Pom-pom Model Research Articles

Numerical simulation of flow past a cylinder using models of XPP type

We present stable and accurate spectral element methods for predicting the steady-state flow of branched polymer melts past a confined cylinder. The fluid is modelled using a modification of the pom-pom model known as the single eXtended Pom-Pom (XPP) model, where we have included a multi-mode model of a commercial low-density polyethylene. We have analyzed the XPP model and found interesting multiple solutions for certain choices of the parameters which indicate possible problems with the model. The operator-integration-factor-splitting technique is used to discretize the governing equations in time, while the spectral element method is used in space. An iterative solution algorithm that decouples the computation of velocity and pressure from that of stress is used to solve the discrete equations. Appropriate preconditioners are developed for the efficient solution of these problems. Local upwinding factors are used to stabilize the computations. Numerical results are presented demonstrating the performance of the algorithm and the predictions of the model. The influence of the model parameters on the solution is described and, in particular, the dependence of the drag on the cylinder as function of the Weissenberg number.

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 W i = 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.

Non-Isothermal Simulation of the Film Blowing Process Using Multi-Mode Extended Pom-Pom Model

Abstract Plastic films are largely produced using the film blowing process. Branched polymers such as LDPE are commonly used in this process to produce a variety of plastic films. In this study, recently developed eXtended Pom-Pom (XPP) model, which focuses on describing rheological behavior of branched polymers, is employed to analyze the non-isothermal flow in the film blowing process. Furthermore, the Nakamura equation for crystallization kinetics is employed to consider crystallization effects and improve prediction of the state of the stress in the film. Results of the numerical analysis show good agreement with the experimental values reported by Tas (1994) for the bubble shape and film velocity of the LDPE films. Predictions for the strain rates in machine and hoop directions which are critical for the estimation of the stress are also compared with the reported data and showed reasonably good agreement.

The effect of viscoelasticity on stress fields within polyethylene melt flow for a cross-slot and contraction–expansion slit geometry

The sensitivity of the principal stress difference (PSD) profiles to material viscoelasticity is demonstrated for two flow geometries using three different polyethylenes. Studies were performed using both experimental optical techniques and computational simulations, in the latter case to evaluate the ability to model these complex flows. The materials were characterised using linear and extensional rheology which was fitted to a multimode POM-POM model implemented in the Lagrangian–Eulerian code flowSolve. A contraction–expansion (CE) slit geometry was used to create a mixed, but primarily simple shear flow, whilst a cross-slot geometry provided a region of high extensional shear and high strain. In both flows, the PSD developed from an initial Newtonian profile to increasing levels of asymmetry between the inlet and the outlet flow. More specific phenomena, such as downstream stress fangs in the CE slit and the formation of centreline cusps and “W”-shaped cusps in the cross-slot, were also observed. The simulations of PSD development within the CE slit geometry quantitatively captured the experimental results. In the case of the cross-slot geometry, the qualitative features of the PSD development were well captured, although the results were quantitatively less accurate.

Pom-pom model predictions on nonlinear stress relaxation in single-step strain flow

A phenomenological pom-pom model, which utilizes linear stress relaxation data as an input, is proposed for simulating the detailed retraction behavior of a full pom-pom chain as well as the associated nonlinear stress relaxation in single-step shear flows. Specifically, the initial, Rouse-like arm and/or backbone retraction as well as the long-time, renormalized (dumbbell-like) backbone retraction is simulated at one time, and a possible coupling between linear orientation relaxation and nonlinear stretch relaxation due to diffusive or convective constraint release is self-consistently accounted for. The model’s predictions are systematically tested against nonlinear stress relaxation data on three nearly monodisperse pom-pom melts, including two six-arm pom-pom polybutadienes (PPBDs) and an H-shaped polyisoprene (PPI). The model can describe the present data reasonably well by incorporating a previously proposed effect of drag–strain coupling. Accounting further for the effect of constraint release, due primarily to arm diffusive motion and its coupling with a known polydispersity in arm molecular weight distribution, appreciably broadens the predicted nonlinear stress relaxation and substantially improves the results of theory/data comparisons. The significance of the current findings is discussed.

