Branched Polymer Melts Research Articles

Numerical prediction of nonlinear rheology of branched polymer melts

In a recent short communication [Read, D. J. et al., Science 333, 1871 (2011)], we showed that a computational scheme can describe the nonlinear flow properties for a series of industrial low-density polyethylene (LDPE) resins starting from the molecular architecture. The molecular architecture itself is determined by fitting parameters of a reaction kinetics model to average structural information obtained from gel-permeation chromatography and light scattering. Flow responses of these molecules in transient uniaxial extension and shear are calculated by mapping the stretch and orientation dynamics of the segments within the molecules to effective pom-pom modes. In this paper, we provide the details of the computational scheme and present additional results on a LDPE and a high-density polyethylene resin to illustrate the dependence of segmental maximum stretch variables on the flow rate.

Nonisothermal analysis of extrusion film casting process using molecular constitutive equations

Extrusion film casting (EFC) is a commercially important process that is used to produce several thousand tons of polymer films and coatings. In a recent work, we demonstrated the influence of polymer chain architecture on the extent of necking in an isothermal film casting operation (Pol et al., J Rheol 57:559–583, 2013). In the present research, we have explored experimentally and theoretically the effects of long-chain branching on the extent of necking during nonisothermal film casting conditions. Polyethylenes of linear and long-chain branched architectures were used for experimental studies. The EFC process was analyzed using the 1-D flow model of Silagy et al. (Polym Eng Sci 36:2614–2625, 1996) in which the energy equation was introduced to model nonisothermal effects, and two multimode constitutive equations, namely the “extended pom-pom” (XPP, for long-chain branched polymer melts) equation and the “Rolie-Poly stretch version” (RP-S, for linear polymer melts) equation, were incorporated to account for the effects of polymer chain architecture. We show that the model does a better job of capturing the qualitative features of the experimental data, thereby elucidating the role of chain architecture and nonisothermal conditions on the extent of necking.

A SPH-based particle method for simulating 3D transient free surface flows of branched polymer melts

In this article we present a particle method based on smoothed particle hydrodynamics (SPH) for simulating three-dimensional (3D) transient free surface flows of branched polymer melts which are governed by the single equation version of the eXtended Pom-Pom (XPP) model. In order to remedy the so-called tensile instability which results in particle clustering and unphysical fracture in the state of fluid stretching, both artificial stress and artificial viscosity are incorporated into the momentum equation. For facilitating the implementation of the wall boundary condition in 3D space, an enhanced treatment of solid boundaries is proposed in this paper. The proposed SPH algorithm is validated by the excellent agreement between the numerical and the analytical solutions of Hagen-Poiseuille flow of an Oldroyd-B fluid. To demonstrate the ability of the numerical method in simulating free surface flows of branched polymer melts, the fall and impact on a rigid plate of a XPP fluid droplet is investigated for a wide range of the relevant parameters. Specifically, the convergence of the numerical results is discussed by three particle sizes of different levels of refinement. To provide some additional applications of 3D free surface flows of branched polymer melts, the challenging jet buckling and rod-climbing effect of viscoelastic fluids are further simulated and analyzed. All numerical results establish the capability of the SPH-based particle method to simulate 3D transient free surface flows of branched polymer melts in an entirely mesh-free framework.

Creep Measurements Confirm Steady Flow after Stress Maximum in Extension of Branched Polymer Melts

We provide conclusive evidence of nonmonotonic mechanical behavior in the extension of long-chain branched polymer melts. While nonmonotonic behavior is known to occur for solids, for the case of polymeric melts, this phenomenon is in direct contrast with current theoretical models. We rule out the possibility of the overshoot being an experimental artifact by confirming the existence of steady flow after a maximum in the ratio of stress to strain rate versus strain under both constant stress and constant strain-rate kinematics. This observation indicates the omission of important physics from current models for these industrially important materials, whose processing properties depend on extreme molecular extension.

Necking in extrusion film casting: The role of macromolecular architecture

Extrusion film casting (EFC) is used on an industrial scale to produce several thousand tons of polymer films and coatings. While significant research has been carried out on necking of films of viscoelastic melts in EFC, the influence of macromolecular chain architecture on the necking behavior is not yet fully understood. In the present research, we have explored experimentally and theoretically the effects of long chain branching and molecular weight distribution on the extent of necking during EFC. Polyethylenes of essentially linear architecture but having narrow and broad molecular weight distributions, and polyethylenes having long chain branching were used for experimental studies. The EFC process was analyzed using the one-dimensional flow model of Silagy et al. [Polym. Eng. Sci. 36(21), 2614–2625 (1996)] in which multimode molecular constitutive equations namely the “extended pom-pom” equation (for long chain branched polymer melts) and the “Rolie–Poly (Rouse linear entangled polymers)” equation (for linear polymer melts) were incorporated. We show that the model qualitatively captures the salient features of the experimental data thereby elucidating the role of chain architecture on the extent of necking.

