Elongational and shear rheology of carbon nanotube suspensions

Rheological behavior of concentrated suspensions of chemical vapor deposition carbon nanotubes in uniaxial elongation and simple shear is studied experimentally and theoretically. Nanotubes are suspended in viscous host liquids—castor oil or its blends with n-decane. The elongational measurements are performed by analyzing self-thinning (due to surface tension effect) liquid threads of nanotube suspensions. A quasi-one-dimensional model is used to describe the self-thinning process, whereas corrections accounting for thread nonuniformity and necking are introduced a posteriori. The effects of nanotube concentration and aspect ratio, viscosity of the suspending liquid, and initial diameter of the self-thinning thread in uniaxial elongation are elucidated. The results for uniaxial elongation are compared with those for simple shear. The correspondence in the results of the shear and elongational measurements is addressed and interpreted. The results conform to the Herschel–Bulkley rheological constitutive equation (i.e., power law fluids with yield stress). However, the yield stress in elongation is about 40% higher than in simple shear flow, which suggests that the original Herschel–Bulkley model need modification with the yield stress being a function of the second invariant of the deviatoric stress tensor. The present effort is the first to study capillary self-thinning of Herschel–Bulkley liquids, which are exemplified here by suspensions of carbon nanotubes.     

Recoverable deformation and morphology after uniaxial elongation of a polystyrene/linear low density polyethylene blend

The transient recoverable deformation ratio after melt elongation at various elongational rates and maximum elongations was investigated for pure polystyrene and for a 85 wt.% polystyrene/15 wt.% linear low density polyethylene (PS/LLDPE 85:15) blend at a temperature of 170 ^oC. The ratio p of the zero shear rate viscosity of LLDPE to that of PS is p = 0.059 ≈ 1:17. Retraction of the elongated LLDPE droplets back to spheres and end-pinching is observed during recovery. A simple additive rule is applied in order to extract the contribution of the recovery of the elongated droplets from the total recovery of the blend. In that way, the recoverable portion of the PS/LLDPE blend induced by the interfacial tension is determined and compared with the results of a theory based on an effective medium approximation. The effective medium approximation reproduces well the time scale of the experimental data. In addition, the trends that the recoverable deformation increases with elongational rate and maximum elongation are captured by the theoretical approach.     

Linear viscoelasticity,simple and planar melt extension of linear polybutadienes with bimodal molar mass distributions

Shear oscillations, simple and planar elongations have been performed with anionically polymerized polybutadienes (PB) and their blends at room temperature. The PB components were of different molar mass averages and of narrow molar mass distributions; the blends had bimodal molar mass distributions and are represented by the weight ratio w of the high molecular component. The crossover G() = G() obtained from oscillatory measurements shows correlations with molecular parameters. For the zero shear viscosity the well-known relation ₀ M _w ^3.4 is found. The recoverable equilibrium shear compliance J _e ⁰ is nearly the same for the components; for the blends it strongly depends on w with a pronounced maximum at small w. In elongation outside the linear region strain hardening is found; its magnitude depends on M _w of the components, the composition w of the blend, the mode of elongation (simple or planar), and the elongational strain rate. The hardening revealed in the increase of the elongational viscosity above the linear viscoelastic limit increases as a function of w up to a maximum similar to J _e ⁰ such that, for both properties, the molecular processes may be the same. The elongational viscosity µ₂ (from the lateral stress in planar elongation) is above the linear viscoelastic limit for bimodal and below this limit for conventional broad molar mass distributions. In general, it can be stated that with a more narrow molar mass distribution of linear polymers the elongational behavior of the melts comes closer to the linear viscoelastic limit.Dedicated to Professor Arthur S. Lodge on the occasion of his 70th birthday and his retirement from the University of Wisconsin.Extended version of a paper presented at the Annual Conf. German Soc. of Rheology, Berlin, May 13–15, 1991.     

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.     

Large amplitude oscillatory elongation flow

A filament stretching rheometer (FSR) was used for measuring the elongation flow with a large amplitude oscillative elongation imposed upon the flow. The large amplitude oscillation imposed upon the elongational flow as a function of the time t was defined as where ε is the Hencky strain, is a constant elongational rate for the base elongational flow, Λ the strain amplitude (Λ ≥ 0), and Ω the strain frequency. A narrow molecular mass distribution linear polystyrene with a molecular weight of 145 kg/mol was subjected to the oscillative flow. The onset of the steady periodic regime is reached at the same Hencky strain as the onset of the steady elongational viscosity ( Λ = 0). The integral molecular stress function formulation within the ‘interchain pressure’ concept agrees qualitatively with the experiments.     

