Linear Viscoelastic Data Research Articles

A technique to infer structural information for low level long chain branched polyethylenes

A technique is proposed to relate the weight average molecular weight of linear chains, M wBL, in low level long chain branched polyethylenes to their linear viscoelastic data. The new method is based on a previously reported empirical technique [Macromolecules 33 (2000) 7481] and was developed through the use of basic molecular theories. The new technique was applied to model systems whose linear viscoelastic properties were simulated using the molecular model of Milner et al. [Macromolecules 31 (1998) 9345] and to long chain branched metallocene polyethylenes. It is applicable to branched polyethylenes with low levels of long chain branching. In the case of a branched metallocene polyethylene, the structural parameter, M wBL, inferred from the rheological data together with the GPC data such as M w or M n of the sample describes all aspects of the structure of the polymer. In the case of highly branched polymers, a possible modification of the technique is also proposed.

Simplified theory for linear rheology of monodisperse linear polymers

A new version of the tube theory based on the de Gennes–Doi–Edwards reptation concept (reported in Likhtman and McLeish's work published in 2002) is evaluated, modified to allow for simplified computations, and used to study the relationship between zero-shear viscosity and molecular weight for monodisperse entangled linear homopolymers. The Likhtman–McLeish model combines self-consistent theories for contour length fluctuations and constraint release with reptation theory for monodisperse linear polymers. Because of the nature of the Rubinstein and Colby approach used for the treatment of constraint release, the related term is probabilistic and requires stochastic simulations for the calculation of the relaxation modulus G(t). This makes the Likhtman–McLeish model computationally difficult to use. In this work we solve this problem by generating an approximate closed-form solution for the stochastic term. Then analytical integration of the relaxation modulus function G(t) provides an expression for the zero-shear viscosity (η0). Results of the computations of the zero-shear viscosity and of the slope of η0 versus molecular weight are compared with available experimental data for monodisperse entangled linear polystyrene and polyethylene (hydrogenated polybutadiene). The model is a major improvement over previous theoretical models, even if there is still some disagreement between the predictions and experimental data of the slope of η0 versus molecular weight. The possibility of inferring monomer chemistry–dependent parameters from the zero-shear viscosity remains a difficult task because of the introduction of a constraint-release parameter. Nevertheless, the model is a useful tool for the prediction of linear viscoelasticity data. © 2004 Wiley Periodicals, Inc. J Appl Polym Sci 94: 569–586, 2004

Fourier-transform rheology experiments and finite-element simulations on linear polystyrene solutions

Large amplitude oscillatory shear strain was applied to anionically synthesized linear polystyrene solutions in dioctylphthalate. The resulting torque was analyzed in Fourier space with respect to frequency, magnitude, and phase (Fourier-transform rheology). The concentration of the solutions was varied to achieve different degrees of entanglement. In addition, numerical simulations were performed using the Giesekus constitutive equation fitted on the basis of linear viscoelastic data. We found a good qualitative agreement between experiments and predictions; a quantitative agreement was reached for intermediate strain amplitudes. Some deviations were observed at very low strain amplitudes. We present a descriptive relation for the relative magnitude of the third harmonic as a function of strain amplitude using a modified damping function. From this relation we obtained a universal parameter that describes the scaling behavior law for the increasing non-linearity (e.g., measured by the relative intensity ...

Definitions of entanglement spacing and time constants in the tube model

Numerous papers have recently appeared in the literature presenting quantitative comparisons of experimental linear viscoelastic data to the most recent versions of “tube” models for entangled polymer melts and solutions. Since these tube models are now being used for quantitative, rather than just qualitative, predictions, it has become important that numerical prefactors for the time constants that appear in these theories be evaluated correctly using literature data for the parameters (i.e., density, plateau modulus, etc.) that go into the theories. However, in the literature two definitions of the entanglement spacing in terms of plateau modulus have been presented, and confusion between these has produced numerous errors in the recent literature. In addition, two different definitions of the “equilibration time,” a fundamental time constant, have also appeared, creating additional potential for confusion. We therefore, carefully review the alternative definitions and clarify the values of the prefactors that must be used for the different definitions, in the hope of helping future authors to avoid such errors.

