Viscoelastic plastic rheological model for particle filled polymer melts

A viscoelastic plastic model for suspension of small particles in polymer melts has been developed. In this model, the total stress is assumed to be the sum of stress in the polymer matrix and the filler network. A nonlinear viscoelastic model along with a yield criterion were used to represent the stresses in the polymer matrix and the filler network, respectively. The yield function is defined in terms of differential equations with an internal parameter. The internal parameter models the evolution of structure changes during floc rupture and restoration. The theoretical results were obtained for steady and oscillatory shear flow and compared with experimental data for particle filled thermoplastic melt. The experimental data included the steady state shear strress over a wide range of shear rates, the transient stress in a start up shear flow, stress relaxation after cessation of a steady state shear flow, the step shear and the oscillatory shear flow at various amplitudes.     

Dynamic and transient rheological properties of glass filled polymer melts

The dynamic and transient rheological properties of a low density polyethylene melt and a plasticized polyvinylchloride melt filled with glass beads were measured at 200 °C and 180 °C respectively in a modified Weissenberg Rheogoniometer R-17. Its main modification consisted of the use of a piezoelectric transducer instead of the conventional torsion bar, and of the interfacing of a microcomputer Apple II plus to the Rheogoniometer for data acquisition and analysis. The glass beads were pretreated with silane and titanate coupling agents to observe the effect of the chemical modification of the polymer filler interface on these properties. It was observed that both the dynamic viscosity and the storage modulus increased with the weight fraction, but this last parameter did appreciably affect the stress growth and stress relaxation curves of the polymeric matrices at low shear rates. The effect of coupling agents on these properties was varied.     

Peculiarities of rheological behavior of filled polymer melts in uniaxial stretching

The peculiarities of theological behavior of filled polymer melts in uniaxial extension in a wide range of strain rates (from 2× 10^–5 to 1 × 10^–1 s^–1) have been studied. Linear polyethylene and 1,4-polybutadiene containing up to 21.5 vol.% of carbon black, silica, calcium carbonate or glass fibers were used. It has been found that the transition from uniform to nonunion stretching due to the neck formation is typical of all specimen compositions, when they approach steady-state straining. Depending on the structure and rheological characteristics of the compositions general conditions for this transition have been established. The general regularities for varying the rheological characteristics of filled polymers in the course of their uniform stretching have been recognized. These regularities depend on the molecular characteristics of the polymer matrix and the presence in the compositions of the structural framework of high disperse filler or the network formed by the entangled fibers. Using polyethylene compositions it has been shown that the introduction of small amounts of disperse or fibrous fillers can give rise to acceleration of the relaxation process in filled polymers.     

Die swell of filled polymer melts

The Barus effect in polypropylene and polystyrene blended with a variety of fillers at various concentrations was investigated using a capillary extrusion rheometer. If the die swell is defined as the square of the ratio of the extrudate diameterd to the die diameterD, it is found to depend on the apparent shear stress _W . Below a certain value of _w the relation =B _B ^A applies. The die swell, _M , of a filled polymer depends on the type, size and volume fraction of the filler. In particular,A increases as the volume fraction increases and is largest for powders, smaller for flakes and smallest for fibres, whereasB shows the opposite trend but to a lesser extent.     

Viscosity of particle filled polymer melts

Summary Viscosities of particle filled polymer melts were measured at fairly low shear rate. Particles were glass beads, glass balloons, and silas balloons. Polymers were polyethylene and polystyrene. Flow curves were superimposed with respect to concentration of filler. The relative viscosity defined as the ratio of viscosity of filled polymer to that of unfilled polymer at the same shear stress is obtained as an asymptotic value even for highly filled material. The relation between relative viscosity and volume fraction of filler was represented by the equations derived byMaron andPierce orMooney. Only the distribution of particle size had influence on relative viscosity at a defined filler concentration. Yield stresses were estimated, and found to increase exponentially in the range of volume fraction from 0.1 to 0.5.

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Zusammenfassung Viskositäten von Polymerschmelzen, die mit Teilchen gefüllt worden waren, wurden bei mittleren Schergeschwindigkeiten bestimmt. Als Teilchen wurden Voll- und Hohlkugeln aus Glas, sowie sog. silas balloons, als Polymere Polyäthylen und Polystyrol verwendet. Die Fließkurven für verschiedene Füllstoffkonzentrationen wurden überlagert. Die relative Viskosität, definiert als das Verhältnis der Viskositäten von gefülltem und ungefülltem Polymer bei gleicher Schubspannung, besitzt einen asymptotischen Wert selbst für hoch gefülltes Material. Die Beziehung zwischen relativer Viskosität und Füllstoff-Volumenkonzentration läßt sich durch eine vonMaron undPierce oder eine vonMooney abgeleitete Gleichung beschreiben. Nur die Teilchengrößenverteilung hat bei einer definierten Füllstoffkonzentration einen Einfluß auf die relative Viskosität. Fließspannungen werden abgeschätzt und dafür in einem Konzentrationsbereich zwischen 0,1 und 0,5 ein exponentieller Anstieg gefunden.

