Quantitative evaluation of extrudate swell from viscoelastic properties of polystyrene

A theory of extrudate swell for short, intermediate or long dies is presented. In our experiment, we consider that the swelling phenomenon is mainly due to the recoverable elongational strain induced by the converging flow at the die entrance, as well as by recoverable shear strain originating within the die. From these concepts, an equation has been derived for the quantitative prediction of extrudate swell from the elastic material properties such as the entrance pressure drop, the relaxation modulus and the recoverable shear strain. Excellent agreement is found between predicted and measured values of extrudate swell obtained on commercial polystyrene melt, using capillaries of length-to-diameter ratios ranging from 1 to 20 and in a wide range of shear rates.     

Three-dimensional study of extrusion processes by Boundary Element Method.

In this paper we report an implementation of a Boundary Element Method (BEM) dealing with three-dimensional extrusion problems of viscous fluid. Die shapes with and without sharp corners and their influence on the final shape of extrudate have been studied: both shrinkage and swelling of the extrudates relative to the original die shapes have been discovered. The first half of this paper deals with the problem of implementing isoparametric boundary elements. It is followed by results of test problems, and finally the results of extrusion study are presented.     

Prediction of extrudate swell in polymer melt extrusion using an Arbitrary Lagrangian Eulerian (ALE) based finite element method

Accurate prediction of extrudate (die) swell in polymer melt extrusion is important as this helps in appropriate die design for profile extrusion applications. Extrudate swell prediction has shown significant difficulties due to two key reasons. The first is the appropriate representation of the constitutive behavior of the polymer melt. The second is regarding the simulation of the free surface, which requires special techniques in the traditionally used Eulerian framework. In this paper we propose a method for simulation of extrudate swell using an Arbitrary Lagrangian Eulerian (ALE) technique based finite element formulation. The ALE technique provides advantages of both Lagrangian and Eulerian frameworks by allowing the computational mesh to move in an arbitrary manner, independent of the material motion. In the present method, a fractional-step ALE technique is employed in which the Lagrangian phase of material motion and convection arising out of mesh motion are decoupled. In the first step, the relevant flow and constitutive equations are solved in Lagrangian framework. The simpler representation of polymer constitutive equations in a Lagrangian framework avoids the difficulties associated with convective terms thereby resulting in a robust numerical formulation besides allowing for natural evolution of the free surface with the flow. In the second step, mesh is moved in ALE mode and the associated convection of the variables due to relative motion of the mesh is performed using a Godunov type scheme. While the mesh is fixed in space in the die region, the nodal points of the mesh on the extrudate free surface are allowed to move normal to flow direction with special rules to facilitate the simulation of swell. A differential exponential Phan Thien Tanner (PTT) model is used to represent the constitutive behavior of the melt. Using this method we simulate extrudate swell in planar and axisymmetric extrusion with abrupt contraction ahead of the die exit. This geometry allows the extrudate to have significant memory for shorter die lengths and acts as a good test for swell predictions. We demonstrate that our predictions of extrudate swell match well with reported experimental and numerical simulations.     

The effect of temperature gradients on the sharkskin surface instability in polymer extrusion through a slit die

The sharkskin surface instability is commonly observed in the extrusion of polymer melts. We present a series of experiments in which a specifically designed rectangular slit die with insulated and independently heated sides and is used to induce precise temperature gradients across a flowing polyethylene melt. Our previous experiments demonstrated that the character of the surface distortions produced by the sharkskin instability was a function of the die wall temperature and therefore the extrudate had viscoelastic properties at the surface. In this paper, we explore the role of temperature and viscoelastic property gradients near the capillary wall. The amplitude of the sharkskin instability is quantified and plotted against apparent shear and extension rates. Analysis of the data demonstrates that the amplitude and frequency of the instability is independent of bulk temperature and temperature gradient and is dependent only on wall temperature. The data are normalized using a dimensionless Weissenberg number based on the extension rate to collapse the data collected over all temperatures and gradients onto a single master curve. We conclude with an example of a rectangular extrudate exhibiting varying surface roughness due to differential die heating and discuss the implications of our observations on the sharkskin surface instability mechanism and on commercial applications.     

