Evaluation on nonisothermal crystallization kinetics of polypropylene/kaolin composites by employing Dobreva and Kissinger methods

The influences of kaolin content, processing temperature, and shear stress on crystallization of all samples were investigated by differential scanning calorimetry (DSC). The crystallization activation energy calculated using Kissinger’s method displayed a decreasing trend with increasing kaolin content, processing temperature, and shear stress. A study of nucleation activity, which could indicate the influence of filler on polymer matrix, revealed that kaolin filler had a slight nucleation effect on polypropylene (PP) matrix. A thorough observation on nucleation effect also revealed that the incorporation of kaolin in tandem with increasing temperature and shear stress have contributed to successive heterogeneous nucleation in the system.     

Applicability of the Kissinger equation in thermal analysis

Pt/WO_x–ZrO₂ bifunctional catalysts were synthesized using impregnation and polymeric precursor methods. After the synthesis process the samples were calcined at 600, 700 and 800°C and characterized by X-ray diffraction, nitrogen adsorption and temperature programmed reduction study by thermogravimetry. DTG-TPR profiles showed between three and five reduction events at different temperatures attributed to platinum reduction and to different stages of tungsten species reduction. A comparative study of the synthesis method influence on the DTG-TPR curves was accomplished.     

Determination of nucleation rate in polymers using isothermal crystallization experiments and computer simulation

Within the frame of overall kinetics of crystallization, polymer crystallization is characterized by only three parameters: the initial density of potential nucleiN ₀, their activation frequencyq and the growth rateG. The growth rateG is rather easy to measure, whereas the nucleation parameters are generally unknown. Our purpose is to determine bothN ₀ andq using experimental isothermal two-dimensional crystallizations and computer simulation. Both these parameters are deduced from the rate of appearance of spherulites expressed as a function of the transformed surface fraction. The activation times of the spherulites are deduced from the shape of the boundaries between spherulites at the end of the transformation. When the growth rateG is known, the evolution of the transformed surface fraction is rebuilt using a computer simulation, so that only one observation of the final stage of the crystallization is needed.     

On the calculation of activation energies using a modified Kissinger method

Augis and Bennett (J. Thermal Anal. 13 (1978) 283.) [6] recently proposed a modified Kissinger method for determining the activation energy of a transformation. It is shown that the proposed method was, in fact, based upon a modification to the equation for the rate of reaction under non-isothermal conditions. The apparent discrepancy between the proposed method and the original Kissinger method is therefore resolved. The modified rate equation appears to have, at best, only a limited application. However, if the equation should be appropriate for a particular transformation, it is demonstrated that Augis and Bennett's method would be the correct method for determining the activation energy.

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Zusammenfassung Von Augis und Bennett (J. Thermal Anal.13, (1978) 283) wurde eine modifizierte Kissinger-Methode zur Bestimmung der Aktivierungsenergie einer Umwandlung vorgeschlagen. Es wird gezeigt, dass die vorgeschlagene Methode tatsächlich in einer Modifizierung der Gleichung für die Reaktionsgeschwindigkeit unter nicht-isothermen Bedingungen ihren Ursprung hat. Die scheinbare Diskrepanz zwischen der vorgeschlagenen Methode und der ursprünglichen Kissinger-Methode wird dadurch behoben. Die modifizierte Geschwindigkeitsgleichung hat bestenfalls nur eine begrenzte Anwendung. Jedoch, bei Eignung dieser Gleichung für eine bestimmte Umwandlung zeigt sich, dass die Methode von Augis und Bennett die richtige Methode zur Bestimmung der Aktivierungsenergie sein kann.

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The author is indebted to the Senate of the University of Queensland for the award of a University Research Fellowship; and to Professor R. R. Stephens for providing laboratory facilities.     

