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.     

The Weissenberg effect in molten polymers

The climb of viscoelastic liquids up a rotating rod, often called the Weissenberg effect, has often been observed in polymer solutions, and several theoretical analyses of this phenomenon have been published. However, no observations of rod-climbing in polymer melts have been reported. In the present work, three commercial polyethylenes were melted in a special chamber under a controlled atmosphere, and a rod was rotated in the pool of molten polymer. The behavior of the free surface was noted as a function of time and rotational speed.In the case of the resin with the lowest molecular weight and the highest density, the free surface reached a steady-state shape within seconds of changing the rotational speed. For the other two resins, the development of the shape of the free surface took place over periods of the order of one hour or more. In the cases where the starting transient was very long-lasting, the melt first climbed the rod and accumulated in a large bulge. Then, the fluid in the bulge flowed down to the bottom of the raised column and a pear-shaped body of liquid was formed whose shape continued to undergo gradual change for periods up to several hours. At high speeds, inertial effects destabilized the flow and led to asymmetry and time unsteadiness in the shape of the raised body of liquid.     

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.     

Rheology and structural changes of plasticized zeins in the molten state

Zeins, storage proteins from maize, are suitable for making biobased thermoplastic materials. The rheological behavior of a commercial zein plasticized with 20 w% glycerol was studied in the molten state by steady-state flow experiments in extrusion conditions and oscillatory rheometry. For low residence times, a shear-thinning viscoelastic behavior was observed, with G″ exceeding G′. After 300 s at 130 °C, the complex viscosity |η ^?|?=?7?×?10³ ω ^?0.46 was found to be similar to that of thermoplastic polymer melts used in fused deposition modeling. However, the ratio between the exponents of the power laws describing G′(ω) and G″(ω) did not meet the typical value of 2 for entangled polymer melts. Moreover, for longer residence times, the viscosity increased and a gelation phenomenon was observed with a crossing over of G′(ω) and G″(ω). Gel times ranged from 6000 s at 120 °C to 1700 s at 150 °C. The evolution of the macromolecular structure assessed by sodium dodecyl sulfate-polyacrylamide gel electrophoresis and high-performance size exclusion chromatography suggested that this gelation phenomenon involves various types of covalent and non-covalent cross-links. Disulfide bonds played a significant role in gelation kinetics despite a very low cysteine residue content in the protein primary structure (about 1 mol%). These results suggested that plasticized zeins initially behave like a low-viscosity non-entangled polymer melt, before cross-linking progressively led to a continuous network.     

Rheology of polymers. Viscous properties of polypropylene melt

Summary A simple capillary viscometer for studying polymer melts is described operating at constant pressures (up to 600 atm) and temperatures up to 350. With this apparatus measurements can be made over a range of rate gradients of five to seven decimal orders. The errors in viscosity determination do not exceed a few per cent.The invariant (with respect to capillary dimensions) flow curves of polypropylene are given for rate gradients ranging from 10^–2 to 10³ sec^–1 and temperatures from 190 to 270C. On the basis of direct measurements and an extrapolational calculation the highest Newtonian. viscosities of polypropylene were determined at various temperature, making it possible to obtain the temperature invariant characteristic of the viscous properties of the polymer. The activation energy of viscous flow varies considerably depending on the shear stresses. dropping from 21 to 9.3 kcal per mole when the shear stress rises from 0 to 1 × 10⁶ dyne/cm².The ratio of the diameter of the extrudate, issuing from the capillary to the capillary inner diameter is a single valued function of the shear stresses. Calculation of the normal stresses revealed them to be related by a power law to the shear rate, this dependence being invariant with respect to the temperature.Although the normal stresses grow much faster with the shear rate than the tangential, they reach the latter in absolute value only at shear rates of about 10³ sec^–1. This is an indirect indication of the fact that high elasticity is relatively weakly expressed in the polypropylene melt. With this are also connected the low entrance losses determined at 190 by the method of two equal-diameter capillaries of different lengths. Measuring the viscosity of a melt at a temperature close the melting point is a sensitive method of determining the degradation of the polymer, which becomes perceptible when it is heated to temperatues above 270.Observations of crystallization of the polymer in the melt stream have shown that this process begins at the flow axis in the non-stressed zone.

