An appraisal of rheological models as applied to polymer melt flow

Summary An attempt is made at giving an appraisal of some representative rheological models of both differential and integral type, using the standard rheological measurements of six polymer melts. Experimental data obtained were the steady shear viscosity and the first normal stress difference by means of aWeissenberg rheogoniometer over the range of shear rates: 10^–2 ~ 10 sec^–1, and by means of aHan slit/capillary rheometer over the range of shear rates: 10 ~ 10³ sec^–1. Also measured by means of theWeissenberg rheogoniometer were the dynamic viscosity and dynamic elastic modulus over the range of frequencies: 0.3 × 10^–2 ~ 3 × 10² sec^–1. Rheological models chosen for an appraisal are theSpriggs 4-constant model, theMeister model, and theBogue model.It is found that the capability of the three models considered is about the same in their prediction of the rheological behavior of polymer melts in simple shearing flow. It is pointed out however that, due to the ensuing mathematical complexities, the usefulness of these models is limited to the study of flow problems associated with simple flow situations. Therefore, in analysing the complex flow situations often encountered with various polymer processings, the authors suggest use of the empirical models of the power-law type for both the viscosity and normal stress functions.With 11 figures, 4 schemas and 1 table     

Longitudinal volume viscosity of poly (vinyl chloride)

The volume flow of poly (vinyl chloride) ( = 45,000,T _g = 350 K) has been measured in an Instron Capillary Rheometer.The elastic modulus in longitudinal compression, the longitudinal volume viscosity and initial longitudinal volume viscosity, and retardation times were determined at temperatures both below (324 – 343 K) and above (403 – 453 K) the glass transition temperatureT _g , and at compression rates between approximately 10^–5 and 200 · 10^–5 s^–1.An increase in the longitudinal volume viscosity was observed for decreases in the volume deformation, increases in the compression rate and increases in temperature.T _g decreased at 0.16 K/MPa. The volume flow activation energy was found to be equal to that for shear flow with a constant value of 91.37 kJ/mol.     

Rheology and birefringence of Fomblin YR at very high shear rates

The rheological and structural properties of perfluoropolyether (PFPE) lubricant films including viscosity, shear stress, and birefringence were measured at relatively low to extremely high shear rates using a rotational optical rheometer. The viscosity of various films with different thicknesses exhibit Newtonian behavior up to a shear rate 1 × 10⁴ s⁻¹, with a transition to shear-thinning behavior obvious at higher shear rates. Birefringence of these films was also measured for the first time, and these results indicate chain alignment with shear in the shear-thinning regime. The shear rate at which alignment occurs is similar to that of the onset of shear thinning. This correlation between chain alignment and shear thinning provides direct evidence that the ability of PFPEs to lubricate hard drives at high shear rates is a direct consequence of the ability of the applied shear field to align the molecules on a molecular level.     

Rheology of oil-in-water emulsions

The effect of interfacial tension on the steady-flow and dynamic viscoelastic behavior of emulsions are studied experimentally. At very low inter-facial tensions and low volume fractions, the viscosity decreases with increasing shear rate and becomes constant at high shear rates. The high-shear-rate Newtonian viscosity is not affected by interfacial tension, but the transition from pseudoplastic to Newtonian flow shifts to lower shear rates as the interfacial tension decreases. At an interfacial tension of 5 × 10^–3 Nm^–1, the viscosity decreases, passes through a minimum, and then increases as the shear rate is increased. The dilatant behavior may be attributed to elastic responses of interfaces during collision of drops. At high volume fractions, the emulsions show remarkable elasticity resulting from the interfacial energy associated with deformation of liquid films. The modulus and viscosity are proportional to interfacial tension and inversely proportional to drop size.     

Elongational flow and rheology of monodisperse polymers in solution

We describe the utilization of idealized stagnation point extensional flows, produced by opposed jets, for birefringence visualization of induced molecular strain and flow resistance measurements. We identify rheological changes associated with the coil---stretch transition which occurs beyond a critical strain-rate in elongational flow-fields. In dilute solutions of monodisperse atactic polystyrene, increases in extensional viscosity are observed as isolated molecules become stretched. The largest increases in extensional viscosity, however, are found only beyond a critical concentration and strain rate, and are associated with the stretching of transient networks of interacting molecules. These results parallel similar effects seen in porous media flow and capillary entrance experiments. We determine the molecular weight dependence of the critical concentration which scales as M^−0.55 in agreement with pairwise interaction of coils, but is much lower than conventional values of the critical polymer concentration, c^*. We believed that polydispersity may play an important role in the development of such transient networks, and in controlling the degradation behaviour during flow.     

