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.     

A small scale capillary viscometer for polymer melts

Summary A small scale capillary viscometer that is capable of measuring the flow curves of 2–3 gms of molten polymer over the shear rate range 0.4 sec^–1 to 20,000 sec^–1 is described. A piezo electric crystal pressure transducer is used to measure the pressure directly above the capillary. Flow curves are given for anionic polystyrenes and polyethylene fractions, and it is shown that the viscosities measured agree with determinations in a cone and plate viscometer.

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Zusammenfassung Es wird ein maßstäblich kleines Kapillarviskosimeter für 2–3 g geschmolzener Polymere beschrieben, das die Durchfiußkurve in Abhängigkeit vom Scherkoeffizienten im Bereich von 0,4 sec^–1 bis 20000 sec^–1 bestimmt. Verwendet wird ein piezoelektrischer Kristall-Druckaufnehmer, der den Druck genau über der Kapillare mißt. Angeführt werden Durchflußkurven für anionische Polystyrole und Polyäthylenanteile. Es wird gezeigt, daß die gemessenen Zähigkeiten mit denen von Kegel- und Plattendichtemessern übereinstimmen.

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Paper presented at the Conference on Experimental Rheology, University of Bradford, April 17–19, 1968.     

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.

     

Inlet instabilities in the capillary flow of polyethylene melts

Inlet instabilities in the capillary flow of polyethylene melts were studied in this work. Extrudate distortions in branched polyethylenes, produced by unstable upstream flow, were found to be accompanied by pressure oscillations that do not have their origin in the slip phenomenon, but on polymer compressibility. The absence of slip was clearly evidenced in the experiments, and the differences between pressure oscillations occurring in linear and branched polymers are shown.Pressure oscillations in the capillary flow of branched polyethylenes were found to be made up of at least two components of different frequency and amplitude. These two components were identified with different bulk defects appearing in the extrudates. Information about the dynamics of vortices upstream of the contraction and extrudate distortions is obtained from the analysis of pressure oscillations.The influence of capillary entrance angle on flow curves was also investigated. From the results, it is concluded that the extensional component of the flow in the contraction is the main factor responsible for the slope change usually found in the log-log flow curves of both linear and branched polyethylenes.     

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.     

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     

Unstable slit flow of a liquid crystalline polymer solution

A lyotopic solution of 27 wt% hydroxypropylcellulose [HPC] in m-cresol has been studied in pressure-driven slit flow. At high flow rates an instability leads to large wavelength disturbances in fluid structure. A combination of image analysis and time signal processing is used to determine the velocity at which the structural disturbances are convected downstream, which is shown to be equal to the independently measured and predicted centerline velocity. This implies that the disturbance structure is confined near the midplane of the slit flow. Upstream of the onset point of the wavy fluid structures, the fluid exhibits unusual optical properties when viewed between crossed polarizers that are rotated relative to the flow direction. Specifically, the optical properties indicate that there must be some variation in the macroscopic optical axis of the sample as light passes through the slit flow. A discrete optical model consisting of birefringent elements twisted away from and back to the flow direction as a function of depth in the sample is able to predict the essential optical characteristics; however, independent x-ray scattering measurements show that the macroscopic molecular alignment is along the flow direction. The wavy textures apparently emerge as a result of an inhomogeneous transition of orientation back to the flow direction, trapping thin bands of fluid in the twisted configuration.     

A calorimetric measurement of frictional heat in capillary rheometry of polymer melts

Summary A viscometer is described in which the capillary wall temperature is controlled by an adjustable low voltage current directly flowing through the thin tube wall.A change in the required electric power input measures a corresponding change in transferred frictional heat. Additional readings of pressure and throughput enable the calculation of various interesting quantities as cup mixing temperature of the extrudate and temperature correction of the flow curve of the polymer.

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Zusammenfassung Ein Viskosimeter wird beschrieben, durch dessen dünnwandige Kapillare ein regelbarer Strom niedriger Spannung fließt, mit dessen Hilfe die Wand auf der gewünschten Temperatur gehalten wird. Eine Veränderung in der erforderlichen elektrischen Leistung mißt eine entsprechende Änderung der übertragenen Reibungswärme. Zusätzliche Messungen von Druck und Ausstoßleistung ermöglichen die Berechnung verschiedener interessanter Größen wie z. B. der mittleren Extrudattemperatur und der Temperaturkorrektur der Strömungskurve des Polymeren.

