An experimental evaluation of the behaviour of mono and polydisperse polystyrenes in Cross-Slot flow

The behaviour of a number of mono and polydisperse polystyrenes are probed experimentally in complex extensional flow within a Cross-Slot geometry using flow-induced birefringence. Polystyrenes with similar molecular weight (M _w) and increasing polydispersity (PD) illustrated the effect of PD on the principal stress difference (PSD) pattern in extensional flow. Monodisperse materials exhibited only slight asymmetry at moderate flowrates, although increased asymmetry and cusping was observed at high flowrates. The response of monodisperse materials of different M _w at various flowrates is presented and characterised by Weissenberg numbers for both chain stretch and orientation using a theory for linear entangled polymers. The comparison of stress profiles against Weissenberg number for each process is used to determine whether the PSD pattern observed is independent of M _w and elucidate which relaxation mechanism is dominant in the flow regimes probed. For monodisperse materials, at equivalent chain orientation Weissenberg number (We _τd), different molecular weight materials were seen to exhibit similar steady state PSD patterns independent of We _τR (chain stretch We). Whilst no obvious critical Weissenberg number (We) was found for the onset of increased asymmetry and cusping, it was found to occur in the “orientating flow without chain stretch” regime.     

Numerical simulation studies of the planar entry flow of polymer melts

This paper is concerned with the numerical simulation of planar entry flow using a penalty finite element method and the comparison of predictions with flow visualization and birefringence data for two polymer melts. The Phan-Thien Tanner (PTT) model was fit to the steady state shear and extensional viscosity data and the transient extensional viscosity data of both polystyrene and low-density polyethylene (LDPE) melts to obtain the parameters λ, ξ, and ϵ in this model. Agreement was found between the flow visualization and birefringence data and the predictions of streamlines and stress. With some modification of the constitutive equation, the vortex growth and intensity observed for LDPE could be predicted by the use of the PTT model and the material parameters fit to the rheological properties. Likewise, the flow behavior of polystyrene, in which only small vortices with no growth were observed, was also predicted. Furthermore, it was found that the size and intensity of the vortex could be affected by the parameter ϵ in the PTT model which controls the predictions of the extensional viscosity. Based on these results it seems that accurate simulation of entry flow behavior requires the use of a constitutive equation which is capable of giving realistic preciction's of a fluid's extentional flow properties.     

Modeling strain hardening of polydisperse polystyrene melts by molecular stress function theory

The strain hardening of blends of polystyrene (PS) and ultra-high molecular weight polystyrene (UHMW-PS) in elongational flow is modeled by the molecular stress function (MSF) theory. Assuming that the ratios of strain energies stored in polydisperse and monodisperse polymers are identical for linear and nonlinear deformations, the value of the only non-linear parameter of the theory in extensional flows, the maximum molecular stress f_max, can be determined and is shown to be related to steady-state compliance J_e0. Using only linear-viscoelastic data, the elongational viscosity of PS/UHMW-PS blends is consistently predicted by the MSF theory.     

Linear viscoelastic behavior of bidisperse polystyrene blends: experiments and slip-link predictions

The linear viscoelastic behavior of three well-entangled linear monodisperse polystyrene melts and their blends is investigated. The monodisperse melts are blended in a 1:1 weight ratio to obtain three polystyrene bidisperse blends for which the linear viscoelastic behavior is also measured. Special attention is paid to controlling sample size and solvent content, and checking for consistency in the high-frequency regime. We also attempt to estimate uncertainty quantitatively. The experimental results agree well with the discrete slip-link model, a robust mesoscopic theory that has been successful in predicting the rheology of flexible entangled polymer liquids and gels. Using recently developed analytic expressions for the relaxation modulus, predictions of the monodisperse samples are made. The parameters for the model are obtained from the low-frequency crossover of one experiment. Using this parameter set without adjustment, predictions over the fully accessible experimental frequency range are obtained for the monodisperse samples and their blends with very good agreement.     

