Cell size distribution in a random tessellation of space governed by the Kolmogorov-Johnson-Mehl-Avrami model: Grain size distribution in crystallization

The space subdivision in cells resulting from a process of random nucleation and growth is a subject of interest in many scientific fields. In this paper, we deduce the expected value and variance of these distributions while assuming that the space subdivision process is in accordance with the premises of the Kolmogorov-Johnson-Mehl-Avrami model. We have not imposed restrictions on the time dependency of nucleation and growth rates. We have also developed an approximate analytical cell size probability density function. Finally, we have applied our approach to the distributions resulting from solid phase crystallization under isochronal heating conditions

Comparação do ajuste dos modelos de morgan-mercer-flodin e de johnson-mehl-avrami na reação de precipitação na liga cu-3%al-5%ag

Em geral, os modelos de crescimento empregados na descrição de processos cinéticos do estado sólido utilizam as funções de Morgan-Mercer-Flodin (MMF) e Johnson-Mehl-Avrami (JMA). Neste trabalho comparou-se o comportamento das estimativas dos parâmetros dessas funções, em duas parametrizações, quando ajustadas aos dados experimentais de variação isotérmica da microdureza da liga Cu-3%Al-5%Ag com o tempo de envelhecimento, em cinco temperaturas diferentes. Na estimativa dos parâmetros aplicou-se o método dos mínimos quadrados à função linearizada do modelo e para o refinamento da solução, os procedimentos da regressão não linear. Foram obtidas as distribuições de freqüências das estimativas dos parâmetros, por simulação, e determinados seus erros relativos. Observou-se que, em uma das parametrizações das funções, a estimativa do parâmetro cinético apresentou maior estabilidade. Apesar dessas funções serem anteriormente consideradas distintas, verificou-se que a função de MMF é uma aproximação da função de JMA, sendo que esta, com a parametrização adequada, é a mais indicada para descrever o processo cinético considerado.

Avrami exponent of crystallization in tellurite glasses

Nucleation process and crystal growth for three samples of the (20-x)Li(2)O-80TeO(2)-xWO(3) glass system were studied using X-ray diffraction and differential scanning calorimetry techniques. X-ray diffraction data confirmed the amorphous characteristic of the as-quenched samples and indicated the growth of crystalline phases formed due to the thermal treatment for annealed samples. These results reveal the presence of three distinct gamma-TeO(2), alpha-TeO(2) and alpha-Li(2)Te(2)O(5) crystalline phases in the TL sample, and two distinct alpha-TeO(2) and gamma-TeO(2) crystalline phases in the TLW5 and TLW10 samples. The activation energy and the Avrami exponent were determined from DSC measurements. The activation energy values X-ray diffraction data of the TLW10 glass sample suggest that gamma-TeO(2) phase occur before the alpha-TeO(2). The results obtained for the Avrami exponent point to that the nucleation process is volumetric and that the crystal growth is two or three-dimensional.

Comparação do ajuste dos modelos de morgan-mercer-flodin e de johnson-mehl-avrami na reação de precipitação na liga cu-3%al-5%ag

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

The stability of the Johnson-Mehl-Avrami equation parameters

The stability of the parameters of the Johnson-Mehl-Avrami equation was studied using two parametrizations of the sigmoidal function and its fit to some kinetic data. The results indicate that one of the forms of the function has more stable parameters and only for this form it is reasonable to use, as an approximation, the linear regression theory to analyse the parameters. © 1995 Chapman & Hall.

