A Mathematical Model for the Kinetics of Crystallization in Crystaf

Summary : A series of ethylene homopolymers and ethylene/1-hexene copolymers with different molecular weight distributions (MWD) and chemical composition distributions (CCD) was analyzed by crystallization analysis fractionation (Crystaf) at several cooling rates to investigate the effect of MWD, CCD, and cooling rate on their Crystaf profiles. Using these results, we developed a mathematical model for Crystaf that considers crystallization kinetic effects ignored in all previous Crystaf models and can fit our experimental profiles very well.     

Application of a crystallization kinetics model to simulate the effect of operation conditions on Crystaf profiles and calibration curves

Crystallization analysis fractionation (Crystaf) is a polymer characterization technique used to estimate chemical composition distributions (CCDs) of semicrystalline copolymers. The Crystaf profile can be transformed into a CCD using a calibration curve that relates average comonomer content to peak crystallization temperature. The calibration curve depends on copolymer molecular properties and Crystaf operation conditions. In this investigation, we applied a crystallization kinetics model to simulate Crystaf calibration curves and to quantify how Crystaf calibration curves depend on these factors. We applied the model to estimate the CCDs of three ethylene/1‐hexene copolymers from Crystaf profiles measured at different cooling rates and showed that our predictions agree well with the CCDs described by Stockmayer's distribution. We have also used this new methodology to investigate the effects of cooling rate, molecular weight, and comonomer type on Crystaf profiles and calibration curves. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47: 866–876, 2009     

Mathematical modeling of crystallization analysis fractionation of ethylene/1‐hexene copolymers

Crystallization analysis fractionation (Crystaf) is a polymer characterization technique for estimating the chemical composition distributions of semicrystalline copolymers. Although Crystaf has been widely used during the recent years, it is still a relatively new polymer characterization technique. More quantitative understanding of its fractionation mechanism is essential for further developments. In this work, three ethylene/1‐hexene copolymers with different 1‐hexene fractions, but similar number‐average molecular weights, were analyzed by Crystaf at several cooling rates. A mathematical model was proposed to describe the effect of comonomer fraction and cooling rate on Crystaf fractionation from a fundamental point of view. The model describes the experimental Crystaf profiles of ethylene/1‐hexene copolymers with different 1‐hexene fractions measured at distinct cooling rates very well. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 1010–1017, 2007     

Effect of Chain Microstructure and Cooling Rate on Crystaf Calibration Curves: An Experimental Study

Summary: Crystallization analysis fractionation (Crystaf) is a polymer characterization technique based on differences in chain crystallizabilities in a dilute solution during non-isothermal crystallization. Crystaf profiles, a weight distribution function of chains crystallized at each temperature, can be used to infer the chemical composition distribution (CCD) of copolymers when a Crystaf calibration curve, a relationship between peak crystallization temperature and average comonomer content, is known. In this investigation, the effect of the number average molecular weight, comonomer type, and cooling rate on Crystaf calibration curves were experimentally investigated. It was found that the cooling rate and comonomer type may strongly affect Crystaf calibration curves, while the influence of molecular weight is relatively subtle.     

Simulation of Crystallization Analysis Fractionation (Crystaf) of Linear Olefin Block Copolymers

Summary: Linear olefin block copolymers (OBCs) have microstructures that are unique among polyolefins and exhibit properties that are different from those of other polyolefin elastomers. Characterizing their chain microstructures is a challenging task, as conventional characterization techniques cannot probe directly block length distribution or composition. In this work, we used a Monte Carlo model to predict the microstructure details of OBCs and a modified version of the Crystaf model previously developed in our groups to describe theoretical Crystaf profiles for model OBCs. This model can be used as a tool to interpret Crystaf results of these interesting new polyolefins and to relate them to OBC microstructures. Effects of polymerization parameters on OBC microstructure and Crystaf profiles were also discussed.     

