Computer simulation of the rheology of concentrated star polymer suspensions

We use particle-based computer simulations to study the rheology of suspensions of high-functionality star polymers with long entangled arms. Such particles have properties which are intermediate between those of soft colloidal particles and entangled polymer chains. In the simulations, each star polymer is coarse-grained to a single particle. In order to faithfully reproduce dynamical properties, it is very important to not only include time-averaged interactions (potentials of mean force) but to also account for transient interactions induced by entanglements between the arms of different star polymers. Using a model which has all these features, it is found that, for sufficiently high shear rates, the start-up shear stress displays an overshoot. With increasing concentration, the core interactions increasingly dominate the initial stress response, leading to a maximum in the stress overshoot at relatively low strain values (0.1 to 0.5). Transient forces start to dominate after this initial stage. In a simulated experiment in which the shear rate is suddenly stepped-down from a high to a lower value, the stress shows a clear undershoot, with the minimum stress again at a relatively low strain value (based on the new shear rate). Finally, it is shown that a stress plateau develops in the flow curve. This plateau is absent when the transient forces between the polymer stars are not taken into account.     

A mesoscopic tube model of polymer/layered silicate nanocomposites

Rheology of isothermal suspensions of completely exfoliated silicate lamellae in polymer melts is investigated. In order to express more faithfully the physics involved in low shear rates and low frequencies, we model the polymer molecules composing the melt as chains whose motion is confined to a tube formed by surrounding chains and lamellae. In the absence of lamellae, the model reduces to the mesoscopic model of reptating chains developed in Eslami and Grmela (Rheol Acta, 2008). If the chains are seen only as FENE-P dumbbells, the model reduces to the model developed in Eslami et al. (J Rheol 51:1189–1222, 2007). Responses to oscillatory, transient, and steady shear flows are calculated and compared with available experimental data. Particular attention is payed to the region of low shear rate and low frequency.     

A single segment differential tube model with interchain tube pressure effect

We develop a single segment differential tube model including interchain tube pressure effect (ITPE) [G. Marrucci, G. Ianniruberto, Interchain pressure effect in extensional flows of entangled polymers, Macromolecules 36 (2004) 3934–3942], able to describe the non-linear behaviour of entangled linear polymers. The model accounts for the effect of flow on the tube length and diameter. It is presented in two versions, depending on which tube dimension is assumed to deform affinely. The classical relaxation mechanisms, i.e., reptation, stretch dynamics, convective constraint release (CCR), as well as finite extensibility, are incorporated in a simple manner; hence the model allows an explicit comparison of the relative importance of various effects. A striking result is the insignificance of finite extensibility and the detrimental influence of CCR for moderately entangled systems when ITPE is taken into account. For highly entangled systems, CCR regains importance to avoid the well-known shear stress instability. The proposed model is able to make quantitative predictions of steady elongational and shear data for monodisperse melts, while transient values are less accurate but within experimental errors.     

Comparison of viscous and elastic properties of polyolefin melts in shear and elongation

Viscous and elastic properties of a linear polypropylene (PP) and a long-chain branched low-density polyethylene (LDPE) have been investigated by creep and creep–recovery experiments in shear and elongation. The data obtained verify the ratios between the linear values of the viscosities and the steady-state elastic compliances in shear and elongation predicted by the theory of linear viscoelasticity. In the nonlinear range, no simple correlation between the viscous behaviour in shear and elongation exists. The elongational viscosity of the PP decreases with increasing stress analogously to the shear thinning observed; the linear range extends to higher stresses in elongation than in shear, however. The LDPE shows thinning in shear and strain hardening in elongational flow. For the LDPE, a linear steady-state elastic tensile compliance corresponding to one third of the linear steady-state elastic compliance in shear was determined. For the PP, this theoretically predicted value is approximately reached. Analogous to the viscous behaviour, the linear range extends to higher stresses in elongation than in shear. For both materials, the steady-state elastic compliances in the nonlinear range decrease with increasing stress in shear as well as in elongation. However, the decrease in elongation is more pronounced.     

