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.

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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.

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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     

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.     

A shear rheometer for measuring shear stress and both normal stress differences in polymer melts simultaneously: the MTR 25

The MTR 25 is a multitask rheometer (for shear and squeeze flow) with 25 kg of normal force and a partitioned plate. Torque and normal force are measured at both, the inner disk and the outer ring of the plate. The first and second normal stress differences can be determined from a single test. The axial stiffness is high (10⁷ N/m) by using rigid springs and strain gauges for the load cell. Monodisperse polystyrene (M _w = 206 kg/mol, 180°C) has been sheared in the range from 0.05 to 47 s^− 1. The viscosity and first normal stress difference are highly reproducible. The second normal stress difference scatters and mirrors the instability at the rim. A critical comparison is made between the MTR 25 method and the single transducer evaluation method (RMS 800 method, Schweizer, Rheol Acta 41:337–344, 2002): Both yield excellent and coinciding viscosity and first normal stress difference data. The RMS 800 method gives more stable second normal stress difference data, since the normal force from the outer ring, which is influenced by edge fracture, is not used. Data for the RMS 800 method can be acquired on the MTR 25. The high normal force capacity permits larger samples and higher shear rates than on the RMS 800.

Exponential shear flow of branched polymer melts

  The behavior of a low-density polyethylene melt in exponential shear strain histories is examined and compared to its behavior in constant rate planar elongation. A new set of shear stress and first normal stress difference data in exponential shear are presented and used in several different material functions that have been previously proposed. Viscosities composed of principal stress differences for the two flows showed no correspondence suggesting that, contrary to previous assertions, exponential shear and constant rate planar elongation flows are fundamentally different. It is further suggested that the presence of vorticity makes exponential shear a weak, rather than strong, flow. Received: 5 March 1999/Accepted: 1 September 1999     

Transient shear flow properties of fiber-filled polyethylene melts

The growth and relaxation of shear and normal stresses have been investigated for glass and carbon fiber-filled polyethylene melts over a wide range of shear rates and temperatures by means of a cone-and-plate rheogoniometer. Flow parameters and flow curves characterizing the stress overshoot and relaxation phenomena of the fiber-filled systems were determined experimentally. The influence of fiber loading, fiber size and temperature on the transient flow parameters are discussed.Predictions by the Meister and Bogue constitutive equations were compared with the experimental data for the transient shear and normal stresses. These equations predict satisfactorily the non-linear transient shear flow of polymer melts and its fiber-filled systems.     

A new elongational rheometer for polymer melts and other highly viscoelastic liquids

In polymer melt elongational rheometry only by the rotary clamp technique large elongations can be obtained homogeneously. However, as described in this paper, there still remain disadvantages that led to the development of a new rheometer with the following main features: The dimensions of the required sample are small (60 × 7 × 2 nun³), the sample is supported by a cushion of inert gas and, after having reached the test temperature of up to well above 300°C, it can be extended by a new type of clamps that make use of metal conveyor belts. The resulting tensile force is measured with a resolution of better than 100 mgf (0.001 N). The strain rate range is 0.001-1 s^–1, and the maximum Hencky strain is 7, corresponding to a maximum stretch ratio of 1100. Within the sample, the temperature variation in time and space is less than 0.1°C. For the evaluation and documentation of the test performance, a video camera records the top and side views of the sample that carries a marking powder to permit the evaluation of the true strain rate. The operation of the instrument is easy, and so is the sample preparation, but care must be taken concerning the necessary isotropy and internal homogeneity. Examples of test results are given for several polymer melts at various temperatures: (1) Polystyrene up to a total Hencky strain larger than 7 at 170°C, (2) several types of polyethylene (LDPE, LLDPE, HDPE) at 150°C, (3) poly(amide) at 250°C, and (4) poly(ethersulfone) at 350°C. The wide applicability of the new rheometer is demonstrated by adding results obtained from samples of bread dough. The surface tension has no influence on the results if an error of 3% can be tolerated. From the results it follows that by means of the newly developed rheometer many problems in polymer melt elongation have been solved.Dedicated to Professor Dr. Hermann Janeschitz-Kriegl on the occasion of his 70th birthday.Extended version of a paper presented at the XIth International Congress on Rheology, Brussels, Belgium, August 17–21, 1992.     

