Molecular theory for moderately concentrated polymer solutions in shear flow

A molecular theory for the rheological properties of moderately concentrated polymer solutions is developed on the basis of a model of interacting dumbbells. The interaction is treated in a mean field approximation, leading to an effective one-particle potential and a Gaussian stationary distribution function. Various rheological functions such as birefringence, shear viscosity and first normal-stress coefficient for simple shear flow and the Trouton viscosity for simple extensional flow are calculated. Good qualitative agreement with experimental observations is found, especially at intermediate flow rates. It is predicted, for example, that the birefringence increases approximately linearly with shear rate at intermediate shear rates and that the concentration dependence of the gradient varies asc ^1/2. The typical non-Newtonian behaviour is obtained for the shear viscosity. For small concentrations the onset of shear rate dependence decreases asc ^–1/2. At intermediate shear rates an apparent power law is obtained with an exponent between – 0.5 and – 1.0, decreasing with concentration.     

The rheological properties of saliva

Summary Saliva is a complex biological fluid that cannot be effectively studied by conventional methods such as theFerranti-Shirley Viscometer. Testing with theWeissenberg Rheogoniometer shows that saliva is viscoelastic and that the dynamic viscosity falls from 400-0.1 Poise over the frequency range 2.5×10^–3 to 10 Hertz. The calculated elastic contribution can be correlated with dry weight content.Dynamic viscosity and shear modulus are both reduced in storage, by proteolytic enzymes. The mucolytic agent Ascoxal also brings about a reduction in these parameters but by a different mechanism. A comparison of the results with those obtained previously for sputum, shows, that saliva may be a suitable model system for assessing mucolytic agentsin vitro.

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Zusammenfassung Speichel ist eine komplexe biologische Flüssigkeit, die mit konventionellen Methoden, wie z. B. demFerranti-Shirley-Viskosimeter, nicht umfassend untersucht werden kann. Studien mit einemWeissenberg-Rheogoniometer zeigen, daß Speichel viskoelastisch ist und daß seine dynamische Viskosität über einen Frequenzbereich von 2,5 · 10^–3 bis 10 Hz von 400 Poise auf 0,1 Poise fällt. Die berechnete elastische Komponente kann mit dem Trockengewichtsanteil korreliert werden.Dynamische Viskosität und Schermodul werden gemeinsam bei Lagerung von proteolytischen Enzymen reduziert. Das mukolytisch aktive Ascoxal reduziert auf andere Art ebenfalls diese Parameter. Ein Vergleich dieser Ergebnisse mit denen, die für Sputum erhalten werden, zeigt, daß Speichel ein günstiges Modellsystem ist, um mukolytisch aktive Stoffe in vitro zu testen.

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Paper presented to the British Society of Rheology Conference on Rheology in Medicine and Pharmacy, London, April 14–15, 1970.     

An improved Couette high shear viscometer

Summary The advantages of theCouette-type concentric cylinder viscometer have been cited for making viscosity measurements at high shear rates. Significant improvements in theBarber design are described which have resulted in the development of an instrument which can make measurements under laminar shear conditions at over several decades of shear rate. The range is over twice that of previous instruments demonstrating the achievement of shear rates to above 1 000 000 sec^–1, and with minimum viscous heating and over a temperature range of over 100°C.

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Zusammenfassung Nachdem auf die Vorteile desCouette-Viskosimeters für Viskositätsmessungen bei hohen Schergeschwindigkeiten hingewiesen worden ist, werden wesentliche Verbesserungen des Geräts nachBarber beschrieben, die zur Entwicklung eines Instruments geführt haben, das Messungen unter laminaren Scherbedingungen über mehrere Dekaden der Schergeschwindigkeit erlaubt. Der Bereich ist mehr als doppelt so groß wie bei früher beschriebenen Instrumenten und reicht bis über 10⁶ sec^–1. Dabei tritt nur eine geringe viskose Erwärmung auf, und es kann noch bei Temperaturen von mehr als 100°C gemessen werden.

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With 4 figures and 2 tables     

Extensional flow of polystyrene melt

Summary The extensional flow of molten polystyrene was studied in the strain rate range of 7.8×10^–4 sec^–1 to 2.2×10^–2 sec^–1 at a test temperature of 300F (149 C) Extensional viscosity was compared to shear viscosity measured at the same degree of stress and temperature and found to be from 3 to 350 times greater in magnitude but much less stress dependent.     

