Molecular theory for moderately concentrated polymer solutions in shear flow

A molecular theory for the rheological properties of moderately concentrated polymer solutions is developed on the basis of a model of interacting dumbbells. The interaction is treated in a mean field approximation, leading to an effective one-particle potential and a Gaussian stationary distribution function. Various rheological functions such as birefringence, shear viscosity and first normal-stress coefficient for simple shear flow and the Trouton viscosity for simple extensional flow are calculated. Good qualitative agreement with experimental observations is found, especially at intermediate flow rates. It is predicted, for example, that the birefringence increases approximately linearly with shear rate at intermediate shear rates and that the concentration dependence of the gradient varies asc ^1/2. The typical non-Newtonian behaviour is obtained for the shear viscosity. For small concentrations the onset of shear rate dependence decreases asc ^–1/2. At intermediate shear rates an apparent power law is obtained with an exponent between – 0.5 and – 1.0, decreasing with concentration.     

Normal stress measurements

A review of the methods for obtaining the normal stress differences in simple shear flow from force measurements on the walls of apparatus in which curvilinear shear flows are generated is given. Indirect methods, for example the flow birefringence method, are considered. Some new work on normal stress measurement using cone and plate, parallel plate pressure distribution and total thrust measurements in conjunction with flow birefringence methods are reviewed. The difficulties found in obtaining consistant results are discussed.Paper presented at the Conference on Experimental Rheology, University of Bradford, April 17–19, 1968. — Original paper published in Rheo. Acta,7, 368 (1968).     

Rheo-optical measurements of the first and third normal stresses of homopolymer poly(vinyl methyl ether) melt

Normal stresses play a key role in polymer processing, yet accurate measurements are still challenging. Simultaneous rheo-optical measurements are conducted on a poly(vinyl methyl ether) homopolymer melt over a wide range of temperatures and oscillatory shear frequencies, in an effort to measure the normal stresses, by using quantitative flow birefringence measurements. The stress optical rule holds well for this polymer as expected, with the value of the stress optic coefficient of (6.38±0.19)×10⁻¹¹ cm²/dyn at 30°C. The first and third normal stress difference coefficients, calculated using a single memory constitutive equation applied to the stress and birefringence data, are in excellent agreement. The ratio of the measured third and first normal stress difference coefficients, (1−β)=0.71±0.05, agrees well with the result of the Doi–Edwards model with independent alignment approximation (β=0.28). The measurement of normal stress difference coefficients with such small deviations proves the robust nature of the improved rheo-optical instrument and its ability to measure complete stress tensor.     

Turbulence production in flow over a wavy wall

Measurements of the spatial and time variation of two components of the velocity have been made over a sinusoidal solid wavy boundary with a height to length ratio of 2a/λ = 0.10 and with a dimensionless wave number of α⁺ = (2π/λ)(v/u ^?) = 0.02. For these conditions, both intermittent and time-mean flow reversals are observed near the troughs of the waves. Statistical quantities that are determined are the mean streamwise and normal velocities, the root-meansquare of the fluctuations of the streamwise and normal velocities, and the Reynolds shear stresses. Turbulence production is calculated from these measurements. The flow is characterized by an outer flow and by an inner flow extending to a distance of about α^?1 from the mean level of the surface. Turbulence production in the inner region is fundamentally different from flow over a flat surface in that it is mainly associated with a shear layer that separates from the back of the wave. Flow close to the surface is best described by an interaction between the shear layer and the wall, which produces a retarded zone and a boundary-layer with large wall shear stresses. Measurements of the outer flow compare favorably with measurements over a flat wall if velocities are made dimensionless by a friction velocity defined with a shear stress obtained by extrapolating measurements of the Reynolds stress to the mean levels of the surface (rather than from the drag on the wall).     

Direct measurement of static yield properties of cohesive suspensions

A technique of yield stress investigation based upon the combined use of two devices (an applied stress rheometer and an instrument for measuring the propagation velocity of small amplitude, torsional shear waves) is described. Investigations into the low shear rate rheological properties of illitic suspensions are reported for shear rates, typically, in the range 10^–4— 10^–1 s^–1 under applied stresses in the range 0.01 — 10 Nm^–2 and involving shear strains between 10^–1 and 10^–4. Results are presented which demonstrate that the technique does not invoke the excessive structural disruption of material associated with applied shear rate based methods (direct and otherwise) and the widely encountered problem of wall slip at the surface of rotational measuring devices is avoided using miniature vane geometries. Results are compared with those obtained using smooth-walled cyclindrical measuring devices in both applied stress and applied shear rate instruments.Yield measurements are considered in relation to the structural properties of the undisturbed material state and shear moduli obtained by studying the propagation of small amplitude (10^–5 rad), high frequency (~ 300 Hz) torsional shear waves through the test materials are reported. Experimental techniques and instrument modifications to permit these measurements are described.     

