Material functions for constant-strain-rate elongational flows

The problem of determination of invariant material functions for elongational flows in which two components of the constant strain-rate tensor are equal is briefly discussed, and a method of its solution described. The method is based on simultaneous modeling of the elongational viscosity as measured in uniform uniaxial elongational flow, and the shear viscosity as measured in steady viscometric flow. A single integral model with a strain-rate dependent memory is used to correlate both viscosities over a given experimental range of strain rates. The procedure has been applied to a set of experimental data obtained for a low-density polyethylene melt by Laun and Münstedt.     

An intriguing empirical rule for computing the first normal stress difference from steady shear viscosity data for concentrated polymer solutions and melts

The Cox–Merz rule and Laun’s rule are two empirical relations that allow the estimation of steady shear viscosity and first normal stress difference, respectively, using small amplitude oscillatory shear measurements. The validity of the Cox–Merz rule and Laun’s rule imply an agreement between the linear viscoelastic response measured in small amplitude oscillatory shear and the nonlinear response measured in steady shear flow measurements. We show that by using a lesser-known relationship also proposed by Cox and Merz, in conjunction with Laun’s rule, a relationship between the rate-dependent steady shear viscosity and the first normal stress difference can be deduced. The new empirical relation enables a priori estimation of the first normal stress difference using only the steady flow curve (i.e., viscosity vs shear rate data). Comparison of the estimated first normal stress difference with the measured values for six different polymer solutions and melts show that the empirical rule provides values that are in reasonable agreement with measurements over a wide range of shear rates, thus deepening the intriguing connection between linear and nonlinear viscoelastic response of entangled polymeric materials.     

Is there a relation between the relaxation time measured in CaBER experiments and the first normal stress coefficient?

We investigate a variety of different semidilute polymer solutions in shear and elongational flow. The shear flow is created in the cone-plate-geometry of a commercial rheometer. We use capillary thinning of a filament that is formed by a polymer solution in the Capillary Breakup Extensional Rheometer (CaBER) as an elongational flow. We compare the relaxation time measured in the CaBER with relaxation times based on the first normal stress difference and the zero shear polymer viscosity that we measure in our rheometer. All of these three measurable quantities depend on different fluid parameters—the viscosity of the solvent, the polymer concentration within the solution, and the molecular weight of the polymers—and on the shear rate (in the shear flow measurements). Nevertheless, we find that the first normal stress coefficient depends quadratically on the CaBER relaxation time. Several scaling laws are presented that could help to explain this empirical relation.     

Solutions of xanthan gum/guar gum mixtures: shear rheology,porous media flow,and solids transport in annular flow

Mixtures of xanthan and guar gum in aqueous solution were studied in two flow situations: simple shear and porous media. In addition, solids transport in vertical annular flow of sand suspensions was explored. The zero shear rate viscosity of the solutions displayed a pronounced synergy: the viscosity of the mixture is higher than that of the polymer solutions in a wide range of relative concentrations of the two polymers, in agreement with previous literature. However, at relatively high shear rates, the viscosity approaches the value of the more viscous xanthan gum solutions at mass fractions of xanthan gum between 0.1 and 0.15, and the degree of synergy substantially decreases. Stress relaxation experiments in simple shear indicate that the polymer mixtures exhibit a well-defined yield stress after relaxation that is absent in solutions of pure polymers. In porous media flow experiments, a synergistic behavior mimicking the shear flow results was obtained for the polymer mixtures at low shear rates. However, at a critical shear rate, the apparent viscosity in porous media flows exceeds the shear viscosity due to the elongational nature of flow in the pores. The solids transport capacity in annular flows is well-represented by trends in shear viscosity and stress relaxation behavior. However, the lack of viscosity synergy at high shear rates limits the applicability of the mixtures as a way to improve solids suspension capacity in annular flows.     