A pseudo-time integral method for non-isothermal viscoelastic flows and its application to extrusion simulation

A pseudo-time integral scheme based on a finite streamline element method is developed to combine variable temperature with viscoelasticity. A specific KBKZ integral model for isothermal flow is transformed to its non-isothermal version by introducing a pseudo-time and applying the Morland-Lee hypothesis. The coupling between momentum and energy equations is through the time-temperature shifting factor by which the pseudo-time is defined. The observer time and the pseudo-time are simultaneously calculated when tracing the strain history for the stress calculation in a non-homogeneous temperature field. Using this scheme, a full non-isothermal numerical simulation of some IUPAC extrusion experiments is carried out. Results show that while the temperature distribution near the die exit plane is an important factor controlling extrudate swell, either self-heating inside the die tube or external cooling on the free surface dominantly determines the temperature distribution near the die exit when the wall temperature is kept constant, depending on whether the Péclet number is large or small. The hot layer effect predicted by the inelastic swell mechanism is confirmed and well illustrated by the computation. Calculations with reasonable thermal boundary conditions further convince us that the isothermal assumption in our earlier numerical simulation is a good approximation in this particular case.     

An experimental study of food melt rheology

The shear viscosity of commercial maize grits, potato powder and a low density polyethylene has been measured under a range of extrusion processing conditions using an extruder-fed slit die viscometer and a capillary rheometer. The results show the strong dependence of the viscosity of food melts on the processing history undergone during extrusion. To this end, the shear viscosity data for the food materials have been fitted to relationships including the effects of temperature, shear rate and moisture. The effect of the shear processing history on the viscosity has been represented by a power-law relationship with extruder screw speed.     

Biaxial orientation of polymers by solid state extrusion through convergent-divergent dies

Solid-state extrusion of crystalline thermoplastics is a well known method for the production of monoaxially oriented filaments exhibiting very high modulus and strength. This is achieved primarily by forcing the polymer through a converging conical die at temperatures below its melting point.In the present study the polymer is subjected to deformations simultaneously along the longitudinal and transverse directions by using dies featuring converging and diverging walls, perpendicular to each other, to produce flat extrudates exhibiting biaxial orientation. Two geometries are examined to determine the factors controlling the relative magnitude of the orientation in the extrusion and transverse directions respectively, which is being assessed by measuring the birefringence and tensile strength in various sections of the extrudates.Experiments have been carried out over a wide range of temperatures on billets of PTFE and UHMWPE, using dies mounted on a compression testing apparatus to measure the extrusion forces.The mechanics of the converging-diverging flow has been analysed to calculate the extrusion forces as function of the yield strength and coefficient of friction of the polymer and to establish the relationship between type of orientation in extruded products and die geometry.     

Effect of processing and particulate fillers on the rheology of a nematic polymer melt

Processing of a nematic HBA/HNA polymer melt increases the capillary diameter dependence of the viscosity and induces structural changes which are evident in oscillatory shear, but cannot be characterized by DSC. The effect of 6-m calcium carbonate particulate fillers is to increase the viscosity uniformly. Low concentrations of sub-micron carbon black particles cause an unexplained viscosity minimum in a large (30-mil) capillary.     

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．     

Boundary element simulations of spheres settling in circular,square and triangular conduits

Numerical simulations of a spherical particle sedimenting in circular, triangular and square conduits containing a viscous, inertialess, Newtonian fluid were investigated using the Boundary Element Method (BEM). Settling velocities and pressure drops for spheres falling along the centre-lines of the conduits were computed for a definitive range of sphere sizes. The numerical simulations for the settling velocities showed good agreement with existing experimental data. The most accurate analytic solution for a sphere settling along the axis of a circular conduit produced results which were almost indistinguishable from the present BEM calculations. For a sphere falling along the centre-line of a square conduit, the BEM calculations for small spheres agreed well with analytic results. No analytic results for a sphere falling along the axis of a triangular conduit were available for comparison. Extrapolation of the BEM predictions for the pressure drops, to infinitely small spheres, showed remarkable agreement with analytic results. For the circular conduit, the sphere's settling velocity and angular velocity were computed as a function of drop position for small, medium and large spheres. Excellent agreement with a reflection solution was achieved for the small sphere. In addition, end effects were investigated for centre-line drops and compared where possible with available experimental data and analytic results.Los Alamos National Laboratory, Los Alamos, New Mexico, USA.     

