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.     

Excluded volume effects on the birefringence and stress of dilute polymer solutions in extensional flow

An algorithm is derived for calculating flow-induced birefringence using a bead-spring model with and without excluded volume effects. The simulation results for the bead-spring model compare well with experimental results for stress and birefringence in extensional flows of dilute solutions of polystyrene molecular weight 2 million in a filament-stretching device in both “theta” and “good” solvents (Orr and Sridhar 1999; Sridhar et al. 2000). In a “good” solvent, both stress and birefringence rise much more rapidly with strain than in a “theta” solvent, making extensional rheology a very sensitive indicator of solvent quality. Received: 7 December 1999 Accepted: 23 May 2000     

Suspensions of monodisperse spheres in polymer melts: particle size effects in extensional flow

Suspensions in polymeric, viscoelastic liquids have been studied in uniaxial extensional flow. The fibre wind-up technique has been used for this purpose. The effects of particle size and particle volume fraction have been investigated, using monodisperse, spherical particles. The results have been compared with shear flow data on the same materials. The values of the relative extensional viscosities at low stretching rates are in agreement with the relative shear viscosities and relative moduli. This indicates that hydrodynamic forces are stronger than the particle interaction forces. At larger strain rates strain hardening occurs; it is suppressed when particles are added. Small aggregating particles reduce the strain hardening more strongly than larger particles; strain hardening can even be totally eliminated. When further increasing the stretching rate, hydrodynamic effects dominate again and the effect of particle size effect on strain hardening disappears.     

Three-dimensional flow in axial flow fans of non-free vortex design

Three-dimensional laser Doppler anemometer (LDA) measurements were carried out downstream of isolated axial fan rotors of non-free vortex design in order to investigate the role of radial velocity components in design. The structure of secondary flows due to non-free vortex operation was studied in detail. It is pointed out that the tangential gradient of radial velocity at midspan is nearly in direct proportion with the spanwise gradient of ideal total head rise prescribed in design. Design criteria have been established for the neglect of torsion of stream surface segments inside the blading. A linear relationship was proposed in order to estimate the pitch-averaged radial velocities at the rotor exit. Using this relationship, a proposal has been put forward for taking the radial velocity components into account in non-free vortex design with the assumption of conical stream surfaces through the blading.     

Resistance to uniaxial extensional flow of fibre suspensions

The tensile stress due to resistance to uniaxial extensional flow of fibre suspensions in Newtonian and non-Newtonian fluids has been measured using the filament stretching technique. It has been found that addition of fibres to a Newtonian fluid increases the extensional viscosity. The steady state results agree with Bachelors theory and the stress growth behaviour is qualitatively predicted by the theory of Dinh and Armstrong. Experimental results from this work have also shown that the behaviour of a fibre suspension in viscoelastic fluid is qualitatively described by Fans equation. The added fibres increase the extensional stress growth coefficient of the viscoelastic fluid at low strain but have marginal effect on the fluid after the onset of strain-hardening.     

Nonequilibrium stretching dynamics of dilute and entangled linear polymers in extensional flow

We propose an extension of the FENE-CR model for dilute polymer solutions [M.D. Chilcott, J.M. Rallison, Creeping flow of dilute polymer solutions past cylinders and spheres, J. Non-Newtonian Fluid Mech. 29 (1988) 382–432] and the Rouse-CCR tube model for linear entangled polymers [A.E. Likhtman, R.S. Graham, Simple constitutive equation for linear polymer melts derived from molecular theory: Rolie–Poly equation, J. Non-Newtonian Fluid Mech. 114 (2003) 1–12], to describe the nonequilibrium stretching dynamics of polymer chains in strong extensional flows. The resulting models, designed to capture the progressive changes in the average internal structure (kinked state) of the polymer chain, include an ‘effective’ maximum contour length that depends on local flow dynamics. The rheological behavior of the modified models is compared with various results already published in the literature for entangled polystyrene solutions, and for the Kramers chain model (dilute polymer solutions). It is shown that the FENE-CR model with an ‘effective’ maximum contour length is able to describe correctly the hysteretic behavior in stress versus birefringence in start-up of uniaxial extensional flow and subsequent relaxation also observed and computed by Doyle et al. [P.S. Doyle, E.S.G. Shaqfeh, G.H. McKinley, S.H. Spiegelberg, Relaxation of dilute polymer solutions following extensional flow, J. Non-Newtonian Fluid Mech. 76 (1998) 79–110] and Li and Larson [L. Li, R.G. Larson, Excluded volume effects on the birefringence and stress of dilute polymer solutions in extensional flow, Rheol. Acta 39 (2000) 419–427] using Brownian dynamics simulations of bead–spring model. The Rolie–Poly model with an ‘effective’ maximum contour length exhibits a less pronounced hysteretic behavior in stress versus birefringence in start-up of uniaxial extensional flow and subsequent relaxation.     