Effect of Surface Modification on the Interfacial Tension between the Melts of High-Density Polyethylene and Nylon 66: Correlation between Rheology and Morphology

This paper presents the effect of a low-energy Ar+ ion-beam irradiation on the interfacial tension between high-density polyethylene (HDPE) and Nylon 66 (Ny66). Ny66 was melt blended in a twin screw extruder with two different kinds of HDPE (pristine HDPE and HDPE with its surface modified by low-energy ion-beam irradiation (IBHDPE)). The viscoelastic dynamic moduli of two kinds of HDPE/Ny66 blends in the molten state have been measured in a cone and plate rheometer as a function of the frequency. The rheology of the blends was fit to three theoretical models (the Palierne model, the Choi−Schowalter (CS) model, and the Lee−Park (LP) model) to infer the values of the equilibrium interfacial tension. For pristine HDPE/Ny66 blends, all three models predict the linear viscoelastic data pretty well despite the relatively high concentration of the dispersed phase. The values of the interfacial tension calculated for the IBHDPE/Ny66 blends were less than those of the unmodified HDPE/Ny66 blends, which indicates the effect of surface functionalization. For IBHDPE/Ny66 blends, predictions of the Palierne and the CS models show some discrepancies from the experimental data in the low-frequency region. The LP model provides a better prediction in the low-frequency region but it shows deviations in the high-frequency region. A modified LP model gives a better description of changes in the interfacial tension with irradiation time. Explanations for the model predictions and modifications of the theoretical models are provided.

Polydispersity index from linear viscoelastic data: unimodal and bimodal linear polymer melts

This article describes a method for determining the polydispersity index Ip2=Mz/Mw of the molecular weight distribution (MWD) of linear polymeric materials from linear viscoelastic data. The method uses the Mellin transform of the relaxation modulus of a simple molecular rheological model. One of the main features of this technique is that it enables interesting MWD information to be obtained directly from dynamic shear experiments. It is not necessary to achieve the relaxation spectrum, so the ill-posed problem is avoided. Furthermore, a determinate shape of the continuous MWD does not have to be assumed in order to obtain the polydispersity index. The technique has been developed to deal with entangled linear polymers, whatever the form of the MWD is. The rheological information required to obtain the polydispersity index is the storage G′(ω) and loss G″(ω) moduli, extending from the terminal zone to the plateau region. The method provides a good agreement between the proposed theoretical approach and the experimental polydispersity indices of several linear polymers for a wide range of average molecular weights and polydispersity indices. It is also applicable to binary blends.

Relaxation Dynamics of Entangled and Unentangled Multiarm Polymer Solutions: Comparisons with Theory

Linear viscoelastic properties of model branched A3−A−A3 1,4-polybutadiene melts and solutions are investigated experimentally, using small-amplitude oscillatory shear measurements, and theoretically, using a recent tube model theory for stress relaxation of entangled H-shaped polymers. This study provides experimental guidance on two key assumptions in the theory: namely, that stress relaxation of multiply branched, entangled polymers is hierarchical and that relaxed portions of a branched molecule accelerate stress relaxation of unrelaxed sections of the same molecule in qualitatively the same manner as a low molecular weight diluent. Experimental results for multiarm A3−A−A3 polymer melts and solutions covering a wide range of arm A and connector A molecular weights provide broad support for a hierarchical relaxation process in these materials. However, linear viscoelastic data for A3−A−A3 polymers strongly disagree with the second assumption. Specifically, we find that the relaxed arms provide a much greater than expected retardation of branch point motion even on time scales well beyond the arm relaxation time. Branch point diffusivities estimated from the experimental data are typically 1−2 orders of magnitude lower than predicted by theory. The theory also underpredicts the breadth and amplitude of the dynamic storage and loss moduli of multiarm melts and solutions, particularly for materials where the number of entanglements per arm is low.

Poly(ethylene oxide)/Silica Nanocomposites: Structure and Rheology

The effects of polymer−particle and particle−particle interactions on viscoelastic properties of nanocomposite materials are investigated using narrow molecular weight distribution poly(ethylene oxide) (PEO) containing isotropic silica nanospheres. Nanocomposites are prepared using a “freeze-drying” method to guarantee homogeneous dispersion of silica. The dispersion state of silica nanospheres is characterized by atomic force microscopy phase contrast imaging, and mechanical rheometry is used to study relaxation dynamics and viscoelastic properties of these materials in the melt state. Linear viscoelastic data indicate a transition to a solidlike response at low oscillation frequencies for particle volume fractions φ as low as 2%, dramatically lower than the theoretical percolation threshold (φ ∼ 30%). Nanoparticle volume fraction, surface chemistry, and PEO molecular weight are all found to strongly influence nanocomposite structure and dynamics. A filler networking mechanism, wherein nanosized silica p...