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With 5 figures     

Pressure dependent viscosity and dissipative heating in capillary rheometry of polymer melts

In high shear rate capillary rheometry the combined effect of pressure dependent viscosity and dissipative heating becomes significant. Analytical expressions are derived to treat curved Bagley plots and throttle experiments. End effects are taken into account by using an effective length over radius ratio. The non-adiabatic case is described using a lump heat transfer coefficient ? following Hay et al. (1999). The latter enters into the dissipative heating coefficient % MathType!MTEF!2!1!+- % feaaeaart1ev0aaatCvAUfKttLearuavTnhis1MBaeXatLxBI9gBam % XvP5wqSXMqHnxAJn0BKvguHDwzZbqegm0B1jxALjhiov2DaeHbuLwB % Lnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFf % euY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9 % q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqaba % WaaqaafaaakeaacqaH1oqzdaWgaaWcbaGaemiCaahabeaakiabg2da % 9iabeg8aYnaaCaaaleqabaGaeyOeI0IaeGymaedaaOWaaeWaaeaacq % WGJbWydaWgaaWcbaGaemiCaahabeaakiabgUcaRmaalyaabaGaeu4M % dWeabaGafmyBa0MbaiaaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacq % GHsislcqaIXaqmaaaaaa!4D6C! e_p = r ^{- 1} ( c_p + L \mathord

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.     

Shear flow rheological properties of vinylon- and glass-fiber reinforced polyethylene melts

Shear viscosity, shear stress and first normal-stress difference have been investigated for glass- and vinylon-fiber filled polyethylene melts over a wide range of shear rate by means of three kinds of instruments. The influence of fiber content and fiber properties on the rheological properties is discussed. The viscosity increases with increasing aspect ratio and fiber content, and the influence of these parameters on the flow properties is evident at low shear rates. The first normalstress difference increases more rapidly with increasing glass fiber content, especially at low shear stresses. The influence of vinylon fibers on the first normal stress-difference vs. shear-stress relationship is different from that of glass fibers.     

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     

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.     

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.     

Laminar and melt fracture flow of polymer melts through successive capillaries

Summary The flow behavior of a polystyrene and a polyethylene melt has been studied during extrusion through successive capillaries arranged to produce converging and diverging flow patterns through the two orifices. Laminar flow and flow instability regions were included in the experiments.The pressure drop at fixed mass flow rate in converging or diverging flow through the combined capillaries equalled the sum of the pressure drops in each die if the ratio of die diameters was 1.43 in this case. The combined pressure drop exceeded the sum of the individual pressure drops, however, during converging melt fracture flow through die combinations with higher contraction ratios. Die swell measurements through successive orifices indicate that extrudate diameter may be influenced by the contraction between the reservoir and capillary. This effect is particularly noticeable at lower shear rates.

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Zusammenfassung Das Fließverhalten einer Polystyrol- und einer Polyäthylen-Schmelze wurde bei der Extrusion durch zwei hintereinandergeschaltete Kapillaren untersucht. Diese ließen sich so anordnen, daß darin entweder konvergente oder aber divergente Strömungsformen erzeugt werden konnten.Der Druckabfall bei konstantgehaltenem Massenstrom war in den kombinierten Kapillaren sowohl bei konvergenter als auch bei divergenter Strömung gleich der Summe der Druckabfälle in den Einzelkapillaren, solange das Durchmesserverhältnis 1,43 blieb. Dagegen überstieg der Druckabfall in der kombinierten Anordnung die Summe der einzelnen Druckabfälle, wenn bei höheren Kontraktionsverhältnissen die konvergente Strömung Schmelzenbruch zeigte. Die Messung der Strangaufweitung (die swell) nach hintereinandergeschalteten Düsen zeigt, daß der Extrudatdurchmesser durch die Verengung zwischen Reservoir und Kapillare beeinflußt werden kann. Dieser Effekt tritt besonders ausgeprägt bei kleineren Schergeschwindigkeiten auf.

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With 4 figures and 1 table     

Dynamic flow properties of vinylon fibre and glass fiber reinforced polyethylene melts

An experimental study of the dynamic shear flow properties of polyethylene melts filled with glass fibers and vinylon fibers was carried out and comparison with the steady shear flow properties was made. The effects of loading level and the characteristics of the fibers on the rheological properties of the fiber-filled systems is discussed. The rigidity and complex viscosity of the fiber-filled systems is sensitive not only to the quantity of fibers but also to their length, distribution and properties. The Cox-Merz empirical law for complex viscosity and steady shear viscosity, and Roscoe's empirical relation for estimating the normal-stress coefficients are both able to be applied to pure polymer melts but not to fiber-filled systems.     

Microstructure and rheology of polymer melts reinforced by long glass fibres: direct simulations

The two stages of classical modelling (formulation of the governing equations and finding their solutions) become particularly difficult, in fact practically unfeasable, in the case of concentrated suspensions of long fibres in polymer melts. In such situations it may be useful to explore a completely different route. One such route is offered by the idea of direct simulations. On this route the model is formulated by specifying the fibre–solvent and the fibre–fibre interactions. The passage from the single-fibre point of view to the macroscopic view on which rheological properties and the microstructure are observed is made by first calculating trajectories of all fibres in a model-system composed of several hundreds of fibres and making appropriate averages. The preliminary results of such simulations are in a good qualitative agreement with experimental observations for a polypropylene melt reinforced by long glass fibres. For examples, dilution and migration of fibres away from die walls are predicted; it is also observed that in concentrated situations the fibres form bundles which could in the entry flow generate large energy dissipation and oscillations.     

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.