Rheological Constitutive Equation Research Articles

Electrorheology of a mesoporous silica having conducting polypyrrole inside expanded pores

A pore-expanded mesoporous silica (MCM-41-E) having particles of conducting polypyrrole (PPY) included inside its uniformly aligned one-dimensional channels was prepared, and their electrorheological (ER) characteristics when dispersed in a silicone oil were investigated. Both N 2-adsorption isotherm and SEM images were employed to confirm the successful formation of PPY/MCM-41-E nanocomposite. The XRD also provided an additional information on the properties of the synthesized nanocomposite material. The nanocomposite showed typical ER characteristics. Shear stress curve was more accurately described by using the rheological constitutive equation (CCJ model) we proposed compared with the Bingham fluid model. Yield stress which increases with applied electric field strength was analyzed based on the universal yield stress equation we suggest as a function of applied electric field strength. It was found that the universal yield stress equation collapses these data onto a single curve well in the case of PPY/MCM-41-E nanocomposite based ER fluid.

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.

Analytical and rheological investigations into selected unifloral German honey

The physicochemical and rheological properties parameters of five German unifloral honey: false acacia, heather, sunflower, lime and rape (39 samples) were analysed to demonstrate the relation between botanical origin and material properties. The validity of the rheological constitutive equations by Herschel-Bulkley, Ostwald-De Waele and Newton was confirmed by varying temperature and crystalline state. Highly structured (crystalline) honey exhibits temperature dependent with all rheological anomalies (shear thinning, yield point and rheodynamic behaviour) and can described an average effective viscosity. Modern oscillation measurements were used to determine the crystallization behaviour of honey. The results showed that crystallization of honey is depended on botanic origin, temperature and storage time.

Modeling Sorption Kinetics of Carbon Dioxide in Glassy Polymeric Films Using the Nonequilibrium Thermodynamics Approach

The nonequilibrium thermodynamics of glassy polymers (NET-GP) approach (Macromolecules 2005, 38, 10299.) has been applied to the development of a one-dimensional transport model aimed at describing the kinetics of sorption and dilation of polymeric films in supercritical carbon dioxide. The NET-GP model was combined with a simple rheological constitutive equation to build a sorption−diffusion−relaxation model able to describe mass uptake and swelling kinetics of polymeric films in contact with carbon dioxide over a wide range of pressures and temperatures. The model calculations are compared with data on mass sorption kinetics for CO2 in supported glassy poly(methyl methacrylate) (PMMA) films, measured in a high-pressure quartz crystal microbalance (QCM).

Material functions of liquid n-hexadecane under steady shear via nonequilibrium molecular dynamics simulations: Temperature, pressure, and density effects

Computer experiments of rheology regarding the effects of temperature (T), pressure (P), and density (rho) on steady shear flow material functions, which include viscosity (eta) and first and second normal stress coefficients (psi(1) and psi(2)) depending on shear rate (gamma), have been conducted via nonequilibrium molecular dynamics simulations for liquid n-hexadecane. Straightforwardly, using both characteristic values of a zero-shear-rate viscosity and critical shear rate, eta-gamma flow curves are well normalized to achieve the temperature-, pressure-, and density-invariant master curves, which can be formulary described by the Carreau-Yasuda rheological constitutive equation. Variations in the rate of shear thinning, obviously exhibiting in eta-gamma, psi(1)-gamma, and -psi(2)-gamma relationships, under different T, P, and rho values, are concretely revealed through the power-law model's exponent. More importantly, at low shear rates, the fluid explicitly possesses Newtonian fluidic characteristics according to both manifestations; first and second normal stress differences decay to near zero, while nonequilibrium states are close to equilibrium ones. Significantly, the tendency to vary of the degree of shear thinning in rheology is qualitatively contrary to that of shear dilatancy in thermodynamics. In addition, a convergent transition point is evidently observed in the -psi(2)/psi(1)-gamma curves undergoing dramatic variations, which should be associated with shear dilatancy, as addressed analytically.

Modelling PTFE paste extrusion: The effect of an objective flow type parameter

In a recent article [P.D. Patil, J.J. Feng, S.G. Hatzikiriakos, Constitutive modelling and flow simulation of polytetrafluoroethylene paste extrusion, J. Non-Newtonian Fluid Mech. 139 (2006), 44–53], a rheological constitutive equation was proposed to model the incompressible flow of poly-tetra-fluoro-ethylene (PTFE) paste and its structural development (fibrillation between the PTFE particles) during flow. The constitutive model was a combination of shear-thinning and shear-thickening viscosity terms, depending on a structural parameter, ξ, which obeyed a convective-transport equation. In the latter, a non-objective flow type parameter, ψ, was employed, which depended on the magnitudes of the rate-of-strain and vorticity tensors. In the present work, an objective flow type parameter is used instead [R.G. Larson, Flows of constant stretch history for polymeric materials with power-law distributions of relaxation times, Rheol. Acta 24 (1985), 443–449]. Despite its complexity and the severe slip at the wall, good convergence has been obtained even for very high apparent shear rates. New results are shown as contours of the ξ-parameter in the flow field as well as contours of the new objective flow type parameter, ψ*, and compared with previous results. It is shown that the corrected model predicts higher structural levels (more fibrils) in PTFE paste flow through extrusion dies and that the new results are more consistent with experimental observations.