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With 13 figures and 2 tables     

Suspension-based rheological modeling of crystallizing polymer melts

The applicability of suspension models to polymer crystallization is discussed. Although direct numerical simulations of flowing particle-filled melts are useful for gaining understanding about the rheological phenomena involved, they are computationally expensive. A more coarse-grained suspension model, which can relate the parameters in a constitutive equation for the two-phase material to morphological features, such as the volume fractions of differently shaped crystallites and the rheological properties of both phases, will be more practical in numerical polymer processing simulations. General issues, concerning the modeling of linear and nonlinear viscoelastic phenomena induced by rigid and deformable particles, are discussed. A phenomenological extension of linear viscoelastic suspension models into the nonlinear regime is proposed. A number of linear viscoelastic models for deformable particles are discussed, focusing on their possibilities in the context of polymer crystallization. The predictions of the most suitable model are compared to direct numerical simulation results and experimental data.     

Stress-induced interfacial failure in filled polymer melts

We study a dynamic interfacial slip phenomenon in filled polymer melts. When a poly(dimethyl siloxane) melt of weight average molecular weight M _w = 93 700 is mixed with glass beads of diameter up to 45 m, the sample shows some evidence of breakdown of interfacial adhesion between the glass beads and the PDMS matrix at a level of oscillatory stress under which the pure PDMS melt exhibits no decay. The decay of viscoelastic properties with time is essentially independent of the amplitude of shear strain as long as the magnitude of the oscillating stress is sufficiently high. It is suggested that much higher local stress than the apparent applied stress may be generated between the narrowly spaced beads. The interfacial slip was observable because it was measured against a natural length scale d in the filled polymer melts which is the filler size or the inter-filler distance and is much smaller than the dimensions of the flow cell. The decay under high stress and healing of interfacial adhesion upon a large reduction in stress may be related to the de-bonding and reformation of hydrogen bonds between the PDMS chains and the glass bead surfaces.     

In-line measurement of rheological properties of polymer melts

Shear viscosity (), first normal stress difference (N ₁), and extensional viscosity ( _E ) of polymer melts measured under processing conditions are important in process modeling, quality control, and process control. A slit rheometer that could simultaneously measure , N ₁, and the planar extensional viscosity ( _p ) was designed and tested by attaching it in-line to a laboratory model single-screw extruder. A tube (circular cross-section) rheometer to measure and the uniaxial extensional viscosity ( _u ) simultaneously was also designed and tested. Two commercial grades of LDPE (low density polyethylene) with melt index values of 6 and 12 were used as test materials for the study. Exit and hole pressure methods were used to estimate N ₁, and the entrance pressure drop method using the analyses of Cogswell, Binding, and Gibson (the last analysis used with the axisymmetric case only) was used to estimate _E .The hole pressure method was considered better than the exit pressure method to estimate N ₁ (due to the greater susceptibility of the latter to experimental errors). From the hole pressure method N ₁ was obtained from 100 kPa to 500 kPa over a range of shear rates from 40 s^–1 to 700 s^–1. Among the analyses used to estimate the extensional viscosity, Cogswell's is recommended due to its simpler equations without loss of much information compared to the other analyses. The range of extension rates achieved was 1 to 30 s^–1. The combination of the hole pressure and entrance pressure drop methods in a slit rheometer is a feasible design for a process rheometer, allowing the simultaneous measurement of the shear viscosity, first normal stress difference and planar extensional viscosity under processing conditions. Similarly, combining the entrance pressure drop measurements with a tube rheometer is also feasible and convenient.     

