Influence of long-chain branches in polyethylenes on linear viscoelastic flow properties in shear

 This contribution presents a survey on the influence of long-chain branching on the linear viscoelastic properties zero shear-rate viscosity and steady-state recoverable compliance of polyethylene melts. The materials chosen are linear and slightly long-chain branched metallocene-catalyzed polyethylenes of narrow molecular mass distribution as well as linear and highly long-chain branched polyethylenes of broad molecular mass distribution. The linear viscoelastic flow properties are determined in shear creep and recovery experiments by means of a magnetic bearing torsional creep apparatus. The analysis of the molecular structure of the polyethylenes is performed by a coupled size exclusion chromatography and multi-angle laser light scattering device. Polyethylenes with a slight degree of long-chain branching exhibit a surprisingly high zero shear-rate viscosity in comparison to linear polyethylenes whereas the highly branched polyethylenes have a much lower viscosity compared to linear samples. Slightly branched polyethylenes have got a higher steady-state compliance in comparison to linear products of similar polydispersity, whereas the highly branched polyethylenes of broad molecular mass distribution exhibit a surprisingly low elasticity in comparison to linear polyethylenes of broad molecular mass distribution. In addition sparse levels of long-chain branching cause a different time dependence in comparison to linear polyethylenes. The experimental findings are interpreted by comparison with rheological results from literature on model branched polymers of different molecular topography and chemical composition. Received: 12 July 2001 Accepted: 30 October 2001     

Influence of molecular structure on secondary flow of polyolefin melts as investigated by laser-Doppler velocimetry

Laser-Doppler velocimetry (LDV) is applied to investigate velocity profiles in the entrance region of a slit die. Due to the high spatial resolution of the device and the accuracy of the velocity measurements the secondary flow patterns of different polyolefins have quantitatively been analyzed for the first time. A linear polyethylene is compared with two long-chain branched polyethylenes and a conventional linear polypropylene with a long-chain branched one. All materials are rheologically characterized with respect to their viscosity functions, elasticity, and elongational properties. For the two linear materials no indication of secondary flow is found, but the three long-chain branched polymers (two polyethylenes and one polypropylene) exhibit pronounced vortices. Neither viscosity nor elasticity seem to be decisive for the occurrence of secondary flow. The viscosity has an influence, however, on the size of the vortices and the velocities within them. All of the three long-chain branched polymers are strongly strain hardening which gives rise to the conclusion that this behavior may be a necessary condition for the formation of vortices. The linear polypropylene does not show any indication of strain hardening. The linear polyethylene, surprisingly, is significantly strain hardening, but it becomes less pronounced with higher strain rates. As most of the deformation in the entrance region takes place at elongational rates at which the strain hardening of the linear polyethylene is not significant, the findings on the linear polyethylene do not contradict the hypothesis that strain hardening and vortex formation in entrance flow may be related to each other. Received: 27 April 2000 Accepted: 30 November 2000     

On the “viscosity overshoot” during the uniaxial extension of a low density polyethylene

An experimental investigation of the viscosity overshoot phenomenon observed during uniaxial extension of a low density polyethylene is presented. For this purpose, traditional integral viscosity measurements on a Münstedt-type extensional rheometer are combined with local measurements based on the in-situ visualization of the sample under extension. For elongational experiments at constant strain rates within a wide range of Weissenberg numbers (Wi), three distinct deformation regimes are identified. Corresponding to low values of Wi (regime I), the tensile stress displays a broad maximum, but such maximum is observed with various polymeric materials deformed at low rates and it should not be confused with the “viscosity overshoot” phenomenon. Corresponding to intermediate values of Wi (regime II), a local maximum of the integral extensional viscosity is systematically observed. Moreover, within this regime, a strong discrepancy between integral measurements and the space average of the local elongational viscosity is observed which indicates large deviations from an ideal uniaxial deformation process. Images of samples within this regime reinforce this finding by showing that, corresponding to the maximum of the integral viscosity, secondary necks develop along the sample. The emergence of a maximum of the integral elongational viscosity is, thus, related to the distinct inhomogeneity of deformation states and most probably not to the rheological properties of the material. In the fast stretching limit (high Wi, regime III), the overall geometric uniformity of the sample is well preserved, no secondary necks are observed and both the integral and the space averaged transient elongational viscosity show no maximum. A detailed but yet incomplete comparison of the experimental findings with results from the literature is presented and several open questions are stated.     

