Brownian dynamics simulation for suspensions of oblong-particles under shear flow

Computer simulations of the shear flow for the suspensions of oblong-particles were performed using nonequilibrium Brownian Dynamics (BD). The model particle is a rigid body made up of linearly connected spheres with the interparticle potential of a repulsive Lennard-Jones potential. The length-over-width ratios of the oblong-particles used in the present calculations are 5/3 and 3. In the concentrated suspensions high orientation is easily induced by shear at low shear rates. The systems of the oblong-particles exhibit the structural transition that causes the significant change in the rheological properties at high shear rates. Furthermore, the dependence of the length-over-width ratio of the particle is examined. Received: 16 June 1997 Accepted: 3 February 1998     

Brownian dynamics simulations of star-branched polymers in dilute solutions in extensional flow

Using Brownian dynamics (BD) simulations of FENE bead-spring models, the dynamics of star-branched polymers in dilute solutions under extensional flow have been investigated. Studies on star polymers in transient extensional flow reveal that the initial transient stress response at low strains is governed by both the number of arms and the shortest arm. On the other hand, the steady-state behavior of star polymers in elongational flow is limited by the maximum effective “contour” length of the molecules. The distribution of arm extension and birefringence of the star-branched molecule are broader and the mean is shifted to lower values, when compared to equivalent linear systems. As a result, the degree of arm extension at steady-state decreases as the number of arms in the star increases. Both an analysis of individual ensembles in Brownian dynamics simulations and a study of a simple force balance indicate that the constraint imposed on the star arms by the central branch point and contributions from “asymmetric” arm arrangements give rise to overall less extended and oriented star-branched molecules with broader arm extension and birefringence distributions. The results obtained from stress-conformation hysteresis simulation indicate that less-stretched arms exhibit more retarded relaxation, as the number of arms increases in star-branched molecules. The effect of excluded volume (EV) interactions, incorporated through the Lennard–Jones potential, on the dynamics of star polymers in extensional flow appears unimportant.     

A terminally anchored polymer chain in shear flow: Self-consistent velocity and segment density profiles

The behavior of a terminally anchored freely-jointed bead-rod chain, subjected to solvent shear flow, was investigated via Brownian dynamics simulations. Previous calculations have been improved by computing the segment density and fluid velocity profiles self-consistently. The segment density distributions, components of the radius of gyration, and chain attachment shear and normal stresses were found to be sensitive to low values of shear rate. Additionally, it was found that the thickness of a model polymer layer was a strong function of the shear rate, and that the functional dependence on shear rate changed dramatically as the chain length increased. For the longest chains studied, the thickness of the model polymer layer first increased as the shear rate increased, passed through a maximum, and then decreased at high shear rates, in accordance with experimental results in theta solvents. These results suggest that a dilute or semi-dilute layer model may explain hydrodynamic behavior previously thought to be due to the entanglements that occur in dense surface bound polymer layers.Nomenclature a _i acceleration of bead i - b radius of the beads - d length of the rods connecting the chain beads - d _i vector from bead i to bead i + 1 - F _i external force applied to bead i - F _i ^b external force on bead i due to Brownian motion of surrounding fluid - F _i ^h external force on bead i due to viscous drag - F _i ^s external force on bead i due to surface interactions - f Stokes drag coefficient - Boltzmann's constant - L _h effective hydrodynamic thickness - m _i mass of bead i - N number of beads on a model chain - n number of chains anchored to the surface per unit surface area - P segment density distribution P pressure - Q flow in a tube with no surface bound polymer layer - Q _a flow in a tube with a surface bound polymer layer - R _g vector representation of the radius of gyration - R tube radius - r radial coordinate in the tube geometry - S _ij pair hydrodynamic interaction tensor for beads i and j - T _i internal chain force in rod i connecting beads i and i + 1 - T _X component of the surface attachment force in the direction of the fluid flow - T _y component of the surface attachment force perpendicular to the surface - T temperature - v _i velocity of the center of mass of bead i - V _if average fluid velocity at the location of bead i - v _if ⁰ fluid velocity in the absence of a polymer chain - v _if perturbation to the fluid velocity due to hydrodynamic interactions - V _b bead volume = 4 b ³/3 - scalar fluid speed in the axial direction down the tube - x axial coordinate in the tube geometry Greek symbols _w apparent shear rate - fluid viscosity - polymer layer permeability - volume fraction of space occupied by chain beads - (_w)_a chain attachment stress perpendicular to the surface - (_w)_a chain attachment stress in the plane of the surface and in the direction of fluid flow     

