Normal Stress Coefficient Research Articles

Flow of granular materials with slip boundary condition: A continuum–kinetic theory approach

We study the steady fully developed flow of granular materials between two horizontal flat plates, subject to slip at the walls. The constitutive model for the stress tensor is based on ideas in continuum mechanics and kinetic theory. The constitutive equation used in our study is a model proposed by Rajagopal et al. (1994) [24], and the material properties such as viscosity and the normal stress coefficients are derived using the kinetic theory approximation proposed by Boyle and Massoudi (1990) [2] which includes the effect of the gradient of volume fraction. The slip boundary condition is based on the particle dynamics simulation results of Rosato and Kim (1994) [30]. The governing equations are non-dimensionalized, and the resulting system of non-linear differential equations is solved numerically. The results for the velocity profiles and the volume fraction profiles are presented.

Normal-stress coefficients and rod climbing in colloidal dispersions

We calculate tractable microscopic expressions for the low-shear normal-stress coefficients of colloidal dispersions. Although restricted to the low rate regime, the presented formulas are valid for all volume fractions below the glass transition and for any interaction potential. Numerical results are presented for a system of colloids interacting via a hard-core attractive Yukawa potential, for which we explore the interplay between attraction strength and volume fraction. We show that the normal-stress coefficients exhibit nontrivial features close to the critical point and at high volume fractions in the vicinity of the reentrant glass transition. Finally, we exploit our formulas to make predictions about rod-climbing effects in attractive colloidal dispersions.

Dynamical and Rheological Properties of Ultrasoft Colloids under Shear Flow

The dynamical and rheological properties of ultrasoft colloids and star polymers are investigated in dilute and semidilute solutions under linear shear flow. We apply a hybrid mesoscale hydrodynamics simulation approach, which combines molecular dynamics simulations for the solute with the multiparticle collision dynamics approach for the solvent. We investigate the effect of concentration on relaxation, diffusion, and the rheological properties of the star polymers. We find that the relaxation time of a star-polymer arm is a universal function of a concentration-dependent Weissenberg number. The center-of-mass mean square displacements of the star polymers are anisotropic under shear flow. At high shear rate, we find shear-induced enhanced center-of-mass displacements along the vorticity and gradient directions. Moreover, we determine the shear viscosity and normal stress coefficients as a function of concentration. The shear viscosity exhibits shear thinning with a weak functionality dependence.

Concentration Dependence of Nonlinear Rheological Properties of Hydrophobically Modified Ethoxylated Urethane Aqueous Solutions

For aqueous solutions of a representative telechelic polymer, hydrophobically modified ethoxylated urethane (HEUR), a recent transient network model suggested that sparse and dense HEUR networks are formed at low and high HEUR concentrations c thereby exhibiting different rheological features [Uneyama, T.; Suzuki, S.; Watanabe, H. Phys. Rev. E2012, 86, 031802]. In this study, we examined those differences for nonlinear rheological behavior of HEUR solutions (with MHEUR = 4.6 × 104 and c = 1–10 wt %) under steady shear at 25 °C to test the model. For rather dilute solutions (c ≤ 3 wt %), the steady state viscosity η exhibited crossover from the linear (zero-shear) behavior to thickening and further to thinning with increasing the shear rate γ̇, whereas the first normal stress coefficient Ψ1 remained in the linear regime up to intermediate γ̇ (where η exhibited the thickening) and then decreased with γ̇ in the thinning regime of η. In contrast, for concentrated solutions (c ≥ 4 wt %), η exhibited no thickening and a direct linear-to-thinning crossover was observed for both η and Ψ1. The critical HEUR concentration for the disappearance of the thickening of η, c* ≅ 4 wt %, was in agreement with that noted for the linear viscoelastic data, suggesting that the HEUR network structure changed from the sparse state to the dense state at c*, as considered in the above transient network model: At low c < c*, linear sequences of HEUR chains (superbridges) connected at the HEUR micellar cores would have served as effective strands to form the sparse network. Those superbridge strands dissociate and reassociate under steady shear, and the orientation of the reassociated strands should be quite sensitive to anisotropy of the spatial distribution of the cores (intra-strand dissociation sites) so that the thickening of η at intermediate γ̇ is accompanied by linear Ψ1, as deduced from the model. In contrast, at large c > c*, the dense network mainly sustained by individual HEUR chains would have been formed. For this case, the anisotropy of core distribution should less significantly affect the orientation of the created strands, which possibly erased the thickening of η at intermediate γ̇, as suggested by the model. Thus, the changes of the nonlinear behavior with c observed in this study were in harmony with the expectation from the model, lending qualitative support to the structures and dynamics of the networks considered in the model. In addition, the flow visualization using tracer particles confirmed that the flow was uniform up to the onset of thinning of η and Ψ1 (which is again in harmony with the model) and that the flow was destabilized to form shear bands in the thinning regime as similar to the behavior of a wide variety of softmatters.