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.     

Die swell of filled polymer melts

The Barus effect in polypropylene and polystyrene blended with a variety of fillers at various concentrations was investigated using a capillary extrusion rheometer. If the die swell is defined as the square of the ratio of the extrudate diameterd to the die diameterD, it is found to depend on the apparent shear stress _W . Below a certain value of _w the relation =B _B ^A applies. The die swell, _M , of a filled polymer depends on the type, size and volume fraction of the filler. In particular,A increases as the volume fraction increases and is largest for powders, smaller for flakes and smallest for fibres, whereasB shows the opposite trend but to a lesser extent.     

Lubricated optical rheometer for the study of two-dimensional complex flows of polymer melts

We describe a novel optical cross-slot channel rheometer generating two-dimensional and isothermal complex flows of polymer melts. This is made possible by lubricating the channel front and back viewing windows. Flow-induced birefringence and particle tracking velocimetry are reviewed and used to investigate the cross-slot flow of a low density polyethylene melt involving mixed shear and planar extensional deformations. This new device solves the issue of end effects in flow birefringence experiments where no variations of the optical properties along the light path are expected. It greatly facilitates the interpretation of stress field data by providing reliable measurements of the polymer melt extinction angle χ$\chi$ and retardation δ$\delta$, with a spatial resolution of one tenth of a millimeter. At the same time, it offers an enhanced temperature control and an increased optical accuracy due to an improved laser beam shaping. The capabilities and performances of this unique type of lubricated rheometer are discussed in detail and compared with previous approaches.     

A shear stress transducer for molten polymers

Present rheometrical techniques are inadequate for the measurement of viscoelastic properties associated with shearing at high rates. A possible solution to this problem is to use a sliding plate rheometer together with a device for measuring the local wall shear stress away from the ends and edges of the plates. Such a device has been constructed, and the results of preliminary tests are encouraging.     

A torsional rheometer with a double slit arrangement suitable for fast gelation

A home built torsional rheometer is presented that consists of a measuring cell with a double slit arrangement. This double slit arrangement guarantees a quick cooling down of the sample which is of major importance for the observation of the start of the sol-gel transition under isothermal conditions. This torsional rheometer is based on an earlier version of a home built dynamic viscometer. The modifications are focusing first and foremost on the miniaturizing of the measuring cell. This results in high cooling rates, small volume of the sample, small weight of the oscillating cylinder, and therefore a relatively large range of frequencies. Received: 28 August 2000 Accepted: 8 January 2001     

Instabilities in shear and simple shear deformations of gold crystals

We use the tight-binding potential and molecular mechanics simulations to study local and global instabilities in shear and simple shear deformations of three initially defect-free finite cubes of gold single crystal containing 3480, 7813, and 58,825 atoms. Displacements on all bounding surfaces are prescribed while studying simple shear deformations, but displacements on only two opposite surfaces are assigned during simulations of shear deformations with the remaining four surfaces kept free of external forces. The criteria used to delineate local instabilities in the system include the following: (i) a component of the second-order spatial gradients of the displacement field having large values relative to its average value in the body, (ii) the minimum eigenvalue of the Hessian of the energy of an atom becoming non-positive, and (iii) structural changes represented by a high value of the common neighborhood parameter. It is found that these criteria are met essentially simultaneously at the same atomic position. Effects of free surfaces are evidenced by different deformation patterns for the same specimen deformed in shear and simple shear. The shear strength of a specimen deformed in simple shear is more than three times that of the same specimen deformed in shear. It is found that for each cubic specimen deformed in simple shear the evolution with the shear strain of the average shear stress, prior to the onset of instabilities, is almost identical to that in an equivalent hyperelastic material with strain energy density derived from the tight-binding potential and the assumption that it obeys the Cauchy-Born rule. Even though the material response of the hyperelastic body predicted from the strain energy density is stable over the range of the shear strain simulated in this work, the molecular mechanics simulations predict local and global instabilities in the three specimens.     