Shear dependence of viscosity for perfluoropolyether fluids

The shear dependence of the bulk viscosities of two structurally different types of perfluoropolyether fluids was determined by two different techniques. The first involved direct measurement in a high shear Couette viscometer, the second utilized the time-temperature superposition principle to establish master curves from viscosity determinations at low shear rates and temperature; the results are comparable. Both fluids begin to show non-Newtonian behavior at shear rates above 10,000 s^–1.     

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.     

Rheologische Eigenschaften gesunder Synovialflüssigkeiten

Zusammenfassung Die rheologischen Eigenschaften gesunder menschlicher Gelenksflüssigkeiten im Geschwindigkeitsgefällebereich vonD = 10^–3-10³ s^–1 wurden untersucht. Es wurden die Scherviskosität und die erste Normalspannungsdifferenz gemessen. Gesunde Synovialflüssigkeiten besitzen hohe Anfangsviskosität (~40 Pa · s) und zeigen eine starke Abhängigkeit vom Geschwindigkeitsgefälle. Der SchermodulG ist im Gegensatz zu pathologischen Proben niedrig und über weite Bereiche konstant. Die längsten Relaxationszeiten betragen 5–10 s. Die kritische Konzentration, bei der Netzwerkbildung einsetzt, beträgt 0,75 10^–3 g/ml. Die Proben lassen sich zu einer Masterkurve vereinigen, die als verallgemeinertes Fließgesetz für gesunde Synovia aufgefaßt werden kann. Eine Untersuchung über die zeitliche Abhängigkeit der post-mortem-Synovia zeigt, daß innerhalb von 12 Stunden keine nennenswerten Veränderungen eintreten.

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The rheology of healthy human synovial fluids has been investigated at a shear-rate between 10^–3-10³ s^–1. Shear viscosity and first normal-stress difference were measured. Healthy synovial fluids show high zero-shear viscosity (about 40 Pa · s) and a strong shear rate dependence. The modulusG is constant over a large range, in contrast to pathological samples. The longest relaxation times are 5–10 s. The critical concentrationc _cr , at which entanglement occurs is about 0.75 10^–3 g/ml. The samples can be represented by a master-curve, which may be regarded as the constitutive equation of healthy synovial fluids. An investigation of the time dependency of synovial fluids indicated no changes within 12 hours post mortem.

Herrn Prof. Dr. DDr. h. c. O. Kratky zum 80. Geburtstag mit den besten Wünschen gewidmet.     

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.     

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.     

Characterization of the flow properties of sodium carboxymethylcellulose via mechanical and optical techniques

Sodium carboxymethylcellulose (NaCMC) in solution represents a complex rheological system, since it forms aggregates and associations and hence higher-level structures and, depending on the synthesis, is only found in a molecularly dispersed form in exceptional cases. Rheo-mechanical investigations of the viscoelasticity showed that the Cox-Merz rule is not fulfilled. The aim was therefore to examine whether rheo-optics could be employed to provide more detailed conclusions about the parameters that influence the flow behavior of NaCMC than has hitherto been available with mechanical methods. The flow birefringence, Δn ^′, rises as the degree of polymerization increases, and exhibits the same dependence on molar mass as does the viscosity: Δn ^′∝M _w ^3.4. As the degree of polymerization increases while the shear rate remains constant, the polymer segments become more distinctly aligned in the direction of shear. Hence increasing the degree of polymerization also affects the solution structure, i.e. the interaction of the molecules with one another. The stress-optical rule only applies to a limited extent for this system. The stress-optical coefficient, C, is almost independent of the shear rate, but is strongly influenced by the concentration and attains a limiting value of 3 × 10⁻⁸ Pa⁻¹. C was determined for a polymer in dilute solution and the curve obtained also enabled transitions in the solution structure to be recognized. Received: 1 May 1998 Accepted: 5 October 1998     

Modeling of blood rheological behavior in unsteady-state shearing flows

Issues of blood flow modeling under unsteady-state conditions at moderate shear rates are considered using a blood rheological model accounting both for the viscoelastic properties and the thixotropy caused by erythrocyte aggregation. The resulting shear stress versus time relations for single shear rate steps and the dependence of the complex viscosity components on the shear rate amplitude in oscillating shear flow show good qualitative correspondence with the experimental data reported in the literature.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 26–30, November–December, 1995.     

Rheological properties of some biphenyl liquid crystals

Rheological measurements have been carried out using a rotational viscometer with a system of coaxial cylinders on four liquid crystalline substances from the group of cyanobiphenyls. On the basis of results of these investigations it was found that in the investigated range of shear rates the nematic phases exhibited Newtonian flow behaviour, while the smectic phases exhibited non-Newtonian behaviour. For shear rates up to ca. 1000 s^–1 the dependence of the shear stress on shear rate is well described by a power-law model.     