Viscosity of suspensions in polymeric solutions

Viscosities of suspended particles in polymeric solutions depend upon dissolved polymer concentration, volume fraction of particles and shear rate. In this analysis of viscosity data, relative viscosity was defined as the ratio of suspension viscosity to solution viscosity at the same shear stress rather than shear rate. These relative viscosities reached asymptotic values at high shear stress for all concentrations of dissolved polymer and for all particle loadings. At a given particle loading, the asymptotic values of relative viscosity were nearly independent of the concentration of dissolved polymer. Realtive viscosities were correlated with volume fraction by the one-constant equation of Maron and Pierce.     

The stability of two concentric non-Newtonian fluids in circular pipe flow

The flow of two concentric non-Newtonian fluids, under constant pressure gradient in a circular tube, is studied by linear stability analysis. The viscosities of the two fluids are different and their dependence on shear stress is described by the Ellis model. It is found that the steady state flow can be unstable, depending on certain combinations of the values of physical parameters, to infinitesimal axisymmetric disturbances of large wavelengths, for any Reynolds number however small. The flow is predominantly stable if the inner fluid is more viscous and predominantly unstable if the outer fluid is more viscous. Stronger dependence of viscosity on shear stress can both stabilize and destabilize the flow. Interfacial tension is also destabilizing when the Weber number is small than about 10⁴.     

Rheological and foaming behavior of linear and branched polylactides

In this work, a chain extender (CE) was added to polylactide (PLA) to improve its foamability. The steady and transient rheological properties of neat PLA and CE-treated PLA revealed that the introduction of the CE profoundly affected the melt viscosity and elasticity. The linear viscoelastic properties of CE-enriched PLA suggested that a long-chain branching (LCB) structure was formed from the reaction with the CE. LCB-PLA exhibited an increased viscosity, more shear sensitivity, and longer relaxation time in comparison with the linear PLA. The LCB structure was also found to affect the transient shear stress growth and elongational flow behavior. LCB-PLA exhibited a pronounced strain hardening, whereas no strain hardening was observed for the linear PLA. Batch foaming of the linear and LCB-PLAs was also examined at foaming temperatures of 130, 140, and 155 °C. The LCB structure significantly increased the integrity of the cells, cell density, and void fraction.     

Relaxation effects of slip in shear flow of linear molten polymers

The transient shear response of a linear molten polymer (linear low-density polyethylene) in the nonlinear domain was studied using a true shear (sliding plate) rheometer with different gap spacings to detect slip effects. It was found that nonlinear viscoelasticity is further complicated by wall slip phenomena. Experimental evidence suggested that static slip models coupled with Wagner’s constitutive equation cannot adequately describe the experimental data at large and fast shear deformations. A new dynamic slip model involving multiple slip relaxation times is proposed in this paper, together with a method to assess the model parameters. Significant improvement in predicting the stress response is demonstrated by several examples of start-up of steady shear and large-amplitude oscillatory tests of a linear low-density polyethylene.     

Dissipative particle dynamics simulation of polymer drops in a periodic shear flow

The steady-state and transient shear flow dynamics of polymer drops in a microchannel are investigated using the dissipative particle dynamics (DPD) method. The polymer drop is made up of 10% DPD solvent particles and 90% finite extensible non-linear elastic (FENE) bead spring chains, with each chain consisting of 16 beads. The channel’s upper and lower walls are made up of three layers of DPD particles, respectively, perpendicular to Z-axis, and moving in opposite directions to generate the shear flow field. Periodic boundary conditions are implemented in the X and Y directions. With FENE chains, shear thinning and normal stress difference effects are observed. The “colour” method is employed to model immiscible fluids according to Rothman–Keller method; the χ-parameters in Flory–Huggins-type models are also analysed accordingly. The interfacial tension is computed using the Irving–Kirkwood equation. For polymer drops in a steady-state shear field, the relationship between the deformation parameter (D_def) and the capillary number (Ca) can be delineated into a linear and nonlinear regime, in qualitative agreement with experimental results of Guido et al. [J. Rheol. 42 (2) (1998) 395]. In the present study, Ca<0.22, in the linear regime. As the shear rate increases further, the drop elongates; a sufficiently deformed drop will break up; and a possible coalescence may occur for two neighbouring drops. Dynamical equilibrium between break-up and coalescence results in a steady-state average droplet-size distribution. In a shear reversal flow, an elongated and oriented polymer drop retracts towards a roughly spherical shape, with a decrease in the first normal stress difference. The polymer drop is found to undergo a tumbling mode at high Schmidt numbers. A stress analysis shows that the stress response is different from that of a suspension of solid spheres. An overshoot in the strain is observed for the polymer drop under extension due to the memory of the FENE chains.     

