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.     

Development of the “binary interaction” theory for entangled polydisperse linear polymers

Based on the theoretical principles previously described in the literature, the development of the “naïve” binary interaction model is detailed in this paper. The new theory is effectively a sweeping generalization of the “Double Reptation” model. The “switch function” has been shown to be an essential feature of any constraint release model for Doi–Edwards type molecular models that invoke the concept of a discrete slip-link tube and is used in our formulation. Using the assumption of a constant entanglement density, a slip link linear density evolution equation is derived to rigorously count matrix entanglements. This function has no counterpart in the conventional Doi–Edwards theory, or its derivatives, and is absolutely required to properly generalize the “Double Reptation” model so that nonlinear flows can be modeled. The binary interaction polydispersity model is complex mathematically but can be rigorously and justifiably simplified by suppressing the tube coordinate dependence using a boundary layer analysis. The simplification process can be continued to the continuum level to create a hierarchy of approximate binary interaction models, thereby making large-scale numerical simulations of complex flows viable, indeed straightforward.     

Predictions of linear viscoelastic properties for polydisperse entangled polymers

Different blending laws have been proposed in the literature to describe the polydispersity effect on the rheological behavior of polymer melts. In this paper predictions of linear viscoelastic properties of entangled polydisperse polymers have been derived from the double reptation mixing rule. The results in terms of the relaxation modulus, the zero shear-rate viscosity, η₀, and the steady-state compliance, J _e ⁰, have been obtained using three different relaxation functions for the monodisperse fractions, namely the Tuminello step function, the single exponential function and the BSW function. Both discrete and continuous molecular weight distributions (MWDs) have been investigated. The Generalized Exponential Function (GEX) has been considered in the continuous case. The results showed that, in systems with a large number of components, the predictions of linear viscoelastic properties mainly depend on the double reptation mixing rule assumption, while the choice of the relaxation function is not crucial. In particular, the mathematical simplicity of the Tuminello step relaxation function has allowed analytical computation of the linear viscoelastic properties in closed form. Indeed, the analytical results indicated a dependence of η₀ on the MWD that could be expressed in terms of (M _z/M _w)^0.8, in agreement with experimental results reported in the literature. In the case of J _e ⁰, the analytical model defines a dependence on (M _z/M _w)^5.5, i.e. as expected a strong dependence on the MWD is predicted for the steady-state compliance. Finally, dynamic moduli have been computed from the relaxation modulus and their predictions have been favorably compared with experimental results from the literature. Received: 19 July 1999/Accepted: 24 November 1999     

A hierarchical algorithm for predicting the linear viscoelastic properties of polymer melts with long-chain branching

The hierarchical model proposed earlier [Larson in Macromolecules 34:4556–4571, 2001] is herein modified by inclusion of early time fluctuations and other refinements drawn from the theories of Milner and McLeish for more quantitative prediction. The hierarchical model predictions are then compared with experimental linear viscoelastic data of well-defined long chain branched 1,4-polybutadienes and 1,4-polyisoprenes using a single set of parameter values for each polymer, which are obtained from experimental data for monodisperse linear and star polymers. For a wide range of monodisperse branched polymer melts, the predictions of the hierarchical model for monodisperse melts are very similar to those of the Milner–McLeish theories, and agree well with experimental data for many, but not all, of the branched polymer samples. Since the modified hierarchical model accounts for arbitrary polydispersity in molecular weight and branching distributions, which is not accounted for in the Milner–McLeish theories, the hierarchical algorithm is a promising one for predicting the relaxation of general mixtures of branched polymers.     

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     

A differential tube-based model for predicting the linear viscoelastic moduli of polydisperse entangled linear polymers

We present a simple tube theory for topologically linear entangled polymers that accounts for reptation, contour-length fluctuations and thermal constraint release. This theory is based on a new differential formulation of the thermal constraint release phenomenon proposed by the authors [A. Leygue, C. Bailly, R. Keunings, A differential formulation of thermal constraint release for entangled polymers, J. Non Newtonian Fluid Mech. 128 (1) (2005) 23–28] which is extended here to account for contour-length fluctuations. We apply the theory to mono- and poly-disperse polystyrene melts and demonstrate its ability to produce quantitative predictions. Additionally, we discuss a mathematically linear approximation of our approach that preserves the structure of the model. While most quantitative tube theories for predicting linear viscoelasticity are mathematically non-linear, our approach allows one to address the linear viscoelastic response of a polydisperse entangled system with a mathematically linear theory.     

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.     

Viscoelasticity of low molecular weight polymers and the transition to the entangled regime

The viscoelasticity of unentangled polystyrene melts has been investigated in terms of terminals parameters: zero-shear viscosity, steady-state compliance and relaxation spectrum. The Rouse model applies well for molecular weights lower than the average molecular weight between entanglements, providing that one takes into account the proper variations of the radius of gyration. Moreover, local motions at the scale of Kuhn segments have to be considered in order to describe correctly the relaxation modes intermediate between the terminal zone and the glassy plateau. On the other hand, reptation models are commonly used for describing the entangled regime. We propose an expression of the shear modulus which accounts not only for the terminal modes (reptation, tube length fluctuations and tube renewal), but also for the relaxation modes responsible for the plateau zone and the transition of the glassy plateau. A crossover region between the unentangled and untangled regimes is located around . When the molecular weight increases, a shift transfer of Rouse modes towards reptation modes occurs. That leads to a continuity of the expression of the shear modulus over the entire range of molecular weights. Received: 29 December 1997 Accepted: 27 July 1998     