Brownian dynamics simulations for suspension of ellipsoids in liquid crystalline phase under simple shear flows

Brownian dynamics simulations of shear flows are carried out for various suspensions of ellipsoids interacting via the Gay-Berne potential. In this simulation all the systems of the suspension are in a liquid crystalline phase at rest. In a shear flow they exhibit various motions of the director depending on the shear rate: the continuous rotation, the intermittent rotation, the wagging-like oscillation, and the aligning. The director is almost always out of the vorticity plane when it rotates, that is the kayaking. The number density of the system and the inter-particle potential intensity significantly affect the shear rate dependence of orientation. In particular, the continuous rotation of director is maintained to higher shear rates for the system with a stronger potential. Furthermore, the rheological properties are examined. The shear-thinning in viscosity is observed, but the negative first normal difference is not obtained.     

Brownian dynamics simulations of shear-thickening in dilute polymer solutions

A versatile model describing the shear thickening behaviour of dilute polymer solutions in high shear flows is presented. The polymer macromolecules are modelled as Hookean elastic dumbbells which deform affinely during flow. In addition, the dumbbells feel a retractive anisotropic hydrodynamic drag and an isotropic Brownian force. Furthermore, it is assumed that high shear rate increases the probability of molecules forming associations and this is described through expressions for the frequencies of association and dissociation, without explicitly accounting for finite extensibility, hydrodynamic interaction or excluded volume effects. Thus, a reversible kinetic process is incorporated into the model, which results in two diffusion equations for the associated and dissociated dumbbells. Numerical simulations predict shear thickening for specific range of parameters, which are physically meaningful and related to molecular characteristics of the polymer. A comparison against experimental data reported in the literature revealed very promising results, thus confirming the ability of this model to predict shear thickening under a wide range of conditions, for various polymer models.Nomenclature A A factor in the frequency of association - B Frequency of dissociation - B ₀ Reference frequency of dissociation - c Concentration of polymer solution - c _i Concentration of singlets (i = 1) and doublets (i = 2) in the solution - c ^* The overlap concentration - D _t Translation coefficient of molecule - F _i (Q) Spring force for a singlet (i = 1) and for a doublet (i = 2) - F Frequency of association - F ₀ Reference frequency of association - H _i Dumbbell spring constant for a singlet (i = 1) and for a doublet (i = 2) - k Boltzman's constant - k _H Huggins constant - MW Molecular weight - MW _c Critical molecular weight for formation of entanglements - n Number density of molecules in the polymer solution - n ₀ Number density of dumbbells at equilibrium - n _i Number density of singlets (i = 1) and doublets (i = 2) - Q Vector defining the size and orientation of a dumbbell - t Time - T Absolute temperature - x Degree of multimerization - W Interaction energy between the two components of a doublet Greek letters a Dimensionless anisotropy parameter - Shear rate - _i Friction coefficient of singlets (i = 1) and doublets (i = 2) - _i Intrinsic viscosity of singlets (i = 1) and doublets (i = 2) - _red Reduced viscosity of solution - _sp Specific viscosity - Viscosity of the polymer solution of concentration c - _s Viscosity of the solvent - (t) White noise - K ^T Velocity gradient tensor - _Hi Time constant of a singlet (i = 1) and a doublet (i = 2) - ₁ Length scale of singlets (standard deviation of singlet lengths at equilibrium) - ₂ Length scale of doublets - T _p Stress tensor - T _xy Shear Stress (xy element of T _p ) - T _pi Contributions to the stress tensor of singlets (i = 1) and doublets (i = 2) - ₀ Equilibrium configuration distribution function of Q - _i Configuration distribution function of singlets (i = 1) and doublets (i = 2)     

Orientation dynamics in commercial thermotropic liquid crystalline polymers in transient shear flows

