The nonlinear strain measure of polyisobutylene melt in general biaxial flow and its comparison to the Doi-Edwards model

The nonlinear strain measure of a polyisobutylene (PIB) melt as determined by analysis of uniaxial, planar, ellipsoidal, and equibiaxial extensions is compared to the predictions of the molecular model of Doi and Edwards. It is found that the universal strain function of the Doi-Edwards model is unable to predict the nonlinear behavior of this polymer melt in general extensional flow. The qualitative agreement between predictions and experimental data for the strain dependence of shear stress and first normal stress difference in shear flow that was considered as powerful evidence for the correctness of the Doi-Edwards model seems to be accidental. The exaggerated strain dependence of the model suggests a need to reconsider the assumptions concerning the chain retraction process.Presented at the Golden Jubilee Conference of the British Society of Rheology and Third European Rheology Conference, Edinburgh, 3–7 September, 1990.Dedicated to Professor F.R. Schwarzl on the occasion of his 65th birthday     

Time-dependent Viscoelastic Behavior of an LDPE Melt

Two differential constitutive equations, i.e. Giesekus model and Johnson–Segalman model were employed here to predict the time-dependent viscoelastic behavior of an LDPE melt in thixotropy-loop experiments and step shear rate experiment. Multiple relaxation modes were adopted, and the parameters used to describe the nonlinear viscoelasticity in the two models were obtained by fitting the shear-thinning viscosity. The predictions on those transient shear characteristics by the two models are found in qualitative agreement with our previous experiments. Johnson– Segalman model predicts oscillation behavior in the thixotropy-loop and step shear rate experiments, whereas Giesekus model does not. Both models predict higher shear stresses than the experimental data in the case of long time shearing, implying that both models are not able to completely characterize the time-dependent shear stress of the melt at high shear rate.The project was supported by the National Natural Science Foundation of China (10402024, 50335010).The English text was polished by Yunming Chen.     

Exponential shear flow of branched polyethylenes in rotational parallel-plate geometry

Exponential shear flow, as a strong flow with the potential to generate a high degree of molecular stretching, has attracted considerable interest in recent years. So far, exponential shear flow has been realized by either sliding-plate or cone-and-plate (CP) geometry. Both geometries guarantee homogeneous shear flow. Here, we present experimental data on exponential shear flow of several long-chain branched polyethylene melts with different degrees of strain hardening obtained by using parallel-plate (PP) geometry in a rotational rheometer. This type of geometry, which is standard in linear-viscoelastic characterization of polymer materials, produces inhomogeneous shear flow. A comparison of exponential shear flow data obtained by PP and CP geometry is made. Additionally, the experimental data are compared to predictions of the rubber-like liquid (RLL) and the molecular stress function (MSF) theories. For this purpose, the relaxation spectra of the polymer melts considered were obtained by standard linear-viscoelastic characterization. In addition, two irrotational parameters and one rotational parameter are required by the MSF theory. While the irrotational parameters were obtained from fitting to elongational viscosity data, the value of the rotational parameter was used as given in the literature. It can be concluded that viable experimental data in exponential shear flow can be obtained by PP geometry. For finite linear-viscoelasticity (RLL theory), predictions of reduced shear stress for CP and PP geometry coincide, but nonlinear material behavior (as modeled by the MSF theory) leads to small differences between both geometries. Furthermore, it is shown that the MSF predictions are in excellent agreement with the experimental data in exponential shear flow and that this type of flow leads to much less chain stretching than elongational flow.Dedicated to the memory of Prof. Arthur S. Lodge (1922–2005).     

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.     

Relaxation dynamics of selected polymer chain segments and comparison with theoretical models

Simultaneous measurement of infrared dichroism and birefringence is used to study selected polymer segment dynamics in isotopically labeled block copolymers. Two different polymers were studied: polybutadiene and poly (ethylene propylene). The first type consisted of a triblock with a short middle block labeled and a diblock with a short end block labeled, while the second type consisted of a triblock with three equal blocks and the end blocks labeled. Results of step strain experiments at –10°C for polybutadiene and at room temperature for poly(ethylene propylene) indicated that segments located at chain ends relax faster than segments located at chain centers. These experimental data were compared to the predictions of two molecular models: the bead-spring model of Rouse and the tube model of Doi and Edwards, and it was found that both models correctly predict the qualitative features of segmental relaxation. However, the tube-model predictions were closer to the experimental results. In addition, when the effects of orientational coupling interactions between segments in the melt were incorporated into this model, its predictions quantitatively agreed with the experimental results. The orientational coupling coefficient for poly(ethylene propylene) was 0.45 as measured from previous work, and for polybutadiene it was found to be 0.4.Delivered as a Keynote Lecture at the Golden Jubilee Conference of the British Society of Rheology and Third European Rheology Conference, Edinburgh, 3–7 September, 1990.Dedicated to Prof. R.S. Stein, University of Massachussets at Amherst, USA, on the occasion of his 65th birthday.     