Cocrystallization of ethylene/1‐octene copolymer blends during crystallization analysis fractionation and crystallization elution fractionation

Blending of ethylene/1‐octene copolymers can be used to achieve a well‐controlled broad chemical composition distribution (CCD) required in several polyolefin applications. The CCD of copolymer blends can be estimated using crystallization analysis fractionation (CRYSTAF) or crystallization elution fractionation (CEF). Unfortunately, both techniques may be affected by the cocrystallization of chains with different compositions, leading to profiles that do not truly reflect the actual CCD of the polymer. Therefore, understanding how the polymer microstructure and the analytical conditions influence copolymer cocrystallization is critical for the proper interpretation of CRYSTAF and CEF curves. In this investigation, we studied the effect of chain crystallizabilities, blend compositions, and cooling rates on cocrystallization during CEF and CRYSTAF analysis. Cocrystallization is more prevalent when the copolymer blend has components with similar crystallizabilities, one of the components is present in much higher amount, and fast cooling rates are used. CEF was found to provide better CCD estimates than CRYSTAF in a much shorter analysis time. © 2011 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2011     

Estimation of Average Comonomer Content of Ethylene/1-Olefin Copolymers Using Crystallization Analysis Fractionation (Crystaf) and Artificial Neural Network (ANN)

Summary: An artificial neural network (ANN) with a 4-3-3-1 architecture was developed to estimate average comonomer content of ethylene/1-olefin copolymers from crystallization analysis fractionation (Crystaf) results. The ANN was trained with a back propagation algorithm. It was found that average comonomer contents predicted from ANN agree well with experimental results for both training and testing data sets. The developed ANN was also used to systematically investigate the effects of chain microstructures and Crystaf operating conditions on Crystaf calibration curves.     

Crystallization and melting of polyethylene copolymers: Differential scanning calorimetry and atomic force microscopy studies

The Hoffman–Lauritzen theory of secondary, surface nucleation and growth was primarily relied upon for about 40 years after its introduction in about 1960 to rationalize the crystallization of flexible chain polymers into lamellar crystals. However, in about 1998, Strobl and coworkers introduced a different model for crystallization, based on the stage‐wise formation of lamellae. Two major components of this model were as follows: (1) the concept of the formation of a mesomorphic melt as a precursor to crystallization and (2) the control of the melting temperature range of lamellar crystals of homogeneous polyolefin copolymers by an inner degree of order or perfection rather than on the crystal thickness. The first concept is in disagreement with the HL theory and the second with the Gibbs‐Thomson theory, which associates melting temperature with lamella thickness. In the present study, differential scanning calorimetry and atomic force microscopy were successfully employed to monitor the in situ quiescent crystallization of polyethylene homopolymer and copolymer. In the present study, evidence was not found to support the concept of lamellae with equal thickness melting over a broad temperature range. Some evidence was found that might be interpreted to support the concept of a mesomorphic melt as a precursor to crystallization. At present, the model promoted by Strobl and coworkers appears to be at an uncertain stage at which strong proof or disproof are not available. However, this alternative model has injected a new vitality into the study of crystallization of flexible chain polymers. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2369–2388, 2006     

Simultaneous Deconvolution of the Molecular Weight and Chemical Composition Distribution of Polyolefins Made with Ziegler-Natta Catalysts

Heterogeneous Ziegler-Natta catalysts produce polyolefins that have broad distributions of molecular weight (MWD) and chemical composition (CCD). For such broad distributions, mathematical models are useful to quantify the information provided by polyolefin analytical techniques such as high-temperature gel permeation chromatography (GPC), temperature rising elution fractionation (TREF), and crystallization analysis fractionation (CRYSTAF). In this paper, we developed a mathematical model to deconvolute the MWD and CCD of polyolefins simultaneously, using Flory's most probable distribution and the cumulative CCD component of Stockmayer's distribution. We have applied this procedure to “model” polyolefin resins and to one industrial linear low-density polyethylene (LLDPE) resin. The proposed methodology is able to deconvolute theoretical distributions even when random noise is added to the MWDs and CCDs, and it can be used to calculate the minimum number of active site types on heterogeneous Ziegler-Natta catalysts.     

