Interchain tube pressure effect in extensional flows of oligomer diluted nearly monodisperse polystyrene melts

We have derived a constitutive equation to explain the extensional dynamics of oligomer-diluted monodisperse polymers, if the length of the diluent has at least two Kuhn steps. These polymer systems have a flow dynamics which distinguish from pure monodisperse melts and solutions thereof, if the solvent has less than two Kuhn steps, e.g. is not a chain. The constitutive equation is based on a phenomenological tube-based model within the methodology of the molecular stress function approach. The nonlinear dynamics have been explained as a consequence of a constant thermal interchain pressure originating from the short polymer chains (e.g. the oligomers) on the wall of the tube containing the long chains. The nonlinear dynamics are uniquely defined by the Rouse time and the maximal extensibility of the long polymer chains. Both are linked to the entanglement length. The relation between the Rouse times and entanglements have been established based on published extensional experiments on nearly monodisperse polystyrene melts. The constitutive equation has shown agreement with the experimental startup of and steady extension data from Huang et al. (Macromolecules 46:5026–5035, 2013a) based on 285 and 545 kg/mol polystyrenes diluted in styrene oligomers containing 3.3 (1.92 kg/mol) and 7.3 (4.29 kg/mol) Kuhn steps.     

Non-isothermal flow of polymer melts in a curved tube

The results of a numerical study (using finite differences) of heat transfer in polymer melt flow is presented. The rheological behaviour of the melt is described by a temperature-dependent power-law model. The curved tube wall is assumed to be at constant temperature. Convective and viscous dissipation terms are included in the energy equation. Velocity, temperature and viscosity profiles, Nusselt numbers, bulk temperatures, etc. are presented for a variety of flow conditions. Br — Brinkman number - c specific heat, J/kg K - De — Dean number - E dimensionless apparent viscosity, eq. (14d) - G dimensionless shear rate, eq. (19) - k parameter of the power-law model, °C^–1, eq. (7) - mass flow rate, kg/s - m ₀ parameter of the power-law model, Pa · s ⁿ , eq. (7) - n parameter of the power-law model, eq. (7) - Nu 2r _p/ — Nusselt number, eqs. (28,31) - p pressure, Pa - Pe — Péclet number - P —(p/)/r _c — pressure gradient, Pa/m - dissipated energy, W, eq. (29) - total energy, W, eq. (30) - r radial coordinate, m - r _c radius of tube-curvature, m, fig. 1 - r _p radius of tube, m, fig. 1 - r _t variable, m, eq. (6) - R dimensionless radial coordinate, eq. (14a) - R _c dimensionlessr _c, eq. (14a) - R _t dimensionlessr _t, eq. (14a) - t temperature, °C - bulk temperature, °C, eq. (27) - t ₀ inlet temperature of the melt, °C - t _w tube wall temperature, °C - T dimensionless temperature, eq. (14c) - T _w dimensionless tube wall temperature - T dimensionless bulk temperature - u ₁ variable, s^–1, eq. (4) - u ₂ variable, s^–1, eq. (5) - U ₁ dimensionlessu ₁, eq. (18) - U ₂ dimensionlessu ₂, eq. (18) - v velocity in-direction, m/s - average velocity of the melt, m/s - V dimensionlessv, eq. (14b) - dimensionless , eq. (15) - z r _c — centre length of the tube, m - Z dimensionlessz, eq. (14e) - heat transfer coefficient, W/m² K - shear rate, s^–1, eq. (8) - — shear rate, s^–1 - apparent viscosity, Pa · s, eq. (7) - ₀ — apparent viscosity, Pa · s - angular coordinate, rad, fig. 1 - thermal conductivity, W/m K - melt density, kg/m³ - axial coordinate, rad, fig. 1 - rate of strain tensor, s^–1, eq. (8) - (—p) pressure drop, Pa     

Stress relaxation of bi-disperse polystyrene melts

We present start-up of uniaxial extension followed by stress relaxation experiments of a bi-disperse 50 % by weight blend of 95k and 545k molecular weight polystyrene. We also show, for comparison, stress relaxation measurements of the polystyrene melts with molecular weight 95k and 545k, which are the components of the bi-disperse melt. The measurements show three separated relaxation regimes: a fast regime, a transition regime, and a slow regime. In the fast regime, the orientation of the long chains is frozen and the stress relaxation is due to stretch relaxation of the short chains primarily. Conversely in the slow regime, the long chains have retracted and undergo relaxation of orientation in fully relaxed short chains.     

