Tensile deformations in molten polymers

Summary This paper describes three complementary methods of measuring the rheological response of polymer melts under tension. A specially designed rheometer maintains constant stress throughout the deformation and gives well defined data, but is limited to high viscosity materials. The two other systems which have been studied are the deformation during the drawdown of an extrudate and that of convergent flow. A wide range of materials and degrees of deformation can be studied though the results are less precise. With empirical assumptions these systems yield data which are consistent with those obtained from the rheometer over a wide range of conditions.Viscous, elastic and plastic deformations are observed. The viscosity is, in general, independent of stress: its value is three times the limiting shear viscosity at low stress. Important departures from this rule occur for polythene and polypropylene. The viscosity of low density polythene increases with stress over a limited range of stress. This behaviour is associated with branching. Theviscosityof polypropylene decreases with stress. The elastic modulus at low stress is commonly in the range 10⁵–10⁶ dyn/cm² and is largely independent of temperature and molecular weight. At high stress there is a limiting elastic strain of about 2.0 units. Low density polythene has a plastic breaking stress of about 10⁷ dyn/cm². The breaking stress increases with temperature.

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Zusammenfassung Die Arbeit beschreibt drei sich ergänzende Methoden zur Messung des Theologischen Verhaltens polymerer Schmelzen unter Zugbeanspruchung. Ein speziell ausgelegtes Rheometer sorgt für eine konstante Spannung während der Deformation und ergibt eindeutige Meßgrößen. Es ist allerdings auf hochviskose Stoffe beschränkt. Die beiden anderen Methoden, die untersucht wurden, sind zum einen die Deformation während des Abziehens eines Extrudates und zum anderen die der konvergenten Strömung. Auf diese Weise können zahlreiche Stoffe und Verformungsgrade untersucht werden, obwohl die Ergebnisse weniger genau sind. Unter empirischen Annahmen ergeben diese Systeme Größen, die mit den mit Hilfe des Rheometers für einen weiten Bereich von Bedingungen ermittelten Ergebnissen übereinstimmen.Viskose, elastische und plastische Deformationen werden beobachtet. Die Viskosität ist im allgemeinen unabhängig von der Spannung; ihr Wert ist dreimal größer als die Grenz-Scherviskosität (bei geringer Spannung). Wichtige Abweichungen von dieser Regel treten bei Polyäthylen und Polypropylen auf. Die Viskosität von Polyäthylen niedriger Dichte wächst mit der Spannung in einem begrenzten Spannungsbereich. Dieses Verhalten ist von Verzweigungen begleitet. Die Viskosität des Polypropylens fällt dagegen mit der Spannung ab. Der Elastizitätsmodul liegt im allgemeinen zwischen 10⁵ und 10⁶ dyn/cm² und ist weitgehend von der Temperatur und dem Molekulargewicht unabhängig. Bei hoher Spannung zeigt sich eine begrenzende elastische Dehnung von etwa 2 Einheiten. Polyäthylen geringer Dichte hat eine plastische Bruchspannung von etwa 10⁷ dyn/cm². Die Bruchspannung wächst mit der Temperatur.

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Paper presented at the Conference on Experimental Rheology, University of Bradford, April 17–19, 1968.     

Rheology of molten polymers

Summary TheCross equation describes the flow of pseudoplastic liquids in terms of an upper and a lower Newtonian viscosity corresponding to infinite and zero shear, and ₀, and of a third material constant related to the mechanism of rupture of linkages between particles in the intermediate, non-Newtonian flow regime, Calculation of of bulk polymers is important, since it cannot be determined experimentally. The equation was applied to the melt flow data of two low density polyethylenes at three temperatures.Using data in the non-Newtonian region covering 3 decades of shear rate to extrapolate to the zero-shear viscosity resulted in errors amounting to about onethird of the measured ₀ values. The extrapolated upper Newtonian viscosity was found to be independent of temperature within the precision of the data, indicating that it has a small activation energy.The ₀ values were from 100 to 1,400 times larger than the values at the corresponding temperatures.The values of were large compared to the values found for colloidal dispersions and polymer solutions, but decreased with increasing temperature. This shows that shear is the main factor in reducing chain entanglements, but that the contribution of Brownian motion becomes greater at higher temperatures.

