Steady and transient shear flows of polystyrene solutions I: Concentration and molecular weight dependence of non-dimensional viscometric functions

Start up from rest and relaxation from steady shear flow experiments have been performed on monodisperse polystyrene solutions with molecular weight ranging from 1.3 × 10⁵ to 1.6 × 10⁶ and concentration c ranging from 5% to 40%. A method of reduced variables based on the use of a characteristic time τ_w is proposed. τ_w is defined as the product of zero shear viscosity with the steady state elastic compliance.Reduced steady and transient viscometric functions so obtained depend on the ratio M/M_e (where M_e is the entanglement molecular weight). Limiting forms are obtained when M/M_e ? 18. In steady flow, a simple correlation is found between shear and normal stresses.In stress relaxation experiments, independent of shear rate, the long-time behaviour can be characterised by a single relaxation time τ₁, which is identical for shear and normal stresses. τ₁ can be simply related to the zero shear rate viscosity and the limiting elastic compliance.     

Microsecond relaxation processes in shear and extensional flows of weakly elastic polymer solutions

In this paper, we introduce an experimental protocol to reliably determine extensional relaxation times from capillary thinning experiments of weakly elastic dilute polymer solutions. Relaxation times for polystyrene in diethyl phthalate solutions as low as 80?μ s are reported: the lowest relaxation times in uniaxial extensional flows that have been assessed so far. These data are compared to the linear viscoelastic relaxation times that are obtained from fitting the Zimm spectrum to high frequency oscillatory squeeze flow data measured with a piezo-axial vibrator (PAV). This comparison demonstrates that the extensional relaxation time reduced by the Zimm time, λ _ext/λ _z, is not solely a function of the reduced concentration c/c*, as is commonly stated in the literature: an additional dependence on the molecular weight is observed.     

Shear viscosity of slightly-elastic concentrated suspensions at low and high shear rates

A quasi-static asymptotic analysis is employed to investigate the elastic effects of fluids on the shear viscosity of highly concentrated suspensions at low and high shear rates. First a brief discussion is presented on the difference between a quasi-static analysis and the periodic-dynamic approach. The critical point is based on the different order-of-contact time between particles. By considering the motions between a particle withN near contact point particles in a two-dimensional “cell” structure and incorporating the concept of shear-dependent maximum packing fraction reveals the structural evolution of the suspension under shear and a newly asymptotic framework is devised. In order to separate the influence of different elastic mechanisms, the second-order Rivlin-Ericksen fluid assumption for describing normal-stress coefficients at low shear rates and Harnoy's constitutive equation for accounting for the stress relaxation mechanism at high shear rates are employed. The derived formulation shows that the relative shear viscosity is characterized by a recoverable shear strain,S _R at low shear rates if the second normal-stress difference can be neglected, and Deborah number,De, at high shear rates. The predicted values of the viscosities increase withS _R , but decrease withDe. The role ofS _R in the matrix is more pronounced than that ofDe. These tendencies are significant when the maximum packing fraction is considered to be shear-dependent. The results are consistent with that of Frankel and Acrivos in the case of a Newtonian suspension, except for when the different divergent threshhold is given as [1 ? (Φ/Φ _m )^1/2] ? 1.     

Shear rheology of polymer solutions near the critical condition for elastic instability

The use of constant viscosity, highly elastic polymer solutions, so called Boger fluids, has been remarkably successful in elucidating the behavior of polymeric materials under flowing conditions. However, the behavior of these fluids is still complicated by many different physical processes occurring within a narrow window of observation time and applied shear rate. In this study, we investigate the long-time shear behavior of an ideal Boger fluid: a well characterized, athermal, dilute, binary solution of high molecular weight polystyrene in oligomeric polystyrene. Rheological measurements show that under an applied steady shear flow, this family of polymer solutions undergoes a transient decay of normal stresses on a timescale much longer than the polymer molecule's relaxation time. Rheological and flow visualization results demonstrate that the observed phenomenon is not caused by polymer degradation, phase separation, viscous heating, or secondary flows from elastic instabilities. Although the timescale is much shorter than that associated with polymer migration in the same solutions (MacDonald and Muller, 1996), the appearance of this phenomenon only at the rates where migration has been observed suggests that it may be a prerequisite for observing migration. In addition, we note that through sufficient preshearing of the sample, the normal stress decrease suppresses the elastic instability. These results show that there is considerable uncertainty in choosing the appropriate measure of the fluid relaxation time for consistently modeling the critical condition for the elastic instability, the decay of normal stresses, and the migration of polymer species.     

