Extensional rheology of a shear-thickening cornstarch and water suspension

A filament-stretching rheometer is used to measure the extensional viscosity of a shear-thickening suspension of cornstarch in water. The experiments are performed at a concentration of 55 wt.%. The shear rheology of these suspensions demonstrates a strong shear-thickening behavior. The extensional rheology of the suspensions demonstrates a Newtonian response at low extension rates. At moderate strain rates, the fluid strain hardens. The speed of the strain hardening and the extensional viscosity achieved increase quickly with increasing extension rate. Above a critical extension rate, the extensional viscosity goes through a maximum and the fluid filaments fail through a brittle fracture at a constant tensile stress. The glassy response of the suspension is likely the result of jamming of particles or clusters of particles at these high extension rates. This same mechanism is responsible for the shear thickening of these suspensions. In capillary breakup extensional rheometry, measurement of these suspensions demonstrates a divergence in the extensional viscosity as the fluid stops draining after a modest strain is accumulated.     

Resistance to uniaxial extensional flow of fibre suspensions

The tensile stress due to resistance to uniaxial extensional flow of fibre suspensions in Newtonian and non-Newtonian fluids has been measured using the filament stretching technique. It has been found that addition of fibres to a Newtonian fluid increases the extensional viscosity. The steady state results agree with Bachelors theory and the stress growth behaviour is qualitatively predicted by the theory of Dinh and Armstrong. Experimental results from this work have also shown that the behaviour of a fibre suspension in viscoelastic fluid is qualitatively described by Fans equation. The added fibres increase the extensional stress growth coefficient of the viscoelastic fluid at low strain but have marginal effect on the fluid after the onset of strain-hardening.     

Extensional rheology of concentrated poly(ethylene oxide) solutions

The shear and extensional rheology of three concentrated poly(ethylene oxide) solutions is examined. Shear theology including steady shear viscosity, normal stress difference and linear viscoelastic material functions all collapse onto master curves independent of concentration and temperature. Extensional flow experiments are performed in fiber spinning and opposed nozzles geometries. The concentration dependence of extensional behavior measured using both techniques is presented. The zero-shear viscosity and apparent extensional viscosities measured with both extensional rheometers exhibit a power law dependence with polymer concentration. Strain hardening in the fiber spinning device is found to be of similar magnitude for all test fluids, irrespective of strain rate. The opposed nozzle device measures an apparent extensional viscosity which is one order of magnitude smaller than the value determined with the fiber spinline device. This could be attributed to errors caused by shear, dynamic pressure, and the relatively small strains developed in the opposed nozzle device. This instrument cannot measure local kinematics or stresses, but averages these values over the non-homogenous flow field. These results show that it is not possible to measure the extensional viscosity of non-Newtonian and shear thinning fluids with this device. Fiber spin-line experiments are coupled with a momentum balance and constitutive model to predict stress growth and diameter profiles. A one-mode Giesekus model accurately captures the plateau values of steady and dynamic shear properties, but fails to capture the gradual shear thinning of viscosity. Giesekus model parameters determined from shear rheology are not capable of quantitatively predicting fiber spinline kinematics. However, model parameters fit to a single spinline experiment accurately predict stress growth behavior for different applied spinline tensions.     

Filament stretching of carbon nanotube suspensions

This paper reports the application of a recently developed filament stretching protocol for the study of the extensional rheology of both treated and untreated carbon nanotubes (CNT) suspended within an epoxy resin. It was experimentally observed that filaments formed by treated and untreated CNT suspensions behaved differently after initial stretching. The filament thinning process of the base epoxy was consistent with a simple Newtonian fluid, whilst the filament of treated CNT suspensions also thinned in a Newtonian way but with an enhanced extensional viscosity. Filaments formed with untreated CNT suspensions behaved in a non-uniform way with local fluctuation in filament diameter, and it was not possible to obtain reliable extensional viscosity data. Irregularity of the untreated CNT filaments was consistent with coupled optical images, where spatial variation in CNT aggregate concentration was observed. In the case of treated CNT suspensions, the enhanced extensional viscosity was modelled in terms of the alignment of CNTs in the stretching direction, and the degree of alignment was subsequently estimated using a simple orientation model.     