A phenomenological model for linear viscoelasticity of monodisperse linear polymers

Although the reptational model of Doi and Edwards gives a successful description of viscoelasticity of flexible linear polymers, the success is restricted to the terminal region.1 There have been several attempts to modify the Doi-Edwards model to describe wider range of time or frequency.2–6 This paper suggests a simple phenomenological model which can describe wider range of molecular weight than such molecular models can. Although our model is a phenomenological one, it is practical and convenient to predict the effect of molecular weight distribution on linear viscoelastic data because of its simple mathematical form.

Polymerization compounding: Epoxy‐montmorillonite nanocomposites

AbstractA strategy to design intercalated montmorillonite nanocomposites has been explored. A commercial organoclay, 1.34 TCN (Nanocor Inc.), with bis(2‐hydroxylethy1) methy1 tallow ammonium, was modified by tolylene 2,4‐diisocyanate (TDI) and bisphenol A (BA). Thermogravimetric analysis (TGA), Fourier transform infrared (FTIR) spectroscopy and X‐ray diffraction (XRD) results of unmodified and modified 1.34 TCN (1.34‐TDI‐BA) indicate that TDI and BA have reacted with hydroxy1 groups on the surface of 1.34 TCN and hydroxy1 groups in the interlayer of 1.34 TCN. Using a classical two‐stage cure process with diamine as curing agent, intercalated epoxy nanocomposites were prepared for both types of organoclays. XRD and TEM results showed that the basal spacing of clay in nanocomposites was 3.68 and 4.42 nm for 1.34 TCN and 1.34‐TDI‐BA, respectively. Dynamic mechanical analysis (DMA) was performed on both modified and unmodified organoclay composites. Modified organoclay composites were found to have enhanced storage moduli, particularly at temperatures higher than the glass transition, Tg, of the matrix. Glass transition temperatures extracted from linear viscoelastic data are found to be slightly higher for modified organoclay nanocomposites, indicating enhanced interactions between the modified organoclay and the epoxy matrix. These results were also confirmed by independent measurements of Tg using differential scanning calorimetry (DSC).

Determination of the molecular weight distribution of entangled linear polymers from linear viscoelasticity data

In a companion paper [Macromolecules 35 (2002) 2689], we have addressed the direct problem of predicting the linear viscoelastic response of entangled linear polymers from their molecular weight distribution, using a suitable model inspired by reptation theory. By comparing the theoretical results to experimental data for a variety of samples of different nature (polystyrene, polycarbonate, high-density polyethylene) and distribution (monomodal and multimodal), we found that the time-dependent diffusion reptation model proposed by des Cloizeaux, suitably modified to treat short chains and to include Rouse processes, is capable of quantitative predictions. In the present paper, we use the modified des Cloizeaux model to address the inverse problem of predicting molecular weight distribution from dynamic moduli. A parametric approach is implemented to deal with the ill-posedness of the problem. Results are given for most of the samples studied in [Macromolecules 35 (2002) 2689]. They are in quantitative agreement with size-exclusion chromatography data. In particular, the proposed methodology is able to resolve small amounts of short chains in bimodal blends containing large amounts of long chains.

Linear Viscoelasticity of a Polymeric Bicontinuous Microemulsion

Ternary polymer blends of immiscible homopolymers and the corresponding diblock copolymer are useful models for studying equilibrium and nonequilibrium behavior of self-assembled fluids. We report linear viscoelastic data for a polymeric bicontinuous microemulsion, experiments that are possible due to the comparatively high viscosities of the principal components. After subtracting a viscous background contribution from the pure constituents, the microemulsion exhibits “excess” viscoelastic behavior similar in character to that resulting from the Rouse model of polymer dynamics. The data are compared to the predictions of a time-dependent Landau−Ginzburg model developed by Pätzold and Dawson, using structural parameters derived from neutron scattering as input. This model captures the essential characteristics of the viscoelastic behavior very well. However, using independent dynamic light scattering measurements of the Onsager coefficient for these blends, it appears that the model fails to predict either the magnitude or the temperature dependence of the zero shear viscosity and relaxation time of the microemulsion accurately. Possible origins for these discrepancies are discussed.