Kinetics of nonisothermal polymer crystallization

We adopt a cluster size distribution model to investigate the kinetics of nonisothermal polymer crystallization. The time dependencies of polymer concentration, number and size of crystals, and crystallinity (in Avrami plots) are presented for different cooling rates. The incubation period is also investigated at different cooling rates and initial temperatures. The relationship between cooling rates and incubation time is presented graphically and compared with experimental measurements. The initial temperature (relative to melting point) has a significant effect on nonisothermal crystallization. A comparison of moment and numerical solutions of the population balance equations shows the influence of Ostwald ripening. Agreement between modeling results and experimental measurements at different cooling rates supports the application of the distribution kinetics model for nonisothermal crystallization.     

Analysis of crystallization kinetics of poly(ether ether ketone) by a nonisothermal method

During cooling at a rate of 10°C/min from the melt state of PEEK we have followed the growth of spherulites using an optical microscope equipped with a camera. The isothermal growth rates of crystallization in the temperature range of 266–308°C could be estimated by means of a differential equation. These continuous growth rate data were used further for kinetic analysis, which indicated that PEEK exhibited an unmistakable regime II → III transition at 296°C. The results compared favorably with those obtained by the traditional isothermal method, which is time consuming. Due to chain folding, the Thomas–Staveley constant should be closer to 0.25 instead of 0.1 or 0.3. © 1998 John Wiley & Sons, Inc. J. Polym. Sci. B Polym. Phys. 36: 2393–2399, 1998     

Interpretation of the nonisothermal crystallization kinetics of polypropylene using a power law nucleation rate function

A nucleation rate function is proposed for use in analyzing the overall crystallization kinetics of polymers. This function allows for the possibility that the nucleation rate varies substantially during the crystallization. This feature is particularly useful in analyzing nonisothermal crystallization, but it can be used to analyze isothermal crystallization as well. The nucleation rate function was used in the derivation of a modified transformation kinetics equation of the Avrami type. The modified Avrami equation was found to be suitable for kinetics analysis for the data obtained from nonisothermal crystallization at rapid cooling rates. Kinetics parameters used to describe nonisothermal crystallization under rapid cooling rates are presented and discussed. These include crystallization induction time, plateau (crystallization) temperature, crystallization half-time, crystallization rate constant, Avrami index, and newly defined quantities called nucleation index, geometric index, and nucleation rate constant. The procedure used to obtain the nucleation rate constant and nucleation index for the nucleation rate function is described and illustrated by application to the analysis of the crystallization kinetics of polypropylene. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35: 1077–1093, 1997     

FEM simulation of nonisothermal crystallization, 1 Crystallinity distribution on 2 D space

This work employs the relaxed Stefan model and Nakamura crystallization kinetics to describe the nonisothermal crystallization process of polymeric materials by finite element discretization method (FEM) simulation, giving the evolution of crystallinity distribution on 2 D space. Numerical results show that the final crystallinity and its distribution are mainly dependent on the cooling rate. Crystallinity decreases with increasing cooling rate, but the influence is negligible as long as the cooling rate is below a critical value (ca. 30°C·min^–1 for poly(ethylene terephthalate) (PET)). If the cooling rate is higher than this critical one, crystallinity drops sharply. It is also concluded that the crystallization behavior of polymeric samples in a mild cooling medium is quite different from that in a strong cooling medium. In the first case (for example, in silicon oil), crystallinity of the article is relatively high and its distribution is fairly uniform. During the initial short period, the crystallinity on the surface is higher than that on the inside. Crystallinity increases slowly with time, and finally, the crystallinity of the internal part exceeds the crystallinity on the surface. In the second case (for instance, in water), crystallinity is relatively low, and there is a serious gradient of crystallinity. The crystallinity on the surface reaches a very low equilibrium value in a short time and changes little afterwards. Although the crystallinity of the inside part can be improved by changing the shape of the polymeric article, the crystallinity on the surface essentially remains constant, which leads to a significant gradient. Geometrical shape and dimension of the article are also important to the crystallinity and its distribution, and the ratio of surface area to volume can be used as a rough index to estimate the shape/dimension influence on crystallinity. Except the coefficient of thermal conductivity, physical parameters of the polymeric material and kinetic parameters of crystallization show only weak effects compared to cooling conditions.     