Influence of chemical interesterification on thermal behavior, microstructure, polymorphism and crystallization properties of canola oil and fully hydrogenated cottonseed oil blends

This work evaluated chemical interesterification of canola oil (CaO) and fully hydrogenated cottonseed oil (FHCSO) blends, with 20%, 25%, 30%, 35% and 40%(w/w) FHCSO content. Interesterification produced reduction of trisaturated and increase in monounsaturated and diunsaturated triacylglycerols contents, which caused important changes in temperatures and enthalpies associated with the crystallization and melting thermograms. It was verified reduction in medium crystal diameter in all blends, in addition crystal morphology modification. Crystallization kinetics revealed that crystal formation induction period and maximum solid fat content were altered according to FHCSO content in original blends and as a result of random rearrangement. Changes in Avrami constant (k) and exponent (n) indicated, respectively, that interesterification decreased crystallization rates and altered crystalline morphology. However, X-ray diffraction analyses showed randomization did not change the original crystalline polymorphism. The original and interesterified blends had significant predominance of beta` polymorph, which is interesting for several food applications. (C) 2009 Elsevier Ltd. All rights reserved.

Thermal Behavior, Microstructure, Polymorphism, and Crystallization Properties of Zero Trans Fats from Soybean Oil and Fully Hydrogenated Soybean Oil

Blends of soybean oil (SO) and fully hydrogenated soybean oil (FHSBO), with 10, 20, 30, 40, and 50% (w/w) FHSBO content were interesterified under the following conditions: 20 min reaction time, 0.4% sodium methoxide catalyst, and 500 rpm stirring speed, at 100 A degrees C. The original and interesterified blends were examined for triacylglycerol composition, thermal behavior, microstructure, crystallization kinetics, and polymorphism. Interesterification produced substantial rearrangement of the triacylglycerol species in all the blends, reduction of trisaturated triacylglycerol content and increase in monounsaturated-disaturated and diunsaturated-monosaturated triacylglycerols. Evaluation of thermal behavior parameters showed linear relations with FHSBO content in the original blends. Blend melting and crystallization thermograms were significantly modified by the randomization. Interesterification caused significant reductions in maximum crystal diameter in all blends, in addition to modifying crystal morphology. Characterization of crystallization kinetics revealed that crystal formation induction period (tau (SFC)) and maximum solid fat content (SFC(max)) were altered according to FHSBO content in the original blends and as a result of the random rearrangement. Changes in Avrami constant (k) and exponent (n) indicated, respectively, that-as compared with the original blends-interesterification decreased crystallization velocities and modified crystallization processes, altering crystalline morphology and nucleation mechanism. X-ray diffraction analyses revealed that interesterification altered crystalline polymorphism. The interesterified blends showed a predominance of the beta` polymorph, which is of more interest for food applications.

Cristalização de Biomateriais Vitrocerâmicos e Mineralização em Meio Fisiológico Simulado

We propose new theoretical models, which generalize the classical Avrami-Nakamura models. These models are suitable to describe the kinetics of nucleation and growth in transient regime, and/or with overlapping of nucleation and growth. Simulations and predictions were performed for lithium disilicate based on data reported in the literature. One re-examined the limitations of the models currently used to interpret DTA or DSC results, and to extract the relevant kinetic parameters. Glasses and glass-ceramics with molar formulation 0.45SiO2? (0.45-x)MgO?xK2O?0.1(3CaO.P2O5) (0?x?0.090) were prepared, crystallized and studied as potential materials for biomedical applications. Substitution of K+ for Mg2+ were used to prevent devritification on cooling, to adjust the kinetics of crystallization and to modify the in vitro behaviour of resulting biomaterials. The crystallization of the glass frits was studied by DTA, XRD and SEM. Exothermic peaks were detected corresponding to bulk crystallization of whitlockite-type phosphate, Ca9MgK(PO4)7, at approximately 900ºC, and surface crystallization of a predominant forsterite phase (Mg2SiO4) at higher temperatures. XRD also revealed the presence of diopside (CaMgSi2O6 in some samples. The predominant microstructure of the phosphate phase is of the plate-type, seemingly crystallizing by a 2-dimensional growth mechanism. Impedance spectroscopy revealed significant changes in electrical behaviour, associated to crystallization of the phosphate phase. This showed that electrical measurements can be used to study the kinetics of crystallization for cases when DTA or DSC experiments reveal limitations, and to extract estimates of relevant parameters from the dependence of crystallization peak temperature, and its width at half height. In vitro studies of glasses and glass-ceramics in acelular SBF media showed bioactivity and the development of apatite layers The morphology, composition and adhesion of the apatite layer could be changed by substitution of Mg2+ by K+. Apatite layers were deposited on the surface of glass-ceramics of the nominal compositions with x=0 and 0.09, in contact with SBF at 37ºC. The adhesion of the apatite layer was quantified by the scratch test technique, having been related with SBF?s immersion time, with composition and structure of the glass phase, and with the morphology of the crystalline phase of the glass-ceramics. The structure of three glasses (x=0, 0.045 and 0.090) were investigated by MAS-NMR ( 29Si and 31P), showing that the fraction of Q3 structural units increases with the contents of Mg, and that the structure of these glasses includes orthophosphate groups (PO43-) preferentially connected to Ca2+ ions. Mg2+ ions show preference towards the silicate network. Substitution of Mg2+ by K+ allowed one to change the bioactivity. FTIR data revealed octacalcium phosphate precipitation (Ca8H2(PO4)6.5H2O) in the glass without K, while the morphology of the layer acquires the shape of partially superimposed hemispheres, spread over the surface. The glasses with K present a layer of acicular hidroxyapatite, whose crystallinity and needles thickness tend to increase along with K content.