Fourier-transform rheology under medium amplitude oscillatory shear for linear and branched polymer melts

Nonlinear response of linear and branched polymers has been investigated under medium strain amplitude oscillatory shear (strain amplitude range from 10% to 100%) with Fourier-transform rheology. A power law relationship was found between the relative third intensity (I3∕I1), which is an indicator of nonlinearity, and the strain amplitude at low and medium strain amplitudes. On a log-log plot, the intercept and slope of I3∕I1 were investigated at different excitation frequencies and temperatures. Simulation results with three different constitutive equations [Giesekus, exponential Phan-Thien Tanner (E-PTT), pom-pom model] were also compared. Experimental results show that the intercept was affected by the excitation frequency and temperature, and the slope of I3∕I1 for linear polymer remained constant regardless of molecular weight, molecular weight distribution, and excitation frequency in accordance with the predictions of the constitutive equations (Giesekus and E-PTT). It should be noted that the slope of I3∕I1 for branched polymer was lower than that of linear polymer, unlike the prediction of the pom-pom model. Among the molecular architecture and processing parameters (e.g., molecular weight, molecular weight distribution, frequency, and temperature), the slope of I3∕I1 under medium amplitude oscillatory shear was found to depend only on the long chain branching, which means that it can be used as a measure of the degree of branching. The failure of the pom-pom model in predicting the nonlinear shear behavior was also pointed out.

Unphysical phenomena associated with the extended pom-pom model in steady flow

The extended pom-pom (XPP) model was first introduced to eradicate perceived deficiencies of the original pom-pom model such as the lack of a second normal stress difference and a discontinuity in the derivative of the extensional viscosity. However, in this paper it is shown that the XPP model itself possesses some disconcerting attributes. In simple steady shear and uniaxial extensional flow, multiple solutions are found. Furthermore, the extended pom-pom model can predict positive as well as negative values for the second normal stress difference. Experimentally, the second normal stress difference is negative for polymer melts.

Sensitivity analysis of low-speed melt spinning of isotactic polypropylene

Linearized sensitivity analysis of fiber spinning has been studied using a two-phase constitutive model that includes the effects of crystallization. The analysis focuses on sensitivity predictions for the low-speed melt spinning of isotactic polypropylene and comparison to the experimental results of Young and Denn [1989, Disturbance propagation in melt spinning. Chemical Engineering Science 44, 1807–1818]. Modifications in an earlier-developed two-phase model enable comparisons of two different constitutive equations for the melt phase, namely the Giesekus and extended pom-pom models. Comparisons with sensitivity data for low-speed spinning conditions demonstrate that the incorporation of crystallization effects leads to improved predictions of the magnitude and trends of the perturbation frequency dependence for both constitutive equations. At low spin speeds where flow-enhanced crystallization effects are negligible, the sensitivity is predicted to decrease with increasing cooling and this trend is also shown to be consistent with increased crystallinity.

Thermodynamic admissibility of the extended Pom-Pom model for branched polymers

The thermodynamic consistency of the eXtended Pom-Pom (XPP) model for branched polymers of Verbeeten et al. [W.M.H. Verbeeten, G.W.M. Peters, F.P.T. Baaijens, Differential constitutive equations for polymer melts: the extended pom-pom model, J. Rheol. 45 (4) (2001) 823–843; W.M.H. Verbeeten, G.W.M. Peters, F.P.T. Baaijens, Differential constitutive equations for polymer melts: the extended pom-pom model (vol 45, pg 823–843, 2001), J. Rheol. 45 (6) (2001) 1489] as well as its modified version [J. van Meerveld, Note on the thermodynamic consistency of the integral pom-pom model, J. Non-Newtonian Fluid Mech. 108 (1–3) (2002) 291–299] is investigated from the perspective of non-equilibrium thermodynamics, namely the General Equation for Non-Equilibrium Reversible–Irreversible Coupling (GENERIC) framework. The thermodynamic admissibility of the XPP model is shown for both its original and modified form. According to the GENERIC formalism, the parameter α introduced by Verbeeten et al. to predict non-zero second normal stress in shear flows must fulfill the condition 0 ≤ α ≤ 1.