Surfactant systems for drag reduction: Physico-chemical properties and rheological behaviour

The first part of the work presents an overview of the physical chemistry of surfactants which in aqueous solutions reduce the frictional loss in turbulent pipe flow. It is shown that these surfactants form rodlike micelles above a characteristic concentraionc _t. The experimental evidence for rodlike micelles are reviewed and the prerequisites that the surfactant system must fulfill in order to form rodlike micelles are given. It is demonstrated by electrical conductivity measurements that the critical concentration for the formation of spherical micelles shows little temperature dependence, whereasc _t increases very rapidly with temperature. The length of the rodlike micelles, as determined by electric birefringence, decreases with rising temperature and increases with rising surfactant concentration. The dynamic processes in these micellar systems at rest and the influence of additives such as electrolytes and short chain alcohols are discussed.In the second part, the rheological behaviour of these surfactant solutions under laminar and turbulent flow conditions are investigated. Viscosity measurements in laminar pipe and Couette flow show the build-up of a shear induced viscoelastic state, SIS, from normal Newtonian fluid flow. A complete alignment of the rodlike micelles in the flow direction in the SIS was verified by flow birefringence. In turbulent pipe flow, drag reduction occurs in these surfactant systems as soon as rodlike micelles are present in the solution. The extent and type of drag reduction, i.e. the shape of the friction factor versus Reynolds number curve, depends directly on the size, number and surface charge of the rodlike micelles. The friction factor curve of each surfactant investigated changes in the same characteristic way as a function of temperature. For each surfactant, independent of concentration, an upper absolute temperature limit,T _L, for drag reduction exists which is caused by the micellar dynamics.T _L is influenced by the hydrophobic chain length and the counter-ion of the surfactant system. A first attempt is made to explain the drag reduction of surfactants by combining the results of these rheological measurements with the physico-chemical properties of the micellar systems.     

Rheological properties of liver actin solutions

G-actin prepared from pork liver and purified to give a product of 95–98% purity was polymerized to F-actin in a rotational dynamic viscometer. The rheological properties were investigated during the course of polymerisation and after polymerisation. G andG measured at frequencies around 0.1 Hz increased strongly during the first 3 hours of the reaction and then slowly approached a constant value in the range of 10–50 mPa. The rise during polymerisation was more than two orders of magnitude. When a solution of F-actin was subjected to steady shear at a high shear rate for a short period of time and subsequently dynamic measurements were performed, then a considerably smallerG (about 70% smaller) was observed immediately after cessation of the steady shear.G then increased with time and approached its initial value.When measuring the viscosity of F-actin solutions as a function of shear rate, a strong shear thinning effect was seen which did not vanish even at shear rates as low as 0.001 s^–1. Even at this low shear rates, a pronounced yield maximum was observed before reaching the steady state. Oscillatory experiments showed a remarkably weak frequency dependence ofG. The results imply that F-actin solutions are largely structured forming a weak temporary network which can be easily destroyed by application of high shear rates. It seems most likely that the destruction occurs by a reversible, shear induced depolymerisation process.     

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.     

A novel approach to high shear rate rheometry

A procedure is presented for converting torque-speed plots obtained from experiments using discs rotating rapidly in shear thinning materials into information on the viscosity function. The method is based on an exact boundary-layer solution for the power-law model and on the concept of pseudo-similarity of non-Newtonian flows. It enables the rheological behaviour to be evaluated at very high shear rates. Experimental data for concentrated shear-thinning kaolin suspensions at shear rates from 400 to 2 10⁵s^–1 are compared with values of the viscosity function obtained from customary viscometers of the Brookfield and Couette type.     

Flow development of xanthan solutions in capillary rheometers

Flow experiments through capillaries with 0.2% xanthan in aqueous solution and 0.1 N NaCl brine were carried out to study the influence of the molecular conformation on the flow development at relatively low shear rates, from 20s^–1 to 400s^–1. Capillaries with a wide range of length-to-diameter ratios, L/D = 4.5 to 1015 were used.The apparent viscosity as a function of L/D at a constant shear rate shows a continuous decrement of the viscosity as L/D increases, until an asymptotic value is reached. The decrement in the apparent viscosity is partially explained in terms of slip. It was found that slip is a function of L/D as well as shear stress, i.e., slip develops during flow, thus inducing spatial anisotropy in the fluid until a stable state is reached. However, the substantial difference in apparent viscosity between short capillaries and capillaries longer than 300 D may be attributed to dominant elongational flow due to the contraction in the small capillaries and slip in long capillaries.The flow in a sufficiently long capillary can be divided in four regions rather than three, as is usually assumed. In the first region, which corresponds to the entry, elongational and shear flow coexist and elongational flow dominant. In the second region, end effects and slip development are coupled. In the third region the flow is fully developed and end effects are negligible. However, the fluid shows physical characteristics different from those of the fluid at rest, as a consequence of prior slip development. The fourth zone is the exit region in which the velocity rearranges due to the change of boundary conditions. The length of each region depends on the conformation of the macromolecules and shear rate. In addition, it was found that the stiffness of xanthan increases with the increase of the ionic strength.Finally, a performance of Bagley's analysis in the whole range of L/D studied showed that the use of the Bagley correction is not a reliable way to correct for end effects when the flow is not fully developed and/or in the presence of slip.Dedicated to Arthur S. Lodge at the occasion of his 70th birthday and his retirement from the University of Wisconsin.     

