Investigation of failure during elongational flow of polymer melts

A basic study of the mechanisms of necking and ductile failure of polymer melts in uniaxial elongational flow has been carried out. A linear stability analysis was carried out using a White—Metzner convected Maxwell model with a deformation-rate-dependent relaxation time, which varies according to τ = τ_o/(1 + aτ_o[2trd²]^{$\text{1}\text{2}$}). It was shown that filament stability and elongation to break depend upon τ_oE, where E is the elongation rate, and a. At fixed τ_oE, filament stability decreases with increasing a. At small a, stability increases with increasing τ_oE while for a > $\text{1}\text{√3}$, stability decreases with increasing τ_oE. For a material with small a, ductile failure can occur for small τ_oE, but cohesive fracture should be the cause of failure at larger τ_oE. For a material with large a, however, ductile failure always dominates the failure mode. These results are used to interpret failure in elongational flow of low density and high density polyethylene and polypropylene melts and describe how the latter two melts exhibit ductile failure.     

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     

Rheo-optical measurements of the elongational flow of aqueous polymer solutions

An opposed jets rheometer was used to investigate the elongational viscosity as a function of the strain rate for dilute aqueous solutions of polyvinylformamide and polyacrylamide. Critical strain rates at which the enhancement in elongational viscosity occurs were determined for both systems. The influence of the polymer concentration on the elongational viscosity was investigated. The measurements were performed with polymer concentrations less than the critical polymer concentration c_p^* c_p^* . In order to assess the deformation and orientation of the macromolecules, flow-induced birefringence was measured simultaneously.     

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     

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).     

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.     

Determination of elongational viscosity of polymer melts by RME and Rheotens experiments

Several linear (LLDPE, HDPE, PS) and long-chain-branched (LDPE, PP) polymer melts were investigated by an elongational rheometer (RME Rheometrics) and by Rheotens (Göttfert). The Molecular Stress Function (MSF) theory is briefly reviewed and used to extrapolate the steady-state elongational viscosity. To evaluate Rheotens experiments, a new process model is introduced which assumes that the elongational viscosity in the Rheotens test is a function of the draw ratio only. The apparent elongational viscosities extracted from Rheotens curves are found to lie in between the steady-state elongational viscosity and three times the shear viscosity.     

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 relaxation after steady shear flow in immiscible model polymer blends

Stress relaxation in immiscible blends is studied for a well defined shear history, i.e. after prolonged steady state shearing. Model systems are used that consist of quasi-Newtonian liquid polymers. Hence the relaxation is dominated by changes in the morphology of the interface. Both shear stress and the first normal stress are considered. The measurements cover the entire concentration range. For dilute blends the interfacial contribution to the stress relaxation compares well with model predictions. Deviations occur when the matrix phase is slightly elastic. In that case the similarity between the relaxation of shear and normal stresses is also lost. The latter is attributed to a wider drop size distribution.Increasing the concentration of the disperse phase results in a complex evolution of the characteristic relaxation times. The normal stresses relax systematically slower than the shear stresses and the concentration curve includes two maxima. Even for equiviscous components the concentration curves are not symmetrical. It is concluded that even a slight degree of elasticity in the matrix phase drastically affects the morphology and the interfacial relaxation of such blends.     

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.     

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     

Constitutive equation for dilute polymer solutions in strong elongational flow

A constitutive equation is proposed, which is constructed using both phenomenological and structural ideas. In this formulation, the kinematics of the fluid is characterized by the deformation rate and a structural vector. The vector follows an evolutionary law which is inspired by known molecular models. The expression for the stress is given by introducing a dissipative term related to the strong hydrodynamic interaction of the distorted molecules and a deformation term for the molecules, as well as by using the second law of thermodynamics.A study of the general properties of the evolutionary equation and its response in a homogeneous two-dimensional flow provides evidence of the performance of the proposed model.     