Time-dependent non-linear dynamics of polymer solutions in microfluidic contraction flow—a numerical study on the role of elongational viscosity

A generalised form of the finitely extensible non-linear elastic (FENE) model for modelling non-linear flow of semi-dilute polymer solutions is proposed. It accounts for conformation-dependent polymer elasticity and predicts shear-thinning shear viscosity, non-linear elongational viscosity and first and second normal stress differences. The rheometric material functions predicted by the model are critically compared with the results of the linear Phan–Thien–Tanner model. The predictabilities of these constitutive models under benchmark flow problems are evaluated by time-dependent simulations, using finite volume method based on a CFD simulation toolbox. The effects of the model parameters, the inertia and the contraction ratio are numerically studied. The modified FENE model qualitatively captures the non-linear flow phenomena of polymer solution in the high elasticity number ( $\mathrm {El}$ ) flow regime observed in experiments. The results show that an accurate growth function of the elongational viscosity is the key to the prediction of the time-dependent highly asymmetric flow patterns.     

Numerical simulation of entry flow of the IUPAC-LDPE melt

Numerical simulations have been undertaken for the creeping entry flow of a well-characterized polymer melt (IUPAC-LDPE) in a 4:1 axisymmetric and a 14:1 planar contraction. The fluid has been modeled using an integral constitutive equation of the K-BKZ type with a spectrum of relaxation times (Papanastasiou–Scriven–Macosko or PSM model). Numerical values for the constants appearing in the equation have been obtained from fitting shear viscosity and normal stress data as measured in shear and elongational data from uniaxial elongation experiments. The numerical solutions show that in the axisymmetric contraction the vortex in the reservoir first increases with increasing flow rate (or apparent shear rate), goes through a maximum and then decreases following the behavior of the uniaxial elongational viscosity. For the planar contraction, the vortex diminishes monotonically with increasing flow rate following the planar extensional viscosity. This kinematic behavior is not in agreement with recent experiments. The PSM strain-memory function of the model is then modified to account for strain-hardening in planar extension. Then the vortex pattern shows an increase in both axisymmetric and planar flows. The results for planar flow are compared with recent experiments showing the correct trend.     

Elongational properties and molecular structure of polyethylene melts

Summary The viscosity and the recoverable strain in the steady state of elongation have been measured on several polyethylenes of different molecular structures. The elongational viscosity as a function of tensile stress runs through a more or less pronounced maximum in the nonlinear range whereas in the linear range the Trouton viscosity is reached. For low density polyethylenes it could be demonstrated that the maximum of the steady-state elongational viscosity and the elasticity expressed by the steady-state compliances in shear and tension sensitively increase if the molecular weight distribution is broadened by the addition of high molecular weight components. A variation of the weight average molecular weight does only shift the elongational viscosity curve but leaves its shape unchanged. Two of the four high density polyethylenes investigated do not show a maximum of the steady-state elongational viscosity, for the others it is less pronounced than in the case of low density polyethylenes. The influence of branching on the elongational behaviour of polyethylene melts in the steady-state and the transient region is qualitatively discussed.With 11 figures and 4 tables     

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.     

液晶高分子-各向异性流体挤出拉伸分岔研究

应用共转导数型本构方程研究了液晶高分子纺丝挤出过程的拉伸黏度,应用计算机符号运算软件 Maple得出解析表达式,拉伸黏度与拉伸率之间关系(随剪切速率变化)表明存在分岔现象,得出拉伸黏度显著高于相应的剪切黏度,解释了液晶高分子熔体挤出时不发生挤出胀大的物理机制．     

Influence of molecular structure on the extensional behaviour of polyethylene melts

The elongational behaviour of polyethylene samples having different molecular structure has been tested. Elongational viscosity measurements have been carried out using the isothermal melt spinning technique. The extensional behaviour of the different samples is analysed as a function of total strain. The effect of long-chain branching on elongational viscosities is described. A comparison is presented between elongational viscosity and melt strength data.Some of the results reported here were presented at the VIIIth International Congress on Rheology, Naples, September 1–5, 1980, cf. [16].     

