On the extensional viscosity of mobile polymer solutions

We describe experimental results on the extensional viscosity of mobile polymer solutions obtained from two instruments, the first being a commercial Spin Line Rheometer and the second a custom-built lubricated-die Converging Flow Rheometer. The interpretation of data in terms of Trouton ratios is facilitated by a simple analysis for the Generalized Newtonian Fluid model.Agreement between data from the two rheometers is satisfactory and we show that polymer solutions can be either tension stiffening or tension thinning. However, the Trouton ratios in both cases are greater than the Newtonian values and we anticipate that this will always be the case for polymer solutions.Invited paper, presented at the 2nd Conference of European Rheologists, Prague, June 17–20, 1986     

Measurement of extensional viscosity of polymer solutions

A filament stretching technique for measuring the extensional viscosity of polymer solutions at constant stretch rate is presented. The liquid sample is held between two coaxial discs and stretched by moving the bottom disc downwards with a speed that increases exponentially with time. This is illustrated using a constant viscosity, elastic fluid consisting of 0.185% polyisobutylene in a solvent of kerosene and polybutene. For the case of this particular fluid, two distinct stretch rate regions are found to arise. The stretch rate in the first region is much higher than in the second, which is, in most cases, close to the overall stretch rate imposed on the sample. Nonetheless, all the results of any given run can be represented using an average extensional rate. The extensional stress growth data, plotted as the Trouton ratio against time, show an initial linear viscoelastic region where T_R rises to a value of 3, independent of extensional rate. Beyond this region, T_R depends on the stretch rate and rises dramatically to values in excess of 10³; the higher the extensional rate, the faster is the increase in T_R. These data do not seem to reach a steady state and appear to be similar to polymer melt data obtained by others in the past. The reproducibility of the results is very good and all this suggests that it is now possible to obtain unambiguous constant-stretch-rate stress-growth data for polymer solutions stretched from a state of rest.     

On the planar extensional viscosity of mobile liquids

An instrument is described in which steady planar orthogonal stagnation flows of the type proposed by Winter [1] are achieved with lubrication. From pressure readings on steady results acceptable Trouton ratios are achieved for a Newtonian sample and results are obtained for a ‘Boger’ fluid showing high levels of extensional viscosity with Trouton ratios approaching 4 in the limit of zero stretch rate.     

Effect of a controlled pre-deformation history on extensional viscosity of dilute polymer solutions

We use a modified filament stretching rheometer to quantify the influence of a known controlled pre-shear history on the transient extensional viscosity of a dilute polymer solution. Two different types of pre-deformation are explored; both influence the subsequent stretching significantly, albeit in opposite ways. Small-amplitude oscillatory straining parallel to the direction of stretching enhances strain hardening and accelerates the tensile stress growth toward the steady-state value. Conversely, steady torsional shearing orthogonal to the direction of stretching retards strain hardening and results in a delayed approach to steady-state elongational flow. In both cases, the final steady-state extensional viscosity is the same as that observed with no pre-shearing. Calculations using a finitely extensible nonlinear elastic Peterlin dumbbell model qualitatively capture the trends observed in experiments, enabling interpretation of these observations in terms of the degree of polymer chain stretching imposed by the flow before extensional stretching.     

On a generalised viscosity equation for polymer solutions

Based on the AFVS concept, a simple equation to predict the viscosity behaviour of polymer solutions is proposed in the following form $$(\ln \eta _r )^{ - 1} = \frac{\alpha }{{\tilde c}} + \delta .$$ This has been verified by comparison with experimental data. The equation appears to have a far more general validity as it appears to correlate the data for a large variety of binary systems including suspensions, liquid mixtures etc.     

Measurement of relaxation times in extensional flow of weakly viscoelastic polymer solutions

The characterization of the extensional rheology of polymeric solutions is important in several applications and industrial processes. Filament stretching and capillary breakup rheometers have been developed to characterize the extensional properties of polymeric solutions, mostly for high-viscosity fluids. However, for low concentration polymer solutions, the measurements are difficult using available devices, in terms of the minimum viscosity and relaxation times that can be measured accurately. In addition, when the slow retraction method is used, solvent evaporation can affect the measurements for volatile solvents. In this work, a new setup was tested for filament breakup experiments using the slow retraction method, high-speed imaging techniques, and an immiscible oil bath to reduce solvent evaporation and facilitate particle tracking in the thinning filament. Extensional relaxation times above around 100 μs were measured with the device for dilute and semi-dilute polymer solutions. Particle tracking velocimetry was also used to measure the velocity in the filament and the corresponding elongation rate, and to compare with the values obtained from the measured exponential decay of the filament diameter.     