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Zusammenfassung Die Gleichung vonCross beschreibt das Fließverhalten von pseudoplastischen Flüssigkeiten durch drei Konstante: Die obereNewtonsche Viskosität (bei sehr hohen Schergeschwindigkeiten), die untereNewtonsche Viskosität ₀ (bei Scherspannung Null), und eine Materialkonstante, die vom Brechen der Bindungen zwischen Partikeln im nicht-Newtonschen Fließbereich abhängt. Die Berechnung von ist wichtig für unverdünnte Polymere, wo man sie nicht messen kann.Die Gleichung wurde auf das Fließverhalten der Schmelzen von zwei handelsüblichen Hochdruckpolyäthylenen bei drei Temperaturen angewandt. Die Werte von ₀, durch Extrapolation von gemessenen scheinbaren Viskositäten im Schergeschwindigkeitsbereich von 10 bis 4000 sec^–1 errechnet, wichen bis 30% von den gemessenen ₀-Werten ab. Die Aktivierungsenergie der war so klein, daß die-Werte bei den drei Temperaturen innerhalb der Genauigkeit der Extrapolation anscheinend gleich waren. Die ₀-Werte waren 100 bis 1400 mal größer als die-Werte.Im Verhältnis zu kolloidalen Dispersionen und verdünnten Polymerlösungen war das der Schmelzen groß, nahm aber mit steigender Temperatur ab. Deshalb wird die Verhakung der Molekülketten hauptsächlich durch Scherbeanspruchung vermindert, aber der Beitrag derBrownschen Bewegung nimmt mit steigender Temperatur zu.

     

The velocity profiles of molten polymers during laminar flow

Summary The velocity profiles of the flow of some polymeric melts (low density polyethylene, two high density polyethylenes and a polydimethyl siloxane) through a rectangular slit have been measured. The velocity profiles measured were in good agreement with those calculated from the flow curves. In agreement with the assumption usually made for laminar flow, the velocities at the wall were found to be zero in all cases. This was true even during melt fracture for a low density polyethylene and a polydimethyl siloxane.     

An instability in the flow of molten polymers

Ohne Zusammenfassung     

An instability in the flow of molten polymers

Summary In the flow of molten high polymers through capillaries, an instability occurs at shear stresses near 10⁶ dynes/cm². The instability results in emerging streams of irregular shape. Several explanations have been proposed:Reynolds' turbulence, buckling during elastic recovery of the emerging stream, and rupture or fracture of the molten polymer at the entrance to the capillary. The first two are found not to apply. The latter appears correct on the bases of experimental observations and rheological considerations.

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Zusammenfassung Das Fließen von geschmolzenen Hochpolymeren zeigt Instabilitäten der Scherspannung in der Nähe von 10⁶ dyn/cm². Die Instabilitäten haben Irregularitäten in der Form der austretenden Stränge zur Folge. Verschiedene Erklärungen wurden vorgeschlagen:Reynoldssche Turbulenz, Aufkrümmung während der elastischen Erholung des Ausströmenden und Zerreißen oder Bruch des geschmolzenen Polymeren bei Eintritt in die Kapillare. Die ersten beiden kommen nicht in Frage, die letztere Erklärung scheint auf Grund der experimentellen Beobachtungen und der rheologischen Betrachtungen das Richtige zu treffen.

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The suggestions ofW. E. Langlois are gratefully acknowledged.     

Weissenberg effect and its dependence upon the experimental geometry

The flow of a second-order fluid with a free surface between two coaxially mounted cylinders of finite length, the inner one of which is rotating, is being studied. In the case of slow flow and small shear rates the flow can be divided into a primary flow in the plane perpendicular to the axis of rotation and a secondary flow in the meridional plane. These flow components are numerically calculated and the results are compared with the analytical results for the semi-infinite cylinder approximation. The influence of the finiteness of the cylinders (end effect) upon the free surface deformation is analysed. The numerical results for the secondary flow are compared with results obtained by flow visualisation.     

A shear stress transducer for molten polymers

Present rheometrical techniques are inadequate for the measurement of viscoelastic properties associated with shearing at high rates. A possible solution to this problem is to use a sliding plate rheometer together with a device for measuring the local wall shear stress away from the ends and edges of the plates. Such a device has been constructed, and the results of preliminary tests are encouraging.     

