Primary Normal Stress Difference Research Articles

Rheological Property Modification of a Molten-State Polyamide through the Addition of an α-Olefin–Maleic Anhydride Copolymer

The rheological properties of a polyamide (PA) resin with low crystallinity were modified by melt-mixing it with a small amount of an alternative α-olefin–maleic anhydride copolymer as a reactive compound. Because PA has a low melting point, rheological characterization was performed over a wide temperature range. Owing to the reaction between PA and the alternative α-olefin–maleic anhydride copolymer, the blend sample behaved as a long-chain branched polymer in the molten state. The thermo-rheological complexity was obvious owing to large flow activation energy values in the low modulus region, i.e., the rheological time–temperature superposition principle was not applicable. The primary normal stress difference under steady shear was greatly increased in the wide shear rate range, leading to a large swell ratio at the capillary extrusion. Furthermore, strain hardening in the transient elongational viscosity, which is responsible for favorable processability, was clear. Because this is a simple modification method, it will be widely employed to modify the rheological properties of various polyamide resins.

Rheological response of entangled isotactic polypropylene melts in strong shear flows: Edge fracture, flow curves, and normal stresses

Isotactic polypropylene (iPP) melts are industrial semicrystalline polymers whose processing typically involves strong shear flows. The study of the rheological response of iPP melts, well beyond the linear viscoelastic limit, is limited by edge fracture, which manifests in rotational rheometers. In this work, we used a reflection polariscope under shear to detect the onset shear rate at which edge fracture is observed for various rotational rheometry fixture diameters. The onset shear rate for edge fracture was found to correlate with the zero-shear viscosity, thereby enabling the prediction of edge fracture by only knowing the zero-shear viscosity; a quantity that is easier to measure compared to the second normal stress difference. Edge fracture is then mitigated by using a cone-partitioned plate, which enabled the study of the first normal stress difference, and in combination with capillary rheometry, allowed the measurement of flow curves with a very well-resolved shear thinning region. For strongly polydisperse iPPs at high shear rates, we found that viscosity scales as the −0.7 power of the shear rate, while primary normal stress difference scales as the square root of the shear rate. The dependence of the shear thinning of iPPs on polydispersity was then unravelled, offering a broad set of data to develop and test molecular models.

Determination of the first normal stress difference from viscometric data for shear flows of polymer liquids

New equation for determination of the primary normal stress difference from viscosity curve is proposed, and the methods of calculations with the use of adequate viscosity models are discussed. Comparison of calculated and measured values of primary normal stress difference has shown that the proposed equation describes quite well experimental data but the calculations are not stable. For this reason, the proposed equation along with other equations of this type known from the literature was transformed into shear stress-dependent form. It was shown that such transformation makes it possible to represent the normal stress-to-shear stress ratio (Weissenberg number) as a unique function of the local slope of the flow curve, which is simultaneously temperature invariant. Such representation was confirmed using experimental data for eight systems comprising linear polymers and various measurement temperatures. It was found that new equation in the shear stress-dependent form is numerically stable and provides excellent description of experimental data. The general structure of expressions, which may be used for description of the elasticity of polymer liquids based on the flow curve shape, was discussed from the point of view of dimensional analysis. Obtained results made it possible to modify other expressions known from the literature in such way that they also provide an excellent fit to experimental data.

Rheological behaviors and molecular motions of semi-diluted Xanthan solutions under shear: Experimental studies

Semi-diluted Xanthan solution has been widely used in various fields, especially in enhancing oil recovery. The oscillatory shear and flow shear behaviors of Xanthan are important to oil flooding. The oscillatory shear relates to molecular motions, while flow shear reflects flowing characterization. In oscillatory shear mode, the storage modulus, loss modulus and tanδ has been measured. Calculating relaxation spectra through storage modulus, we found that the peak of segments’ relaxation heads to smaller relaxation time side. Also, the quantity of relaxation units increases as concentration increases. However, the relaxation time spectra are less affected by salinity. In flow shear mode, the relationship between shear rate and viscosity has been investigated. As concentration or salinity increases, the pseudoplastic of Xanthan solutions becomes more obvious. Furthermore, primary normal stress differences of Xanthan semi-diluted solutions lightly increase at first then sharply decrease as shear rate increases. This abnormal phenomenon may refer to wall slip.

