Rheo-PIV analysis of the slip flow of a metallocene linear low-density polyethylene melt

The continuous extrusion of a metallocene linear low-density polyethylene through a transparent capillary die with and without slip was analyzed in this work by rheometrical measurements and particle image velocimetry (PIV). For this reason, a comparison was made between the rheological behaviors of the pure polymer and blended with a small amount of fluoropolymer polymer processing additive. Very good agreement was found between rheometrical and PIV measurements. The pure polymer exhibited stick-slip instabilities with nonhomogeneous slip at the die wall, whereas the blend showed stable flow. The slip velocity was measured directly from the velocity profiles and was negligible for the pure polymer before the stick-slip but increased monotonously as a function of the shear stress for the blend. The flow curves and the slip velocity as a function of the shear stress deviated from a power law and were well fitted by continuous “kink” functions. Comparison of PIV data with rheometrical ones permitted a direct proof of the basic assumption of the Mooney theory. Finally, the analysis of the velocity profiles showed that there is a maximum in the contribution of slip to the average fluid velocity, which is interpreted as the impossibility for the velocity profile to become plug like in the presence of shear thinning.     

Enhanced melt strength and stretching of linear low-density polyethylene extruded under strong slip conditions

The influence of extrusion under strong slip conditions on the extensional properties of linear low-density polyethylene was studied in this work. The material was extruded at two different temperatures under strong slip and no slip conditions, and was subsequently subjected to uniaxial elongational flow by means of a Rheotens device. Strong slip was evident through the elimination of sharkskin distortions and the stick-slip instability, as well as by the electrification of the extrudates. The extrudate swell was smaller in the presence of slip when comparing with no slip conditions at constant apparent shear rate, but it was found to be a unique function of the shear stress if comparison was performed at constant stress. The draw ratio and melt strength of the filaments obtained under slip conditions were larger compared to those without slip. In addition, draw resonance was postponed to higher draw ratios during the extrusion with strong slip at constant apparent shear rate. It is suggested that slip of the polymer at the die wall decreases the shear stress in the bulk, and therefore, restricts the disentanglement and orientation of macromolecules during flow, which subsequently produces the increase in draw ratio and melt strength during stretching.     

Comparison of viscous and elastic properties of polyolefin melts in shear and elongation

Viscous and elastic properties of a linear polypropylene (PP) and a long-chain branched low-density polyethylene (LDPE) have been investigated by creep and creep–recovery experiments in shear and elongation. The data obtained verify the ratios between the linear values of the viscosities and the steady-state elastic compliances in shear and elongation predicted by the theory of linear viscoelasticity. In the nonlinear range, no simple correlation between the viscous behaviour in shear and elongation exists. The elongational viscosity of the PP decreases with increasing stress analogously to the shear thinning observed; the linear range extends to higher stresses in elongation than in shear, however. The LDPE shows thinning in shear and strain hardening in elongational flow. For the LDPE, a linear steady-state elastic tensile compliance corresponding to one third of the linear steady-state elastic compliance in shear was determined. For the PP, this theoretically predicted value is approximately reached. Analogous to the viscous behaviour, the linear range extends to higher stresses in elongation than in shear. For both materials, the steady-state elastic compliances in the nonlinear range decrease with increasing stress in shear as well as in elongation. However, the decrease in elongation is more pronounced.     