Least‐squares finite element processes in h,p, k mathematical and computational framework for a non‐linear conservation law

This paper considers numerical simulation of time‐dependent non‐linear partial differential equation resulting from a single non‐linear conservation law in h, p, k mathematical and computational framework in which k=(k₁, k₂) are the orders of the approximation spaces in space and time yielding global differentiability of orders (k₁?1) and (k₂?1) in space and time (hence k‐version of finite element method) using space–time marching process. Time‐dependent viscous Burgers equation is used as a specific model problem that has physical mechanism for viscous dissipation and its theoretical solutions are analytic. The inviscid form, on the other hand, assumes zero viscosity and as a consequence its solutions are non‐analytic as well as non‐unique (Russ. Math. Surv. 1962; 17 (3):145–146; Russ. Math. Surv. 1960; 15 (6):53–111). In references (Russ. Math. Surv. 1962; 17 (3):145–146; Russ. Math. Surv. 1960; 15 (6):53–111) authors demonstrated that the solutions of inviscid Burgers equations can only be approached within a limiting process in which viscosity approaches zero. Many approaches based on artificial viscosity have been published to accomplish this including more recent work on H(Div) least‐squares approach (Commun. Pure Appl. Math. 1965; 18 :697–715) in which artificial viscosity is a function of spatial discretization, which diminishes with progressively refined discretizations. The thrust of the present work is to point out that: (1) viscous form of the Burgers equation already has the essential mechanism of viscosity (which is physical), (2) with progressively increasing Reynolds (Re) number (thereby progressively reduced viscosity) the solutions approach that of the inviscid form, (3) it is possible to compute numerical solutions for any Re number (finite) within hpk framework and space–time least‐squares processes, (4) the space–time residual functional converges monotonically and that it is possible to achieve the desired accuracy, (5) space–time, time marching processes utilizing a single space–time strip are computationally efficient. It is shown that viscous form of the Burgers equation without linearizing provides a physical and viablemechanism for approaching the solutions of inviscid form with progressively increasing Re. Numerical studies are presented and the computed solutions are compared with published work. Copyright © 2008 John Wiley & Sons, Ltd.     

In-line measurement of rheological properties of polymer melts

Shear viscosity (), first normal stress difference (N ₁), and extensional viscosity ( _E ) of polymer melts measured under processing conditions are important in process modeling, quality control, and process control. A slit rheometer that could simultaneously measure , N ₁, and the planar extensional viscosity ( _p ) was designed and tested by attaching it in-line to a laboratory model single-screw extruder. A tube (circular cross-section) rheometer to measure and the uniaxial extensional viscosity ( _u ) simultaneously was also designed and tested. Two commercial grades of LDPE (low density polyethylene) with melt index values of 6 and 12 were used as test materials for the study. Exit and hole pressure methods were used to estimate N ₁, and the entrance pressure drop method using the analyses of Cogswell, Binding, and Gibson (the last analysis used with the axisymmetric case only) was used to estimate _E .The hole pressure method was considered better than the exit pressure method to estimate N ₁ (due to the greater susceptibility of the latter to experimental errors). From the hole pressure method N ₁ was obtained from 100 kPa to 500 kPa over a range of shear rates from 40 s^–1 to 700 s^–1. Among the analyses used to estimate the extensional viscosity, Cogswell's is recommended due to its simpler equations without loss of much information compared to the other analyses. The range of extension rates achieved was 1 to 30 s^–1. The combination of the hole pressure and entrance pressure drop methods in a slit rheometer is a feasible design for a process rheometer, allowing the simultaneous measurement of the shear viscosity, first normal stress difference and planar extensional viscosity under processing conditions. Similarly, combining the entrance pressure drop measurements with a tube rheometer is also feasible and convenient.     

Uniaxial deformation and relaxation of polymer blends: relationship between flow and morphology development

Uniaxial elongational flow followed by stress relaxation of a dilute mixture of polystyrene/polymethylmethacrylate) PS/PMMA with PS (5 wt%) as a dispersed phase was investigated. The behavior of the blend was found to be dominated by the PMMA matrix during elongation and by the interface during the relaxation at long time. Such a behavior was related to drop deformation and shape recovery during the relaxation process as was confirmed by morphological analyses on samples quenched within the rheometer just after elongation and at various times during the relaxation process. The morphology and the rheological material functions variation were compared to the Yu model (Yu W, Bousmina M, Grmela M, Palierne JF, Zhou C (2002) Quantitative relationship between rheology and morphology in emulsions. J Rheol 46(6):1381–1399).     