Probing the use of small‐angle light scattering for characterizing structure of titanium dioxide/low‐density polyethylene nanocomposites

Small‐angle light scattering (SALS) measurements were used to study the structure of titanium dioxide (TiO₂)/low‐density polyethylene (LDPE) nanocomposites. The results showed that the scattering from LDPE crystalline structures and the scattering from TiO₂ nanoparticles can be resolved and separated. It is shown that the independent effects of crystallization conditions and the presence of nanoparticle aggregates on the spherulitic structure of the LDPE matrix can be determined by analyzing the scattering patterns using the methods proposed. From the SALS results, we conclude that the nanoparticle surface chemistry affects both nucleation of spherulites and their structure particularly under rapid cooling conditions. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 1084–1095, 2006     

An experimental and numerical study on crystallization analysis fractionation (Crystaf)

Crystallization analysis fractionation (Crystaf) is a new technique used to estimate the chemical composition distribution (CCD) of semi-crystalline copolymers. In this study, the effect of chain microstructure and operation parameters on Crystaf profiles was investigated using a series of ethylene/1-hexene copolymers and their blends. The Crystaf profiles were also modeled via stochastic simulation based on the distribution of average ethylene sequence lengths.     

Nonisothermal bulk crystallization studies of high density polyethylene using light depolarizing microscopy

The quiescent nonisothermal bulk crystallization kinetics of two high-density polyethylene resins were investigated by a modified light-depolarizing microscopy (LDM) technique. The technique allows studies at average cooling rates up to 2500°C/min. The polymer was found to crystallize at a pseudo-isothermal temperature even at these very high cooling rates. The overall bulk crystallization rate increased rapidly as the cooling rate and supercooling increased. Crystallization kinetics was analyzed by Avrami analysis. Avrami exponents near 3 suggested spherical growth geometry and instantaneous nucleation at predetermined sites. Observation of spherulites by optical microscopy together with a number density of spherulites that changed little with increase in cooling rate or supercooling supported this model of crystallization behavior. Analysis of the half-time of crystallization based on the Lauritzen and Hoffman secondary nucleation theory indicated that the regime II-III transition was found to occur at a degree of supercooling of approximately 22°C. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36: 681–692, 1998     

Effect of operation parameters on temperature rising elution fractionation and crystallization analysis fractionation

Crystallization analysis fractionation and temperature rising elution fractionation are two techniques used to estimate the chemical composition distributions of semicrystalline copolymers. This study investigates the cooling rate and cocrystallization effects for both techniques with a series of ethylene/1‐olefin copolymers and their blends. Ideally, both techniques should operate in the vicinity of thermodynamic equilibrium so that crystallization kinetic effects are avoided. The results show that, in fact, crystallization kinetic effects play an important role at the typical cooling rate used with both techniques. Cocrystallization is significant when fast cooling rates are used. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1762–1778, 2003     

Flash DSC crystallization study of blown film grade bimodal high density polyethylene (HDPE) resins. Part 2. Non‐isothermal kinetics

Non‐isothermal ultra‐fast cooling crystallization tests were conducted on three blown film grade bimodal high density polyethylene (HDPE) resins using a fast differential scanning calorimeter, the Flash DSC. Non‐isothermal tests were performed at cooling rates between 50 and 4000°K/s, and the data were analyzed using the modified Avrami model by Jeziorny (Polymer, 1978 , 19, 1142). Non‐isothermal data were used to propose a new method named crystallization–time–temperature–superposition, and the two activation energies were obtained for each of the resins. This is very useful for obtaining theoretical crystallization kinetics data at different cooling rates, allowing its use in ultra‐fast cooling polymer processes such as blown film. © 2017 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2017 , 55, 1822–1827     

Ethylene/1‐hexene copolymers synthesized with a single‐site catalyst: Crystallization analysis fractionation,modeling, and reactivity ratio estimation

A series of poly(ethylene‐co‐1‐hexene) samples made with rac‐ethylene bis(indenyl)zirconium dichloride/methylaluminoxane were analyzed by crystallization analysis fractionation (CRYSTAF). The nine samples had comonomer contents of 0–4.2 mol % 1‐hexene with a narrow range of molecular weights (34,000–39,000 g/mol). Because all the copolymer samples had narrow, unimodal chemical composition distributions, they were ideal as calibration standards for CRYSTAF. A linear calibration curve was constructed relating the peak crystallization temperature from CRYSTAF operated at a cooling rate of 0.1 °C/min and the comonomer content as determined by ¹³C NMR. Reactivity ratios for ethylene and 1‐hexene were estimated by the fitting of reactant liquid‐phase compositional data to the Mayo–Lewis equation. It was found that a value of the 1‐hexene reactivity ratio could not be unequivocally determined from the set of samples analyzed because the range of comonomer incorporation was too narrow. Stockmayer's bivariate distribution was used to model the fractionation process in CRYSTAF, and although a good fit to experimental CRYSTAF profiles was attained, the model did not fully describe the underlying crystallization phenomena. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 2595–2611, 2002     