Elastic properties of polyethylene melts at high shear rates with respect to extrusion

At high shear rates a steady state of shear flow with constant shear rate, constant shear stress, and constant recoverable shear strain is observed in the short-time sandwich rheometer after some few shear units already. The melt exhibits rather high elastic shear deformations and the recovery occurs at much higher speed than it is observed in the newtonian range. The ratio of first normal stress difference and twice the shear stress, being equal to the recoverable strain in the second-order fluid limit, significantly underestimates the true elastic shear strains at high shear rates. The observed shear rate dependence of shear stress and first normal stress difference as well as of the (constrained) elastic shear strain is correctly described on the basis of a discrete relaxation time spectrum. In simple shear a stick-slip transition at the metal walls is found. Necessary for the onset of slip is a critical value of shear stress and a certain amount of elastic shear deformation or orientation of the melt.     

Assessment of nonlinear strain measures for extensional and shearing flows of polymer melts

From stress-strain experiments in extensional and shearing flows, nonlinear strain measures and effective damping functions are derived for a polyisobutylene melt. The strain measures determined in planar extensional flow and in simple shear flow coincide. Experimental results are compared with predictions of two molecular theories, the Doi-Edwards model and the molecular stress function approach of Wagner and Schaeffer. Discrepancies between theories and experiment lead to a reconsideration of the classification of extensional flows. The symmetry of the flow field is identified and quantified as an important parameter influencing the strain measure, and a unifying strain measure for general extensional and shearing flows of polymer melts is presented.     

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 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 specific slit die rheometer for extruded starchy products. Design,validation and application to maize starch

A novel in-line rheometer, called Rheopac, has been designed and built in order to study the rheological behaviour of starchy products or, more generally, of products sensitive to a thermomechanical treatment. It is based on the principle of a twin channel, using a balance of feed rate between each of them, in order to make local shear rate vary in the measuring section without changing the flow conditions into the extruder. A wide range of shear rate could be reached and measurements were performed more swiftly than with a classical slit die. The viscous behaviour of maize starch was studied by taking into account the influence of the thermomechanical history, which modified the starch degradation and thus led to important variations in the viscosity. Experimental results were satisfactorily compared to previously published models.Nomenclature E activation energy (J · mol^–1) - h channel depth (m) - h ₁ depth under the piston valve in channel 1 (m) - h ₂ depth under the piston valve in channel 2 (m) - K consistency (Pa·s ⁿ ) - K ₀ reference consistency (Pa·s ⁿ ) - L total channel length (m) - L _p length of the piston valve (m) - MC moisture content (wet basis) - n power law index - N screw rotation speed (rpm) - P ₀ entrance pressure (Pa) - P _e pressure at the entry of the piston valve (Pa) - Q ₁ flow rate in channel 1 (m3 · s^–1) - Q ₂ flow rate in channel 2 m³·s^–1) - Q _T total flow rate (m3 · s^–1) - R constant of perfect gas (8.314 J·mol^–1·K^–1) - SME specific mechanical energy (kWh · t^–1) - T temperature (°C) - T _a absolute temperature (K) - T _b barrel temperature (°C) - T _d die temperature (°C) - T _p product temperature (°C) - w channel width (m) - W energetical term (J·m^–3) - viscosity (Pa · s) - [gh ₀] intrinsic viscosity of native starch (ml·g^–1) - [] intrinsic viscosity (ml·g^–1) - shear rate (s^–1) - shear rate in measuring section (s^–1) - maximum shear rate (s^–1)     

The third-normal stress difference in entangled melts: Quantitative stress-optical measurements in oscillatory shear

Stress-optical measurements are used to quantitatively determine the third-normal stress difference (N ₃ = N ₁ + N ₂) in three entangled polymer melts during small amplitude (<15%) oscillatory shear over a wide dynamic range. The results are presented in terms of the three material functions that describe N ₃ in oscillatory shear: the real and imaginary parts of its complex amplitude ₃ ^* = ₃ - i ₃ , and its displacement ₃ ^d . The results confirm that these functions are related to the dynamic modulus by ² ₃ ^* ()=(1-)[G ^*())– G ^*(2)] and ² ₃ ^d ()=(1- )G() as predicted by many constitutive equations, where = –N ₂/N ₁. The value of (1-) is found to be 0.69±0.07 for poly(ethylene-propylene) and 0.76±0.07 for polyisoprene. This corresponds to –N ₂/N ₁ = 0.31 and 0.24±0.07, close to the prediction of the reptation model when the independent alignment approximation is used, i.e., –N ₂/N ₁ = 2/7 – 0.28.