An encapsulated dumbbell model for concentrated polymer solutions and melts II. Calculation of material functions and experimental comparisons

The consitutive equation derived in Part I [1] is used to compute steady-state and time-dependent material functions. A procedure is suggested for determining the four independent model parameters (λ_H, b, σ, β) from experimental data.Material functions have been calculated using parameters determined by experimental data for concentrated solutions of nearly monodisperse linear polymers. The theoretical curves for steady shear properties (η and Ψ₁) are in quantitative agreement with the data well into the range of nonlinear behavior. The calculated time-dependent properties (η*, η⁺, Ψ⁺, η^?) are in qualitative agreement with observed behavior. Theoretical curves for elongational viscosity are reasonable, but suitable experimental data are not available for comparison.     

Rheology of suspensions of spherical particles in a newtonian and a non-newtonian fluid

Properties of suspensions of spherical glass beads (25–38 μm dia.) in a Newtonian fluid and a non-Newtonian (NBS Fluid 40) fluid were measured at volume fractions, φ, of 0%, 10%, 20% and 30%. Measurements were made using a modified and computerized Weissenberg Rheogoniometer. Properties measured included steady shear viscosity, η($\text{γ}\text{.}$), first normal stress difference, N₁($\text{γ}\text{.}$), linear viscoelastic properties, η′(ω) and G′(ω), shear stress relaxation, σ^? ($\text{γ}\text{.}$, t), and growth, σ⁺($\text{γ}\text{.}$, t) and normal stress relaxation, N₁^?($\text{γ}\text{.}$, t).For a the Newtonian fluid, increasing φ causes both η and η′ to increase, with η′ showing a slight frequency dependence. Both N₁ and G′ are zero and stress relaxation and growth occur essentially instantaneously. For the NBS fluid, both η and η′ increse with φ at all $\text{γ}\text{.}$ and ω, respectively, the increase being greater as $\text{γ}\text{.}$ and ω approach zero. N₁ and G′ are less affected by the presence of the particles than η and η′ with the effect on G′ being more pronounced than on N₁. For fixed $\text{γ}\text{.}$, stress relaxation and growth exhibit greater non-linear effects as φ is increased. A model for predicting a priori the linear viscoelastic properties for suspensions was found to yeild reasonable estimates up to φ = 20%.     

Nonlinear viscoelasticity of PP/PS/SEBS blends

The nonlinear viscoelastic behavior of polypropylene/polystyrene (PP/PS) blends compatibilized or not with the linear triblock copolymer (styrene-ethylene-/butylene-styrene, SEBS) was investigated. Start-up of steady-shear at rates from 0.1 to 10 s^–1 was carried out using a controlled strain rotational rheometer and a sliding plate rheometer for strain histories involving one or several shear rates. The shear stress and first normal shear stress difference were measured as functions of time, and the morphologies of the samples before and after shearing were determined. For each strain history except that involving a single shear rate of 0.1 s^–1 the blends showed typical non-linear viscoelastic behavior: a shear stress overshoot/undershoot, depending on the history, followed by a steady state for each step. The first normal stress difference increased monotonically to a steady-state value. The values of the stresses increased with the addition of SEBS. The shear stress overshoot and undershoot and the times at which they occurred depended strongly on the strain history, decreasing for a subsequent shear rate step performed in the same direction as the former, and the time at which stress undershoot occurred increased for a subsequent shear rate step performed in the opposite direction, irrespective of the magnitude of the shear rate. This behavior was observed for all the blends studied. The time of overshoot in a single-step shear rate experiment is inversely proportional to the shear rate, and the steady-state value of N₁ scaled linearly with shear rate, whereas the steady-state shear stress did not. The average diameter of the dispersed phase decreased for all strain histories when the blend was not compatibilized. When the blend was compatibilized, the average diameter of the dispersed phase changed only during the stronger flows. Experimental data were compared with the predictions of a model formulated using ideas of Doi and Ohta (1991), Lacroix et al. (1998) and Bousmina et al. (2001). The model correctly predicted the behavior of the uncompatibilized blends for single-step shear rates but not that of the compatibilized blends, nor did it predict morphologies after shearing.     