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     

Tensile properties of an extruded polyethylene-melt

Summary Isothermal continuous stretching experiments with a Rheotens apparatus were carried out on an extruded polyethylene melt. Measurements of the tensile force, the thread velocity and the thread profiles for various mass flow rates resulted in the determination of the elongational viscosity as a function of the strain rate and the deformation time. It was found that the deformation time is a more suitable general parameter than the strain rate for the characterization of the elongational viscosity of extruded polymer melts.

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Zusammenfassung Experimente zum isothermen Schmelzspinnen wurden mit dem Rheotens an einer extrudierten Polyäthylenschmelze durchgeführt.Messungen von Abzugskraft, Abzugsgeschwindigkeit und Querschnittsprofil des extrudierten Stranges bei variierenden Ausstößen führten zur Bestimmung der Dehnviskosität als Funktion von Dehngeschwindigkeit und Deformationszeit. Es konnte nachgewiesen werden, daß die Deformationszeit besser als die Dehngeschwindigkeit zur Charakterisierung der Dehnviskosität von extrudierten Schmelzen geeignet ist.

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Vortrag, gehalten auf der Jahrestagung der Deutschen Rheologischen Gesellschaft e.V., Universität Dortmund, 9. bis 11. März 1977.

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With 8 figures     

Design and operation of the free jet elongational rheometer

The controlled free jet instability is used for elongational rheometry of polymer solutions. The instantaneous elongational viscosity represents a convenient comparative parameter. Its dependence on the operating parameters and the conditions of operation will be discussed.     

Prediction of steady-state viscous and elastic properties of polyolefin melts in shear and elongation

The linear and nonlinear steady-state viscosities and elastic compliances were measured in shear and elongational flows for two low-density polyethylenes, a linear polypropylene, and two metallocene catalyzed polyethylenes (one linear and one long-chain branched) by Wolff et al. (Rheol Acta 49:95?C103, 2010) and Resch (dissertation, 2010). Comprehensive data of this type are rarely found in the literature, and comprehensive modeling of both viscous and elastic effects is even rarer. In this contribution, the reliability of a modeling approach proposed by Laun (J Rheol 30(3):459?C501, 1986) and based on the damping function concept is tested. The strain hardening seen for the long-chain branched polymers and its absence in the case of the linear polymer, the stronger decrease of the tensile compliance in comparison to the shear compliance with increasing stress, as well as the extended linear-viscoelastic regime of the shear viscosity in contrast to the shear compliance are correctly modeled. While the modeling of the nonlinear response in shear can be achieved with only one material parameter for most of the polymers considered here, the nonlinear modeling in elongation is achieved with two parameters. The same parameter values are shown to describe viscous as well as elastic properties of the melts, and thus the relations of Laun can be used to test the consistency of viscosity and compliance measurements.     

Experimental estimation of elongational stresses of dilute polymer solutions and a related examination of some constitutive equations

Elongational stresses of dilute polymer solutions have been estimated by utilizing the flow through small orifices under the condition of no vortex upstream of the orifice plane. The flow was approximated with a linearly converging flow towards an apex of a cone, its validity being partially confirmed by the measured center velocities, and the elongational stresses are determined from the measured thrusts of dilute polymer solutions. On the other hand, elongational stresses were theoretically obtained with the modified Maxwell model and the second order fluid. A comparison was made between the experimental and the theoretical results and the following points were clarified; below an elongational rate of 2 × 10⁴ s⁻¹ the modified Maxwell model gives elongational stresses close to the experimentally determined ones, but above that elongational rate it deviates from the experimental results. The second order fluid is not sufficient to describe the stresses in this kind of elongational flow and an acceleration term such as δ²e^ij/δt² may be necessary in this case.     