Helical flow of molten polymers in a cylindrical annulus

Summary The paper is concerned with an analytical investigation of helical flow of a non-Newtonian fluid through an annulus with a rotating inner cylinder. The shear dependence of viscosity is described by a power law and the temperature dependence by an exponential function.Velocity and temperature profiles, energy input and shear along the stream lines, pressure drop, and torque are presented for the range of input parameters encountered in polymer extrusion.The results of the study can be applied to a mixing element in a screw extruder and for a device to control extrudate temperature and output.Nomenclature a thermal diffusivity [m²/s] - b temperature coefficient [K^–1], see eq. [4] - c heat capacity [J/kg K] - h slot width [m] - I ₁,I ₂,I ₃ invariants of the rate of deformation tensor, see eq. [5] - k thermal conductivity [J/m s K] - l, L = 1/h length of the slot - l _T ,l _K thermal and kinematic entrance length - m power law exponent, see eq. [3] - M torque [m N] - p pressure [N/m²] - P dimensionless pressure gradient, see eq. [24] - P _R,P _RZ dimensionless components of the shear stress tensor, see eq. [25] and eq. [26] - r, R = r/r _wa radial coordinate - r _wa, r_wi outer and inner radius of annulus [m] - t time [s]; dwell time in the annulus - T temperature [K] - v , v_r, V_z velocity components [m/s] - v ₀ angular velocity at inner wall [m/s] - average velocity inz-direction [m/s] - V , V_R, V_Z dimensionless velocity components,v /v₀, v_r/v₀, v_z/v₀ - V _z velocity ratio, helical parameter - Y coordinate inr-direction, see eq. [20] - z, Z = z/h Pe axial coordinate - deformation - rate of deformation tensor [s^–1] - apparent viscosity [N s/m²], see eq. [3] - dimensionless temperature,b (T – T ₀) - azimuth coordinate - ratio of radii,r _wi/r_wa - density [kg/m³] - , kl shear stress tensor [N/m²] - fluidity [m²w/Nw s], see eq. [4] - Gf Griffith number, see eq. [12] - Pe Péclet number, see eq. [13] - Re Reynolds number, - 0 initial state, reference state - equilibrium state - e entrance - wi, wa at surface of inner or outer wall - r, R, z, Z, coordinates - i, j radial and axial position of nodal point in the grid - k, l tensor components Presented at Euromech 37, Napoli 6. 20–23. 1972.With 15 figuresDedicated to Prof. Dr.-Ing. G. Schenkel on his 60th birthday     

The dynamic performance of the Weissenberg Rheogoniometer

The dynamic performance of a standard Model R18 Weissenberg Rheogoniometer has been studied in detail. The Rheogoniometer was carefully calibrated and used to measure accurately the rheological behaviour of a highly nonlinear viscoelastic polymer solution (1% polyacrylamide in 50% glycerol/water).In this paper the elaborate procedures that were used to calibrate the electronic signal processing equipment are described. The various static and dynamic calibration/correction factors are defined and incorporated into a computer implemented calculation scheme for evaluating the linear dynamic properties from the raw digital transfer function analyser readings.The linear dynamic properties of the polymer solution are presented together with the corresponding steady shearing properties. Both cone and plate and parallel plates geometries were used and good agreement was obtained over the wide range (six decades) of frequencies and shear rates employed.Fluid inertia effects were found to become important when the modified Reynolds number,Re _c ² orRe(H/R)², exceeded a value of about 0.1. These effects had a strong influence on the phase angle() which could readily be detected by varying the gap angle/width. The Walters-Kemp equations were found to give consistently accurate values for the linear dynamic properties for modified Reynolds numbers up to 11.6 which was the highest reached.     

Excess pressure losses in the capillary flow of molten polymers

The excess pressure losses due to end effects in the capillary flow of a linear low-density polyethylene resin (LLDPE) were studied both experimentally and numerically. First, they were determined experimentally by using two methods: i) by extrapolating experimental data of pressure drop versus length-to-radius ratios (L/R) to zero capillary length and ii) by means of using orifice dies (L/R 0). Both methods resulted in about the same end corrections. Numerical simulation was also used to model this important aspect of experimental rheology. The constitutive equation used in the simulations is a multimode K-BKZ equation proposed by Papanastasiou et al. (1983, J. Rheol. 27:387) and further modified by Luo and Tanner (1988, Int. J. Num. Meth. Eng., 25:9). It was found that the numerical predictions agreed qualitatively, but underestimated the experimental data for the various geometries used to determine the end effects.Dedicated to the memory of Professor Tasos C. Papanastasiou     