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Zusammenfassung Es wird ein einfaches Kapillar-Viskosimeter zur Untersuchung von Polymerschmelzen beschrieben, das in Bereichen mit konstantem Druck (bis 600 at) und bei Temperaturen bis 350 arbeitet. Messungen mit Hilfe dieses GerÄtes gestatten im Bereich der Geschwindigkeitsgradienten Änderungen um 5... 7 Zehnerpotenzen. Die Fehler bei der ViskositÄtsbestimmung betragen höchstens einige Prozente.In dem Artikel werden die in bezug auf die Kapillar-abmessungen invarianten Flie\kurven des Polypropylens bei Gradientengeschwindigkeiten von 10^–2 bis 10³ sec^–1 und bei Temperaturen von 190 bis 270C angegeben. Durch direkte Messungen und Extrapolationsrechnung wurden die grö\ten Newtonschen ViskositÄten des Polypropylens bei unterschiedlichen Temperaturen bestimmt, wodurch eine in bezug auf die Temperatur invariante Charakteristik der ViskositÄts-eigenschaften dieses Polymers erhalten wurde. Die Aktivierungsenergie der zÄhflüssigen Strömung hÄngt erheblich von der Schubspannung ab und sinkt von 21 auf 9,3 kcal/Mol, wenn die Schubspannung von 0 auf 1,10⁶ Dyn/cm² zunimmt.Das VerhÄltnis des aus der Kapillare ausströmenden Polymerstrahls zum Durchmesser des Kapillar-innenquerschnittes ist eine monotone Funktion der Schubspannung. Die Normalspannungen sind potenz-abhÄngig von der Deformationsgeschwindigkeit. Diese AbhÄngigkeit ist invariant in bezug auf die Temperatur.Die Normalspannungen wachsen mit dem Geschwin-digkeitsgefÄlle wesentlich schneller als die Schubspannungen. Sie erreichen deren Absolutwert nur bei einem GeschwindigkeitsgefÄlle von der Grö\enordnung 10³ sec^–1. Dies ist ein indirekter Hinweis darauf, da\ in der Polypropylenschmelze die ElastizitÄt verhÄltnis-mÄ\ig schwach ausgeprÄgt ist. Hiermit sind geringe Eingangsverluste verbunden, die bei 190 unter Verwendung von zwei Kapillaren gleichen Durchmessers aber unterschiedlicher LÄngen bestimmt werden.Das Me\verfahren zur Bestimmung der Schmelz-viskositÄt bei Temperaturen, die nahe dem Schmelzpunkt liegen, ist eine empfindliche Methode zur Bestimmung des Polymerabbaus, die bei ErwÄrmen des Polymers auf Temperaturen über 270 bemerkbar wird.Beobachtungen der Polymerkristallisation im Schmelzestrahl haben gezeigt, da\ dieser Vorgang an der Strömungsachse in der spannungsfreien Zone beginnt.

     

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.     

An instability in the flow of molten polymers

Ohne Zusammenfassung     

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.     

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     

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     

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     

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.     

Convective linear stability analysis of two-layer coextrusion flow for molten polymers

The interface instability of the coextrusion flow of a polyethylene and a polystyrene is studied both experimentally and theoretically in a slit geometry. For prototype industrial conditions, we have found a stable/unstable transition which bounds the occurrence of stable/unstable sheets at die exit. By investigating a large range of processing conditions, we have shown that this transition is controlled by both temperature and flow rate ratios. Close to the transition, we used a transparent die to measure spatial amplification of different controlled perturbations at die inlet and pointed out the convective nature of the instability which exhibits a dominant mode (for which the instability is the most severe). We have then found that a convective stability analysis, using the White–Metzner constitutive equation, is able to account for the spatial amplification rate experimentally measured on controlled perturbation experiments. By considering that the instability is controlled by its dominant mode, we performed a convective stability analysis for all studied prototype industrial conditions and showed that such an analysis is able to forecast the occurrence of defects at die exit.