Bending and buckling of a class of nonlinear fiber composite rods

An investigation of the mechanics of bending and buckling is carried out for a class of nonlinear fiber composite rods composed of embedded unidirectional fibers parallel to the rod axis. The specific class of composite considered is one in which the fibers interact with the matrix through a nonlinear Needleman-type cohesive zone [Needleman, A., 1987. A continuum model for void nucleation by inclusion debonding. ASME J. Appl. Mech. 54, 525-531; Needleman, A., 1992. Micromechanical modelling of interfacial decohesion. Ultramicroscopy 40, 203-214]. The primary decohesive mechanism active in bending and buckling of these composite rods is shear slip along the fiber-matrix interfaces allowing the use of a previously developed constitutive relation for antiplane shear response [Levy, A.J., 2000b. The fiber composite with nonlinear interface—part II: antiplane shear. ASME J. Appl. Mech. 67, 733-739]. The formulation requires the specification of a potential interface force-slip law that is assumed to permit interface failure in shear.Four cases of the bending and shearing of beams (concentrated or uniform load on a cantilever or a simply supported beam) are analyzed, each of which exhibits qualitatively distinct response. For certain values of interface parameters, the beam deflection or its gradient at a fixed location can change discontinuously with load. Furthermore, for interface parameter values within a certain range, singular surfaces will exist in uniformly loaded beams where there is a non-uniform distribution of shear stress along the beam length. These singular surfaces divide the beam into regions of maximal and minimal fiber slip and propagate with a rate that varies inversely as the square of the applied load. For other parameter values, singular surfaces will not exist and fiber slip will be diffuse.For the class of nonlinear composite considered, bifurcation and imperfection buckling of pinned-pinned columns is analyzed. For bifurcation buckling, a nonlinear eigenvalue problem is derived and the solution is obtained by Galerkin's method. It is demonstrated that critical loads are influenced by the initial slope, and hence the linear portion, of the interface force-slip relation but the post-buckling response, which in some sense resembles that of plastic buckling, is affected by the entire interface constitutive relation. Imperfection buckling is analyzed in a similar manner by assuming a slight initial curvature of the rod. Sensitivity of the response to imperfection magnitude is discussed as well.     

Comparison of the melt fracture behavior of metallocene and conventional polyethylenes

The influence of sparse long-chain branching and molecular weight distribution on the melt fracture behavior of polyethylene melts was investigated. Four commercial polyethylene resins were employed for this study: a conventional low-density polyethylene, a conventional linear low-density polyethylene, a linear metallocene polyethylene, and a sparsely branched metallocene polyethylene. Rheological measurements were obtained for both shear and extensional deformations, and melt fracture experiments were carried out using a controlled rate capillary rheometer. A single capillary geometry was used to focus on the effects of material properties rather than geometric factors. For the linear polyethylenes, surface melt fracture, slip-stick fracture, and gross melt fracture were all observed. Conversely, the branched PE resins did not exhibit a slip-stick regime and the degree of gross fracture was observed to be much more severe than the linear resins. These variations can be explained by the effects that long-chain branching has on the onset of shear-thinning behavior (slip-stick fracture) and the degree of extensional strain hardening (gross melt fracture). Although there is some indication that the breadth of molecular weight distribution indirectly influences surface melt fracture, the results remain inconclusive.     

Upshot of ohmically dissipated Darcy-Forchheimer slip flow of magnetohydrodynamic Sutterby fluid over radiating linearly stretched surface in view of Cash and Carp method

The present work concerns the momentum and heat transmission of the electro-magnetohydrodynamic(E-MHD) boundary layer Darcy-Forchheimer flow of a Sutterby fluid over a linear stretching sheet with slip. The nonlinear equations for the proposed model are analyzed numerically. Suitable techniques are used to transform the coupled nonlinear partial differential equations(PDEs) conforming to the forced balance law, energy, and concentration equations into a nonlinear coupled system of ordinary differential equations(ODEs). Numerical solutions of the transformed nonlinear system are obtained using a shooting method, improved by the Cash and Carp coefficients. The influence of important physical variables on the velocity, the temperature, the heat flux coefficient, and the skin-friction coefficient is verified and analyzed through graphs and tables. From the comprehensive analysis of the present work, it is concluded that by intensifying the magnitude of the Hartmann number, the momentum distribution decays,whereas the thermal profile of fluid increases. Furthermore, it is also shown that by augmenting the values of the momentum slip parameter, the velocity profile diminishes. It is found that the Sutterby fluid model shows shear thickening and shear thinning behaviors.The momentum profile shows that the magnitude of velocity for the shear thickening case is dominant as compared with the shear thinning case. It is also demonstrated that the Sutterby fluid model reduces to a Newtonian model by fixing the fluid parameter to zero.In view of the limiting case, it is established that the surface drag in the case of the Sutterby model shows a trifling pattern as compared with the classical case.     