Rheological properties of some biphenyl liquid crystals

Rheological measurements have been carried out using a rotational viscometer with a system of coaxial cylinders on four liquid crystalline substances from the group of cyanobiphenyls. On the basis of results of these investigations it was found that in the investigated range of shear rates the nematic phases exhibited Newtonian flow behaviour, while the smectic phases exhibited non-Newtonian behaviour. For shear rates up to ca. 1000 s^–1 the dependence of the shear stress on shear rate is well described by a power-law model.     

Transient MHD stagnation flow of a non-Newtonian fluid due to impulsive motion from rest

The transient boundary layer flow and heat transfer of a viscous incompressible electrically conducting non-Newtonian power-law fluid in a stagnation region of a two-dimensional body in the presence of an applied magnetic field have been studied when the motion is induced impulsively from rest. The non-linear partial differential equations governing the flow and heat transfer have been solved by the homotopy analysis method and by an implicit finite-difference scheme. For some cases, analytical or approximate solutions have also been obtained. The special interest are the effects of the power-law index, magnetic parameter and the generalized Prandtl number on the surface shear stress and heat transfer rate. In all cases, there is a smooth transition from the transient state to steady state. The shear stress and heat transfer rate at the surface are found to be significantly influenced by the power-law index N except for large time and they show opposite behaviour for steady and unsteady flows. The magnetic field strongly affects the surface shear stress, but its effect on the surface heat transfer rate is comparatively weak except for large time. On the other hand, the generalized Prandtl number exerts strong influence on the surface heat transfer. The skin friction coefficient and the Nusselt number decrease rapidly in a small interval 0<t^*<1 and reach the steady-state values for t^*≥4.     

Rheological properties of poly(dimethylsiloxane) filled with fumed silica: I. Hysteresis behaviour

The apparent viscosity and primary normal stress difference were measured for dispersions of fumed silica in poly(dimethylsiloxane). Dispersions with less than 4.75% by weight of filler exhibit hysteresis in both the viscosity and normal stress, when the shear rate was increased and then decreased in discrete steps. The shape of the hysteresis loops were sensitive to the details of the deformation history. By using the appropriate deformation history, the material properties determined during the increasing shear rate part of the hysteresis experiment compare favourably with the steady-state rheological properties. The rheological properties of the dispersion were quite sensitive to the age of the fluid with no hysteresis behaviou exhibited by dispersions less than three days old. For dispersions with at least 4.75% by weight of fumed silica, neither the apparent viscosity nor the primary normal stress coefficient exhibited significant hysteresis behaviour. The relationship between the observed rheological behaviour and the dispersion's microstructure is discussed.     

Two new methods for the approximate determination of the initial coefficient of the first normal stress difference

Two methods for determining the initial coefficient of the first normal stress difference are presented. They are based on the evaluation of the steady viscosity function η($\text{γ}\text{.}$) and the viscosity function η⁺($\text{γ}\text{.}$, t) at the start-up of a flow with a very small rate of deformation $\text{γ}\text{.}$ < $\text{γ}\text{.}$₀. For the functions η($\text{γ}\text{.}$) and η⁺($\text{γ}\text{.}$), equations are given which can be used for a simple evaluation of the integral relationships obtaiend for ψ₁₀. The values for ψ₁₀ calculated by the two methods are compared with values obtained by the well-known methods via measurement of the ψ₁($\text{γ}\text{.}$) or η″(ω)/ω functions and extrapolation to zero). Both methods give values which are in satisfactory agreement with the experimental values.     