Unusual Scaling in the Rheology of Branched Wormlike Micelles Formed by Cetyltrimethylammonium Bromide and Sodium Oleate

Natural fatty acids such as sodium oleate form highly viscous supramolecular complexes with long-chain cationic surfactants through cooperative self-assembly. Here we report the rheological behavior of linear and branched wormlike micelles formed in mixtures of cetyltrimethylammonium bromide (CTAB) and sodium oleate (NaOL). Addition of sodium oleate induces an increase in the axial ratio of the mixed micelles. At a constant mole fraction of the mixture, an increase in total surfactant concentration leads to a transition from linear to branched micelles. Both linear and branched micelles impart viscoelastic behavior to the fluid, and the low-frequency data can be approximated to the Maxwell model. Scaling of the rheological parameters of CTAB-NaOL catanionic mixtures, as a function of concentration, employing dynamic rheological measurements has been determined and compared with the predictions of existing scaling laws. The structural transition from linear micelles to the branched micelles in the CTAB-NaOL micellar system greatly influences the scaling behavior of shear modulus. The scaling exponent is lower for the branched micelles compared to linear micelles, analogous to those of linear and branched polymer melts. The structural evolution is probed by light scattering and small-angle neutron scattering measurements as well.

Cross-slot extensional rheometry and the steady-state extensional response of long chain branched polymer melts

Stress-optical measurements at a flow stagnation point in confined geometries such as the cross-slot provide an elegant way to perform extensional testing for polymer melts. This technique is especially useful for samples which have a steady-state that cannot be reached (easily) in standard elongational rheometry, for example, highly branched polymers which show a non-homogeneous deformation that occurs in stretching experiments for Hencky strains above 4. In contrast to filament stretching, the cross-slot provides one point at which steady-state extensional flow may be sustained indefinitely. In this study, a Cambridge multi-pass rheometer [Coventry, K. D., and M. R. Mackley, J. Rheol. 52, 401–415 (2008)] is used to generate planar elongational flow in a cross-slot geometry for different polyethylene melts. The experimental results are compared to finite element flow simulations using the multi-mode Pompom constitutive equations. The steady-state elongational viscosity at the stagnation point is computed...

Numerical Predictions of Viscoelastic Fluid in Pipe Flow Using Extended Pom-Pom Model

In this study prediction of the steady-state flow of branched polymer melts in pipe geometry with finite volume method is presented. The fluid is modelled using a modification of the Pom-Pom model known as the single eXtended Pom-Pom (XPP) where viscoelastic fluid, is typically a commercial low-density polyethylene. In finite volume method, the operator-integration is used to discretize the governing equations in space or control volume. An iterative solution algorithm that decouples the computation of momentum from that of stress is used to solve the discrete equations. Numerical results are presented; include the profiles of all relevant stresses, the axial velocity, stretch and the viscosity across the gap, demonstrating the performance of the model predictions. The influence of the model parameters on the solution and flow behaviour is described and, in particular, the dependence of velocity and stresses distribution as a function of Weissenberg number is analyzed. Finally, for rectitude and accuracy, the verification of this model is performed with the Gieskus model that shows good conformity between the results.

Analytical rheology of branched polymer melts: Identifying and resolving degenerate structures

Recently, a computational algorithm based on Bayesian data analysis was presented to invert the linear rheology of branched polymers [Shanbhag, S., Rheol. Acta 49, 411–422 (2010)]. When rheological data of an unknown polymer mixture are supplied, the algorithm produces an exhaustive distribution of structures and compositions, consistent with the rheology. Frequently, it identifies multiple or degenerate structures. In this proof-of-concept paper, a resolution of the degeneracy is sought by appealing to the concept of combinatorial rheology [Larson, R. G., Macromolecules 34, 4556–4571 (2001)], where the unknown sample is strategically blended with a well-characterized fraction. Experimental load is alleviated by identifying the optimal type, molecular weight, and composition of the polymer fraction to blend with the unknown sample, to discriminate between the degenerate structures, most conclusively. Two methods are proposed and tested on a particular mixture that is characterized by severe degeneracy. Both proposed methods are able to correctly identify the structure and composition of the original mixture by requiring two additional rheological experiments.