Time-dependent,non-Newtonian behavior of viscoelastic materials

Summary The present paper is a further development in our effort to describe non-Newtonian flow behavior. As in the past, the thixotropic shear stress or shear rate is characterized by the kinetic theory and currently a modified form ofOldroyds model provides the elastic contribution. Presented here are further modifications of the model in order to predict the first normal stress difference and logical changes to make the model more internally consistent.In order to test the approach, transient and steadystate data have been obtained with aWeissenberg rheogoniometer on a solution of polymethylmethacrylate in diethylphthalate. Both constant stress and constant shear rate data have been taken over a broad range of the parameters involved. To facilitate data acquisition, a PDP-15, model 30 computer was programmed to gather data at rates of 1000 points per second. Of the data obtained there is a great deal of uncertainty about the normal stress results.By optimization methods, the constants of the model were evaluated from steady-state data and transient results on stress growth and shear rate growth. Comparisons of the model with stress growth results as well as the prediction of the first normal stress difference are given. The inadequacies of the analysis are emphasized and possible points for improvement are made.With 8 figures     

Transient rheological behaviour of poly-para-henylenetherephthalamide (PpPTA) in sulphuric acid

Nearly all the available information on the transient flow behaviour of liquid crystalline polymers has been obtained on model systems, especially on solutions of polybenzylglutamate (PBG) and hydroxypropylcellulose (HPC). The assessment of rheological models has been based almost entirely on these model systems. It is not clear how much of the available theoretical and experimental knowledge can be applied to systems of industrial relevance, which have quite different molecular structures. Here, an industrial lyotropic system, poly(p-phenylenetherephthalamide) (PpPTA) in sulphuric acid (TWARON from AKZO), is investigated. Various techniques to study transient behaviour are used, these include measurements of transient shear and normal stresses after sudden changes in shear rate, dynamic moduli and stress relaxation after cessation of flow and elastic recoil. At all shear rates studied the PpPTA solution is shear thinning, and the first normal stress difference remains positive. For the stress transients a strain scaling applies reasonably well as it did in model systems. The moduli increase with time upon cessation of flow, indicating that the molecules become less oriented in the previous flow direction. This particular behaviour is similar to that of HPC. Transients also resemble more closely those of HPC rather than those of PBG. This latter difference might be attributed to the higher flexibility of HPC and PpPTA chains as compared with PBG molecules.     

Rheology and flow-birefringence from viscoelastic polymer-clay solutions

 The shear orientation of viscoelastic clay-polymer solutions was investigated by means of rheology and flow birefringence (Δn). The polymer chains are in dynamic adsorption/desorption equilibrium with the clay particles to form a “network”. The elastic behavior of the network was characterized by constant stress, oscillatory shear, and stress relaxation experiments. Constant stress experiments indicated a yield stress upon which shear flow started and no strain recovery could be observed. Oscillatory shear experiments showed a broad elastic region followed by flow when a critical strain was reached. Stress relaxation experiments showed several relaxation times when the same critical strain was reached. Experiments under steady flow characterized the transient behavior of the network. With increasing steady shear rate a pronounced minimum in birefringence was observed at a critical shear rate. The shear rate dependent viscosity showed near power law behavior and no corresponding critical feature. While birefringence detects orientational effects on a microscopic length scale, rheology averages over macroscopic changes in the sample. The same degree of orientation could be achieved under constant shear rate or constant stress conditions. Received: 25 January 2001 Accepted: 22 May 2001     

Rheological properties of branched polystyrenes: nonlinear shear and extensional behavior