Drag characteristics of longitudinal and transverse riblets at low dimensionless spacings

A surface grooved with microscopic riblets aligned parallel to the flow is an effective means to reduce the turbulent skin friction up to 10% compared to a smooth surface. The maximum drag reduction is found for a dimensionless rib spacing s ⁺ in the range of 15–17. For s ⁺ < 10, a linear behaviour of the drag reduction curve is predicted by viscous theory. This linear slope of the drag reduction curve is in contradiction to Schlichting’s postulation of a hydraulically smooth behaviour of small-scale roughness in a turbulent flow. This regime of evanescent dimensionless rib spacings is investigated experimentally by direct wall shear stress measurements in a fully developed channel flow. Additionally, a numerical calculation of the viscous flow over riblets was carried out to predict the drag reducing behaviour. The experimental results show a linear drag reducing behaviour down to s ⁺ = 0.3, which is in good agreement with the numerical results of the viscous simulation. The postulation of Schlichting’s hydraulically smooth regime of a rough surface was not confirmed, neither for a riblet surface nor for a surface geometry with grooves oriented perpendicular to the flow. In the latter case, the drag increases as a quadratic function of the roughness height.     

Non-linear behaviour of asphalts in steady and transient shear flow

Steady-state and transient shear stress and normal stress data were obtained for four asphalts with a modified Weissenberg Rheogoniometer. Interest was specially related to non-linear behaviour at high shear-rates. The time-temperature superposition principle was found to hold in non-linear behaviour. Moreover, steady-state and transient data could be plotted as master curves irrespective of the nature of the asphalts. In particular, the master curve of steady-state viscosity could be extended to results published in the literature. In the nonlinear region the shear stress relaxation after cessation of a steady shear rate becomes a function of t only and is related to the primary normal-stress coefficient, as predicted by the Yamamoto equation. In the shear stress growth experiment an overshoot is obtained at a constant strain close to 1.5, independent of the rate of strain.     

The effect of 1,3:2,4-<Emphasis Type="Italic">bis</Emphasis>-<Emphasis Type="Italic">O</Emphasis>-(<Emphasis Type="Italic">p</Emphasis>-methylbenzylidene)-<Emphasis Type="SmallCaps">d</Emphasis>-sorbitol (PDTS) on uniaxial elongational viscosity of polypropylene

We investigated the dynamic viscoelasticity and elongational viscosity of polypropylene (PP) containing 0.5 wt% of 1,3:2,4-bis-O-(p-methylbenzylidene)-d-sorbitol (PDTS). The PP/PDTS system exhibited a sol–gel transition (T _gel) at 193 °C. The critical exponent n was nearly equal to 2/3, in agreement with the value predicted by a percolation theory. This critical gel is due to a three-dimensional network structure of PDTS crystals. The elongational viscosity behavior of neat PP followed the linear viscosity growth function 3η ⁺ (t), where η ⁺ (t) is the shear stress growth function in the linear viscoelastic region. The elongational viscosity of the PP/PDTS system also followed the 3η ⁺ (t) above T _gel but did not follow the 3η ⁺ (t) and exhibited strong strain-softening behavior below T _gel. This strain softening can be attributed to breakage of the network structure of PDTS with a critical stress (σ _c) of about 10⁴ Pa.     

An evaluation of the Larson-Doi model for liquid crystalline polymers using recoil

The mesoscopic models for the rheological properties of liquid crystalline polymers proposed by Larson and Doi in 1991 and Kawaguchi and Denn in 1999 are based on phenomenological expressions that describe the evolution of the defect density and the contribution of the “texture” to the stress. In the present work, we attempt to assess some of these assumptions by monitoring how the energy stored in the texture of liquid crystalline materials evolves during shear flows. For that purpose, strain recovery is measured as a function of the applied strain for flow reversal and intermittent flow. Solutions of poly-benzylglutamate in m-cresol, hydroxypropylcellulose in water and a nematic surfactant solution are used as model systems. Although the behaviour is described qualitatively by the model, discrepancies between the predictions and the experiments are observed, especially when the shear history includes rest periods. Received: 14 July 1999 /Accepted: 30 August 1999     

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 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.     