Modeling spurt and stress oscillations in flows of molten polymers

The paper presents an approach for modeling polymer flows with non-slip, slip and changing non-slip — slip boundary conditions at the wall. The model consists of a viscoelastic constitutive equation for polymer flows in the bulk, prediction of the transition from non-slip to sliding boundary conditions, a wall slip model, and a model for the compressibility effects in capillary polymer flows. The bulk viscoelastic constitutive equation contains a hardening parameter which is solely determined by the polymer molecular characteristics. It delimits the conditions for the onset of solid, rubber-like behavior. The non-monotone wall slip model introduced for polymer melts, modifies a slip model derived from a simple stochastic model of interface molecular dynamics for cross-linked elastomers. The predictions for the onset of spurt, as well as the numerical simulations of hysteresis, spurt, and stress oscillations are demonstrated. They are also compared with available data for a high molecular weight, narrow distributed polyisoprene. By using this model beyond the critical conditions, many of the qualitative features of the spurt and oscillations observed in capillary and Couette flows of molten polymers, are described.Notations upper convected derivative of elastic strain tensor - f, f_m, f_min dimensionless (sliding) shear friction characteristics, and its maximum and minimum - G Hookean elastic modulus - G_p plateau modulus - G, G storage and loss moduli - I₁, I₂ first and second invariant of strain tensor - I₁, I₀ capillary and barrel lengths - M non-dimensional mass flow rate - M_C critical molecular weight - M^*, M_e molecular weights of a statistical segment, and of polymer chain between entanglements - M_n, M_W number average and weight average molecular weights - m, k two fitting parameters of slip model - _s , _s ^o nominal and characteristic sliding velocities - u non-dimensional sliding velocity - u _sc initial (infinitesimal) slip velocity - u ₁ upper limit of u on the lower branch - u ₂ lower limit of u on the upper branch - u _max value of u corresponding to f_min - u _min value of u corresponding to f_max - U piston speed - Q nominal volumetric flow rate - q non-dimensional volumetric flow rate - R, R_o capillary and barrel radii - M non-dimensional mass flow rate     

Resistance to flow of molten polymers through granular beds

Summary The resistance to flow of molten polymers (exhibiting memory effects) through granular beds was investigated. The data were correlated using simple methods of rheological characteristics of polymer melts.

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Zusammenfassung Es wird der Strömungswiderstand von Polymerschmelzen (welche Gedächtnis-Effekte zeigen) in Kornschüttungen untersucht. Bei der Korrelation der Meßwerte werden einfache Methoden zur rheologischen Charakterisierung der Schmelzen verwendet.

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Notation a ^* regression coefficient - b constant in eq. [22] - b ^* regression coefficient - B swell ratio, defined by eq. [2] - B ₁ swell ratio, defined by eq. [17a] - B ₂ swell ratio, defined by eq. [18a] - B ₃ swell ratio, defined by eq. [19a] - B ₀ swell ratio during the extrusion of a Newtonian fluid from the capillary - B _exp swell ratio determined according to the method described byCogswell (22) - B _exp swell ratio determined according to the method described byMendelson andFinger (23) - C constant in eq. [22] - d _p particle diameter, m - D extrudate diameter, m - D ₀ capillary diameter, m - f _CM friction factor, defined by eq. [12] - F() function occurring in eq. [15] - k fluid consistency factor, Ns ⁿ /m² - l bed height, m - L ₁,L ₂ length scales, m - n flow behaviour index - p pressure, N/m² - p ₁₁ –p ₂₂ first normal-stress difference, N/m² - r correlation coefficient - S recoverable shear strain, defined by eq. [3] - S average value of recoverable shear strain - S _w recoverable shear strain at the wall - s bed permeability, defined by eq. [8], m² - v ₀ mean linear velocity of fluid, related to an empty cross-section of the column, m/s - w ₀ mass flow rate, kg/m² s - level of probability of the correlation - shear rate, s^–1 - shear rate in a porous bed, defined by eq. [14], s^–1 - porosity - coefficient of dynamic fluid viscosity, Ns/m² - relaxation time, defined by eq. [1], s - friction factor, defined by eq. [5] - density of molten polymer, kg/m³ - ₀ density of polymer at room temperature, kg/m³ - ₁₂ shear stress, N/m² - parameter, defined by eq. [7], Ns ⁿ /m¹⁺ⁿ - function occurring in eq. [4] - De Deborah number, defined by eq. [16] - Re_BK Reynolds number, defined by eq. [25] - Re_BK generalized Reynolds number, defined by eq. [6] - Re_CM generalized Reynolds number, defined by eq. [13] With 10 figures and 3 tables     

