Institute of Fundamental Technological Research

Kinetic model of polymorphic crystallization of uniaxially oriented amorphous poly(L-lactide) under isothermal conditions is formulated basing on the Hoffman-Lauritzen theory. Entire ranges of the crystallization temperatures and amorphous orientation factor are considered. The inverse transformation half-times of the amorphous phase to the individual polymorphs, between the unstable α’ and stable α polymorphs, as well as of the overall crystallization are discussed. The kinetic effects are assigned to the decrease in the amorphous phase entropy caused by deformation and orientation of the flexible chain macromolecules. The model predicts that crystallization to the stable α form is controlled by the rate of transformation of the amorphous phase to α’ Concentration of heterogeneous nuclei typical for commercial polymer is assumed and in this case the role of homogeneous nucleation in the transformation kinetics is negligible. The role of homogeneous ucleation at different hypothetical concentrations of heterogeneous nuclei is estimated vs. the Hermans amorphous orientation factor and crystallization temperature. Ranges of domination of heterogeneous and homogeneous nucleation are predicted as dependent on the heterogeneous nuclei concentration. The increase of the overall oriented crystallization rate is predicted for the entire crystallization temperature range as resulting from the increase of the amorphous-to-α′ transformation rate at increasing amorphous orientation. The model provides a view on the mechanisms for possible control of the development of α’/α composition by adjusting the crystallization time vs. orientation and temperature. composition by adjusting the crystallization time vs. orientation and temperature.

Modeling and simulation, Oriented crystallization

Development of polymorphic composition during isothermal crystallization of unstressed amorphous PLLA is considered in the introduced model. The model analysis of the overall crystallization half-time indicates involvement of both α′ and α polymorphs in the crystallization kinetics at substantial impact of the α′ form strongly dependent on temperature. Above 120 ◦C effective creation of α form takes place at much slower amorphous-to-α transformation and proceeds via α′ -to-α transformation which occurs through orders of magnitude faster direct solid-to-solid transition mechanism. The model allows to predict development of indi-vidual α′ and α phases during isothermal crystallization of PLLA basing only on the overall temperature-dependent crystallization half-time, t1/2. The predictions are in good agreement with the experimental data reported by different authors in the literature. The model indicates that the transformation degrees to the individual α′ and α polymorphs vs. crystallization temperature, as well as the overall transformation degree are independent of the nuclei concentration in terms of the multiples of the crystallization half-time, because the concentration of nuclei is directly involved in t1/2.

modeling of crystallization,crystallization kinetics,polymorphic crystallization of PLLA

Closed-form analytical formulas describing kinetics of oriented crystallization under constant or variable amorphous orientation and isothermal or non-isothermal conditions are derived, valid in the whole range of orientation. Master relation for the deformation free energy vs. orientation factor, or tensile stress, is derived accounting for non-linear effects of finite chain extensibility. The Avrami-Evans model is expanded to account for the effects of orientation in thermodynamic driving force of nucleation and crystal growth. Involvement of predetermined and spontaneous nucleation varies strongly with the orientation and leads to domination of spontaneous nucleation at high orientations. Crystallization half-time involving separated or coexisting predetermined and spontaneous nucleation is discussed. A formula predicting equal contribution of both nucleation modes vs. orientation factor and temperature is derived and ranges of domination of the modes are discussed. Example computations illustrate the model predictions for an example polymer (PLLA) and are in good agreement with the experimental results.

amorphous orientation, oriented crystallization kinetics, nucleation rate

The thermoplastic polymers present amorphous or semi-crystalline structures which are very important factors in describing volumetric shrinkage. The thermoplastic materials are commonly used for production of daily life products, industrial or as the prototypes. Different techniques of manufacturing polymer structures are considered like: injection molding, extrusion, milling, additive manufacturing (AM). AM is a very fast developing field in the manufacturing and research. Unfortunately, components or prototypes made using the thermoplastic semi-crystalline materials in 3D techniques have quite low mechanical strength compared to the parts made by injection molding processes. It is caused by porosity obtained during the processing, as well as by fraction of crystallinity in the volume of the components. Additionally, the volumetric shrinkage is hard to predict without knowledge of its origin. Therefore, it is necessary to consider crystallization kinetics and the melting of the analysed materials. The investigations presented in this work concern the crystallization and melting model to be implemented in the finite element (FE) analyses. With use of the model, one can predict development of the structure during the real processes and, in the future, to control the warpage of the manufactured components.

additive manufacturing, Avrami model, crystallization, differential scanning calorimetry (DSC), glass transition temperature, Hoffman-Lauritzen theory, melting, PA2200, van Krevelen empirical model