Viscous properties of polymer melts and elastomers exemplified by ethylene-propylene copolymer

Summary The copolymer of ethylene and propylene possesses a sufficiently high thermo-oxidative resistance, making it possible to study its viscous properties, determine the appearance of elastic turbulence and wall slippage and to measure the rate of the latter over a wide interval of temperatures ranging from room temperature to 260 °C.At low shear stresses and rates the copolymer behaves like aNewtonian liquid with a viscosity of about 10⁸ poise at room temperatures.Elastic turbulence and wall slippage are displayed in sharp form when the viscosity of the copolymer is lowered to its critical value, which depends very little on the temperature and may be accepted as averaging 2.2×10⁴ poise. The corresponding critical shear stress values vary about 10-fold. The criteria of appearance of elastic turbulence suggested in (12, 14) do not agree with experimental data. The entrance losses during the flow of the copolymer through capillaries are low right until elastic turbulence sets in, after which it becomes practically impossible to measure them by the method of capillaries of different length. The average wall slippage rate values of the copolymer at shear stresses above 10⁶ dyne/cm² amount to tens of cm/sec. They increase very abruptly with rising temperature.The temperature dependence of the viscosity and the dynamic characteristics of the copolymer indicate that it has a phase transition at temperatures of about 100–120 °C, which must be related to melting of blocks contained in the copolymer macromolecules, having a structure close to that of high-pressure polyethylene. This shows that the rheological method of studying block-type polymer and grafted polymers is promising.     

Break-Up of Aerosol Agglomerates in Highly Turbulent Gas Flow

Agglomerate aerosols in a turbulent flow may be subjected to very high turbulent shear rates which through the generation of lift and drag can overcome the adhesive forces binding the constituents of an agglomerate together and cause it to break-up. This paper presents an analysis of the experimental measurements of the breakup of agglomerates between 0.1?C10???m in size, in a turbulent pipe flow followed by an expansion zone with a Reynolds numbers in the range 10⁵ to 10⁷. The analysis shows that even in wall bounded turbulence, the high turbulent shear stresses associated with the small scales of turbulence in the core can be the main source of breakup preceding any break-up that may occur by impaction at the wall. More importantly from these results, a computationally fast and efficient solution is obtained for the General Dynamic Equation (GDE) for agglomerate transport and breakup in highly turbulent flow. Furthermore the solution for the evolution of the aerosol size distribution is consistent with the experimental results. In the turbulent pipe flow section, the agglomerates are exposed continuously to turbulent shear stresses and experience more longer term breakup than in the expansion zone (following the pipe flow) where the exposure time is much less and break-up occurs instantaneously under the action of very high local turbulent shear stresses. The validity of certain approximations made in the model is considered. In particular, the inertia of the agglomerates characterised by a Stokes Number from 0.001 for the smallest particles up to 10 for 10???m particles and the fluctuations of the turbulent shear stresses are important physical phenomena which are not accounted for in the model.     

A reassessment of the wall effect of non-newtonian flow of polymer solutions

An apparatus for checking slip interpretation of flow anomalies in the laminar capillary flow of macromolecular solutions is described. It consists of a two-dimensional flow channel, having a uniform width of 1.5 inch and an alternative, convergent taper that is adjustable. Dilute aqueous solutions of the polymers Carbopol, Natrosol, and Polyox are recirculated in steady laminar flow from a large reservoir. Velocities, pressures, and wall shear stresses are measured. Local velocities are obtained by the local injection of conductive tracer fluid, whose passage is sensed by sets of electrodes stationed along the flow. Wall shear stresses are measured on a small, freely displaceable, traction surface flush with the channel wall. The tests cover concentrations of Carbopol of 0.1% to 0.4%, Natrosol of 0.5% to 1%, and Polyox of 0.5% to 1%. Approximate viscosities range from about 10 to 1000 cP. Wall shear rates up to 1000 s^–1 are attained.The results are in good agreement with the established viscometric properties of the tested materials. Actual shear stresses agree with those calculated from pressure drops, and velocities exhibit no detectable anomaly near the wall. Any velocity anomaly in the experiment would have to be less than 3% of the mean flow velocity.An order of magnitude analysis, based on particulate behavior, is made in an attempt to delineate an underlying mechanism.     