Flexible ring polymers in an obstacle environment: Molecular theory of linear viscoelasticity

We formulate a coarse-grained mean-field approach to study the dynamics of the flexible ring polymer in any given obstacle (gel or melt) environment. The similarity of the static structure of the ring polymer with that of the ideal randomly branched polymer is exploited in formulating the dynamical model using aspects of the pom-pom model for branched polymers. The topological constraints are handled via the tube model framework. Based on our formulation we obtain expressions for diffusion coefficient D, relaxation times tau, and dynamic structure factor g(k,t). Further, based on the framework we develop a molecular theory of linear viscoelasticity for ring polymers in a given obstacle environment and derive the expression for the relaxation modulus G(t). The predictions of the theoretical model are in agreement with previously proposed scaling arguments and in qualitative agreement with the available experimental results for the melt of rings.

Channel, tube, and Taylor–Couette flow of complex viscoelastic fluid models

We show how to formulate two-point boundary value problems to compute laminar channel, tube, and Taylor–Couette flow profiles for some complex viscoelastic fluid models of differential type. The models examined herein are the Pom-Pom Model [McLeish and Larson 42:81–110, (1998)] the Pompon Model [Ottinger 40:317–321, (2001)] and the Two Coupled Maxwell Modes Model (Beris and Edwards 1994). For the two-mode Upper-Convected Maxwell Model, we calculate analytical solutions for the three flow geometries and use the solutions to validate the numerical methodology. We illustrate how to calculate the velocity, pressure, conformation tensor, backbone orientation tensor, backbone stretch, and extra stress profiles for various models. For the Pom-Pom Model, we find that the two-point boundary value problem is numerically unstable, which is due to the aphysical non-monotonic shear stress vs shear rate prediction of the model. For the other two models, we compute laminar flow profiles over a wide range of pressure drops and inner cylinder velocities. The volumetric flow rate and the nonlinear viscoelastic material properties on the boundaries of the flow geometries are determined as functions of the applied pressure drop, allowing easy analysis of experimentally measurable quantities.

Extensional response of the pom-pom model through planar contraction flows for branched polymer melts

One objective of this study is to assess the influence of the number of dangling arms at each end of the pom-pom molecule on flow characteristics in a 4:1 planar rounded-corner contraction, where varying the number of arms affects the level of entanglement in the system. For these complex flows, we evaluate the major influence of extensional viscosity and Trouton ratio when comparing kinetic-based as opposed to phenomenological network-based models. The stability of the proposed extended pom-pom model (XPP) is investigated for a range of Weissenberg numbers utilising a hybrid finite-element/volume scheme. For this scheme, we solve the momentum-continuity equations by a fractional-staged Taylor–Galerkin/pressure-correction procedure and invoke a cell-vertex fluctuation distribution finite volume scheme for the constitutive law. Distinction may be drawn between fluid response in the extension-hardening regime and those displaying a degree of softening. In particular, trends in salient-corner vortex-intensity response can be associated with the extensional properties of the extended version of the pom-pom model. Critical levels of solution elasticity attainable depend on precise extensional viscosity characteristics. The influence of various model parameters upon field response is described through rates of deformation, stress and stretch. This would include consideration for parameters governing anisotropy ( α ), and the ratio of stretch to orientation relaxation times ( ɛ ).

Fourier Transform Rheology of Branched Polyethylene: Experiments and Models for Assessing the Macromolecular Architecture

Fourier transform rheology (FTR) consists of analyzing the frequency spectrum of the nonlinear torque response obtained during large-amplitude oscillatory shear. In the present work, the analysis has been carried out in the whole complex space and was especially focused on the study of the third complex harmonic I*3/I1 (= I*(3ω1)/I1, where I1 is the spectrum intensity at the fundamental frequency ω1), the intensity I3/I1 (= |I*3/I1|), and the phase shift Φ3 of I*3/I1. The accuracy of FTR to characterize the architecture of linear, sparsely, and densely branched polyethylene regarding the number of branches and their length has been shown. A modified Wagner integral model and a differential multimode pompom model were used to simulate FTR experiments. Nonlinear parameters of each model were determined by fitting the curve of the third complex harmonic I*3/I1 versus the strain amplitude γ0 with the experimental data. Good results have been obtained with the modified Wagner model. Whereas, the pompom model accounts for the experimental intensity of the third harmonic, discrepancies remain for the phase shift behavior Φ3 for the polymers showing the highest nonlinear behavior regarding the level of I3/I1. Nevertheless, when fitting was possible, the results have been correlated with the supposed structure expected from the type of synthesis.