Effect of radial flow in the die entrance region on gross melt fracture of PDMS extrudate

The gross melt fracture defect is related to the flow instabilities developed in the contraction region. To mitigate these upstream instabilities, a convergent radial flow in the die entrance has been created. In fact, the ultimate objective of the present work is to examine the effect of the clearance width of radial flow on the appearance and development of gross melt fracture defect. So, capillary rheometer experiments were performed with linear polydimethylsiloxane (PDMS) oil.As for the influence of radial flow width on the morphology of gross melt fracture defect, extrudate photographs show that this imperfection can be mitigated since its frequency is higher and amplitude smaller when the gap of radial flow decreases. Such results may be related both to shear and elongational components of radial flow. Actually, when gap width is very small compared to the external diameter of radial flow, shear deformations become more enhanced with respect to the elongational deformations and thus the helical gross melt fracture becomes more like a surface defect than volume defect.     

Numerical simulation of delayed die swell

In a recent paper, Joseph et al. showed that, for a number of viscoelastic fluids, one can observe the phenomenon of delayed die swell beyond a critical extrusion velocity, or beyond a critical value of the viscoelastic Mach number. Giesekus had also observed that delayed die swell is a critical phenomenon.In the present paper, we find a set of material and flow parameters under which it is possible to simulate delayed die swell. For the viscoelastic flow calculation, we use the finite element algorithm with sub-elements for the stresses and streamline upwinding in the discretized constitutive equations. For the free surface, we use an implicit technique which allows us to implement Newton's method for solving the non-linear system of equations. The fluid is Oldroyd-B which, in the present problem, is a singular perturbation of the Maxwell fluid. The results show very little sensitivity to the size of the retardation time. We also show delayed die swell for a Giesekus fluid.This paper is dedicated to Professor Hanswalter Giesekus on the occasion of his retirement as Editor of Rheologica Acta.     

Einsatz von DMS- und Piezomeßtechnik zur direkten Schubspannungsermittlung in der Kapillarrheometrie

Zusammenfassung Der Stand der Kapillarheometrie mit den Vor- und Nachteilen der einzelnen Systeme und Meßverfahren wird dargestellt. Ausgehend von diesem Stand werden zwei alternative Meßverfahren, nämlich Piezo-Einzelkraftund DMS-Meßverfahren, zur direkten Schubspannungsermittlung besprochen und eine nach diesem Prinzip konzipierte Versuchsdüse vorgestellt. Die mit dieser Düse erhaltenen Meßergebnisse zeigen im Vergleich zu konventionell, d. h. unter Anwendung des Bagley-Verfahrens, ermittelten Fließkurven im Falle der Einzelkraftmessung gute Übereinstimmung. Das Problem der Einlaufdruckkorrektur ist bei diesem Konzept von vornherein nicht vorhanden, und der ununterbrochene Fließkanal erlaubt, bei entsprechender Erweiterung auf zwei Kraftaufnehmer, einen unproblematischen Einsatz im On-line- oder sogar im In-line-Betrieb in der Polymerverarbeitung.Schlüsselwörter Rheometrie; Kapillarrheometrie; Schubspannungsermittlung; Piezokraftmessung; DMS-Anwendung; Kunststoffschmelzen; On-line-Rheometrie

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The state-of-the-art of capillary rheometry, especially the advantages and disadvantages of different systems, is presented. Two measuring methods, a piezo single-force transducer, and a strain-gauge device combined in a testing die are compared. The strain gauges are fixed at the outside of a small capillary to detect the resulting wall shear stress between inner surface of the die and polymer flow by measuring the axial strain in the wall of the die. The single force method uses an elastic section of the die which uncouples the die from the supporting structure without changing the cross-section of the flow. The uncoupled part of the die is supported by a stiff, piezo-force transducer that detects the integral effect of wall shear stress. The resulting flow curves compare well with those using the Bagley method. With the novel die no entrance pressure-loss correction is needed. In principle, it also could be used as an on-line device for polymer processing.

Vortrag, gehalten auf der Jahrestagung der Deutschen Rheologischen Gesellschaft, TH Darmstadt, 19.–21. April 1989.     