Viscous properties of calcium carbonate filled polymer melts

Summary The viscous properties of calcium carbonate filled polyethylene and polystyrene melts were examined. The relative vircosity _r defined in the previous paper gave an asymtptotic value( _r)_l in the range of the shear stress below 10⁵ dyne/cm².( _r)_l of the calcium carbonate filled system was higher than that of the glass beads or glass balloons filled system at the same volume fraction of the filler. Maron-Pierce equation with ₀ = 0.44 was able to approximate the( _r)_l — relationship. However, it was deduced here that the high value of( _r)_l of calcium carbonyl filled system was due to the apparent increase of and this increase was attributed to the fixed polymer layer formed on the powder particle. By assuming the particle as a sphere with a diameter of 2 µm, the thickness of the fixed polymer layer was estimated as about 0.17 µm. The yield stress estimated from the Casson's plots increased exponentially with.

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Zusammenfassung Es wurden die viskosen Eigenschaften von Polyäthylen-und Polystyrol-Schmelzen untersucht, die mit Kalziumkarbonat-Teilchen gefüllt waren. Für die relative Viskosität _r, wie sie in einer vorangegangenen Veröffentlichung definiert worden war, ergab sich bei Schubspannungen unterhalb 10⁵ dyn/cm² ein asymptotischer Wert( _r)_l. Dieser war bei den mit Kalziumkarbonat gefüllten Schmelzen höher als bei Schmelzen, die bis zur gleichen Volumenkonzentration mit Glaskugeln oder Glasballons gefüllt waren. Die ( _r) _l —-Abhängigkeit ließ sich durch eine Gleichung nachMaron und Pierce mit ₀ = 0,44 beschreiben. Es wurde jedoch geschlossen, daß der hohe( _r)_l-Wert der mit Kalziumkarbonat gefüllten Schmelzen auf eine scheinbare Zunahme von zurückzuführen ist, verursacht durch eine feste Polymerschicht auf der Teilchenoberfläche. Unter Annahme kugelförmiger Teilchen mit einem Durchmesser von 2 µm ließ sich die zugeordnete Schichtdicke zu 0,17 µm abschätzen. Die mittels der Casson-Beziehung geschätzte Fließspannung ergab eine exponentielle-Abhängigkeit.

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With 7 figures and 1 table     

Relative viscosities of polymer melts filled with inorganic fillers

Summary Viscosities of polyethylene melts filled with natural calcium carbonate (NC), glass fiber (GF), and carbon fiber and polypropylene melts filled with NC, talc, and GF were examined. Constant relative viscosities _r were obtained on these filled systems if use was made of the definition of _r suggested in an earlier paper. The relationships between _r and the volume fraction of the filler were discussed qualitatively in conjunction with earlier data. The magnitude of _r for the material filled with fiber increases with the increase of the average aspect ratio of the fiber.

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Zusammenfassung Es werden die Viskositäten von Polyäthylenschmelzen untersucht, die mit natürlichem Kalziumkarbonat (NC), Glasfasern (GF) und Karbonfasern, sowie von Propylenschmelzen, die mit Kalziumkarbonat, Kreide und Glasfasern gefüllt sind. Man findet bei diesen gefüllten Systemen konstante relative Viskositäten _r , wenn man dafür die in einer früheren Untersuchung eingeführte Definition zugrundelegt. Die Beziehungen zwischen _r und dem Volumenanteil des Füllstoffs werden unter Bezugnahme auf frühere Ergebnisse qualitativ diskutiert. Die Größe von _r wächst bei den mit Fasern gefüllten Systemen mit zunehmendem Verhältnis von Länge zu Dicke der Fasern.

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With 6 figures and 3 tables     

Constrained inversion of rheological data to molecular weight distribution for polymer melts

In this paper an inverse method for converting linear viscoelastic properties of polymer melts into molecular weight distribution (MWD) is presented. It relies on the use of the double reptation mixing rule and the Generalized Exponential function (GEX) describing the MWD; i.e. an a priori knowledge of the nature of the solution is postulated following previous attempts (Nobile et al. 1996b; Nobile and Cocchini 1999, 2001) where the Tuminello kernel was adopted to get analytical results. Here a MWD dependent kernel, recently proposed in the literature (Thimm et al. 1999), is applied, still obtaining analytical results but avoiding the consequences of the crude step-like approximation of the Tuminello kernel. In particular, the steady-state compliance and the elastic modulus at small frequencies do not suffer from the underestimation discussed in the cited papers. At high frequencies, the Rouse modes contribution has also been considered in the model. The present approach turns out to be simpler, more accurate and robust than the previous one. Moreover, the results can be more clearly stated in the frame of the theory. Careful and easy-to-use formulas for the zero-shear-rate viscosity, the steady-state compliance, the modulus and frequency at the G'-G' crossover, have been obtained as a function of the MW averages. As expected, the comparison among the results obtained through the two kernels explicitly confirms the minor effects of the kernel on the rheology of polydisperse polymers. Concerning the molecular weight averages values, the predictions obtained from the dynamic moduli measurements typically differ less than 10% from those obtained from the reference measurements obtained with other methods.     

