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

Obliczono i przedyskutowano wpływ orientacji molekularnej oraz temperatury na szybkość nukleacji i wzrostu kryształów ze wskazaniem na rolę entropii w kinetyce krystalizacji polimerów. W celu określenia efektów entropowych w całym zakresie orientacji molekularnej wywołanej naprężeniami rozciągającymi zastosowano dwa przybliżenia statystyki molekularnej. Badania doświadczalne kinetyki krystalizacji orientowanej przeprowadzono dla orientowanej włókniny z polilaktydu (PLLA) oraz nieorientowanego granulatu PLLA jako przykładowego polimeru, z wykorzystaniem różnicowego kalorymetru skaningowego DSC. Przedstawiono znaczny wpływ orientacji molekularnej na szybkość nukleacji, wzrostu kryształów i szybkość krystalizacji, wynikający ze spadku entropii fazy amorficznej w warunkach orientacji. Ujęte jest to w rozszerzonych modelach Hoffmana i Lauritzena oraz Avramiego i Evansa dla orientowanej krystalizacji i w eksperymentalnych wynikach badań kalorymetrycznych DSC.
Effect of mol. orientation and temp. on the nucleation and crystal growth rates of polymers were calcd. and discussed to indicate the role of entropy in the kinetics of its crystn. Two approximations of mol. statistics were used to det. the entropic effects in the entire range of mol. orientation produced by tensile stresses. The exp. investigations on the oriented crystn. kinetics of polylactide nonwoven fibres and unoriented pellets were carried out by differential scanning calorimetry. High effects of the mol. orientation in the entropy strongly affecting the nucleation and crystal growth rates, as well as the crystn. rate were shown based on the Hoffman-Lauritzen and Avrami-Evans models.

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

A kinetic model of crystallization based on two-dimensional nucleation and growth of plate-like crystals with constant thickness is analyzed. It is shown that plate thickness required for nucleation is limited. The lower limit is determined by zero Gibbs’ free energy of transition, the upper one corresponds to the conditions when the critical cluster volume of nucleation is equal to two elementary kinetic units. Effects of plate thickness on crystallization kinetics are discussed. In the lower temperature range, creation of thicker plates is preferred. For a given plate thickness, frequency of the phase transition decreases with increasing temperature. Numerical calculations for α-polypropylene concern kinetics of primary nucleation and global phase transition in a system of one or several fractions of plate-like kinetic elements.

crystallization kinetics, crystal nucleation, lamellar crystals, crystal growth rate

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 kinetics of isothermal crystallization of various polymers was investigated by light depolarization technique (LDT) using the new setup with direct registration of depolarization ratio. Experimental data were analyzed using new method proposed by Ziabicki who shown that degree of crystallinity is a non-linear function of degree of depolarization, crystal thickness, and its birefringence. Other experimental methods were involved providing supplementary information on crystal thickness (SAXS) and allowing comparison of crystallization kinetics (WAXS, DSC). The advantage of LDT relies on high sampling rate allowing on-line measurements and lack of inertia effects that exist in other methods like calorimetry. The limitations of the applicability of the method are discussed. The method needs supplementary information not only on crystal thickness but also on variable optical birefringence of real crystals. Our results show that LDT can be used in a simple way for investigation of crystallization kinetics at relatively high temperatures, providing large and perfect crystals. In such a case it is sufficient to use crystal intrinsic birefringence and final crystal thickness typical at particular temperature of crystallization. On the other hand, depolarization ratio combined with measurements by other methods (crystallinity and crystal thickness) can be used for estimation of crystal birefringence.

Polymer, Crystallization kinetics, Light depolarization, Crystal birefringence

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

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

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

Light depolarization method was tested on the investigations of fast crystallization kinetics. Isothermal crystallization of isotactic polypropylene is disussed as an example. Theoretical model of statistical assembly of many birefringent crystals is presented as basis of the method, as well as an optical setup for fast registration of the depolarized light intensity during crystallization.

light depolarization, crystallization kinetics, birefringent crystals, isotactic polypropylene

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