The state of the art in the rheology of polymers: Achievements and challenges

The state of the art in the rheology of polymer fluids (polymer solutions and melts) and filled composites is reviewed. This review includes two parts: analysis of the basic principles for the construction of rheological constitutive equations in terms of the continuum mechanics and finding correlations between the rheological characteristics and molecular structure of polymers on the basis of molecular models. Possible approaches to the formulation of constitutive equations are discussed. Special attention is focused on the correct selection of the form of the elastic potential for rubbery deformations induced under the flow of polymer fluids. The use of a power-law potential leads to the best results. To gain unequivocal results and minimize the number of free constants, viscoelastic characteristics of polymer fluids should be described in terms of a continuous relaxation spectrum as a power-law function limited by the maximum relaxation time. To solve the boundary problems by the selected constitutive equation, analysis of the dynamic stability is required, because the combination of viscosity and elasticity controls the limits of flow upon shear and tensile. Deformation can also lead to changes in the phase state of a polymer system. Furthermore, correct formulation of the boundary conditions is necessary because, in many cases, polymer fluids and, in particular, filled materials tend to efficient slip along walls. The existing molecular models adequately describe the characteristics of monodisperse polymers; however, on passing to polydisperse polymers, the additional use of semiempirical approaches is required. The modern level of experimental studies allows test measurements over a wide range of deformation rates, frequencies, and temperatures. However, in this field, the mainstream tendency in experimental studies is concerned with hybrid methods, which combine direct rheological measurements with optical observations of local structure and its evolution in the material. In this case, various physical principles of measurements are applied. In recent years, much interest has been focused on studying polymer compositions containing nanosized fillers, which are able to produce their structures in melt.

Prediction and experimental verification of bubble and processing characteristics in blown‐film extrusion

AbstractBlown‐film modeling is useful to the flexible packaging industry for predicting process and bubble characteristics, such as freeze line height (FLH), bubble diameter, and film thickness. The use of a suitable rheological equation to describe material properties is critical in simulating the blown‐film process. In this article, we present an improved rheological constitutive equation, which incorporates more realistic parameters of stress and deformation properties of the materials by combining the Hookean model with the Phan‐Thien Tanner (PTT) model. The proposed PTT–Hookean model is aimed at enhancing the viscoelastic behavior of the melt during biaxial stretching in the blown‐film extrusion. Predictions of the blown‐film bubble characteristics and FLH obtained with the PTT–Hookean model agreed well with the experimental data of this study and previous studies with different materials and different die geometries. The justification for combining the Hookean model with the PTT model in the blown‐film process is also reported here. © 2008 Wiley Periodicals, Inc. J Appl Polym Sci, 2009

An improved constitutive model for concentrated suspensions accounting for shear-induced particle migration rate dependence on particle radius

Several rheological constitutive equations for the modeling of dense suspensions in nonlinear shear flows have been developed over the last three decades. Although these models have been able to predict the correct steady-state solid-phase concentration profile, none have been able to follow the transient experimentally measured concentration profile over a range of suspended particle radii with a consistent set of diffusion coefficients. In this research, two improvements are made to the diffusive-flux model, namely, modeling the diffusion coefficients as linear functions of the so-called nonlinearity parameter and adding slip boundary conditions at the wall. A particle-level explanation for the linear dependence of the diffusion coefficients on the nonlinearity parameter is provided. With these two improvements, it is shown that the modified diffusive flux model can accurately predict the transient solid-phase concentration profile in a Couette device over a wide range of particle radii.

Steady flow simulations of compressible PTFE paste extrusion under severe wall slip

In a recent article [P.D. Patil, J.J. Feng, S.G. Hatzikiriakos, Constitutive modelling and flow simulation of polytetrafluoroethylene paste extrusion, J. Non-Newtonian Fluid Mech., 139 (2006) 44–53], a rheological constitutive equation was proposed to model the flow of polytetrafluoroethylene (PTFE) paste. Steady-state flow simulations were carried out, governed by the conservation equations of mass and momentum under isothermal, incompressible conditions, and taking into account a linear slip law at the wall due to the presence of lubricant in the paste. The constitutive model was a combination of shear-thinning and shear-thickening viscosity terms, depending on a structural parameter, ξ, which obeyed a convective-transport equation. The present work re-examines the flow problem, taking into account the significant compressibility of the paste and implementing the slip law based on the consistent normal-to-the-surface unit vector. New results are shown as contours of the ξ-parameter in the flow field as well as contours of the flow type parameter, ψ, used in the model. They offer a better understanding of the flow behaviour of PTFE in flow through extrusion dies.

