Rheological characterization of commercial highly viscous alginate solutions in shear and extensional flows

The rheological properties of sodium alginate in salt-free solutions were studied by steady shear, dynamic oscillatory and extensional measurements. This biopolymer consists of mannuronic and guluronic acid residues that give a polyelectrolyte character. We applied the scaling theories and checked their accordance with polyelectrolyte behaviour for low concentrations with a shift to neutral polymer behaviour at larger concentrations. This nature was supported by the effect of the concentration on the specific viscosity, the relaxation times from steady shear and the longest relaxation times from small amplitude oscillatory shear (SAOS) measurements. To analyze the extensional behaviour of the samples, we conducted a study of dimensionless numbers and time scales where filament thinning driven by viscous, capillary or elastic forces is at play. We conclude that an exponential filament thinning followed by breakup results in the best regimes that describe the experimental data. Besides, the data pointed out that alginate in salt-free concentrated solutions shows strain thinning of the extensional viscosity and chain rigidity, behaviours that cannot be inferred from the shear rheometry.     

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.     

The effect of pre-shear on the extensional rheology of wormlike micelle solutions

The effect of initial microstructural deformation, alignment, and morphology on the response of wormlike micelle solutions in transient uniaxial extensional flows is investigated using a pre-shear device attached to a filament stretching rheometer. In filament stretching experiments, increasing the strength and the duration of the pre-shear just before stretch is found to delay the onset of strain hardening. In these experiments, the wormlike micelle solution filaments fail through a rupture near the axial midplane. The value of the elastic tensile stress at rupture is found to decrease with increasing pre-shear rate and duration. The most dramatic effects are observed at shear rates for which shear banding has been independently observed. The reduction in the strain hardening suggests that pre-shear before filament stretching might break down the wormlike micelles reducing their size before stretch. Strain hardening is also observed in capillary breakup rheometry experiments; however, the pre-sheared wormlike micelle solutions strain harden faster, achieve larger steady-state extensional viscosities and an increase in the extensional relaxation time with increasing shear rate and duration. The difference between the response of the wormlike micelles in filament stretching and capillary breakup experiments demonstrates the sensitivity of these self-assembling micelle networks to pre-conditioning.     

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.     

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.     

Experimental investigations into an isothermal spinning threadline: Extensional rheology of a Separan AP 30 solution in glycerol and water

Summary Experimental observations on a steady isothermally extending filament of a water/glycerol solution of Separan AP 30 are presented. Photographic records were analysed to give filament diameter (and hence filament speed) as a function of distance below the extrusion die (a glass capillary). Measurements of inline tension were also made. When effects of weight, surface tension and air drag were accounted for, the extensional stress at every point along the filament could be calculated. Results for stress versus extension rate are presented for various flow situations.Independent rheogoniometric measurements of simple shear viscosity, first and second normal stress differences, and of a crude relaxation time were also made at comparable rates of deformation.Comparison shows that apparent extensional viscosities are several orders of magnitude larger than corresponding simple shear viscosities. After discussion, no conclusion can be drawn about what constitutive equation is most suitable to describe the results.An analysis to predict air drag is given.With 18 figures     

Extracting extensional properties through excess pressure drop estimation in axisymmetric contraction and expansion flows for constant shear viscosity,extension strain-hardening fluids

In this study, hyperbolic contraction–expansion flow (HCF) devices have been investigated with the specific aim of devising new experimental measuring systems for extensional rheological properties. To this end, a hyperbolic contraction–expansion configuration has been designed to minimize the influence of shear in the flow. Experiments have been conducted using well-characterized model fluids, alongside simulations using a viscoelastic White–Metzner/FENE-CR model and finite element/finite volume analysis. Here, the application of appropriate rheological models to reproduce quantitative pressure drop predictions for constant shear viscosity fluids has been investigated, in order to extract the relevant extensional properties for the various test fluids in question. Accordingly, experimental evaluation of the hyperbolic contraction–expansion configuration has shown rising corrected pressure drops with increasing elastic behaviour (De=0～16), evidence which has been corroborated through numerical prediction. Moreover, theoretical to predicted solution correspondence has been established between extensional viscosity and first normal stress difference. This leads to a practical means to measure extensional viscosity for elastic fluids, obtained through the derived pressure drop data in these HCF devices.     

A new look at extensional rheology of low-density polyethylene

The nonlinear rheology of three selected commercial low-density polyethylenes (LDPE) is measured in uniaxial extensional flow. The measurements are performed using three different devices including an extensional viscosity fixture (EVF), a homemade filament stretching rheometer (DTU-FSR) and a commercial filament stretching rheometer (VADER-1000). We show that the measurements from the EVF are limited by a maximum Hencky strain of 4, while the two filament stretching rheometers are able to probe the nonlinear behavior at larger Hencky strain values where the steady state is reached. With the capability of the filament stretching rheometers, we show that LDPEs with quite different linear viscoelastic properties can have very similar steady extensional viscosity. This points to the potential for independently controlling shear and extensional rheology in certain rate ranges.     