A rheological measurement of three synovial fluids

Summary Using aWeissenberg Rheogoniometer, three samples of bovine synovial fluid were tested at room temperature (60 °F) at shear rates from 1.0 × 10^–2 to 9.0 × 10³ sec^–1.There is no apparent explanation for the absence (or small value outside the instruments' range) of normal stress in ankle fluid A. No information was available as to the rearing environment of the animals.It is clear, however, that synoval fluid can exhibit a high elastic component. The shear rate to which the fluid is subjected during the normal action of the joint is probably quite as high as the maximum value used in these tests.     

A Rheological comparison of three gloss paints

Summary Using aWeissenberg Rheogoniometer, three gloss paints were tested in rotation, using cone and plate geometry, at shear rates from 1.0 to 2.0×10⁴ secs^–1. The moduli of viscosity and elasticity are calculated. A practical method of correcting the reduction of normal stress measurements due to secondary flows at high platen speeds in described.Presented at the Joint Meeting of the British Society of Rheology and Research Association of British Paint, Colour and Varnish Manufacturers at Teddington, April 29, 1964.     

Viscoelastic properties of polymer solutions

Summary Measurements were taken of the thrust of liquid jets ejecting from a long capillary into air for aqueous solutions of polyacrylamide (ET 597) of various concentrations. The measurements were then used to determine axial normal stresses for two capillary diameters: 1.52 mm (L/D = 201.5) and 2.44 mm (L/D = 207.2). The results show that the calculated values of the axial normal stress are higher for the larger capillary diameter than for the smaller, as recently reported byPowell andMiddleman. It has been further found that this diameter effect becomes more pronounced as the concentration of solute is increased. For the same materials, we also measured the primary normal stress difference by means of aWeissenberg rheogoniometer. Comparison between the two kinds of measurements shows that the magnitude of axial normal stresses is much smaller than that of normal stress differences over the range of shear rates studied (200 20,000 sec^–1) for the materials investigated. This result seems to point out the necessity of measuring the wall normal stresses, which are believed to depend not only on the capillary diameter, but also on the concentration. The authors therefore contend that, in general, the measurement of axial normal stresses alone is not sufficient to completely determine the elastic properties of viscoelastic solutions.     

Constitutive relationships for polymeric materials with power-law distributions of relaxation times

The linear relaxation modulus of polydisperse polymer melts and solutions can often be approximated by a power law,ct ^–m over some range of time,t. If, in addition, the nonlinear rheology is given by a separable integral equation, with a strain-dependent factor typical of those observed experimentally, then some commonly observed empirical rules and equations can be readily derived as approximations, namely the Cox-Merz relationship between complex viscosity and steady-state shear viscosity, Bersted's predictions of steady shear stress and first normal-stress difference from a truncated spectrum of linear relaxation times, and the observation of Koyama and coworkers that the ratio of the nonlinear to the linear time-dependent elongational viscosity is independent of strain rate, over a range of strain rates outside the linear regime.     

Second-order strain accumulation in workhardening media

Summary Second-order or cross effects are the result of quadratic tensor terms in the constitutive equations of isotropic elastic, viscous and visco-elastic media, which are required by the condition of tensor invariance of those relations. These effects are most pronounced when they are clearly separable from the first-order deformation, as in the case of second-order elongation and volume change of an elastic cylinder subject to a twisting moment (Poynting effect, dilatancy) or of second-order normal stress in the case of shear flow of polymeric liquids (Weissenberg effect).An accumulating second-order effect (Ronay effect) has been discovered in experiments on strain-hardening metal specimens in reversed torsion. While thePoynting effect vanishes at zero strain in elastic solids and theWeissenberg effect at zero velocity in polymeric fluids, the second-order strain increments accumulate in strainhardening media with the number of repeated torsion cycles. Hence their observation is simple and does not require the elaborate procedures necessary for the observation od second-order effects in elastic solids, viscous fluids and visco-elastic substances.It can be shown that the observed second-order strain accumulation (Ronay effect) is implied by thePrager-Hill stress strain-increment relation for strain-hardening media, combined with theKadshevich-Novozhilov formulation of kinematic hardening, provided that the arbitrary condition that strain-increment and stress change sign simultaneously is not imposed.Paper read at the Annual Meeting of the German Rheologists, Berlin-Dahlem June 7–10, 1966.     