Van Gurp-Palmen Plot II - classification of long chain branched polymers by their topology

Rheological data on monodisperse model-type polymers with a certain topology are available in the literature. A comparison between these series has not been made yet, mainly because the different chemical nature of these polymers inhibited a direct comparison. With the so-called reduced van-Gurp-Palmen plot (rvGP) this task has become possible and this article applies this plot to these model-type polymers. We check whether these polymers exhibit features allowing one to classify them on basis of the corresponding linear viscoelastic data. We define a characteristic point PC(Gred,c, δc) that represents the sample. By drawing all available points in rvGP-coordinates we obtain a topology map on which certain topologies cover distinct areas on this map. Unfortunately these areas overlap to some extent. For the unambiguous characterization of an unknown sample additional information is needed. We apply the topology map to metallocene poly(olefin)s.

Numerical Simulation of Extrudate Swell Problem and Evaluation of Applicability of Viscoelastic Constitutive Models. 1. A Study on Axisymmetric Extrudate Swell from a Straight Die.

Numerical simulations of axisymmetric extrudate swell for a low density polyethyrene (LDPE IUPAC-A) melt were carried out using the K-BKZ model as an integral type and the PTT and Larson models as a differential type. We employed both the reversible and irreversible models for the K-BKZ and Larson models to investigate the effect of reversing deformation on the extrudate swell from a straight die. First, relaxation spectrum and nonlinear parameters in the constitutive equation were determined from dynamic data of linear viscoelasticity and steady shear flow data by using the nonlinear regression, and then they were applied to the simulation of extrudate swell problem. We evaluated the applicability of these models by comparing experiments and simulation for both rheological data and extrudate swell ratio. For the K-BKZ model, the reversible model over-estimates the swell ratio in high shear rate, whereas the irreversible model predicts the experimental data with high accuracy. The irreversible Larson model also shows the good prediction of swell, on the other hand, the reversible one over-estimates the swell and there exists the multiple solutions. For the PTT model, we carried out the simulation using the models with different values of nonlinear parameter, e , which has the same steady shear flow properties and the different properties of stress relaxation under reversing double step deformation. The model having the larger stress in the stress relaxation under reversing double step deformation shows the larger swell. From these results, we find that the consideration of reversing deformation characteristics is required for the extrudate swell from the straight die.

Rheological models based on the double reptation mixing rule: The effects of a polydisperse environment

We present a comparison of various rheological models based on the double reptation concept, which relates linear viscoelastic data to molecular-weight distribution in order to determine the most efficient way to describe polydispersity effects. We have used for this study a two-step process. First, we have performed a systematic comparison of the predictions of various models with experimental data already published in the literature for binary blends of polystyrene (PS) and polymethylmethacrylate (PMMA). Then, we have compared the values of the complex shear modulus derived from these molecular models with rheological data obtained on “commercial” polydisperse polymers (PS, PMMA, and high-density polyethylene). It is shown that only the two models which explicitly take into account the effects of the polydisperse surrounding of a macromolecular chain through “tube renewal” effects are able to describe correctly the polydispersity effects on zero-shear viscosity and steady-state compliance.

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.

Molecular Drag-Strain Coupling in Branched Polymer Melts

ABSTRACTThe “pom-pom” model of McLeish and Larson (J. Rheol. 42, 81, 1998) provides a simple molecular theory for the nonlinear rheology of long chain branched polymer melts. The Edwards-de Gennes tube concept is used to derive a constitutive equation for a simple branched molecule composed of two star polymers linked by a single backbone chain. A feature of this model is that the backbone section of tube can stretch up to maximum length given by the maximum entropic drag-force from the arms, after which the star arms are withdrawn into the backbone tube. This produces a sharp transition in the extensional viscosity at this maximum stretch. This unphysical feature results from an over-simplification of the behaviour near the branch points.In this paper we introduce a simple treatment of the coupling between relaxed and unrelaxed polymer segments at branch-points. This allows for localised displacements of branch-point within a quadratic potential before maximum extension is reached. Displacing the branch-point reduces the length of arm outside the tube and so reduces in the drag on the star arms. This smoothes out the sharp transitions in extensional viscosity in the original “pom-pom” model at the cost of introducing an extra unknown parameter.This modification improves the prediction of the nonlinear rheology of H-polymers whose molecular structure is known. Alternatively, for polymers of unknown structure such as commercial Low Density Polyethylene, the model parameters may be fitted from linear viscoelastic and uniaxial extension data, to provide predictions for the behaviour in transient nonlinear shear and planar extension. By including local branch-point displacement we find improved agreement with the data for Low-Density Polyethylene.