Modeling Die Swell of Second-Order Fluids Using Smoothed Particle Hydrodynamics

This work presents the development of a weakly compressible smoothed particle hydrodynamics (WCSPH) model for simulating two-dimensional transient viscoelastic free surface flow which has extensive applications in polymer processing industries. As an illustration for the capability of the model, the extrudate or die swell behaviors of second-order and Olyroyd-B polymeric fluids are studied. A systematic study has been carried out to compare constitutive models for second-order fluids available in literature in terms of their ability to capture the physics behind the swelling phenomenon. The effects of various process and rheological parameters on the die swell such as the extrusion velocity, normal stress coefficients, and Reynolds and Deborah numbers have also been investigated. The models developed here can predict both swelling and contraction of the extrudate successfully. The die swell of a second-order fluid was solved for a wide range of Deborah numbers and for two different Reynolds numbers. The numerical approach was validated through the solution of fully developed Newtonian and non-Newtonian viscoelastic flows in a two-dimensional channel as well as modeling the die swell of a Newtonian fluid. The results of these three benchmark problems were compared with analytic solutions and numerical results in literature when pertinent, and good agreements were obtained.

Density dependence of the stress relaxation function of a simple fluid

We present accurate molecular dynamics calculations of the shear stress relaxation modulus of a simple atomic fluid over a wide range of densities. The high accuracy of the data enables us to study changes in the functional form of the shear relaxation modulus, and the properties that are derived from it, as the density is increased from the ideal gas limit to the upper limit of the fluid range. We show that the shear relaxation modulus of a dilute atomic fluid can be accurately described with a simple functional form consisting of a Gaussian plus a single exponential, whereas the dense fluid exhibits a more complicated relaxation function. The infinite-frequency shear modulus, the zero-shear viscosity, and the zero-shear first normal stress coefficient are calculated from the stress autocorrelation function. The ratio of the first normal stress coefficient to the viscosity is used to calculate the viscoelastic relaxation time. While the viscosity and the infinite-frequency shear modulus both increase monotonically with increasing density, the first normal stress coefficient and the viscoelastic relaxation time both decrease to a minimum at intermediate densities before increasing again. Our results for the viscosity and the first normal stress coefficient in the low-density limit both agree well with the predictions of kinetic theory.

Multiparticle collision dynamics simulations of viscoelastic fluids: Shear-thinning Gaussian dumbbells

The structural, dynamical, and rheological properties are studied of a multiparticle collision dynamics (MPC) fluid composed of shear-thinning Gaussian dumbbells. MPC is a mesoscale hydrodynamic simulation technique, which has successfully been applied in simulations of a broad range of complex fluids with Newtonian solvent. The MPC particles are replaced by Gaussian dumbbells, where we enforce a constant mean square length even under nonequilibrium conditions, which leads to shear thinning. This conserves the simplicity and efficiency of the original MPC fluid dynamics, since the analytical solution is known of Newton's equations of motion of the Gaussian dumbbells. Moreover, analytically obtained nonequilibrium structural, dynamical, and rheological properties are presented of Gaussian dumbbells under shear flow within the preaveraging approximation of hydrodynamic interactions. The comparison of the analytical and simulation results shows good agreement, with small deviations only due to the preaveraging approximation. In particular, we observe shear thinning and a nonzero second normal stress coefficient.

