Two-dimensional numerical analysis of non-isothermal melt spinning with and without phase transition

A model and simulation method are developed for two-dimensional non-isothermal melt spinning of a visco elastic melt. The visco elastic stress is evaluated from a non-isothermal Giesekus constitutive equation developed by application of the pseudo-time method to the isothermal form of the model [J. Non-Newt. Fluid Mech. (2001)]. The crystallization kinetics is described with the model proposed by Nakamura et al. [J. Appl. Polym. Sci. 17 (1973) 1031], whereas the crystallization rate, which is a function of both temperature and molecular orientation, is evaluated according to the equation proposed by Ziabicki [Fundamentals of Fiber Formation, Wiley, New York, 1976]. The set of non-linear governing equations is solved by using the DEVSS-G/SUPG finite element method. Melt spinning is simulated for two different polymers: amorphous polystyrene and fast-crystallizing Nylon-6,6. The analysis demonstrates that although the kinematics in the thread-line are approximately one-dimensional, the radially non-uniform thermal history, caused by the leading order variation of the temperature gradient ∂T/∂r, gives rise to radially non-uniform visco elastic stresses. This stress gradient results in radially non-uniform molecular orientation and a strong radial variation in crystallinity for Nylon-6,6. The radially non-uniform stress profiles obtained from the simulations are in good agreement with experimental results for melt spinning of polystyrene. Simulations of Nylon-6,6 show that the thermally-induced crystallization depends strongly on the choice of the Avrami index n, and a sharp increase in crystallinity due to stress-induced crystallization is predicted only when the molecules are highly oriented in the drawing direction at high drawing speeds. The significant influences of visco elasticity, air drag, and operating conditions on non-isothermal melt spinning dynamics also are predicted.     

Likelihood and expected-time statistics of monodomain attractors in sheared discotic and rod-like nematic polymers

Employing a mesoscopic Doi tensor model, we develop transient statistical properties of sheared nematic polymer monodomains consistent with typical experimental protocols. Our goal is to convey to the experimentalist a list of expected outcomes, based not only on properties of the nematic liquid and imposed flow rate, but also on the timescale of the experiment and variability in the initial conditions. Step 1 is deterministic: we solve the model equations completely, then compile the flow-phase diagram of all monodomain attractors and phase transitions versus nematic concentration and Peclet number (shear rate normalized by molecular relaxation rate). Step 2 is to overlay on the phase diagram a statistical diagnostic of the expected time, t_A, to reach a small neighborhood of every attractor A. The statistics are taken over the arbitrary quiescent director angle on the sphere, modeling experiments that begin from rest. Step 3 is to explore parameter regimes with multiple attractors, where we statistically determine the likelihood of convergence to each attractor. These statistical properties are critical for any application of theoretical models to the interpretation of experimental data. If t_A is longer than the timescale of the experiment, attractor A is never fully resonated and the relevant stress and scattering predictions are those of the transients, not the attractor. In bi-stable and tri-stable parameter regimes, which are typical of nematic polymers, a distribution of monodomains of each type will populate the sample, so experimental data must be compared with weighted averages based on the likelihood of each attractor (see Grosso et al (2003) Phys Rev Lett 90:098304). The final step is to give statistics of shear stress and normal stress differences during the approach to each attractor type, as well as typical paths of the major director that are contrasted with the results of Van Horn et al (Rheol Acta (2003) 42(6):585–589) with Leslie-Ericksen theory.     

1H NMR spectroscopy of polymers under shear and extensional flow

We discuss the potential insights gained from ¹H nuclear magnetic resonance (NMR) spectroscopy experiments on polymeric systems under both shear and extension, and we show in particular that ¹H spin-spin relaxation is sensitive to both molecular conformation and to molecular interactions. Rheo-NMR ¹H spectroscopy studies on semi-dilute solutions of polyacrylamide demonstrate that the chain protons exhibit a marked T ₂ reduction under shear and that the recovery on shear cessation is indicative of slow reorganisational dynamics. Studies of the wheat flour protein, gluten, indicate marked spectroscopic changes in the vicinity of the amidic resonances associated with glutamine residues, an effect we attribute to the disruption of hydrogen bonding. Received: 23 June 1999/Accepted: 23 August 1999     