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     

Unsteady flow of polymer melts: Polypropylene

Summary The unsteady flow effect has been studied experimentally in cylindrical capillaries for polypropylene melts with melt indices differing over an approximately 30-fold range. The entrance losses were negligibly small. The flow curves obtained on capillaries of different diameters, coincided, indicating the absence of any considerable near-wall slippage; hence the attainment of unsteady flow is not necessarily due to or accompanied by wall slippage.Two critical regimes can be distinguished distinctly in the flow of polypropylene melts, the first corresponding to the appearance of mattness on the extrudate surface, and the second to the appearance of defects such as a regular spiral with a constant longitudinal pitch. No unsteady flow of the melt fracture type was observed.The shear stresses in the first critical flow regime increase by 40 per cent when the temperature is raised from 180 to 240 °C. Within the same temperature range the stresses corresponding to attainment of the second critical flow regime change by 25 per cent.The critical flow parameters of polypropylene melts grow with increasing capillary length-to-diameter ratio, this effect not being damped even with big capillary lengths.The elastic deformations corrrsponding to attainment of the first critical flow regime of polypropylene melts_e2.7, while that corresponding to the second critical regime _e 3.3 with considerably changed critical stress values.

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Zusammenfassung Der Effekt des nicht-stationären Fließens wurde experimentell in zylindrischen Kapillaren für Polypropylen-Schmelzen mit Schmelz-Indizes zwischen 1 und 30 untersucht. Die Eintrittsverluste waren vernachlässigbar klein. Die Fließkurven, die bei Kapillaren verschiedenen Durchmessers ermittelt wurden, stimmten überein und zeigten das Fehlen irgendwelcher Gleitung in der Nähe der Wand. Demnach ist das Entstehen eines nicht-stationären Fließens nicht notwendigerweise abhängig bzw. begleitet von einer Wandgleitung.Zwei kritische Bereiche können eindeutig beim Fließen von Polypropylen-Schmelzen unterschieden werden: der erste im Zusammenhang mit dem Auftreten einer matten Extrudat-Oberfläche und der zweite mit dem Auftreten von Fehlern wie einer regelmäßigen Spirale mit konstanter Steigung. Ein nicht-stationäres Fließen vom Schmelzbruchtyp wurde nicht beobachtet.Die Schubspannung in dem ersten kritischen Fließbereich wächst bei einem Temperaturanstieg von 180 auf 240 °C um 40%. Im gleichen Temperaturbereich wachsen die Spannungen bei Erreichen des zweiten kritischen Fließgebietes um 25%.Die kritischen Fließparameter von Polypropylen-Schmelzen wachsen mit steigendem Verhältnis Kapillarlänge zu Durchmesser, wobei dieser Effekt auch durch große Kapillarlängen nicht gedämpft wird. Die elastischen Deformationen, die bei Erreichen des ersten kritischen Fließgebietes von Polypropylen-Schmelzen auftreten, betragen _e 2,7, während für das zweite kritische Gebiet bei deutlich geänderten kritischen Spannungswerten sich Deformationen von _e 3,3 ergeben.

     

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.     

Testing of a constitutive equation for entangled networks by elongational and shear data of polymer melts

Summary An entangled network such as a polymer melt or a concentrated solution is here described by a set of two simultaneous equations. One of them is a balance of entanglements, the other gives the stress in the classical form of aMaxwell equation.The balance of entanglements contains both an entanglement generation term which depends on the distance from equilibrium and an entanglement destruction term which depends on the stress level. The parameters appearing in theMaxwell and the balance equations are made to depend in a specified way on the existing number of entanglements.The model is here tested by comparison with existing data of steady-state elongational and shear viscosity of polymer melts.With 6 figures     

The relevance of entry flow measurements for the estimation of extensional viscosity of polymer melts

The extensional viscosity of some flexible chain polymers and a thermotropic liquid crystalline polymer was measured in uniaxial extensional flow at constant extension rate. Power law functions were found for the dependence of the extensional viscosity at constant accumulated strain on strain rate. The stress growth curves were compared with measurements in axisymmetric entry flow, where both elongation and shear occur. The comparison showed that the values of the extensional viscosity calculated from the measurements in the entry flow correspond to the ones calculated from the viscosity growth measured in uniaxial elongation and averaged over extensional strain equal to what is accumulated on the fluid as it flows from the barrel into the capillary.