Stress relaxation of a polymer melt after extensional flow

The relaxation behaviour of a stretched polymer melt is described with the aid of the semi-empirical constitutive equation of Wagner. The theoretical predictions based on this model, are in fair agreement with the experimental results, as far as the period of stress decay following the stress build-up is not too long.     

On the burst of branched polymer melts during inflation

Two molten low-density polyethylene melts, shaped as plates, have been inflated into a circular cylinder during isothermal conditions. Lowering the inflation rates allow the plates to be inflated into a larger volume of the cylinder before bursting. Numerical simulations of the inflations have been performed, using a time-strain separable constitutive K-BKZ equation based on the potential function from the Doi–Edwards theory. The material parameters in the constitutive model are based on liner viscoelastic and time dependent uniaxial elongational viscosities. The numerical calculations show quantitative agreement with the experiments, including the appearance of the burst, for a wide range of experimental conditions. This strongly suggests that the initiation of the burst in the polymer melts is a hydrodynamic phenomenon.     

A method for calculating rheological and morphological properties of constant-volume polymer blend models in inhomogeneous shear fields

We show how to formulate two-point boundary-value problems in order to compute fully-developed laminar channel and tube flow profiles for viscoelastic fluid models. The formulation is applied to Couette and pressure-driven flows separately, or a combination of both. The application of this methodology is illustrated analytically for the Upper-Convected Maxwell Model, and it is applied computationally for the Phan-Thien/Tanner and Giesekus Models. Numerical solutions exist for the last two models [J.Y. Yoo, H.C. Choi, On the steady simple shear flows of the one-mode Giesekus fluid, Rheol. Acta 28 (1989) 13–24; P.J. Oliveira, F.T. Pinho, Analytical solution for fully developed channel and pipe flow of Phan-Thien–Tanner fluids, J. Fluid Mech. 387 (1999) 271–280; M.A. Alves, F.T. Pinho, P.J. Oliveira, Study of steady pipe and channel flows of a single-mode Phan-Thien–Tanner fluid, J. Non-Newtonian Fluid Mech. 101 (2001) 55–76], allowing verification of the computational technique. Subsequently, the computational algorithm is applied to the constant-volume polymer blend models of Maffettone and Minale [P.L. Maffettone, M. Minale, Equation of change for ellipsoidal drops in viscous flow, J. Non-Newtonian Fluid Mech. 84 (1999) 105–106 (Erratum), J. Non-Newtonian Fluid Mech. 78 (1998) 227–241] and Dressler and Edwards [M. Dressler, B.J. Edwards, The influence of matrix viscoelasticity on the rheology of polymer blends, Rheol. Acta 43 (2004) 257–282; M. Dressler, B.J. Edwards, Rheology of polymer blends with matrix-phase viscoelasticity and a narrow droplet size distribution, J. Non-Newtonian Fluid Mech. 120 (2004) 189–205]. Rheological and morphological properties of the model blends are thus obtained as functions of the spatial position within the channel, applied pressure drop, and shear rate at the wall.     

Lubricated cross-slot flow of a low density polyethylene melt

Flow-induced birefringence and particle tracking velocimetry are used to investigate the lubricated flow of a low density polyethylene melt in a cross-slot geometry. The numerical predictions of the extended Pom-Pom (XPP) model in its original and modified (mXPP) version as well as those of the Giesekus model are analyzed along selected streamlines in two-dimensional (2D) complex flows involving a mixture of shear and planar extensional deformations at two Weissenberg numbers of Wi=21 and 29. Oil film light reflections perturbing the particle tracking velocimetry data analysis together with multiple orders of retardation occurring within the laser beam close to the stagnation point prevent a conclusive discrimination between the mentioned models. Although the agreement with the experimental data is mostly qualitative, the Pom-Pom model does not overestimate the data along the cross-slot symmetry lines contrary to what was observed in other cross-slot experiments without lubrication. This is a clear indicator that end effects play a central role in unlubricated cross-slot geometries having a large aspect ratio.     