The measurement of melt stress relaxation in the Weissenberg Rheogoniometer

Summary It has been suggested that stress relaxation after steady rotational shear flow of a polymer melt can be conveniently studied in theWeissenberg Rheogoniometer. To carry out such experiments, it is claimed that rotation can be stopped in a time interval of less than 10 milliseconds. Careful experiments reveal that neither of these statements is correct.The drive to the main spindle of the Rheogoniometer is effected through a clutch disc held in close contact with a driving plate by means of a magnetic coil. Braking is effected by a similar friction disc and coil attached to the body of the apparatus. With the standard rotary switch the time interval between deenergising the drive coil and energising the brake coil is arbitrarily determined by the speed of rotation of the switch. The replacement of the rotary by a toggle switch eliminates this variation and gives a reproducible time of braking of less than 10 milliseconds.In the Rheogoniometer the shear stress is determined by the deflection of a torsion bar. After steady shearing, the bar is displaced from its mean position, and the stress relaxation characteristics which are observed correspond to the changing position of the torsion bar. Consequently the relaxation of stress in the system does not occur under conditions of fixed boundaries as demanded by the theory, and the non-zero rate of shear can be readily obtained from the plate displacement-time relation. The fixed points in the system are the stationary cone and the top of the torsion bar. Hence the bar constant influences the relaxation behaviour, as has been demonstrated experimentally with an elastoviscous fluid. With aNewtonian fluid, the system behaves as a rotaryMaxwell element, with a time constant determined by the viscosity of the fluid and the elastic constant of the torsion bar.

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Zusammenfassung Angeblich sei es angebracht, die nach stetiger Kreisscherung einer Polymerschmelze eintretende Spannungsrelaxation in einemWeissenberg-Rheogoniometer zu bestimmen. Es heißt, für diesen Zweck könne die Umdrehung innerhalb 10 Millisekunden beendet werden. Sorgfältige Versuche haben jedoch gezeigt, daß keine der zwei Behauptungen zutrifft.Die Hauptwelle des Rheogoniometers wird von einer Kupplungsscheibe, die mittels eines Elektromagnets mit einer Antriebsscheibe in engem Kontakt gehalten wird, angetrieben. Das Abbremsen findet infolge des Einschaltens einer in ähnlicher Weise betätigten Reibungsscheibe, die mit dem Gehäuse des Apparates unbeweglich verbunden ist, statt. Der Zeitverlust zwischen dem Moment des Ausschaltens der Antriebsscheibe und des Einschaltens der Bremsscheibe hängt von der Drehgeschwindigkeit des Schalters ab. Die Ersetzung des Drehschalters durch einen Kniehebelschalter eliminiert diese Abhängigkeit: Bremszeiten von weniger als 10 Millisekunden wurden somit wiederholt erzielt.Im Rheogoniometer wird die Scherspannung in der Gestalt des Ausschlages einer Welle gemessen. Nach einer stetigen Scherung wird die Welle aus ihrer Ruhelage gedreht und die beobachteten Spannungsrelaxationserscheinungen entsprechen der Bewegung der Torsionswelle. Die Spannungsrelaxation findet also nicht unter konstanten Begrenzungen statt wie es die Theorie bedingt; eine einfache Berechnung auf Grund der Scheibenbewegung-Zeitfunktion zeigt das Ungleich-Null — Schergeschwindigkeitsgefälle an. Unbeweglich im System sind der stillstehende Konus und der obere Teil der Torsionswelle. Es wird also das Relaxationsverhalten von der Wellenkonstante beeinflußt, wie es mittels des Versuches mit einer elastoviskosen Flüssigkeit veranschaulicht wurde. Bei einer Newtonschen Flüssigkeit benimmt sich das System wie ein drehendesMaxwell-Element, worin die Zeitkonstante von der Viskosität der Flüssigkeit und der Elastizitätskonstante der Torsionswelle bestimmt wird.

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Paper presented at the Conference on Experimental Rheology, University of Bradford, April 17–19, 1968.     

Relaxation effects of slip in shear flow of linear molten polymers

The transient shear response of a linear molten polymer (linear low-density polyethylene) in the nonlinear domain was studied using a true shear (sliding plate) rheometer with different gap spacings to detect slip effects. It was found that nonlinear viscoelasticity is further complicated by wall slip phenomena. Experimental evidence suggested that static slip models coupled with Wagner’s constitutive equation cannot adequately describe the experimental data at large and fast shear deformations. A new dynamic slip model involving multiple slip relaxation times is proposed in this paper, together with a method to assess the model parameters. Significant improvement in predicting the stress response is demonstrated by several examples of start-up of steady shear and large-amplitude oscillatory tests of a linear low-density polyethylene.