Correlation between the gel time and quiescent/quasi-quiescent crystallization onset time of poly(butene-1) as determined from rheological methods

A comparison between two quantities concerning quiescent crystallization of polymer melts, namely the quiescent crystallization onset time (t _on,q) and gel time (t _gel), is performed using rheological methods. It was found that t _gel as measured from the evolution of loss angle (tan δ) occurs slightly earlier than t _on,q, defined as the time required for the viscosity to reach twice its steady-state value. The change of viscosity with time was measured with Small Amplitude Oscillatory Shear (SAOS) method. Two alternative methods to measure t _on,q are studied: by continuous shear at a very small shear rate and by creep method. A very good agreement from all the three methods was achieved, indicating the robust nature of this quantity. Finally, the gel times for different PB-1 samples are considered. When the difference in crossover angular frequencies (ω _x ) of the samples is taken into account in performing t _gel experiments, it can be shown that the temperature dependence of t _gel for the samples are the same. However, for samples having high molecular weight t _gel measurements are problematic. In this case, t _on,q serves as an easier tool to compare the crystallization behaviour of different samples.     

Modeling flow induced crystallization in film casting of polypropylene

Data from iPP film casting experiments served as a basis to model the effect of flow on polymer crystallization kinetics. These data describe the temperature, width, velocity and crystallinity distributions along the drawing direction under conditions permitting crystallization along the draw length.In order to model the effect of flow on crystallization kinetics, a modification of a previously defined quiescent kinetic model was adopted. This modification consisted in using a higher melting temperature than in the original quiescent model. The reason for the modification was to account for an increase of crystallization temperature due to entropy decrease of the flowing melt. This entropy decrease was calculated from the molecular orientation on the basis of rubber elasticity theory applied to the entangled and elongated melt. The evolution of molecular orientation (elongation) during the film casting experiments was calculated using a non-linear dumbbell model which considers the relaxation time, obtained from normal stress difference and viscosity functions, to be a function of the deformation rate.The comparison between experimental distributions and model based crystallinity distributions was satisfactory.     

Transient shear rheology of carbon nanofiber/polystyrene melt composites

We characterize the transient shear rheology of polystyrene/carbon nanofiber composites. Our experimental measurements of the composites show increasing stress overshoot responses to transient shear as the carbon nanofiber concentration increases. We also find the steady state viscosity reached at long times during application of a constant shear rate increases with increasing carbon nanofiber concentration. Flow reversal experiments show the effects of nanofiber orientation and structural evolution on the composite's rheological response.We present a microstructurally based constitutive model where all but two parameters are determined by rheological characterization of the pure polymer and the shape and concentration of the nanoparticles. The Folgar-Tucker constant, C_I, is treated as a fitting parameter, while several definitions for the shape factors A, B, C and F are evaluated. We make note of the effects each parameter has on the model's predictions. We find that the constitutive model is in agreement with our experimentally measured transient shear rheology of the PS/CNF melt composites for the CNF concentrations and shear rates presented.     

Hall effects on the magnetohydrodynamic shear flow past an infinite porous flat plate subjected to uniform suction or blowing

An analysis is made of Hall effects on the steady shear flow of a viscous incompressible electrically conducting fluid past an infinite porous plate in the presence of a uniform transverse magnetic field. It is shown that for suction at the plate, steady shear flow solution exists only when S²<Q, where S and Q are the suction and magnetic parameters, respectively. The primary flow velocity decreases with increase in Hall parameter m. But the cross-flow velocity first increases and then decreases with increase in m. Similar results are obtained for variation of the induced magnetic field with m. It is further found that for blowing at the plate, steady shear flow solution exists only when , where S₁ is the blowing parameter.     

Steady and transient shear flows of polystyrene solutions I: Concentration and molecular weight dependence of non-dimensional viscometric functions

Start up from rest and relaxation from steady shear flow experiments have been performed on monodisperse polystyrene solutions with molecular weight ranging from 1.3 × 10⁵ to 1.6 × 10⁶ and concentration c ranging from 5% to 40%. A method of reduced variables based on the use of a characteristic time τ_w is proposed. τ_w is defined as the product of zero shear viscosity with the steady state elastic compliance.Reduced steady and transient viscometric functions so obtained depend on the ratio M/M_e (where M_e is the entanglement molecular weight). Limiting forms are obtained when M/M_e ? 18. In steady flow, a simple correlation is found between shear and normal stresses.In stress relaxation experiments, independent of shear rate, the long-time behaviour can be characterised by a single relaxation time τ₁, which is identical for shear and normal stresses. τ₁ can be simply related to the zero shear rate viscosity and the limiting elastic compliance.     