Elongational-flow predictions of the Leonov constitutive equation

The basic thermodynamic ideas from rubber-elasticity theory which Leonov employed to derive his constitutive model are herein summarized. Predictions of the single-mode version are presented for homogeneous elongational flows including stress growth following start-up of steady flow, stress decay following sudden stretching and following cessation of steady flow, elastic recovery following cessation of steady flow, energy storage in steady-state flow, and the velocity profile in constantforce spinning. Using parameters of the multiple-mode version which fit the linearviscoelastic data, the Leonov-model predictions of elongational stress growth during, and elastic recovery following, steady elongation are calculated numerically and compared to the experimental results for Melt I and to the Wagner model. It is found that the Leonov model, as originally formulated, agrees qualitatively with the data, but not quantitatively; the Wagner model gives quantitative agreement, but requires much nonlinear data with which to fit model parameters. Quantitative agreement can be obtained with the Leonov model, if the nonequilibrium potential which relates recoverable strain to strain rate is adjusted empirically. This can most simply be done by making each relaxation time dependent upon the recoverable strain. The Leonov model, unlike the Wagner model, is derived from an entropic constitutive equation, which is advantageous for calculating stored elastic energy or viscous dissipation. The Leonov model also has an appealingly simple differential form, similar to the upper-convected Maxwell model, which, in numerical calculations, may be an important advantage over the integral Wagner model.     

A modification of the convective constraint release mechanism in the molecular stress function model giving enhanced vortex growth

The molecular stress function model with convective constraint release (MSF with CCR) constitutive model [M.H. Wagner, P. Rubio, H. Bastian, The molecular stress function model for polydisperse polymer melts with dissipative convective constraint release, J. Rheol. 45 (2001) 1387] is capable of fitting all viscometric data for IUPAC LDPE, with only two adjustable parameters (with difference found only on reported “steady-state” elongational viscosities). The full MSF with CCR model is implemented in a backwards particle-tracking implementation, using an adaptive method for the computation of relative stretch that reduces simulation time many-fold, with insignificant loss of accuracy. The model is shown to give improved results over earlier versions of the MSF (without CCR) when compared to well-known experimental data from White and Kondo [J.L. White, A. Kondo, Flow patterns in polyethylene and polystyrene melts during extrusion through a die entry region: measurement and interpretation, J. Non-Newtonian Fluid Mech. 3 (1977) 41]; but still to under-predict contraction flow opening angles. The discrepancy is traced to the interaction between the rotational dissipative function and the large stretch levels caused by the contraction flow. A modified combination of dissipative functions in the constraint release mechanism is proposed, which aims to reduce this interaction to allow greater strain hardening in a mixed flow. The modified constraint release mechanism is shown to fit viscometric rheological data equally well, but to give opening angles in the complex contraction flow that are much closer to the experimental data from White and Kondo. It is shown (we believe for the first time) that a constitutive model demonstrates an accurate fit to all planar elongational, uniaxial elongational and shear viscometric data, with a simultaneous agreement with this well-known experimental opening angle data. The sensitivity of results to inaccuracies caused by representing the components of the deformation gradient tensor to finite precision is examined; results are found to be insensitive to even large reductions in the precision used for the representation of components. It is shown that two models that give identical response in elongational flow, and a very similar fit to available shear data, give significantly different results in flows containing a mix of deformation modes. The implication for constitutive models is that evaluation against mixed deformation mode flow data is desirable in addition to evaluation against viscometric measurements.     

Analytical elasto-creep model of interfacial thermal stresses and strains in trilayer assemblies

A two-dimensional model has been developed for thermal stresses, elastic strains, creep strains, and creep energy density at the interfaces of short and long trilayer assemblies under both plane stress and plane strain conditions. Both linear (viscous) and non-linear creep constitutive behavior under static and cyclic thermal loading can be modeled for all layers. Interfacial stresses and strains are approximated using a combination of exact elasticity solutions and elementary strength of materials theories. Partial differential equations are linearized through a simple finite difference discretization procedure. The approach is mathematically straightforward and can be extended to include plastic behavior and problems involving external loads and a variety of geometries. The model can provide input data for thermal fatigue life prediction in solder or adhesive joints. For a typical solder joint, it is demonstrated that the predicted cyclic stress–strain hysteresis shows shakedown and a rapid stabilization of the creep energy dissipation per cycle in agreement with the predictions of finite element analysis.     