Shear-induced nonisothermal crystallization of two grades of PLA

Shear-induced nonisothermal crystallization of two commercial polylactides (PLAs) differing in optical purity was studied. The molten polymers were sheared at selected temperatures (T_s) and subsequently cooled. The crystallization was followed by a light depolarization method, whereas the specimens were analysed ex-situ by DSC, 2D-WAXS and SEM after etching. It was found that the effect of shear, especially on the crystallinity developed during post-shearing cooling, intensified with a decrease of T_s from 160 to 146 ^°C, and with increasing shear rate and strain. Moreover, the effect of shear on PLA1.5 with d-lactide content of 1.5% was stronger than PLA2.8 with 2.8% of d-lactide, although maximum crystallinity of both polymers was practically the same. A decrease of cooling rate from 30 to 10 ^°C/min increased crystallinity of both PLAs, except for those shearing conditions which induced high crystallinity even during faster cooling. Although SEM examination revealed some row-nucleated forms, no significant crystal orientation was detected by 2D-WAXS, indicating that, under the experimental conditions, the shear induced predominantly point-like nuclei.     

Prediction and analysis of nonisothermal crystallization of polymers

The crystallization behavior of polyetheretherketone (PEEK), polyoxymethylene (POM), polyethyleneterephtalate (PET), and polypropylene (PP) under nonisothermal conditions has been studied. Differential scanning calorimetry was used to monitor crystallization from the melt and a kinetic model has been proposed to describe three-dimensional spherulitic crystal growth. The model, which accounts for crystalline growth rate, uses two modified Avrami equations to represent both heterogeneous and homogeneous nucleation and growth processes. The model parameters are all associated with physical constants. The predicted evolution of absolute crystallinity showed good agreement with experimentally obtained values for a wide range of cooling rates. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys, 35 : 875–888, 1997     

An experimental method for studying nonisothermal crystallization of polymers at very high cooling rates

A new technique based on light depolarizing microscopy was developed for studying non-isothermal crystallization of polymers at average cooling rates up to about 5000°C/min. The polymer is cooled down by a gaseous cooling medium supplied at a constant temperature. The temperature of polymer is measured by a thermocouple imbedded directly in the sample. A heat transfer analysis was used to establish appropriate sample geometry to assure that, under the applied cooling condition, the temperature distribution along the sample thickness can be neglected. A light-scattering effect, which occurs when crystallization is carried out under high cooling rates, was observed. This required the development of a method to correct the depolarized light intensity for the effect of light scattering. An appropriate correction method was developed based on both a theoretical and an experimental analysis of the light intensity measurement. This provided a means to measure the overall crystallization kinetics. Examples of such measurements for iPP, HDPE, and LDPE are presented. In addition to the overall crystallization kinetics, the developed technique includes a video camera and VCR system used for measurements of spherulite growth rates during crystallization under high cooling rates. Constant spherulite growth rates were observed for isotactic polypropylene crystallized under very non-isothermal conditions. © 1996 John Wiley & Sons, Inc.     

A computer simulation of model discotic dimers

We set up a model for discotic liquid crystal dimers and study, by means of Monte Carlo simulations, their phase behaviour and self–assembling properties, in comparison with the simpler monomeric case. Each discotic dimer is described by two oblate Gay–Berne ellipsoids connected by a flexible spacer, modelled by a harmonic “spring” of three different lengths. We find that dimerization in general yields produces a significant change on the phase behaviour, with an increase of the columnar–nematic transition temperature, a widening of the nematic region and the apparent suppression of the crystalline phase in favour of the columnar phase up to very low temperatures. Longer spacers prove to ease the formation of columns and to increase the orientational order. Contribution to the Fernando Bernardi memorial issue.     