Imperfect lubricated squeezing flow viscometry for foods

Commercial mayonnaise and mustard samples placed in a wide, shallow Teflon container were compressed by a wide Teflon plate to induce an imperfect lubricated squeezing flow. A dominant squeezing flow regime could be clearly identified as a linear region in the log F(t) vs log H(t) relationship, F(t) and H(t) being the momentary force and specimen height respectively. The slope of the relationship enabled the estimation of the flow index, n, and the consistency coefficient K. The n values of the mayonnaise were on the order of 0.6–0.85 and those of the mustard about 0.7. The corresponding K values were on the order of 6–13 and 4–5 kPasⁿ respectively. Considering the crudeness of the array the measurements were highly reproducible and sensitive enough to detect differences (mayonnaise) or establish similarities (mustard) in products of different brands. The calculated flow index was practically independent of the plate's radius and of the consistency coefficient, which had a weak dependency on the latter. The calculated elongational viscosity vs biaxial strain rate relationship could also be used to compare the different products and brands. At 0.01 s^–1 the elongational viscosity of the maynonnaise was on the order of 150 kPas, and of the mustard 60 kPas.     

The shear viscosity dependence on concentration,molecular weight,and shear rate of polystyrene solutions

The solution viscosity of narrow molecular weight distribution polystyrene samples dissolved in toluene and trans-decalin was investigated. The effect of polymer concentration, molecular weight and shear rate on viscosity was determined. The molecular weights lay between 5 10⁴ and 24 10⁶ and the concentrations covered a range of values below and above the critical valuec ^*, at which the macromolecular coils begin to overlap. Flow curves were generated for the solutions studied by plotting log versus log . Different molecular weights were found to have the same viscosity in the non-Newtonian region of the flow curves and follow a straight line with a slope of – 0.83. A plot of log ₀ versus logM _w for 3 wt-% polystyrene in toluene showed a slope of approximately 3.4 in the high molecular weight regime. Increasing the shear rate resulted in a viscosity that was independent of molecular weight. The sloped (log)/d (logM _w ) was found to be zero for molecular weights at which the corresponding viscosities lay on the straight line in the power-law region.On the basis of a relation between _sp and the dimensionless productc · [], simple three-term equations were developed for polystyrene in toluene andt-decalin to correlate the zero-shear viscosity with the concentration and molecular weight. These are valid over a wide concentration range, but they are restricted to molar masses greater than approximately 20000. In the limit of high molecular weights the exponent ofM _w in the dominant term in the equations for both solvents is close to the value 3.4. That is, the correlation between _sp andc · [] results in a sloped(log _sp)/d(logc · []) of approximately 3.4/a at high values ofc · [] wherea is the Mark-Houwink constant. This slope of 3.4/a is also the power ofc in the plot of ₀ versusc at high concentrations. a Mark-Houwink constant - B ₁,B ₂,B _n constants - c concentration (g · cm^–3) - c ^* critical concentration (g · cm^–3) - K, K constants - K _H Huggins constant - M molecular weight - M _c critical molecular weight - M _n number-average molecular weight - M _w weight-average molecular weight - n sloped(log _sp)/d (logc · []) at highc · [] - PS polystyrene - T temperature (K) - shear rate (s^–1) - critical shear rate (s^–1) - viscosity (Pa · s) - ₀ zero-shear viscosity (Pa · s) - _s solvent viscosity (Pa · s) - _sp specific viscosity - [] intrinsic viscosity (cm³ · g^–1) - dynamic viscosity (Pa · s) - | ^*| complex dynamic viscosity (Pa · s) - angular frequency (rad/s) - density of polymer solution (g · cm^–3) - ₁₂ shear stress (Pa) Dedicated to Prof. Dr. J. Schurz on the occasion of his 60^th birthday.Excerpt from the dissertation of Reinhard Kniewske: Bedeutung der molekularen Parameter von Polymeren auf die viskoelastischen Eigenschaften in wäßrigen und nichtwäßrigen Medien, Technische Universität Braunschweig 1983.