Prediction of steady-state viscous and elastic properties of polyolefin melts in shear and elongation

The linear and nonlinear steady-state viscosities and elastic compliances were measured in shear and elongational flows for two low-density polyethylenes, a linear polypropylene, and two metallocene catalyzed polyethylenes (one linear and one long-chain branched) by Wolff et al. (Rheol Acta 49:95?C103, 2010) and Resch (dissertation, 2010). Comprehensive data of this type are rarely found in the literature, and comprehensive modeling of both viscous and elastic effects is even rarer. In this contribution, the reliability of a modeling approach proposed by Laun (J Rheol 30(3):459?C501, 1986) and based on the damping function concept is tested. The strain hardening seen for the long-chain branched polymers and its absence in the case of the linear polymer, the stronger decrease of the tensile compliance in comparison to the shear compliance with increasing stress, as well as the extended linear-viscoelastic regime of the shear viscosity in contrast to the shear compliance are correctly modeled. While the modeling of the nonlinear response in shear can be achieved with only one material parameter for most of the polymers considered here, the nonlinear modeling in elongation is achieved with two parameters. The same parameter values are shown to describe viscous as well as elastic properties of the melts, and thus the relations of Laun can be used to test the consistency of viscosity and compliance measurements.     

SQUEEZE FILM FLOW WITH NONLINEAR BOUNDARY SLIP

A nonlinear boundary slip model consisting of an initial slip length and a critical shear rate was used to study the nonlinear boundary slip of squeeze fluid film confined between two approaching spheres. It is found that the initial slip length controls the slip behavior at small shear rate, but the critical shear rate controls the boundary slip at high shear rate. The boundary slip at the squeeze fluid film of spherical surfaces is a strongly nonlinear function of the radius coordinate. At the center or far from the center of the squeeze film, the slip length equals the initial slip length due to the small shear rate. However, in the high shear rate regime the slip length increases very much. The hydrodynamic force of the spherical squeeze film decreases with increasing the initial slip length and decreasing the critical shear rate. The effect of initial slip length on the hydrodynamic force seems less than that of the critical shear rate. When the critical shear rate is very small the hydrodynamic force increases very slowly with a decrease in minimum film thickness. The theoretical predictions agree well with the experiment measurements.     

Detection and quantification of industrial polyethylene branching topologies via Fourier-transform rheology,NMR and simulation using the Pom-pom model

The significance of sparse long-chain branching in polyolefines towards mechanical properties is well-known. Topology is a very important structural property of polyethylene, as is molecular weight distribution. The method of Fourier-transform rheology (FTR) and melt state nuclear magnetic resonance (NMR) is applied for the detection and quantification of branching topology (number of branches per molecule), for industrial polyethylenes of various molecular weight and molecular weight distributions. FT rheology consists of studying the development of higher harmonics contribution of the stress response to a large amplitude oscillatory shear deformation. In particular, when applying large-amplitude oscillatory shear (LAOS), one observes the development of mechanical higher harmonic contributions at 3ω ₁, 5ω ₁,..., in the shear stress response. We correlate the relative intensity, I _3/1, and phase Φ ₃ of these harmonics with structural properties of industrial polyethylene, i.e. polymer topology and molecular weight distribution. Experiments are complemented by numerical simulations, using a multimode differential Pom-pom constitutive model (DCPP formulation), by fitting to the experimental linear and nonlinear viscoelastic behaviours. Simulation results in the nonlinear regime are correlated with molecular properties of the “pom-pom” macromolecular architecture. Qualitative agreement is found between predicted and experimental FT rheology results.     