大深度立坑の時間依存性挙動および破壊挙動の有限要素解析

Shaft plays an important role in accessing or ventilating to deep underground, so it is necessary to assess its long-term stability. But little knowledge about the long-term behavior of shaft has been obtained from in-situ measurement, theoretical study and numerical simulation. In this study, time-dependent behavior and failure process of deep shaft was simulated with the aid of non-linear rheological constitutive equation of rock. This simulation focused on the effect of gravity and the deformation behavior parallel to the direction of excavation, which has hardly been examined with horizontal roadway or tunnel.Simulation with the model of 1000m deep shaft was carried out, and the effect of rock properties and rock stress conditions was examined. It was found that time-dependency, ductility, Poisson's ratio of rock mass and the ratio between horizontal and vertical rock stresses had an effect on failure process around the bottom of shaft. In addition, simulation with vertically and horizontally loaded disc-shaped model was carried out to investigate the effect of vertical rock stress and gravity in detail. It was found that the vertical deformation of shaft wall was much smaller than the radial one just before failure, but that the vertical rock stress and gravity had a large effect on failure process and long-term stability of shaft.

Steady laminar flow of non-Newtonian bubbly suspensions in pipes

The steady laminar flow of concentrated bubbly suspensions in pipes is modeled using the Pal–Oldroyd rheological constitutive equation for bubbly suspensions [R. Pal, Rheological constitutive equation for bubbly suspensions, Ind. Eng. Chem. Res. 43 (2004) 5372–5379]. Equations are developed to predict the following: (a) wall shear stress ( τ w) versus apparent wall-shear rate (8 V/ D, where V is the average velocity and D is the pipe diameter); (b) velocity profile; and (c) friction factor. The τ w versus 8 V/ D plot (log–log scale) is linear with a slope of unity at low and high values of wall stress ( τ w/( σ/ R) < 0.4 and τ w/( σ/ R) > 1.0, where σ is the interfacial tension and R is the bubble radius). At intermediate values of wall stress (0.4 < τ w/( σ/ R) < 1.0), the slope of τ w versus 8 V/ D plot on a log–log scale is less than unity. The velocity profile is parabolic at low and high values of wall stress; at intermediate values of τ w, the velocity profile is significantly flatter as compared with the parabolic profile. At low values of Reynolds number ( N Re ), the friction factor ( f) follows the usual equation for Newtonian fluids ( f = 16/ N Re ) provided that the Reynolds number is evaluated using the zero shear-rate viscosity of bubbly suspension. At high values of Reynolds number too, the friction factor follows the usual relationship ( f = 16/ N Re ) provided that the Reynolds number is evaluated using the high shear-rate limiting viscosity of bubbly suspension. At intermediate values of Reynolds number, the friction factor deviates from the standard relationship, f = 16/ N Re .

Using multiple‐mode models for fitting and predicting the rheological properties of polymeric melts. II. Single and double step‐strain flows

AbstractSeveral classes of multiple‐mode rheological constitutive equations are examined for predicting the viscoelastic flow properties of a typical polymer melt in single and double step‐strain flows. The phenomenological parameters appearing in these models have been obtained by the fitting of experimental data taken in small‐amplitude oscillatory shear and steady shear flows. The performance of the models for predicting the experimental data in the step‐strain experiments is examined in detail. Specifically, we examine whether or not mode coupling is necessary to describe the experimental behavior under step‐strain flows. Furthermore, it is demonstrated that the reversing double step‐strain experiment is a very powerful tool for testing viscoelastic constitutive equations. © 2007 Wiley Periodicals, Inc. J Appl Polym Sci 2007

A simple constitutive equation for immiscible blends

A simple rheological constitutive equation for immiscible blend is suggested in this work. The model is based on the ellipsoidal description of droplets. Droplet deformation is described by the ellipsoidal model, and droplet breakup∕coalescence is considered to affect the droplet size only. The discrete breakup∕coalescence process is approximated by a continuous dynamic equation. A simple rescaling method is suggested to integrate the volume-preserving ellipsoidal deformation model and the droplet size evolution dynamics. The predictions of the model are compared with experiments in start up of shear and shear rate sweep modes. The agreement is quite satisfactory on transient and steady rheological properties, as well as the morphology of droplets.

