Index of content:
Volume 52, Issue 1, January 2008

Large scale dynamic simulation of platelike particle suspensions. Part I: NonBrownian simulation
View Description Hide DescriptionRheology and microstructure of platelike particle suspensions in linear shear flows are studied through numerical simulations for a range of volume fraction up to 0.30. Particles with aspect ratio are modeled as planar assemblages of spheres. Numerical methods are developed to calculate the hydrodynamic interactions based on an extension of the Stokesian dynamics method for spheres. At low , suspensions exhibit a degree of particle alignment consistent with the orientation distribution predicted by Jeffery orbits. At high , hydrodynamic interactions produce a high degree of ordering with particles aligned in horizontal layers perpendicular to the gradient direction. This allows sufficient free volume for shearing of the suspension with moderate viscosity at high . A second ordering mechanism is the formation of transient stacks of platelike particles which move as rigid assemblies. The two mechanisms—particle alignment and particle stacking—reduce the effects of hydrodynamic interaction. Over the range of considered, viscosity is not a strong function of aspect ratio for nonBrownian suspensions, because the increased hydrodynamic interactions of high aspect ratio particles are offset by the increased degree of particle ordering.

Large scale dynamic simulation of platelike particle suspensions. Part II: Brownian simulation
View Description Hide DescriptionRheology and microstructure of platelike particle suspensions in linear shear flows are studied through Stokesian dynamics simulations with Brownian motion for a range of volume fraction up to 0.30 and Peclet number Pe ranging from 0.01 to . As in Part I [Meng and Higdon, J. Rheol.52, 1 (2008)], particles are modeled as planar assemblages of spheres. The effects of Brownian motion on the suspensions microstructure are studied with special attention to two ordering mechanisms observed for nonBrownian systems: (1) the formation of sliding planes of aligned layers of particles and (2) the formation of transient stacks of platelike particles which move as rigid assemblies. At low Pe, strong Brownian motion yields random particle orientations, however the aligned particle layers are recovered at Pe as low as 0.1 for low and 0.4 for . Brownian motion acts more effectively in disrupting particle stacks with measurable reduction in stack formation up to Pe of 1000. The platelike particle suspensions exhibit both shear thinning and shear thickening behavior as a function of Pe, however, the Pe dependence differs from that for suspensions of spheres. The effect of Brownian motion on particle alignment introduces an additional factor enhancing shear thinning, and the shear thickening is weaker than for suspensions of spheres.

A constitutive analysis of transient and steadystate elongational viscosities of bidisperse polystyrene blends
View Description Hide DescriptionThe transient and steadystate elongational viscosity data of three bidisperse polystyrene blends were investigated recently by Nielsen et al. [J. Rheol.50, 453–476 (2006)]. The blends contain a monodisperse high molar mass component in a matrix of a monodisperse small molar mass component (either or at two different weight fractions). The experimental data are analyzed in the framework of the molecular stress function model of Wagner et al. [J. Rheol.49, 1317–1327 (2005)], which is based on the assumption of a straindependent tube diameter and the interchain pressure term of Marrucci and Ianniruberto [Macromolecules37, 3934–3942 (2004)]. The dilution of the long chains in the matrix of the short chains is identified as the origin of a drastic increase in the tubediameter relaxation time of the long chains, leading to a large stretching potential of the longchain component and an increasing steadystate elongational viscosity with increasing strain rate. In addition, in the dilution regime, a transition from affine chain stretch to nonaffine tube squeeze with decreasing strain rate is identified. The dilution regime ends at a critical strain rate, when the tube diameter of the supertubes created by the interaction of the long chains among themselves, is reduced by deformation to the tube diameter of the bulk. A nonlinear extension of the basic double reptation concept is developed comprising all of these different phenomena, and allowing (albeit by use of empirical linearviscoelastic shift factors to correct the linearviscoelastic predictions) for a quantitative description of the transient and steadystate elongational viscosities of the bidisperse polystyrene blends.