The peculiar rheo-optical behavior of bisphenol A-polycarbonate and polymethylmethacrylate

The rheological and stress-optical behavior of the melts of several grades ob bisphenol-A-polycarbonate (PC) and polymethylmethacrylate (PMMA) is investigated. Pertinent flow birefringence measurements are carried out in a remodelled cone-plate apparatus [1]. The shear stress in the polymer melt is calculated from the dynamic moduli, which are determined separately. It is shown that the linear stress optical rule is obeyed. In this way, the stress-optical coefficient C of the melt can be determined. The low-Mw polycarbonates all behave as Maxwellian fluids. The main stress direction does not deviate significantly from 45°. In the temperature range from 160° to 260°C the stress-optical coefficients of the different grades lie between 3 and 4×10^–9 Pa^–1 and show a weak temperature dependence. The stress-optical coefficient of PMMA is about a factor of 100 lower and shows a peculiar temperature-dependence, changing its sign at 144°C. The results are discussed in terms of the anisotropy of the polarizability of the polymer chain.     

The instron rheometer

The rheometer is based on an original design by the Monsanto Chemical Co., for the study of flow properties of polymer melts. Shear rates over the range 10¹ to 10⁶ reciprocal seconds are obtained by driving a plunger at constant rates through a barrel with interchangeable capillary tubes at the exit. The resulting shear stress is measured by the accurate load measuring system of the basic instrument.Errors due to the length to diameter ratio of the capillaries, plunger friction, polymer compressibility, temperature rise due to friction and pressure drop in the barrel, are discussed and correction methods suggested.Standard and special applications of the rheometer are discussed indicating that it can be used to determine factors other than apparent viscosity.Paper presented at the Conference on Experimental Rheology, University of Bradford, April 17–19, 1968. — Original paper published in Instron Application Series SA-3     

The influence of pressure on the capillary flow of poly(methyl methacrylate)

Summary Pressure effects in the capillary flow of a single sample of poly(methyl methacrylate),M _v = 1.33 · 10⁵, were evaluated. The length/diameter ratios of the different capillaries used varied from 4 to 100. The tests were made with an Instron rheometer in the range 160–250 °C. The pressure-viscosity model, derived from the free volume-WLF equation, was used to pressure correct the capillary data. The corrected data agreed well with data obtained at atmospheric pressure using aWeissenberg rheogoniometer. A derived expression to calculate an increase in the flow activation energy with increasing stress predicts the observed increase in activation energy.

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Zusammenfassung Druckeffekte bei der Kapillarströmung einer einzelnen Polymethylmethacrylat-ProbeM _v = 1,33 · 10⁵ werden abgeschätzt. Die Längen/Durchmesserverhältnisse der verschiedenen verwendeten Kapillaren variierten zwischen 4 und 100. Im Temperaturbereich zwischen 160 und 250 °C wurden die Versuche mit einem Instron-Rheometer durchgeführt. Das Druck-Viskositätsmodell, das aus der WLF-Gleichung abgeleitet war, wurde zur Druckkorrektur der Kapillardaten verwendet. Die korrigierten Werte zeigten eine gute Übereinstimmung mit den Werten, die bei Atmosphärendruck mit Hilfe einesWeissenberg-Rheogoniometers gewonnen worden waren. Ein abgeleiteter Ausdruck zur Berechnung des Anstiegs der Strömungs-Aktivierungsenergie mit steigender Spannung sagt den beobachteten Anstieg in der Aktivierungsenergie voraus.

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On leave from Montecatini Edison, Milan (Italy).     

Viskositätsmessungen bei kleinen Schergefällen

Zusammenfassung Große, stark strukturierte Moleküle können Konformationswechselzeiten haben, die in der Größenordnung der Deformationsperiode bei der Scherung der Moleküle in Lösung liegen, wenn das Geschwindigkeitsgefälle kleine Werte annimmt. Es werden einige Viskosimeter beschrieben, die zur Untersuchung verdünnter Lösungen in einem Geschwindigkeitsgefälle-Intervall von 10^–2–10⁴ sec^–1 geeignet sind. Anhand von Messungen an DNA wird gezeigt, daß bei Geschwindigkeitsgefällen von 10 sec^–1 noch starke Strukturviskosität auftreten kann.