An analytical formula is derived for the oriented crystallization coefficient governing kinetics of oriented crystallization under uniaxial amorphous orientation in the entire temperature range. A series expansion approach is applied to the free energy of crystallization in the Hoffman-Lauritzen kinetic model of crystallization at accounting for the entropy of orientation of the amorphous chains. The series expansion coefficients are calculated for systems of Gaussian chains in linear stress-orientation range. Oriented crystallization rate functions are determined basing on the 'proportional expansion' approach proposed by Ziabicki in the steady-state limit. Crystallization kinetics controlled by separate predetermined and sporadic primary nucleation is considered, as well as the kinetics involving both nucleation mechanisms potentially present in oriented systems. The involvement of sporadic nucleation in the transformation kinetics is predicted to increase with increasing amorphous orientation. Example computations illustrate the dependence of the calculated functions on temperature and amorphous orientation, as well as qualitative agreement of the calculations with experimental results.

modeling and simulation, kinetics of oriented crystallization, amorphous orientation, sporadic nucleation, predetermined nucleation

Closed-form analytical formulas are proposed for non-linear stress-orientation relation valid in the entire range of tensile stresses, intermediate and high, basing on the inverse Langevin chain statistics. Tensile force and orientation characteristics for single, flexible chain macromolecule are reconsidered in a closed-form Padè approximation of the inverse Langevin function in the entire range of chain extensions, as well as the dynamics of molecular deformation and orientation for systems of chains subjected to uniaxial elongational flow. Average stress and orientation tensors, not collinear in the non-linear range, as well as the tensile stress and axial orientation factor of the chain segments in the system are calculated. For the intermediate tensile stresses, the closed-form formula is derived in the first non-Gaussian term approximation of the inverse Langevin function with the assumption of dominating longitudinal elongation of the chains. The high-stress non-linearity formula is derived with the Padè and Peterlin modulus approximations and aligned chain end-to-end vectors by the flow. Both formulas are validated by exact numerical calculations without the assumptions and influence of the approximations is estimated. Ranges of applicability of the formulas are illustrated and examples of their application are presented.

Molecular orientation, Tensile stress, Non-linear stress-orientation behavior, Elongational flow, Langevin chain statistics

A kinetic model of primary homogeneous non-isothermal crystal nucleation with transient and athermal effects is developed. For comparison, steady-state and transient isothermal nucleation rates are considered. Kinetic equation for the development of cluster size distribution provides the basis for the model. Transient effects are characterized by the longest relaxation time which increases with temperature at low and moderate undercooling. In isothermal conditions, nucleation rate is controlled by thermal mechanism; in non-isothermal conditions, there appears also athermal mechanism. Closed-form analytical formula for the development of transient cluster size distribution in single-relaxation-time approximation is derived for non-isothermal processes, as well as thermal and athermal nucleation rates and total number of nuclei produced in a cooling or heating run. The transient term contributes to isothermal nucleation kinetics the more the higher is temperature. Under non-isothermal conditions, the relaxation time contributes to the nucleation kinetics by the product with the cooling/heating rate. Considerable transient effects should be expected for the relaxation times as long as 102–105 s. Contribution of thermal nucleation to the concentration of nuclei is inversely proportional to the temperature rate, while the contribution of athermal nucleation depends on the temperature interval of cooling or heating. Our kinetic model indicates similarities in the nucleation mechanisms in polymers and metals undergoing crystallization. Example computations are presented for molten indium and a linear polymer—polyhydroxybutyrate (PHB). A low-temperature limit is predicted for the nucleation mechanism in PHB, while for indium the mechanism is active in the entire temperature range.

nucleation kinetics, non-isothermal nucleation, transient nucleation, athermal nucleation, nucleation relaxation time

Kinetic model of development of lamellar thickness distribution and average lamellar thickness during melt crystallization is proposed. The model bases on the Avrami–Evans and Hoffman–Lauritzen approaches to the crystallization and crystal growth kinetics and involves a thermal thickening model proposed in this paper which accounts for two processes – lamellar doubling transformation and logarithmic thickening of the doubled crystals. In the thickening model, continuity equation for the lamellar thickness distribution is proposed which includes a term responsible for fast creation of doubled crystals during a doubling time introduced as phenomenological parameter and a term which accounts for slow, logarithmic with time thickening subsequent to the doubling transformation. The model concerns melt crystallization in systems where the undercooling in the crystallization and melting processes refers to the same thermodynamic point – the thermodynamic equilibrium melting temperature.

Analytical formulae for time-dependent lamellar thickness distribution vs. crystallization time are obtained by solving the continuity equation for the case of isothermal crystallization, as well as for the development of the average lamellar thickness of the doubled and un-doubled crystals during the crystallization time.

The model predictions are illustrated by example computations for isothermal crystallization of PE grades characterized by high differences in the crystallization half-time resulting in high differences in the development of lamellar thickness distribution. Relative contributions of doubled and un-doubled crystals are predicted during the conversion and influence of the doubling time and logarithmic thickening parameters is discussed.

