Double Reptation Model Research Articles

Phase morphology and melt viscoelastic properties in blends of ethylene/vinyl acetate copolymer and metallocene-catalysed linear polyethylene

Abstract The aim of this study was to determine the linear viscoelastic properties of a series of ethylene/vinyl acetate copolymer/metallocene-catalysed polyethylene (mPEs) blends. Newtonian viscosity showed a pronounced positive deviation from the double reptation model, which assumes miscibility or, at least, cooperative relaxation between the mixed species. Enhanced values of steady-state compliance and elastic indices with respect to those of the pure components were also noted. These features are typical of emulsion-like polymer blends and are thought to arise from additional relaxation processes associated with dispersed phase deformability. Application of the Palierne model for emulsions of two viscoelastic liquids showed good agreement with our experimental dynamic results at both ends of the phase diagram. However, the model failed at intermediate compositions. Through the application of several rheological criteria we were able to locate the phase inversion concentration at a weight fraction of w =0.60 in the mPEs. It is suspected that, in this composition range, a fully co-continuous phase develops due to the phase inversion mechanism, which has considerable effects on the viscoelastic properties of the blends.

Viscoelastic behaviour of metallocene-catalysed polyethylene and low density polyethylene blends: Use of the double reptation and Palierne viscoelastic models

The linear viscoelastic behaviour of a series of mPE/LDPE blends was evaluated. Newtonian viscosity showed slight positive deviation from the double reptation model, which assumes miscibility or, at least, cooperative relaxation between the mixed species. This feature is typical of emulsion-like polymer blends and is attributable to additional relaxation processes that occur at low frequencies associated with deformability of the dispersed phase. Close agreement between data obtained by applying the Palierne model for emulsions of two viscoelastic liquids and experimental data was only shown for the blend of lowest LDPE content. Good predictions were derived for the higher LDPE contents, when partial miscibility of the components was assumed. In these cases, the viscoelastic response can be modelled through a hybrid model that takes into account the double reptation approach in the miscible phase models. Contrary to other heterogeneous blends, the blends examined here do not show an enhanced elastic character. This behaviour is probably due both to the extremely different relaxation times of the phases in the blends rich in mPE, and to the partial miscibility of the components in the blends of high LDPE content.

Linear rheology of binary melts from a phenomenological tube model of entangled polymers

We develop a simple phenomenological theory to describe linear viscoelasticity in bidisperse linear polymer melts. We describe the single-chain relaxation spectrum using a local description of relaxation times along the chain, which includes contour-length fluctuations as well as reptative motion. The complex modulus is calculated by summing the contributions from all the segments along the chain, and weighting the contributions by the entangled volume fraction remaining at each time. We find that the resulting predictions for the modulus fit data of binary blends of polybutadiene better than those of the widely used double reptation model.

Evaluation of Reptation Models for Predicting the Linear Viscoelastic Properties of Entangled Linear Polymers

The ability of reptation theories to quantitatively predict the linear viscoelastic response of linear polymer melts is systematically tested for various combinations of almost monodisperse, bidisperse, and polydisperse polystyrene (PS), high-density polyethylene (PE), and polycarbonate (PC) samples. Broad industrial HDPE and PC samples have been fractionated according to molar weight by preparative methods. The experimental master curves for dynamic moduli G‘ and G‘‘ are compared with predictions obtained from the experimentally determined molecular weight distribution (MWD) (by size exclusion chromatography) with the help of reptation models proposed in the literature. We focus on the Doi and Edwards relaxation function, with and without fluctuation effects, and the des Cloizeaux time-dependent diffusion relaxation function. These relaxation functions are combined with double reptation. We conclude that the des Cloizeaux relaxation function, associated with the double reptation model, gives the best results for well-entangled polymers, which are in quantitative agreement with experimental data. Indeed, it is the only model that is able to correctly predict the intermediate region between reptation relaxation and Rouse relaxation. We study the influence of short chains (slightly longer than the critical molecular weight) on the relaxation moduli of blends with long chains. We conclude that the model predicts that the chains smaller than 4Me relax too quickly in comparison with the experimental data. After a modification of the model, quantitative predictions are also obtained.

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 τe 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 τe) are determined from data for monodisperse polymers, the double reptation model predicts that for fixed weight-average molecular weight Mw, the zero-shear viscosity η0 increases slightly with increasing polydispersity Mw/Mn for log normal distributions, while for the dual constraint model η0 is almost independent of Mw/Mn. Experimental data for polystyrenes show no increase (or even a slight decrease) in η0 with increasing Mw/Mn at fixed Mw, 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.