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Summary Large strongly structured molecules may have change of constellation times comparable to the deformation period in shearing the molecules in solution, if the shear rate assumes low values. Several viscometers are described, which are suitable for the investigation of diluted solutions in a shear rates range of 10^–2–10⁴ sec^–1. With the help of measurements with DNA, it is shown that strong shear dependence is observed for shear rates of 10 sec^–1.

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Vorgetragen auf der Rheologen-Tagung in Darmstadt am 7. und 8. Juni 1971

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Diese Arbeiten wurden von der DFG im Rahmen des SFB 41 Mainz-Darmstadt unterstützt, wofür wir herzlich danken.     

A small scale capillary viscometer for polymer melts

Summary A small scale capillary viscometer that is capable of measuring the flow curves of 2–3 gms of molten polymer over the shear rate range 0.4 sec^–1 to 20,000 sec^–1 is described. A piezo electric crystal pressure transducer is used to measure the pressure directly above the capillary. Flow curves are given for anionic polystyrenes and polyethylene fractions, and it is shown that the viscosities measured agree with determinations in a cone and plate viscometer.

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Zusammenfassung Es wird ein maßstäblich kleines Kapillarviskosimeter für 2–3 g geschmolzener Polymere beschrieben, das die Durchfiußkurve in Abhängigkeit vom Scherkoeffizienten im Bereich von 0,4 sec^–1 bis 20000 sec^–1 bestimmt. Verwendet wird ein piezoelektrischer Kristall-Druckaufnehmer, der den Druck genau über der Kapillare mißt. Angeführt werden Durchflußkurven für anionische Polystyrole und Polyäthylenanteile. Es wird gezeigt, daß die gemessenen Zähigkeiten mit denen von Kegel- und Plattendichtemessern übereinstimmen.

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

Nonequilibrium stretching dynamics of dilute and entangled linear polymers in extensional flow

We propose an extension of the FENE-CR model for dilute polymer solutions [M.D. Chilcott, J.M. Rallison, Creeping flow of dilute polymer solutions past cylinders and spheres, J. Non-Newtonian Fluid Mech. 29 (1988) 382–432] and the Rouse-CCR tube model for linear entangled polymers [A.E. Likhtman, R.S. Graham, Simple constitutive equation for linear polymer melts derived from molecular theory: Rolie–Poly equation, J. Non-Newtonian Fluid Mech. 114 (2003) 1–12], to describe the nonequilibrium stretching dynamics of polymer chains in strong extensional flows. The resulting models, designed to capture the progressive changes in the average internal structure (kinked state) of the polymer chain, include an ‘effective’ maximum contour length that depends on local flow dynamics. The rheological behavior of the modified models is compared with various results already published in the literature for entangled polystyrene solutions, and for the Kramers chain model (dilute polymer solutions). It is shown that the FENE-CR model with an ‘effective’ maximum contour length is able to describe correctly the hysteretic behavior in stress versus birefringence in start-up of uniaxial extensional flow and subsequent relaxation also observed and computed by Doyle et al. [P.S. Doyle, E.S.G. Shaqfeh, G.H. McKinley, S.H. Spiegelberg, Relaxation of dilute polymer solutions following extensional flow, J. Non-Newtonian Fluid Mech. 76 (1998) 79–110] and Li and Larson [L. Li, R.G. Larson, Excluded volume effects on the birefringence and stress of dilute polymer solutions in extensional flow, Rheol. Acta 39 (2000) 419–427] using Brownian dynamics simulations of bead–spring model. The Rolie–Poly model with an ‘effective’ maximum contour length exhibits a less pronounced hysteretic behavior in stress versus birefringence in start-up of uniaxial extensional flow and subsequent relaxation.     