Extrudate swell of Newtonian fluids from converging and diverging annular dies

Finite element results are presented for the extrudate swell of Newtonian fluids from converging and diverging annular dies. Numerical calculations for a variety of diameter ratios and taper angles show the dependance of diameter and thickness swell on the angle. For diverging dies a thickness contraction occurs for angles greater than 30 degrees, while the diameter swell increases rapidly. For converging dies the design is limited to angles that do not allow contact of the inner free surfaces. The present results show that the diameter swell is highest for the diverging, followed by the straight and then the converging dies.     

The effect of die exit curvature,die surface roughness and a fluoropolymer additive on sharkskin extrusion instabilities in polyethylene processing

This paper reports experimental observations and numerical simulations relating to sharkskin extrusion instabilities for two different types of polyethylene, a metallocene high-density polyethylene (HDPE) and a linear low-density polyethylene (LLDPE). Experimental results are presented for both the effect of die exit curvature and die surface roughness for slit die geometry. Matching polyflow numerical simulations are also reported and are shown to be qualitatively consistent with experimental observations. The onset of the sharkskin instability is correlated with the magnitude of the stress concentration at the die exit, and is found to be sensitive to both the melt/wall separation point for a curved exit die, and the level of partial slip at the die wall. Additional observations on the effect of a fluoropolymer additive also support the sensitivity of the sharkskin instability to partial slip at the wall.     

Zum Einfluß von Füllstoffen auf das rheologische Verhalten von hochmolekularen Polyethylenschmelzen

Zusammenfassung Zur Sichtbarmachung der Strömungsvorgänge im Bereich der konvergenten Strömung vor einer Kapillaren wurde ein Verfahren entwickelt, das die Eigenschaften bestimmter teilkristalliner Polymere bei der Kristallisation unter Scherung bzw. aus einer vorgescherten Schmelze nutzt. Diese zeigen im erstarrten Zustand sogenannte Fließlinien, die aus in bestimmter Weise auf reihenförmig angeordnete Keime aufgewachsenen Kristallitstrukturen bestehen. Solche linienförmigen Aggregate sind in der Lage, ein Stromlinienbild der Strömung in der Schmelze im erstarrten Polymer wiederzugeben. Für ein LDPE vom Spritzgußtyp konnte die Entwicklung der Einlaufwirbel mit wachsender Schergeschwindigkeit dargestellt werden. Die hochmolekularen Polymertypen erwiesen sich schon bei geringsten Durchsätzen als instabil, die Einlaufwirbel verschwanden dabei vollständig, und die Bilder des erstarrten Polymers legen einen Mechanismus der Instabilitäten nahe, bei dem jeweils unterschiedliche Teilvolumina zeitweise in die Kapillare einströmen, bei Überschreiten einer kritischen Deformation jedoch abreißen und zurückschnellen, während andere Volumenelemente ihren Platz einnehmen. Das Verhalten der Suspensionen unterscheidet sich diesbezüglich nicht von dem der ungefüllten Schmelzen.Ein signifikanter Einfluß der Füllstoffe zeigt sich jedoch beim Einströmen in Kapillaren. Während die ungefüllten Schmelzen nach sehr kurzer Zeit ein laminares Strömungsprofil ausbilden, findet man mit zunehmender Füllung im Bereich stabiler Einströmbedingungen einen sehr hohen Einlaufdruckverlust bei kurzen Kapillaren, nicht jedoch bei Lochdüsen. Dies deutet auf einen Verfestigungsmechanismus bei der Ausbildung des laminaren Strömungsprofils hin, der durch Aggregierungsvorgänge im Bereich der hohen Dehnbeanspruchung in der konvergenten Einlaufströmung bewirkt wird.

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For the visualization of polymer flow in the entry region of a capillary, a method has been developed that uses distinct crystallization phenomena of the partially crystalline polymer itself solidifying from a presheared melt. The solid polymer shows so-called flow lines made up of transcrystalline overgrowth on rows of nuclei capable of showing a picture of the past melt-flow streamlines. The development of circulating flow in the entry region of capillaries could be shown with an injection-molding type LDPE. The higher molecular-weight types appeared to show unstable flow with disappearing circulating flow in the corners even at very low flow rates. The bulk polymer seems to flow into the capillary from certain parts of the reservoir until a critical strain is reached, then snaps back, giving way for the same flow type from another part of the reservoir. The behaviour of the suspensions does not differ in this respect from that of the pure melts.A significant filler influence is, however, shown within the capillary. While unfilled polymer melts reach steady flow conditions only one or two diameters downstream of the capillary entrance, an entry pressure loss increasing with increasing filler content is found in short capillaries (l/d = 5 – 15), but not in no-land dies (l/d 1). This seems to indicate a flow hardening mechanism in the region where the steady laminar flow profile develops and which might result from the very high extensional strain (next to instability) in the converging flow of the capillary entrance.