Nonisothermal flow of polymer melts

The Curtiss-Bird theory for polymer melts is modified to allow the temperature to be a function of time. The resulting constitutive equation involves integrals over the temperature history of the polymer. The predictions of this constitutive equation for the inception of uniaxial elongation with simultaneous cooling at a constant rate are calculated and compared with the experimental data of Matsumoto and Bogue. Qualitative agreement between the theory and the data is obtained when the link tension coefficient in the Curtiss-Bird theory is not equal to zero, but such agreement cannot be obtained when this coefficient is equal to zero. The extension of the theory to include spatially varying temperature is also discussed.     

Elongational flow opto-rheometry for polymer melts

Polymer melt elongation is one of the most important procedures in polymer processing. To understand its molecular mechanisms, we constructed an elongational flow opto-rheometer (EFOR) in which a high precision birefringence apparatus of reflection-double path type was installed into a Meissner's new elongational rheometer of a gas cushion type (commercialized as RME from Rheometric Scientific) just by mounting a small reflecting mirror at the center of the RME's sample supporting table. The EFOR enabled us to achieve simultaneous measurements of tensile stress (t) and birefringence n(t) as a function of time t under a given constant strain rate within the range of 0.001 to 1.0s^–1. (t) can be monitored upto the maximum Hencky strain (t) of 7 as attained, in principle, with RME, while the measurable range of the phase difference in the birefringence was 0 to 250 (0 to 79 100 nm for He-Ne laser light) within the accuracy of ±0.1 (±31.6 nm) up to (t) 4. The performance was tested on an anionically polymerized polystyrene (PS) and a low density polyethylene (LDPE). For both polymers (t) first followed the linear viscoelasticity rule in that the elongational viscosity, , is three times the steady shear viscosity, 3 _o(t), at low shear rate , but the _E (t) tended to deviate upward after a certain Hencky strain was attained. The birefringence n(t) was a function of both Hencky strain and strain rate in such a way that the stress-optical law holds with the stress-optical coefficient C(t) = n(t)/(t) being equal to the ones reported from shear flow experiments. Interestingly, however, for PS elongated at low strain rates the C(t) vs (t) relation exhibited a strong nonlinearity as soon as (t) reached steady state. This implies that the tensile stress reaches the steady state but the birefringence continues to increase in the low strain-rate elongation. For the PS melt elongated at high strain rates, on the other hand, C(t) was nearly a constant in the entire range observed. For LDPE with long-chain branchings, (t) exhibited tendency of strain-induced hardening after certain critical strain, but C(t) was nearly a constant in the entire range of (t) observed.     

Exponential shear flow of branched polymer melts

  The behavior of a low-density polyethylene melt in exponential shear strain histories is examined and compared to its behavior in constant rate planar elongation. A new set of shear stress and first normal stress difference data in exponential shear are presented and used in several different material functions that have been previously proposed. Viscosities composed of principal stress differences for the two flows showed no correspondence suggesting that, contrary to previous assertions, exponential shear and constant rate planar elongation flows are fundamentally different. It is further suggested that the presence of vorticity makes exponential shear a weak, rather than strong, flow. Received: 5 March 1999/Accepted: 1 September 1999     

Unsteady flow of polymer melts: Polypropylene

Summary The unsteady flow effect has been studied experimentally in cylindrical capillaries for polypropylene melts with melt indices differing over an approximately 30-fold range. The entrance losses were negligibly small. The flow curves obtained on capillaries of different diameters, coincided, indicating the absence of any considerable near-wall slippage; hence the attainment of unsteady flow is not necessarily due to or accompanied by wall slippage.Two critical regimes can be distinguished distinctly in the flow of polypropylene melts, the first corresponding to the appearance of mattness on the extrudate surface, and the second to the appearance of defects such as a regular spiral with a constant longitudinal pitch. No unsteady flow of the melt fracture type was observed.The shear stresses in the first critical flow regime increase by 40 per cent when the temperature is raised from 180 to 240 °C. Within the same temperature range the stresses corresponding to attainment of the second critical flow regime change by 25 per cent.The critical flow parameters of polypropylene melts grow with increasing capillary length-to-diameter ratio, this effect not being damped even with big capillary lengths.The elastic deformations corrrsponding to attainment of the first critical flow regime of polypropylene melts_e2.7, while that corresponding to the second critical regime _e 3.3 with considerably changed critical stress values.