Fingerprinting the processing behavior of polyethylenes from transient extensional flow and peel experiments in the melt state

The flow curves of linear (linear-low and high density) and branched polyethylenes are known to differ significantly. At increasing shear rates, the linear polymers exhibit a surface melt fracture or sharkskin region that is followed by an unstable oscillating or stick-slip flow regime when a constant piston speed capillary rheometer is used. At even higher shear rates, gross melt fracture appears. Unlike their linear counterparts, branched polyethylenes rarely exhibit sharkskin melt fracture and although gross melt fracture appears at high shear rates there is no discontinuity in their flow curve. The various flow regimes of these two types of polyethylenes are examined by performing experiments in the melt state using a unique extensional rheometer (the SER by Xpansion Instruments) that is capable of performing accurate extensional flow and peel experiments at very high rates not previously realized. The peel strength curves of these linear and branched polyethylenes exhibit all of the distinct flow regimes exhibited in their respective flow curves, thereby providing a fingerprint of their melt flow behavior. Moreover, these extensional flow and peel results in the melt state provide insight into the origins and mechanisms by which these melt flow phenomena may occur with regard to rapid tensile stress growth, melt rupture, and adhesive failure at the polymer wall interface.     

Three-dimensional flow pattern visualization and bubble size distributions in stationary and transient upward flashing flow

For the first time, an experimental three-dimensional reconstruction and visualization of stationary and transient flashing flow in a vertical pipe (47 mm diameter) is presented. The measurements have been performed by means of wire-mesh sensors. This type of sensor delivers two-dimensional void-fraction distributions in the pipe cross-section where it is mounted with a maximum sampling rate of 10,000 frames per second. A sampling rate of 1200 frames per second has been used in this work. Steam bubbles have been identified from the wire-mesh data and their complete three-dimensional reconstruction has been performed by taking into account the steam bubble velocity. For the estimation of the bubble velocity, two wire-mesh sensors positioned at a small axial distance from each other have been used. The velocity has been determined by cross-correlation of the two wire-mesh signals, by direct identification of the traveling time of the steam bubbles between the two sensors and by means of a drift-flux model. A comparison between the three methods of bubbles velocity measurement is reported. Stationary and time-dependent bubble size distributions have been derived. The stationary radial void-fraction profiles have been decomposed according to bubble size classes and compared with the results obtained with an equilibrium model.     

Effect of vinyl acetate content and silicate loading on EVA nanocomposites under shear and extensional flow

In this study, three EVAs (ethylene-vinyl acetate co-polymers) with different vinyl contents (VA) ranging from 9 wt% to 28 wt% (EVA9, EVA18 and EVA28) were melt blended with organo-clay to obtain polymer layered silicate nanocomposites. Filler intercalation and exfoliation were evidenced by X-ray diffraction. The melt state viscoelastic properties of EVA nanocomposites were studied to examine the influence of clay in altering the flow properties of these polymeric nanocomposites. The EVA18 and EVA28 nanocomposites exhibited remarkable difference in dynamic and steady shear properties compared to neat polymers. On the other hand, EVA9-5% nanocomposite did not exfoliate and exhibited rheological behaviour very similar to that of the neat polymer. Furthermore, the first normal stress difference was found to be dependent on the silicate loadings when measured at low shear stresses. The uniaxial extensional viscosity measurement indicated that the strain hardening was weaker in EVA nanocomposites compared to neat polymers. Environmental scanning electron (ESE)-microscopy elucidated a possible reason for reduced strain hardening in these systems.     

A new constitutive equation that models extensional flow strain hardening based on evolving natural configurations: stability analysis

Polymer melt viscoelastic fluids often exhibit in elongational flows a significant increase in the elongational viscosity known as strain hardening. This phenomenon could be related to polydispersity, e.g. the presence of a small fraction of very high molecular weight chains whose time frame relaxation spectrum is different from the small chains one. In the present work, we present a fully objective constitutive equation (CE) to primarily model extensional strain hardening based on the new concept of multiple configuration materials. Next, we analyze the CE stability properties with respect to small perturbations about the rest state.     