Isothermal and nonisothermal crystallization kinetics of irradiated nylon 1212

The crystallization behavior of nylon 1212, irradiated at ⁶⁰Co γ‐rays (50 kGy), was studied by a rheometer, polarized optical microscopy (POM), and differential scanning calorimeter (DSC). The results showed that irradiated nylon 1212 samples exhibited abnormal crystallization behavior during the crystallization process: The Avrami exponent n was calculated and was found to be in the range from 2.06–2.41 for isothermal crystallization, and from 2.67–4.91 for nonisothermal crystallization; the spherulite morphology also changed largely by polarized optical microscopy (POM); the crystallization activation energy ΔE for isothermal and nonisothermal crystallization process of irradiated nylon 1212 are determined to be 57.4 kJ/mol and 78.65 kJ/mol, respectively, which are lower than that of nonirradiated nylon 1212. At the same time, a new method by a combination of the Avrami and Ozawa equations was successfully applied to analyze the noncrystallization process of irradiated nylon 1212. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 2326–2333, 2005     

A stochastic simulation of nonisothermal nucleation

The results of stochastic simulations of growth and evaporation of small clusters in vapor are reported. Energy dependent growth rates are determined from the monomer-cluster collision rate and decay rates are found from a detailed balance, with the equilibrium size and energy distribution of clusters calculated using the capillarity approximation and the equilibrium vapor pressure. These rates are used in simulations of two-dimensional random walks in size and energy space to determine the fraction of clusters in supersaturated vapor of size (i(min)+1) that reach a size i(max). By assuming that clusters of size i(min) are in equilibrium, this fraction can be related to the nonisothermal nucleation rate. The simulated rates show good agreement with the previously published analytical results. In the absence of an inert carrier gas, the nonisothermal nucleation rates are typically between 1% and 5% of the isothermal rates.     

Calculation of the Avrami parameters for heterogeneous solid state reactions using a modification of the Kissinger method

Several isothermal experiments are generally needed to determine the parameters of the Avrami equation which describe most of the heterogeneous solid state reactions. Differential scanning calorimeters are suitable for such experiments. While most differential thermal analysis (DTA) apparatus cover a wider temperature range than DSC apparatus they cannot be used to perform isothermal determinations. However, Kissinger has already shown how activation energy and frequency factor can be calculated from DTA experiments for the case of homogeneous reactions following first order kinetics. We derive in this paper an extension of the Kissinger method and show its applicability to heterogeneous reactions described by an Avrami expression. The new method will allow the study of the kinetics of metallic reactions at the higher temperature range obtainable with DTA. The transformation kinetics of the metastable equiatomic tin-nickel alloy are given as an example.

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Zusammenfassung Im Allgemeinen werden zur Bestimmung der Parameter der Avrami-Gleichung, welche sich zur Beschreibung der meisten heterogenen Festphasenreaktionen eignet, mehrere isotherme Versuche benötigt. Differential-Scanning-Kalorimeter (DSC) eignen sich für solche Versuche. Während die meisten Geräte der Differential-Thermoanalyse (DTA) ein weiteres Temperaturgebiet umfassen als DSC-Geräte, können sie zu isothermen Bestimmungen nicht eingesetzt werden. Kissinger hatte jedoch bereits gezeigt, wie die Aktivierungsenergie und der Frequenzfaktor aus DTA-Versuchen im Falle homogener Reaktionen erster Ordnung errechnet werden können. In dieser Veröffentlichung wird eine Erweiterung der Methode von Kissinger abgeleitet und ihre Anwendbarkeit auf mit einem Avrami-Ausdruck beschriebene heterogene Reaktionen gezeigt. Die neue Methode ermöglicht das Studium der Kinetik von Metallreaktionen im mittels DTA zu erhaltenden höheren Temperaturbereich. Die Umwandlungskinetik einer metastabilen äquiatomaren Zinn-Nickel-Legierung wird als Beispiel angeführt.

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Résumé Plusieurs expériences isothermes sont en général nécessaires pour déterminer les paramètres de l'équation d'Avrami qui décrit la plupart des réactions hétérogènes en phase solide. Les analyseurs calorimétriques différentiels (DSC) conviennent pour ces expériences. La plupart des appareils d'analyse thermique différentielle (ATD) couvrent un intervalle de température plus grand que les appareils DSC mais ils ne peuvent pas être utilisés pour des études isothermes. Cependant Kissinger a déjà montré comment l'énergie d'activation et le facteur de fréquence pouvaient être calculés à partir d'expériences ATD dans le cas des réactions homogènes suivant une cinétique du premier ordre. La présente publication décrit une extension de la méthode de Kissinger et montre qu'il est possible de l'appliquer aux réactions hétérogènes décrites par une expression d'Avrami. La nouvelle méthode permet d'étudier la cinétique des réactions des phases métalliques dans un domaine de température plus grand qu'avec l'ATD. La cinétique de transformation d'un alliage métastable équiatomique étain-nickel est donnée comme exemple.