Internal parameters and regime map for soft polydispersed granular materials
View Description Hide DescriptionSeveral internal parameters are studied for a 3D simple shear flow of soft polydispersed granular materials consisting of viscoelastic spheres. These internal parameters include the contact time, the multiple collision group size, and the coordination number. It is found that the contour plots of the contact time and the coordination number in the plane defined by the concentration (solid fraction) and dimensionless stiffness are similar. These contours are qualitatively the same as the regime chart/flowmap proposed in earlier studies. The resulting constitutive relation shows different rate dependency at different concentrations and shear rates. Based on a simple dimensional analysis and a power law formulation, the rate dependency may be expressed by an index. The two extreme values of this index, 0 and 1, correspond to solidlike and gaslike granular materials, respectively. The contour map of this rate index (the power of the dimensionless shear rate in the constitutive relation) resembles those of the shaped curve typical of a phase diagram for ordinary materials.

Energetic and entropic elasticity of nonisothermal flowing polymers: Experiment, theory, and simulation
View Description Hide DescriptionThe thermodynamical aspects of polymeric liquids subjected to nonisothermal flow are examined from the complementary perspectives of theory, experiment, and simulation. In particular, attention is paid to the energetic effects, in addition to the entropic ones, that occur under conditions of extreme deformation. Comparisons of experimental measurements of the temperature rise generated under elongational flow at high strain rates with macroscopic finite element simulations offer clear evidence of the persistence and importance of energetic effects under severe deformation. The performance of various forms of the temperatureequation are evaluated with regard to experiment, and it is concluded that the standard form of this evolution equation, arising from the concept of purely entropic elasticity, is inadequate for describing nonisothermal flow processes of polymeric liquids under high deformation. Complete temperatureequations, in the sense that they possess a direct and explicit dependence on the energetics of the microstructure of the material, provide excellent agreement with experimental data.

Particle pressure in sheared Brownian suspensions
View Description Hide DescriptionThe isotropic contribution of the particle phase to the bulk stress, or the particle pressure, is studied for Brownian hard sphere suspensions in computationally simulated shear flow. The particle pressure is mechanically defined as the negative mean normal stress exerted by the particles, i.e., for a viscometric flow where 1, 2, and 3 refer to the flow, velocity gradient, and vorticity directions, respectively. Analysis is provided to relate the particle pressure to the equilibrium osmotic pressure and to show the relation of to particle migration phenomena. Utilizing existing hydrodynamic functions and simulating the flow by the Stokesian Dynamics technique, the particle pressure is evaluated for particle volume fractions in the range for monodisperse spherical particles. The relative strength of Brownian to shearing motion is given by the Péclet number , where is the shear rate of a simple shear flow, is the spherical particle radius, and with the thermal energy and the suspending fluid viscosity. For each , the range has been studied. The particle pressure at , where it is given predominantly by a Brownian contribution, is found to approach the exact results for the osmotic pressure of an equilibrium hardsphere dispersion, , where is the particle number density and is the pair distribution function evaluated at contact. The hydrodynamic contribution to grows with and dominates the Brownian contribution at . The particle pressure scales as at elevated . The relative contributions to of Brownian and hydrodynamic stress are similar as a function of to the normal stress differences of the suspension, and , but, for , exceeds and for all .