Evaluation of molecular weight distribution from dynamic moduli

A method to evaluate molecular weight distribution (MWD) from dynamic moduli is presented here. It relies on the least-square fitting of the dynamic data to a model whose parameters depend on the MWD. In particular, the analytical solution for the relaxation modulus previously obtained from the double reptation model, with the Tuminello step relaxation function and the Generalized Exponential Function (GEX) describing the MWD (Nobile and Cocchini 2000), has been used. A Finite Element Approximation (FEA) has been applied to calculate dynamic moduli from the relaxation modulus as a function of MWD. The sensitiveness of the GEX-double reptation dynamic moduli on the model parameters has also been investigated and the results show that large changes of the Mw/Mn ratio weakly affect the dynamic moduli, while small changes of the Mz/Mw ratio significantly deform the dynamic moduli curves. The use of rheological data to obtain MWD, by the model used in this paper, will, therefore, be able to give rather well defined Mz/Mw ratios, while more uncertainty will be presented in the Mw/Mn results. The so-called GEX-rheological model for the dynamic moduli was applied to fit the experimental data of different polymers in order to obtain the best-fit parameters of the MWD of these polymers, without the need for the inversion of the double reptation integral equation. The stability of the results has been confirmed through the evaluation of the 90% confidence intervals for the first molecular weight averages. Finally, concerning the Mw and Mz values, the predictions obtained from the dynamic moduli measurements differ by less than 10% from those obtained from GPC measurements while, as expected, more uncertainty is present in the Mn predictions.

On the Rouse spectrum and the determination of the molecular weight distribution from rheological data

Maier et al. [Maier, D., et al., J. Rheol. 42, 1153–1173 (1998)] examined the reconstruction of binary molecular weight distributions from rheological data for a series of polystyrene mixtures using a generalized mixing rule recently introduced [Anderssen, R. S. and D. W. Mead, J. Non-Newtonian Fluid Mech. 76, 299–306 (1998)]. They found an unexpected high value for β(β=3.84±0.1), the mixing parameter, which is 1 for the reptation and 2 for the double reptation or entanglement models. This result can be understood when the relaxation time spectrum is decomposed into a Rouse and an entanglement part and only the latter is used for determination of the molecular weight distribution. Applying a procedure which separates the Rouse processes from the spectrum determined from measured dynamic shear moduli, β values are found which are in accordance with the double reptation theory and which give very good agreement between molecular weight distributions determined by size-exclusion chromatography and by rheological data evaluation.

Determination of Molecular Weight Distributions from Rheological Data: An Application to Polystyrene, Polymethylmethacrylate and Isotactic Polypropylene

Abstract Based on a recently introduced generalized mixing rule, which contains the results of the reptation and double reptation model as special cases, it is possible to determine the molecular weight distribution (MWD) from rheological data. By evaluating data from bimodal PS-mixtures Maier et al. (1998) have shown how the MWD can be estimated from the relaxation shear modulus, G(t), using an inversion method. Thimm et al. (1999) derived an analytical relation between the relaxation time spectrum and the MWD, which is able to reproduce the result of Maier et al. (1998) with less computational effort. In this article we compare both methods by evaluating data from three different series of polymer mixtures: Polystyrene (PS), Polymethylmethacrylate (PMMA) and isotactic Polypropylene (iPP). We compare the MWD obtained from rheological data with results from size exclusion chromatography (SEC) and discuss differences.

On the recovery of molecular weight functionals from the double reptation model

Recently, Mead (J. Rheology, 38 (1994) 1797–1827) has shown, for the double reptation model with a monodisperse relaxation function, characterized by a single time constant, how moments for the molecular weight distribution (MWD) can be determined directly from measurements of the linear viscoelastic relaxation modulus. The purpose of this paper is to show that this result is a special case of a more general result which allows linear inner-product functionals defined on the MWD, such as its moments, to be determined directly from the measurements. The applicability of this strategy is verified and validated using synthetic data.

Evolution of the molecular weight distribution and linear viscoelastic rheological properties during the reactive extrusion of polypropylene

AbstractHomogeneous and nonhomogeneous reactive extrusion of polypropylene is modeled using random chain scission statistics coupled with the double reptation mixing rule. In this manner, the evolution of both the molecular weight distribution and the linear viscoelastic material properties is quantitatively predicted for the reactive extrusion–pelletization process. Dispersion in the level of random chain scission has little impact on the MFI for a given average level of chain scission; however, dispersion does generate a marked increase in the recoverable compliance (melt elasticity) relative to the ideal homogeneous random chain scission case. Methods to quantitatively determine the degree of cracking dispersion in processing equipment are identified. Quality control issues such as blending materials of known linear viscoelastic properties to obtain a desired property set is considered in the context of known empirical relations consistent with the double reptation model. Simple mixing rules for the melt flow index and the steady‐state recoverable compliance involving only single component MFI and Je information are derived. © 1995 John Wiley & Sons, Inc.