Experimental study on the capillary thinning of entangled polymer solutions

The transient elongation behavior of entangled polymer and wormlike micelles (WLM) solutions has been investigated using capillary breakup extensional rheometry (CaBER). The transient force ratio X = 0.713 reveals the existence of an intermediate Newtonian thinning region for polystyrene and WLM solutions prior to the viscoelastic thinning. The exponential decay of X(t) in the first period of thinning defines an elongational relaxation time λ _x which is equal to elongational relaxation time λ _e obtained from exponential diameter decay D(t) indicating that the initial stress decay is controlled by the same molecular relaxation process as the strain hardening observed in the terminal regime of filament thinning. Deviations in true and apparent elongational viscosity are discussed in terms of X(t). A minimum Trouton ratio is observed which decreases exponentially with increasing polymer concentration leveling off at Tr_min = 3 for the solutions exhibiting intermediate Newtonian thinning and Tr_min ≈ 10 otherwise. The relaxation time ratio λ _e/ λ _s, where λ _s is the terminal shear relaxation time, decreases exponentially with increasing polymer concentration and the data for all investigated solutions collapse onto a master curve irrespective of polymer molecular weight or solvent viscosity when plotted versus the reduced concentration c[ η], with [ η] being the intrinsic viscosity. This confirms the strong effect of the nonlinear deformation in CaBER experiments on entangled polymer solutions as suggested earlier. On the other hand, λ _e ≈λ _s is found for all WLM solutions clearly indicating that these nonlinear deformations do not affect the capillary thinning process of these living polymer systems.     

Molecular interpretation of the behaviour of polyisobutylene in different solvents

The effects of solvent environment on the behaviour of a high molecular weight polyisobutylene dissolved in kerosene and various grades of poly-1-butene solvent mixtures are investigated. The dependence of various molecular parameters such as zero-shear viscosity, intrinsic viscosity, specific viscosity, relaxation time and molecular expansion factor, on the polymer concentration, type of solvent and solvent viscosity is studied in the vicinity of dilute and semidilute regions (near the critical concentrationc ^*). The dependence of these parameters on solvent environment follows qualitatively Zimm's molecular model. The dependence on the polymer concentration deviates from this dilute solution theory. The effects of temperature on the zero-shear viscosity and the Maxwell relaxation time are also presented for two PIB solutions.     

Mesoscopic dynamics of polymer chains in high strain rate extensional flows

We take a step towards accessing the physics of viscoelastic liquid breakup in high speed, high strain rate flows by performing Brownian dynamics computations of dilute uniaxial, equibiaxial, and ellipsoidal polymeric extensional flows. Our computational implementation of the bead-spring model, when tailored to the DNA molecule, consistently with recent works of Larson and co-workers, is shown: (a) to predict a coil-stretch transition at Deborah number De≈0.5, and (b) to reproduce the experimental longest relaxation time. Furthermore, after adapting the model parameters to represent the polyethylene oxide (PEO) chain (for M=10⁶ Da), we find it possible to reproduce our own experimental data of the longest relaxation time, the transient extensional viscosity of dilute solutions at small Deborah numbers, and a coil-stretch transition at Deborah number De≈0.5. Extended to large Deborah numbers, the model predicts that polymer stretching is controlled by: (a) the randomness of the initial conditions that, in combination with rapid kinematically imposed compression, lead to the formation of initially frozen chain-folds, and (b) the speed with which thermo-kinematic processes relax these folds. The slowest fold relaxation occurs during uniaxial extension. As expected, the introduction of stretching along a second direction enhances the efficiency of fold relaxation mechanisms. Even for Deborah numbers (based on the chain longest relaxation time) of the order of one thousand, there is a large variation in the time a polymer needs in order to extend fully, and the effects of Brownian motion cannot be ignored. The computed Trouton ratios and polymer contributions to the total stress as functions of Hencky strain provide information about the relative importance of elastic effects during polymeric liquid stretching. At high strain rates, the steady state elastic stresses increase linearly with the Deborah number, resembling at that stage an anisotropic Newtonian fluid (constant extensional viscosity).     