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Simulation of nonisothermal reactive liquid chromatography using two-dimensional lumped kinetic model

A nonisothermal two-dimensional lumped kinetic model of reactive liquid chromatography is formulated and applied to simulate the separation of multicomponent mixtures in a fixed-bed cylindrical column operating under nonisothermal condition. The axial and radial variations of concentration and temperature as well as reversibility of the chemical reactions are incorporated in the model equations. The model comprises a system of convection-diffusion-reaction partial differential equations coupled with algebraic and differential equations. Due to the nonlinearity of adsorption and reaction kinetics, it is required to apply an accurate numerical scheme for solving the model equations. In this study, an efficient and accurate high-resolution flux-limiting finite-volume scheme is proposed to solve the model equations. A number of stoichiometrical reactions are numerically simulated to determine the level of coupling between the temperature and concentration profiles. Moreover, the effects of various critical parameters on the process performance are examined. The results obtained are beneficial for understanding reaction and separation processes inside a liquid chromatographic reactor and to improve its performance.     

Understanding nucleation and growth using computer simulation

Atomistic simulation has been used to examine two distinct problems: (1) the growth of calcium carbonate and how growth inhibitors may operate at the atomic scale and (2) the external surface properties of a zeolite and how these influence the transport of reactants or products at the surface. In the first study we have examined how the class of monophosphonate growth inhibitors disturbs the assembly of calcite at a growing surface. We show that a retarding growth inhibition mechanism is that of kink-blocking, where the poison irreversibly binds to a kink site preventing kink–kink annihilation or step assembly in the vicinity of the impurity. Secondly, we show that two models proposed by Terasaki (O. Terasaki, J. Elec. Micro. 43 (1994) 337), are stable and provide an external surface with characteristic steric and chemical properties. To assess the question of transport, we used a probe molecule of benzene to navigate the energy barrier that separates the external surface and internal surface. We find that the barrier is smaller than that in bulk which suggests that for molecules with relatively small kinetic radii, the rate determining step in the transport process will be in the host medium or crystal bulk.     

Kinetic method of analysis by analogue computer simulation

The simulation by an analogue computer of slow reactions used for the purpose of chemical analysis is described. Bromination of maleic and fumaric acids is used for illustration. The concentration of bromine is traced spectrophotometrically as a function of time and the trace simulated with an analogue computer by the trial and error method. The initial concentration of the substance in question and the rate constant of the reaction are determined from the values given to the computer variables to obtain the best fit. Only a single experiment is required for the analysis. The procedure can be followed easily without laborious mathematical or graphical treatment.     

High-performance liquid chromatographic method-development using computer simulation

Computer-assisted method-development schemes for high-performance liquid chromatography have taken another step forward with the introduction of 'computer simulation'. Software for the application of this technique is described and illustrated.     

Isothermal and nonisothermal crystallization kinetics of nylon‐46

Isothermal and nonisothermal crystallization kinetics of nylon‐46 were investigated with differential scanning calorimetry. The equilibrium melting enthalpy and the equilibrium melting temperature of nylon‐46 were determined to be 155.58 J/g and 307.10 °C, respectively. The isothermal crystallization process was described by the Avrami equation. The lateral surface free energy and the end surface free energy of nylon‐46 were calculated to be 8.28 and 138.54 erg/cm², respectively. The work of chain folding was determined to be 7.12 kcal/mol. The activation energies were determined to be 568.25 and 337.80 kJ/mol for isothermal and nonisothermal crystallization, respectively. A convenient method was applied to describe the nonisothermal crystallization kinetics of nylon‐46 by a combination of the Avrami and Ozawa equations. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 1784–1793, 2002