Normal Stress Coefficient Research Articles

A new ALS model for dilute polymer solutions in flows with strong shear components

Ghosh et al. (J Rheol 46:1057–1089, 2002) developed a new model for dilute polymer solutions in flows with strong extensional components. The model based on introducing an adaptive length scale (ALS) as an internal variable was developed to reproduce the fine-scale physics of the Kramers chain. The ALS model describes the polymer molecule as a set of identical segments in which each segment represents a fragment of the polymer that is short enough so that it can sample its entire configuration space on the time scale of an imposed deformation and therefore stretch reversibly. As the molecule unravels, the number of the segments decreases, but the maximum length of each segment increases, so that the constant maximum contour length of the molecule is preserved. Though the single-segment-based ALS model accounts for the orientability of the polymer molecules, it cannot describe the internal motions of the molecules due to the lack of internal nodes. Hence, in this work we consider the more realistic chain (multi-segments) model composed of Nseg springs connected linearly. The model presented in this work is an extension of the ALS model developed by Ghosh et al. (J Rheol 46:1057–1089, 2002). We demonstrate that the ALS varies with the flow strength. Specifically, it is found that as the flow strength increases, the ALS decreases. This implies that as the flow strength increases, the polymer molecule is required to divide into finer and finer segments such that each segment can locally equilibrate with the imposed flow. However, there is a critical number of such subdivisions beyond which further subdivision of the polymer molecule is not required to capture the polymer dynamics for a given flow strength. Both shear viscosity and first normal stress coefficient predictions from ALS model show shear thinning behavior with Weissenberg number. In weak flows, the ALS model and the finitely extensible non-linear elastic model exhibit the same behavior.

A mean-field anisotropic diffusion model for unentangled polymeric liquids and semi-dilute solutions: Model development and comparison with experimental and simulation data

A coarse-grained mesoscopic model was developed based on the ansatz that a specific polymer molecule diffuses through the nearby neighboring chains more easily in the direction parallel to its molecular axis than perpendicular to it. This idea is modeled using a mean-field approach in terms of an anisotropic diffusion matrix, which represents enhanced diffusion along the chain background once a significant degree of molecular extension and orientation has developed in response to an applied flow field. The rheological and microstructural characteristics of this model are examined and compared with atomistic nonequilibrium molecular dynamics (NEMD) simulation data of short-chain polyethylene liquids and experiments of semi-dilute DNA solutions under shear flow. Rheological and microstructural properties examined include the viscosity, normal stress coefficients, conformation tensor, etc., to gauge the usefulness of the model. In addition, this model was further coarse-grained to the continuum level through pre-averaging, and was also compared with the simulation and experimental data to examine the relationships between different levels of description on the rheological and structural properties of unentangled polymeric materials under shear flow. At the mesoscopic level, the polymer molecules are modeled as bead-spring chains using the finitely extensible nonlinear elastic (FENE) force law. Brownian dynamics (BD) simulations of this coarse-grained model displayed remarkable quantitative agreement with NEMD simulations of dense liquids and experiments of semi-dilute DNA solutions for system properties with a single adjustable parameter representing the relative magnitude of diffusive enhancement along the chain backbone. Furthermore, the BD simulations revealed the dependence of system response on the chain stretching at low values of Weissenberg number ( Wi) and on the rotational motion of individual chains induced by shear flow at high values of Wi, similarly to the NEMD simulation data. The continuum model matched the mesoscopic model at low shear rates, but greatly diverged at high values of Wi where the tumbling dynamics of the individual chains dominated the system response. This provides direct evidence that the onset of rotational motion under shear in these liquids is responsible for the well-known breakdown in pre-averaged constitutive equations at the continuum level of description. Furthermore, a possible explanation of the shear stress plateau at intermediate ranges of shear rate is offered for experimental data of semi-dilute solutions, wherein this phenomenon occurs with the onset of chain rotation within these fluids.

Semidilute Polymer Solutions at Equilibrium and under Shear Flow

The properties of semidilute polymer solutions are investigated at equilibrium and under shear flow by mesoscale simulations, which combine molecular dynamics simulations and the multiparticle collision dynamics approach. In semidilute solution, intermolecular hydrodynamic and excluded volume interactions become increasingly important due to the presence of polymer overlap. At equilibrium, the dependence of the radius of gyration, the structure factor, and the zero-shear viscosity on the polymer concentration is determined and found to be in good agreement with scaling predictions. In shear flow, the polymer alignment and deformation are calculated as a function of concentration. Shear thinning, which is related to flow alignment and finite polymer extensibility, is characterized by the shear viscosity and the normal stress coefficients.