Thermally and flow induced crystallization of polymers at low shear rate

In semi-crystalline thermoplastic products, final properties are strongly dependent on the thermo-mechanical history experienced by the polymer melt during processing. More precisely, structural heterogeneities such as rigidity gradients and shrinkage anisotropy are directly related to the crystalline microstructure. Therefore, accurate prediction of part properties by a processing computer simulation code requires the implementation of an appropriate crystallization kinetics model, including both the effects of thermally and flow induced structure development. One issue is the necessity to improve the modeling of shear/extensional experimental data by relating the crystallization accelerating factors to an easily accessible material related variable. Several authors modeled the effect of the flow on the crystallization kinetics by using the isokinetic approach of Nakamura. Often, the resulting kinetic equations of these models account only for the evolution of the crystallinity fraction α leaving the influence of crystalline morphology aside. We may quote the work of Guo and Narh [1], which connects the flow influence on the crystallization rate to the increase in the thermodynamic melting temperature in the Nakamura model. In 2005, R.I. Tanner presented a comparison of some models describing the polymer crystallization at low shear deformation rates under isothermal conditions. Based on Tanner's study, we developed a model of crystallization at low shearing, applied to non-isothermal flows, using only macroscopic measurable parameters. The key features of the concentrated suspension theory were used to characterize the effect of crystallization on the viscosity. In addition, we assumed that the flow generates additional crystallization nuclei via a parameter which combines the deformation and the deformation rate. The concept of germination-growth is introduced using the fundamentals of the Avrami–Kolmogorov theory, coupled with a modified Schneider's approach. The model is applied to a polypropylene, in a cooled Couette flow configuration. The results show the enhancement of the crystallization kinetics due to the shearing. The definition of global parameters simplifies the type and the number of experiments needed for the model parameter identification. The use of Schneider's approach leads to a new way of discriminating the relative roles of the flow and the temperature on the crystallization phenomenon. The competition between the two driving causes is presented and discussed: at low cooling rate or at high temperature, the shearing effect predominates. Other interesting results show the size distribution of the spherulites as well as the volume proportion for each crystalline size in the polymer.     

Rheological scaling and modeling of shear-enhanced crystallization rate of polypropylene

Shear-induced isothermal crystallization of a commercial isotactic polypropylene (iPP) has been investigated by using a rotational rheometer at the steady shear rates ranging from 0.00012 s⁻¹ to 1 s⁻¹, and the temperatures from 135 to 145 °C. Two time scales can be utilized to characterize the crystallization rates: one is the level-upturn onset time of the viscosity; another is that of the normal force. Plotting the onset times against the corresponding onset strain, a common critical value for all the undercooling temperatures can be identified, below which the shear flows have no significant effect on the crystallization rates. Furthermore, we propose a concept of dimensionless onset work; this parameter can make the normalized onset times approximately temperature-invariant in the range of our experiment. Our modeling of the quiescent crystallization is based on the nucleation theory of Ziabicki; the results indicate two-dimensional crystallite growth on pre-existing nuclei. The shear enhanced crystallization is modeled by estimating the excess free energy induced by the flow, and using the rheological model recently proposed by Marrucci, in which the required relaxation times are derived from our rheological measurements. The results imply that the crystallization under the present low shear rates is still two-dimensional crystallite growth on pre-existing nuclei, thus supporting the athermal nucleation theory proposed by Janeschitz-Kriegl. Compared with the experimental data, the modeling is only partially successful. Further improvements encompassing the effects of shear flows on the non-linear increase of the number density of athermal nuclei and on the acceleration of polymer chain disentanglement are needed.     