Zweiter Teil einer vom Fachbereich Chemietechnik der Universität Dortmund genehmigten Dissertation.     

Nonlinear viscoelasticity of polymer melts in different types of flow

The nonlinear viscoelastic properties of a fairly large class of polymeric fluids can be described with the factorable single integral constitutive equation. For this class of fluids, a connection between the rheological behaviour in different flow geometries can be defined if the strain tensor (or the damping function) is expressed as a function of the invariants of a tensor which describes the macroscopic strain, such as the Finger tensor. A number of these expressions, proposed in the literature, are tested on the basis of the measuring data for a low-density polyethylene melt. In the factorable BKZ constitutive equation the strain-energy function must be expressed as a function of the invariants of the Finger tensor. The paper demonstrates that the strain-energy function can be calculated from the simple shear and simple elongation strain measures, if it is assumed to be of the shape proposed by Valanis and Landel. The measuring data for the LDPE melt indicate that the Valanis-Landel hypothesis concerning the shape of the strainenergy function is probably not valid for polymer melts.     

Rheological characteristics of glass-fibre-filled polypropylene melts

The rheological properties of glass fibre-filled polypropylene melts have been investigated. A high pressure capillary rheometer has been used for the experimental study. The effect of shear rate, temperature, and fibre concentration on the melt viscosity and viscoelastic properties have been studied. An equation has been proposed to correlate the melt viscosity with shear rate, temperature and fibre content. A master curve relation on this basis has been brought out using the shift factora _T . a _T shift factor (=/ _r ) - A _i coefficients of the polynomical of eq. (1) (i = 0, 1, 2, ,n) - B constant in the AFE equation (eq. (2)) (Pa s) - B constant in eq. (3) - D extrudate diameter - d capillary diameter - activation energy at constant shear rate (kcal/mole) - E activation energy at constant shear stress (kcal/mole) - T melt temperature (K) - X fraction glass fibre by weight - shear rate (s^–1) - shear viscosity (Pa s) - normal stress coefficient (Pa s²) - ₁ – ₂ first normal-stress difference (Pa) - shear stress (Pa) - r at reference temperature     

Enhanced melt strength and stretching of linear low-density polyethylene extruded under strong slip conditions

The influence of extrusion under strong slip conditions on the extensional properties of linear low-density polyethylene was studied in this work. The material was extruded at two different temperatures under strong slip and no slip conditions, and was subsequently subjected to uniaxial elongational flow by means of a Rheotens device. Strong slip was evident through the elimination of sharkskin distortions and the stick-slip instability, as well as by the electrification of the extrudates. The extrudate swell was smaller in the presence of slip when comparing with no slip conditions at constant apparent shear rate, but it was found to be a unique function of the shear stress if comparison was performed at constant stress. The draw ratio and melt strength of the filaments obtained under slip conditions were larger compared to those without slip. In addition, draw resonance was postponed to higher draw ratios during the extrusion with strong slip at constant apparent shear rate. It is suggested that slip of the polymer at the die wall decreases the shear stress in the bulk, and therefore, restricts the disentanglement and orientation of macromolecules during flow, which subsequently produces the increase in draw ratio and melt strength during stretching.     

An experimental study of exit pressures for polymer melts

A slit die apparatus is used to measure exit pressures for five different polymer melts. Viscosity data obtained from the same apparatus agree well with values obtained from a cone-and-plate rheometer or a capillary rheometer. Except for a PVC sample where thermal degradation was found to occur, the exit pressures obtained by linear extrapolation of the measured pressure profiles are all positive, and increase with increasing shear stress. The values of the first normal stress difference calculated according to the exit pressure theory are of the right order of magnitude and in some cases correlate satisfactorily with values measured in a cone-and-plate rheometer. However, the high sensitivity of the exit pressure values to the method of extrapolation and the wild scatter of exit pressure data for some materials make it difficult to use the exit pressure method as a routine procedure for accurate determination of the first normal stress difference.