Normal Force as Moisture Feel for Lipsticks ∼The Control of Spinnability∼

We have been focusing on the normal force difference in the high shear rate region as moisture feel for lipsticks. It is widely known that the use of polymeric materials is effective to increase the normal force. However the control of spinnability has been a problem for ordinary polymers like polyether. A cellulose derivative with low spinnability has been developed. Spinnability and viscoelasticity were evaluated for a polyether (POE) and a long-alkyl acylated cellulose derivative (RCE). Although spinnability was easily evaluated by using the tack mode of rheometer, it was difficult to measure 10 cm or more. Since dynamic moduli of both samples could not be expressed with a single Maxwell mode, one of these samples is shifted by the correspondence of their tan δ. The storage modulus, G', of RCE with low spinnability is greater than that of POE with high spinnability. For the non-linear rheology, we found that the correspondence between the first normal stress coefficient, Ψ1, and G'/ω2 is different between two samples. POE with high spinnability has no correspondence of this relationship.

Non-Equilibrium Properties of Semidilute Polymer Solutions under Shear Flow

We review results of the non-equilibrium properties of semidilute polymer solutions under shear flow. A hybrid simulation approach is adopted, combing molecular dynamics simulations and the multiparticle collision dynamics method, which has been shown to fully capture hydrodynamic interactions. The polymers exhibit a shear rate dependent deformation and alignment. In addition, shear thinning is observed, which is related to the finite polymer extensibility. It is characterized by the shear-rate dependent viscosity and the first normal-stress coefficient. The conformational and rheological properties are universal functions of a concentration dependent Weissenberg number. The cyclic dynamics of an individual polymer is described in terms of a shear-rate dependent tumbling time. The tumbling time is tightly linked to the polymer end-to-end vector relaxation time and depends on concentration, which is attributed to screening of hydrodynamic interactions in semidilute solutions.

Effect of kinetic and configurational thermostats on calculations of the first normal stress coefficient in nonequilibrium molecular dynamics simulations

Thermostats for homogeneous nonequilibrium molecular dynamics simulations are usually designed to control the kinetic temperature, but it is now possible control any combination of many different types of temperature, including the configurational and kinetic temperatures and their directional components. It is well known that these temperatures can become unequal in homogeneously thermostatted shearing steady states. The microscopic expressions for these temperatures are all derived from equilibrium distribution functions, and it is pertinent to ask, what are the consequences of using these equilibrium microscopic expressions for temperature in thermostats for shearing nonequilibrium steady states? Here we show that the answer to this question depends on which properties are being investigated. We present numerical results showing that the value of the zero shear rate viscosity obtained by extrapolating results of nonequilibrium molecular dynamics simulations of shearing steady states is the same, regardless of the type of temperature that is controlled. It also agrees with the value obtained from the equilibrium stress autocorrelation function via the Green-Kubo relation. However, the values of the limiting zero shear rate first normal stress coefficient obtained from nonequilibrium molecular dynamics simulations of shearing steady states are strongly dependent on the choice of temperature being controlled. They also differ from the value of the first normal stress coefficient that is calculated from the equilibrium stress autocorrelation function. We show that even when all of the directional components of the kinetic and configurational temperatures are simultaneously controlled to the same value, the agreement with the result obtained from the equilibrium stress autocorrelation function is poor.

Parameter Free Prediction of Rheological Properties of Homopolymer Melts by Dynamic Monte Carlo Simulation

A biased Monte Carlo method is developed to simulate shear flow of linear homopolymers between neutral hard walls. Simulations are conducted on a face- centered cubic (FCC) lattice at full occupancy (φ = 1) using a modified version of Pakula's cooperative motion (COMOTION) algorithm. As chain lengths are increased under quiescent conditions, a clear crossover in chain dynamics from Rouse behavior to reptation-like behavior is revealed in characteristic correlation functions. Shear flow is simulated through the use of a biasing technique that favors segmental movement in the direction of flow; this combination of cooperative motion with flow is referred to as the COMOFLO algorithm. Using this technique, rheological properties are calculated as a function of shear rate and chain length. The viscosity and the first and second normal stress coefficients are calculated and found to be consistent with well-established experimental facts. The zero-shear viscosity scales linearly with low molecular weights (η0 ∼ N 1.0 ) and scales to the 3.5 power at higher molecular weights (η0 ∼ N 3.5 ). This crossover in viscosity scaling occurs at the same chain lengths (N ∼ 150) as the transition in chain dynamics. Because the simulations are conducted in the athermal limit, there are no parameter inputs. The known observable linear viscoelastic rheological properties of polymer melts are thus predicted in an a priori fashion.