Crystallization kinetics, Nucleation, Crystal growth, Thickness distribution, Lamellar doubling, Thermal thickening

An air-drawing model of poly-l-lactide melt under a supersonic air jet in the Laval nozzle is presented. The aerodynamic fields are computed using the k–ω model. The pneumatic process is considered based on the mathematical model of melt spinning in single-, thin-filament approximation. Simultaneous acceleration of the air and the melt within the nozzle leads to fast attenuation of the filament. Air velocity dominates velocity of the filament and results in continual air-drawing on the entire spinning line. Oriented crystallization and nonlinear viscoelasticity effects under fast uniaxial elongation of the polymer melt are considered. The filament velocity at the collector increases significantly with increasing air compression, from the values typical for high-speed melt spinning up to values by two folds higher. The increase in filament velocity is limited by the effects of online oriented crystallization at higher air compressions. Influence of the inlet air compression, melt extrusion temperature and weight-average molecular weight on the axial profiles of the melt spinning functions is discussed, as well as on the development of amorphous orientation and online oriented crystallization.

Laval nozzle, Pneumatic melt spinning, Super-thin fibers, Oriented crystallization, Computer simulation, Polylactides

Melt spinning of the fibers by supersonic air jet in the Laval nozzle is a novel, efficient and energy saving method of formation of super-thin fibers. In the process, polymer melt is extruded from a row of orifices and fast drawn by the pneumatic forces. In the modelling, air velocity, temperature and pressure distributions are computed from the k-! aerodynamic model. Computations of the polymer air-drawing dynamics are based on the mathematical model of melt spinning in a single-, thin-filament approximation and Phan-Thien/Tanner non-linear viscoelasticity of the polymer melt. Axial profiles of the polymer velocity, temperature, tensile stress and rheological extra-pressure are computed. Influence of the Laval nozzle geometry, initial air compression, an initial melt temperature, a polymer mass output and the diameter of the melt extrusion die is discussed. The role of the polymer molecular weight, melt viscosity and relaxation time is considered. Example computations show the influence of important processing and material parameters. In the supersonic process, a high negative internal extra-pressure is predicted in the polymer melt under high elongation rates which may lead to cavitation and longitudinal burst splitting of the filament into a high number of sub-filaments. A hypothetical number of sub-filaments at the splitting is estimated from an energetic criterion. The diameter of the sub-filaments may reach the range of nano-fibers. A substantial influence of the Laval nozzle geometry is also predicted.

air-drawing, Laval nozzle, pneumatic melt spinning, superthin fibers

Computer simulation of the pneumatic processes of fiber formation from the polymer melts is discussed. The dynamics of air-drawing of thin polymer streams in supersonic air jets formed in the Laval nozzle is presented versus the melt blowing process. In the Laval nozzle process, the air flow takes place with high Reynolds number and the k–omega model is used which considers kinetic energy of the air flow and the specific dissipation rate of the kinetic energy. For melt blowing, the air fields are simulated with the use of the k–epsilon turbulent model. The air velocity, temperature, and pressure distributions along the centerline of the air jets are considered in the modeling of both pneumatic processes. The air fields are predetermined at the absence of the polymer streams for several air compression values in the Laval nozzle inlet and several initial air velocities in the melt blowing process. Each polymer stream in a usual configuration of a single row of the filaments in the process is considered as non-interacting aerodynamically with other streams, and the air jet is assumed to be undisturbed by the polymer streams. Airdrawing of the polymer filaments is simulated as controlled by the distribution of air velocity, temperature, and pressure on the air jet centerline with the use of a stationary model of melt spinning in a single-, thin-filament approximation. Effects of non-linear viscoelasticity of the polymer melt subjected to fast uniaxial elongation are accounted for in the modeling. Strong influence of the air jet velocity, the melt viscosity which controls response of the polymer melt on the air-drawing forces, and the dieto-collector distance has been predicted. Influence of initial air temperature, geometry of the air die, initial velocity and temperature of the melt, extrusion orifice diameter can be also predicted from the model. The example computations concern air-drawing of isotactic polypropylene with the use of the Laval nozzle are compared with the predictions for the melt blowing process.

computer modeling, fibers, melt blowing, supersonic melt spinning, polypropylene

The dynamics of stationary air drawing in the melt blowing of nonwovens were determined on the basis of a single-filament model in a thin-filament approximation that accounts for polymer viscoelasticity, heat of viscous friction in the polymer bulk, and surface energy. Predetermined distributions of the air velocity and temperature along the melt blowing axis were assumed. Axial profiles of the polymer velocity, temperature, elongation rate, filament diameter, tensile stress, and extra-pressure were computed for the melt blowing of isotactic polypropylene. The effects of the air-jet velocity, die-to-collector distance, and polymer molecular weight are discussed. We predicted that the filament attenuation and velocity at the collector located in the air-drawing zone would increase with increasing die-to-collector distance. The air-drawing zone was shorter for higher air velocities and lower molecular weights. No online crystallization was predicted before the achievement of the collector, and melt bonding of the filament in the web should have occurred during cooling on the collector, accompanied by spherulitic crystallization. Significant online extrapressure in the filament was predicted in the case of supersonic air jets as resulting from polymer viscoelasticity, which could have led to longitudinal splitting of the polymer into sub-filaments.