Effect of Elasticity Parameter on Viscoelastic Fluid in Pipe Flow Using Extended Pom-Pom Model

In this study prediction of the steady-state flow of branched polymer melts in pipe geometry with finite volume method is presented. Our analysis in this study revealed that; for normal-stress τθθ , the XPP model can predict this tensor unlike the other viscoelastic models such as PTT or Gieskus which can not predict τθθ for viscoelastic fluid in two dimensional pipe flows. The fluid is modelled using a modification of the Pom-Pom model known as the single eXtended Pom-Pom (XPP) where viscoelastic fluid is typically a commercial low-density polyethylene. In finite volume method, the operator-integration is used to discretize the governing equations in space or control volume. An iterative solution algorithm that decouples the computation of momentum from that of stress is used to solve the discrete equations. Numerical results are presented, including the profiles of all relevant stresses, the axial velocity, stretch and the viscosity across the gap, demonstrating the performance of the model predictions. The influence of elasticity parameter on flow behaviour is studied, which demonstrates in particular, the dependence of velocity and stresses distribution as a function of Weissenberg number is analyzed. Also, the effect of Weissenberg number on pressure gradient has been considered. Finally, verification of the present model was made by comparing to the Generalized Newtonian Fluid (GNF) model.

A numerical study of constitutive models endowed with Pom-Pom molecular attributes

The branched polymer melts are modeled respectively in this investigation by the existing XPP and PTT–XPP models, along with the proposed S-MDCPP (Single/Simplified Modified Double Convected Pom-Pom) model developed on the basis of the existing MDCPP model. A pressure stabilized mass equation is formulated with the finite increment calculus (FIC) process to restrain and further eliminate spurious oscillations of pressure field due to the incompressibility of fluids. The discrete elastic viscous stress splitting (DEVSS) technique is employed, in order to retain an elliptic contribution in the weak form of the momentum equation. An inconsistent streamline-upwind (SU) method is applied to spatially discretize the constitutive equations. The mass, momentum conservation and constitutive equations are discretized and solved by the iterative stabilized fractional step algorithm along with the Crank–Nicolson implicit difference scheme. Thus the finite elements with equal low-order interpolation approximations for velocity–pressure–stress variables can be devised to numerically simulate the viscoelastic contraction flows for branched LDPE melts. The influences of the three viscoelastic constitutive models and the branched arms at the end of the Pom-Pom molecule on the rheological behaviors occurring in this complex flow are discussed. The numerical results demonstrate that the proposed S-MDCPP model is capable of reproducing some properties similar to those predicted by the XPP model in high shear flow and, on the other hand, reproducing some properties similar to those predicted by the PTT–XPP model in high elongational flow. Furthermore, the proposed S-MDCPP model is capable of well identifying the macromolecule topological structures of branched polymer melts.

Comparing tube models for predicting the linear rheology of branched polymer melts

The hierarchical and “bob” (or branch-on-branch) models are tube-based computational models recently developed for predicting the linear rheology of general mixtures of polydisperse branched polymers. These two models are based on a similar tube-theory framework but differ in their numerical implementation and details of relaxation mechanisms. We present a detailed overview of the similarities and differences of these models and examine the effects of these differences on the predictions of the linear viscoelastic properties of a set of representative branched polymer samples in order to give a general picture of the performance of these models. Our analysis confirms that the hierarchical and bob models quantitatively predict the linear rheology of a wide range of branched polymer melts but also indicate that there is still no unique solution to cover all types of branched polymers without case-by-case adjustment of parameters such as the dilution exponent α and the factor p2 which defines the hopping distance of a branch point relative to the tube diameter. An updated version of the hierarchical model, which shows improved computational efficiency and refined relaxation mechanisms, is introduced and used in these analyses.

Numerical study of the rheology of rigid fillers suspended in long-chain branched polymer under planar extensional flow

We report a detailed numerical study of the rheology of two-dimensional rigid fillers suspended in branched polymer melt under planar extensional flow. The polymer melt is modelled using the pom-pom constitutive equation. The numerical method uses a finite element solution of the flow in a unit cell within the self-replicating lattice for planar extensional flow identified by Kraynik and Reinelt [Int. J. Multiphase flow 18 (1992) 1045]. This is implemented using a quotient space representation that maps all points in space onto points within the unit cell. We show that the Kraynik and Reinelt cell allows simulations of suspensions under planar extensional flow to be conducted to large strains in a truly periodic cell. The influence of both the pom-pom parameters and the particle area fraction on the rheology of the suspension are investigated. We find a reduction in the degree of extension-rate thickening with the increase of both particles concentration and Weissenberg numbers in agreement with published experimental and numerical findings on other viscoelastic models. This reduction is found to be due to flow disturbance created by the particles which disrupts the alignment of backbone tube segments with the extensional axis.