In situ X-ray scattering measurements of molecular orientation under shear are reported for two commercial thermotropic liquid crystalline polymers (TLCPs), Vectra A950® and Vectra B950®. Transient shear flow protocols (reversals, step changes, and flow cessation) are used to investigate the underlying director dynamics. Synchrotron X-ray scattering in conjunction with a high-speed area detector provides sufficient time resolution to limit the total time spent in the melt during testing, whereas a redesigned X-ray capable shear cell provides a more robust platform for working with TLCP melts at high temperatures. The transient orientation response upon flow inception or flow reversal does not provide definitive signatures of either tumbling or shear alignment. However, the observation of clear transient responses to step increases or step decreases in shear rate contrasts with expectations and experience with shear-aligning nematics and suggests that these polymers are of the tumbling class. Finally, these two polymers show opposite trends in orientation following flow cessation, which appears to correlate with the evolution of dynamic modulus during relaxation. Specifically, Vectra B shows an increase in orientation upon flow cessation, an observation that can only be rationalized by the assumption of tumbling dynamics in shear. Together with prior observations of commercial LCP melts in channel flows, these results suggest that this class of materials, as a rule, exhibits director tumbling.     

Computer simulation of polymer brushes under shear

 Recently two different methods were used to simulate the stationary properties of polymer brushes under strong shear: stochastic dynamics of a multi-chain brush model, and self-consistent Brownian dynamics of a one-chain model. The former explicitly describes volume interactions (VI) between polymer segments but does not take into account hydrodynamic interactions (HI) inside the brush. In the latter the self-consistent molecular field method has been chosen to calculate VI, and HI were accounted for using the Brinkman equation. Despite a significant difference between models a collapse of the brush under shear was observed in both studies. In particular, the density profile changes from parabolic to step-like and the free ends of the chains become concentrated in a narrow region at the periphery of the brush. However, when HI are taken into account much higher shear rates are necessary to attain the same brush deformation because the shear flow only slightly penetrates into the brush in contrast to the free-draining case. The inner brush structure is also found to be different for the two models. In the first model all chains are inclined approximately at the same angle when shear is applied. In the second model chains with the free ends found in the inner sublayer of the brush do not feel the flow at all whereas those in the upper sublayer are stretched and inclined by the flow. Received: 24 June 1999 Accepted: 8 February 2000     

Hydrodynamic interactions of dilute polymer solutions under shear flow in a narrow channel

Hydrodynamic interactions on dilute solutions of spherical beads under shear flow are calculated with the method of induced forces. The Navier-Stokes equation is considered in the Stokes approximation. Hydrodynamic interactions cause the drag to be anisotropic in space.Numerical solutions are obtained for the added stress, caused by polymeric molecules in solution in a narrow channel under shear flow. The polymeric molecules are considered as Hookean spring-dumbbells.Slip velocity and the effective viscosity are obtained taking different dumbbells' bead radii. Transversal migration in the channel is obtained for different bead radii.     

Ordering effects in shear flows of discotic polymers

The tensorial mechanical model of Farhoudi and Rey (1993) for uniaxial, rodlike, spatially homogeneous and monodomain nematics is modified to describe the microstructural response of discotic nematic network polymers in rectilinear simple shear flow. The particular topological features of the discotic phase are taken into account by a proper modification of the phenomenological parameters. Asymptotic and numerical solutions of the microstructural balance equations indicate the appearance of tumbling, oscillating, and stationary flow regimes as the strength of shear increases, as is the case for rod-like nematic polymers (Marrucci, 1991). The tumbling-oscillating transition is characterized by a diverging tumbling function , while the oscillating-stationary transition is characterized by a single steady value smaller than —1. The stable steady states of the stationary regime are shown to belong to the family of unstable isotropic solutions that exist at small shear rates, and are characterized by a director angle close to, but less than +90° to the flow direction.     

Rheology of rodlike polymers in the nematic phase with tumbling or shear orientation

The theory of rodlike polymers in the nematic phase is now sufficiently well developed as to allow predictions of the rheological behaviour that qualitatively compare well with the experimental observations. One of the main results of the theory is the prediction that rodlike polymers are nematics of the tumbling type at low shear rates, whereas they become shear oriented at high rates: a nonlinear effect, which is absent in low molecular weight nematics. This aspect is here reviewed in an effort to highlight the intuitive aspects of the matter. First, the low shear-rate situation is discussed in order to investigate the conditions that determine the existence of a stationary solution as opposed to a periodic one (tumbling). Then, the high shear-rate range is considered, where the shear-oriented situation prevails under all conditions. The intermediate range of shear rates is the most interesting one for its peculiar rheological behavior.Delivered as a Keynote Lecture at the Golden Jubilee Conference of the British Society of Rheology and Third European Rheology Conference, Edinburgh, 3–7 September, 1990.     

