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.     

Elongational flow and rheology of monodisperse polymers in solution

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

Elongational rheology of polyethylene melts — temporary network constitutive laws

The uniaxial elongational properties of various polyethylenes have been evaluated using an elongational rheometer and a melt-strength apparatus. It is possible to derive the data obtained in elongation from the distribution of relaxation times obtained from oscillatory shearing measurements (linear viscoelasticity), using a Wagner constitutive equation. The effects of the molecular parameters of the samples have been studied, in particular the effect of polydispersity on the shape of the damping function.     

Elongational viscosity by fiber spinning

Isothermal melt, fiber-spinning was recently analyzed by means of a nonlinear, integral, constitutive equation that incorporates shear history effects, spectrum of relaxation times, shear-thinning, and extension thinning or thickening when either the drawing force or the draw ratio is specified. The predictions agreed with experimental data on spinning of polystyrene, low-density polyethylene, and polypropylene melts. The predicted apparent elongational viscosity along the threadline (which, as shown in this work, must be identical to that measured experimentally by fiber spinning type of elongational rheometers) is compared with the true elongational viscosity predicted by the same constitutive equation under well-defined experimental conditions of constant extension rate, independent of any strain history. It is concluded that the apparent elongational viscosity, as measured by fiber-spinning, approaches the true elongational viscosity at low Weissenberg numbers (defined as the product of the liquid's relaxation time multiplied by the extension rate). At moderate Weissenberg numbers, the two viscosities may differ by an order of magnitude and their difference grows even larger at high Weissenberg numbers.     

Elongational flows of some non-colloidal suspensions

The rheology of a system must be explored not only in viscometric flows, but also in other flow classes, and so, we present some results for the axisymmetric elongational flow of non-colloidal suspensions of spheres. We compare our results with data from shear flows using the same matrices and spheres. We have experimented with non-colloidal suspensions of 40-μm diameter polystyrene spheres with volume fractions (?) varying from 0.3 to 0.5. Two matrix fluids were used—one was a near-Newtonian polydimethyl siloxane of 12 Pa-s viscosity and the other was a variant of the M1 Boger fluid sample of Sridhar which we call M1*. We did not find that the Trouton ratio for either of these fluids was 3; generally, the ratio was larger. We investigated the role of sphere roughness using spheres roughened to 5.3 % of the radius in a 50 % suspension in silicone oil and found an increase of elongational viscosity of about 65 % which is comparable with the 60 % increase in shear viscosity with roughness noted previously. For the silicone oil matrix, we found no rate effect, with very little strain-hardening. By contrast, the M1-type matrix suspensions showed strain-hardening and an increase of elongational viscosity with elongation rate.     

Elongational flow opto-rheometry for polymer melts

Polymer melt elongation is one of the most important procedures in polymer processing. To understand its molecular mechanisms, we constructed an elongational flow opto-rheometer (EFOR) in which a high precision birefringence apparatus of reflection-double path type was installed into a Meissner's new elongational rheometer of a gas cushion type (commercialized as RME from Rheometric Scientific) just by mounting a small reflecting mirror at the center of the RME's sample supporting table. The EFOR enabled us to achieve simultaneous measurements of tensile stress (t) and birefringence n(t) as a function of time t under a given constant strain rate within the range of 0.001 to 1.0s^–1. (t) can be monitored upto the maximum Hencky strain (t) of 7 as attained, in principle, with RME, while the measurable range of the phase difference in the birefringence was 0 to 250 (0 to 79 100 nm for He-Ne laser light) within the accuracy of ±0.1 (±31.6 nm) up to (t) 4. The performance was tested on an anionically polymerized polystyrene (PS) and a low density polyethylene (LDPE). For both polymers (t) first followed the linear viscoelasticity rule in that the elongational viscosity, , is three times the steady shear viscosity, 3 _o(t), at low shear rate , but the _E (t) tended to deviate upward after a certain Hencky strain was attained. The birefringence n(t) was a function of both Hencky strain and strain rate in such a way that the stress-optical law holds with the stress-optical coefficient C(t) = n(t)/(t) being equal to the ones reported from shear flow experiments. Interestingly, however, for PS elongated at low strain rates the C(t) vs (t) relation exhibited a strong nonlinearity as soon as (t) reached steady state. This implies that the tensile stress reaches the steady state but the birefringence continues to increase in the low strain-rate elongation. For the PS melt elongated at high strain rates, on the other hand, C(t) was nearly a constant in the entire range observed. For LDPE with long-chain branchings, (t) exhibited tendency of strain-induced hardening after certain critical strain, but C(t) was nearly a constant in the entire range of (t) observed.     