Uniaxial elongational flow effects and morphology development in LDPE/phosphate glass hybrids

Shear and elongational viscosity measurements were performed on low-density polyethylene/phosphate glass (LDPE/Pglass) hybrid materials in the liquid state. Under shear deformation, the hybrids with low concentrations of Pglass showed a Newtonian region at low frequencies, followed by shear-thinning behavior at high frequencies. High Pglass concentrations displayed shear-thinning behavior over the whole range of frequencies studied. Deviations from the log-additivity rule for viscosity were found to be compositionally dependent and generally indicated an immiscible mixture. The elongational viscosity of the hybrids increased at very low Pglass concentrations (1–2 vol.% Pglass) and then was drastically reduced at higher concentrations (i.e., >10 vol.% Pglass). In addition, elongational flow was found to induce the formation of Pglass fibrils in hybrids containing at least 10 vol.% Pglass. This was correlated to the elongational capillary number; the critical elongational capillary number was estimated to be 0.22. The elongational deformation was also found to greatly increase the overall crystallinity of the system due to molecular orientation of the LDPE polymer chains as confirmed by wide angle X-ray diffraction. A critical composition of 5 vol.% Pglass was found to be the point at which LDPE hybrid rheological properties, molecular orientation, and morphology changed drastically.     

Modeling strain hardening of polydisperse polystyrene melts by molecular stress function theory

The strain hardening of blends of polystyrene (PS) and ultra-high molecular weight polystyrene (UHMW-PS) in elongational flow is modeled by the molecular stress function (MSF) theory. Assuming that the ratios of strain energies stored in polydisperse and monodisperse polymers are identical for linear and nonlinear deformations, the value of the only non-linear parameter of the theory in extensional flows, the maximum molecular stress f_max, can be determined and is shown to be related to steady-state compliance J_e0. Using only linear-viscoelastic data, the elongational viscosity of PS/UHMW-PS blends is consistently predicted by the MSF theory.     

A comprehensive experimental study of the rheological behaviour of HDPE.

Extensional properties of four high density polyethylenes with different molecular weights and molecular weight distributions are presented. The samples have already been well characterized in shear and non-isothermal extensional flow. The data were collected at 180 °C for elongational rates between 3 · 10^–1 and 10^–4s^–1. Some qualitative and quantitative generalizations of the results are given.     

Correlation between rheological behaviour in uniaxial elongation and film blowing properties of various polyethylenes

Six various low density polyethylenes and one blend were rheologically characterized in elongation. Their different strain-hardening behaviour could qualitatively be related to their molecular structure. All the materials were blown to films on laboratory equipment under various conditions. The take-up force and the film homogeneity were determined quantitatively, the bubble stability was visually assessed. The bubble stability increased with growing take-up force. The take-up force was found to be the stronger the higher the elongational viscosity was. The homogeneity of film thickness is not related to the bubble stability but to the occurrence of strain hardening in the uniaxial elongational experiment at high Hencky strains. Measurements of the uniaxial elongational behaviour of polyethylene melts are a valuable and promising way to assist the development and optimisation of film blowing materials.

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.     

Comparison of viscous and elastic properties of polyolefin melts in shear and elongation

Viscous and elastic properties of a linear polypropylene (PP) and a long-chain branched low-density polyethylene (LDPE) have been investigated by creep and creep–recovery experiments in shear and elongation. The data obtained verify the ratios between the linear values of the viscosities and the steady-state elastic compliances in shear and elongation predicted by the theory of linear viscoelasticity. In the nonlinear range, no simple correlation between the viscous behaviour in shear and elongation exists. The elongational viscosity of the PP decreases with increasing stress analogously to the shear thinning observed; the linear range extends to higher stresses in elongation than in shear, however. The LDPE shows thinning in shear and strain hardening in elongational flow. For the LDPE, a linear steady-state elastic tensile compliance corresponding to one third of the linear steady-state elastic compliance in shear was determined. For the PP, this theoretically predicted value is approximately reached. Analogous to the viscous behaviour, the linear range extends to higher stresses in elongation than in shear. For both materials, the steady-state elastic compliances in the nonlinear range decrease with increasing stress in shear as well as in elongation. However, the decrease in elongation is more pronounced.     

Branched viscoelastic surfactant solutions and their response to elongational flow

Several years ago, Münstedt and Laun reported on the influence of branching on the elongational flow properties of polymer chains (Münstedt and Laun, 1981). They concluded that, in addition to the molecular weight distribution, the degree of branching strongly affects the degree of strain thickening of the elongational viscosity in such a way that the maximum in this material function increases with branching. In a recent paper by Lin, a ternary system of dodecyldimethylamine oxide-sodium laureth sulphate-sodium chloride surfactant solutions was investigated by CryoTEM and rheology (Lin, 1996). He reported a linear relation between the added sodium chloride and the branching of the wormlike micelles. In this paper we present an investigation of these surfactant solutions in elongational flow. Our results indicate that for branched micellar systems the presence of branching enhances the maximum of the elongational viscosity in the same manner as in the case of polymer melts.     