The relevance of entry flow measurements for the estimation of extensional viscosity of polymer melts

The extensional viscosity of some flexible chain polymers and a thermotropic liquid crystalline polymer was measured in uniaxial extensional flow at constant extension rate. Power law functions were found for the dependence of the extensional viscosity at constant accumulated strain on strain rate. The stress growth curves were compared with measurements in axisymmetric entry flow, where both elongation and shear occur. The comparison showed that the values of the extensional viscosity calculated from the measurements in the entry flow correspond to the ones calculated from the viscosity growth measured in uniaxial elongation and averaged over extensional strain equal to what is accumulated on the fluid as it flows from the barrel into the capillary.     

Numerical simulations of Boger fluids through different contraction configurations for the development of a measuring system for extensional viscosity

This paper reports the flow behaviour of Newtonian and Boger fluids through various axisymmetric contraction configurations by means of numerical predictions. A principal aim has been to evaluate the geometrical design choice of the hyperbolic contraction flow. The FENE-CR model has been used to reflect the behaviour of Boger fluids, with constant shear viscosity, finite (yet large) extensional viscosity and less than quadratic first normal stress difference. Numerical calculations have been performed on six different contraction configurations to evaluate an optimized geometry for measuring extensional viscosity in uniaxial extensional flow. The influence of a sharp or rounded recess-corner on the nozzle has also been investigated. Few commercial measuring systems are currently available for measurement of the extensional rheology of medium-viscosity fluids, such as foods and other biological systems. In this context, a technique based on the hyperbolic contraction flow would be a suitable alternative. The pressure drop, the velocity field, the first normal stress difference and the strain rate across the geometry have each been evaluated for Newtonian and Boger fluids. This numerical study has shown that the hyperbolic configuration is superior to the other geometry choices in achieving a constant extension rate. In this hyperbolic configuration, no vortices are formed, the measuring range is broader and the strain rate is constant throughout the geometric domain, unlike in the alternative configurations tested. The difference between sharp and rounded recess-corner configurations proved to be negligible and a rise in excess pressure drop (epd) for increasing deformation rates has been observed.     

Excluded volume effects on the birefringence and stress of dilute polymer solutions in extensional flow

An algorithm is derived for calculating flow-induced birefringence using a bead-spring model with and without excluded volume effects. The simulation results for the bead-spring model compare well with experimental results for stress and birefringence in extensional flows of dilute solutions of polystyrene molecular weight 2 million in a filament-stretching device in both “theta” and “good” solvents (Orr and Sridhar 1999; Sridhar et al. 2000). In a “good” solvent, both stress and birefringence rise much more rapidly with strain than in a “theta” solvent, making extensional rheology a very sensitive indicator of solvent quality. Received: 7 December 1999 Accepted: 23 May 2000     

On the flow of a yield strength fluid through a contraction

This paper presents a lubrication-type analysis of the pressure gradient-driven flow in a contraction connecting two parallel plate (Poiseuille) flow regions for a Bingham fluid. The solution is complete in the streamfunction at lowest order in the slope of the contraction. I find a floating core on the downstream side of the contraction, growing to match the floating core in the downstream Poiseuille flow. This is unusual. In most contracting or distorting flows the necessary deviatoric shear requires yielding everywhere, accompanied by deviatoric normal stresses.     

Stress relaxation of a polymer melt after extensional flow

The relaxation behaviour of a stretched polymer melt is described with the aid of the semi-empirical constitutive equation of Wagner. The theoretical predictions based on this model, are in fair agreement with the experimental results, as far as the period of stress decay following the stress build-up is not too long.     

The unraveling of a polymer chain in a strong extensional flow

By assuming that in a strong extensional flow a polymer molecule in dilute solution is quickly driven into a folded or kinked state in which drag and entropic elastic forces dominate over the Brownian force, we derive kink dynamics equations that describe the unraveling of the molecule in the extensional flow. Solving these equations numerically, we find that although the ends of the chain move, on average, affinely in the flow field until the chain is unfolded to about 1/3 of its fully extended length, large viscous stresses are produced because the solvent must flow around nonextending strands of polymer that lie between neighboring kinks. These predictions are compared with available experimental data and with other theoretical models.     