The peculiar rheo-optical behavior of bisphenol A-polycarbonate and polymethylmethacrylate

The rheological and stress-optical behavior of the melts of several grades ob bisphenol-A-polycarbonate (PC) and polymethylmethacrylate (PMMA) is investigated. Pertinent flow birefringence measurements are carried out in a remodelled cone-plate apparatus [1]. The shear stress in the polymer melt is calculated from the dynamic moduli, which are determined separately. It is shown that the linear stress optical rule is obeyed. In this way, the stress-optical coefficient C of the melt can be determined. The low-Mw polycarbonates all behave as Maxwellian fluids. The main stress direction does not deviate significantly from 45°. In the temperature range from 160° to 260°C the stress-optical coefficients of the different grades lie between 3 and 4×10^–9 Pa^–1 and show a weak temperature dependence. The stress-optical coefficient of PMMA is about a factor of 100 lower and shows a peculiar temperature-dependence, changing its sign at 144°C. The results are discussed in terms of the anisotropy of the polarizability of the polymer chain.     

Steady and transient shear flows of polystyrene solutions I: Concentration and molecular weight dependence of non-dimensional viscometric functions

Start up from rest and relaxation from steady shear flow experiments have been performed on monodisperse polystyrene solutions with molecular weight ranging from 1.3 × 10⁵ to 1.6 × 10⁶ and concentration c ranging from 5% to 40%. A method of reduced variables based on the use of a characteristic time τ_w is proposed. τ_w is defined as the product of zero shear viscosity with the steady state elastic compliance.Reduced steady and transient viscometric functions so obtained depend on the ratio M/M_e (where M_e is the entanglement molecular weight). Limiting forms are obtained when M/M_e ? 18. In steady flow, a simple correlation is found between shear and normal stresses.In stress relaxation experiments, independent of shear rate, the long-time behaviour can be characterised by a single relaxation time τ₁, which is identical for shear and normal stresses. τ₁ can be simply related to the zero shear rate viscosity and the limiting elastic compliance.     

An experimental investigation into the kinematics of a concentrated hard-sphere colloidal suspension during Hopkinson bar evaluation at high stresses

The behavior of a concentrated, hard-sphere colloidal suspension is evaluated using the split Hopkinson pressure bar (SHPB) experimental technique. The composition of the suspension is measured using thermograviometric analysis before and after loading. This, combined with recorded pressure distributions, result in the conclusion that the suspension undergoes high rate squeezing flow. Experimental results demonstrate that the suspension exhibits shear thickening consistent with that observed in standard rotational rheometry. At sufficiently high stresses the suspension exhibits a second regime of shear thinning behavior that is consistent with elastohydrodynamic theory that incorporates the shear modulus of the particles themselves. Further increases in stress result in irreversible behavior, i.e., rather than fracturing or crushing, the particles form non-reversible agglomerates during testing. This fact is demonstrated through the use of dynamic light scattering and electron microscopy. This behavior occurs within a regime of viscous material response which is seen to occur at normal strain rates and stresses over 10⁴ s^?1 and 40 MPa, respectively.     

Structure and dynamics of a polymer solution subject to flow-induced phase separation

Experiments combining mechanical rheometry with polarimetry (birefringence and scattering dichroism) have been conducted on a 6% solution of polystyrene (1.86x10⁶ molecular weight) in dioctyl phthalate. Birefringence is used to measure the extent of segmental orientation, whereas the dichroism is sensitive to orientation and deformation of concentration fluctuations associated with the process of flow-induced phase separation. The results indicate that these fluctuations grow predominately along the neutral (or vorticity axis) of a simple shear flow. At higher rates of shear, orientation in the flow direction is favored. The transition in orientation direction is accompanied by time-dependent behavior in the optical properties of the solution during shear and the onset of shear thickening of the viscosity and the first normal stress difference coefficient.     

The flow of polymer-thickened oils in convergent jet-thrust nozzles at a temperature of 84°C

It is shown that an existing form of jet-thrust device may be modified for satisfactory use at elevated temperatures. Jets are produced from a straight capillary tube and from three nozzles designed to provide different rates of uniaxial extension in the flowing oil, with shear present near the nozzle walls.The behaviour of three simulated multigrade motor oils with additives of different chemical type is compared with that of Newtonian oils at temperature of 84°C. In straight-tube flow, no measurable normal stress is detected in one of the oils (that with an alkylmethacrylate as polymer additive), but the other two oils give stresses which are measurable and, in one case, as high as those obtained at ambient temperature (that with the styrene-butadiene copolymer additive). For these oils the normal stresses measured in tubes are much closer to the axial stresses measured in nozzles than was the case at ambient temperature.In nozzle flow, axial stresses are detected in each oil which are rather lower than those measured at ambient temperature, the deviation increasing with increased jet velocity. The relative importance of axial stress, compared with shear stress, is shown to increase with increasing temperature and shear rate. The ration of axial stress to shear may reach a value of 3 or 4 at a shear rate of 10⁵set^?1, the oils with styrene-butadiene and styrene-isoprene copolymer additives being somewhat better performers than that with the alkylmethacrylate copolymer additive.It is suggested that the presence of normal, or axial, stresses might improve lubrication performance in those situations where normal load is applied with little relative movement of the bearing surfaces.     