fibers, polymer melt, modeling of melt spinning, polypropylene

Dynamics of a novel pneumatic process of superthin fibres formation from polymer melts in supersonic air jets in the Laval nozzle is studied using computer simulation. The approach bases on the mathematical k-w models of air flow in the nozzle and air drawing of polymer filaments in the coaxial air jet. The aerodynamic fields can be considered as undisturbed by presence of a single row of thin polymer filaments and predetermined air conditions are used in the modeling. The air fields are simulated for several values of the air compressions in the nozzle inlet and two nozzle geometries. Driving force of the Laval nozzle process results form air drag forces acting onto the filament surface. Mathematical model of stationary melt spinning in single-, thin-filament approximation is applied with the effects of non-linear viscoelasticity of the polymer melt accounted for. The model allows also to discuss non-linear stress-optical relationship reflecting online molecular orientation, as well as online crystallization of the polymer filament if it occurs. Negative rheological extra-pressure in the air-drawn filament is predicted, as resulting from non-linear viscoelasticity of the polymer melt subjected to high elongation rates. The negative extra-pressure could lead to cavitation and longitudinal burst splitting of each filament into a high number of superthin sub-filaments. A hypothetical mean diameter of the sub-filaments is estimated from an energetic criterion. Example computations of the dynamic profiles of air drawing and discussion concern isotactic polypropylene (iPP) subjected to the Laval nozzle process.

melt spinning, polymer air drawing, Laval nozzle process, superthin fibres

Computer simulation of the pneumatic melt spinning of isotactic polypropylene based on the mathematical model of the process is presented. Two dynamic zones of the air jet-filament interactions along the melt blowing axis are predicted – a zone with a drawing activity of the air jets with aligned filaments in this zone, and a passive zone with the bending and coiling of the filaments. The diameter of the fibres and structure of the nonwoven should depend on the die-to-collector distance, and the zone in which the collector is located. Ranges of the zones are discussed as dependent on the initial velocity of the air jets, the melt extrusion temperature, and the molecular weight of the polymer. Axial profiles of the polymer velocity, diameter and temperature of the filament, the tensile force, tensile stress, and rheological pressure along the melt blowing axis are presented for a process with the collector located within the air-drawing zone at a fixed take-up distance. The dynamic profiles indicate a narrow axial range of air-drawing next to the spinneret.

melt blowing, air-drawing, nonwovens, polypropylene, modelling

A single-, thin-filament model for stationary meltblowing of nonwovens from isotactic polypropylene is proposed. The Phan-Thien, Tanner equation of viscoelasticity is used, as well as the effects of stress-induced crystallisation on polymer viscosity and relaxation time during the processing are accounted for. The predetermined air velocity, temperature and pressure fields are assumed which are computed for different initial air velocities, as well as a fixed initial temperature, and approximated along the melt blowing axis by analytical formulae. The model is more general and can be applied to melt blowing of nonwovens from other crystallising polymers and air fields. Axial profiles of polymer velocity, temperature, tensile stress, pressure, amorphous molecular orientation and the degree of crystallinity are computed using the model.

melt blowing of nonwovens, modeling of air drawing, dynamics of melt spinning, air jet dynamics

Crystallization is a structural transformation influencing dynamics of industrial polymer processing and controlling physical and mechanical properties. Adequate modle of the transformation kinetics is needed in computer modeling of industrial processes. Specific feature of polymeric materials is strong increase of the crystallization rate caused by deformation and orientation of polymer molecules. In melt processing, the chain deformation and orientation is controlled by flow deformation and cooling. In high-speed melt spinning, supersonic melt blowing, electro-spinning of fibers, the polymer fluid is often subjected to elongation rates of several hundred times per second and fast deformations do not allow for chain relaxation during the processing time. In case of crystallizing polymers, it results in fast stress-induced crystallization during the processing. Consequence of high molecular orientation is strong enhancement of mechanical properties, often by orders of magnitude. In our research on this subject we have focused on (a) modeling of molecular deformation and orientation under fast, time-dependent uniaxial or biaxial elongationa flow, as well as on kinetics of stress-induced crystallization, (b) flow behavior of the polymer during crystallization (crystallinity-dependent viscosity) and (c) effects of stress-induced crystallization in modeling of melt spinning of fibers, in particular on limitation in the spinning speed and fiber thickness.

polymer processing, flow deformation, high-speed melt spinning, molecular orientation, stress-induced crystallization

A numerical simulation of melt spinning reveals bifurcation of dynamic solutions leading to limited spinning conditions. The bifurcation phenomenon is controlled by stress-oriented crystallization and crystallinity dependent polymer viscosity. Under the conditions of bifurcation, the space of the spinning conditions (take-up velocity filament thickness) splits into three regions corresponding to amorphous fibers, partially crystalline fibers, and inaccessible conditions. Major factors affecting the maximum spinning speed and minimum filament thickness for melt-spun poly(ethylene terephthalate) are analyzed.

crosslinking, crystallization, computer simulation, tensile stress, polymer viscosity