Nonlinear shear and uniaxial extensional measurements on a series of graft-polystyrenes with varying average numbers and molar masses of grafted side chains are presented. Step-strain measurements are performed to evaluate the damping functions of the melts in shear. The damping functions show a decreasing dependence on strain with an increase in mass fraction of grafted side chains. Extensional viscosities of the melts of graft-polystyrenes exhibit a growing strain hardening with increasing average number of grafted side chains as long as the side branches have a sufficient molar mass to be entangled. Graft-polystyrenes with side arms below the critical molar mass M _c for entanglements of linear polystyrene but above the entanglement molar mass M _e of linear polystyrene (M _e < M _w,br < M _c) still show a distinct strain hardening. With decreasing molar mass of the grafted side chains (M _w,br < M _e) the nonlinear-viscoelastic properties of the graft-polystyrene melts approach the behavior for a linear polystyrene with comparable polydispersity.Electronic Supplementary Material Supplementary material is available for this article at     

Electrorheologial properties of hematite/silicone oil suspensions under DC fields

Electrorheological (ER) fluids composed of α-Fe₂O₃ (hematite) particles suspended in silicone oil are studied in this work. The rheological response has been characterized as a function of field strength, shear rate and volume fraction. Rheological tests under DC electric fields elucidated the influence of the electric field strength, E, and volume fraction, ϕ, on the field-dependent yield stress, τ_y. It was found that this quantity scales with E and ϕ with a linear and parabolic dependence, respectively. The viscosities of electrified suspensions were found to increase by several orders of magnitude as compared to the unelectrified suspension at low shear rates, although at high-shear rates hydrodynamic effects become dominant and no effects of the electric field on the viscosity are observed. The work is completed with the analysis of microscopic observations of the structure acquired by the ER fluid upon application of a constant electric field. Electrohydrodynamic convection is found to be the origin of the ER response rather than the commonly admitted particle fibrillation. This fact can provide an explanation to the relationship between yield stress and electric field strength as well as the pattern of periodic structures observed in the measurement geometries.     

Relaxation time spectra of star polymers

 The linear viscoelastic data for model star polymer melts with varying functionality and arm molecular weight were represented by means of a modified Baumgaertel-Schausberger-Winter (BSW) relaxation time spectrum, based on data analysis with the parsimonius model of Baumgaertel et al., reported in 1990. In the case of high arm functionality, the second slow terminal relaxation observed by Vlassopoulos et al. in 1997, was captured with a straightforward extension of the BSW model using broad cut-off functions. This study represents a potentially promising attempt to extend the applicability of this representation of viscoelastic data to more complex architectures, beyond simple linear chains which are characterized by self-similarity. The casting of linear viscoelastic data into spectra allows the exploration of star polymer behavior. It is a necessary step in preparation for large scale complex flow calculations in conjunction with constitutive models and for material databases. Received: 18 November 1998/Accepted: 12 August 1999     

The interchain pressure effect in shear rheology

Recently, the tube diameter relaxation time in the evolution equation of the molecular stress function (MSF) model (Wagner et al., J Rheol 49: 1317–1327, 2005) with the interchain pressure effect (Marrucci and Ianniruberto, Macromolecules 37:3934–3942, 2004) included was shown to be equal to three times the Rouse time in the limit of small chain stretch. From this result, an advanced version of the MSF model was proposed, allowing modeling of the transient and steady-state elongational viscosity data of monodisperse polystyrene melts without using any nonlinear parameter, i.e., solely based on the linear viscoelastic characterization of the melts (Wagner and Rolón-Garrido 2009a, b). In this work, the same approach is extended to model experimental data in shear flow. The shear viscosity of two polybutadiene solutions (Ravindranath and Wang, J Rheol 52(3):681–695, 2008), of four styrene-butadiene random copolymer melts (Boukany et al., J Rheol 53(3):617–629, 2009), and of four polyisoprene melts (Auhl et al., J Rheol 52(3):801–835, 2008) as well as the shear viscosity and the first and second normal stress differences of a polystyrene melt (Schweizer et al., J Rheol 48(6):1345–1363, 2004), are analyzed. The capability of the MSF model with the interchain pressure effect included in the evolution equation of the chain stretch to model shear rheology on the basis of linear viscoelastic data alone is confirmed.     