Extensional flow of polymer solutions through porous media

The flow behaviour of various polymer solutions of non-hydrolyzed polyacrylamide, hydrolyzed polyacrylamide, polyox and Xanthan was investigated in a plexiglass column having a succession of enlargements and constrictions, and compared with the flow behaviour and mechanical degradation of a solution of non-hydrolyzed polyacrylamide in a packed column of non-consolidated sand. The flow behaviour of this solution was found to be very similar in both the sand pack and plexiglass pore.Apart from the Xanthan solution, all other polymer solutions showed a viscoelastic behaviour in the plexiglass pore. The onset of viscoelastic behaviour, which has previously been defined using the shear rate ( ), stretch rate ( _s ) and Ellis number (E ₁), could be more precisely evaluated using a modified stretch rate (S _G). The pressure losses across the plexiglass pore for different polymer solutions of the same type were found to follow a unique curve provided the suggested group (S _G) was used, a situation which was not achieved with the other rheological parameters.The multipass mechanical degradation of the non-hydrolized polyacrylamide was tested through the sand pack against the suggested group (S _G) and Maerker's group (M _a). It was found that the loss of the solution viscoelasticity due to multipass mechanical degradation was better represented usingS _G thanM _a. A cross-sectional area (cm²) - C ^* critical concentration of polymer (ppm) - d plexiglass pore enlargement diameter - D average sand grain diameter (cm) - e equivalent width for the plexiglass pore - E ₁ Ellis number (a Deborah number) - F _R resistance factor - F _Ri resistance factor at the first pass - h height of the flow path of the plexiglass pore - K power-law constant - K _h,K _w effective permeability to hydrocarbon and water, respectively (10^–8 cm²) - M _a Maerker's group for a given porosity (s^–1) - M _ai value ofM _a at the first pass - N _D Deborah number - n power-law index - Q flow rate (cm³/s) - R capillary radius (cm) - R _g radius of gyration - S _G suggested group of rheological parameters representing a modified maximum stretch rate (s^–1) - S _Gi value ofS _G at the first pass - T _R,t characteristic time for the fluid (s) - t _s residence time (s) - V ₀ superficial velocity (cm/s) - V mean velocity of flow through a porous medium (cm/s) - average axial velocity in the enlargement section of the plexiglass pore (cm/s) - V ₁,V ₂ maximum velocity at a plexiglass enlargement neck and centre - [] intrincis viscosity - viscosity (mPa s) - _r relative viscosity (ratio of the viscosity of the polymer solution to that of the solvent) - shear rate (s^–1) - _s stretch rate (s^–1) - characteristic time for the polymer solution (s)     

Shear and extensional flow of polyacrylamide solutions

As part of an EEC Science Stimulation programme on extensional viscosity two major conferences were organised on the subject. The second of these was devoted to the results obtained on a standard fluid, M 1. The data obtained in shear flow was remarkably consistent from laboratory to laboratory. Extensional flow results presented quite a different picture. Using a series of nonequilibrium techniques such as the spinline rheometer, opposing jet, falling drop and converging flow, extensional viscosity results were obtained which differed by as much as two to three orders of magnitude. Nevertheless, it was apparent that consistancy did exist between similar techniques. It is in the context of this information that the measurements described below have been made.The shear and extensional flow properties of partially ionised polyacrylamide in solution at concentrations ranging from 5 ppm were measured. The method of solution preparation was found to have a profound effect on the behaviour of the solutions in shear flow. The influence of salt concentration and pH was investigated and is discussed in the context of molecular shape in solution.Extensional flow measurements, using the spinline rheometer, show that the solutions are strongly strain thickening even at concentrations as low as 5 ppm. These results are considered in the light of polymer entanglement and association in the strong flow field.Delivered as a Keynote Lecture at the Golden Jubilee Conference of the British Society of Rheology and Third European Rheology Conference, Edinburgh, 3–7 September, 1990.     

Non-Newtonian viscosity of concentrated polymer solutions

Summary The results of investigations of Newtonian and Non-Newtonian viscosity of solutions of various polymers in wide range of concentrations, temperatures, molecular weight of polymers in different solvents are presented here.These results show that the nature of solvent has a much greater effect on the viscosity of polar polymer solutions with very strong specific interaction than on the viscosity of non-polar polymer solutions.The nature of solvent affects not only the absolute value of viscosity, which as a rule, in the whole range of investigated shear stress is greater in poor solvent solutions than in good ones, but also determines the type of the flow curves.Solutions of flexible polymer (polyisobutylene) are characterized by complete flow curves regardless of the solvent nature. Solutions of rigid-chain polymer (acetyl-cellulose) are characterized by complete flow curves, both in good and in poor solvents. For the polystyrene solutions in good solvents incomplete flow curves and in poor solvents complete flow curves are observed.All these observations show that the anomaly of viscosity of polymer solutions is caused by two processes which occur simultaneously: the process of desintegration of structures and the process of orientation of chains and elements of the desintegrated structures.The structures presented in the solution dissociate under heat motion and activation heat and its changes with temperature and shear stress reflect very well the changes in the structure of solutions.The nature of solvent influences the value of critical molecular weight which in the case of good solvent is greater than in a poor one.