Towards a rheological classification of flow induced crystallization experiments of polymer melts

Departing from molecular based rheology and rubber theory, four different flow regimes are identified associated to (1) the equilibrium configuration of the chains, (2) orientation of the contour path, (3) stretching of the contour path, and (4) rotational isomerization and a deviation from the Gaussian configuration of the polymer chain under strong stretching conditions. The influence of the ordering of the polymer chains on the enhanced point nucleation, from which spherulites grow, and on fibrous nucleation, from which the shish-kebab structure develops, is discussed in terms of kinetic and thermodynamic processes. The transitions between the different flow regimes, and the associated physical processes governing the flow induced crystallization process, are defined by Deborah numbers based on the reptation and stretching time of the chain, respectively, as well as a critical chain stretch. An evaluation of flow induced crystallization experiments reported in the literature performed in shear, uniaxial and planar elongational flows quantitatively illustrates that the transition from an enhanced nucleation rate of spherulites towards the development of the shish-kebab structure correlates with the transition from the orientation of the chain segments to the rotational isomerization of the high molecular weight chains in the melt. For one particular case this correlation is quantified by coupling the wide angle X-ray diffraction and birefringence measurements of the crystallization process to numerical simulations of the chain stretch of the high molecular weight chains using the extended Pom-Pom model in a cross-slot flow.     

Measurement of rheological and ultrasonic properties of food and synthetic polymer melts

A slit die viscometer has been used in conjunction with a co-rotating twin screw extruder to study the rheological behaviour of maize grits, potato powder and low density polyethylene, as a function of feed rate, screw speed and temperature. The shear viscosity of both maize and potato decreased with increasing feed rate. Increasing the temperature or screw speed at any given feed rate also reduced the viscosity. The ultrasonic velocity through the material has also been shown to be sensitive to the extruder operating conditions. Overall, the ultrasonic velocity decreased as screw speed and temperature increased and feed rate decreased.     

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.     

Representation of the rheological properties of polymer melts in terms of complex fluidity

In dynamic rheological experiments melt behavior is usually expressed in terms of complex viscosity ^* () or complex modulusG ^* (). In contrast, we attempted to use the complex fluidity ^* () = 1/µ ^* () to represent this behavior. The main interest is to simplify the complex-plane diagram and to simplify the determination of fundamental parameters such as the Newtonian viscosity or the parameter of relaxation-time distribution when a Cole-Cole type distribution can be applied. ^* () complex shear viscosity - () real part of the complex viscosity - () imaginary part of the complex viscosity - G ^* () complex shear modulus - G() storage modulus in shear - G() loss modulus in shear - J ^* () complex shear compliance - J() storage compliance in shear - J() loss compliance in shear - shear strain - rate of strain - angular frequency (rad/s) - shear stress - loss angle - ^* () complex shear fluidity - () real part of the complex fluidity - () imaginary part of the complex fluidity - ₀ zero-viscosity - ₀ average relaxation time - h parameter of relaxation-time distribution     

An extension of the Curtiss-Bird theory for the rheological properties of concentrated polymer solutions and melts

The constitutive equation for the elastic reptating rope model is considered in the limit at which the rope becomes inextensible. It is shown that in this limit a constitutive equation is obtained which differs essentially from the Curtiss-Bird theory if the correlation length is much smaller than the contour length of the rope. This constitutive equation contains an extra term related to correlations between segments. The differences between this result and the Curtiss-Bird theory are considered, and consequences are indicated.     

Assessing the practical utility of the hole-pressure method for the in-line rheological characterization of polymer melts

The most promising method capable of providing accurate measurements of the first and second normal-stress differences in shear flows at shear rates typical of polymer processing is the so-called hole-pressure method, but its use has not been as widespread as would be expected, namely due to the experimental difficulties associated with performing such experiments accurately. In this work, we use a small-scale modular slit die to assess the practical utility of the method for in-line monitoring of polymer melt flow. We provide a quantitative analysis of intrinsic error sources and use state-of-the-art data acquisition tools to minimize errors associated with pressure transducers. Our results demonstrate that the method can be used to accurately measure the viscosity and first normal-stress difference in melts but probably not the second normal-stress difference because the intrinsic errors are too high, even when the influence of all the potential error sources is minimized or eliminated.     