Modelling elongational and shear rheology of two LDPE melts

Experimental data of two low-density polyethylene (LDPE) melts at 200°C for both shear flow (transient and steady shear viscosity as well as transient and steady first normal stress coefficient) and elongational flow (transient and steady-state elongational viscosity) as published by Pivokonsky et al. (J Non-Newtonian Fluid Mech 135:58–67, 2006) were analysed using the molecular stress function model for broadly distributed, randomly branched molecular structures. For quantitative modelling of melt rheology in both types of flow and in a very wide range of deformation rates, only three nonlinear viscoelastic material parameters are needed: Whilst the rotational parameter, a ₂, and the structural parameter, β, are found to be equal for the two melts considered, the melts differ in the parameter describing maximum stretch of the polymer chains.     

Elastic response from pressure drop measurements through planar contraction–expansion geometries by molecular dynamics: structural effects in melts and molecular origin of excess pressure drop

In this work, we use nonequilibrium molecular dynamics to simulate a contraction–expansion flow of various systems, namely melts with molecules of various conformations (linear, branched, and star), linear molecules in solution, and a reference Lennard–Jones fluid. The equations for Poiseuille flow are solved using a multiple time scale algorithm extended to nonequilibrium situations. Simulations are performed at constant temperature using the Nose–Hoover dynamics. The main objective of this analysis is to investigate the molecular origin of pressure drop along planar contraction–expansion geometry, varying the length of the contraction, and the effect that different molecular conformations have on the resulting pressure drop along the geometry. Pressure drop is closely related to mass distribution (in neutral and gradient directions) and branching index of molecules. Also, it is shown that remarkable increases of pressure drops are also possible in planar geometries, provided large extensional viscosities combined with moderate values of the first normal stress difference in shear are considered, in addition to considerable reductions of the flow area at the contraction region.     

Stability analysis of unbounded uniform shear flows of dense, slightly inelastic spheres based on a frictional-kinetic theory

Asymptotic and transient stability analyses of unbounded uniform shear flows of dense, slightly inelastic, spherical particles were carried out using a frictional-kinetic theory. This model proposed for describing dense flows is based on a critical state plasticity theory and a simplified kinetic theory. In this model, the bulk and shear viscosities, the “thermal” conductivity, and the energy dissipation rate are proportional to a “mean pressure” which is composed of a quasistatic-frictional-contribution pressure considered for slow, plasticity deformations and a granular-kinetic-theorycollisional-contribution pressure. We studied two-dimensional stability analyses of layering disturbances (i.e., the perturbations whose wave number vectors are aligned only in the gradient) as well as nonlayering disturbances (the wave number vectors have nonzero streamwise components). Although this model has a simpler framework, it predicted similar results to those obtained using a more elaborate frictional-kinetic model. For instance, nonlayering disturbances are asymptotically stable at large time; the maximum transient growth of disturbances increases as the solids fraction or the friction coefficient is increased; and transient growths of disturbances can be significant due to the non-normality of the system. However, the prediction of the asymptotic stability of layering disturbances may be questionable because the collisional-contribution terms of the present model were over-simplified.     

Relationships between linear and nonlinear shear response of polymer nano-composites

Rheological analysis was used to understand the structure?Cproperty relations of polymer nano-composites based on ethylene vinyl acetate. Two geometrically different nano-particles (sphere of CaCO₃ and platelet of montmorillonite) having the same energetic attractions with ethylene vinyl acetate were studied for concentrations between 2.5 and 15 wt%. Three phenomena were studied: the appearance of a solid-like behavior in the linear viscoelastic domain, the limits of linear viscoelasticity, and the presence of stress overshoot in step shear tests. In particular, stress overshoot was investigated based on the tube concept of polymeric chains. Also, differences related to nano-particle geometry (platelet vs. spherical) were investigated based on a filler-network mechanism. Due to higher physical contacting probability, platelet particles can better interact and create a network structure, which dominates the rheological response. On the other hand, although spherical particles can limit the motion of polymeric chains under flow, a strong physical network was not formed. For platelets, scaling behavior was well described by fractal model which considers direct aggregation, and such scaling was not observed for spherical particles. The filler-network mechanism was validated by image analysis.