Rheological modelling of the peeling of pressure-sensitive adhesives and other elastomers

The transient extensional viscosity of a polyisobutylene (PIB)-based pressure-sensitive adhesive (PSA) is measured in a filament stretching instrument at three stretch rates, and compared with theoretical predictions based on the linear viscoelastic (LVE) properties of the PSA and the use of linear and non-linear constitutive equations. Based on previously reported adhesive stretch rate profiles across the peel front, a model predicting peel force from peel rate is proposed and evaluated for each constitutive equation. A new expression for the relative contributions of the individual relaxation modes of the polymeric material to the total measured peel force is derived and the result is discussed with regard to correlation of LVE data to peel performance. Comparison of predicted and measured peel force values reveals that peeling of this PSA is only qualitatively described by linear viscoelasticity; quantitative agreement is obtained when the strain-hardening behaviour of the adhesive is properly modelled.

Stress Relaxation and Oscillatory Tests to Distinguish Between Doughs Prepared from Wheat Flours of Different Varietal Origin

ABSTRACTThe relaxation properties of flour‐water‐salt doughs prepared from four different flour types (weak, medium, strong, and extra strong) at different water absorption levels from 58 to 66% with protein contents of 10.0, 10.9, 13.2, and 11.8%, respectively, were studied by imposing varying strain amplitudes of 0.1–29%. Oscillatory tests in the linear viscoelastic region of the 66% absorption strong and weak dough cannot distinguish between the two types of dough. The inability to differentiate between dough types also applied to oscillatory tests on 58% absorption weak and 66% absorption strong doughs. However, the relaxation modulus of dough (extending over time) behaved quite distinctively at high strains, where dough samples experience large deformations. At strain amplitudes of ≤0.1% (i.e., in the linear viscoelastic region), different dough types behaved similarly. Likewise, the relaxation modulus completely relaxed at sufficiently long times. The magnitude of the modulus at intermediate‐ and high‐strain amplitudes were in the order: extra strong > strong > medium > weak, indicating a higher level of elasticity in the extra strong dough samples despite its lower protein content. The relaxation times spectrum of the weak flour, extracted from the relaxation modulus data, reveals a broad relaxation process. The stress relaxation data are very reproducible at high‐strain amplitudes (≈1–15% for up to 3 × 103 sec). This work demonstrated, for the first time, the consistency in oscillatory and relaxation measurements for dough. It also clearly showed that linear viscoelastic data, although important in the characterization of time scales in dough, are largely irrelevant in differentiating between dough types. Furthermore, without proper care, a false steady‐state behavior can be obtained with standard viscometric measurements due to slippage at the dough‐plate interface.

An intermediate model method for obtaining a discrete relaxation spectrum from creep data

A convenient method is described for obtaining a discrete stress relaxation spectrum from linear viscoelastic creep data by means of a three-stage process. In stage one, a discrete retardation spectrum is fitted to the creep data using a least squares procedure, subject to the constraint that the discrete spectrum must be a specified order of polynomial function of the retardation time. In stage two, the resulting generalised Voigt model is solved numerically for an imposed step in strain, to determine the stress relaxation modulus function of time. In stage three, a discrete relaxation spectrum is fitted to the calculated stress relaxation modulus function. Although three stages are involved instead of the usual two, the procedure has been found to have certain practical advantages. These advantages make it suitable for the generation of relaxation spectra needed in viscoelastic stress analyses of solids, for example by the finite element method. In order to illustrate the proposed procedure it is applied to both artificial data and experimental creep data for poly(methyl methacrylate) at 70°C and at the glass transition.