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Zusammenfassung Es werden die Untersuchungsergebnisse der Newtonschen und Nicht-Newtonschen Viskosität von konzentrierten Polymerlösungen in verschiedenen Lösungsmitteln innerhalb eines weiten Bereichs von Konzentration, Temperatur und Molekulargewicht der Polymeren mitgeteilt. Diese Resultate zeigen, daß die Art des Lösungsmittels einen wesentlich größeren Einfluß auf die Viskosität polarer Polymerlösungen mit starker spezifischer Wechselwirkung hat als auf die Viskosität nichtpolarer Polymerlösungen.Die Natur des Lösungsmittels beeinflußt nicht nur den Absolutwert der Viskosität, der üblicherweise im gesamten untersuchten Schubspannungsgebiet für schlechte Lösungsmittel größer ist als für gute, sondern auch den Fließkurvenverlauf. Unabhängig vom Lösungsmittel sind Lösungen von Polymeren mit flexiblen Ketten (Polyisobutylen) durch unvollständige Fließkurven gekennzeichnet. Die Lösungen von Polymeren mit steifen Ketten (Acetylcellulose) sind durch geschlossene Fließkurven sowohl in schlechten als auch in guten Lösungsmitteln charakterisiert. Die Polystyrollösungen ergeben bei guten Lösungsmitteln offene und bei schlechten Lösungsmitteln geschlossene Fließkurven.Diese Ergebnisse zeigen, daß die Viskositätsanomalie von Polymerlösungen von zwei gleichzeitig auftretenden Prozessen abhängig ist; dem Prozeß des Strukturzerfalls und dem Prozeß der Orientierung von Ketten und Elementen der zerfallenen Strukturen. Die in der Lösung existierenden Strukturen zerfallen durch die Wärmebewegung. Die Aktivierungswärme und ihre Änderungen in Abhängigkeit von Temperatur und Schubspannung geben die Strukturänderung der Lösungen sehr gut wieder. Die Art des Lösungsmittels beeinflußt die Größe des kritischen Molekulargewichts, das bei guten Lösungsmitteln größer ist als bei schlechten.

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Paper presented at the Conference on Advances in Rheology, Glasgow, September 16–18, 1969.     

Wall effects in the flow of flexible polymer solutions through small pores

Effective viscosities of dilute and semidilute flexible solutions flowing through small cylindrical pores were determined in the Newtonian regime for various pore diameters. The low viscosities relative to the bulk were associated with a depletion phenomenon due to a steric exclusion of macromolecules from the pore wall. Using a two-fluid flow model, the depletion layer thickness was determined and discussed as a function of polymer concentration, ionic strength, and molecular weight. This thickness, which was constant and close to the macromolecule gyration radius in dilute regime, was found to decrease rapidly with polymer concentration in the semidilute regime.     

The rheology of polymer solution elastic strands in extensional flow

We apply micro-oscillatory cross-slot extensional flow to a semi-dilute poly(ethylene oxide) solution. Micro-particle image velocimetry (μPIV) is used to probe the real local flow field. Extreme flow perturbation is observed, where birefringent strands of extended polymer originate from the stagnation point. This coincides with a large increase in the extensional viscosity. The combination of stagnation point flow and μPIV enables us to investigate directly the stress and strain rates in the strand and so determine the true extensional viscosity of the localised strand alone. The Trouton ratio in the strand is found to be ~4000, amongst the highest values of Trouton ratio ever reported. Consideration of the flow in the exit channels surrounding the highly elastic strand suggests a maximum limit for the pressure drop across the device and the apparent extensional viscosity. This has implications for the understanding of high Deborah number extensional thinning reported in other stagnation point flow situations.     

Measurement of pressure loss in the flow of polymer solutions through wavy channels

In order to examine the flow behavior of polymer solutions through porous media, the measurement of pressure loss and the experiment for flow visualization were carried out with wavy channels as one of the model channels of porous media. The test fluids used are aqueous solutions of polyacrylamide (PAA) with two different concentrations. The occurrence of the excess pressure loss, which was not due to the effect of the centrifugal force, was found for the PAA solutions. The relations between the friction factor ratio and the Deborah number were similar to that obtained for the flow through porous media. Furthermore, the results of the flow visualization suggest that the elongational property of the PAA solutions is connected with the occurrence of the excess pressure loss.