Institute of Fundamental Technological Research

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

Formation of different polymorphic structures plays important role in crystallization of some polymers, like polypropylene, polycaproamide, poly(vinylidene difluoride), and others. Physical properties of such materials depend on phase structure, that is, fractions of different polymorphs. Basing on the model of many-phase transitions (Ziabicki, J Chem Phys 2005, 123, 174103; Ziabicki and Misztal-Faraj, J Mater Res, 2011, 26, 1585) a model system consisting of an amorphous phase, stable solid phase, and metastable polymorphic phase was analyzed. Two effects in the kinetics of polymorphic crystallization have been analyzed: effect of the presence of universal and/or selective predetermined nuclei and limited crystallizability resulting from molecular constraints in high-molecular systems. Conditions in which different phase compositions (i.e., different proportions of individual phases) have been discussed.

polymorphic phase transitions, phase composition, sporadic nucleation, predetermined nucleation, limited degree of transformation

Development of phase composition in one-component, three-phase systems containing a liquid phase (melt) and two polymorphic solids has been discussed. Two types of polymorphic systems have been analyzed: enantiotropic systems composed of three thermodynamically stable phases and monotropic systems with two stable and one metastable phase. Detailed relations between transition rates, molecular characteristics, and external conditions have been derived. Simulation of isothermal crystallization of a model system has been performed and discussed.

isothermal crystallization, phase composition, polymorhic systems, enantiotropic systems, monotropic systems, metastable phase

Nonequilibrium phase composition in multiphase systems affects physical properties of many materials. Development of phase composition is controlled by external conditions and material characteristics. Based on the model presented in Part I [A. Ziabicki and B. Misztal-Faraj, J. Mater. Res. 26(13), 1585 (2011)], rates of phase transitions in a three-phase model monotropic system composed of an amorphous (liquid) phase and two solid polymorphs have been analyzed. Effects of material characteristics including activation energy of molecular mobility, heat and entropy of the transitions, interface tensions, and concentration of predetermined nuclei have been discussed.

development of phase composition, monotropic three-phase system, predetermined nucleation, influence of material parameters

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

Phase transitions in isotactic polypropylene were investigated during isothermal crystallization and heating after isothermal crystallization using various experimental techniques. The results obtained by wide angle x-ray scattering (WAXS), light depolarization technique (LDT), differential scanning calorimetry (DSC) and optical microscopy show that crystallization of isotactic polypropylene can result in simultaneous formation of two crystal modifications, alpha and beta. There is clear experimental evidence that beta phase tends to convert into alpha modification during crystallization as well as during subsequent heating. Experimental results are compared with numerical simulation performed according to the model of nucleation-controlled phase transitions in multiphase systems. The results of simulation show that beta phase is not thermodynamically stable in any temperature range. The reason for the appearance of beta phase is related to low interfacial tension of melt vs. beta. It has been also shown that maximum crystallinity reached in experiments does not exceed 40–50% in agreement with the concept of constrained amorphous phase.

polypropylene, polymorphism, metastability, crystallization

A new setup for light depolarization measurements was designed. Two innovative elements have been introduced. The first is an electronic system which enables depolarization ratio to be registered directly. The second is a system of temperature control allowing effective implementation of a temperature–time program according to the particular requirements. Direct registration of depolarization ratio instead of intensity of depolarized light for individual components (parallel and perpendicular), as is performed in the usual apparatus, allows elimination of light scattering effects because of the insensitivity of depolarization ratio to the scattering level. Application of the new setup was shown for crystallization and melting of isotactic polypropylene (i-PP). Comparison of phase transitions in i-PP, as registered by light depolarization and DSC, indicates some differences. Possible sources of the observed differences are discussed.

Light depolarization, Polymers, Crystallization, Melting, Kinetics of phase transitions

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

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

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

Theoretical principles of light depolarization technique (LDT) have been analysed. It has been shown that the traditional way of interpretation assuming a linear relationship between depolarization ratio, average optical retardation of a single birefringent plate, D, and volume fraction of crystalline material (crystallinity) is incorrect. An exact non-linear relationship between depolarization ratio and the product DE (E is average number of plates in the light path) has been derived. The parameter DE has been shown to be proportional not to the crystallinity alone but to the product of crystallinity and average crystal thickness, (x). A method of decoupling light depolarization data, based on measurements at different sample orientations and different wavelengths, has been outlined.

optical retardation, theory of compensators, depolarization, depolarization ratio, polymer crystallization, degree of crystallinity, degree of crystal orientation

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

Critical analysis of light depolarization technique (LDT) widely used in polymer crystallization kinetics is presented. It is demonstrated, that intensity of light depolarized on a system of birefringent crystals cannot be treated as a measure of volume fraction of crystalline phase (degree of crystallinity). Intensity of light depolarized in a polycrystalline system is a non-linear function of the product of the degree of crystallinity and function of average crystal dimensions. Closed-form expression for depolarization ratio is derived and the range of conditions where linear approximation is acceptable is discussed. In spite of interpretational weaknesses, further development of light depolarization technique seems to be justified by potentially short response time. LDT may appear useful for studying rapid crystallization processes (above 1 kHz sampling frequency), which cannot be followed by measurements of density, X-ray diffraction or calorimetry. However, the LDT data either have to be combined with independent measurements of crystal thickness, or treated as a ‘crystallization characteristic’ per se, quantitatively inconsistent with calorimetric, volumetric or X-ray diffraction data.

crystallization kinetics, crystallinity degree, optical retardation, light depolarization

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

Flow induced crystallization of polymer systems exhibits strong memory effects. Crystalline structures gradually change when the flow is switched off and the polymer is relaxed prior to crystallization. A simple model based on the multidimensional theory of crystal nucleation[1] is proposed. Steady, potential flow applied to a polymer fluid above melting temperature (Tp > Tm) results in molecular orientation of crystallizing units. The flow controls formation of molecular clusters which convert into athermal nuclei when the system is cooled down to crystallization temperature, Tc < Tm. Orientation effects gradually disappear when the melt is relaxed above Tm in the absence of flow or stress.

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.

The program of studies carried out in the Institute of Fundamental Technological Research concerning fundamentals of polymer crystallization and orientation is briefly described. Discussions center on studies of polymer texture formation caused by complex external conditions.

Phenomenological analysis of the effects of molecular orientation and deformation on crystallization kinetics is given. It was shown that nucleation- and growth rates can be represented by functions of six invariants of the molecular displacement gradient tensor and related tensors. For some simple geometries of deformation (including simple shear and uniaxial extension), crystallization rate functions can be also expressed as functions of orientation factorsf (for simple shear) orf * (for uniaxial extension).

The kinetic equations of simultaneous orientation and crystallization were derived. The integral equation ofKolmogoroff-Avrami-Evans was used with nucleation-and growth rates being functions of time-dependent orientation factor f In the orientation equation the relaxation time was assumed to be an increasing function of crystallinity as the result of intermolecular “cross-linking” of macromolecules by crystallites. Simple example solutions have been given and more general equations analyzed.

The transition parameterδ equal to the ratio of orientation half-periodt or 1/2 to crystallization half-period,t or 1/2 was analyzed and discussed. The kinetics of isothermal oriented crystallization differs qualitatively from that for unoriented processes. At very low, or very high values of the transition parameter S however, the process can be approximated by simple kinetic equations (“Avrami equations”) with modified rate constants.Transition parameters δ as calculated from the available experimental data for several polymers were discussed.