Influence of the normal stresses to direction of shear flow in the rheology of concentrated kaolin suspensions

Due to the multiple applications that kaolin has, including the ceramic industry where it is used as a slurry with high solid concentrations, its rheological behavior is quite important in mixing, agitation and pumping stages. After characterizing the kaolin by applying XRD, XRF, SEM and particle size methods, a rheological evaluation was performed in a rotational rheometer which consisted of measuring the primary normal stress difference N1 by a method that delivered good reproducibility results. With N1 data it was obtained a kind of viscosity known as dilatant viscosity which was found only for those suspensions with a volumetric concentration higher than 0.13. This dilatant viscosity depends on the shear rate and the CV, and it was modeled with the power law structure having good coefficients of determination. Finally it was found that the dilatant viscosity with a CV= 0.2 is almost five times the value of the shear viscosity, thus demonstrating that for a better understanding of the deformation stage must be taken into account the normal stresses in shear flow happening in the kaolin concentrated suspension.

A Note on the Elasticity of Polymer Melts Described by Primary Normal Stress Difference (N1)

AbstractA method is proposed to characterize the elastic behavior of polyolefin melts, by calculating the first normal stress difference, N1 as a function of shear rate. The method is predicated on Cogswell's abrupt contraction flow analysis.1 These results are compared with experimental measurements and calculated values with a variety of methods, such as: i) Laun's rule2 for obtaining viscometric functions from linear viscoelastic data, ii) methods proposed by A. Khalik, et. al.3,4 and Wagner,5 for calculating N1 from viscosity data, and iii) N1 values calculated with the Johnson & Segalman constitutive equation.6 These comparisons yield acceptable agreement.

Wall effects on terminal falling velocity of spherical particles moving in a Carreau model fluid

AbstractExperimental verification of our previous numerical simulation of wall effects on the terminal falling velocity of spherical particles moving slowly along the axis of a cylindrical vessel filled with a Carreau model fluid is presented. Dependences of the wall correction factor F W on the sphere to tube ratio d/D and on the dimensionless Carreau model parameters m, Λ, and η r were obtained using a finite element method. Calculated data of the wall correction factor were compared with the results of our new falling sphere experiments. The experiments were carried out in six types of cylindrical Perspex columns (16 mm, 21 mm, 26 mm, 34 mm, 40 mm, and 90 mm in diameter) filled with aqueous solutions of polymers exhibiting different degrees of shear thinning and elasticity. Seventeen types of spherical particles (1–8 mm in diameter) made of glass, ceramics, steel, lead, and tungsten carbide were used for the drop tests. Measurements of the liquid flow curves, primary normal stress differences, oscillatory, creep and recovery, stress relaxation, and stress growth tests were carried out on the rheometer Haake MARS (Thermo Scientific). A good agreement between numerically and experimentally obtained F W data was found.

A numerical study of the effect of fibre stiffness on the rheology of sheared flexible fibre suspensions

A recently developed particle-level numerical method is used to simulate flexible fibre suspensions in Newtonian simple shear flow. In this method, the flow is computed on a fixed regular ‘lattice’ using the lattice Boltzmann method, where each solid particle, or fibre in this case, is mapped onto a Lagrangian frame moving continuously through the domain. The motion and orientation of the fibre are obtained from Newtonian dynamics equations. The effect of fibre stiffness on the rheology of flexible fibre suspensions is investigated and a relation for the relative viscosity is obtained. We show that fibre stiffness (bending ratio,BR) has a strong impact on rheology in the rangeBR< 3. The relative viscosity increases significantly asBRdecreases. These results show that the primary normal stress difference has a minimum value atBR~ 1. The primary normal stress difference for slightly deformable fibres reaches a minimum and increases significantly asBRdecreases below one. The results are explained based on Batchelor's relation for non-Brownian suspensions.

Steady‐state viscoelastic flow simulation of polymer melts in a rotational‐type rheometer

AbstractPolymer samples in the jigs begins to protrude when the heat is turned up when we measure the rheological characteristics of polymer melts using rotational‐type rheometers, such as parallel and cone‐and‐plate types. To clarify the effects of this protruding part on the obtained rheological data, we tried to evaluate the rotational‐type rheometer by a non‐isothermal viscoelastic flow simulation using the finite element method. The multiple mode Phan‐Thien Tanner (PTT) model was employed as the constitutive equation. As a result, the shear viscosity in the steady state increases with the size of the protruding part of the polymer melt specimen at the same shear rate in case with a parallel plate and a cone‐and‐plate type rheomters. In contrast, the deviation of the primary normal stress difference between the estimated value from the simulation results and the data from the PTT model is almost independent of the size of the protruding part with the cone‐and‐plate type rheometer. In addition, the deviations of the primary normal stress difference with a parallel plate rheometer increase with the protruding part size. However, these deviations are smaller than those of shear viscosity. POLYM. ENG. SCI., 2009. © 2009 Society of Plastics Engineers