Fiber spinning of very dilute solutions of polyacrylamide in water

The extensional viscometer developed earlier by the authors was refined and used to extend very dilute (50 ppm) solutions of polyacrylamide in distilled water. A slender liquid filament was stretched by the use of a suction device, and this resulted in the spinning of the fiber. By varying the volumetric flow rate and the filament length, stretch rates in the 100–1000 s⁻¹ range were easily obtained. The corresponding tensile stresses were very large, and these gave apparent extensional viscosities of the order of 200 P (20 Pa s). In contrast to this, the material functions in shear were difficult to measure, except for the shear viscosity which showed pronounced shear thinning. It was found that all the measurements, in shear as well as extension, could be explained based on the four constant Johnson-Segalman constitutive equation.     

A simple extensional viscometer

An extensional viscometer is described in which the liquid filament leaving a capillary is subjected to a stretching deformation. In order to keep the flow rate through the capillary unaltered upon inception of stretching, the pressure head at the capillary entrance has to be reduced by an amount equal to the extensional viscoelastic stress at the capillary exit. This affords a simple means of measuring small fluid forces such as those that occur in the stretching of dilute polymer solutions. Since stretch rates can be obtained from a knowledge of the mass flow rate and the filament diameter profile, extensional viscosities can be computed. The efficacy of the technique is demonstrated by obtaining the anticipated results for Newtonian liquids.     

Constitutive equation of co-rotational derivative type for anisotropic-viscoelastic fluid

A constitutive equation theory of Oldroyd fluid B type, i.e. the co-rotational derivative type, is developed for the anisotropic-viscoelastic fluid of liquid crystalline (LC) polymer. Analyzing the influence of the orientational motion on the material behavior and neglecting the influence, the constitutive equation is applied to a simple case for the hydrodynamic motion when the orientational contribution is neglected in it and the anisotropic relaxation, retardation times and anisotropic viscosities are introduced to describe the macroscopic behavior of the anisotropic LC polymer fluid. Using the equation for the shear flow of LC polymer fluid, the analytical expressions of the apparent viscosity and the normal stress differences are given which are in a good agreement with the experimental results of Baek et al. For the fiber spinning flow of the fluid, the analytical expression of the extensional viscosity is given. The project supported by the National Natural Science Foundation of China (19832050 and 10372100)     

Determination of the elongational viscosity of polymer melts by melt spinning experiments. A comparison with different experimental techniques

In this work, melt spinning experiments were tentatively used for the determination of the elongational viscosity of polymer melts at different levels of tensile strain and strain rate. The materials examined were two high-density polyethylene grades for blow moulding with similar number-average molecular mass but different polydispersity index. The data from melt spinning tests were compared with transient extensional viscosity data obtained by uniform isothermal tensile tests, performed by means of an extensional rheometer, as well as with those produced by converging flow tests (Cogswell model). The results showed that for high strain and strain rate levels, the melt spinning experiments provide elongational viscosity data quite close to the transient extensional viscosity values obtained from the tensile tests.     

The effect of step-stretch parameters on capillary breakup extensional rheology (CaBER) measurements

Extensional rheometry has only recently been developed into a commercially available tool with the introduction of the capillary breakup extensional rheometer (CaBER). CaBER is currently being used to measure the transient extensional viscosity evolution of mid to low-viscosity viscoelastic fluids. The elegance of capillary breakup extensional experiments lies in the simplicity of the procedure. An initial step-stretch is applied to generate a fluid filament. What follows is a self-driven uniaxial extensional flow in which surface tension is balanced by the extensional stresses resulting from the capillary thinning of the liquid bridge. In this paper, we describe the results from a series of experiments in which the step-stretch parameters of final length, and the extension rate of the stretch were varied and their effects on the measured extensional viscosity and extensional relaxation time were recorded. To focus on the parameter effects, well-characterized surfactant wormlike micelle solutions, polymer solutions, and immiscible polymer blends were used to include a range of characteristic relaxation times and morphologies. Our experimental results demonstrate a strong dependence of extensional rheology on step-stretch conditions for both wormlike micelle solutions and immiscible polymer blends. Both the extensional viscosity and extensional relaxation time of the wormlike micelle solutions were found to decrease with increasing extension rate and strain of the step-stretch. For the case of the immiscible polymer blends, fast step-stretches were found to result in droplet deformation and an overshoot in the extensional viscosity which increased with increasing strain rates. Conversely, the polymer solutions tested were found to be insensitive to step-stretch parameters. In addition, numerical simulations were performed using the appropriate constitutive models to assist in both the interpretation of the CaBER results and the optimization of the experimental protocol. From our results, it is clear that any rheological results obtained using the CaBER technique must be properly considered in the context of the stretch parameters and the effects that preconditioning has on viscoelastic fluids.     