Dynamics of viscoelastic liquid filaments: Low capillary number flows

Many applications of viscoelastic free surface flows requiring formation of drops from small nozzles, e.g., ink-jet printing, micro-arraying, and atomization, involve predominantly extensional deformations of liquid filaments. The capillary number, which represents the ratio of viscous to surface tension forces, is small in such processes when drops of water-like liquids are formed. The dynamics of extensional deformations of viscoelastic liquids that are weakly strain hardening, i.e., liquids for which the growth in the extensional viscosity is small and bounded, are here modeled by the Giesekus, FENE-P, and FENE-CR constitutive relations and studied at low capillary numbers using full 2D numerical computations. A new computational algorithm using the general conformation tensor based constitutive equation [M. Pasquali, L.E. Scriven, Theoretical modeling of microstructured liquids: a simple thermodynamic approach, J. Non-Newtonian Fluid Mech. 120 (2004) 101–135] to compute the time dependent viscoelastic free surface flows is presented. DEVSS-TG/SUPG mixed finite element method [M. Pasquali, L.E. Scriven, Free surface flows of polymer solutions with models based on conformation tensor, J. Non-Newtonian Fluid Mech. 108 (2002) 363–409] is used for the spatial discretization and a fully implicit second-order predictor–corrector scheme is used for the time integration. Inertia, capillarity, and viscoelasticity are incorporated in the computations and the free surface shapes are computed along with all the other field variables in a fully coupled way. Among the three models, Giesekus filaments show the most drastic thinning in the low capillary number regime. The dependence of the transient Trouton ratio on the capillary number in the Giesekus model is demonstrated. The elastic unloading near the end plates is investigated using both kinematic [M. Yao, G.H. McKinley, B. Debbaut, Extensional deformation, stress relaxation and necking failure of viscoelastic filaments, J. Non-Newtonian Fluid Mech. 79 (1998) 469–501] and energy analyses. The magnitude of elastic unloading, which increases with growing elasticity, is shown to be the largest for Giesekus filaments, thereby suggesting that necking and elastic unloading are related.     

Elastic–plastic near field solution of an eccentric crack under shear in a finite width plate

The near crack line analysis method is used to investigate an eccentric crack loaded by shear forces in a finite width plate, and the analytical solution is obtained in this paper. The solution includes: the unit normal vector of the elastic–plastic boundary near the crack line, the elastic–plastic stress fields near crack line, variations of the length of the plastic zone along the crack line with an external loads, and the bearing capacity of a finite plate with a centric crack loaded by shear stress in the far field. The results obtained in this paper are sufficiently precise near the crack line because the assumptions of small scale yielding theory have not been made and no other assumptions have been taken. Subsequently, the present results are compared with the traditional line elastic fracture mechanical solutions and elastoplastic near field solutions under small scale yielding condition. On the basis of the minimum strain energy density (SED) theory, the minimum values of SED in the vicinity of the crack tip are determined, the initial growth orientation of crack are determined. It is found that the normalized load under large scale yielding condition is higher than those under small scale yielding condition when the length of the plastic zone is the same.     

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 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.     

The triple jet: influence of shear history on the stretching of polymer solutions

A kerosene-based aircraft safety fuel and aqueous solutions of poly (ethylene oxide) and polyacrylamide are examined using the “triple jet” system. This device allows the solution to be stretched as it flows from a capillary tube and the axial stress, strain and strain rate in the liquid are measured.The shear history of the solution is altered by placing cylindrical inserts in the capillary tube. This is shown to have a large effect on the extensional behaviour of aircraft safety fuel, a moderate effect on the extensional behaviour of poly (ethylene oxide) solution and little effect on the behaviour of polyacrylamide solution. The extensional viscosity of the aircraft fuel is raised by an order of magnitude when a long period of high shear is used; the effects last for periods of up to one second, though traditional methods suggest a relaxation time of the order of 10^?3 seconds. A liquid of shear viscosity 4 centipoise may have an extensional viscosity of over 100 poise.Plots of the extensional modulus of the jet as a function of distance along the jet emphasize the importance of shear history for the first two types of solution and suggest that the latter stages of the stretching process are elastic in character. Typical extensional moduli for the solutions tested are in the range 1.3–5.0 × 10⁴ dyn.cm^?2.The relevance of the interplay between shearing and stretching flow to the phenomena of lubrication and turbulence suppression is mentioned.     

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.     