Large eddy simulation of bluff body wake in planar shear flow

Large eddy simulation of planar shear flow past a square cylinder has been investigated. Dynamic Smagorinsky model has been used to model subgrid scale stress. The shear parameter, K, namely the nondimensional streamwise velocity gradient in the lateral direction, is 0.0, 0.1 and 0.2. Reynolds number based on the centerline velocity is fixed at Re=21400. The time and span‐averaged velocity components, pressure coefficient, Reynolds stresses for uniform are in good agreement with the literature. In shear flow the calculated flow structure and mean velocity components are shown to be markedly different from those of the uniform flow. With increasing shear parameter, the cylinder wake is dominated by clockwise vortices. Both the velocity components in shear flow are compared with respective components in uniform flow. Comparison of normal and shear stresses between shear and no shear case have also been presented. Copyright © 2009 John Wiley & Sons, Ltd.     

Monte Carlo simulation of self-avoiding lattice chains subject to oscillatory shear flow in polymer solution

 This paper has introduced a pseudo-potential in bond-fluctuation model to simulate oscillatory shear flow of multiple self-avoiding chains in three dimensions following our previous work under simple shear flow. The oscillatory flow field was reasonably reproduced by lattice Monte Carlo simulation using this pseudo-potential neglecting hydrodynamic interaction. By sampling the configuration distribution functions, the macroscopic viscoelasticity of semi-concentrated polymer solution was determined. Both Newtonian and non-Newtonian regimes were studied. The complex modulus and dynamic viscosity exhibit a reasonable power relation with oscillatory frequency, which is consistent with present theories and experiments. Consequently, lattice Monte Carlo simulation has been extended to model free-draining self-avoiding multi chains subject to oscillatory shear flow and to investigate associated viscoelasticity on the molecular level. Received: 1 October 1999 Accepted: 19 October 1999     

Molecular dynamics simulation of liquid argon flow at platinum surfaces

The micro Poiseuille flow for liquid argon flowing in a nanoscale channel formed by two solid walls was studied in the present paper. The solid wall material was selected as platinum, which has well established interaction potential. We consider the intermolecular force not only among the liquid argon molecules, but also between the liquid argon atoms and the solid wall particles, therefore three regions, i.e. the liquid argon computation domain, the top and bottom solid wall regions are included for the force interaction. The present MD (Molecular Dynamics) simulation was performed without any assumptions at the wall surface. The objective of the study is to find how the flow and the slip boundaries at the wall surface are affected by the applied gravity force, or the shear rate. The MD simulations are performed in a nondimensional unit system, with the periodic boundary conditions applied except in the channel height direction. Once the steady state is reached, the macroscopic parameters are evaluated using the statistical mechanics approach. For all the cases tested numerically in the present paper, slip boundaries occur, and such slip velocity at the stationary wall surface increases with increasing the applied gravity force, or the shear rate. The slip length, which is defined as the distance that the liquid particles shall travel beyond the wall surfaces to reach the same velocity as the wall surface, sharply decreases at small shear rate, then slightly decreases with increasing the applied shear rate. We observe that the liquid viscosity remains nearly constant at small shear rates, and the Newtonian flow occurs. However, with increasing the shear rate, the viscosity increases and the non-Newtonian flow appears.     

Transient shear flow properties of fiber-filled polyethylene melts

The growth and relaxation of shear and normal stresses have been investigated for glass and carbon fiber-filled polyethylene melts over a wide range of shear rates and temperatures by means of a cone-and-plate rheogoniometer. Flow parameters and flow curves characterizing the stress overshoot and relaxation phenomena of the fiber-filled systems were determined experimentally. The influence of fiber loading, fiber size and temperature on the transient flow parameters are discussed.Predictions by the Meister and Bogue constitutive equations were compared with the experimental data for the transient shear and normal stresses. These equations predict satisfactorily the non-linear transient shear flow of polymer melts and its fiber-filled systems.     

Orientation and rheology of rodlike particles with weak Brownian diffusion in a second-order fluid under simple shear flow

An analysis of particle orientation in a dilute suspension of rodlike particles in a second-order fluid was performed to examine the effects of the elasticity of the fluid and of weak Brownian diffusion of the particle on its orientation. Distributions of particle orientation under a simple shear flow with rate of shearg have been obtained as a function of a single nondimensional parameter, ^* =/r _e ² (D/g), which combines the effects of the particle aspect ratior _e , the weak fluid elasticity, and the weak Brownian rotation diffusion coefficientD of the particle. In the limit of larger _e , when the fluid elasticity is strong enough to overcome the rotational diffusion effect on the particle motion, most of the particles will orient close to the vorticity axis. A new shear-thinning mechanism of the shear viscosity of such systems is predicted by the theory.     