Predicting the linear viscoelastic properties of monodisperse and polydisperse polystyrenes and polyethylenes

 For linear homopolymers the linear viscoelastic predictions of the double reptation model are compared to those of a recent, more detailed model, the “dual constraint model” and to experimental data for monodisperse, bidisperse, and polydisperse polystyrene melts from several laboratories. A mapping procedure is developed that links the empirical parameter K of the double reptation model to the molecular parameter τ_e of the dual constraint model, thereby allowing the parameter K to be related to molecular characteristics such as the monomeric friction coefficient ζ. Once K (or τ_e) are determined from data for monodisperse polymers, the double reptation model predicts that for fixed weight-average molecular weight M_w, the zero-shear viscosity η₀ increases slightly with increasing polydispersity M_w/M_n for log normal distributions, while for the dual constraint model η₀ is almost independent of M_w/M_n. Experimental data for polystyrenes show no increase (or even a slight decrease) in η₀ with increasing M_w/M_n at fixed M_w, indicating a deficiency in the double reptation model. The dual constraint theory is also applied to hydrogenated polybutadienes and commercial high-density polyethylenes, where we believe it can be used to indicate the presence of long side branches, which are difficult to detect by other analytic methods. Received: 11 October 2000 Accepted: 17 May 2001     

Elongational flow and rheology of monodisperse polymers in solution

We describe the utilization of idealized stagnation point extensional flows, produced by opposed jets, for birefringence visualization of induced molecular strain and flow resistance measurements. We identify rheological changes associated with the coil---stretch transition which occurs beyond a critical strain-rate in elongational flow-fields. In dilute solutions of monodisperse atactic polystyrene, increases in extensional viscosity are observed as isolated molecules become stretched. The largest increases in extensional viscosity, however, are found only beyond a critical concentration and strain rate, and are associated with the stretching of transient networks of interacting molecules. These results parallel similar effects seen in porous media flow and capillary entrance experiments. We determine the molecular weight dependence of the critical concentration which scales as M^−0.55 in agreement with pairwise interaction of coils, but is much lower than conventional values of the critical polymer concentration, c^*. We believed that polydispersity may play an important role in the development of such transient networks, and in controlling the degradation behaviour during flow.     

Bidisperse polymeric mixtures for independent control of viscosity and elasticity

 It is known that the zero shear viscosity of a polydisperse melt of linear polymers depends only on its weight-average molecular weight, whereas its recoverable compliance increases with polydispersity. These facts can be exploited to design model viscoelastic fluids using mixtures of short and long chains of the same homopolymer (bidisperse mixtures). The composition required to obtain a bidisperse mixture with the desired viscosity can be calculated from the molecular weights of the components, and the known relationship between viscosity and weight-average molecular weight. The terminal viscoelastic properties of such a bidisperse mixture are estimated from theoretical predictions for the compliance of bidisperse mixtures available in the literature. These predictions suggest that the elasticity of bidisperse mixtures can be varied independent of their viscosity by appropriately choosing the molecular weights of their components and their composition. This strategy is applied here on bidisperse mixtures of monodisperse 1,4-polyisoprene, which are shown to display second-order fluid behavior over a reasonable range of accessible shear rates. The same procedure is also applied to mixtures of PDMS polymers which are not particularly monodisperse. Rheological measurements show that the elasticity of these polyisoprene and PDMS mixtures can indeed be varied without changing their viscosity. Such materials are ideally suited to study structure-rheological properties relationships in blends of immiscible viscoelastic fluids. Received: 12 April 2001 Accepted: 28 August 2001     

Determining polymer molecular weight distributions from rheological properties using the dual-constraint model

Although multiple models now exist for predicting the linear viscoelasticity of a polydisperse linear polymer from its molecular weight distribution (MWD) and for inverting this process by predicting the MWD from the linear rheology, such inverse predictions do not yet exist for long-chain branched polymers. Here, we develop and test a method of inverting the dual-constraint model (Pattamaprom et al., Rheol Acta 39:517–531, 2000; Pattamaprom and Larson, Macromolecules 34:5229–5237, 2001), a model that is able to predict the linear rheology of polydisperse linear and star-branched polymers. As a first step, we apply this method only to polydisperse linear polymers, by comparing the inverse predictions of the dual-constraint model to experimental GPC traces. We show that these predictions are usually at least as good, or better than, the inverse predictions obtained from the Doi–Edwards double-reptation model (Tsenoglou, ACS Polym Prepr 28:185–186, 1987; des Cloizeaux, J Europhys Lett 5:437–442, 1988; Mead, J Rheol 38:1797–1827, 1994), which we take as a “benchmark”—an acceptable invertible model for polydisperse linear polymers. By changing the predefined type of molecular weight distribution from log normal, which has two fitting parameters, to GEX, which has three parameters, the predictions of the dual-constraint model are slightly improved. These results suggest that models that are complex enough to predict branched polymer rheology can be inverted, at least for linear polymers, to obtain molecular weight distribution. Further work will be required to invert such models to allow prediction of the molecular weight distribution of branched polymers.     

