatively describes the steady elongational viscosity of the IUPAC polymer melt A (including the strain softening of the melt), but it cannot describe the elongational and shear viscosities simultaneously. It is also shown that the constitutive equation satisfies the Lodge-Meissner relation for shear strains less than unity.     

A multidirectional shear rheometer for polymer melts

A parallel plate shear rheometer for polymer melts was built for superposed shear tests in two orthogonal directions. To examine its performance, a standard test was developed, including a change of the shear direction by angles between 0° and 180°. The results were analyzed to obtain preliminary information on the multidirectional shear behavior of a low-density polyethylene melt. The measured orthogonal components of the shear force were added and then split again into components of the original and of the changed shear direction. It was found, within the range of error, that these components depend only on time and on the shear rate, but not on the angle by which the shear direction is changed. The total shear rate was 0.56 s^–1.Dedicated to Prof. Dr. Joachim Meissner on the occasion of his 60th birthday.     

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.     

The modified imbedded disc retraction method for measurement of interfacial tension in polymer melts

The imbedded disc retraction method is used to estimate interfacial tension in LLDPE/PS system with PS as the imbedded disc. Shape evolution of a disc of one material (PS) imbedded into the matrix of another material is observed (LLDPE). Three to five repetitions at three different temperature levels are observed. The Newtonian model of Rundqvist et al. (1996) for the imbedded disc retraction is modified to include elastic effects. The modified model is derived assuming uniaxial extension, starting with the lower convected Maxwell model. Both the original model and modified imbedded disc retraction model are used in data analysis. The mean values of interfacial tension at 190 °C, 200 °C, and 210 °C are 6.8 ± 0.7 mN/m, 3.9 ± 0.3 mN/m, and 3.7 ± 0.2 mN/m, respectively. A method of estimating whether elastic effects will significantly affect the estimated interfacial tension value during retraction for the given polymer pair is provided. Received: 6 August 1999 Accepted: 2 January 2001     

The imbedded disk retraction method for measurement of interfacial tension between polymer melts

A method for measuring interfacial tension of high viscosity polymer melts at elevated temperatures is described. The method involves the tracking of the shape evolution of a disk of one material imbedded in a second one. This makes it possible to determine the interfacial tension over a relatively short time period. The technique of preparing the samples makes it possible to measure on practically any combination of polymer melts without restrictions on viscosities and melting temperature, as long as one of the materials is transparent in the molten state.The retraction of the disk is observed by using a microscope with a high resolution video camera. The camera is connected to a video frame grabber in a personal computer which is programmed to collect images with preselected time intervals. Data of the retraction is acquired by using an image analysis software, measuring the average radius of the disk.The driving force for the shape evolution is interfacial tension and it is balanced by viscous forces. The analysis of the retraction process is done analytically with a simplified one-dimensional model. The model has been compared to experiments with the system PS/PMMA at 210 °C, covering viscosity ratios over a range of six decades and five different molecular weight values of PS. It is shown that interfacial tension can be determined over the whole range and a value of 1.1±0.2 mN/m was obtained for all samples.     

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.     

Stress optical measurement of the third normal stress difference in polymer melts under oscillatory shear

Results are reported for the dynamic moduli,G andG, measured mechanically, and the dynamic third normal stress difference, measured optically, of a series bidisperse linear polymer melts under oscillatory shear. Nearly monodisperse hydrogenated polyisoprenes of molecular weights 53000 and 370000 were used to prepare blends with a volume fraction of long polymer, _L, of 0.10, 0.20, 0.30, 0.50, and 0.75. The results demonstrate the applicability of birefringence measurements to solve the longstanding problem of measuring the third normal stress difference in oscillatory flow. The relationship between the third normal stress difference and the shear stress observed for these entangled polymer melts is in agreement with a widely predicted constitutive relationship: the relationship between the first normal stress difference and the shear stress is that of a simple fluid, and the second normal stress difference is proportional to the first. These results demonstrate the potential use of 1,3-birefringence to measure the third normal stress difference in oscillatory flow. Further, the general constitutive equation supported by the present results may be used to determine the dynamic moduli from the measured third normal stress difference in small amplitude oscillatory shear. Directions for future research, including the use of birefringence measurements to determineN ₂/N ₁ in oscillatory shear, are described.