The role of elongational viscosity in the mechanism of drag reduction by polymer additives

The role of elongational viscosity in the mechanism of drag reduction by polymer additives is investigated qualitatively by means of direct numerical simulations of a turbulent pipe flow. For the polymer solution, a generalised Newtonian constitutive model is utilised in which the viscosity depends on the second and third invariant of the rate-of-strain tensor via an elongation parameter. This elongation parameter is capable of identifying elongational type of regions within the flow. The simulations show that complementary to stretching of the polymers, also compression must be incorporated to have drag reduction, contrary to many suggestions done in the literature on the mechanism which assume that stretching of the polymers is most important.     

Channel flow of a liquid crystal polymer around an obstacle: Weld line structure and strain field

 We have studied by in situ microscopy the flow of a lyotropic liquid crystal polymer, hydroxypropylcellulose (HPC) in water, around an obstacle placed in a rectangular flow channel. The obstacle separates the flow into two parts which rejoin downstream of the obstacle, resulting in the formation of a `weld-line'. Measuring the velocity field in the vicinity of the weld-line beyond the obstacle, we find as expected a positive elongational strain (acceleration) along the weld (parallel to the flow direction). For an anisotropic (concentrated) HPC solution we observe in addition a significant shear strain in the weld-line region, there being an important velocity gradient perpendicular to the plane of the weld line. Isotropic (lower concentration) solutions of the same polymer demonstrate no visible weld line, a larger elongational strain rate near the obstacle, and no shear component of strain downstream of the obstacle. These results are similar to observations reported for fluids reinforced by macroscopic fibres. Polarised light observations of the anisotropic solution show that the strain field generates a generally increased degree of orientation of the liquid crytalline polymer near the weld (generally reduced crossed-polariser transmitted intensity when the polariser is parallel to the flow direction), however there is also a striking fine birefringent colour variation in the weld-line region, reminiscent of the structure observed at the channel side walls in rectangular channel flow (Haw and Navard 2000). The results show that the simple concept of weld-line structure as confined to an enhanced alignment along the weld due to elongational strain is incomplete; the two-dimensional shear strain field must also be taken into account for the anisotropic fluid. Received: 22 December 1999/Accepted: 4 January 2000     

Effect of a controlled pre-deformation history on extensional viscosity of dilute polymer solutions

We use a modified filament stretching rheometer to quantify the influence of a known controlled pre-shear history on the transient extensional viscosity of a dilute polymer solution. Two different types of pre-deformation are explored; both influence the subsequent stretching significantly, albeit in opposite ways. Small-amplitude oscillatory straining parallel to the direction of stretching enhances strain hardening and accelerates the tensile stress growth toward the steady-state value. Conversely, steady torsional shearing orthogonal to the direction of stretching retards strain hardening and results in a delayed approach to steady-state elongational flow. In both cases, the final steady-state extensional viscosity is the same as that observed with no pre-shearing. Calculations using a finitely extensible nonlinear elastic Peterlin dumbbell model qualitatively capture the trends observed in experiments, enabling interpretation of these observations in terms of the degree of polymer chain stretching imposed by the flow before extensional stretching.     

Experiments on elongational flow of dilute polymer solutions: Part II: velocity field for the flow through small apertures

Following Part I which reported on the jet reaction and the excess pressure drop, this paper describes the results of flow velocities measured upstream of small apertures for dilute polymer solutions. The results suggest the existence of the two flow regimes reorted by Boger and Cable: the vortex growth regime and the divergent flow regime. The former corresponds to the regime of zero jet reaction and the latter to the regime of positive jet reaction. The axial velocity development for PEO solutions along the center line upstream of the aperture is found to yield an extremely high velocity rise at the onset of the vortex region and to give a nearly constant elongational rate within the vortex region. The constant elongational rate takes values such that the product with the relaxation time is about 0.5. Separan solutions do not provide so high a velocity rise at the onset of the vortex region but show an exponential rate of elongation in the vortex region.It is shown, by using the upper convected Maxwell model, that the steep velocity rise at the onset of the vortex region is given in the simplified flow model and the constant elongational rate within the vortex region holds the elongational stress nearly constant.     

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.