Interplay Between Inertia and Elasticity in Film Casting

The influence of inertia and boundary conditions on the steady state and stability of isothermal film casting of viscoelastic fluids is examined using a Phan-Thien–Tanner rheological model. The elongational flow between the die exit and the take-up point is investigated. In general, the steady-state film tends to contract for low-inertia flow; this contraction, however, is significantly diminished by inertia. The polymeric normal stresses and primary normal stress difference decrease in the most of the air gap as inertia increases. In contrast, the stress and stress difference increase considerably near the take-up point due to a dramatic increase in the elongation rate. The linear stability analysis for two-dimensional disturbances is carried out. For a polymer with no degradation, and in the absence of inertia (Re=0), the analysis predicts critical draw ratios that form an envelope to an unstable region. This region of unstable conditions reduces as inertia increases. Two branches of neutral stability curve are observed for higher-inertia flow as opposed to a single curve for Re=0. The unstable region expands as α increases, where α is a measure of polymer degradation. When α becomes sufficiently large, the elasticity tends to destabilize the flow. It is also found that boundary conditions have an important influence on the steady-state profiles and stability region, particularly for high-elasticity fluids.

Rheological behavior of fiber-filled polymers under large amplitude oscillatory shear flow

Small and large amplitude oscillatory shear measurements (SAOS and LAOS) were used to investigate the rheological behavior of short glass fibers suspended in polybutene and molten polypropylene. Raw torque and normal force signals obtained from a strain-controlled instrument (ARES rheometer) were digitized using an analog to digital converter (ADC) card to allow more precise data analysis. The fiber concentration did not affect the torque signal in the SAOS mode, except for its magnitude, whereas the normal force signal was too low to be measurable. With increasing strain amplitude, the magnitude of the torque became a function of time. Depending on the applied frequency and strain rate, the stress in the filled polybutene increased with time, whereas for reinforced polypropylene (viscoelastic matrix), the behavior was opposite, i.e. the stress decreased with time. These effects were more pronounced at high fiber content. In addition the primary normal stress differences were no longer negligible at large deformation amplitude and exhibited a non-sinusoidal periodic response. Fast Fourier transform (FFT) analysis was performed and the resulting spectra, along with Lissajous figures of the shear stress and the primary normal stress differences, are explained in terms of fiber orientation. The experimental results for the suspensions in polybutene are well predicted by the Folgar-Tucker-Lipscomb (FTL) model.

Primary and secondary normal stress differences of a magnetorheological fluid (MRF) up to magnetic flux densities of 1 T

First and second normal stress differences of a 50 vol.% magnetorheological fluid (MRF) are investigated by using a commercial plate–plate magneto-rheometer (Anton Paar GmbH) with plate–plate and cone–plate geometry. The manufacturer modified the instrument to achieve higher normal force (60 N) and torque (295 mNm) capacity. An additional modification by us allows an online determination of the true magnetic flux density B in the MRF by means of a Hall probe. FEM Maxwell 2D simulations quantitatively verify the Hall probe results and give detailed insight into the radial flux density profile within the MRF sample. Without shear, the static normal force F N for plate–plate increases as a power law: F N ∝ B 2.4. A similar magnitude is found for cone–plate geometry, in contrast to the expectation. For steady shear at 10 s −1, the plate–plate normal force built-up limits the experiments at high flux densities rather than the torque generated. The normal forces increase linearly with the shear stress at high flux density. The first normal stress difference N 1 is positive and about five times larger than the shear stress. The second normal stress difference N 2 is also positive. The experimentally derived N 2/ N 1 ratio of 1/4 distinctly deviates from theoretical predictions ( N 2/ N 1 = −1) for a semi-dilute MRF. Improvements of the radial flux density profiles are required to verify the normal stress difference ratio and to support the conjecture that the positive but small N 2 is a consequence of the densely packed MRF, which does not allow to create extended chain-like structures. As shown in the appendix, the experimentally determined N 2/ N 1 ratio is favorable to stabilize concentricity in concentric cylinder arrangements, relevant for the MRF application in clutches.

A Model for Primary Normal Stress Difference of Dynamic Extrusion of a Polymer Melt through a Capillary

The dynamic extrusion process of a polymer melt through a capillary under a superimposed vibration has been was researched deeply, and a mathematical model for the primary normal stress difference of a polymer melt under the vibration force field was set up. Accordingly, the calculation steps of above primary normal stress difference were established based on a rheological measuring equipment which was disigned by the authors. The primary normal stress difference of polymer melt under a vibration force field can be calculated by measuring the instantaneous data of capillary die swell, capillary entry pressure, capillary volume flux, and their phase difference under vibration with different frequencies and amplitudes.