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.     

THE FLOW-INDUCED VIBRATION OF A FLEXIBLE STRIP HANGING VERTICALLY IN A PARALLEL FLOW PART 1: TEMPORAL AEROELASTIC INSTABILITY

The effect of several parameters in a fluid–strip system are studied for linear/nonlinear models in detail. Such parameters are: the number of modes in the Galerkin discretization, the length of the strip, and the flow velocity. The present simulation clearly shows that when nonlinear forces are considered the response approaches a flutter-type limit cycle at supercritical flow speeds. The Reynolds number at which the strip begins to oscillate is about 10⁴–10⁵. With a further increase of the flow velocity the strip oscillates regularly. At higher flow velocities the oscillation becomes violent and irregular. The amplitude, frequency and drag coefficient at the limit cycle are presented as functions of the flow velocity for a given strip. The numerical predictions are in qualitative agreement with previous experimental data.     

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.     

Polymeric liquids in extension: fluid mechanics or rheometry?

We use a transient 3D free surface finite element method to simulate flow of entangled polymer fluids in the dual cylinder wind-up extensional rheometer. The constitutive equations are K-BKZ integral representations of the Doi–Edwards models with and without the independent alignment approximation (IA). It is demonstrated that the actual kinematics in this rheometer is a mixture of planar and uniaxial extension. Moreover, the ratio of planar to uniaxial deformation is highly dependent upon whether IA is invoked. Without IA, the flow has a tendency toward planar extension, while it tends to be more uniaxial with IA invoked. As a second illustration of the techniques, we simulate the phenomenon of delayed rupture after rapid extension of entangled polymer systems. It is demonstrated that this phenomenon can be explained on the basis of the Doi–Edwards model in terms of a Considere-type instability after chain stretch relaxation.     

Two-dimensional numerical analysis of non-isothermal melt spinning with and without phase transition

A model and simulation method are developed for two-dimensional non-isothermal melt spinning of a visco elastic melt. The visco elastic stress is evaluated from a non-isothermal Giesekus constitutive equation developed by application of the pseudo-time method to the isothermal form of the model [J. Non-Newt. Fluid Mech. (2001)]. The crystallization kinetics is described with the model proposed by Nakamura et al. [J. Appl. Polym. Sci. 17 (1973) 1031], whereas the crystallization rate, which is a function of both temperature and molecular orientation, is evaluated according to the equation proposed by Ziabicki [Fundamentals of Fiber Formation, Wiley, New York, 1976]. The set of non-linear governing equations is solved by using the DEVSS-G/SUPG finite element method. Melt spinning is simulated for two different polymers: amorphous polystyrene and fast-crystallizing Nylon-6,6. The analysis demonstrates that although the kinematics in the thread-line are approximately one-dimensional, the radially non-uniform thermal history, caused by the leading order variation of the temperature gradient ∂T/∂r, gives rise to radially non-uniform visco elastic stresses. This stress gradient results in radially non-uniform molecular orientation and a strong radial variation in crystallinity for Nylon-6,6. The radially non-uniform stress profiles obtained from the simulations are in good agreement with experimental results for melt spinning of polystyrene. Simulations of Nylon-6,6 show that the thermally-induced crystallization depends strongly on the choice of the Avrami index n, and a sharp increase in crystallinity due to stress-induced crystallization is predicted only when the molecules are highly oriented in the drawing direction at high drawing speeds. The significant influences of visco elasticity, air drag, and operating conditions on non-isothermal melt spinning dynamics also are predicted.     

The Giesekus fluid in ω–D form for steady two-dimensional flows

Numerical results of the simulation of the Giesekus model in ω–D form, which has previously been introduced in Part I of this study, are presented. The model has been applied to the flow of a concentrated polymer solution through a planar 3.97:1 contraction. To obtain an accurate fit of the rheological properties of the fluid a four-mode model is used. The predictions of the numerical simulations are directly compared with the experimental results published by Quinzani et al. in 1994. For the velocity fields a good quantitative agreement is reached, especially in the upstream channel. Regarding the shear stress and first normal stress difference, qualitative predictions of the experimental profiles are obtained.