Modelization of extensional experiments under constant force using memory-integral constitutive equations and stream-tube analysis

In this paper, we propose a numerical simulation of axisymmetric extensional experiments on a viscoelastic polydimethylsiloxane (PDMS) material, using a falling-weight extensional rheometer. The polymer behaviour is represented by a K-BKZ memory-integral constitutive equation, involving a damping function of the Wagner type. Under the assumption of a homogeneous flow zone in the sample, a numerical model is set up, using the stream-tube method and approximating functions. The governing equations of the problem, associated to a limited number of unknowns, are solved by means of the Levenberg-Marquardt optimization algorithm. The numerical results are found to be consistent with the experimental data and reveal the importance of the non-homogeneous flow zone, in relation to the estimation of the extensional strain rate. The calculations involve the sensitivity of the model on the fluid parameters and those concerning the size of the initial column of fluid. The limited computing (CPU) time of the code is also to be underlined.     

Analysis of entry flow to determine elongation flow properties revisited

The minimisation technique proposed by Binding (J. Non-Newtonian Fluid Mech., 27 (1988) 173) was used in our Generalised Engineering Bernoulli Equation framework to relate the entry pressure and stress power. We arrived at a final result similar to Binding's using assumed kinematics. Through subsequent assumptions to the kinematics we finally arrive at a result exactly equivalent to Cogswell's technique (Trans. Soc. Rheol., 16 (1972) 383). Thus, these two techniques are related in this general framework. The techniques were used to predict elongation flow properties of a polymer melt and polymer solution. The results for the polymer melt clearly show Cogswell's technique is adequate at high elongation rates. All these techniques require minimisation of the stress power with respect to the flow volume and discussion is given as to the validity of this minimisation technique. In addition, the approximate variational technique we propose gives clears limits as to when a technique, such as Cogswell's, can be applied.     

Extensional flow of liquid jets formed by bubble collapse in oils under cavitation-generated pressure waves

We report a study of liquid jets which are formed by bubble collapse under cavitation-generated pressure waves. The results obtained for jets formed from samples of a multigrade motor oil provide the first evidence that such jets experience a significant degree of extensional deformation, at high rates of extension. The results support the conclusion that the reduced velocity and final length of such jets, relative to their Newtonian counterparts, is due to an increased resistance to extensional flow. Insofar as the multigrade oils studied here are made viscoelastic by polymer additives and evidently possess significant levels of resistance to extension, the results provide evidence in support of a mitigating effect of viscoelasticity on a cavitation damage mechanism, as mooted by Berker et al. (J Non Newton Fluid Mech 56:333, 1995).     

Structure and orientation dynamics of sepiolite fibers–poly(ethylene oxide) aqueous suspensions under extensional and shear flow,probed by in situ SAXS

The structure and orientation dynamics of sepiolite clay fibers about 1,000 nm long and 10 nm thick, suspended in an aqueous poly(ehtylene oxide) matrix of 10⁵ g/mol molecular mass, have been studied under control extensional and shear flow. A new extensional flow cell developed at the “Laboratoire de Rhéologie” and the combined rheology and small angle X-ray scattering (Rheo-SAXS) setup available at the European Synchrotron Radiation Facility have allowed access to in situ and time-resolved fiber orientations and structure properties in the volume of suspensions under flow. In the volume fractions and shear rate domains for which the suspensions exhibit shear-thinning properties, two regimes of orientation separated by a critical strain rate have been identified under extensional flow.     

Extensional flow of a viscoelastic filament governed by the FENE-CR model

A thin filament model is used to analyze the extensional flow of a viscoelastic thread governed by the FENE-CR model. The problem is solved numerically by finite differences using a third-order upwind scheme in space and a second order Runge-Kutta scheme in time. The behavior of the filament is controlled by the competing effects of surface tension and axial normal stresses which are characterized in terms of three-dimensional groups, the Deborah number De, the extensibility parameter L and the capillary number Ca. Surface tension has a destabilizing effect causing the filament to thin in the mid-section leading to a rupture. On the other hand normal stresses tend to stabilize the filament. If axial normal stresses are sufficiently large the filament deforms almost uniaxially due to strain hardening.     

Three-dimensional flow structure in highly buoyant jet by scanning stereo PIV combined with POD analysis

The flow characteristics and the structure of highly buoyant jet of low density fluid issuing into a stagnant surrounding of high density fluid is studied by scanning stereo PIV combined with proper orthogonal decomposition (POD) analysis. The experiment is carried out at Froude number of 0.3 and Reynolds number of 200, which satisfies the inflow condition due to the unstable density gradient near the nozzle exit. An increase in the maximum mean velocity occurs and the vertical velocity fluctuation is highly amplified near the nozzle exit, which suggests the influence of inflow due to the unstable density gradient. The POD analysis indicates that the vertical velocity fluctuation is the major source of fluctuating energy contributing to the development of the highly buoyant jet. The examination of the POD modes show that the longitudinal structure of the vertical velocity fluctuation is generated along the jet axis having the opposite sign of velocity fluctuation on both sides of the jet axis. The vertical scale of the POD mode decreases with increasing the mode number and results in the frequent appearance of cross-flow across the buoyant jet. The reconstruction flow from the POD modes indicates that the vortex structure is caused by the highly sheared layer between the upward and downward velocity and the inflow is induced by the vortex structure. The magnitude of the vortex structure seems to be weakened with an increase in the distance from the nozzle and the buoyant jet approaches to an asymptotic state in the further downstream.     