Linear viscoelastic model for elongational viscosity by control theory

Flows involving different types of chain branches have been modelled as functions of the uniaxial elongation using the recently generated constitutive model and molecular dynamics for linear viscoelasticity of polymers. Previously control theory was applied to model the relationship between the relaxation modulus, dynamic and shear viscosity, transient flow effects, power law and Cox–Merz rule related to the molecular weight distribution (MWD) by melt calibration. Temperature dependences and dimensions of statistical chain tubes were also modelled. The present study investigated the elongational viscosity. We introduced earlier the rheologically effective distribution (RED), which relates very accurately and linearly to the viscoelastic properties. The newly introduced effective strain-hardening distribution (RED_H) is related to long-chain branching. This RED_H is converted to real long-chain branching distribution by melt calibration and a simple relation formula. The presented procedure is very effective at characterizing long-chain branches, and also provides information on their structure and distribution. Accurate simulations of the elongational viscosities of low-density polyethylene, linear low-density polyethylene and polypropylene, and new types of MWDs are presented. Models are presented for strain-hardening that includes the monotonic increase and overshoot effects. Since the correct behaviour at large Hencky strains is still unclear, these theoretical models may aid further research and measurements.     

The effect of die exit curvature,die surface roughness and a fluoropolymer additive on sharkskin extrusion instabilities in polyethylene processing

This paper reports experimental observations and numerical simulations relating to sharkskin extrusion instabilities for two different types of polyethylene, a metallocene high-density polyethylene (HDPE) and a linear low-density polyethylene (LLDPE). Experimental results are presented for both the effect of die exit curvature and die surface roughness for slit die geometry. Matching polyflow numerical simulations are also reported and are shown to be qualitatively consistent with experimental observations. The onset of the sharkskin instability is correlated with the magnitude of the stress concentration at the die exit, and is found to be sensitive to both the melt/wall separation point for a curved exit die, and the level of partial slip at the die wall. Additional observations on the effect of a fluoropolymer additive also support the sensitivity of the sharkskin instability to partial slip at the wall.     

Sharkskin instabilities and the effect of slip from gas-assisted extrusion

This paper is concerned with a polymer extrusion instability and the effect of introducing slip by means of a thin lubricating gas layer between the extrusion die wall and the flowing polymer melt. Gas-assisted extrusion (GAE) experiments were carried out using high-density polyethylene (HDPE) and linear low-density polyethylene (LLDPE) for a number of different gas injection die geometries. The stress distributions within the polymer melt were monitored during extrusion using flow birefringence. Polyflow numerical simulations were used to calculate the local stress concentrations in the melt at the die exit, as these were believed to be related to the occurrence of sharkskin. Simulations were also used to observe the effect of a full slip boundary condition as imparted by GAE. A key finding of the paper is that GAE in the parallel section of the die wall simply moved the local exit stress concentration upstream to the point of gas injection, and therefore did not reduce sharkskin. Simulations indicated that for correctly designed dies, the local surface stress concentration would be reduced. However, it was found experimentally that it was not possible to obtain a stable gas layer for this die design with upstream gas injection. A numerical investigation, involving simulations of varying levels of partial slip along the die wall, indicated an optimum level of slip where the stress concentrations were reduced. It is speculated that this is the reason that coatings such as PTFE, which impart a partial slip, can reduce sharkskin while GAE does not. The findings show that a controlled level of partial slip lowers the overall stress concentrations.     

On the Torsion of a Class of Nonlinear Fiber Composite Cylinders

This paper presents an analysis of the torsion of a solid or annular circular cylinder consisting of nonlinear material in the form of an elastic matrix with embedded unidirectional elastic fibers parallel to the cylinder axis. The specific class of composite considered is one for which nonlinear fiber-matrix interface slip is captured by uniform cohesive zones of vanishing thickness. Previous work on the effective antiplane shear response of this material leads to a stress–strain relation depending on the interface slip together with an integral equation governing its evolution. Here, we obtain an approximate single mode solution to the integral equation and utilize it to solve the torsion problem. Equations governing the radial distributions of shear stress and interface slip are obtained and formulae for torque–twist rate are presented. The existence of singular surfaces, i.e., surfaces across which the slip and the shear stress experience jump discontinuities are analyzed in detail. Specific results are presented for an interface force law that allows for interface failure in shear.     