Microrheology of colloidal dispersions: Shape matters
View Description Hide DescriptionWe consider a “probe” particle translating at constant velocity through an otherwise quiescent dispersion of colloidal “bath” particles, as a model for particletracking microrheology experiments in the active (nonlinear) regime. The probe is a body of revolution with major and minor semiaxes and , respectively, and the bath particles are spheres of radii . The probe’s shape is such that when its major or minor axis is the axis of revolution the excludedvolume, or contact, surface between the probe and a bath particle is a prolate or oblate spheroid, respectively. The moving probe drives the microstructure of the dispersion out of equilibrium; counteracting this is the Brownian diffusion of the bath particles. For a prolate or oblate probe translating along its symmetry axis, we calculate the nonequilibrium microstructure to first order in the volume fraction of bath particles and over the entire range of the Péclet number , neglecting hydrodynamic interactions. Here, is defined as the nondimensional velocity of the probe. The microstructure is employed to calculate the average external force on the probe, from which one can infer a “microviscosity” of the dispersion via Stokes drag law. The microviscosity is computed as a function of the aspect ratio of the probe, , thereby delineating the role of the probe’s shape. For a prolate probe, regardless of the value of , the microviscosity monotonically decreases, or “velocity thins,” from a Newtonian plateau at small until a second Newtonian plateau is reached as . After appropriate scaling, we demonstrate this behavior to be in agreement with microrheology studies using spherical probes [Squires and Brady, “A simple paradigm for active and nonlinear microrheology,” Phys. Fluids17(7), 073101 (2005)] and conventional (macro)rheological investigations [Bergenholtz et al., “The nonNewtonian rheology of dilute colloidal suspensions,” J. Fluid. Mech.456, 239–275 (2002)]. For an oblate probe, the microviscosity again transitions between two Newtonian plateaus: for (to two decimal places) the microviscosity at small is greater than at large (again, velocity thinning); however, for the microviscosity at small is less than at large , which suggests it “velocity thickens” as is increased. This anomalous velocity thickening—due entirely to the probe shape—highlights the care needed when designing microrheology experiments with nonspherical probes.

Coilstretch transition and the breakdown of computations for viscoelastic fluid flow around a confined cylinder
View Description Hide DescriptionThe breakdown of finite element(FEM) computations for the steady symmetric twodimensional flow of dilute and ultradilute OldroydB fluids around a cylinder in a channel, at Weissenberg numbers , is shown to arise due to a coilstretch transition experienced by polymer molecules in the wake of the cylinder in the vicinity of the location of the stress maximum on the centerline. In dilute OldroydB fluids, due to the modification of the flow caused by the presence of the polymer, the coilstretch transition leads to the stress maximum diverging toward infinity at a finite value of . On the other hand, in ultradilute solutions, the stress maximum approaches infinity only as . In FENEP fluids, the coilstretch transition leads to the mean extension of the molecules saturating to a value close to the fullyextended length, with the maximum stress remaining bounded with increasing . An estimation of the number of finite elements required to achieve convergence for ultradilute OldroydB fluids reveals that obtaining solutions at is not feasible.

Transient behavior of electrorheological fluids in shear flow
View Description Hide DescriptionThe transient response of electrorheological suspensions in shear flow subjected to a suddenly imposed electric field is investigated experimentally. Barium titanate∕silicone oil and alumina∕mineral oil suspensions are employed. The evolution of both the rheological properties and the suspension structure are investigated. Results are compared with predictions from a twofluid continuum model reported previously. Transient responses appear above a critical field strength, and the critical Mason number for the onset of a transient rheological response is equivalent to the critical Mason number for the onset of lamella formation, within experimental uncertainty. These results are consistent with predictions. The experimentally determined values of the critical Mason number agree with those predicted, with differences of the order of the experimental uncertainty. However, we find that the critical Mason number depends on shear rate, rather than being independent of shear rate as predicted.

Chaintopologycontrolled hyperbranched polyethylene as effective polymer processing aid (PPA) for extrusion of a metallocene linearlowdensity polyethylene (mLLDPE)
View Description Hide DescriptionHyperbranched polymers have recently received extensive interest for applications as polymer processing aids (PPAs). In this work, we present the first study on the performance of novel hyperbranched polyethylene as a processing aid for a metallocene linearlowdensity polyethylene (mLLDPE). Moreover, we investigated for the first time the unique role that chain topology plays in determining the efficiency of the polymers as PPA. Two polymers of drastically different chain topologies, a hyperbranched polyethylene (HBPE) and a linear branched polyethylene (LBPE) having a linear chain topology, were prepared by ethylene polymerization using chain walking Pddiimine catalysis. The performances of these two polymers of different chain topologies in improving the processability of the mLLDPE were investigated and compared by using twostep capillary extrusion experiments, including constantshear testing at and shearrate sweep testing. For comparison purposes, a commercial fluoropolymerbased PPA, Viton Free Flow Z100, together with a polyethylene wax sample, were also evaluated for their performances. It was found that, at a concentration , the presence of HBPE in mLLDPE significantly reduced the apparent shear stress, improved extrudate surface morphology, and postponed the onset shear rate for sharkskin instability. However, LBPE did not improve the processability of mLLDPE and it worked merely as a plasticizer, slightly reducing the viscosity of polymer melts. Morphology studies conducted on the cross sections of the blends showed that chain topology greatly affected the polymer additive’s miscibility with mLLDPE; the mLLDPE/HBPE blend was a phaseseparated system, whereas LBPE appeared to be miscible with mLLDPE. The hyperbranched polymer, HBPE, forms phaseseparated droplets, which can migrate to the die surface and form a lubricating layer promoting extrudate slippage. By identifying the effect of chain topology on the polymers’ performance, this work may provide a chaintopology guideline for designing polymerbased processing aids.