Elastic Behavior Analysis of Polymer Melt Extruding through Capillary with An Additional Sinusoidal Vibration

A novel rheological measuring apparatus was designed by the authors, which introduced an additional sinusoidal vibration in parallel on the extruding direction of polymer melt and was called as Constant Velocity Dynamic Capillary Rheometer in this article. After introducing different melt extrusion speeds and different intensities of vibration, the rheological measuring experiments for a LDPE melt were done respectively by using above-mentioned apparatus, and a mathematical model for the primary normal stress difference of polymer melt under the action of vibration was set up accordingly. Then, the change laws of die swell ratio and those of the primary normal stress difference under different frequencies and amplitudes of vibration were obtained. Compared with the stable state extrusion, after introducing an additional sinusoidal vibration, the die swell ratio and the primary normal stress difference of polymer melt are all decreased obviously.

Inception of shear flow for ferromagnetic dispersions

Rheological data are presented for shear flows of concentrated dispersions of acicular ferromagnetic particles; such dispersions are widely used in the manufacture of data storage materials. In steady shear flow these dispersions are strongly shear thinning. Upon the inception of shear flow, they show stress overshoots for both the shear stress and the primary normal stress difference. The data can be rationalized in terms of a temporary junction network, resulting from the strong magnetic interactions between the particles, if the network consists of relatively weak magnetic links connecting strongly bound flocs of particles. Application of a shear deformation results in disruption of the initially sample spanning network. The overshoot in the shear stress occurs when the flocs become arranged in the shear planes. The overshoot in the primary normal stress difference occurs as a result of deformation of the strongly bound flocs.

Dynamic Rheological Properties of Concentrated Chitosan Soltions

AbstractA detailed analysis of the dynamic flow properties of chitosan in solution at different temperatures (25 - 45°C), chitosan concentration (0.5% - 2.0%), solvent type (acetic, lactic, and hydrochloric acid), and ionic strength (0 and 0.2M NaCl) has been undertaken. The storage modulus, G’, loss modulus, G’’ and complex viscosity, η* have been determined over a wide range of frequencies and the results are presented using master curves. For the conditions studied, at low frequencies chitosan solutions show a constant complex viscosity which decreases as frequency increases. Likewise, storage modulus, G’ and loss modulus, G’’ increase as frequency increases with G’’ being always greater than G’ (η’ > η’’) indicating that viscous effects are more important than elastic effects. For modelling the oscillatory-shear results we used the generalized Maxwell model. Two empirical equations were used to correlate the data: Cox-Merz rule for viscosity and Laun's rule for primary normal stress difference. Both relations were found to represent our data for the experimental conditions studied.

Separating the effects of sparse long-chain branching on rheology from those due to molecular weight in polyethylenes

The effects of sparse (<1 branch per chain) long-chain branching (LCB), molecular weight (MW), and molecular weight distribution on the shear rheological properties of commercial polyethylenes are often convoluted. In this paper a method for separating the effects of sparse LCB in metallocene-catalyzed polyethylenes (mPE) from those of molecular weight and its distribution based on time–molecular weight superposition is proposed. Four metallocene polyethylenes with degrees of long-chain branching [i.e., M of the arm (Ma) is greater than that for the onset of entanglements, Mc] as determined from dilute solution measurements ranging from zero (linear) to 0.79 LCB/104 CH2, along with a conventional Ziegler–Natta polymerized linear low-density polyethylene (LDPE), and a tubular free-radical polymerized LDPE are investigated. In general, it is observed that sparse LCB (for levels < 1.0 LCB/104 CH2) increases the zero shear viscosity, η0 (e.g., by a factor of 7) and decreases, but even to a greater degree, the critical shear rate (γ̇c) for the onset of shear thinning (e.g., by a factor of 100). The breadth of the molecular weight distribution just affects γ̇c but not η0 for the range of data used in this study. Furthermore, the dynamic storage modulus G′ shows similar enhancement at low frequencies as viscosity does, while the primary normal stress difference coefficient, Ψ1,0, exhibits a greater dependence on long-chain branching than that predicted from the zero-shear viscosity enhancement. The results for the mPEs are consistent with recent molecular theories for randomly branched molecules in that it is the spacing between branch points and not the number of branches at a point that is important. Furthermore, the results are consistent with the idea that the branches are located on the longest chains, and hence, have the greatest effects on the longest relaxation modes.