Elastico-viscous squeeze films: Part 3. the torsional-balance rheometer

The observation made in Part 2 that squeezing flow with a superimposed rotation results in an equilibrium situation with the applied load just balancing the normal stresses generated in the test fluid is used to develop a new technique (the Torsional-Balance Rheometer) for measuring the viscometric functions of elastic liquids.The Rheometer utilizes conventional torsional flow and its novel feature is that the applied load is fixed and the associated shear rate at the rim determined, in contrast to the usual situation where the shear rate is fixed and the total normal force measured.It is argued that the Torsional Balance has significant advantages over other rheometers in the very high shear-rate range, since the normal stresses being measured themselves supply a mechanism for keeping the top plate (which is free to float on the test fluid) at a constant separation from the rotating bottom plate, hence allowing very small gaps to be considered. Consistent data are shown to be possible for shear rates in excess of 10⁵ s^?1.     

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.     

Simultaneous measurements of velocity and stress distributions in polyisobutylenes using laser-Doppler velocimetry and flow induced birefringence

An experimental device was set up for the synchronous measurement of velocities and stresses in polyisobutylenes using laser-Doppler velocimetry (LDV) and the two-colour flow-induced birefringence method (FIB). The materials investigated are three low molecular polyisobutylenes. Velocity (LDV) and stress (FIB) measurements are performed in the flow entrance region and inside a slit die with a contraction ratio of 1:10. The behaviour of the polyisobutylenes is Newtonian under the flow conditions applied. Therefore, the stresses inside the fluids can be calculated and compared to the stresses experimentally determined. A good agreement in shear and elongational flows was found between the calculated (LDV) and directly measured stresses (FIB). This result demonstrates the applicability of the experimental setup as an optical rheometer that can preferentially be used to measure elongational properties of low viscous fluids.

The stresses of an upper convected Maxwell fluid in a Newtonian velocity field near a re-entrant corner

We investigate the stresses of an upper convected Maxwell fluid in the neighborhood of a re-entrant 270° corner. It is assumed (incorrectly, of course) that the velocity field is Newtonian. Both asymptotic analysis and numerical solutions are presented. It is found that, for a fixed angle, the stresses behave approximately as r^−0.74, which contrasts with a behavior as r^−0.91 at the walls (the latter is simply the square of the Newtonian shear rate at the wall, where the flow is viscometric). The analysis shows that there are boundary layers near the walls, in which there is a transition from the viscometric behavior at the wall to a core region which the behavior is dominated by the convected derivative in the constitutive equation. Moreover, our computations show large spurious stresses downstream resulting from numerical errors.     

用电化学方法测试动脉模型壁面剪应力

应用电化学方法,对动脉模型Ｔ型分叉部位流场壁面剪应力进行测试研究。测试了对于现有理论分析和数值计算都比较困难的高雷诺数（ＲＥ＝１０００－２０００）流动流场的壁面剪应力,并且对苦干不同雷诺数及不同支管分流情况进行了系列测试。通过实验发现,此部痊同时存在高剪应力区和低剪应力区,确定了它们的位置和剪应力的大上。系列测试还显示：随着雷诺数的变化,无量纲管应力有一定的变化;而当支管分流变化时,无量纲剪应力的     

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.     

Flow birefringence and rheological measurements on shear induced micellar structures

Aqueous solutions of cationic surfactant systems with strongly binding counterions show the striking phenomenon of shear induced phase transitions. At low shear rates or angular frequencies, the solutions exhibit Newtonian flow. At high rates of shear, however, the rheological properties change dramatically. Above a well defined threshold value of the velocity gradient, a supermolecular structure can be formed from micellar aggregates. This shear induced structure (SIS) behaves like a gel and exhibits strong flow birefringence. The formation of the shear induced structure is very complicated and depends on the specific conditions of the surfactant system. In this paper we discuss new results which have been obtained from rheological measurements and from flow birefringence data. We examine the stability of the shear induced state as a function of temperature, surfactant concentration and salt concentration and we analyse the effect of solubilisation of alcohols and hydrocarbons. The results are interpreted in terms of a kinetic model which accounts for the observed behavior.Dedicated to the 60. birthday of Prof. H. Harnisch, Hoechst AGPartly presented at the 2nd Conference of European Rheologists, Prague, June 17–20, 1986