Modeling the microstructural evolution of metallic polycrystalline materials under localization conditions

In general, the shear localization process involves initiation and growth. Initiation is expected to be a stochastic process in material space where anisotropy in the elastic-plastic behavior of single crystals and inter-crystalline interactions serve to form natural perturbations to the material's local stability. A hat-shaped sample geometry was used to study shear localization growth. It is an axi-symmetric sample with an upper “hat” portion and a lower “brim” portion with the shear zone located between the hat and brim. The shear zone length was 870-890 μm with deformation imposed through a Split-Hopkinson Pressure Bar system at maximum top-to-bottom velocity in the range of 8-25 m/s. The deformation behavior of tantalum tophat samples is modeled through direct polycrystal simulations. An embedded Voronoï-tessellated two-dimensional microstructure is used to represent the material within the shear zone of the sample. A thermo-mechanically coupled elasto-viscoplastic single crystal model is presented and used to represent the response of the grains within the aggregate shear zone. In the shoulder regions away from the shear zone where strain levels remain on the order of 0.05, the material is represented by an isotropic J₂ flow theory based upon the elasto-viscoplastic Mechanical Threshold Stress (MTS) model for flow strength. The top surface stress versus displacement results were compared to those of the experiments and over-all the simulated stress magnitude is over-predicted. It is believed that the reason for this is that the simulations are two-dimensional. A region within the numerical shear zone was isolated for statistical examination. The vonMises stress state within this isolated shear zone region suggests an approximate normal distribution with a factor of two difference between the minimum and maximum points in the distribution. The equivalent plastic strain distribution within this same region has values ranging between 0.4 and 1.5 and is not symmetric. Other material state distributions are also given. The crystallographic texture within this isolated shear zone is also compared to the experimental texture and found to match reasonably well considering the simulations are two-dimensional.     

Numerical study of chain conformation on shear banding using diffusive Rolie-Poly model

Shear-banding phenomenon in the entangled polymer systems was investigated in a planar Couette cell with the diffusive Rolie-Poly (ROuse LInear Entangled POLYmers) model, a single-mode constitutive model derived from a tube-based molecular theory. The steady-state shear stress ?? _s was constant in the shear gradient direction while the local shear rate changed abruptly, i.e., split into the bands. We focused on the molecular conformation (also calculated from the Rolie-Poly model) around the band boundary. A band was found also for the conformation, but its boundary was much broader than that for the shear rate. Correspondingly, the first normal stress difference (N ₁) gradually changed in this diffuse boundary of the conformational bands (this change of N ₁ was compensated by a change of the local pressure). For both shear rate and conformation, the boundary widths were quite insensitive to the macroscopic shear rate but changed with various parameters such as the diffusion constant and the relaxation times (the reptation and the Rouse times). The broadness of the conformational banding, associated by the gradual change of N ₁, was attributed to competition between the molecular diffusion (in the shear gradient direction) and the conformational relaxation under a constraint of constant ?? _s.     

Effect of gravity on the mechanical properties of lunar regolith tested using a low gravity simulation device

Simulations of the bearing capacity and shear strength of regolith under Earth’s gravity produce different results from those under low gravity. A low-gravity simulation device was developed in this study, and an internal stress model of regolith simulant was established to correct the errors. The model revealed additional force on both shear plane in the shear test and the press plate area in the pressure–sinkage test. The sinkage and shear test results showed that low gravity decreased the deformable index n, frictional modulus k_φ and cohesion c, whereas there were no obvious changes to the cohesive modulus k_c and internal friction angle φ. The sinkage generally increased as the gravity decreased under a consistent normal load larger than 50 N, but when the wheel load was lower than 50 N, the sinkage of the TYII-1 simulant was larger under 1 G than 1/6 G. Gravity had little effect on the shear strength of the regolith. However, the tractive thrust of the TYII-1 simulant was lower under 1/6 G than 1 G. The smaller difference was due to differences in the way the soils responded to changes in the gravity level for the TYII-2 simulant.     