A model long-discontinuous-fiber filled thermoplastic melt in extensional flow

The shear cell model works for dilute fiber filled systems in extensional flow. This research investigates the suitability of the idea for highly aligned fibers in a concentrated suspension. A model fiber-filled polymer system made from nylon fibers in low-density polyethylene provided a means of controlling the material parameters. Two systems, with fiber aspect ratios of 20 and 100, containing 50% 0.5 mm fibers by volume are investigated. The thickness of the polymer layer, i.e. with fibers this size, allows bulk viscosity data to be compared with the data from the filled fluid. A weaving process created the discontinuous fiber/polyethylene preforms with high alignment of the fibers and with control of the fiber to fiber overlap. Testing the polyethylene in simple shear and extending the nylon/polyethylene provided the data needed to check the micro mechanics. A cone and plate rheometer and a capillary instrument produced the viscosity/strain rate data that characterized the specific polyethylene used in the composite. A furnace inset placed in an Instron hydraulic test machine allowed extension of the filled system at strain rates from 0.002 to 0.4 s⁻¹. The shear experiments show that the low-density polyethylene is a simple shear-thinning melt that provides a good model fluid. The extension of the filled systems shows an increase of the apparent extensional viscosity from that of neat polyethylene. Apparent viscosity rises two to three orders of magnitude for the systems investigated. The micromechanics allowed the conversion of the extensional data from the two filled systems to the shear viscosity of the polymer surrounding the fibers. The calculated polyethylene viscosity compares well with the data from the standard rheometers. The shear cell approach may be applied to highly aligned, high fiber-volume-fraction suspensions when the viscosity of the polymer is known at the scale of the film surrounding each fiber.     

Step planar extension of polymer melts using a lubricated channel

A technique to do step planar extension on polymer melts has been developed using a rectangular channel with lubricated walls and the linear motor of the Rheometrics System Four rheometer. Using this method we probe the stress relaxation of two polymer melts, a linear low density polyethylene (LLDPE) and a highly branched low density polyethylene (IUPAC X), and compare the step planar extensional data to step shear data. Since a step planar deformation is theoretically equivalent to a step shear in a rotating frame of reference, we expect that the nonlinear modulus for step planar extension should be equivalent to that for step shear. Although we find the time dependence of the stress relaxation modulus to be the same in both shear and planar extension, the strain dependence is surprisingly different for the two experiments.     

Capillary flow of linear polyethylene melt: sudden increase of flow rate

During the oscillatory flow of linear polyethylene (HDPE) melt through a capillary, the shape of the extrudate varies periodically with time: sharkskin, plug and rough. This paper deals with the transition between increasing and decreasing pressure. At that transition, the flow rate through the tube is suddenly and intensively increased. We present a theoretical analysis which is in good agreement with experiment. The “slip” at the wall is held to be a consequence of and not the cause of phenomenon. The radial profile of the shear rate is described by means of Dirac's delta function.     