A new constitutive equation that models extensional flow strain hardening based on evolving natural configurations: stability analysis

Polymer melt viscoelastic fluids often exhibit in elongational flows a significant increase in the elongational viscosity known as strain hardening. This phenomenon could be related to polydispersity, e.g. the presence of a small fraction of very high molecular weight chains whose time frame relaxation spectrum is different from the small chains one. In the present work, we present a fully objective constitutive equation (CE) to primarily model extensional strain hardening based on the new concept of multiple configuration materials. Next, we analyze the CE stability properties with respect to small perturbations about the rest state.     

Melt rheology of long-chain-branched polypropylenes

Rheological properties of long-chain-branched isotactic polypropylene (PP) via copolymerization with a very small amount of nonconjugated α,ω-diene monomer using metallocene catalyst system in both linear and nonlinear regions were investigated, comparing with conventional linear and long-chain-branched PP modified at postreactor. Although comonomer incorporation was equal to 0.05 mol% or less, it caused high molecular weight, broad molecular weight distribution, and long-chain branching. A detailed study on the effect of diene incorporation on the polymer properties was conducted, comparing with modified PP in postreactor. Polymer chain microstructures were characterized by gel permeation chromatography with multiangle laser light scattering (MALLS), differential scanning calorimetry, and rheological means: dynamic viscoelasticity, step-strain, uniaxial elongational flow measurements, and large amplitude oscillatory shear. The PP, which incorporated a small amount of diene monomer, showed significantly improved viscoelastic behaviors. The diene-propylene copolymer containing long-chain branches showed extremely long relaxation mode under shear and outstanding viscosity increase under elongational flow, so-called strain hardening. The difference in microstructure of diene-propylene copolymer with modified PP with long-chain branches is investigated by MALLS and rheological characterizations.     

Rheological properties of long glass fiber filled polypropylene

Long glass fiber-filled polypropylene (PP) composites are produced by pultrusion, and the extrudate is cut at different lengths producing composites containing long fibers of controlled length. The rheological properties of such composites in the molten state have been studied using different rheometers. A capillary rheometer has been constructed and mounted on a molding-injection machine. The shear viscosity of filled PP determined from the capillary rheometer, after corrections for entrance effects, was found to be very close to that of unfilled PP. However, large excess pressure losses at the capillary entrance were observed and these data have been used to obtain an apparent elongational viscosity. The apparent elongational viscosity was shown to be considerably larger than the shear viscosity for PP and filled PP, and it increased markedly with fiber length and fiber content. Rotational rheometers with a parallel-plate geometry were used to investigate the viscoelastic properties of these composites and their behavior was found to be non-linear, exhibiting a yield stress. A model is proposed to describe the shear viscosity from a solid-like behavior at low stresses to fluid-like behavior at high shear stresses taking into account fiber content and orientation. A modified model, proposed for elongational flow, describes relatively well the apparent elongational data.     

Comparison of the elongational behaviour of a polyethylene melt at constant stress and constant strain rate

Summary Two different apparatuses for measuring the elongational behaviour of polymer melts at constant tensile stress and constant elongational strain rate are described. Measurements on a low density polyethylene were carried out up to stretching ratios of 400. The homogeneity of sample deformation in both test methods was sufficient to reach a steady-state of elongational flow where the tensile stress and the strain rate as functions of time are constant. By unloading the molten rod the recoverable strain can be determined at any state of deformation. The recoverable strain increases with growing deformation and reaches a constant value in the steady-state. The elongational viscosity calculated from the rate of viscous flow agrees with the Trouton viscosity in the case of very small deformations only. With growing deformation the elongational viscosity increases up to a constant value in the steady-state which is greater than the Trouton viscosity by about a factor of six at measured strain rates of 0.03 s^–1 and 0.1 s^–1, respectively. The elongational viscosity and the recoverable strain in the steady-state measured with the two different test methods under the same experimental conditions are in good agreement.

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Zusammenfassung Zwei verschiedene Apparaturen zur Untersuchung des Dehnverhaltens von Polymerschmelzen unter konstanter Zugspannung und konstanter Dehngeschwindigkeit werden beschrieben. An einem Polyäthylen niedriger Dichte wurden Messungen bis zu Verstreckgraden von 400 durchgeführt. Die ausreichend homogene Probenverformung erlaubt in beiden Versuchsführungen das Erreichen eines stationären Dehnfließens mit zeitlich konstanter Zugspannung und Dehngeschwindigkeit. Durch Entlasten des Schmelzestranges ist der reversible Dehnungsanteil für jeden Verformungszustand direkt zu messen. Die reversible Dehnung steigt mit wachsender Dehnverformung an, bis sich im stationären Bereich ein konstanter Wert einstellt. Die aus der Geschwindigkeit des viskosen Dehnfließens berechnete Dehnviskosität stimmt nur für den Grenzfall kleiner Deformationen mit der Trouton-Viskosität überein. Mit wachsender Dehndeformation steigt die Dehnviskosität bis zu einem Gleichgewichtswert an, der bei den gemessenen Dehngeschwindigkeiten von 0,03 s^–1 und 0,1 s^–1 um etwa einen Faktor 6 über dem Wert der Trouton-Viskosität liegt. Die unter gleichen Versuchsbedingungen mit beiden Apparaturen bestimmten Dehnviskositäten und reversiblen Dehnungen im stationären Bereich stimmen überein.