Modelling pom-pom type models with high-order finite volume schemes

We are concerned with the numerical solution of viscoelastic flows using two contrasting high-order finite volume schemes. We take our earlier work for transient start-up flow in a channel and extend this beyond Oldroyd-B modelling to consider a different fluid model of the pom-pom class. This includes Single Extended form of the pom-pom model (SXPP), comparing the results of two different finite volume schemes. The numerical techniques employed are time-stepping algorithms, one of hybrid finite element/volume type, the other of pure finite volume form. The pure finite volume scheme is a staggered-grid cell-centred scheme based on area-weighting and a semi-Lagrangian formulation. This may be implemented on structured or unstructured rectangular grids, utilising backtracking along the solution characteristics in time. For the hybrid scheme, we solve the momentum/continuity equations by a fractional-staged Taylor–Galerkin/pressure-correction procedure and invoke a cell-vertex finite volume scheme for the constitutive law. This draws upon fluctuation distribution schemes (upwinding), different combinations of ‘flux’ and ‘median-dual-cell’ spatial discretisations and time-term treatments. Here, unstructured and structured meshes may be used, based largely on triangular grids. A comparison of the two finite volume approaches will be presented, concentrating upon the new features posed by the pom-pom class of models.

Numerical Analysis of Elongation Behavior of Melt Flow in Contraction Channels with Cylindrical Barriers

The flow of polymer melt in contraction channels with cylindrical barriers was numerically simulated. A commercial LDPE melt was considered, and the rheological property of the melt was represented using the Extended Pom-Pom model (XPP). The simulation was performed with various barrier configurations, and the effects of the barriers were investigated extensively in relation to the efforts to clarify the phenomenon of melt fracture mitigation by the addition of filter or porous medium. It can be observed that the addition of barriers modifies significantly the flow fields in the region near the contraction. Investigation of the elongation stress shows that the addition of barriers increases the stress in the region near the contraction, but decreases the elongation rate and elongation stress in the region at the upstream of the contraction and induces the undershoots of elongation rate and elongation stress in the region at the downstream of the contraction. With the increase in Weissenberg numbers, these effects are more significant for the channels with two barriers as compared to that with single barrier. The reduction of elongation rate and elongation stress should play an important role in the mitigation of the upstream instability, which in turn mitigates the gross melt fracture.

Modelling of flow-enhanced crystallisation in fibre spinning

A modified version of an earlier two-phase model and algorithm developed for melt spinning of semi-crystalline polymers is discussed. Modifications include coupling of the flow-enhancement factor in the Avrami crystallisation kinetics based on the melt phase free energy, and introduction of both phases at the start of the integration algorithm to produce a 'seamless' transition. Introduction of the extended Pom-Pom model for the melt phase is used to incorporate chain entanglement effects These modifications provide a framework for modelling the melt spinning of PLA fibres. Excellent fits of literature data for the low and high speed spinning of PLA and racemic mixtures with single sets of parameters are shown. In the latter case, prediction of the co-crystal form is also demonstrated.

The prediction of complex flows of polymer melts using spectral elements

The paper is concerned with the numerical prediction of the flow of polymer melts using a pom-pom model [J. Rheol. 42 (1998) 81]. The pom-pom model is a coarse-grained molecular model that was developed for describing branched polymers. A description of the configuration distribution is given in terms of the orientation and stretch of the pom-pom molecule. The variant of the pom-pom model used in this paper is the extended pom-pom model [J. Rheol. 45 (4) (2001) 823]. The model has been rewritten to make it suitable for implementation in an existing spectral element algorithm, which solves the constitutive equation together with the momentum and continuity equations The temporal discretization of the convective terms in the momentum equation and the constitutive equation is performed using a second-order time-splitting technique. Spatial discretization is performed using spectral element techniques. The system of discretized equations is solved using a preconditioned conjugate gradient method, and details of the preconditioners are given. Results are presented for the start-up of Poiseuille flow in a planar channel, and flow past a confined cylinder. The influence of the model parameters on the solution is described.