Flow birefringence of polymer melts: Application to the investigation of time dependent rheological properties

Summary Transient stresses including normal stresses, which are developed in a polymer melt by a suddenly imposed constant rate of shear, are investigated by mechanical measurement and, indirectly, with the aid of the flow birefringence technique. For the latter purpose use is made of the so-called stress-optical law, which is carefully checked.It appears that the essentially linear model of the rubberlike liquid, as proposed byLodge, is capable of describing the behaviour of polymer melts rather well, if the applied total shear does not exceed unity. In order to describe also steady state values of the stresses successfully, one should extend measurements to extremely low shear rates.These statements are verified with the aid of a method which was originally designed bySchwarzl andStruik for the practical calculation of interrelations between linear viscoelastic functions. In the present paper dynamic shear moduli are used as reference functions.

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Zusammenfassung Mit der Zeit anwachsende Spannungen, darunter auch Normalspannungen, wie sie sich nach dem plötzlichen Anlegen einer konstanten Schergeschwindigkeit in einer Polymerschmelze entwickeln, werden mit Hilfe mechanischer Messungen und indirekt mit Hilfe der Strömungsdoppelbrechung untersucht. Für den letzteren Zweck wird das sogenannte spannungsoptische Gesetz herangezogen, dessen Gültigkeit sorgfältig überprüft wird.Es ergibt sich, daß das im Wesen lineare Modell der gummiartigen Flüssigkeit, wie es vonLodge vorgeschlagen wurde, sich recht gut zur Beschreibung des Verhaltens von Polymerschmelzen eignet, solange der im ganzen angelegte Schub den Wert Eins nicht überschreitet. Um auch stationäre Werte der Spannungen in die Beschreibung erfolgreich einzubeziehen, sollte man die Messungen bis zu extrem niedrigen Schergeschwindigkeiten ausdehnen.Die gemachten Feststellungen werden mit Hilfe einer Methode verifiziert, die vonSchwarzl undStruik ursprünglich für die praktische Berechnung von Beziehungen zwischen Zustandsfunktionen entwickelt wurde, die dem linear viskoelastischen Verhalten entsprechen. In der vorliegenden Veröffentlichung dienen die dynamischen Schubmoduln als Bezugsfunktionen.

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a _T shift factor - B _ij Finger deformation tensor - C stress-optical coefficient, (m²/N) - f (p _jl ) undetermined scalar function - G shear modulus, (N/m²) - G(t) time dependent shear modulus, (N/m²) - G() shear storage modulus, (N/m²) - G() shear loss modulus, (N/m²) - G _r reduced shear storage modulus, (N/m²) - G _r reduced shear loss modulus, (N/m²) - H() shear relaxation time spectrum, (N/m²) - k Boltzmann constant, (Nm/°K) - n _ik refractive index tensor - p undetermined hydrostatic pressure, (N/m²) - p _ij ,p _ik stress tensor, (N/m²) - p ₂₁ shear stress, (N/m²) - p ₁₁ –p ₂₂ first normal stress difference, (N/m²) - p ₂₂ –p ₃₃ second normal stress difference, (N/m²) - q shear rate, (s^–1) - t, t time, (s) - T absolute temperature, (°K) - T ₀ reference temperature, (°K) - x the ratiot/ - x position vector of a material point after deformation, (m) - x position vector of a material point before deformation, (m) - ₀, ₁ constants in eq. [37] - ₀, ₁ constants in eq. [37] - shear deformation - (t, t) time dependent shear deformation - _ij unity tensor - n flow birefringence in the 1–2 plane - (q) non-Newtonian shear viscosity, (N s/m²) - ^* () complex dynamic viscosity, (N s/m²) - | ^* ()| absolute value of complex dynamic viscosity, (N s/m²) - () real part of complex dynamic viscosity, (N s/m²) - () imaginary part of complex dynamic viscosity, (N s/m²) - (t — t) memory function, (N/m² · s) - v number of effective chains per unit of volume, (m^–3) - temperature dependent density, (kg/m³) - ₀ density at reference temperatureT ₀, (kg/m³) - relaxation time, (s) - integration variable, (s) - (x) approximate intensity function - ₁ (x) error function - extinction angle - _m orientation angle of the stress ellipsoid - circular frequency, (s^–1) - 1 direction of flow - 2 direction of the velocity gradient - 3 indifferent direction - t time dependence The present investigation has been carried out under the auspices of the Netherlands Organization for the Advancement of Pure Research (Z. W. O.).North Atlantic Treaty Organization Science Post Doctoral Fellow.Research Fellow, Delft University of Technology.With 11 figures and 2 tables