Numerical simulation studies of the planar entry flow of polymer melts

This paper is concerned with the numerical simulation of planar entry flow using a penalty finite element method and the comparison of predictions with flow visualization and birefringence data for two polymer melts. The Phan-Thien Tanner (PTT) model was fit to the steady state shear and extensional viscosity data and the transient extensional viscosity data of both polystyrene and low-density polyethylene (LDPE) melts to obtain the parameters λ, ξ, and ϵ in this model. Agreement was found between the flow visualization and birefringence data and the predictions of streamlines and stress. With some modification of the constitutive equation, the vortex growth and intensity observed for LDPE could be predicted by the use of the PTT model and the material parameters fit to the rheological properties. Likewise, the flow behavior of polystyrene, in which only small vortices with no growth were observed, was also predicted. Furthermore, it was found that the size and intensity of the vortex could be affected by the parameter ϵ in the PTT model which controls the predictions of the extensional viscosity. Based on these results it seems that accurate simulation of entry flow behavior requires the use of a constitutive equation which is capable of giving realistic preciction's of a fluid's extentional flow properties.     

Equibiaxial extensional flow of polymer melts via lubricated squeezing flow. II. Flow modeling

A model for lubricated squeezing flow of a viscoelastic fluid is developed in order to study the viability of this flow as a rheological technique for generating equibiaxial extensional deformations in polymer melts. In this simple flow model, the melt, described by an upper-convected Maxwell fluid, is squeezed between thin films of a Newtonian fluid. Comparisons of the model predictions for constant strain rate and constant stress flows are made with experimental results presented in the first paper. Predictions from the model are able to describe the effects of lubricant viscosity and experimental configuration and indicate the technique fails for these flows at Hencky strains of approximately one. The cause for this failure is lubricant thinning, which leads to significant errors in both the measured stress difference and the strain. Received: 31 January 2000 Accepted: 31 May 2000     

Equibiaxial extensional flow of polymer melts via lubricated squeezing flow. I. Experimental analysis

The technique of lubricated squeezing flow is evaluated using experiments in both constant strain rate and constant stress flows of two low-density polyethylene melts. Experimental parameters that include the lubricant viscosity and sample aspect ratio are systematically varied to examine their effect on the viability of the technique in the linear viscoelastic regime by comparing measured quantities to those predicted by finite linear viscoelastic theory. An evaluation is also made by comparing viscosities measured using the lubricated squeezing flow technique and Meissner's rotating clamp (MAD) rheometer in the non-linear regime. In both cases, deviations between the expected and measured viscosities were observed, indicating that the technique is not applicable to large strains for constant strain rate and constant stress flows. It is suggested that this limit is the result of lubricant thinning. Received: 22 March 2000 Accepted: 31 May 2000     

Measurement of extensional viscosity of polymer solutions

A filament stretching technique for measuring the extensional viscosity of polymer solutions at constant stretch rate is presented. The liquid sample is held between two coaxial discs and stretched by moving the bottom disc downwards with a speed that increases exponentially with time. This is illustrated using a constant viscosity, elastic fluid consisting of 0.185% polyisobutylene in a solvent of kerosene and polybutene. For the case of this particular fluid, two distinct stretch rate regions are found to arise. The stretch rate in the first region is much higher than in the second, which is, in most cases, close to the overall stretch rate imposed on the sample. Nonetheless, all the results of any given run can be represented using an average extensional rate. The extensional stress growth data, plotted as the Trouton ratio against time, show an initial linear viscoelastic region where T_R rises to a value of 3, independent of extensional rate. Beyond this region, T_R depends on the stretch rate and rises dramatically to values in excess of 10³; the higher the extensional rate, the faster is the increase in T_R. These data do not seem to reach a steady state and appear to be similar to polymer melt data obtained by others in the past. The reproducibility of the results is very good and all this suggests that it is now possible to obtain unambiguous constant-stretch-rate stress-growth data for polymer solutions stretched from a state of rest.     

On the extensional viscosity of mobile polymer solutions

We describe experimental results on the extensional viscosity of mobile polymer solutions obtained from two instruments, the first being a commercial Spin Line Rheometer and the second a custom-built lubricated-die Converging Flow Rheometer. The interpretation of data in terms of Trouton ratios is facilitated by a simple analysis for the Generalized Newtonian Fluid model.Agreement between data from the two rheometers is satisfactory and we show that polymer solutions can be either tension stiffening or tension thinning. However, the Trouton ratios in both cases are greater than the Newtonian values and we anticipate that this will always be the case for polymer solutions.Invited paper, presented at the 2nd Conference of European Rheologists, Prague, June 17–20, 1986     

Suspensions of monodisperse spheres in polymer melts: particle size effects in extensional flow

Suspensions in polymeric, viscoelastic liquids have been studied in uniaxial extensional flow. The fibre wind-up technique has been used for this purpose. The effects of particle size and particle volume fraction have been investigated, using monodisperse, spherical particles. The results have been compared with shear flow data on the same materials. The values of the relative extensional viscosities at low stretching rates are in agreement with the relative shear viscosities and relative moduli. This indicates that hydrodynamic forces are stronger than the particle interaction forces. At larger strain rates strain hardening occurs; it is suppressed when particles are added. Small aggregating particles reduce the strain hardening more strongly than larger particles; strain hardening can even be totally eliminated. When further increasing the stretching rate, hydrodynamic effects dominate again and the effect of particle size effect on strain hardening disappears.     