The importance of downstream events in microfluidic viscoelastic entry flows: Consequences of increasing the constriction length

Polymer solutions and melts can exhibit large upstream corner and lip vortices, unstable and diverging flow and an enhanced pressure drop when flowing through a geometry containing a constriction. In the present work, we use a planar microfluidic device to show that the length of the downstream constriction plays an important role in the upstream kinematics and the extra pressure drop. That is, the elastic flow phenomena observed upstream of a constriction during entry flows of polymer solutions are not exclusively a result of the stretching dynamics induced by the converging flow—the downstream relaxation events are, at least, equally important. Flow visualization experiments with semi-dilute solutions of a high molecular weight polymer showed that large stable symmetric vortices could be reduced to highly chaotic asymmetric flow, merely by increasing the length of the constriction—the Reynolds number and elasticity number were both held constant. This was accompanied by a higher extra pressure. These results support the hypothesis that elastic flow instabilities originate downstream of the constriction (at the expansion) and move progressively upstream with time and/or flowrate. These findings may also partly explain the discrepancies commonly observed between the results of entry flow experiments and numerical simulations, in which the downstream geometry is very rarely considered. Lastly, we illustrate how to minimize the occurrence of unstable flow upstream of a constriction, which is a necessary condition for closed microrheometry devices used to characterize low viscosity elastic fluids.     

An efficient and accurate fully discrete finite element method for unsteady incompressible Oldroyd fluids with large time step

This paper proposes a second‐order accuracy in time fully discrete finite element method for the Oldroyd fluids of order one. This new approach is based on a finite element approximation for the space discretization, the Crank–Nicolson/Adams–Bashforth scheme for the time discretization and the trapezoid rule for the integral term discretization. It reduces the nonlinear equations to almost unconditionally stable and convergent systems of linear equations that can be solved efficiently and accurately. Here, the numerical simulations for L², H¹ error estimates of the velocity and L² error estimates of the pressure at different values of viscoelastic viscosities α, different values of relaxation time λ₁, different values of null viscosity coefficient μ₀ are shown. In addition, two benchmark problems of Oldroyd fluids with different solvent viscosity μ and different relaxation time λ₁ are simulated. All numerical results perfectly match with the theoretical analysis and show that the developed approach gives a high accuracy to simulate the Oldroyd fluids under a large time step. Furthermore, the difference and the connection between the Newton fluids and the viscoelastic Oldroyd fluids are displayed. Copyright © 2015 John Wiley & Sons, Ltd.     

Linear and non-linear rheology of linear polydisperse polyethylene

Rheological techniques, size-exclusion chromatography, and molecular spectroscopy are the most widely used tools for describing polymer molecular structure in polyolefins. The detection of long-chain branching, and to some extent, its quantification, have been based on quantifying the deviation of polyethylene??s (PE) rheological behavior from that of a linear reference. Although metallocene-based PE has been extensively studied, linear polydisperse originating from Ziegler or Chromium-based catalysts are not often thoroughly considered, despite their high industrial importance. Within this work, we study the linear and non-linear rheology of a set of polydisperse PEs, for which the topological linearity is confirmed by GPC-MALLS measurements. Thus, we can safely quantify the effect of broad molecular weight distribution, high and ultra-high molecular weight fractions on rheological quantities and model parameters. Specifically, the zero-shear viscosity, ?? ₀ vs. M _w, relaxation spectra, phase lag vs. the complex modulus plot (van Gurp?CPalmen method) were applied and significant deviations from the ??rheologically linear?? behavior were observed, attributed only to M _w, M _z and polydispersity. Since the elongational viscosity was typical of linear PE, large-amplitude oscillatory shear and FT-Rheology were applied to quantify the non-linear rheological behavior. The latter was described by a single parameter, $Q=I_{3/1}/\gamma_0^2$ , which for linear polydisperse PE was correlated to the high molecular weight fraction and was constant over a broad range of applied Deborah numbers for the respective excitation frequencies. Since we need to correlate structural features such as broad MWD and HMW to polymer performance under processing conditions, we have to extend the analysis of linear rheological parameters, such as zero-shear viscosity, to non-linear parameters, e.g., the Q parameter quantified and used here.     