Swirling flow of viscoelastic fluids. Part 2. Elastic effects

A torsionally driven cavity has been used to examine the influence of elasticity on the swirling flow of constant-viscosity elastic liquids (Boger fluids). A wealth of phenomena is observed as the degree of inertia, elasticity and viscous forces are varied by using a range of low- to high-viscosity flexible polyacrylamide Boger fluids and a semi-rigid xanthan gum Boger fluid. As the inertia is decreased and elasticity increased by using polyacrylamide Boger fluids, the circulation rates for a ‘Newtonian-like’ secondary flow decreases until flow reversal occurs owing to the increasing magnitude of the primary normal stress difference. For each polyacrylamide fluid, the flow becomes highly unstable at a critical combination of Reynolds number and Weissenberg number resulting in a new time-dependent elastic instability. Each fluid is characterized by a dimensionless elasticity number and a correlation with Reynolds number is found for the occurrence of the instability. In the elasticity dominated flow of the polyacrylamide Boger fluids, the instability disrupts the flow dramatically and causes an increase in the peak axial velocity along the central axis by as much as 400%. In this case, the core vortex spirals with the primary motion of fluid and is observed in some cases at Reynolds numbers much less than unity. Elastic ‘reverse’ flow is observed for the xanthan gum Boger fluid at high Weissenberg number. As the Weissenberg number decreases, and Reynolds number increases, counter-rotating vortices flowing in the inertial direction form on the rotating lid. The peak axial velocity decreases for the xanthan gum Boger fluid with decreasing Weissenberg number. In addition, several constitutive models are used to describe accurately the rheological properties of the fluids used in this work in shear and extensional flow. This experimental investigation of a complex three-dimensional flow using well-characterized fluids provides the information necessary for the validation of non-Newtonian constitutive models through numerical analysis of the torsionally driven cavity flow.

Rheological properties of the mucilage gum ( Opuntia ficus indica)

The rheological properties of aqueous solutions of the mucilage isolated from Opuntia ficus indica have been examined. Steady-shear viscosities in a range of shear rate from 1 to 300 s −1 were observed as a function of mucilage concentration, temperature, pH and ionic strength. A non-Newtonian shear-thinning behavior was observed. A quite important increase in pseudoplasticity (or shear thinning) due to increase in the mucilage concentration, in the range from 1 to 10% (w/w) was showed. In the range of shear rates examined, viscosity values of a mucilage solution at 10% are similar to those shown by a xanthan 3% aqueous solution, both at 25°C. The Ostwald–de Waele or power law model successfully correlated the viscosity–shear rate data. The viscosity was dependent on ionic strength, as in the case of polyelectrolytes, and decreased with ionic strength. This behavior was more pronounced when using divalent cations. A marked dependence of viscosity on pH was also observed, as pH was increased from acidic to alkaline conditions, the viscosity increased. Elastic steady-shear properties were evaluated as the primary normal stress difference. Mucilage aqueous solutions showed high elastic properties, similar to high elastic synthetic polymers like polyisobutylene. At constant shear stress, the normal stresses decreased as the mucilage concentration was increased. The mechanical spectra in the linear viscoelasticity range were observed at a frequency range from 1 to 50 rad/s, in the temperature range from 5 to 35°C. At low mucilage concentration (<3%) the behavior was typical of a dilute solution. However, with increasing concentration, the solution tended toward a weak gel behavior. A cross-over point between G′ and G″ was observed at low frequencies (≈10 rad/s) and 35°C, for a mucilage concentration of 5%, suggesting a change in the conformation of the molecule from entanglements to a random coil configuration. Results from steady-shear measurements also suggested a random coil configuration.

An algebraic preclosure theory for the Reynolds stress

An algebraic preclosure theory for the Reynolds stress 〈u′u′〉 is developed based on a smoothing approximation which compares the space–time relaxation of a convective-diffusive Green’s function with the space–time relaxation of turbulent correlations. The formal preclosure theory relates the Reynolds stress to three distinct statistical properties of the flow: (1) a relaxation time τR associated with the temporal structure of the turbulence; (2) the spatial gradient of the mean field; and, (3) a prestress correlation related to fluctuations in the instantaneous Reynolds stress and the pressure field. Closure occurs by using an isotropic model for the prestress. For simple shear flows, the theory predicts the existence of a nonzero primary normal stress difference and an eddy viscosity coefficient which depends on the temporal relaxation of the turbulent structure and a characteristic time scale associated with the mean field. The asymptotic state of homogeneously sheared turbulence shows that τRS∼1, where S represents the mean shear rate. The Reynolds stress model and a set of recalibrated k−ε transport equations predict that the relaxation of homogeneously sheared turbulence to an asymptotic state requires development distances larger than 20 ×〈uz〉(0)/S, a theoretical result consistent with experimental observations.