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Zusammenfassung Der Effekt des nicht-stationären Fließens wurde experimentell in zylindrischen Kapillaren für Polypropylen-Schmelzen mit Schmelz-Indizes zwischen 1 und 30 untersucht. Die Eintrittsverluste waren vernachlässigbar klein. Die Fließkurven, die bei Kapillaren verschiedenen Durchmessers ermittelt wurden, stimmten überein und zeigten das Fehlen irgendwelcher Gleitung in der Nähe der Wand. Demnach ist das Entstehen eines nicht-stationären Fließens nicht notwendigerweise abhängig bzw. begleitet von einer Wandgleitung.Zwei kritische Bereiche können eindeutig beim Fließen von Polypropylen-Schmelzen unterschieden werden: der erste im Zusammenhang mit dem Auftreten einer matten Extrudat-Oberfläche und der zweite mit dem Auftreten von Fehlern wie einer regelmäßigen Spirale mit konstanter Steigung. Ein nicht-stationäres Fließen vom Schmelzbruchtyp wurde nicht beobachtet.Die Schubspannung in dem ersten kritischen Fließbereich wächst bei einem Temperaturanstieg von 180 auf 240 °C um 40%. Im gleichen Temperaturbereich wachsen die Spannungen bei Erreichen des zweiten kritischen Fließgebietes um 25%.Die kritischen Fließparameter von Polypropylen-Schmelzen wachsen mit steigendem Verhältnis Kapillarlänge zu Durchmesser, wobei dieser Effekt auch durch große Kapillarlängen nicht gedämpft wird. Die elastischen Deformationen, die bei Erreichen des ersten kritischen Fließgebietes von Polypropylen-Schmelzen auftreten, betragen _e 2,7, während für das zweite kritische Gebiet bei deutlich geänderten kritischen Spannungswerten sich Deformationen von _e 3,3 ergeben.

     

Unstable capillary flow of reinforced polymer melts

The injection capillary flow of various unfilled and glass fibre or calcium carbonate filled polypropylene and nylon 6.6 melts is studied using either a single capillary of five capillaries in series, separated by small reservoirs. Only unfilled nylon 6.6 yields instability during flow through a single capillary due to mechanochemical degradation in the capillary at extremely high shear rates above 5 × 10⁵ s^?1. It is found that only short glass fibre reinforced polypropylene yields high frequency oscillations in the reservoir pressure and extrudate diameter and has discontinuity in the flow curve when the apparent shear rate is above 4 × 10⁵ s^?1 and the flow is through multiple capillaries. Further increase in the shear rate restores the stable flow. The intensity of the oscillations and the range of shear rate during which unstable flow occurs are increased with increasing melt temperature. The mechanism of this unstable flow is investigated by studying fibre orientation at the capillary entrance and exit using mouldings simulating capillary entry-exit flows.     

Nonlinear effects in the dynamics of concentrated polymer solutions and melts

The dependence of the environmental characteristics on the induced anisotropy is found by means of self-consistency for large velocity gradients. It is shown that this method of taking the anisotropy into account is consistent with the model previously employed [11–13] and leads to results similar to those obtained in [14, 15]. Steady shear flow and steady uniaxial stretching are considered on the basis of the equations of state obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 88–96, July–August, 1990.     