Elasticity and dynamics of particle gels in non-Newtonian melts

We investigate the relation between the structure and the viscoelastic behavior of a model polymer nanocomposite system based on a mixture of titanium dioxide (TiO₂) nanoparticles and polypropylene. Above a critical volume fraction, Φ _c, the elasticity of the hybrids dramatically increases, and the frequency dependence of the elastic and viscous moduli reflects the superposition of the independent responses of the suspending polymer melt and of an elastic particle network. In addition, the elasticity of the hybrids shows critical behavior around Φ _c. We interpret these observations by hypothesizing the formation of a transient network, which forms due to crowding of particle clusters. Consistent with this interpretation, we find a long-time, Φ-dependent, structural relaxation, which emphasizes the transient character of the structure formed by the particle clusters. For times below this characteristic relaxation time, the elasticity of the network is Φ-independent and reminiscent of glassy behavior, with the elastic modulus, G^′, scaling with frequency, ω, as G^′∼ω ^0.3. We expect that our analysis will be useful for understanding the behavior of other complex fluids where the elasticity of the components could be superimposed.     

Diffusionally assisted grain-boundary sliding and viscoelasticity of polycrystals

Motivated by recent attenuation experiments on finely grained samples, we reanalyse the Raj-Ashby model of grain-boundary sliding. Two linearly elastic layers having finite thickness and identical elastic constants are separated by an interface (grain boundary) whose location is a given periodic function of position. Dissipation is confined to that interfacial region. It is caused by two mechanisms: a slip (boundary sliding) viscosity, and grain-boundary diffusion, with corresponding Maxwell relaxation times t_v and t_d. Owing to the assumption of a given, time-independent interface, the resulting boundary-value problem (b.v.p.) is linear and time-separable. The response to time-periodic forcing depends on angular frequency ω, on the ratio M=t_v/t_d of Maxwell times, and on the characteristic interface slope. The b.v.p. is solved using a perturbation method valid for small slopes. To relate features of the mechanical loss spectrum previously studied in isolation, we first discuss the solution as a function of M. Motivated by experiments, we then emphasize the case M?1 in which the relaxation times are widely separated. The loss spectrum then always has two major features: a frequency band 1?ωt_d?M^-1 within which the loss varies relatively weakly with ω; and a loss maximum at ωt_d∼M^-1 due to the slip viscosity. If corners on the interface are sufficiently rounded, those two universal features are separated by a third feature: between them, there is a strong minimum whose location is (entirely) independent of slip viscosity. The existence of that minimum has not previously been reported. These features are likely to occur even in solutions for finite interface slopes, because they are a consequence of the separation of timescales. The precise form of the spectrum in the weakly varying band must, however, be slope-dependent because it is controlled by stress singularities occurring at corners, and the strength of those singularities depends on the angle subtended by the corner.     