Ageing and collapse of bentonite gels—effects of Li,Na, K and Cs ions

The state of bentonite gels at the start of the ageing experiment must be well-defined, and this required the gels to be at a constant surface chemistry condition. This is achieved by allowing the freshly prepared gels to rest for a day. At this state, the yield stress is constant, provided that the gel is at an equilibrium breakdown state after stirring prior to each measurement. This point is also the yield stress at zero aged time. Ageing study then commenced, and the behaviour is generally characterised by an increasing yield stress with wait time. Alkali metal ion type and concentration affect the gel ageing and stability behaviour significantly. The ageing behaviour is most pronounced at low salt concentrations for the smallest and most strongly hydrated cations, Li ⁺and Na ⁺. The yield stress at any given aged time and its rate of increase are generally larger. Coarsening of these suspensions was observed. The opposite is true for the weakly hydrated K ⁺and Cs ⁺ions. At high concentrations of 0.5 and 1.0 M Cs, K and Na ions, the gels became unstable over time and phase-separated. The stability time of these weak gels was found to increase with decreasing cation size, Na > K > Cs. This stability time displayed a very strong quantitative correlation with the hydration bond length. Coarsening was also expected, but not observed due to the lack of integrity of these weak aggregates during particle size measurement. The recovery or ageing behaviour was fitted with both the Nguyen–Boger and Leong models.     

Rheology of lyotropic and thermotropic lamellar phases

We investigate the steady-state rheological behaviour of the lamellar phase of a lyotropic system (CpCl, hexanol, brine) and of a thermotropic system (8CB). Power laws characterize the behaviour of the imposed stress as a function of the measured shear rate and similarities are observed for both systems; the same regime γ˙∼σ ^m with m≈1.7 is obtained at low shear stresses corresponding to a texture of oily streaks oriented in the direction of the flow, as shown by microscopic observations. The “onion state” only exists in the case of dilute samples of the lyotropic lamellar phase; the stress then varies as γ˙∼σ ^m with m≈4.8, as already observed by other groups with different systems. Rheological measurements at different temperatures allow determination of different activation energies relating to the still badly understood processes involved in the different rheological regimes. We propose a model which reproduces the experimental power laws and which is based on an analogy with the theory of high-temperature creep in metals and alloys. Received: 19 October 1999/Accepted: 1 November 1999     

Rheological properties of disperse systems at low shear stresses

Three-dimensional network structures can be built up in disperse systems due to long-range colloidal interactions between the dispersed particles. The rheological behaviour of such coagulation structures has been studied by means of creep and recovery experiments at low shear stresses, i.e. by measuring the shear strain as a function of time under constant stress and after removal of stress. Measurements of this type give insight into the elastic and viscous deformations and the retardation times necessary to reach equilibrium or steady-state conditions.Results obtained with dispersions of pigments in polymer solutions and with monodisperse polymer latexes indicate the existence of an equilibrium state at low shear stresses with a predominant elastic deformation and a high viscosity suggesting that the disperse systems investigated do not behave exactly as rigid gels but apparently exhibit a dynamic equilibrium of structural break-down and formation under applied stress. This behaviour is approximately described by a 4-parameter-model with an instantaneous and a steady-state compliance, one retardation time, and a viscosity.At higher shear stresses thixotropic structural break-down occurs resulting in a transition from the rheological behaviour described here to a liquid-like state with a comparatively low viscosity. In this stress range the viscoelastic properties become strongly time-dependent.These measurements give evidence of the presence of two types of deformation: an instantaneous, purely elastic deformation attributable to the unperturbed coagulation structure and the creep-recovery behaviour of an elastic liquid apparently related to the breaking and re-forming of bonds.     

Measurement method of complex viscoelastic material properties

A measurement technique of viscoelastic properties of polymers is proposed to investigate complex Poisson’s ratio as a function of frequency. The forced vibration responses for the samples under normal and shear deformation are measured with varying load masses. To obtain modulus of elasticity and shear modulus, the present method requires only knowledge of the load mass, geometrical characteristics of a sample, as well as both the amplitude ratio and phase lag of the forcing and response oscillations. The measured data were used to obtain the viscoelastic properties of the material based on a 2D numerical deformation model of the sample. The 2D model enabled us to exclude data correction by the empirical form factor used in 1D model. Standard composition (90% PDMS polymer + 10% catalyst) of silicone RTV rubber (Silastic^® S2) were used for preparing three samples for axial stress deformation and three samples for shear deformation. Comprehensive measurements of modulus of elasticity, shear modulus, loss factor, and both real and imaginary parts of Poisson’s ratio were determined for frequencies from 50 to 320 Hz in the linear deformation regime (at relative deformations 10^?6 to 10^?4) at temperature 25 °C. In order to improve measurement accuracy, an extrapolation of the obtained results to zero load mass was suggested. For this purpose measurements with several masses need to be done. An empirical requirement for the sample height-to-radius ratio to be more than 4 was found for stress measurements. Different combinations of the samples with different sizes for the shear and stress measurements exhibited similar results. The proposed method allows one to measure imaginary part of the Poisson’s ratio, which appeared to be about 0.04–0.06 for the material of the present study.     