Badania dynamicznych charakterystyk formowania włókien PET ze stopu z zastosowaniem grzania strefowego przeprowadzono z zastosowaniem matematycznego modelu stacjonarnego procesu przędzenia w przybliżeniu pojedynczego cienkiego włókna z cieczy polimerowej krystalizującej pod naprężeniem rozciągającym. Układ równañ modelu rozwiązywano przy użyciu standardowych procedur numerycznych. Obliczono osiowe profile prędkości i gradientu prędkości, temperatury, naprężenia rozciągającego i stopnia krystaliczności strugi polimeru. Obliczenia przewidują wystąpienie maksymalnej prędkości odbioru wskutek silnego wzrostu lepkości polimeru w wyniku szybkiej krystalizacji orientowanej w akresie większych prędkości przędzenia. Prędkość maksymalna i związany z tym zakres niedostępnych prędkości odbioru włókien zależą od temperatury w strefie grzania. Osiowe profile prędkości ulegają silnej zmianie wskutek grzania strefowego, a zakres rozciągania strugi, z maksimum szybkości rozciagania, ulega znacznemu przesunięciu do strefy grzania. W konsekwencji następuje znaczne zmniejszenie przewidywanego naprężenia odbioru włókien wskutek skrócenia zestalonego odcinka włókna. Wnioskuje się, że grzanie strefowe przekraczające 30—40 oC temperaturę zeszklenia (Tg) prowadzi do krystalizacji na osi przędzenia już przy małych prędkościach odbioru, podobnej do krystalizacji w szybkim przędzeniu bez strefy grzejnej. Przewidywane zmniejszenie prędkości odbioru odpowiadające krystalizacji następuje w wyniku powtórnego przechodzenia polimeru przez zakres temperatury krystalizacji. Krystalizacja wywołana szybką krystalizacją orientowaną pod wpływem dużego naprężenia rozciągającego jest przewidywana na stosunkowo krótkim odcinku osi procesu i skorelowana z silnym wzrostem naprężenia i zestaleniem polimeru na osi przędzenia. Korelacja orientacji amorficznej obliczonej przed punktem zestalenia się strugi z doświadczalnymi wartościami czynnika orientacji amorficznej włókien PET świadczy o kształtowaniu się orientacji amorficznej odbieranych włókien przez naprężenie rozciągające w punkcie zestalenia. Porównanie doświadczalnych danych orientacji amorficznej i stopnia krystaliczności z przewidywaniami modelowymi wskazuję, że parametr kinetyczny krystalizacji orientowanej A zależy od temperatury. W badanym zakresie temperatury strefy grzania pomiędzy temperaturą zeszklenia i temperaturą maksymalnej szybkości krystalizacji powinien on wzrastać ze wzrostem temperatury.

przedzenie włókien, model matematyczny przędzenia, grzanie strefowe, bifurkacja, krystalizacja orientowana, poli(tereftalan etylenu)

Melt-blowing is an industrial method for rapid production of nonwovens. In melt-blowing, a polymer melt is extruded through capillary dies while hot air is blown through a longitudinal air nozzle. The air drag forces subjected on the polymer streams cause fast attenuation of the polymer filaments into fine diameter fibres at the take-up. A modified mathematical model of melt spinning for a pneumatic process is presented which accounts for the effects of structural transformation under viscoelastic behavior of the polymer. The model is applied to a novel method of nonwoven formation under supersonic air jet.

mathematical modeling, pneumatic melt spinning, supersonic air jet, nonwoven formation

In the mathematical model of melt spinning of fibres from crystallizing polymers the set of conservation equations is completed with structure-controlled constitutive equations and structure evolution equations describing kinetics of stress-induced crystallization. In a definite range of conditions, bifurcation of solutions is observed. Maximum filament velocity is limited and the same boundary conditions yield different steady-state dynamic and structure profiles. Bifurcation is observed when stress-induced crystallization leads to rapid solidification of the material. Critical conditions for bifurcation in melt spinning are analyzed and physical mechanism of such a behaviour is discussed.

mathematical model, melt spinning of fibers, crystallizing polymers, stress-induced crystallization, bifurcation

Hot-tube effects in melt spinning from crystallizing polymer melts are dicussed by mathematical modeling of the process dynamics. The model accounts for oriented crystallization, melt viscosity influenced by crystallinity and the crystallization heat. The range of admissible spinning speeds and bifurcation of the model solution for amorphous and crystalline fibers is influenced by crystallinity-dependent viscosity and the crystallization heat. Example solutions of the model are presented for melt spinning of PET fibers.

mathematical modeling, melt spinning, zone-heating, structure development, PET fibers