Crossover behavior in crystal growth rate from n‐alkane to polyethylene

A crystal growth rate equation, parameterized from molecular dynamics simulations of n‐alkanes, is compared to recent experiments on growth rates for polyethylene at high undercooling. The analysis reveals that the growth rate of alkanes and polyethylene can both be described by the same relationship. The appropriate relaxation time is used to describe the kinetic barrier to crystallization. For chains shorter than the entanglement length, this is the Rouse time. For chains longer than the entanglement molecular weight, kinetic limitations are modeled by the local relaxation of an entangled segment at the interface. This model supports a different mechanism for fast crystal growth at high undercooling than that usually inferred from slow growth data near the melting temperature. Use of the crystal growth rate model is illustrated for polyethylene crystallizing under conditions of slow cooling and fast cooling. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 2468–2473, 2005     

A query on crystallization temperature‐dependent cooling function under nonisothermal condition

The cooling function (κ) in Ozawa model was investigated through theoretic analysis and experimental method. Different from the fact accepted by researchers over past decades that κ(T) depends only on the crystallization temperature (T) and consequently the parameters for nonisothermal crystallization kinetics could be obtained by plotting ln[? ln(1 ? X(T))] versus ln λ at a given T, we found that κ at a given T was also dependent on onset temperature (T₀) of crystallization process. Because T₀ varies with cooling rate (λ) in nonisothermal crystallization, we conclude that κ is a binary function of T and λ, which was validated by our kinetic data from differential scanning calorimetry measurement in a wide λ range from 1 to 80 °C/min. It is suggested that the conventional method for calculating kinetic parameters based on Ozawa model, by plotting ln[? ln(1 ? X(T))] versus ln λ, might not be exact for nonisothermal crystallization. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44:795–800, 2006     

Correlation of the melting behavior and copolymer composition distribution of Ziegler–Natta‐catalyst and single‐site‐catalyst polyethylene copolymers

The multimodal differential scanning calorimetry melting endotherms observed for commercial linear low‐density polyethylenes are due to broad and multimodal short‐chain‐branching distributions. Multiple peaks, observed in melting endotherms of isothermally melt‐crystallized and compositionally homogeneous polyethylene copolymers are due to intrachain heterogeneity. This intrachain heterogeneity is quantified by the distribution of ethylene sequence lengths within the chains. These compositionally homogeneous copolymers undergo a primary crystallization, which produces a population of thicker lamellae, creating a network that places severe restrictions on segment transport in subsequent secondary crystallization, which produces a population of thinner crystals. The restrictions on segment transport imposed by the initial network created by the primary crystallization of thicker lamellae severely limits the total crystallinity achieved in the random copolymers studied. The solution crystallization of such copolymers produces a continuous distribution due to more facile segment transport in a dilute solution, in contradistinction to the multimodal distribution produced in the melt crystallization. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2800–2818, 2001     

Ellipsometry as a probe of crystallization kinetics in thin diblock copolymer films

We present results on the use of ellipsometry as a novel probe for the crystallization kinetics in thin films of a diblock copolymer. Ellipsometry makes use of the change in polarization induced upon the reflection of light from a film-covered substrate to enable the calculation of the refractive index and thickness of the film. The information obtained with these measurements can be compared with information from differential scanning calorimetry, with the additional advantages that small sample volumes and slow cooling rates can be employed and that expansion coefficients can be determined. By studying the temperature dependence of these quantities, we are able to measure the crystallization kinetics within very small volumes (∼10⁻¹⁰ L) of a poly(butadiene-b-ethylene oxide) diblock copolymer. Through a comparison of two different poly (ethylene oxide) block lengths, we demonstrate a reduction in both the crystallization and melting temperatures as the domain volume is reduced. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 3448–3452, 2006