Active Microrheology: A Proposed Technique to Measure Normal Stress Coefficients of Complex Fluids

We propose a microrheological technique to measure normal stress coefficients (NSCs) of complex fluids, which would represent the first quantitatively accurate measurement of a nonlinear rheological property by microrheology. Specifically, the mechanical response of almost all complex fluids to "weakly nonlinear" deformations is described by the second-order fluid model. Two microrheological probes pulled with equal velocities through a second-order fluid experience a relative force that is linear in the first and second NSCs of the complex fluid. We compute the coupling matrix between NSCs and relative forces for probes translating parallel and perpendicular to their line of centers, which can be inverted to yield NSCs from measured relative forces. There exists an optimum probe separation for inversion of the coupling matrix and, hence, experimental recovery of NSCs.

Is there a relation between the relaxation time measured in CaBER experiments and the first normal stress coefficient?

We investigate a variety of different semidilute polymer solutions in shear and elongational flow. The shear flow is created in the cone-plate-geometry of a commercial rheometer. We use capillary thinning of a filament that is formed by a polymer solution in the Capillary Breakup Extensional Rheometer (CaBER) as an elongational flow. We compare the relaxation time measured in the CaBER with relaxation times based on the first normal stress difference and the zero shear polymer viscosity that we measure in our rheometer. All of these three measurable quantities depend on different fluid parameters—the viscosity of the solvent, the polymer concentration within the solution, and the molecular weight of the polymers—and on the shear rate (in the shear flow measurements). Nevertheless, we find that the first normal stress coefficient depends quadratically on the CaBER relaxation time. Several scaling laws are presented that could help to explain this empirical relation.

One-dimensional viscoelastic fluid model where viscosity and normal stress coefficients depend on the shear rate

We study the unsteady motion of a viscoelastic fluid modeled by a second-order fluid where normal stress coefficients and viscosity depend on the shear rate by using a power-law model. To study this problem, we use the one-dimensional nine-director Cosserat theory approach which reduces the exact three-dimensional equations to a system depending only on time and on a single spatial variable. Integrating the equation of conservation of linear momentum over the tube cross-section, with the velocity field approximated by the Cosserat theory, we obtain a one-dimensional system. The velocity field approximation satisfies both the incompressibility condition and the kinematic boundary condition exactly. From this one-dimensional system we obtain the relationship between average pressure and volume flow rate over a finite section of the tube with constant and variable radius. Also, we obtain the correspondent equation for the wall shear stress which enters directly in the formulation as a dependent variable. Attention is focused on some numerical simulation of unsteady/steady flows for average pressure, wall shear stress and on the analysis of perturbed flows.

Non-linear viscoelastic models for semi-flexible polysaccharide solution rheology over a broad range of concentrations

The experimental linear viscoelastic and steady-state shear data over a broad concentration range of several aqueous semi-flexible polysaccharide solutions can be quantitatively predicted by the multimode Phan-Thien Tanner, Giesekus, and extended Pom-Pom non-linear viscoelastic models using a single set of parameters. Experimental data from an aqueous solution of the polysaccharide guar galactomannan (Mw=1.12×106 g/mol), aqueous phosphopolysaccharide “viilian” solution excreted by Lactococcus lactis subspecies cremoris SBT 0495 (Mw=2×106 g/mol), and aqueous Propionibacterium acidi-propionici polysaccharide solution (Mw≈106 g/mol) are taken from literature. A master curve can be constructed by shifting data from different concentrations over the horizontal and vertical axes. A simple model to describe the concentration dependence of the relaxation time and plateau modulus over the entire concentration range is presented. Transient shear viscosity data are quantitatively and first normal stress coefficient are qualitatively predicted for the guar galactomannan solution. Due to their proven performance in finite element simulations, these non-linear viscoelastic constitutive equations could help us to improve predictive modeling of time-dependent complex flow problems for polysaccharide solutions with variations in concentration and temperature in the spatial domain using only a single set of parameters. However, as a previous step, the performance of additional rheological experiments in simple flows, i.e., transient and steady-state extensional measurements, is recommended.