Affine evolution of chain end-to-end vectors distribution function is derived analytically for non-linear polymer liquids subjected to uniaxial elongational flow, controlled by time-evolution of chain deformation coefficients. Peterlin approximation for non-Gaussian chain elasticity is applied, with Pade` approximation for the inverse Langevin function. The approach enables calculations of transient molecular deformation coefficients in entire range of elongation rates and times. Equations controlling time evolution of the molecular deformation coefficients in elongational flow are solved analytically with an assumption of dominating elongational component. The approach allows to decouple evolution equations and obtain an approximate closed form analytical formula describing time evolution of the molecular deformation with high accuracy, in particular at higher elongation rates, above the Gaussian limit. Predictions of the analytical formula are compared with numerical computations to evaluate the approximation and ranges of its validity. The analytical formula enables predicting evolution of average functions in non-linear systems, such as free energy, tensile stress, molecular orientation, etc. The formula is used to discuss molecular vs. macroscopic deformation in wide range of elongation rates and times, as well as evolution of stress, axial orientation factor, apparent elongational viscosity.

transient molecular deformation, uniaxial elongational flow, non-linear stress-orientation formula

Badano wpływ temperatury komory grzejnej i szybkości przędzenia na stopień krystaliczności, dwójłomność optyczną, orientację amorficzną i krystaliczną włókien PET otrzymanych jednostopniową metodą przędzenia ze stopu z zastosowaniem komory grzejnej w zakresie temperatury termostatowanej komory z gorącym powietrzem w zakresie 100—210oC. Szybkość przędzenia dotyczyła zakresu konwencjonalnego i szybkiego przędzenia z prędkością odbioru włókna powyżej 4000 m/min. Wykazano, że włókna PET o wysokim stopniu krystaliczności można otrzymać w zakresie konwencjonalnych prędkości przędzenia przy temperaturze komory powyżej 135oC, podczas gdy w szybkim przędzeniu wzrost krystaliczności włókien otrzymywanych następuje przy wyższych temperaturach komory bliskich 190oC. Orientacja molekularna charakteryzuje się wysokim czynnikiem orientacji krystalicznej bliskim 0.9, niezależnie od warunków przędzenia. Grzanie strefowe z zastosowaniem komory powietrznej umożliwia krystalizację polimeru na osi przędzenia z małymi szybkościami odbioru, podczas gdy w szybkim przędzeniu czynnikiem wywołującym krystalizację jest wysoka orientacja amorficzna. Wysoka temperatura komory prowadzi do obniżenia orientacji fazy amorficznej wskutek relaksacji. Jednakże włókna o najwyższej dwójłomności otrzymywane były z zastosowaniem wysokiej temperatury komory.

przędzenie włókien PET, szybkie przędzenia, grzanie strefowe, orientacja amorficzna, orientacja krystaliczna, dwójłomność, krystalizacja orientowana

Transient distribution function of chain ends in non-linear polymer fluids subjected to a constant biaxial flow deformation is approximated by an affine evolution of initial Gaussian distribution function. A non-linear elastic dumbbell potential is used in the evolution equation for the distribution function, with the Peterlin and Pade` approximations of inverse Langevin function. With the approximations, the evolution equation reduces to a system of ordinary first order differential equations for axial components of affine molecular deformation tensor. Numerical and a self-consistent analytical method of solving the system of evolution equations are proposed. Example computations are performed for uniaxial, incompressible elongational flow. The non-linear model covers entire range of deformation rates, and predicts molecular deformation tangential to macroscopic deformation at the beginning of the process, and asymptotically converging to the equilibrium chain deformation in the limit of infinite time. The model describes time evolution of the chain distribution function between the macroscopic affine limit at the beginning of the process and the equilibrium asymptote. The evolution deviates from the asymptotes the more, the lower is the deformation rate. For slow processes, linear Gaussian model is valid, and for very fast ones, solid-like behavior takes place with minor deviation between the molecular and macroscopic deformations, up to the level of full chain extension.

biaxial deformation, transient chain distribution, non-linear molecular orientation

Flow-induced crystallization in the course of melt spinning changes properties of as-spun fibres and modifies dynamics of the process. Velocity, stress and temperature profiles depend on the speed of on-line crystallization. Very important but little studied is coupling between crystallinity and rheological properties of the polymer melt. Effects of crystallization on the dynamics of melt spinning and on the rheological behaviour have been discussed and compared with model calculations and available experimental data.

model calculations, flow-induced crystallization, melt spinning, crystallinity-rheological properties coupling

The role of polymer viscosity in the dynamics of melt spinning of a polymer crystallizing under tensile stress is investigated using mathematical model method. The viscosity is assumed to be dependent on the local polymer temperature and degree of crystallinity along the spinning axis. Example computations concern melt spinning of PET in the range from low to high spinning speeds. Strong effects of stress-induced crystallization on the viscosity are predicted which lead to limitations in the spinning speed, with a maximum of the take-up velocity. The limitations in the spinning speed are consequence of strong rheological effects of fast oriented crystallization predicted on the spinning line which leads to physical crosslinking of the spun melt. Temperature influence on the viscosity are predicted to be not resposible for the limitations of the spinning speed.

mathematical modeling, melt spinning, oriented crystallization, crystallinity-dependent viscosity, PET