Nonlinear Branch-Point Dynamics of Multiarm Polystyrene

Two branched polystyrene melts with narrow molar mass distribution have been synthesized: a multiarm An−C−C−An pom-pom polystyrene and an An−C asymmetric star polystyrene where n is the number of arms. The pom-pom and the asymmetric star have molar masses of Mw = 300 kg/mol and Mw = 275 kg/mol, respectively. The pom-pom was estimated to have 2.5 arms on average, while the estimate is 3.3 for the asymmetric star. The molar mass of each arm is about 27 kg/mol. The melts were characterized in the linear viscoelastic regime and by elongational rheometry in the nonlinear regime. For asymmetric star polystyrene, the measured transient elongational viscosity is not consistent with a rheological constitutive equation that is separable in time and strain. Contrary to this situation, however, for pom-pom polystyrene, the transient elongational viscosity may be described by a time−strain separable constitutive equation for elongation rates larger than the inverse reptation time. Up to a Hencky strains of 2.6, the corresponding stress tensor component for the pom-pom is closely approximated by a model that assumes the arms to be fully relaxed while the cross-bar is part of an unrelaxed entanglement network model. At Hencky strains above 2.6 a saturation of stress occurs that we interpret as withdrawal of the arms into the cross-bar tube. The observed strain associated with arm withdrawal is significantly larger than that predicted from an equilibrium force balance on the branch points while it corresponds well with an estimate of the maximum stretchability of the cross-bar. At the highest elongation rate investigated, the transient elongational viscosity for pom-pom went through a reproducible maximum as a function of time.

Arthur B. Metzner

Arthur B. Metzner

Rheology of omphacite at high temperature and pressure and significance of its lattice preferred orientations

We have investigated the rheology of hot-pressed polycrystalline omphacite (Di 58Jd 42, space group C2/c) at strain rates of 10 − 4 –10 − 5 /s, temperatures of 1300–1500 K and a pressure of 3 GPa, using a 5 GPa Griggs-type deformation apparatus. The rheological constitutive equation of omphacite is determined with a stress exponent of 3.5 ± 0.2 and an activation energy of 310 ± 50 kJ/mol. Our study shows that: (1) the creep strength of omphacite falls between those of diopside and jadeite; (2) experimental omphacite microfabrics are indistinguishable from natural ones: S-type fabric and its mesoscopic foliation develop under conditions of axially symmetric shortening, whereas L-type fabric and its mesoscopic lineation develop under conditions of general strain, with L parallel to the greatest elongation; (3) as in naturally deformed omphacite, the deformation substructure is characterized by the dominant slip systems {110}1/2〈110〉, {110}[001] and (100)[001]. We conclude that both S- and L-type omphacite fabrics are produced by these slip systems; fabric differences arise solely from variation of the geometry and orientation of the finite strain ellipsoid. We also confirm the previous suggestion that Na-bearing pyroxenes are significantly weaker than Na-free pyroxenes.

Numerical modelling of the effect of operating parameters in the plastic blown film process

The blown film process with polymer melts is modelled using non-isothermal viscoelastic rheological constitutive equations that are suitable for polyolefins. The model correlates the operating parameters such as mass flows, extruder temperature, tensile axial force and take-up force on processes such as bubble geometry and bubble temperature profile. Unlike Luo and Tanner [ Poly. Eng. Sci. , Vol. 25, 1985] who used the Maxwell constitutive equation, this study considers the viscoelastic Kelvin model that avoids any assumption regarding stress at the die exit. Like Luo and Tanner, the model uses shooting techniques to match the initial and boundary conditions from the freeze line and the die. The pressure drop and the take-up force are estimated as parameters that are optimized with the boundary conditions using the Nelder--Mead optimization method with Matlab. Effects of varying elasticity parameters and heat transfer coefficients on the bubble geometry, the pressure drop and take-up force are investigated.

Shear stress analysis of a semiconducting polymer based electrorheological fluid system

The electrorheological characteristics of a fluid system with semiconducting poly(naphthalene quinone) (PNQR) particles, synthesized from a Friedel–Craft acylation, were investigated via a rotational rheometer equipped with a high voltage generator. The flow curves of these ER fluids under several applied electric field strengths and particle concentrations were constructed and their flow characteristics were examined via three different rheological constitutive equations of Bingham model, De Kee–Turcotte model and our proposed model. Our proposed equation was found to fit the data very accurately.

Using multiple‐mode models for fitting and predicting rheological properties of polymeric melts

AbstractSeveral classes of multiple‐mode rheological constitutive equations are tested for fitting and predicting viscoelastic flow properties of a typical low‐density polyethylene melt. An optimization procedure is used to fit the phenomenological parameters of each model under consideration to experimental data taken in small‐amplitude oscillatory shear flow and steady shear flow. These parameter values are then used to generate predictions for transient shear and uniaxial elongational flow experiments, and the predictions are then compared to experimental data. Model successes and failures are discussed, and the outlook for using rheological equations in real design processes is addressed. © 2005 Wiley Periodicals, Inc. J Appl Polym Sci 99: 405–423, 2006