Elongational behaviour of a low density polyethylene melt

Summary In addition to earlier findings ofMeißner on a low density polyethylene that in the linear range of deformation the time-dependent elongational viscosity is three times the shear viscosity the validity of the relationship could be demonstrated where and describe the compliances in elongation and shear, respectively. In the nonlinear rangeJ(t) was found to increase with stress whereasD(t) decreases. The nonlinear viscosity-time behaviour is different in shear and elongation, too. The shear viscosity obtained by relating the stress to the rate of viscous strain decreases with shear from the zero shear viscosity to its steady-state value. The elongational viscosity goes up with time from 3 ₀ to a higher steady-state value.The temperature dependence which was determined by measuring the steady-state elongational viscosities at different constant stresses as a function of temperature and by shifting the tensile creep compliancesD(t) measured at constant stress with respect to time, was found to be exactly the same as in shear. This result could indirectly be verified by some constant stretching rate tests. It was shown experimentally that the recoverable strain in the steady-state is independent of temperature if measured at constant stress.

---

Zusammenfassung In Ergänzung früherer Ergebnisse vonMeißner an einem Polyäthylen niedriger Dichte, die im linearen Bereich der Verformung für die zeitabhängige Dehnviskosität das Dreifache der Scherviskosität ergaben, konnte die Gültigkeit der Beziehung gezeigt werden. und sind die Nachgiebigkeiten in Dehnung und Scherung. Im nichtlinearen Bereich steigtJ(t) mit zunehmender Spannung an, währendD(t) abfällt. Die Zeitabhängigkeit der Viskosität, definiert als der Quotient aus Spannung und Geschwindigkeit des viskosen Deformationsanteils, ist in Scherung und Dehnung ebenfalls unterschiedlich. Die Scherviskosität fällt von dem Wert der Nullviskosität ₀ mit wachsender Scherung auf einen stationären Wert ab, die Dehnviskosität dagegen steigt von 3 ₀ auf einen höheren stationären Wert an.Die Temperaturabhängigkeit wurde aus Messungen des stationären Wertes der Dehnviskosität bei konstantgehaltener Zugspannung und aus einer Verschiebung der bei unterschiedlichen Temperaturen, aber konstanten Spannungen gemessenen Nachgiebigkeiten in bezug auf die Zeit bestimmt. Sie ist dieselbe wie in Scherung. Dieses Ergebnis konnte indirekt durch Messungen mit konstanter Dehngeschwindigkeit bestätigt werden. Die reversible Dehnung im stationären Zustand wurde bei konstanter Zugspannung als von der Temperatur unabhängig gefunden.

---

---

With 17 figures     

Elongational properties and molecular structure of polyethylene melts

Summary The viscosity and the recoverable strain in the steady state of elongation have been measured on several polyethylenes of different molecular structures. The elongational viscosity as a function of tensile stress runs through a more or less pronounced maximum in the nonlinear range whereas in the linear range the Trouton viscosity is reached. For low density polyethylenes it could be demonstrated that the maximum of the steady-state elongational viscosity and the elasticity expressed by the steady-state compliances in shear and tension sensitively increase if the molecular weight distribution is broadened by the addition of high molecular weight components. A variation of the weight average molecular weight does only shift the elongational viscosity curve but leaves its shape unchanged. Two of the four high density polyethylenes investigated do not show a maximum of the steady-state elongational viscosity, for the others it is less pronounced than in the case of low density polyethylenes. The influence of branching on the elongational behaviour of polyethylene melts in the steady-state and the transient region is qualitatively discussed.With 11 figures and 4 tables     

Elongational,shear and volume viscosities of polymer melts

Volume viscosity of polypropylene, polystyrene, and low density polyethylene are described. Shear, elongational, and volume viscosities may be interrelated with the help of scaling equations. They explain the effect on elongational viscosity of strain rates, stresses, and strains.     