Steady shear rheometry of dissipative particle dynamics models of polymer fluids in reverse Poiseuille flow

Polymer fluids are modeled with dissipative particle dynamics (DPD) as undiluted bead-spring chains and their solutions. The models are assessed by investigating their steady shear-rate properties. Non-Newtonian viscosity and normal stress coefficients, for shear rates from the lower to the upper Newtonian regimes, are calculated from both plane Couette and plane Poiseuille flows. The latter is realized as reverse Poiseuille flow (RPF) generated from two Poiseuille flows driven by uniform body forces in opposite directions along two-halves of a computational domain. Periodic boundary conditions ensure the RPF wall velocity to be zero without density fluctuations. In overlapping shear-rate regimes the RPF properties are confirmed to be in good agreement with those calculated from plane Couette flow with Lees-Edwards periodic boundary conditions (LECs), the standard virtual rheometer for steady shear-rate properties. The concentration and the temperature dependence of the properties of the model fluids are shown to satisfy the principles of concentration and temperature superposition commonly employed in the empirical correlation of real polymer-fluid properties. The thermodynamic validity of the equation of state is found to be a crucial factor for the achievement of time-temperature superposition. With these models, RPF is demonstrated to be an accurate and convenient virtual rheometer for the acquisition of steady shear-rate rheological properties. It complements, confirms, and extends the results obtained with the standard LEC configuration, and it can be used with the output from other particle-based methods, including molecular dynamics, Brownian dynamics, smooth particle hydrodynamics, and the lattice Boltzmann method.

Theoretical Predictions on Normal Stresses under Shear Flow in Transient Networks of Telechelic Associating Polymers

This paper studies the behavior of the normal stresses in associated networks composed of telechelic polymers under steady shear flow on the basis of a transient network theory. We numerically show that the first and second normal stress coefficients reveal thickening as a function of shear rate, and that the sign of the second normal stress coefficient changes depending on the nonlinearity in the chain tension, the dissociation rate of the associative groups from junctions, and the shear rate. By analytic calculation, we show that, in the limit of small shear-rate, the sign inversion occurs by the competition between the nonlinear stretching and dissociation of associative groups. Thus, the molecular mechanism of the sign inversion is shown to be similar to that of thickening of the shear viscosity. We also show that thickening of the first normal stress coefficient has a similar molecular origin. The theoretical predictions on thickening of the first normal stress coefficient, and on the positivity of t...

A molecular simulation study of shear viscosity and thermal conductivity of liquid carbon disulphide

We have calculated the shear viscosity and the thermal conductivity of liquid carbon disulphide at temperature T = 193 K and density ρ = 1420 kg/m 3 near the liquid–vapour curve and temperature T = 309 K and density ρ = 1427 kg/m 3 using a three-centre model due to Tildesley and Madden. Both transport coefficients were calculated by equilibrium molecular dynamics using the Green–Kubo relations and for the 193 K system the shear rate dependent viscosity and the first normal stress coefficient were also calculated by non-equilibrium molecular dynamics methods. The shear rate dependence of the viscosity was well described by both the Cross and Carreau models, which are commonly used rheological constitutive equations. The values of the zero shear rate viscosity obtained from both equilibrium and non-equilibrium computations agreed to within statistical uncertainty. Both the viscosity and thermal conductivity values were in good agreement with a correlation of experimental data based on the rough–hard-sphere model.

Microstructures and rheology of micellar surfactant solution by Brownian dynamics simulation

In this paper, three-dimensional Brownian dynamics simulation has been conducted for dilute micellar surfactant solution under a steady shear flow. The rodlike micelle in surfactant solution is assumed as a rigid rod made up of lined-up beads. The Lennard–Jones potential and soft-sphere potential are employed and taken as the inter-bead potentials for end–end beads and interior–interior beads, respectively. The motion of the rodlike micelles is determined by solving the translational and rotational equations for each rod under hydrodynamic drag force, Brownian force and inter-rod potential force. Velocity Verlet algorithm has also been exerted in the simulation. The micellar network structure is formed at low shear rates and destroyed by high shear rates. The computed shear viscosities and the first normal stress coefficient represent shear thinning characteristics. The paper reveals the relation between rheology and microstructure of surfactant solution at different shear rates. The effect of surfactant solution concentration rested on the micellar structures and rheological properties has also been investigated.