Orientation and rheology of rodlike particles with strong Brownian diffusion in a second-order fluid under simple shear flow

An analysis of orientation in a dilute suspension of rodlike macromolecules in a second-order fluid is presented and the effect of the elasticity of the fluid on the orientation of the suspended particles is examined. Distributions of particle orientation under a simple shear flow have been obtained for small β where β is the ratio of the intrinsic relaxation time of the fluid to the rotational relaxation time of the particle, the latter being inversely proportional to the Brownian rotation diffusion coefficient D_r of the particle. The parameter β represents also the ratio of the Weissenberg number of the fluid to the non-dimensional shear rate, g/D_r. An expression of the stress tensor of the suspension is derived and used in conjuntion with the orientation distribution to obtain the rheological properties of the mixture subjected to a simple shear.     

Director dynamics of uniformly aligned nematic liquid crystals in transient shear flow

We have developed a modular rheo-optical apparatus to study the flow properties of liquid crystals. Its main components are shearing device, strong magnetic field, and optical microscope. We performed experiments on well defined initial morphologies with uniform molecular alignment. The monodomains were achieved with strong magnetic fields (4.7T). Time-resolved conoscopy is the primary optical technique in our investigation. We propose a simple relation between the distribution of alignment angles over the sample thickness and the conoscopically measured angle, to quantitatively measure the alignment angle in shear flow.We followed the relaxation of a shear-induced splay deformation in small molecule model systems (N-(p-methoxybenzylidene) p-butylaniline (MBBA), pentyl-cyano-biphenyl (5 CB) and a commercially available mixture OM14244). We define a rotational director diffusivity (K _s splay elastic constant.i7s splay viscosity) from the relaxation process and devised a model, based on the diffusion equation to determine their values.The director alignment behavior of the small molecule liquid crystals (SMLC's) in shear flow is well described by the two-dimensional Leslie-Ericksen model. The effect of director elasticity can clearly be seen in our experiments, resulting in a decrease of the steady state alignment angle at smaller Ericksen numbers.We found that there is no strain rate dependence of the director vorticity from 0.002/s to 2/s for poly-(-benzyl-D/L-glutamate) (PBG). We determined ₂/₃ = –44 for a 2007o solution of 280000 molecular weight PBG in m-cresol at 20°C. The conoscopic interference pattern vanished after 8 strain units from an initially planar alignment and shearing could be reversed up to 10 strain units to completely recover the initial monodomain.Presented at 4th Meeting of European Rheologists, Sept. 4–9, 1994, Seville, SpainDedicated to Prof. H. Janeschitz-Kriegl at the occasion of his 70th birthday.     

Laminar shear flow of plastic viscoelastic fluids

Summary The theory of plastic viscoelastic fluids was developed by the author to represent the rheological behavior of polymer melts and solutions with high loading of small particles. The present paper develops an asymptotic formulation of the general theory which applies to laminar shear flows. The formulation is analogous to Criminale, Ericksen and Filbey's theory for viscoelastic fluids. We apply this to study plane Poiseuille and Couette flow.With 2 tables     

Evaluation of rheological constitutive equations for branched polymers in step shear strain flows

The pom-pom rheological constitutive equation for branched polymers proposed by McLeish and Larson is evaluated in step shear strain flows. Semianalytic expressions for the shear-stress relaxation modulus are derived for both the integral and approximate differential versions of the pom-pom model. Predictions from the thermodynamically motivated differential pompon model of ?ttinger are also examined. Single-mode integral and differential pom-pom models are found to give qualitatively different predictions, the former displays time–strain factorability after the backbone stretch is relaxed, while the latter does not. We also find that the differential pompon model gives quantitatively similar predictions to the integral pom-pom model in step strain flows. Predictions from multimode integral and differential pom-pom models are compared with experimental data on a widely characterized, low-density polyethylene known as 1810H. The experiments strongly support time–strain factorability, while the multimode pom-pom model predictions show deviations from this behavior over the entire range of time that is experimentally accessible.