Fibre spinning of a weakly elastic liquid

This paper presents a study of a silicone oil (poly(dimethyl siloxane)) in extensional deformation using an instrument developed recently by the authors. Data from steady shear and low amplitude sinusoidal deformation of this liquid clearly establish that it is weakly elastic. The viscometric data, for shear rates less than 100 s ⁻¹, are best represented by either the Maxwell model or the Jeffrey's model, the latter being marginally superior. The extensional data show that at low deformation rates, this fluid exhibits a Newtonian behavior with an apparent extensional viscosity equal to three times the shear viscosity. Under these conditions the velocity profiles along the spinline are also well represented by the Newtonian model. However, at higher deformation rates better predictions of the velocity profiles are obtained from the Jeffrey's and Maxwell models. At deformation rates above 100 s ⁻¹ none of these simple models is adequate. Under the conditions used in these experiments, the fractional increase in tensile stress along the fiber is shown both theoretically and experimentally to be a unique function of the total strain. Furthermore, the apparent extensional viscosity at any point on the spinline can be calculated from steady state expressions if allowance is made for the variation of stretch rates by defining a time averaged stretch rate.The results obtained here show that elasticity must be considered if these model liquids are used to conduct rheological experiments at high deformation rates. Additionally, it is found that elastic effects in extension can be predicted using simple constitutive equations provided viscometric data can be represented properly in the deformation rate range of interest. Finally, the present research further substantiates the utility of the extensional viscometer developed by the authors.     

An innovative extensional viscometer for low-viscosity and low-elasticity liquids

A spinline-type extensional viscometer is described in which an innovative method of tensile stress measurement is employed. A limited amount of liquid flows through a vertical capillary at a constant flow rate under the influence of a constant pressure head. The drainage time decreases when the liquid stream leaving the capillary is stretched by the application of vacuum. These drainage times are measured in a manner similar to that used for intrinsic viscosity measurements. The measured difference in drainage times, with and without stretching, is trivially related to the extensional stress at the capillary exit, and this provides a very simple method of accurately determining fluid stretching forces having a magnitude as low as 10^-4 N; stresses at other axial locations in the stretched liquid jet are obtained by means of a force balance in the usual manner. The validity of the proposed technique is demonstrated by obtaining the expected results for a Newtonian oil having a shear viscosity of 56.2 mPa-s. Also presented are preliminary data on polyethylene oxide-in-water solutions having an even lower shear viscosity.     

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.     

Birefringence and stress growth in uniaxial extension of polymer solutions

Simultaneous measurements of extensional stresses and birefringence are rare, especially for polymer solutions. This paper reports such measurements using the filament stretch rheometer and a phase modulated birefringence system. Both the extensional viscosity and the birefringence increase monotonically with strain and reach a plateau. Estimates of this saturation value for birefringence, using Peterlin’s formula for birefringence of a fully extended polymer chain are in agreement with the experimental results. However, estimates of the saturation value of the extensional viscosity using Batchelor’s formula for suspensions of elongated fibres are much higher than observed. Reasons for the inability of the flow field to fully unravel the polymer chain are examined using published Brownian dynamics simulations. It is tentatively concluded that the polymer chain forms a folded structure. Such folded chains can exhibit saturation in birefringence even though the stress is less than that expected for a fully extended molecule.Simultaneous measurements of stress and birefringence during relaxation indicate that the birefringence decays much more slowly than the stress. The stress-birefringence data show a pronounced hysteresis as predicted by bead-rod models. The failure of the stress optic coefficient in strong flows is noted.Experiments were also performed wherein the strain was increased linearly with time, then held constant for a short period before being increased again. The response of the stress and birefringence in such experiments is dramatically different and can be traced to the different configurations obtained during stretching and relaxation. The results cast doubt on the appropriateness of pre-averaging the non-linear terms in constitutive equations.     