The effect of interbranch spacing on structural and rheological properties of hyperbranched polymer melts

Nonequilibrium molecular dynamics simulations were performed for a family of hyperbranched polymers of the same molecular weight but with different chain lengths between branches. Microscopic structural properties including mean squared radius of gyration, distribution of beads from the center of mass and from the core and the interpenetration function of these systems were characterized. A relationship between the zero shear rate mean squared radius of gyration and the Wiener index was established. The molecular and bond alignment tensors were analyzed to characterize the flow birefringence of these hyperbranched polymers. The melt rheology was also studied and the crossover from the Newtonian to non-Newtonian behavior was captured for all polymer fluids in the considered range of strain rates. Rheological properties including the shear viscosity and normal stress coefficients obtained from constant pressure simulations were found to be the same as those from constant volume simulations except at high strain rates due to shear dilatancy. A linear dependence of zero shear rate viscosities on the number of spacer units was found. The stress optical rule was shown to be valid at low strain rates with the stress optical coefficient of approximately 3.2 independent of the topologies of polymers.

Rheology, Microstructure and Migration in Brownian Colloidal Suspensions

We demonstrate that suspended spherical colloidal particles can be effectively modeled as single dissipative particle dynamics (DPD) particles provided that the conservative repulsive force is appropriately chosen. The suspension model is further improved with a new formulation, which augments standard DPD with noncentral dissipative shear forces between particles while preserving angular momentum. Using the new DPD formulation we investigate the rheology, microstructure and shear-induced migration of a monodisperse suspension of colloidal particles in plane shear flows (Couette and Poiseuille). Specifically, to achieve a well-dispersed suspension we employ exponential conservative forces for the colloid-colloid and colloid-solvent interactions but keep the conventional linear force for the solvent-solvent interactions. Our simulations yield relative viscosity versus volume fraction predictions in good agreement with both experimental data and empirical correlations. We also compute the shear-dependent viscosity and the first and second normal-stress differences and coefficients in both Couette and Poiseuille flow. Simulations near the close packingvolume-fraction (64%) at low shear rates demonstrate a transition to flow-induced string-like structures of colloidal particles simultaneously with a transition to a nonlinear Couette velocity profile in agreement with experimental observations. After a sufficient increase ofthe shear rate the ordered structure melts into disorder with restoration of the linear velocity profile. Migration effects simulated in Poiseuille flow compare well with experiments and model predictions. The important role of angular momentum and torque in nondilute suspensions is also demonstrated when compared with simulations by the standard DPD, which omits the angular degrees of freedom. Overall, the new method agrees very well with the Stokesian Dynamics method but it seems to have lower computational complexity and is applicable to general complex fluids systems.

An empirical constitutive law for concentrated colloidal suspensions in the approach of the glass transition

Concentrated, non-crystallizing colloidal suspensions in their approach of the glass state exhibit distinct dynamics patterns. These patterns suggest a powerlaw rheological constitutive model for near-glass viscoelasticity, as presented here. The rheological parameters used for this model originate in the mode-coupling theory. The proposed constitutive model provides explicit expressions for the steady shear viscosity, the steady normal stress coefficient, the modulus-compliance relation, and the α peak of G″. The relaxation pattern distinctly differs from gelation.

Rheology of hyperbranched polymer melts undergoing planar Couette flow

The melt rheology of four hyperbranched polymer structures with different molecular weights has been studied using nonequilibrium molecular dynamics (NEMD). Systems were simulated over a wide range of strain rates to capture the crossover behavior from Newtonian to non-Newtonian regimes. Rheological properties including shear viscosity and first and second normal stress coefficients were computed and the transition to shear thinning was observed at different strain rates for hyperbranched polymers of different sizes. The results were consistent with previous findings from NEMD simulation of linear and dendritic polymers. Flow birefringence was characterized by taking into account both form and intrinsic birefringences, which result from molecular and bond alignment, respectively. The stress optical rule was tested and shown to be valid only in the Newtonian regime and violated in the strong flow regime where the rule does not take into account flow-induced changes of the microstructure.