Analysis of the magnetic rod interfacial stress rheometer
View Description Hide DescriptionThe magnetic interfacial needle stress rheometer is a device capable of sensitive rheological interfacial measurements. Yet even for this device, when measuring interfaces with low elastic and viscousmoduli, the system response of the instrument contributes significantly to the measured response. To determine the operation limits of the magnetic rod rheometer, we analyze the relative errors that are introduced by linearly subtracting the instrument contribution from the measured response. An analysis of the fluid mechanics demonstrates the intimate coupling between the flow field at the twodimensional interface and in the bulk at low Boussinesq number. A nonzero Reynolds number is observed to have a similar order of magnitude effect. The resulting nonlinear interfacial deformation profiles lead to an error, which depends on the magnitude of the interfacial modulus, as well as on the phase angle. The conditions under which reliable measurements can be obtained are identified. Based on the analysis of the effects of system response, the Boussinesq and Reynolds numbers, small modifications to the measurement probe are proposed. A reduction of the mass and localization of the magnetic material result in even further improved instrument sensitivity. This is demonstrated experimentally for two cases, i.e., a purely viscous interface of known viscosity, as produced by spreading thin silicon oil films on a water layer, and a timedependent viscoelastic interface, generated by the surface gelation of a lysozyme solution.

Yield stress and elastic modulus of suspensions of noncolloidal particles in yield stress fluids
View Description Hide DescriptionWe study experimentally the behavior of isotropic suspensions of noncolloidal particles in yield stress fluids. This problem has been poorly studied in the literature, and only on specific materials. In this paper, we manage to develop procedures and materials that allow us to focus on the purely mechanical contribution of the particles to the yield stress fluid behavior, independently of the physicochemical properties of the materials. This allows us to relate the macroscopic properties of these suspensions to the mechanical properties of the yield stress fluid and the particle volume fraction, and to provide results applicable to any noncolloidal particle in any yield stress fluid. We find that the elastic modulusconcentration relationship follows a KriegerDougherty law, and we show that the yield stressconcentration relationship is related to the elastic modulusconcentration relationship through a very simple law, in agreement with results from a micromechanical analysis.

A general criterion for viscoelastic secondary flow in pipes of noncircular cross section
View Description Hide DescriptionThis paper investigates the mechanism for secondary recirculations in nonNewtonian flows in a noncircular pipe, and develops a general criterion on the direction of the secondary flow based on the fluid rheology and the crosssectional geometry of the pipe. Although the secondary flow is usually attributed to the second normal stress difference , the relationship between the two turns out to be more involved than previously assumed. By theoretical analysis and numerical computations using the Giesekus model, we show that produces an effective body force that, if nonconservative, gives rise to secondary flows in the transverse direction. From this understanding, we propose a criterion for the direction of the secondary flow based on the second normal stress coefficient and the shear viscosity: if is an increasing function of the strain rate , the fluid flows from high shear regions to low shear regions along the walls and vice versa. This criterion accounts for all the prior computational work and resolves some inconsistencies in the literature. It is also consistent with all experimental observations to date.