A model long-discontinuous-fiber filled thermoplastic melt in extensional flow

The shear cell model works for dilute fiber filled systems in extensional flow. This research investigates the suitability of the idea for highly aligned fibers in a concentrated suspension. A model fiber-filled polymer system made from nylon fibers in low-density polyethylene provided a means of controlling the material parameters. Two systems, with fiber aspect ratios of 20 and 100, containing 50% 0.5 mm fibers by volume are investigated. The thickness of the polymer layer, i.e. with fibers this size, allows bulk viscosity data to be compared with the data from the filled fluid. A weaving process created the discontinuous fiber/polyethylene preforms with high alignment of the fibers and with control of the fiber to fiber overlap. Testing the polyethylene in simple shear and extending the nylon/polyethylene provided the data needed to check the micro mechanics. A cone and plate rheometer and a capillary instrument produced the viscosity/strain rate data that characterized the specific polyethylene used in the composite. A furnace inset placed in an Instron hydraulic test machine allowed extension of the filled system at strain rates from 0.002 to 0.4 s⁻¹. The shear experiments show that the low-density polyethylene is a simple shear-thinning melt that provides a good model fluid. The extension of the filled systems shows an increase of the apparent extensional viscosity from that of neat polyethylene. Apparent viscosity rises two to three orders of magnitude for the systems investigated. The micromechanics allowed the conversion of the extensional data from the two filled systems to the shear viscosity of the polymer surrounding the fibers. The calculated polyethylene viscosity compares well with the data from the standard rheometers. The shear cell approach may be applied to highly aligned, high fiber-volume-fraction suspensions when the viscosity of the polymer is known at the scale of the film surrounding each fiber.     

An unsymmetric constitutive equation for anisotropic viscoelastic fluid

A continuum constitutive theory of corotational derivative type is developed for the anisotropic viscoelastic fluid–liquid crystalline (LC) polymers. A concept of anisotropic viscoelastic simple fluid is introduced. The stress tensor instead of the velocity gradient tensor D in the classic Leslie–Ericksen theory is described by the first Rivlin–Ericksen tensor A and a spin tensor W measured with respect to a co-rotational coordinate system. A model LCP-H on this theory is proposed and the characteristic unsymmetric behaviour of the shear stress is predicted for LC polymer liquids. Two shear stresses thereby in shear flow of LC polymer liquids lead to internal vortex flow and rotational flow. The conclusion could be of theoretical meaning for the modern liquid crystalline display technology. By using the equation, extrusion–extensional flows of the fluid are studied for fiber spinning of LC polymer melts, the elongational viscosity vs. extension rate with variation of shear rate is given in figures. A considerable increase of elongational viscosity and bifurcation behaviour are observed when the orientational motion of the director vector is considered. The contraction of extrudate of LC polymer melts is caused by the high elongational viscosity. For anisotropic viscoelastic fluids, an important advance has been made in the investigation on the constitutive equation on the basis of which a series of new anisotropic non-Newtonian fluid problems can be addressed. The project supported by the National Natural Science Foundation of China (10372100, 19832050) (Key project). The English text was polished by Yunming Chen.     

Normal stress distribution in highly concentrated suspensions undergoing squeeze flow

Using pressure-sensitive films, the normal stress distribution is measured in suspensions of glass spheres in a Newtonian liquid undergoing constant-force squeeze flow. At volume fractions of solids up to 0.55, the normal stress distribution is independent of volume fraction and almost identical to the parabolic pressure distribution predicted for Newtonian fluids. However, at higher volume fractions, the normal stresses become an order of magnitude larger near the center and very low beyond that region. At these high volume fractions, the normal stresses decrease in the outer regions and increase in the inner regions as the squeezing proceeds. The normal stress distribution that results when the glass spheres without any fluid are subjected to squeeze flow is very similar to that for suspensions with volume fractions above 0.55, suggesting that the cause for the drastic changes in the normal stress distribution is the jamming of the particles in the suspension.     