Discrepancies between linear viscoelastic measurements and double-reptation predictions for commercial polystyrene samples

We have found discrepancies between the predictions of two double-reptation models and the corresponding linear viscoelasticity measurements for commercial, polydisperse polystyrene samples with weight-average molecular weights ranging from 140 to 300kg/mol. The accuracy of the experimental data has been confirmed by conducting viscoelasticity measurements in different laboratories using different types of rheometers and by verifying that small distortions in chromatographic measurements cannot account for the discrepancies seen in the viscoelastic response. In addition, we show that the discrepancies between predictions and measurements are systematic, suggesting that gaps in the theory are responsible for the mismatch. Is it concluded that commercial polystyrene resins may pose additional challenges for rheological modeling because of their relatively high polydispersity and low degree of entanglement. The experimental data given here can be used to validate future modeling efforts.     

Shear flow properties of concentrated solutions of linear and star branched polystyrenes

Summary The linear viscoelastic and viscometric functions have been determined for solutions of wellcharacterized monodisperse linear and star-branched polystyrenes and for commercial, polydisperse polystyrene. The value of the product c for these solutions was large and was obtained by using both high and low The effect of structure on the rheological properties was determined by examining how parameters in a modified Carreau viscosity equation (used to fit the data) varied with c, , and branching. No enhancement effects on the rheological properties were observed because of branching.The Cox-Merz rule was observed to describe the similarities between the viscosity and complex viscosity for most of the monodisperse samples studied. The broad molecular weight distribution polystyrene solutions did not follow this empiricism.With 17 figures and 4 tables     

Extensional rheology of dilute polymer solutions in oscillatory cross-slot flow: the transient behaviour of birefringent strands

Birefringent strands are key to understanding polymeric non-Newtonian flows, especially in extension. Utilising microfluidic extensional flow oscillatory rheometry coupled with microvelocimetry (μ-PIV), we report experiments on the genesis, steady state and decay of such strands, together with rheological consequences. For closely monodisperse atactic polystyrene, we report massive effects of the polymer on flow even at low concentrations. The often observed startup “overshoot” in stress and birefringence is observed at unprecedented dilution and discussed in terms of the local strain rate. Strand decay shows pronounced hysteresis. These factors are most important in modelling real flows such as cyclic and capillary entrance flows. Even with the closely monodisperse and well-characterised samples used, residual polydispersity plays a vital role in flow behaviour.     

Molecular orientation in non-Newtonian flow of dilute polymer solutions around spheres

A novel approach is presented to study the benchmark problem of flow around spheres in model dilute solutions of monodisperse samples of atactic polystyrene in di-octyl phthalate. Spheres are held stationary on flexible cantilevers of known spring-constant, k, while the polymer solutions are pumped past at controlled flow rates, allowing access to a wide range of Deborah number. In this way the non-Newtonian forces experienced by the spheres can be measured as a function of Deborah number, while detailed observations and measurements of birefringence are made, enabling assessment of macromolecular strain and orientation. In addition the flow field around a sphere has been measured in an a-PS solution. Experiments have been performed on a single sphere and on two spheres axially aligned in the direction of flow. The extensional flow around the downstream stagnation point of the single sphere is found to play a pivotal role in the development of molecular strain and stress, resulting in flow modification and subsequent non-Newtonian behaviour. The flow birefringence in the wake is found to modify severely the flow around a second, downstream, sphere, affecting the non-Newtonian forces encountered by the second sphere. This provides an explanation for the time interval dependent terminal velocity often observed when two spheres follow the same path through viscoelastic liquids.     

Transport of Neutrally Buoyant and Dense Variably Sized Colloids in a Two-Dimensional Fracture with Anisotropic Aperture

The transport of monodisperse as well as polydisperse colloid suspensions in a two-dimensional, water saturated fracture with spatially variable and anisotropic aperture is investigated with a particle tracking model. Both neutrally buoyant and dense colloid suspensions are considered. Although flow and transport in fractured subsurface formations have been studied extensively by numerous investigators, the transport of dense, polydisperse colloid suspensions in a fracture with spatially variable and anisotropic aperture has not been previously explored. Simulated snapshots and breakthrough curves of ensemble averages of several realizations of a log-normally distributed aperture field show that polydisperse colloids exhibit greater spreading than monodisperse colloids, and dense colloids show greater retardation than neutrally buoyant colloids. Moreover, it is demonstrated that aperture anisotropy oriented along the flow direction substantially increases colloid spreading; whereas, aperture anisotropy oriented transverse to the flow direction retards colloid movement.     