Concentration dependence of the linear viscoelastic properties of particle suspensions

The response under small amplitude oscillatory deformations of a suspension of non-Brownian spheres dispersed in a viscoelastic fluid is investigated. The correspondence principle of linear viscoelasticity is used to derive a simple constitutive model from a model for a suspension in a Newtonian liquid. The theory predicts that for a specific particulate system the concentration dependence of the viscoelastic properties should collapse to a single master curve when the values are normalized with those of the carrier fluid alone. Measurements with the micro-Fourier rheometer using oscillatory squeeze flow are carried out on two suspensions of 60 and 80 μm sized particles dispersed in polymeric fluid and in silicon oil, and the master curve is verified. Received: 27 April 1999/Accepted: 15 October 1999     

Application of melt flow index for rheological characterization of branched and linear polymers

A procedure for evaluating rheological characteristics, such as the master curves log/ ₀ vs. log % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xd9Gqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieGaceWFZo% Gbaiaaaaa!3B59!\[\dot \gamma \] ₀ and flow curves, using the melt flow index is described for branched and linear polymers. Experimental data on the melt flow index and branching degree are needed for this purpose, as well as some polymer constants, i.e. coefficients of the ₀ vs. MFI relation and coefficients of fluidity dependence on molecular characteristics. An example is given for bisphenol A polycarbonate.     

Dependence of linear viscoelastic critical strain and stress values on extent of gelation for a thermally activated gelling system

A large body of literature is focused on the accurate determination of a gel point for systems undergoing a sol-gel phase transition. Investigation into the limiting strain and stress for linear viscoelastic behaviour at various stages of a phase transition such as gelation is a subject that is rarely commented on. The small amplitude oscillatory rheological behaviour of a biopolymer cross-linker system through a thermally activated sol-gel transition is presented. Mechanical spectra were interpreted through application of the gelation criteria of Chambon and Winter (Winter and Chambon 1986; Chambon and Winter 1987), where the (so-called gel strength) parameter S, and relaxation exponent, n are obtained. A detailed study of the limit of linear viscoelasticity yields important trends in critical stress (σ°_c) and critical strain (γ°_c) limits highlighting the possible experimental difficulties associated with mechanical measurements obtained in close proximity to the gel point. Received: 17 March 2000 Accepted: 2 October 2000     

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     

Relaxation and viscoelastic properties of complex polymer systems

Dynamic behavior of various polymer melts is studied on the basis of a comparison of viscoelastic properties with the information obtained from dielectric spectroscopy. The experimental observations are compared with results of computer simulation of corresponding systems. The studies include simple melts of linear chains, block copolymer systems of miscible components, as well as the behavior of melts with molecular objects of complex topology-like stars or microgels. In the case of polyisoprene linear chain melts an equivalence of terminal relaxation times determined from mechanical and dielectric measurements is demonstrated. Using linear block copolymers of isoprene and butadiene, relaxation times of chain fragments (isoprene blocks) in relation to relaxation times of whole copolymer chains are determined and compared with theory and simulation. Both the experimentally determined block relaxation times and relaxation times of chain fragments in simulated linear chain melts show a disagreement with predictions of the reptation theory. In the case of multiarm star polymers and microgel melts, the slow relaxation modes observed in viscoelastic spectra are assigned to cooperative translational motions detected in corresponding simulated systems in which an ordering of such molecules is demonstrated. This suggests that the terminal relaxation in multiarm star or microgel melts is governed by another relaxation mechanism than in linear chain melts. High efficiency of the Cooperative Motion Algorithm in simulation of dense systems of complex molecules is demonstrated.Dedicated to the memory of Professor Tasos C. Papanastasiou     

Integral and differential constitutive equations for entangled polymers with simple versions of CCR and force balance on entanglements

The theory of Doi and Edwards for entangled polymers has been recently modified for the case of fast flows to account for convective contributions to molecular dynamics. The flow-induced relative motion between neighboring chains removes constraints and speeds up relaxation. Convective constraint release (CCR) may thus explain why the shear stress is seen to approach a plateau at high shear rates instead of decreasing as predicted by the basic theory. In slow flows, as well as in step strain, another discrepancy between theory and observations can be found in the normal stress ratio in shear Ψ=−N₂/N₁. The theoretical value for Ψ at low deformations is 1/7 whereas measured values for well-entangled systems are systematically larger. We have recently considered the possibility that this discrepancy arises because force balance requirements at the entanglement nodes are ignored in the classical theory. Accordingly, we have proposed a change in the orientational tensor Q. Here, we sum up on these recent findings by proposing single-relaxation-time constitutive equations of the integral or rate type incorporating those concepts in a simple way. Such equations should be suitable for numerical simulation of complex flows. Received: 1 January 2000 Accepted: 8 August 2000     

Micromechanics of semicrystalline polymers: Towards quantitative predictions

An initially qualitative two-phase elasto-viscoplastic micromechanical model for the mechanical performance of semicrystalline materials has previously been developed. In the last decade, a series of extensions to this model have been aimed towards quantitative predictions of the response of semi-crystalline polymers based on their microstructure. These developments included extensive experimental characterization and modelling of the yield kinetics, time-to-failure, creep and thermal shrinkage and expansion. This paper gives an overview of the route from the initially qualitative model to a model that quantitatively captures these complex aspects of the mechanical response of a semicrystalline polymer.