Drop formation dynamics of constant low-viscosity,elastic fluids

The dynamics of drop formation under gravity has been investigated as a function of elasticity using a set of low-viscosity, ideal elastic fluids and an equivalent Newtonian glycerol–water solution. All solutions had the same shear viscosity, equilibrium surface tension, and density, but differed greatly in elasticity. The minimum drop radius in the early stages of drop formation (necking) was found to scale as expected from potential flow theory, independent of the elasticity of the solutions. Thus, during this stage of drop formation when viscous force is still weak, the dynamics are controlled by a balance between inertial and capillary forces, and there is no contribution of elastic stresses of the polymer. However, upon formation of the pinch regions, there is a large variation in the drop development to break-off observed between the various solutions. The elastic solutions formed secondary fluid threads either side of a secondary drop from the necked region of fluid between the upper and lower pinches, which were sustained for increasing amounts of time. The break-off lengths and times increase with increasing elasticity of the solutions. Evolution of the filament length is, however, identical in shape and form for all of the polymer solutions tested, regardless of differing elasticity. This de-coupling between filament growth rate and break-up time (or equivalently, final filament length at break-up) is rationalised. A modified force balance to that of Jones and Rees [48] is capable of correctly predicting the filament growth of these low-viscosity, elastic fluids in the absence of any elastic contributions due to polymer extension within the elongating filament. The elongation of the necked region of fluid (which becomes the filament) is dominated by the inertia of the drop, and is independent of the elasticity of the solution. However, elasticity does strongly influence the resistance of the pinch regions to break-off, with rapid necking resulting in extremely high rates of surface contraction on approach to the pinch point, initiating extension of the polymer chains within the pinch regions. This de-coupling phenomenon is peculiar to low-viscosity, elastic fluids as extension does not occur prior to the formation of the pinch points (i.e. just prior to break-up), as opposed to the high viscosity counterparts in which extension of polymers in solution may occur even during necking. Once steady-state extension of the polymers is achieved within the pinch at high extension rates, the thread undergoes elasto-capillary break-up as the capillarity again overcomes the viscoelastic forces. The final length at detachment and time-to-break-off (relative to the equivalent Newtonian fluid) is shown to be linearly proportional to the longest relaxation time of the fluid.     

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.     

Rheological behaviour of nano-composites based on polyamide 6 under shear and elongational flow at high strain rates

In this work, the rheological behaviour of high molecular mass polyamide 6 (PA6)/organo-montmorillonite nano-composites, obtained via melt blending, was investigated under shear and extensional flow. Capillary rheometry was used for the measurement of high shear rate steady state shear viscosity and die entrance pressure losses; further, by the application of a converging flow method (Cogswell model) to these experimental results, elongational viscosity data were indirectly calculated. The extensional behaviour was directly investigated by means of melt spinning experiments, and data of apparent elongational viscosity were determined. The results evidenced that the presence of the organo-clay in filled PA6 melts modifies the rheological behaviour of the material, with respect to the unfilled polymer, in dependence on the type of flow experienced by the fluid. In shear flow, the nano-composites showed a slightly lower viscosity than neat PA6, whereas in elongation, they appeared much more viscous, in dependence on the organo-clay content.     

Flow of a quasi-Newtonian fluid through a planar contraction

Flows through abrupt contractions are dominated by the rapid extension experienced in passing through the contraction. Thus, it is useful to employ a fluid model which considers the extensional viscosity explicitly in its constitutive equation. In this paper, the quasi-Newtonian fluid model, which admits shear thinning and extension thickening of the viscosity depending on the local type of flow as proposed by Schunk and Scriven [P. Schunk, L. Scriven, J, Rheol 34 (1990) 1085], is applied to the numerical simulation of the flow of a dilute polyacrylamide solution through a planar 4 : 1 contraction. In this theory the extra stress tensor does not only depend on the rate of strain tensor but also on the relative rate of rotation of the fluid. The material function – the viscosity function – is allowed to depend on the invariants of these two kinematic tensors yielding a local distinction between extensional, shear or rotation dominated flow. The governing equations are discretized using a finite volume method. Different model parameters are varied and the simulation results are compared with the generalized Newtonian fluid and experimental data.