Rheological behavior of silica suspensions flocculated by bridging

The viscosity and stress relaxation behavior of silica suspensions in polyacrylamide (PAAm) solutions have been studied as a function of particle concentration, particle diameter, and molecular weight of PAAm by the use of a coaxial cylinder type rheometer. The effects of polymer adsorption on the flocculation of particles and the rheological behavior are discussed in terms of bridging. The suspensions of 10-nm silica are remarkably pseudoplastic because the particles are easily flocculated by bridging. The ability of PAAm to flocculate silica particles is very extensive at a molecular weight of 5.5 × 10⁶. For suspensions of 20-nm silica in a solution of PAAm with M_W = 5.5 × 10⁶ − 1 × 10⁷, the apparent viscosity irreversibly increases with shearing time at shear rates beyond a certain value. This may be due to the flocculation by the shear-induced bridging. The suspensions of 40-nm silica show similar flow behavior to the medium irrespective of molecular weight of PAAm. The bridging flocculation is not expected for large particles as one polymer molecule cannot bridge through many particles.     

Numerical simulation of a bubble rising in shear-thinning fluids

The motion of a single bubble rising freely in quiescent non-Newtonian viscous fluids was investigated experimentally and computationally. The non-Newtonian effects in the flow of viscous inelastic fluids are modeled by the Carreau rheological model. An improved level set approach for computing the incompressible two-phase flow with deformable free interface is used. The control volume formulation with the SIMPLEC algorithm incorporated is used to solve the governing equations on a staggered Eulerian grid. The simulation results demonstrate that the algorithm is robust for shear-thinning liquids with large density (ρ₁/ρ_g up to 10³) and high viscosity (η₁/η_g up to 10⁴). The comparison of the experimental measurements of terminal bubble shape and velocity with the computational results is satisfactory. It is shown that the local change in viscosity around a bubble greatly depends on the bubble shape and the zero-shear viscosity of non-Newtonian shear-thinning liquids. The shear-rate distribution and velocity fields are used to elucidate the formation of a region of large viscosity at the rear of a bubble as a result of the rather stagnant flow behind the bubble. The numerical results provide the basis for further investigations, such as the numerical simulation of viscoelastic fluids.     

Measurement of extensional viscosity of polymer solutions

A filament stretching technique for measuring the extensional viscosity of polymer solutions at constant stretch rate is presented. The liquid sample is held between two coaxial discs and stretched by moving the bottom disc downwards with a speed that increases exponentially with time. This is illustrated using a constant viscosity, elastic fluid consisting of 0.185% polyisobutylene in a solvent of kerosene and polybutene. For the case of this particular fluid, two distinct stretch rate regions are found to arise. The stretch rate in the first region is much higher than in the second, which is, in most cases, close to the overall stretch rate imposed on the sample. Nonetheless, all the results of any given run can be represented using an average extensional rate. The extensional stress growth data, plotted as the Trouton ratio against time, show an initial linear viscoelastic region where T_R rises to a value of 3, independent of extensional rate. Beyond this region, T_R depends on the stretch rate and rises dramatically to values in excess of 10³; the higher the extensional rate, the faster is the increase in T_R. These data do not seem to reach a steady state and appear to be similar to polymer melt data obtained by others in the past. The reproducibility of the results is very good and all this suggests that it is now possible to obtain unambiguous constant-stretch-rate stress-growth data for polymer solutions stretched from a state of rest.     

Shear motions of elastic fluids for a given tangential stress

Stationary and nonstationary modes of elastic fluid motion for a given constant strain rate =const were studied under simple shear conditions, theoretically in [1, 2] as compared with experiment; time dependences of the normal and tangential stresses were examined for the emergence into stationary flow and their relaxation from steady flow. These results permitted a study of the relaxation characteristics of elastic fluids. However, no less interesting are the lagging (retardation) effects in elastic fluids, which can be studied in modes giving the shear stress ₁₂(). In this paper, the two most widespread shear modes in practice are examined theoretically and experimentally for a given ₁₂: the mode of arrival at the stationary flow from the state of rest for ₁₂=const and the mode of retardation (elastic recovery) from stationary flow. Theoretical computations are performed on a model describing large elastic strains. The experiment was performed on a concentrated polymer solution. Quantitative correspondence between theory and experiment is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 9–13, July–August, 1976.     