Modeling Turbulent Droplet Motion in Vaporizing Sprays

The present article is concerned with the influence of turbulent gas-velocity fluctuations on both droplet dispersion and droplet-gas slip velocity in the context of spray simulation. The role of turbulence in generating slip and thus enhancing interphase heat and mass transfer has so far received little attention and is investigated in this work. A model for turbulent gas-velocity fluctuations along droplet trajectories is presented and is first tuned to reproduce elementary dispersion phenomena. It is then shown to give good results for more general dispersion problems as well as for slip velocities. As a fundamental source of information and for the purpose of model validation and comparison, direct numerical simulation (DNS) of droplet motion in homogeneous isotropic steady turbulence (HIST) is used. Dispersion of “injected” droplets (i.e. droplets under the influence of drift due to high injection velocity) as well as slip velocities for linear and nonlinear droplet drag are studied, and reasonable agreement is found with the model. The distributions of the slip velocity are found to be very similar for linear and highly nonlinear drag law. The present model is also used to investigate the influence of turbulence on droplet penetration. Comparison is made with an eddy-interaction model (the KIVA-2 model), which reveals various weaknesses of this model, in particular the underprediction of average slip velocity. The influence of slip due to turbulence on vaporization is shown for a fuel spray injected into a premix gas-turbine combustor. The classical eddy-interaction model is seen to underestimate the rate of vaporization due to the underprediction of slip. This revised version was published online in July 2006 with corrections to the Cover Date.     

Constitutive equations in finite viscoplasticity of semicrystalline polymers

Three series of uniaxial tensile tests with constant strain rates are performed at room temperature on isotactic polypropylene and two commercial grades of low-density polyethylene with different molecular weights. Constitutive equations are derived for the viscoplastic behavior of semicrystalline polymers at finite strains. A polymer is treated as an equivalent network of strands bridged by permanent junctions. Two types of junctions are introduced: affine whose micro-deformation coincides with the macro-deformation of a polymer, and non-affine that slide with respect to their reference positions. The elastic response of the network is attributed to elongation of strands, whereas its viscoplastic behavior is associated with sliding of junctions. The rate of sliding is proportional to the average stress in strands linked to non-affine junctions. Stress–strain relations in finite viscoplasticity of semicrystalline polymers are developed by using the laws of thermodynamics. The constitutive equations are applied to the analysis of uniaxial tension, uniaxial compression and simple shear of an incompressible medium. These relations involve three adjustable parameters that are found by fitting the experimental data. Fair agreement is demonstrated between the observations and the results of numerical simulation. It is revealed that the viscoplastic response of low-density polyethylene in simple shear is strongly affected by its molecular weight.     

Non Linear Rheology for Long Chain Branching characterization,comparison of two methodologies : Fourier Transform Rheology and Relaxation.

In this study we compare three rheological ways for Long Chain Branching (LCB) characterization of a broad variety of linear and branched polyethylene compounds. One method is based on dynamical spectrometry in the linear domain and uses the van Gurp Palmen plot. The two other methods are both based on non linear rheology (Fourier Transform Rheology (FTR) and chain orientation/relaxation experiments). FTR consists in the Fourier analysis of the shear stress signal due to large oscillatory shear strains. In the present work we focus on the third and the fifth harmonics of the shear stress response. Chain orientation/relaxation experiment consists in the analysis of the polymer relaxation after a large step strain obtained by squeeze flow. In this method, relaxation is measured by dynamical spectrometry and is characterized by two relaxation times related to LCB. All methods distinguish clearly the group of linear polyethylene from the group of branched polyethylene. However, FTR and Chain orientation/relaxation experiments show a better sensitivity than the van Gurp Palmen plot. Non linear experiments seem suitable to distinguish long branched polyethylene between themselves.     

On the stability of the shear flow of a viscoelastic fluid with slip along the fixed wall

In this paper we solve the time-dependent shear flow of an Oldroyd-B fluid with slip along the fixed wall. We use a non-linear slip model relating the shear stress to the velocity at the wall and exhibiting a maximum and a minimum. We assume that the material parameters in the slip equation are such that multiple steady-state solutions do not exist. The stability of the steady-state solutions is investigated by means of a one-dimensional linear stability analysis and by numerical calculations. The instability regimes are always within or coincide with the negative-slope regime of the slip equation. As expected, the numerical results show that the instability regimes are much broader than those predicted by the linear stability analysis. Under our assumptions for the slip equation, the Newtonian solutions are stable everywhere. The interval of instability grows as one moves from the Newtonian to the upper-convected Maxwell model. Perturbing an unstable steady-state solution leads to periodic solutions. The amplitude and the period of the oscillations increase with elasticity.