Biaxial deformation of freely jointed chain macromolecules in a solid state is considered. Biaxial molecular orientation is directly related to the applied deformation. Segmental orientation and stress are considered using non-Gaussian inverse Langevin statistics of the chain end-to-end vectors. The Pade approximation and series expansion of the inverse Langevin function are used. Global orientation of the chain segments and stress are analyzed for affine biaxial deformation of non-Gaussian chains. Molecular anisotropy is characterized by the norm of the average orientation tensor, and the global anisotropy of the stress tensor is characterized by the norm of the stress tensor. Non-linear behaviour of the orientation vs. stress characteristics for isochoric uniaxial deformation, calendering and biaxial deformation are discussed.

flexible-chain polymer, non-Gaussian chain statistics, biaxial orientation, biaxial stress

The development of biaxial segmental orientation and stress in a flexible-chain polymer fluid subjected to steady biaxial extensional flow is analyzed. Closed-formula model based on Pade approximation of the inverse Langevin function in the non-Gaussian distribution of the chain end-to-end vectors is considered. The approach is free from the limitations related to the finite chain extensibility and slow convergence of the series expansion of the formulations at higher chain deformations. Segmental orientation is characterized by the average orientation tensor, related axial orientation factors and global orientation anisotropy. Orientation behaviour and corresponding stresses in the biaxial elongational potential flow are discussed in a wide range of elongation rates. Orientation characteristics calculated for the biaxial flow deformation are much higher than those predicted for the affine biaxial stretch deformation in polymer solids.

flexible-chain polymer, biaxial molecular orientation, axial orientation factor, orientation anisotropy

Development of biaxial segmental orientation and stresses in flexible chain polymers subjected to affine deformation of end-to-end vectors or to steady biaxial extensional flow is discussed. A closed-formula theory with non-Gaussian chain statistics and a Padè approximation of the inverse Langevin function is considered. The approach accounts for finite chain extensibility and is free from the problems of weak convergence of series -expansion expressions at higher molecular deformations. Average orientation tensor, global anisotropy tensor, and axial orientation factors characterise segmental orientation. Axial orientation factors and normal stress differences, in the deformation and normal planes, are discussed for biaxial affine deformation and steady biaxial elongational flow in a wide range of molecular deformations using inverse Langevin chain statistics. Orientation characteristics predicted for biaxial flow deformation are higher, and change in a wider range, than those in affine biaxial stretch. Also sensitivity to transversal deformation is different in both types of deformation.

flexible chain polymers, inverse Langevin chain statistics, Padè approximation, affine deformation, biaxial flow deformation, molecular orientation, stress tensor

Scharakteryzowano prace badawcze przedstawione na sekcji Kongresu IUPAC MACRO 2000 poświęconej teorii i modelowaniu układów polimerowych. Prace te obejmują szeroki zakres tematyczny - od syntezy polimerów, poprzez ich strukturę, do właściwości. Szereg prac doświadczalnych przedstawionych na sekcji dotyczyło w gruncie rzeczy badań modelowych układów rzeczywistych o ściśle określonych cechach struktury. Mimo stosowania metod doświadczalnych, prace te można traktować jako dotyczące modelowania. Połączenie metod doświadczalnych z modelowaniem okazało się stymulujące dla sformułowań teorii i badań modelowych, a przedstawione prace są w tym zakresie komplementarne. Zaprezentowane prace można podzielić na cztery grupy tematyczne: zagadnienia dynamiki molekularnej, polimery w zewnętrznych polach sił, struktury w cienkich warstwach i na powierzchniach, reakcje polimeryzacji. W niniejszym opracowaniu scharakteryzowano wymienione grupy tematyczne.

Kongres IUPAC Macro 2000, dynamika molekularna, polimery w polach zewnętrznych, struktura cienkich warstw, struktura powierzchni, reakcje polimeryzacji

Thermodynamics is one of the factors which control the orientation distribution of polymer crystals. The present paper deals with crystal orientation in uncrosslinked polymer systems, in which small, isolated crystals are embedded in a viscous matrix. With transient effects neglected, and in the absence of the production of new crystals, orientation is controlled by the orientation-dependent free energy of an anisotropic crystal, F(), and a hydrodynamic potential of the velocity field, Φ (). Example distributions for uniaxially stressed polyethylene are discussed. It has been shown that different mechanisms control crystal orientation depending on the stress difference Δϱ = ϱ33 − ϱ11 applied, and the crystal shape factor, ø. At low stresses, Δϱ and high assymetry factors, ø, crystal orientation is practically controlled by the hydrodynamic potential. At high stresses and/or low asymmetry ratios it is the strain energy of anisotropic crystals, , which is responsible for orientation distribution. In the intermediate range both mechanisms have to be considered.