Viscosity and dynamics of nanorod (carbon nanotubes, cellulose whiskers, stiff polymers and polymer fibers) suspensions

The zero shear viscosity and the dynamic behaviors of different nanorod dispersions (carbon nanotubes (CNTs), cellulose whiskers, polymer nanofibers, crosslinked polymer nanofibers, and stiff polymers such as poly(γ-benzyl-α-l-glutamate) (PBLG)) were compared and discussed from literature data. Their Brownian dynamic behaviors have always been discussed in the frame of the Doi–Edwards theory. In agreement with this theory, the straight rigid rods (CNTs, cellulose whisker, polymer nanofibers) obey a master curve in the reduced viscosity (or rotary diffusivity) c power laws on viscosity (η ₀ ∝ φ ³) and diffusivity (D _r ∝ ? ^?2). On the contrary, stiff polymer chains and crosslinked polymer fibers at temperature above T _g exhibit different and two distinct dynamic behaviors. Despite their deviation from the ideal rigidity, surprisingly it can be noted that stiff polymers such as PBLG have been extremely used in the literature to verify the Doi–Edwards theory. Finally, flexible crosslinked chains at T > T _g , do not obey the Doi–Edwards theory, and their dynamics are close to the physics of polymer solutions in terms of power laws.     

A network model with free-strand dynamics for polymer melts

 A network model for polymer melts is presented in which disentangled strands relax under flow conditions and may rejoin the network before complete relaxation. For simplicity, we study Gaussian strands that move affinely when incorporated in the network. Network strands are created and lost according to a time constant λ. Free strands have their dynamics given by the Bird-DeAguiar model as a crude representation of reptation and the hindered rotation experienced by polymer strands in melts. The model yields a shear-thinning viscosity with overshoot in the start-up viscosity η⁺ (t). The double-step strain results compare well with available experimental data. Received: 10 July 2000 Accepted: 10 July 2001     

A FENE-P k–ε turbulence model for low and intermediate regimes of polymer-induced drag reduction

A new low-Reynolds-number k–ε turbulence model is developed for flows of viscoelastic fluids described by the finitely extensible nonlinear elastic rheological constitutive equation with Peterlin approximation (FENE-P model). The model is validated against direct numerical simulations in the low and intermediate drag reduction (DR) regimes (DR up to 50%). The results obtained represent an improvement over the low DR model of Pinho et al. (2008) [A low Reynolds number k–ε turbulence model for FENE-P viscoelastic fluids, Journal of Non-Newtonian Fluid Mechanics, 154, 89–108]. In extending the range of application to higher values of drag reduction, three main improvements were incorporated: a modified eddy viscosity closure, the inclusion of direct viscoelastic contributions into the transport equations for turbulent kinetic energy (k) and its dissipation rate, and a new closure for the cross-correlations between the fluctuating components of the polymer conformation and rate of strain tensors (NLT_ij). The NLT_ij appears in the Reynolds-averaged evolution equation for the conformation tensor (RACE), which is required to calculate the average polymer stress, and in the viscoelastic stress work in the transport equation of k. It is shown that the predictions of mean velocity, turbulent kinetic energy, its rate of dissipation by the Newtonian solvent, conformation tensor and polymer and Reynolds shear stresses are improved compared to those obtained from the earlier model.     

Some remarks on “Drag coefficients of a slowly moving sphere in Non-Newtonian fluids”

Viscoelastic solutions were ejected vertically downwards into air and various Newtonian fluids. The measured swell increased significantly when ejected into a liquid rather than air. The observed increase is considered a result of both bouyancy and drag forces on the solution. The following dimensions expression relating the ratio of the swell diameter in liquid and air D_L/D_A to the elastic shear compliance of the ejected solution J_e was experimentally observed.(D_L/D_A)⁶-1=30(Δ?/?_s)^{?$\text{1}\text{2}$}([g²η²_N?_s]$\text{1}\text{3}$J_e)^{$\text{3}\text{5}$}, where Δ? is the density difference between the extruded and Newtonian fluid, ?_s is the solution density, g is the gravitational constant, and η_N is the Newtonian fluid viscosity. Thus with this expression a simple extrudate swell technique exists to estimate the elastic shear compliance of a viscoelastic solution.