Stress boundary layers in the viscoelastic flow past a cylinder in a channel： limiting solutions

The flow past a cylinder in a channel with the aspect ratio of 2:1 for the upper convected Maxwell (UCM) fluid and the Oldroyd-B fluid with the viscosity ratio of 0.59 is studied by using the Galerkin/Least-square finite element method and a p-adaptive refinement algorithm. A posteriori error estimation indicates that the stress-gradient error dominates the total error. As the Deborah number, D_e, approaches 0.8 for the UCM fluid and 0.9 for the Oldroyd-B fluid, strong stress boundary layers near the rear stagnation point are forming, which are characterized by jumps of the stress-profiles on the cylinder wall and plane of symmetry, huge stress gradients and rapid decay of the gradients across narrow thicknesses. The origin of the huge stress-gradients can be traced to the purely elongational flow behind the rear stagnation point, where the position at which the elongation rate is of 1/2D_e approaches the rear stagnation point as the Deborah number approaches the critical values. These observations imply that the cylinder problem for the UCM and Oldroyd-B fluids may have physical limiting Deborah numbers of 0.8 and 0.9, respectively.The project supported by the National Natural Science Foundation of China (50335010 and 20274041) and the MOLDFLOW Comp. Australia.     

Use of a pressless multianvil high-pressure split-sphere apparatus to measure the silicate melt viscosity

The possibility of using a pressless multianvil high-pressure split-sphere apparatus to measure the viscosity of silicate melts was studied experimentally using as an example diopside melt (CaMgSi ₂ O ₆ ) at a pressure of 4.0 GPa and a temperature of 1800°C. The viscosity was calculated by the Stokes formula. The measurement error of the melt viscosity was estimated by the falling ball method. Prospects for using devices of this type to estimate the viscosity of silicate melts at high pressures and temperatures are shown.     

Rheotens-mastercurves and elongational viscosity of polymer melts

In a Rheotens experiment, the tensile force needed for elongation of an extruded filament is measured as a function of the draw ratio. For thermo-rheologically simple polymer melts, the existence of Rheotens-mastercurves was proved by Wagner, Schulze, and Göttfert (1995). Rheotens-mastercurves are invariant with respect to changes in melt temperature and changes in the average molar mass. By use of purely viscous models, we convert Rheotens-mastercurves of a branched and a linear polyethylene melt to elongational viscosity as a function of strain rate. The resulting elongational viscosity from constant force extension experiments is found to be in general agreement with what is expected as steady-state viscosity of polyethylene melts measured in either constant strain-rate or constant stress mode.Dedicated to Prof. Dr. J. Meissner on the occasion of his retirement from the chair of Polymer Physics at the Eidgenössische Technische Hochschule (ETH) Zürich, Switzerland     

Comparison of viscous and elastic properties of polyolefin melts in shear and elongation

Viscous and elastic properties of a linear polypropylene (PP) and a long-chain branched low-density polyethylene (LDPE) have been investigated by creep and creep–recovery experiments in shear and elongation. The data obtained verify the ratios between the linear values of the viscosities and the steady-state elastic compliances in shear and elongation predicted by the theory of linear viscoelasticity. In the nonlinear range, no simple correlation between the viscous behaviour in shear and elongation exists. The elongational viscosity of the PP decreases with increasing stress analogously to the shear thinning observed; the linear range extends to higher stresses in elongation than in shear, however. The LDPE shows thinning in shear and strain hardening in elongational flow. For the LDPE, a linear steady-state elastic tensile compliance corresponding to one third of the linear steady-state elastic compliance in shear was determined. For the PP, this theoretically predicted value is approximately reached. Analogous to the viscous behaviour, the linear range extends to higher stresses in elongation than in shear. For both materials, the steady-state elastic compliances in the nonlinear range decrease with increasing stress in shear as well as in elongation. However, the decrease in elongation is more pronounced.     

The role of elongational viscosity in the mechanism of drag reduction by polymer additives

The role of elongational viscosity in the mechanism of drag reduction by polymer additives is investigated qualitatively by means of direct numerical simulations of a turbulent pipe flow. For the polymer solution, a generalised Newtonian constitutive model is utilised in which the viscosity depends on the second and third invariant of the rate-of-strain tensor via an elongation parameter. This elongation parameter is capable of identifying elongational type of regions within the flow. The simulations show that complementary to stretching of the polymers, also compression must be incorporated to have drag reduction, contrary to many suggestions done in the literature on the mechanism which assume that stretching of the polymers is most important.     