Strain localization analysis using a large deformation anisotropic elastic–plastic model coupled with damage

Sheet metal forming processes generally involve large deformations together with complex loading sequences. In order to improve numerical simulation predictions of sheet part forming, physically-based constitutive models are often required. The main objective of this paper is to analyze the strain localization phenomenon during the plastic deformation of sheet metals in the context of such advanced constitutive models. Most often, an accurate prediction of localization requires damage to be considered in the finite element simulation. For this purpose, an advanced, anisotropic elastic–plastic model, formulated within the large strain framework and taking strain-path changes into account, has been coupled with an isotropic damage model. This coupling is carried out within the framework of continuum damage mechanics. In order to detect the strain localization during sheet metal forming, Rice’s localization criterion has been considered, thus predicting the limit strains at the occurrence of shear bands as well as their orientation. The coupled elastic–plastic-damage model has been implemented in Abaqus/implicit. The application of the model to the prediction of Forming Limit Diagrams (FLDs) provided results that are consistent with the literature and emphasized the impact of the hardening model on the strain-path dependency of the FLD. The fully three-dimensional formulation adopted in the numerical development allowed for some new results – e.g. the out-of-plane orientation of the normal to the localization band, as well as more realistic values for its in-plane orientation.     

Characterization of polyacrylamide gels as an elastic model for food gels

Serving as an elastic model system for food gels, characteristics of polyacrylamide (PAAm) gels were investigated using small amplitude and large deformation rheological tests. The PAAm gels displayed elastic or viscoelastic behavior depending on network crosslink density. For elastic PAAm gels, the rheological properties obeyed the theory of rubber elasticity; whereas for viscoelastic PAAm gels, shear modulus depended on temperature. The elastic PAAm gel fracture parameters did not change with deformation rate (0.2–5.5 s^–1), indicating insignificant viscous flow during deformation. Fracture stress was correlated with gel monomer concentration, whereas the fracture strain remained constant regardless of the monomer concentration. In addition, the stress was linearly proportioned with strain up to fracture, indicating that PAAm gels did not experience finite network chain extensibility during large deformation. Consequently, the fracture of PAAm gels did not result from the extensional limitation of network chains, nor did gel fracture result from the nonlinear force–distance relationship between polymer connections. Purportedly, the fracture of PAAm gels was caused by external force overcoming the gel cohesive forces, and low strength of PAAm gels compared to rubbers caused fracture prior to experiencing nonlinear stress-strain deformation.Paper No. FSR04-20 of the Journal Series of the North Carolina Agricultural Research Service, Raleigh, NC 27695-7643. The use of trade names does not imply endorsement by the North Carolina Agricultural Research Service of products named, nor criticisms of similar ones not mentioned.     

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.     

Torsional flow stability of highly dilute polymer solutions

We examine the torsional flow stability, to axisymmetric disturbances, of a variety of multimode and non-linear constitutive models in a bounded parallel plate geometry. The analysis is facilitated by the construction of a regular perturbation scheme in the ratio of polymer to total viscosity. As a model for Boger fluids this corresponds to the assumption that the Boger fluid is highly dilute. The consequent simplification provided by the perturbation scheme allows us to examine the effects of a discrete spectrum of relaxation times, shear thinning, second normal stress difference, and finite extensional viscosity on the torsional instability.     

Effect of a spectrum of relaxation times on the capillary thinning of a filament of elastic liquid

The capillary thinning of a filament of viscoelastic liquid, which is the basis of a microrheometer, is analyzed in terms of a multi-mode FENE fluid. After a short time of viscous adjustment, the stress becomes dominated by the elastic contribution and the strain-rate takes on a value equal to two-thirds the rate at which the stress would relax at fixed strain. This strain-rate decreases as the dominant mode changes. At late times, modes reach their finite extension limit. The fluid then behaves like a suspension of rigid rods with a large extensional viscosity, and the liquid filament breaks. Predictions are compared with the experiments of Liang and Mackley (1994).     

Generation of inkjet droplet of non-Newtonian fluid

In this study, the generation of inkjet droplets of xanthan gum solutions in water–glycerin mixtures was investigated experimentally to understand the jetting and drop generation mechanisms of rheologically complex fluids using a drop-on-demand inkjet system based on a piezoelectric nozzle head. The ejected volume and velocity of droplet were measured while varying the wave form of bipolar shape to the piezoelectric inkjet head, and the effects of the rheological properties were examined. The shear properties of xanthan gum solutions were characterized for wide ranges of shear rate and frequency by using the diffusive wave spectroscopy microrheological method as well as the conventional rotational rheometry. The extensional properties were measured with the capillary breakup method. The result shows that drop generation process consists of two independent processes of ejection and detachment. The ejection process is found to be controlled primarily by high or infinite shear viscosity. Elasticity can affect the flow through the converging section of inkjet nozzle even though the effect may not be strong. The detachment process is controlled by extensional viscosity. Due to the strain hardening of polymers, the extensional viscosity becomes orders of magnitude larger than the Trouton viscosities based on the zero and infinite shear viscosities. The large extensional stress retards the extension of ligament, and hence the stress lowers the flight speed of the ligament head. The viscoelastic properties at the high-frequency regime do not appear to be directly related to the drop generation process even though it can affect the extensional properties.