Analytic derivation of the Cox�CMerz rule using the MLD ��toy�� model for polydisperse linear polymers

A general constitutive formalism, the ??na?ve?? polydisperse MLD model, has been developed by Mead et al. (Macromolecules 31:7895?C7914, 1998) and Mead (Rheol Acta 46:369?C395, 2007) at both the tube coordinate level and the mathematically simplified ??toy?? level independent of the tube coordinate. The model includes constraint release generated by convection-driven chain retraction (which is equivalent to ??convective constraint release?? (CCR)), reptation, and tube contour length fluctuations. The properties of the mathematically simplified na?ve polydisperse ??toy?? MLD model are explored in linear and nonlinear steady shear flows where we analytically derive the Cox?CMerz rule relating the steady shear viscosity to the modulus of the linear viscoelastic dynamic viscosity. The Cox?CMerz rule relating the linear viscoelastic material properties and the nonlinear material properties is shown to be a direct consequence of convective constraint release. The specific feature of CCR that leads to this result is that the relaxation rate due to convective constraint release is proportional to the shear rate, $\dot{{\gamma }}$ , independent of molecular weight. The viability of this well-known empirical relationship is a direct consequence of a coincidence in the mathematical structure of the linear viscoelastic material properties and convective constraint release. There is no physical analogy or relationship between the molecular relaxation mechanisms operative in linear (diffusive relaxation) and nonlinear (convective relaxation) flow regimes. The polydisperse MLD model predictions of the individual molecular weight component contributions to the flow curve, and interpretations thereof, are effectively identical to those first postulated by Bersted (J Appl Polym Sci 19:2167?C2177, 1975, J Appl Polym Sci 20:2705?C2714, 1976). Following the theoretical developments, a limited experimental study is executed with a commercial polydisperse polystyrene melt. Nearly quantitative agreement between the polydisperse MLD theory and experimental measurements of steady-shear viscosity and dynamic moduli is achieved over a wide range of shear rates.     

Flow of branched polymer melts in a lubricated cross-slot channel: a combined computational and experimental study

Numerical simulations have been performed to evaluate the accuracy of the multimode Giesekus model in predicting the flow behavior of a rheologically well characterized low-density polyethylene melt in a lubricated cross-slot channel. Specifically, the fidelity of the numerical results is established by detailed comparison with flow-induced birefringence measurements in a new optical rheometer with lubricated side walls that allows the creation of ideal two-dimensional flow kinematics that lead to the elimination of end effects commonly encountered in flow birefringence measurements. Based on these comparisons, the ability of the multimode Giesekus model to capture the flow characteristics with reasonable accuracy in the experimentally available Wi range of 21 to 29 has been established. However, it should be noted that the model predictions are, at best, qualitative in the vicinity of the stagnation point. The discrepancy between numerically predicted and experimentally observed stresses in this region is mainly attributed to the inaccuracy of the experimental data that stem from the occurrence of multiple orders of retardation within the measurement volume. Overall, these studies have paved the way for the development of a hi-fidelity lubricated cross-slot channel rheometer.     

Comparison of experimental data with the numerical simulation of planar entry flow: Role of the constitutive equation

The goal of this research was to determine whether there is any interaction between the type of constitutive equation used and the degree of mesh refinement, as well as how the type of constitutive equation might affect the convergence and quality of the solution, for a planar 4:1 contraction in the finite eiement method. Five constitutive equations were used in this work: the Phan-Thien–Tanner (PTT), Johnson–Segalman (JS), White–Metzner (WM), Leonov-like and upper convected Maxwell (UCM) models. A penalty Galerkin finite element technique was used to solve the system of non-linear differential equations. The constitutive equations were fitted to the steady shear viscosity and normal stress data for a polystyrene melt. In general it was found that the convergence limit based on the Deborah number De and the Weissenberg number We varied from model to model and from mesh to mesh. From a practical point of view it was observed that the wall shear stress in the downstream region should also be indicated at the point where convergence is lost, since this parameter reflects the throughput conditions. Because of the dependence of convergence on the combination of mesh size and constitutive equation, predictions of the computations were compared with birefringence data obtained for the same polystyrene melt flowing through a 4:1 planar contraction. Refinement in the mesh led to better agreement between the predictions using the PTT model and flow birefringence, but the oscillations became worse in the corner region as the mesh was further refined, eventually leading to the loss of convergence of the numerical algorithm. In comparing results using different models at the same wall shear stress conditions and on the same mesh, it was found that the PTT model gave less overshoot of the stresses at the re-entrant corner. Away from the corner there were very small differences between the quality of the solutions obtained using different models. All the models predicted solutions with oscillations. However, the values of the solutions oscillated around the experimental birefringence data, even when the numerical algorithm would not converge. Whereas the stresses are predicted to oscillate, the streamlines and velocity field remained smooth. Predictions for the existence of vortices as well as for the entrance pressure loss (ΔP_ent) varied from model to model. The UCM and WM models predicted negative values for ΔP_ent.     