Birefringence measurements on polymer melts in an axisymmetric flow cell

 The stress-optical rule relates birefringence to stress. Consequently, measurement of flow birefringence provides a non-intrusive technique of measuring stresses in complex flows. In this investigation we explore the use of an axisymmetric geometry to create a uniaxial elongational flow in polymer melts. In axisymmetric flows both birefringence and orientation angle change continuously along the path of the propagating light. The cumulative influence of the material's optical properties along the light's integrated path makes determination of local birefringence in the melt impossible. One can nevertheless use birefringence measurements to compare with predictions from computer simulations as a means of evaluating the constitutive equations for the stress. More specifically, in this investigation we compare the light intensity transmitted through the experimental set-up vs entry position, with the theoretically calculated transmitted intensity distribution as a means of comparing experiment and simulation. The main complication in our experiments is the use of a flow cell that necessarily consists of materials of different refractive indices. This introduces refraction and reflection effects that must be modeled before experimental results can be correctly interpreted. We describe how these effects are taken into account and test the accuracy of predictions against experiments. In addition, the high temperatures required to investigate polymer melts mean that a further complication is introduced by thermal stresses present in the flow cell glass. We describe how these thermal-stresses are also incorporated in the simulations. Finally, we present some preliminary results and evaluate the success of the overall method. Received: 2 April 2001 Accepted: 27 August 2001     

Rheotens-mastercurves and elongational viscosity of polymer melts

In a Rheotens experiment, the tensile force needed for elongation of an extruded filament is measured as a function of the draw ratio. For thermo-rheologically simple polymer melts, the existence of Rheotens-mastercurves was proved by Wagner, Schulze, and Göttfert (1995). Rheotens-mastercurves are invariant with respect to changes in melt temperature and changes in the average molar mass. By use of purely viscous models, we convert Rheotens-mastercurves of a branched and a linear polyethylene melt to elongational viscosity as a function of strain rate. The resulting elongational viscosity from constant force extension experiments is found to be in general agreement with what is expected as steady-state viscosity of polyethylene melts measured in either constant strain-rate or constant stress mode.Dedicated to Prof. Dr. J. Meissner on the occasion of his retirement from the chair of Polymer Physics at the Eidgenössische Technische Hochschule (ETH) Zürich, Switzerland     

Elongational flow opto-rheometry for polymer melts

Polymer melt elongation is one of the most important procedures in polymer processing. To understand its molecular mechanisms, we constructed an elongational flow opto-rheometer (EFOR) in which a high precision birefringence apparatus of reflection-double path type was installed into a Meissner's new elongational rheometer of a gas cushion type (commercialized as RME from Rheometric Scientific) just by mounting a small reflecting mirror at the center of the RME's sample supporting table. The EFOR enabled us to achieve simultaneous measurements of tensile stress (t) and birefringence n(t) as a function of time t under a given constant strain rate within the range of 0.001 to 1.0s^–1. (t) can be monitored upto the maximum Hencky strain (t) of 7 as attained, in principle, with RME, while the measurable range of the phase difference in the birefringence was 0 to 250 (0 to 79 100 nm for He-Ne laser light) within the accuracy of ±0.1 (±31.6 nm) up to (t) 4. The performance was tested on an anionically polymerized polystyrene (PS) and a low density polyethylene (LDPE). For both polymers (t) first followed the linear viscoelasticity rule in that the elongational viscosity, , is three times the steady shear viscosity, 3 _o(t), at low shear rate , but the _E (t) tended to deviate upward after a certain Hencky strain was attained. The birefringence n(t) was a function of both Hencky strain and strain rate in such a way that the stress-optical law holds with the stress-optical coefficient C(t) = n(t)/(t) being equal to the ones reported from shear flow experiments. Interestingly, however, for PS elongated at low strain rates the C(t) vs (t) relation exhibited a strong nonlinearity as soon as (t) reached steady state. This implies that the tensile stress reaches the steady state but the birefringence continues to increase in the low strain-rate elongation. For the PS melt elongated at high strain rates, on the other hand, C(t) was nearly a constant in the entire range observed. For LDPE with long-chain branchings, (t) exhibited tendency of strain-induced hardening after certain critical strain, but C(t) was nearly a constant in the entire range of (t) observed.     