Numerical prediction of extrudate swell of branched polymer melts

This paper is concerned with the numerical prediction of the extrudate swell behaviour of branched polymer melts in a planar configuration. The multi-mode extended pom-pom (XPP) model is used to describe the polymer dynamics. A second-order operator-integration-factor splitting scheme is used for the temporal discretisation of the problem, whilst a spectral element scheme is used in space. The free surface is evolved in a Lagrangian manner using the third-order conditionally stable Adams–Bashforth method. A thorough mesh convergence study is performed with respect to the temporal and spatial discretisation parameters. The influence of the nature of the discrete relaxation spectrum on the swelling ratio and as an indicator of polydispersity is investigated. The predictions of numerical simulations are also compared with a selection of experimental results from the literature. The parameters in the XPP model are determined from rheological data. Good agreement is obtained for branched low-density polyethylenes. The ability to model a melt with a high molecular weight tail using a discrete relaxation spectrum for which the largest relaxation time is isolated from the others is also investigated.

Equal Low-Order Finite Element Simulation of the Planar Contraction Flow for Branched Polymer Melts

The rheological characteristics of branched polymer melts is described by the modified XPP model, which is discretized by inconsistent streamline-upwind method. A finite increment calculus procedure is introduced to reformulate the mass equation and to overcome oscillations of the pressure field. Moreover, the governing equations are discretized and solved by the iterative fractional step algorithm. Thus the equal low-order finite elements for velocity-pressure-stress variables are adopted to calculate the planar contraction viscoelastic flows. The influences of the Weissenberg number and the amount of branched-arms on the rheological behavior of the Pom-Pom molecule are discussed. Results demonstrate good agreement with those given in the literatures.

Non-linear step strain of branched polymer melts

Long-chain branched polymer melts such as low density polyethylene (LDPE) and branched metallocene polyethylenes show strong time-strain separability in step strain. Constitutive models of the multi-mode Pom-pom form are highly successful in modeling the stress generated by general flow histories for these materials. However, a single Pom-pom mode is not time-strain separable and reconciling this to the step-strain phenomenon has been a challenge. We investigate multi-mode integral Pom-pom models and a differential approximation to compare time-strain separation, with respect to mode density. Here we show that for a wide class of branched distributions, a family of damping functions can be derived with a response that is very close to separable. We evaluate the family for both LDPE and branched high density polyethylene melts and show that a damping function derived from the multi-mode Pom-pom model gives an accurate prediction of the damping behavior in step-strain experiments.

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.

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.

Viscoelasticity of Monodisperse Comb Polymer Melts

The experimentally determined zero-shear viscosity of entangled branched polymers shows dramatic variation due to the topological arrangements of the branches in branched polymer melts. The position of the branch points, the arm length, and number of the arms are essential to defining the rheological behavior. Recent advances in molecular tube models have led to a much greater understanding of the linear rheology of linear, star, H-shaped, pom-pom, and comb polymers. We correct and extend existing molecular theories for the linear viscoelasticity of comb polymer melts, especially in accounting for (1) polydispersity and (2) the path length of backbone extremities. We compare the predictions with linear rheological data of nearly monodisperse polybutadiene combs. We then predict the zero-shear viscosity for monodisperse comb polyethylenes with varying arm lengths, backbone lengths, and number of arms. For a fixed molecular weight, we find that combs with the longest arms but few branch points give the highest predicted zero-shear viscosities and that they obey an exponential dependence on the length of the arms in the same way as star polymers. We find that combs with short arms, under four entanglements, lie below the 3.4 power law obeyed by linear polymers. All other comb topologies are bounded by these extremes.

The Rheology of Linear and Long‐chain Branched Polymer Melts

AbstractBy generalising the Doi‐Edwards tube model to the Molecular Stress Function theory, the non‐linear rheology of polymer melts can be described quantitatively. The strain‐hardening of linear polymer melts in extensional flows can be accounted for by a strain energy function, which reflects the increase of strain energy due to tube squeeze. In comparison to linear polymer melts, long‐chain branched polymer melts show enhanced strain‐hardening. This is due to the fact that while the backbone of the branched macromolecule is stretched by deformation, side chains are compressed. It is demonstrated that the experimentally observed slope of the elongational viscosity after inception of strain‐hardening depends on the ratio β of total molar mass to backbone molar mass as predicted by the model. The steady‐state (plateau) value of the elongational viscosity depends on the maximum relative stretch, $f_{MAX}^2$, which can be supported by chain segments and which represents the maximum elastic energy storable in the polymeric system.