Elongational flow behavior of polymeric fluids according to the Leonov model

The viscoelastic behavior of polymeric systems based upon the Leonov model has been examined for (i) the stress growth at constant strain rate, (ii) the stress growth at constant speed and (iii) the elastic recovery in elongational flow. The model parameters have been determined from the available rheological data obtained either in steady shear flow (shear viscosity and first normal-stress difference as a function of shear rate) or oscillatory flow (storage and loss moduli as a function of frequency in the linear region) or from extensional flow at very small strain rates (time-dependent elongation viscosity in the linear viscoelastic limit). In addition, the effect of the parameter characterizing the strain-hardening of the material during elongation has also been studied. The estimation of this parameter has been based upon the structural characteristics of the polymer chain which include the critical molecular weight and molecular weight of an independent segment. Five different polymer melts have been considered with varying number of modes (maximum four modes). Resulting predictions are in fair agreement with corresponding experimental data in the literature.     

Smectic rheology

We have studied the oscillatory shear response of three thermotropic smectic-A liquid crystalline materials with no external aligning field (other than the necessary presence of rheometer plates). Two are polymers (one main-chain and one side-chain) and the other is a small molecule smectic. All three exhibit the classical linear response to oscillatory shear characteristic of a viscoelastic solid at sufficiently small strain amplitudes and frequencies. However, for strain amplitudes exceeding a small critical value, these materials exhibit a strongly nonlinear response to strain, which is characterized in detail. While the low-strain moduli and the critical strain of the three smectics are considerably different, the nonlinear response has some universal character which is presumably related to the low energy for the formation of defects in smectic liquid crystals.     

Fourier-transform rheology

Oscillatory shear of polymeric liquids in the non-linear regime generates higher harmonic contributions in the shear stress response. These non-linear contributions are analyzed in Fourier space with respect to the different frequencies and intensities. Simulated and experimental Fourier rheology spectra for atactic poly(propylene) melts are shown. Received: 28 January 1998 Accepted: 26 May 1998     

流变学漫谈

介绍流变学的定义,Bingham模型,判断真实材料中弹性和黏性中哪一种性质占优势的无量纲数组组成的力学参数Deborah数,简要回顾流变学的发展,流变学已经、正在和即将在一些重要工业中起到革命性的影响作用。     

Surface rheology

This review article approaches the subject of surface rheology from a phenomenological point of view. Operational definitions are given for the four surface rheological parameters: modulus and viscosity in either dilation or shear. Results of recent measurements are discussed. Observed non-Newtonian or viscoelastic behaviour has in some cases been interpreted in terms of known relaxation processes. The surface rheological behaviour affects the flow of bulk liquid near the surface; coupling occurs through the tangential stress boundary condition.     

Elongational effects in the flow of viscoelastic fluid through a wavy channel

The numerical simulation of the viscoelastic flow through a wavy channel was carried out using the modified Giesekus model. It was found that the excess pressure loss relates to the stretch-thickening properties of elongational viscosity and the geometry of the wavy channel through a large elongational component of the flow at the winding part of the channel. The profiles of the axial component of the velocity become significantly asymmetric when the excess pressure loss occurs. Furthermore, the velocity profiles of a 0.1 wt% solution of polyacrylamide were measured using laser Doppler velocimetry. The results of these measurements are compared to the numerical results. Received: 30 June 1998 Accepted: 20 May 1999