Bead-spring chain model for the dynamics of dilute polymer solutions : Part 2. Comparisons with experimental data

Comparisons are made between experimental rheological data and theoretical predictions obtained from a recently developed algorithm which incorporates three major molecular concepts in a theory for dilute polymer solutions (hydrodynamic interaction, excluded volume and nonlinear springs). These predictions include the radius expansion factor, the apparent chain expansion factor, the molecular weight dependence of the intrinsic viscosity, the frequency dependence of oscillatory flow birefringence, and the shear rate dependence of the intrinsic viscosity. This paper shows that a bead-spring chain model quantitatively predicts these quantities when the relevant molecular concepts are incorporated, suggesting that the rheological properties of dilute polymer solutions can be explained and predicted in terms of these molecular parameters.     

Comparison of optical and mechanical measurements of second normal stress difference relaxation following step strain

Relaxation of the second normal stress difference (N ₂) following step strain of a concentrated monodisperse polystyrene solution has been studied using mechanical and optical rheometry. Measurements of normal thrust in a parallel plate geometry are corrected for strain inhomogeneity and combined with independent measurements of the first normal stress difference (N ₁) to determine N ₂. Optical experiments were performed using a novel configuration where flow birefringence data collected using multiple light paths within the shear plane are combined with the stress-optical law to determine all three independent stress components for shearing deformations. This technique eliminates end effects, and provides an opportunity to oversample the stress tensor and develop consistency checks of experimental data. N ₂ is found to be nonzero at all accessible times, and relaxes in roughly constant proportion to N ₁. This reflects nonaffine distribution of chain segments, even well within the regime of chain retraction at short times. Data collected with the two techniques are reasonably consistent with each other, and with results of previous studies, generally lying between the predictions of the Doi-Edwards model with and without the independent alignment approximation. The normal stress ratio –N ₂/N ₁ shows pronounced strain thinning in the nonlinear regime.     

Three dimensional finite element analysis of the flow of polymer melts

The finite element simulation of a selection of two- and three-dimensional flow problems is presented, based upon the use of four different constitutive models for polymer melts (Oldroyd-B, Rolie-Poly, Pom-Pom and XPP). The mathematical and computational models are first introduced, before their application to a range of visco-elastic flows is described. Results demonstrate that the finite element models used here are able to re-produce predictions made by other published numerical simulations and, significantly, by carefully conducted physical experiments using a commercial-grade polystyrene melt in a three-dimensional contraction geometry. The paper also presents a systematic comparison and evaluation of the differences between two- and three-dimensional simulations of two different flow regimes: flow of an Oldroyd-B fluid around a cylinder and flow of a Rolie-Poly fluid into the contraction geometry. This comparison allows new observations to be made concerning the relatively poor quality of two-dimensional simulations for flows in even quite deep channels.     

Spectrum of density fluctuations in a particlefluid system—II. Polydisperse spheres

This paper presents an extension of the analysis shown in Part I to a polydisperse particle-fluid system. The density autocorrelation is shown to be a function of two quantities, a generalized Overlap function for which an analytical expression is derived, and the radial distribution function (RDF). In Fourier transform space, the density spectrum again appears to be a strong function of the mean particle size, and secondarily the mean particle separation distance. One unusual result is previously observed oscillations in the density spectrum of a monodisperse system of particles are severely dampened or even eliminated in the polydisperse case, depending on the width of the particle size distribution. Apparently contributions from different particle correlations interfere with each other, thereby reducing the coherent oscillations seen in the monodisperse particle-fluid system. Furthermore at large wavenumbers, the spectrum decays with a −2 power-law, independent of the shape of the particle size distribution. This behavior can be traced to the Overlap function which controls the behavior of the spectrum beyond the first peak. Remarkably the −2 power-law spectrum is determined by the shape of the particles (i.e. spheres) rather than their spatial distribution (RDF).

The effect of an asymptotically large pressure gradient on the correlation of several important higher-order moments is revisited for the polydisperse system. The relatively simple relationships developed for the monodisperse system are lost in the polydisperse case because particles of different sizes will be influenced differently by an applied pressure gradient. The result is moments that are of different order in velocity can no longer be related to each other (as they were in the monodisperse system), even in this idealized flow. A more comprehensive understanding of this phenomenon can only be achieved through direct numerical simulation or experiment.