Transient shear flow behavior of concentrated long glass fiber suspensions in a sliding plate rheometer

In order to eventually predict the behavior of long fiber suspensions in complex flows commonly found in processing operations, it is necessary to understand their rheology and its connection to the evolution of fiber orientation and configuration in well defined flows. In this paper we report the transient behavior at the startup of shear flow of a polymer melt containing long glass fibers with a length (L) >1 mm, using a sliding plate rheometer (SPR). The operation of the SPR was confirmed by comparing the transient shear viscosity (η⁺) for a polymer melt and a melt containing short glass fibers (L < 1 mm) with measurements obtained from a cone-and-plate device, using a modified sample geometry that was designed to avoid wall effects. For the long fiber systems, measurements could only be obtained in the SPR because these systems would not stay within the gap of the rotational rheometer. Transient stress growth behavior of the long fiber systems was obtained as a function of shear rate and fiber concentration for samples prepared with three different initial orientations. Results showed that, unlike short fiber systems (with a random planar initial orientation) that usually exhibit a single overshoot peak followed by a steady state, η⁺ of the long fiber suspensions often passed through multiple transient regions, depending on the fiber concentration and applied shear rate. Additionally, η⁺ of the long fiber suspensions was found to be highly dependent on the initial orientation of the sheared samples. Finally, the initial and final fiber orientations of the long glass fiber samples were measured and used to initiate an explanation of the viscosity behavior. The results obtained in this research will be useful for future assessment of a quantitative correlation between transient rheology and the evolution of fiber orientation.     

Numerical study of chain conformation on shear banding using diffusive Rolie-Poly model

Shear-banding phenomenon in the entangled polymer systems was investigated in a planar Couette cell with the diffusive Rolie-Poly (ROuse LInear Entangled POLYmers) model, a single-mode constitutive model derived from a tube-based molecular theory. The steady-state shear stress ?? _s was constant in the shear gradient direction while the local shear rate changed abruptly, i.e., split into the bands. We focused on the molecular conformation (also calculated from the Rolie-Poly model) around the band boundary. A band was found also for the conformation, but its boundary was much broader than that for the shear rate. Correspondingly, the first normal stress difference (N ₁) gradually changed in this diffuse boundary of the conformational bands (this change of N ₁ was compensated by a change of the local pressure). For both shear rate and conformation, the boundary widths were quite insensitive to the macroscopic shear rate but changed with various parameters such as the diffusion constant and the relaxation times (the reptation and the Rouse times). The broadness of the conformational banding, associated by the gradual change of N ₁, was attributed to competition between the molecular diffusion (in the shear gradient direction) and the conformational relaxation under a constraint of constant ?? _s.     

Constitutive relationships for polymeric materials with power-law distributions of relaxation times

The linear relaxation modulus of polydisperse polymer melts and solutions can often be approximated by a power law,ct ^–m over some range of time,t. If, in addition, the nonlinear rheology is given by a separable integral equation, with a strain-dependent factor typical of those observed experimentally, then some commonly observed empirical rules and equations can be readily derived as approximations, namely the Cox-Merz relationship between complex viscosity and steady-state shear viscosity, Bersted's predictions of steady shear stress and first normal-stress difference from a truncated spectrum of linear relaxation times, and the observation of Koyama and coworkers that the ratio of the nonlinear to the linear time-dependent elongational viscosity is independent of strain rate, over a range of strain rates outside the linear regime.