Micropropulsion and microrheology in complex fluids via symmetry breaking

Many biological fluids have polymeric microstructures and display non-Newtonian rheology. We take advantage of such nonlinear fluid behavior and combine it with geometrical symmetry-breaking to design a novel small-scale propeller able to move only in complex fluids. Its propulsion characteristics are explored numerically in an Oldroyd-B fluid for finite Deborah numbers while the small Deborah number limit is investigated analytically using a second-order fluid model. We then derive expressions relating the propulsion speed to the rheological properties of the complex fluid, allowing thus to infer the normal stress coefficients in the fluid from the locomotion of the propeller. Our simple mechanism can therefore be used either as a non-Newtonian micro-propeller or as a micro-rheometer.

A single‐mode rheological model for steady flows of polymer melts

AbstractThe steady shear viscosity (ηs), the steady first normal stress coefficient (Ψ1), the steady second normal stress coefficient (Ψ2), and extensional viscosity (ηe) are four important parameters for polymer melts during polymer processing. In this article, we propose a stress and rate‐dependent function to describe creation and destruction of polymer junctions. Moreover, we also introduce a movement expression to describe nonaffine movement of network junctions. Based on network theory, a nonaffine single‐mode rheological model is presented for the steady flow of polymeric melts, and the equations of ηs, Ψ1, Ψ2, and ηe are derived from the model accordingly. Furthermore the dependences of ηs and ηe on model parameters are discussed for the model. Without a complex statistical simulation, the single‐mode model with four parameters yields good quantitative predictions of the steady shear and extensional flows for two low density polyethylene melts reported from previous literature in very wide range of deformation rates. © 2012 Wiley Periodicals, Inc. J Appl Polym Sci, 2012

A simple constitutive model describing the steady state shear viscosity and its prediction of the first normal stress function in shear flow

To describe the nonlinear behavior of polymer melts with few parameters, a simple three-parameter model for the shear viscosity function is proposed and applied to the polymer melt viscosity prediction as well as the first normal stress coefficient prediction using Wagner’s relationship. Its predictions were compared with experimental data. The nonlinear form of the model correlates the viscosity very well with the experimental data over many decades of shear rate. It demonstrates that a simple nonlinear term can replace the sum of the series of multiple linear spectra. From the first normal stress coefficient correlation with the experimental data, the damping constant of Wagner’s equation was obtained. Another applicability is demonstrated by the description of the elongational viscosity.

The stress in a dilute suspension of liquid spheres in a second-order fluid

AbstractWe use an ensemble averaging technique to calculate the average stress for a dilute suspension of liquid drops that are instantaneously spherical. The solvent and the drops consist of second-order fluids with differing properties. The suspension is itself a second-order fluid and its viscosity and normal stress coefficients are determined. For the special case of a rigid sphere suspension the results agree with Koch &amp; Subramanian (J. Non-Newtonian Fluid Mech., vol. 138, 2006, p. 87, and vol. 153, 2008, p. 202). Differences from other results in the literature are discussed.

Nonlinear Rheology of Telechelic Associative Polymer Networks: Shear Thickening and Thinning Behavior of Hydrophobically Modified Ethoxylated Urethane (HEUR) in Aqueous Solution

Flow behavior was examined for a 1.0 wt % aqueous solution of hydrophobically modified ethoxylated urethane (HEUR; Mw = 4.6 × 104). In the linear viscoelastic regime, the solution exhibited single-Maxwellian behavior attributable to thermal reorganization of the transient network composed of strings of HEUR flower micelles. Under shear flow at intermediate shear rates γ just above the equilibrium relaxation frequency 1/τ, the solution exhibited thickening characterized by monotonic increase of the viscosity growth function η+(t;γ) with time t above the linear η+(t) and by the steady-state viscosity η(γ) larger than the zero-shear viscosity η0. However, at those γ, the first normal stress coefficient growth function Ψ1+(t;γ) and its steady-state value Ψ1(γ) remained very close to the linear Ψ1+(t) and Ψ1,0 and exhibited no nonlinearity. In addition, the relaxation times of the viscosity and normal stress coefficient decay functions η–(t;γ) and Ψ1–(t;γ) measured after cessation of steady flow agreed...