Molecular deformation and orientation in the amorphous phase subjected to tensile stresses strongly accelerates crystallization of polymers by orders of the magnitude. Closed-form analytical formula is derived for the crystallization rate under high tensile stresses in the entire range of crystallization temperature and the temperature rates. Such formula, lacking in the literature, is needed for modelling of the dynamics of crystallizing polymers processing and predicting structure development in obtaining highly oriented materials of enhanced tensile modulus and tenacity. Rapid online crystallization strongly influences the rheological behaviour during the processing and introduces coupling of crystallization with the processing dynamics. Tensile stress affects the crystallization rate mainly by influencing the configurational entropy of the chain macromolecules in amorphous phase. None of the actually available models are capable to account for the effects of high molecular orientation in the crystallization kinetics. In the present approach, non-Gaussian chain statistics is considered to account for finite extensibility of real macromolecules in the amorphous phase under high tensile stresses which results in non-linear effects in the model. The Hoffman-Lauritzen model of crystallization kinetics is extended to account for free energy of deformation of the amorphous component under uniaxial molecular orientation produced by the tensile stresses. The crystallization rate is considered as controlled by predetermined and sporadic nucleation present in real systems. The involvement of both nucleation mechanisms in the crystallization kinetics varies strongly with the level of tensile stress and amorphous orientation, with domination of sporadic nucleation at high orientations. The closed-form analytical formulas are validated by computations. Example numerical calculations illustrate influence of amorphous orientation on the crystallization free energy and the crystallization rate function involving both nucleation mechanisms.

polymer crystallization, crystallization kinetics, molecular orientation, non-Gaussian chain statistics

Additive Manufacturing (AM) process is a very fast and promising technique to build various very complex prototypes and components directly in the industry. One can choose different techniques of AM like Selective Laser Sintering (SLS), Fused Filament Fabrication (FFF) dedicated for thermoplastic materials or Direct Laser Metal Sintering (DMLS) for powder metals, or Stereolitography Apparatus (SLA) for thermosets. One of the most common techniques in AM are SLS and FFF for thermoplastic materials. The complexity of the processes and the behaviour of the materials in specific environment have a strong influence on the quality, strength and warpage of the obtained structures. The state of the art of the studies indicates that morphology of the material and the crystallization processes influence the aforementioned characteristics of the created components. The knowledge on the crystallization kinetics of polymers is known since many years but it is still developing in order to get an adequate description of the behaviour of the materials in isothermal and non-isothermal conditions. Furthermore, it is needed to predict the warpage of manufactured components based on the virtual AM process in order to decrease the costs. The available tools dedicated for FE analyses allow to increase functionality and implementation of own material models and techniques to perform the customize simulations. Based on the theory and Differential Scanning Calorimetry (DSC) results it is possible to predict the behaviour of the materials and start working on simulation of the virtual AM process [1-4]. The extracted curves of the velocity of material crystallization in temperature domain with different cooling rate obtained in FE simulations are shown in Fig. 1. The simulated curves are confronted with the DSC experimental results.

crystallization kinetics, Differential Scanning Calorimetry (DSC), polyamide 2200, additive manufacturing, FE analysis

Pneumatic melt spinning of the fibers by supersonic air jet is an efficient, energy saving method of super-thin fibers formation. In the process with the use of Laval nozzle, the polymer extruded from a raw of orifices in the spinning beam undergoes fast drawn by the pneumatic friction forces. Air velocity, temperature and pressure distributions are computed using k-omega aerodynamic model. Air-drawing dynamics of the polymer bases on the melt spinning model in a single-, thin-filament approximation with the assumption of Phan-Thien/Tanner non-linear viscoelasticity of the melt. Axial velocity, temperature, tensile stress and rheological extra-pressure profiles are computed along the spinning line. Influence of the nozzle geometry, initial air compression, melt extrusion temperature, polymer mass output and the extrusion die diameter is discussed. Influence of the weight average molecular weight, viscosity and relaxation time of the melt is accounted for. Example computations present the role of processing and material characteristics. High cavitating internal extra-pressure is predicted in the melt at high melt elongation rates which may lead to longitudinal burst splitting of the filament. The number of sub-filaments produced by the splitting is estimated reaching nano-fibers thickness range.

superthin fibers, air drawing, pneumatic melt spinning, Laval nozzle

Computer modeling of pneumatic melt spinning of super-thin fibers from crystallizing polymers is presented. Air drawing dynamics of thin polymer streams in melt blowing technology as well as under supersonic air jet in the Laval nozzle is discussed. Hot air jet is used in melt blowing while in the Laval nozzle process cold air is compressed in the nozzle inlet and accelarated to supersonic velocity. Predetermined air fields are simulated using the k-epsilon turbulence model for melt blowing and the k-omega model for the supersonic Laval nozzle processes, with an assumption that the thin polymer filaments do not disturb the air fields substantially. A single-, thin-filament mathematical model of stationary melt spinning is modified for the pneumatic processes and dynamics of the processes is controlled by the axial distributions of the air velocity, temperature and pressure. Effects of non-linear viscoelasticity important for fast flow elongation of polymer melts are accounted for in the model, as well as non-linear stress-orientation relationship and on-line stress induced crystallization of the filament. Example computations are performed for pneumatic formation of polypropylene nonwovens.

pneumatic melt spinning, melt blowing, supersonic air jet, super-thin fibers, oriented crystallization