Nonlinear viscoelasticity of polymer melts in different types of flow

The nonlinear viscoelastic properties of a fairly large class of polymeric fluids can be described with the factorable single integral constitutive equation. For this class of fluids, a connection between the rheological behaviour in different flow geometries can be defined if the strain tensor (or the damping function) is expressed as a function of the invariants of a tensor which describes the macroscopic strain, such as the Finger tensor. A number of these expressions, proposed in the literature, are tested on the basis of the measuring data for a low-density polyethylene melt. In the factorable BKZ constitutive equation the strain-energy function must be expressed as a function of the invariants of the Finger tensor. The paper demonstrates that the strain-energy function can be calculated from the simple shear and simple elongation strain measures, if it is assumed to be of the shape proposed by Valanis and Landel. The measuring data for the LDPE melt indicate that the Valanis-Landel hypothesis concerning the shape of the strainenergy function is probably not valid for polymer melts.     

Thermodynamically consistent coarse graining the non-equilibrium dynamics of unentangled polymer melts

A thermodynamically consistent strategy of coarse-graining microscopic models for complex fluids is illustrated for low-molecular polymeric melts subjected to homogeneous flow fields. The systematic coarse-graining method is able to efficiently bridge the time- and length scale gap between microscopic and macroscopic dynamics. A projection operator derivation is employed within the framework of nonequilibrium thermodynamics. From an alternating Monte-Carlo-molecular dynamics iteration scheme we obtain the thermodynamic building blocks of the macroscopic model. We investigate a number of imposed shear and elongational flows of interest and find that the coarse-grained model predicts structural as well as material functions beyond the regime of linear response. The elimination of fast degrees of freedom gives rise to dissipation, which we analyse in terms of the Rouse model. The results are in quantitative agreement with those obtained via standard nonequilibrium molecular dynamics (NEMD) simulations for planar shear and elongation. The coarse-graining method is able to deal with more general flows, which are not accessible by standard NEMD simulations.     

Investigation of failure during elongational flow of polymer melts

A basic study of the mechanisms of necking and ductile failure of polymer melts in uniaxial elongational flow has been carried out. A linear stability analysis was carried out using a White—Metzner convected Maxwell model with a deformation-rate-dependent relaxation time, which varies according to τ = τ_o/(1 + aτ_o[2trd²]^{$\text{1}\text{2}$}). It was shown that filament stability and elongation to break depend upon τ_oE, where E is the elongation rate, and a. At fixed τ_oE, filament stability decreases with increasing a. At small a, stability increases with increasing τ_oE while for a > $\text{1}\text{√3}$, stability decreases with increasing τ_oE. For a material with small a, ductile failure can occur for small τ_oE, but cohesive fracture should be the cause of failure at larger τ_oE. For a material with large a, however, ductile failure always dominates the failure mode. These results are used to interpret failure in elongational flow of low density and high density polyethylene and polypropylene melts and describe how the latter two melts exhibit ductile failure.     

A network model with free-strand dynamics for polymer melts

 A network model for polymer melts is presented in which disentangled strands relax under flow conditions and may rejoin the network before complete relaxation. For simplicity, we study Gaussian strands that move affinely when incorporated in the network. Network strands are created and lost according to a time constant λ. Free strands have their dynamics given by the Bird-DeAguiar model as a crude representation of reptation and the hindered rotation experienced by polymer strands in melts. The model yields a shear-thinning viscosity with overshoot in the start-up viscosity η⁺ (t). The double-step strain results compare well with available experimental data. Received: 10 July 2000 Accepted: 10 July 2001     

Three dimensional finite element analysis of the flow of polymer melts

The finite element simulation of a selection of two- and three-dimensional flow problems is presented, based upon the use of four different constitutive models for polymer melts (Oldroyd-B, Rolie-Poly, Pom-Pom and XPP). The mathematical and computational models are first introduced, before their application to a range of visco-elastic flows is described. Results demonstrate that the finite element models used here are able to re-produce predictions made by other published numerical simulations and, significantly, by carefully conducted physical experiments using a commercial-grade polystyrene melt in a three-dimensional contraction geometry. The paper also presents a systematic comparison and evaluation of the differences between two- and three-dimensional simulations of two different flow regimes: flow of an Oldroyd-B fluid around a cylinder and flow of a Rolie-Poly fluid into the contraction geometry. This comparison allows new observations to be made concerning the relatively poor quality of two-dimensional simulations for flows in even quite deep channels.     