Semidilute solutions of ultra-soft colloids under shear flow

We study semidilute star-polymer solutions under shear flow by hybrid mesoscale simulations. Hydrodynamic interactions are modeled by two particle-based simulation techniques, multiparticle collision dynamics (MPC) and dissipative particle dynamics (DPD). Star polymers are considered as a paradigmatic model for ultra-soft colloids with variable softness. The influence of concentration and shear rate on their structural and rheological properties is investigated. Under flow, a star polymer elongates and displays a well-defined alignment angle with respect to the flow direction. Moreover, the structural and rheological properties exhibit a universal behavior as a function of a concentration-dependent Weissenberg number for various concentrations at a given arm length. The rheological properties are characterized by the shear viscosity and the normal-stress coefficients. In dilute solution, the zero-shear viscosity follows the Einstein relation with an effective radius given by the hydrodynamic radius of a star polymer. At high shear rates, the solutions exhibit shear-thinning behavior, where the viscosity decreases faster with increasing shear rate at higher concentrations. We demonstrate that the results obtained from MPC and DPD agree in all scaling properties, with minor quantitative deviations in the numerical values.

Rheology of Aqueous Solution of Hydrophobically Modified Ethoxylated Urethane (HEUR) with Fluorescent Probes at Chain Ends: Thinning Mechanism

Nonlinear flow behavior and the corresponding fluorescent behavior were examined for a 3.0 wt% aqueous solution of hydrophobically modified ethoxylated urethane (HEUR; Mw = 1.1×105) having fluorescent pyrenyl (Py) groups at the chain ends. These end groups associated with each other to serve as cores of HEUR flower micelles, and these micelles were further connected into super-bridges to form a transient network. Correspondingly, single-Maxwellian relaxation reflecting the thermal reorganization of this network was observed in the linear viscoelastic regime. Under steady shear flow, the HEUR solution exhibited thinning of both viscosity η(γ) and first normal stress coefficient Ψ1(γ) at shear rates γ above the equilibrium relaxation frequency 1/τ. Fluorescent emission intensity IE(γ) from excimers formed by Py groups associated in a hydrophobic environment of the micellar cores and the intensity IM(γ) from dissociated Py groups (referred to as monomers) isolated in the aqueous phase were measured under flow simultaneously with η and Ψ1. The IE/IM ratio was found to decrease only slightly (by a factor of ∼4 %) on an increase of γ up to 6/τ well in the thinning regime. (For γ = 6/τ, η and Ψ1 decreased from respective zero-shear values by factors of ∼50 % and ∼75 %, respectively) Thus, the significant thinning at those γ was accompanied by a negligible change in the HEUR core structure that corresponds to just a slight shift of the association/dissociation balance of the Py groups to the dissociated state. From this result, the thinning is quite possibly attributed to flow-induced disruption of the network connectivity through conversion of the super-bridges into super-loops, both having the same core structure.

Mesoscale hydrodynamic modeling of a colloid in shear-thinning viscoelastic fluids under shear flow

In order to study the dynamics of colloidal suspensions with viscoelastic solvents, a simple mesoscopic model of the solvent is required. We propose to extend the multiparticle collision dynamics (MPC) technique--a particle-based simulation method, which has been successfully applied to study the hydrodynamic behavior of many complex fluids with Newtonian solvent--to shear-thinning viscoelastic solvents. Here, the normal MPC particles are replaced by dumbbells with finite-extensible nonlinear elastic (FENE) springs. We have studied the properties of FENE-dumbbell fluids under simple shear flow with shear rate ̇γ. The stress tensor is calculated, and the viscosity η and the first normal-stress coefficient Ψ(1) are obtained. Shear-thinning behavior is found for reduced shear rates Γ= ̇γτ>1, where τ is a characteristic dumbbell relaxation time. Here, both η and Ψ(1) display power-law behavior in the shear-thinning regime. Thus, the FENE-dumbbell fluid with MPC collisions provides a good description of viscoelastic fluids. As a first application, we study the flow behavior of a colloid in a shear-thinning viscoelastic fluid in two dimensions. A slowing down of the colloid rotation in a viscoelastic fluid compared to a Newtonian fluid is obtained, in agreement with recent numerical calculations and experimental results.

Analysis on Shear Lag Effect of Single Box and Double Cell Box Girders for Wei River Grand Bridge

In order to study the influence of shear lag effect of single box and double cell box girder on structure，to make the rigid frame continuous bridge of Wei River grand bridge as the research object, through the establishment of finite element model, the normal stress and shear lag coefficient of the control sections in various working conditions is calculated, and the distribution of the box girder normal stress under shear lag effect is obtained.