Viscosity of some clay-based coating colors at high shear rates

The shear viscosity of clay-based coating colors containing latex and carboxymethyl cellulose (CMC) has been measured over a relatively large shearrate region. In the shear-rate range of 50–1500 s^–1 the measurements were performed using a rotational viscometer and, at higher shear rates extending into the region 10⁵ – 10⁶ s^–1, a high pressure capillary viscometer was employed. The viscosity of the clay colors increased with increasing CMC-concentration, but the influence of the CMC-content was less pronounced at higher shear rates. The apparent shear-thinning behavior of the investigated colors could, in part, be attributed to the shear-thinning of the corresponding polymer (CMC) solution constituting the liquid phase of the color, but the influence of another factor was also indicated. At low shear rates, the interaction between the color components can produce relatively high viscosity levels, but in the high shear rate region these interactions appear to be less important for the viscosity level. It is also of interest to note that the viscosity dependence on the solids content in the high shear-rate region could be described with reasonable accuracy using an empirical equation neglecting interactions between the color components.     

Viscous properties of polymer melts and elastomers exemplified by ethylene-propylene copolymer

Summary The copolymer of ethylene and propylene possesses a sufficiently high thermo-oxidative resistance, making it possible to study its viscous properties, determine the appearance of elastic turbulence and wall slippage and to measure the rate of the latter over a wide interval of temperatures ranging from room temperature to 260 °C.At low shear stresses and rates the copolymer behaves like aNewtonian liquid with a viscosity of about 10⁸ poise at room temperatures.Elastic turbulence and wall slippage are displayed in sharp form when the viscosity of the copolymer is lowered to its critical value, which depends very little on the temperature and may be accepted as averaging 2.2×10⁴ poise. The corresponding critical shear stress values vary about 10-fold. The criteria of appearance of elastic turbulence suggested in (12, 14) do not agree with experimental data. The entrance losses during the flow of the copolymer through capillaries are low right until elastic turbulence sets in, after which it becomes practically impossible to measure them by the method of capillaries of different length. The average wall slippage rate values of the copolymer at shear stresses above 10⁶ dyne/cm² amount to tens of cm/sec. They increase very abruptly with rising temperature.The temperature dependence of the viscosity and the dynamic characteristics of the copolymer indicate that it has a phase transition at temperatures of about 100–120 °C, which must be related to melting of blocks contained in the copolymer macromolecules, having a structure close to that of high-pressure polyethylene. This shows that the rheological method of studying block-type polymer and grafted polymers is promising.     

Rheological behaviour of a filled wax system

The temperature dependent rheological behaviour of a pigment filled wax system is investigated in a cone-and-plate viscometer over a range of shear rates from 60 to 10 000 s^–1. A strong influence of water adsorbed by the pigment on rheological properties of the filled system can be found. The increase of the yield stress and the viscosity at low shear rates can be related to a build-up of pigment structures due to growing water content. The flow behaviour can be described by the Casson equation as well as by the Herschel-Bulkley equation.Both formulations are compared and discussed. The Casson model is evaluated more closely by the calculation of characteristic structural parameters of the suspension which are critically discussed.Dedicated to Prof. Dr. Joachim Meissner on the occasion of his retirement from the chair of polymer physics at the Eidgenössische Technische Hochschule (ETH), Zürich.     

Direct measurement of static yield properties of cohesive suspensions

A technique of yield stress investigation based upon the combined use of two devices (an applied stress rheometer and an instrument for measuring the propagation velocity of small amplitude, torsional shear waves) is described. Investigations into the low shear rate rheological properties of illitic suspensions are reported for shear rates, typically, in the range 10^–4— 10^–1 s^–1 under applied stresses in the range 0.01 — 10 Nm^–2 and involving shear strains between 10^–1 and 10^–4. Results are presented which demonstrate that the technique does not invoke the excessive structural disruption of material associated with applied shear rate based methods (direct and otherwise) and the widely encountered problem of wall slip at the surface of rotational measuring devices is avoided using miniature vane geometries. Results are compared with those obtained using smooth-walled cyclindrical measuring devices in both applied stress and applied shear rate instruments.Yield measurements are considered in relation to the structural properties of the undisturbed material state and shear moduli obtained by studying the propagation of small amplitude (10^–5 rad), high frequency (~ 300 Hz) torsional shear waves through the test materials are reported. Experimental techniques and instrument modifications to permit these measurements are described.