Towards a rheological classification of flow induced crystallization experiments of polymer melts

Departing from molecular based rheology and rubber theory, four different flow regimes are identified associated to (1) the equilibrium configuration of the chains, (2) orientation of the contour path, (3) stretching of the contour path, and (4) rotational isomerization and a deviation from the Gaussian configuration of the polymer chain under strong stretching conditions. The influence of the ordering of the polymer chains on the enhanced point nucleation, from which spherulites grow, and on fibrous nucleation, from which the shish-kebab structure develops, is discussed in terms of kinetic and thermodynamic processes. The transitions between the different flow regimes, and the associated physical processes governing the flow induced crystallization process, are defined by Deborah numbers based on the reptation and stretching time of the chain, respectively, as well as a critical chain stretch. An evaluation of flow induced crystallization experiments reported in the literature performed in shear, uniaxial and planar elongational flows quantitatively illustrates that the transition from an enhanced nucleation rate of spherulites towards the development of the shish-kebab structure correlates with the transition from the orientation of the chain segments to the rotational isomerization of the high molecular weight chains in the melt. For one particular case this correlation is quantified by coupling the wide angle X-ray diffraction and birefringence measurements of the crystallization process to numerical simulations of the chain stretch of the high molecular weight chains using the extended Pom-Pom model in a cross-slot flow.

Hydrodynamic stability of unidirectional shear flow of linear and branched polymeric melts

In this study, the linear stability of homogeneous shear flow of polymer melts has been investigated by using a prototypical reptation-based model, namely, the Pompom model [1]. The hydrodynamic stability characteristics of the flow have been determined by using an eigenvalue analysis. Particular attention has been paid to accurate determination of the discrete and continuous eigenvalues as well as identification and isolation of the spurious eigenvalues. Specifically, it has been shown that the eigenspectrum for the Pompom model has four continuous spectra, three of which are associated with the orientation tensor, S, i.e., one regular and two branch-cuts. One spectrum arises as a result of the stretch, λ evolution equation, which always occurs as the leftmost spectrum. The discrete modes are classified as centered and non-centered eigenvalues, depending on their imaginary parts. It is shown that there is only one pair of non-centered eigenvalues (a non-centered eigenvalue possesses σ i ≠ − α/2, where σ i denotes the imaginary part of the eigenvalue and α denotes the wavenumber in the streamwise direction). For large deformations rates, the real parts of these eigenvalues decay in a similar fashion as the Gorodstov–Leonov eigenvalue pair [2], observed in planar flows of the Upper Convected Maxwell model (UCM). The number of centered eigenvalues however, depends strongly on the choice of parameters. We observe that the leading eigenvalues always belong to the right-most continuous mode (branch-cut), which possess highly singular eigenfunctions. Although, the flow is linearly stable when a maximum in the shear stress versus shear rate curve is not observed, the ballooning of the right most eigenspectrum that arises due to the singular nature of its eigenfunctions could lead to erroneous determination of onset conditions for the instability.

On the evaluation of some differential formulations for the pom-pom constitutive model

In recent years, progress has been made in modelling long chain branched polymers by the introduction of the so-called pom-pom model. Initially developed by McLeish and Larson (1998), the model has undergone several improvements or alterations, leading to the development of new formulations. Some of these formulations however suffer from certain mathematical defects. The purpose of the present paper is to review some of the formulations of the pom-pom constitutive model, and to investigate their possible mathematical defects. Next, an alternative formulation is proposed, which does not appear to exhibit mathematical defects, and we explore its modelling performance by comparing the predictions with experiments in non-trivial rheometric flows of an LDPE melt. The selected rheometric flows are the double step strain, as well as the large amplitude oscillatory shear experiments. For LAOS experiments, the comparison involves the use of Fourier-transform analysis.