On the use of flow through a contraction in estimating the extensional viscosity of mobile polymer solutions

We consider the use of pressure measurements in contraction flows in the determination of the extensional viscosity behaviour of polymer solutions. The experimental data are interpreted on the basis of the recent theory of Binding. The resulting extensional viscosities are compared with those obtained from a commercial Spin Line Rheometer.We conclude that contraction flows provide a convenient means of determining the extensional viscosity of shear-thinning polymer solutions. The case is not so clear for constant viscosity Boger fluids.In the course of the experiments, it is shown that excess pressure losses in the contractions can be brought about by two distinct flow mechanisms in the case of Boger fluids. In the axisymmetric case, both vortex enhancement and excess pressure loss are observed, although there is not a strict one-to-one correlation between these phenomena. In the planar case, vortex enhancement is not conspicuously present, although there is still a substantial excess pressure loss at high flow rates. This excess must be associated with the ‘bulb’ flow field which essentially replaces the vortex-enhancement regime of the axisymmetric case.     

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.     

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.     

A comparison of extensional viscosity measurements from various RME rheometers

The transient uniaxial extensional viscosity η _e of linear low density polyethylene (LLDPE) has been measured using the commercial Rheometric Scientific RME and the Münstedt Tensile Rheometer in an effort to compare the performance of available extensional rheometers. The RME indicated a significant strain hardening of the LLDPE, especially at a strain rate of 1 s⁻¹. In contrast, the Münstedt rheometer showed the LLDPE to be only slightly strain hardening. This artificial strain hardening effect in the RME resulted from the strain rate applied to the sample, determined from the sample deformation, being up to 20% less than the set strain rate. These results initiated a round-robin experiment in which the same LLDPE was tested on several RMEs in various locations around the world. All but one of the RMEs indicated a deviation between set and applied strain rates of at least 10%, especially at strain rates above 0.1 s⁻¹. The strain rate deviation was found to depend strongly on the value of the basis length L ₀ , and may result from the upper pair of belts not properly gripping the sample during extension. Thus visual inspection of the sample deformation is necessary to determine the applied strain rate. The most accurate measurements of η _e with respect to the strain rate deviation were obtained when the correct L ₀ value and belt arrangement were used. A list of recommendations for running an RME test is provided. Future work focusing on the fluid mechanics during the test may identify fully the cause of the strain rate deviation, but from a practical point of view the problem can be corrected for in the determination of η _e . Received: 27 September 2000/Accepted: 5 February 2001     

The rheology of polymer solution elastic strands in extensional flow

We apply micro-oscillatory cross-slot extensional flow to a semi-dilute poly(ethylene oxide) solution. Micro-particle image velocimetry (μPIV) is used to probe the real local flow field. Extreme flow perturbation is observed, where birefringent strands of extended polymer originate from the stagnation point. This coincides with a large increase in the extensional viscosity. The combination of stagnation point flow and μPIV enables us to investigate directly the stress and strain rates in the strand and so determine the true extensional viscosity of the localised strand alone. The Trouton ratio in the strand is found to be ~4000, amongst the highest values of Trouton ratio ever reported. Consideration of the flow in the exit channels surrounding the highly elastic strand suggests a maximum limit for the pressure drop across the device and the apparent extensional viscosity. This has implications for the understanding of high Deborah number extensional thinning reported in other stagnation point flow situations.     

Stress and neutron scattering measurements on linear polymer melts undergoing steady elongational flow

We use small-angle neutron scattering to measure the molecular stretching in polystyrene melts undergoing steady elongational flow at large stretch rates. The radius of gyration of the central segment of a partly deuterated polystyrene molecule is, in the stretching direction, increasing with the steady stretch rate to a power of about 0.25. This value is about half of the exponent observed for the increase in stress value σ, in agreement with Gaussian behavior. Thus, finite chain extensibility does not seem to play an important role in the strongly non-linear extensional stress behavior exhibited by the linear polystyrene melt.     

Dynamic measurements on polymer solutions and melts

Summary An apparatus for the measurement of the dynamic viscosity is described. In a coaxial cylinder geometry forced vibrations are executed in the frequency region from 2·10^–4 to 50 cycles/sec. Fluids with a wide range of viscosities (10^–1 to 10⁷ poise) can be measured. A number of experimental results obtained on dilute solutions and melts of polymers are given and compared with results of the molecular theories.

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Zusammenfassung Es wird ein Gerät für die Messung der dynamischen Viskosität beschrieben. Innerhalb eines koaxialen Zylindersystems werden im Frequenzgebiet von 2 · 10^–4 bis 50 Hz erzwungene Schwingungen ausgeführt. Messungen an Flüssigkeiten sind in einem weiten Viskositätsbereich (10^–1 bis 10⁷ Poise) möglich. Einige Messungen an verdünnten Lösungen oder Schmelzen von Hochpolymeren werden mitgeteilt und mit Voraussagen molekularer Theorien verglichen.

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Paper read at the Annual Meeting of the German Rheologits, Berlin-Dahlem, May 20–21, 1968).