Structure development during crystallization of polycaprolactone

In this paper, the quiescent crystallization of polycaprolactone (PCL) melts is studied by rheological measurements coupled to calorimetry and optical microscopy. Based on a comparison between the different techniques, we find that the increase in viscoelastic properties during crystallization starts only when a relatively high degree of crystallinity is reached, which corresponds to a much developed crystalline microstructure. Like other semicrystalline thermoplastic polymers, the crystallization of PCL can be seen as a gelation process. In this case, however, we find a peculiar critical gel behavior, as the liquid-to-solid transition takes place at a very high (~20%) relative crystallinity, and this value is independent of temperature. These facts, and the comparison with optical microscopy observations, suggest that the microstructure at the gel point is controlled by the interactions between the growing crystallites. The gel time (from rheometry) and the half-crystallization time [from differential scanning calorimetry (DSC)] both show an Arrhenius-like behavior and have the same pseudoactivation energy. A practical implication of this parallel behavior of t _gel and t _0.5 is that the rheological measurements can be used to extend to higher temperatures the study of crystallization kinetics where DSC is not sufficiently sensitive.This paper was presented at the second Annual European Rheology Conference (AERC) held in Grenoble, France, 21–23 April 2005.     

Asymptotic Behaviour of a Pile-Up of Infinite Walls of Edge Dislocations

We consider a system of parallel straight edge dislocations and we analyse its asymptotic behaviour in the limit of many dislocations. The dislocations are represented by points in a plane, and they are arranged in vertical walls; each wall is free to move in the horizontal direction. The system is described by a discrete energy depending on the one-dimensional horizontal positions x _i > 0 of the n walls; the energy contains contributions from repulsive pairwise interactions between all walls, a global shear stress forcing the walls to the left, and a pinned wall at x = 0 that prevents the walls from leaving through the left boundary. We study the behaviour of the energy as the number of walls, n, tends to infinity, and characterise this behaviour in terms of Γ-convergence. There are five different cases, depending on the asymptotic behaviour of the single dimensionless parameter β _n , corresponding to ${\beta_n \ll 1/n, 1/n \ll \beta_n \ll 1}$ , and ${\beta_n \gg 1}$ , and the two critical regimes β _n ~ 1/n and β _n ~ 1. As a consequence we obtain characterisations of the limiting behaviour of stationary states in each of these five regimes. The results shed new light on the open problem of upscaling large numbers of dislocations. We show how various existing upscaled models arise as special cases of the theorems of this paper. The wide variety of behaviour suggests that upscaled models should incorporate more information than just dislocation densities. This additional information is encoded in the limit of the dimensionless parameter β _n .     

Extensional-flow-induced crystallization of isotactic polypropylene

A filament stretching extensional rheometer with a custom-built oven was used to investigate the effect of uniaxial flow on the crystallization of polypropylene. Prior to stretching, samples were heated to a temperature well above the melt temperature to erase their thermal and mechanical histories and the Janeschitz-Kriegl protocol was applied. The samples were stretched at extension rates in the range of 0.01 s^-1 ￡ [(e)\dot] ￡ 0.75 s^-10.01\,\mbox{s}^{-1}\le \dot{{\varepsilon }}\le 0.75\,{\rm s}^{-1} to a final strain of ε = 3.0. After stretching, the samples were allowed to crystallize isothermally. Differential scanning calorimetry was applied to the crystallized samples to measure the degree of crystallinity. The results showed that a minimum extension rate is required for an increase in percent crystallization to occur and that there is an extension rate for which percent crystallization is maximized. No increase in crystallization was observed for extension rates below a critical extension rate corresponding to a Weissenberg number of approximately Wi = 1. Below this Weissenberg number, the flow is not strong enough to align the contour path of the polymer chains within the melt and as a result there is no change in the final percent crystallization from the quiescent state. Beyond this critical extension rate, the percent crystallization was observed to increase to a maximum, which was 18% greater than the quiescent case, before decaying again at higher extension rates. The increase in crystallinity is likely due to flow-induced orientation and alignment of contour path of the polymer chains in the flow direction. Polarized light microscopy verified an increase in number of spherulites and a decrease in spherulite size with increasing extension rate. In addition, small angle X-ray scattering showed a 7% decrease in inter-lamellar spacing at the transition to flow-induced crystallization. Although an increase in strain resulted in a slight increase in percent crystallization, no significant trends were observed. Crystallization kinetics were examined as a function of extension rate by observing the time required for molten samples to crystallize under uniaxial flow. The crystallization time was defined as the time at which a sudden increase in the transient force measurement was observed. The crystallization time was found to decrease as one over the extension rate, even for extension rates where no increase in percent crystallization was observed. As a result, the onset of extensional-flow-induced crystallization was found to occur at a constant value of strain equal to ε _c  = 5.8.     