Numerical simulation studies of the planar entry flow of polymer melts

This paper is concerned with the numerical simulation of planar entry flow using a penalty finite element method and the comparison of predictions with flow visualization and birefringence data for two polymer melts. The Phan-Thien Tanner (PTT) model was fit to the steady state shear and extensional viscosity data and the transient extensional viscosity data of both polystyrene and low-density polyethylene (LDPE) melts to obtain the parameters λ, ξ, and ϵ in this model. Agreement was found between the flow visualization and birefringence data and the predictions of streamlines and stress. With some modification of the constitutive equation, the vortex growth and intensity observed for LDPE could be predicted by the use of the PTT model and the material parameters fit to the rheological properties. Likewise, the flow behavior of polystyrene, in which only small vortices with no growth were observed, was also predicted. Furthermore, it was found that the size and intensity of the vortex could be affected by the parameter ϵ in the PTT model which controls the predictions of the extensional viscosity. Based on these results it seems that accurate simulation of entry flow behavior requires the use of a constitutive equation which is capable of giving realistic preciction's of a fluid's extentional flow properties.     

A 2D model for tube orientation and tube squeezing in fast flows of polymer melts

Motivated by recent data of Hassager and coworkers [A. Bach, K. Almdal, H.K. Rasmussen, O. Hassager, Elongational viscosity of narrow molar mass distribution polystyrene, Macromolecules, 36 (2003) 5174–5179], we develop a new tube model describing the non-linear behaviour of entangled monodisperse linear polymers. Within the context of well established tube theories, the model accounts for the effect of flow on tube diameter, thus somehow picking up a long standing suggestion by Wagner and Schaeffer [M.H. Wagner, J. Schaeffer, Non-linear strain measures for general biaxial extension of polymer melts, J. Rheol., 32 (1992) 1–26] among others. Since tubes with a deformation dependent cross section are rather difficult to deal with, we here limit model development to an artificial two-dimensional situation. The simple 2D model has the advantage of illustrating the new physical assumptions more transparently, and it already proves to correctly predict most qualitative features typically shown by shear and elongational data in the non-linear range.     

Birefringence measurements on polymer melts in an axisymmetric flow cell

 The stress-optical rule relates birefringence to stress. Consequently, measurement of flow birefringence provides a non-intrusive technique of measuring stresses in complex flows. In this investigation we explore the use of an axisymmetric geometry to create a uniaxial elongational flow in polymer melts. In axisymmetric flows both birefringence and orientation angle change continuously along the path of the propagating light. The cumulative influence of the material's optical properties along the light's integrated path makes determination of local birefringence in the melt impossible. One can nevertheless use birefringence measurements to compare with predictions from computer simulations as a means of evaluating the constitutive equations for the stress. More specifically, in this investigation we compare the light intensity transmitted through the experimental set-up vs entry position, with the theoretically calculated transmitted intensity distribution as a means of comparing experiment and simulation. The main complication in our experiments is the use of a flow cell that necessarily consists of materials of different refractive indices. This introduces refraction and reflection effects that must be modeled before experimental results can be correctly interpreted. We describe how these effects are taken into account and test the accuracy of predictions against experiments. In addition, the high temperatures required to investigate polymer melts mean that a further complication is introduced by thermal stresses present in the flow cell glass. We describe how these thermal-stresses are also incorporated in the simulations. Finally, we present some preliminary results and evaluate the success of the overall method. Received: 2 April 2001 Accepted: 27 August 2001     

A modified tube dilatation theory for the relaxation of entangled linear polymer melts

In this paper, the “tube dilatation” or “tube Enlargement” concept introduced by Marrucci, is revisited in the case of broad entangled linear polymer melts. Using the tube fluctuation relaxation function of Doi and a linear mixing rule, the model implicitly contains both the features of double reptation and time modification by tube renewal. It has been shown that theoretical arguments of both double reptation with tube renewal and tube dilatation can be used to take into account the effect of polydispersity on the distribution of relaxation times. The model has been tested for some polymers with various N/Ne values. However, experiments indicate that the loss of only one entanglement does not systematically induce a change of the relaxation times through a constraint release and tube renewal process. The freeing of a critical volume, larger than the volume of a tube segment, is required to induce an efficient dilution effect on relaxation times.