Modelling elongational and shear rheology of two LDPE melts

Experimental data of two low-density polyethylene (LDPE) melts at 200°C for both shear flow (transient and steady shear viscosity as well as transient and steady first normal stress coefficient) and elongational flow (transient and steady-state elongational viscosity) as published by Pivokonsky et al. (J Non-Newtonian Fluid Mech 135:58–67, 2006) were analysed using the molecular stress function model for broadly distributed, randomly branched molecular structures. For quantitative modelling of melt rheology in both types of flow and in a very wide range of deformation rates, only three nonlinear viscoelastic material parameters are needed: Whilst the rotational parameter, a 2, and the structural parameter, β, are found to be equal for the two melts considered, the melts differ in the parameter $f_{\rm max}^2 $ describing maximum stretch of the polymer chains.

Accurate determination of normal stress differences via transient-time correlation function – non-equilibrium molecular dynamics (TTCF–NEMD) simulations

When a non-Newtonian liquid undergoes shear flow, differences between the values taken by the diagonal elements of the pressure tensor arise. These differences, known as normal stress differences, are notoriously difficult to compute accurately. In this work, we propose to apply the transient-time correlation function (TTCF) formalism together with non-equilibrium molecular dynamics (NEMD) simulations to determine normal stress differences, as well as the corresponding normal stress coefficients, in an atomic fluid undergoing shear flow. Our results shed light on the dependence of normal stress differences and coefficients on the applied shear.

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.

Interfacial slip‐stick transition induced by reactive compatibilization in ethylene‐co‐butyl acrylate elastomer dispersed polyamide 6 blend subjected to high shear flow and extensional flow

AbstractThe effect of phase interaction induced by reactive compatibilization during high shear and extensional flow in polyamide (PA6) and ethylene‐co‐butyl acrylate (EBA) blends was studied using advanced dual bore capillary rheometer. The viscosity‐composition behavior of the uncompatibilized PA6/EBA blends exhibited negative deviation behavior from log‐additivity rule. The interfacial slip mechanism, operative between the matrix PA6 and dispersed EBA during shear flow was studied by the use of Lin's and Bousmina‐Palierne‐Utracki (BPU) model for viscosity for the blends under the processing conditions. On the other hand, the compatibilized PA6/EBA‐g‐MAH0.49/EBA blends with varying dispersed phase volume fraction show positive deviation behavior. The reactive compatibilizers EBA‐g‐MAH0.49 and EBA‐g‐MAH0.96 increased the phase interaction with adequate reduction in the dynamic interfacial tension, which favored the particle break‐up and stabilized the morphology in the compatibilized blends. The extensional viscosity of the blends has enhanced because of the inclusion of EBA in all the uncompatibilized and compatibilized blends. The melt elasticity and elasticity function were systematically studied from first normal stress coefficient functions (ψ1). The variation in the recoverable shear strain (γR), shear rate dependent relaxation time (λ) and shear compliance (Jc) under various shear rates were thoroughly analyzed for all the blend compositions. POLYM. ENG. SCI., 2008. © 2008 Society of Plastics Engineers

Two spheres in a free stream of a second-order fluid

The forces acting on two fixed spheres in a second-order uniform flow are investigated. When α1+α2=0, where α1 and α2 are fluid parameters related to the first and second normal stress coefficients, the velocity field for a second-order fluid is the same as the one predicted by the Stokes equations while the pressure is modified. The Stokes solutions given by Stimson and Jeffery [Proc. R. Soc. London, Ser. A 111, 110 (1926)] for the case when the flow direction is along the line of centers and Goldman et al. [Chem. Eng. Sci. 21, 1151 (1966)] for the case when the flow direction is perpendicular to the line of centers are utilized and the stresses and the forces acting on the particles in a second-order fluid are calculated. For flow along the line of centers or perpendicular to it, the net force is in the direction that tends to decrease the particle separation distance. For the case of flow at arbitrary angle, unequal forces are applied to the spheres perpendicularly to the line of centers. These forces result in a change of orientation of the sedimenting spheres until the line of centers aligns with the flow direction. In addition, the potential flow of a second-order fluid past two fixed spheres in a uniform flow is investigated. The normal stress at the surface of each sphere is calculated and the viscoelastic effects on the normal stress for different separation distances are analyzed. The contribution of the potential flow of a second-order fluid to the force applied to the particles is an attractive force. Our explanations of the aggregation of particles in viscoelastic fluids rest on three pillars; the first is a viscoelastic “pressure” generated by normal stresses due to shear. Second, the total time derivative of the pressure is an important factor in the forces applied to moving particles. The third is associated with a change in the normal stress at points of stagnation which is a purely extensional effect unrelated to shearing.