Theoretical confirmation of the crystallization of a compound alloy using the AHP crystal growth method

The crystallization of In_xGa_1−xSb for $x=0.06$ by the AHP-method (Axial Heat flux, close to the Phase interface) is considered. Heavy indium is rejected during crystal growth close to the interface. Indium significantly decreases the crystallization temperature, and has an influence on the melt convection. The AHP-heater serves as a partition; due to this partition a small well-mixed liquid zone with high In concentration exists and causes a stable crystal growth with high composition after the crucible is moved down. The grown crystal is very homogeneous. Numerical modelling has also been performed, using finite difference schemes. To cite this article: M. Marchenko, C. R. Mecanique 335 (2007).     

Shear-induced crystallization of PB-1 up to processing-relevant shear rates

Two experimental methods to study shear-induced crystallization of poly(butene-1) (PB-1) in the high shear rate region are presented: one using a concentric cylinder rheometer and the other a capillary rheometer equipped with a cylindrical die. The crystallization onset time (t _on) is used as the parameter to monitor crystallization progress through the output signal from each device. By combining the new data with the results from a previous paper (2005) in which a plate–plate rheometer was used, onset time data covering a shear rate range from 10^-4 to 500 s^-1 at temperatures 99–107°C are obtained. In this range, a decrease in onset time spanning five decades is observed. The onset times obtained from the capillary rheometer are larger compared to those from the other two methods, which can be explained from the different type of flow. The data also confirm the procedure to construct a temperature-invariant curve, which can be extended to high shear rates for three PB-1 samples having different molecular weight distributions. The slope of the fitted curve for all three cases is −1, which suggests that a critical value is required for shear-induced crystallization. The morphology of the formed crystals (spherulitic or rod-like) depends on the molecular weight, but this does not affect the validity of the T-invariant curve. Above the melting point, it is shown that the amount of long chains influences the temperature limit where shear-induced crystallization can still take place.The paper was presented at AERC 2005, Grenoble, with the title “Shear-Induced Crystallization of Polybutene-1 Covering a Wide Shear Rate Range.”An erratum to this article can be found at     

Conoscopic observations of shear-induced rotations in nematic liquid crystals

Orientational changes in monodomains of flow-aligning liquid crystals, 4-n-pentyl-4′-cyanobiphenyl and N-(4-methoxybenzylidene)-4-butylaniline, were studied during shear and recovery in a linear shearing device fitted to an optical microscope. Planar alignment (director in the shear plane) allows the study of twist effects and was generated by strong planar anchoring at the walls with orientations in a range of 0–90° with the shear direction. While being held back by the anchoring walls, shear caused the bulk director to rotate towards a steady-state alignment angle in the shear direction (Leslie angle θ_L). The transient director rotation was observed with conoscopy. It was found that increasing the initial alignment towards the vorticity direction increased the measured θ_L. Upon stopping the flow, the bulk director returned to its initial state. With initial alignment orientation changing from parallel to perpendicular to the flow direction, the rate of the twist-driven recovery process increases. This rate increase is not seen in the splay-driven recovery which is constant and consistently faster than twist-driven recovery at all orientations studied. Received: 10 December 1998/Accepted: 7 June 1999     

Quiescent and shear-induced crystallization of polyprophylenes

In this paper, the effect of shear on the flow-induced crystallization (FIC) of several polypropylenes of various macrostructures was studied using rheometry combined with polarized microscopy. Generally, an increase in strain and strain rate or decrease of temperature is found to decrease the thermodynamic barrier for crystal formation and thus enhancing crystallization kinetics at temperatures between the melting and crystallization points. Secondly, popular models based on suspension theory which are used to relate the degree of crystallinity to normalized rheological functions (such as viscosity) are validated experimentally. For this purpose, the space filling of crystals in the polarized micrographs determined from image processing was plotted as a function of normalized viscosity under various shear rates. It is found that the constant(s) of various suspension models should be dependent on the flow parameters in order for the suspension models to describe the effect of shear on FIC, particularly at higher shear rates.