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Parts of this paper were presented at the VII^th Internat. Congress on Rheology, Gothenburg, Sweden.

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With 10 figures and 2 tables     

Modeling elongational viscosity of blends of linear and long-chain branched polypropylenes

To enhance the melt strength of a conventional linear polypropylene (L-PP), blends with a long-chain branched polypropylene (LCB-PP) were produced by adding 2, 5, 10, 25, 50, and 75 wt% of LCB-PP to L-PP and mixing in a twin screw extruder. It was found that, already, an addition of 10% or less of LCB-PP to L-PP leads to significant strain hardening. Elongational viscosity data of L-PP and LCB-PP and those of their blends were analyzed by the use of the molecular stress function (MSF) theory. While L-PP is characterized by the MSF parameter, β=1 (typical for linear melts), and shows very little chain stretch (), melt elongational behavior of LCB-PP is characterized by the MSF parameters, β=2 (typical of LCB melts), and (which corresponds to a maximum stretch of molecular chains by a factor of 15). By extruding LCB-PP, a refining effect is observed similar to the refining effects seen in low density polyethylene (LDPE), which reduces the steady-state elongational viscosity and reduces to 121. A second-order mixing rule for the fractional relaxation moduli, g _i , was found to show good agreement with the linear-viscoelastic data of the blends. To simulate the elongational viscosities of the L-PP/LCB-PP blends, a similar second-order mixing rule was used for the MSF parameter, β, while a first-order mixing rule was found to be appropriate for . This allows for a quantitative prediction of the time-dependent elongational viscosities of all L-PP/LCB-PP blends on the basis of the linear and nonlinear parameters of the mixing components L-PP and LCB-PP only. Comparison between the steady-state elongational viscosities as obtained from creep experiments shows good agreement with predictions.     

Modelling elongational and shear rheology of two LDPE melts

Experimental data of two low-density polyethylene (LDPE) melts at 200°C for both shear flow (transient and steady shear viscosity as well as transient and steady first normal stress coefficient) and elongational flow (transient and steady-state elongational viscosity) as published by Pivokonsky et al. (J Non-Newtonian Fluid Mech 135:58–67, 2006) were analysed using the molecular stress function model for broadly distributed, randomly branched molecular structures. For quantitative modelling of melt rheology in both types of flow and in a very wide range of deformation rates, only three nonlinear viscoelastic material parameters are needed: Whilst the rotational parameter, a ₂, and the structural parameter, β, are found to be equal for the two melts considered, the melts differ in the parameter describing maximum stretch of the polymer chains.     

Elongational rheology of polyethylene melts — temporary network constitutive laws

The uniaxial elongational properties of various polyethylenes have been evaluated using an elongational rheometer and a melt-strength apparatus. It is possible to derive the data obtained in elongation from the distribution of relaxation times obtained from oscillatory shearing measurements (linear viscoelasticity), using a Wagner constitutive equation. The effects of the molecular parameters of the samples have been studied, in particular the effect of polydispersity on the shape of the damping function.     

Rheological behaviour of nano-composites based on polyamide 6 under shear and elongational flow at high strain rates

In this work, the rheological behaviour of high molecular mass polyamide 6 (PA6)/organo-montmorillonite nano-composites, obtained via melt blending, was investigated under shear and extensional flow. Capillary rheometry was used for the measurement of high shear rate steady state shear viscosity and die entrance pressure losses; further, by the application of a converging flow method (Cogswell model) to these experimental results, elongational viscosity data were indirectly calculated. The extensional behaviour was directly investigated by means of melt spinning experiments, and data of apparent elongational viscosity were determined. The results evidenced that the presence of the organo-clay in filled PA6 melts modifies the rheological behaviour of the material, with respect to the unfilled polymer, in dependence on the type of flow experienced by the fluid. In shear flow, the nano-composites showed a slightly lower viscosity than neat PA6, whereas in elongation, they appeared much more viscous, in dependence on the organo-clay content.     

Elongational,shear and volume viscosities of polymer melts

Volume viscosity of polypropylene, polystyrene, and low density polyethylene are described. Shear, elongational, and volume viscosities may be interrelated with the help of scaling equations. They explain the effect on elongational viscosity of strain rates, stresses, and strains.