Extensional flow of polymer solutions through porous media

The flow behaviour of various polymer solutions of non-hydrolyzed polyacrylamide, hydrolyzed polyacrylamide, polyox and Xanthan was investigated in a plexiglass column having a succession of enlargements and constrictions, and compared with the flow behaviour and mechanical degradation of a solution of non-hydrolyzed polyacrylamide in a packed column of non-consolidated sand. The flow behaviour of this solution was found to be very similar in both the sand pack and plexiglass pore.Apart from the Xanthan solution, all other polymer solutions showed a viscoelastic behaviour in the plexiglass pore. The onset of viscoelastic behaviour, which has previously been defined using the shear rate ( ), stretch rate ( _s ) and Ellis number (E ₁), could be more precisely evaluated using a modified stretch rate (S _G). The pressure losses across the plexiglass pore for different polymer solutions of the same type were found to follow a unique curve provided the suggested group (S _G) was used, a situation which was not achieved with the other rheological parameters.The multipass mechanical degradation of the non-hydrolized polyacrylamide was tested through the sand pack against the suggested group (S _G) and Maerker's group (M _a). It was found that the loss of the solution viscoelasticity due to multipass mechanical degradation was better represented usingS _G thanM _a. A cross-sectional area (cm²) - C ^* critical concentration of polymer (ppm) - d plexiglass pore enlargement diameter - D average sand grain diameter (cm) - e equivalent width for the plexiglass pore - E ₁ Ellis number (a Deborah number) - F _R resistance factor - F _Ri resistance factor at the first pass - h height of the flow path of the plexiglass pore - K power-law constant - K _h,K _w effective permeability to hydrocarbon and water, respectively (10^–8 cm²) - M _a Maerker's group for a given porosity (s^–1) - M _ai value ofM _a at the first pass - N _D Deborah number - n power-law index - Q flow rate (cm³/s) - R capillary radius (cm) - R _g radius of gyration - S _G suggested group of rheological parameters representing a modified maximum stretch rate (s^–1) - S _Gi value ofS _G at the first pass - T _R,t characteristic time for the fluid (s) - t _s residence time (s) - V ₀ superficial velocity (cm/s) - V mean velocity of flow through a porous medium (cm/s) - average axial velocity in the enlargement section of the plexiglass pore (cm/s) - V ₁,V ₂ maximum velocity at a plexiglass enlargement neck and centre - [] intrincis viscosity - viscosity (mPa s) - _r relative viscosity (ratio of the viscosity of the polymer solution to that of the solvent) - shear rate (s^–1) - _s stretch rate (s^–1) - characteristic time for the polymer solution (s)     

On the extensional viscosity of mobile polymer solutions

We describe experimental results on the extensional viscosity of mobile polymer solutions obtained from two instruments, the first being a commercial Spin Line Rheometer and the second a custom-built lubricated-die Converging Flow Rheometer. The interpretation of data in terms of Trouton ratios is facilitated by a simple analysis for the Generalized Newtonian Fluid model.Agreement between data from the two rheometers is satisfactory and we show that polymer solutions can be either tension stiffening or tension thinning. However, the Trouton ratios in both cases are greater than the Newtonian values and we anticipate that this will always be the case for polymer solutions.Invited paper, presented at the 2nd Conference of European Rheologists, Prague, June 17–20, 1986     

Birefringence measurements in extensional flow of polyisobutylene around an expanding bubble

Birefringence in liquid polymers offers the possibility of obtaining information about stress in complex flows. In this work, this is done for extensional flows of polyisobutylene in a “breathing bubble” rheometer. In this type of rheometer, a bubble consisting of an incompressible, low-viscosity fluid (usually water) is injected into the sample with a nozzle. Expanding or collapsing the bubble by adding or removing water induces biaxial or uniaxial extension in the surrounding sample. The pressure difference between the bubble and the surroundings can be measured and compared to the predictions of constitutive equations. This measurement only gives one integral value for a complex flow history. In this paper, the birefringence around the bubble is measured in order to learn more about the flow. This is done by comparing pressure and birefringence results to those of standard constitutive equations for a polyisobutylene sample. A good agreement between the pressure and optical measurements and the theory is found with a single value of the stress-optical constant. Received: 25 June 1997 Accepted: 12 November 1997