Measurements and model predictions of transient elongational rheology of polymeric nanocomposites

Transient elongational rheology of two commercial-grade polypropylene (PP) and the organoclay thermoplastic nanocomposites is investigated. A specifically designed fixture consisting of two drums (SER Universal Testing Platform) mounted on a TA Instruments ARES rotational rheometer was used to measure the transient uniaxial extensional viscosity of both polypropylene and nanoclay/PP melts. The Hencky strain rate was varied from 0.001 to 2 s^− 1, and the temperature was fixed at 180°C. The measurements show that the steady-state elongational viscosity was reached at the measured Hencky strains for the polymer and for the nanocomposites. The addition of nanoclay particles to the polymer melt was found to increase the elongation viscosity principally at low strain rates. For example, at a deformation rate of 0.3 s^− 1, the steady-state elongation viscosity for polypropylene was 1.4 × 10⁴ Pa s which was raised to 2.8 × 10⁴ and 4.5 × 10⁴ Pa s after addition of 0.5 and 1.5 vol.% nanoclay, respectively. A mesoscopic rheological model originally developed to predict the motion of ellipsoid particles in viscoelastic media was modified based on the recent developments by Eslami and Grmela (Rheol Acta 47:399–415, 2008) to take into account the polymer chain reptation. We show that the orientation states of the particles and the rheological behavior of the layered particles/thermoplastic hybrids can be quantitatively explained by the proposed model.     

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.     

A constitutive equation for a dilute solution of Hookean dumbbells with approximate hydrodynamic interaction

In a recent series of papers, Öttinger's consistently averaged hydrodynamic interaction has been shown to yield shear-rate dependent viscosity and normal stress coefficients in steady shear flow for dilute solutions of elastic dumbbells and chains. Even more recently, Fan has numerically solved the diffusion equation for the Hookean dumbbell with complete hydrodynamic interaction and he has compared his results with those of the Öttinger model.In this paper, a new approximation¹ for the Oseen—Burgers tensor is proposed where the configuration-dependent terms are replaced by appropriate averages rather than averaging the Oseen—Burgers tensor as a whole as in the Öttinger model. The proposed model leads to a differential constitutive equation which at low shear rates is similar to the Giesekus constitutive equation for a Hookean dumbbell with anistropic drag and anisotropic Brownian motion. The steady shear viscosity and normal stress coefficients for the proposed model are shear-rate dependent and are in close agreement with Fan's numerical calculations. Elongational viscosity for both positive and negative elongation rates are calculated.     

Configuration-dependent friction coefficients and elastic dumbbell rheology

The kinetic theory of nonlinear elastic dumbbells, with bead friction coefficients that depend linearly on the interbead distance, is used to obtain the elongational viscosity and the dumbbell stretching as a function of elongation rate. The results are obtained by solving numerically the “diffusion equation” for the configurational distribution function. No S-shaped curves were found for the elongational viscosity or for the mean-square end-to-end distance. Previous investigators did report S-shaped curves and related “hysteresis” effects. However, their results were based on using mathematical approximations that now appear to be inappropriate.     

Three-dimensional viscoelastic numerical analysis of the encapsulation phenomena in coextrusion

In recent years coextrusion process had gained wide recognition as an approach to achieving high quality or quantity and low costs by multilayered system as well as an polymer alloy or blend. We performed a strict 3-D numerical simulation with viscoelastic model on the encapsulation phenomena which is one of the problems in the coextrusion process. The effects of material properties such as viscosity ratio, extensional viscosity and second normal stress difference and an effect of flow rate ratios on the encapsulation phenomena were examined. Numerical results showed that encapsulation phenomena were affected by not only viscous properties but also elastic properties or non-linear properties. It was found that the difference of second normal stress differences working on the interface between fluid I and fluid II (we defined it as DN ₂) had a correlation with the interface shape. Received: 31 August 1998 Accepted: 15 September 1998     

Viscous limits for piecewise smooth solutions to systems of conservation laws

In this paper we study the zero dissipation problem for a general system of conservation laws with positive viscosity. It is shown that if the solution of the problem with zero viscosity is piecewise smooth with a finite number of noninteracting shocks satisfying the entropy condition, then there exist solutions to the corresponding system with viscosity that converge to the solutions of the system without viscosity away from shock discontinuities at a rate of order as the viscosity coefficient goes to zero. The proof uses a matched asymptotic analysis and an energy estimate related to the stability theory for viscous shock profiles.