Crystallization behaviour of some melt spun Nd-Fe-B alloys

The kinetics of crystallization of four amorphous (or partially amorphous) melt spun Nd-Fe-B alloys induced by thermal treatment is studied by means of differential scanning calorimetry and scanning electron microscopy, In the range of temperatures explored experimentally, the crystallization process is thermally activated and generally proceeds in various stages. The Curie temperature and the crystallization behavior have been measured. The apparent activation energy of crystallization of most of the crystallization stages has been determined for each melt spun alloy. The explicit form of the kinetic equation that best describes the first stage of crystallization has been found. It follows in general the Johnson-Mehl-Avrami-Erofe'ev model, but clear deviations to that model occur for one alloy. Scanning electron microscopy demonstrates that preferentially hetereogeneous nucleation occurs at the ribbon surface which was in contact with the wheel. From crystallization kinetics results the lower part of the experimental time-temperature-transformation curves for all studied alloys are deduced and extrapolated to the high temperature limit of their range of validity, also deduced.

Cristalização e taxa crítica de resfriamento para vitrificação do poli(sebacato de decametileno)

We synthesized Poly(decamethylene sebacate) - P10MS - and studied its overall crystallization rates in a range of temperatures using Differential Scanning Calorimetry in isothermal conditions, which enabled us to identify the crystallization mechanism by means of the Johnson-Mehl-Avrami-Kolmogorov equation. The critical cooling rate (Rc) to vitrify the P10MS was determined using a non-isothermal method proposed by Barandiarán & Colmenero (BC). The value of Rc is around 50-250 K/s, which confirms the experimentally observed difficulty to vitrify this polymer.

Linear block copolymers of L–lactide and 2–dimethylaminoethyl methacrylate : synthesis and properties

Part of the research described in this thesis is conducted in collaboration with Centre d' étude et de Recherche sur les Macromolécules (CERM), Université de Liège, Sart-Tilman, Belgium

Numerical model of solid phase transformations governed by nucleation and growth: microstructure development during isothermal crystallization

A simple numerical model which calculates the kinetics of crystallization involving randomly distributed nucleation and isotropic growth is presented. The model can be applied to different thermal histories and no restrictions are imposed on the time and the temperature dependences of the nucleation and growth rates. We also develop an algorithm which evaluates the corresponding emerging grain-size distribution. The algorithm is easy to implement and particularly flexible, making it possible to simulate several experimental conditions. Its simplicity and minimal computer requirements allow high accuracy for two- and three-dimensional growth simulations. The algorithm is applied to explore the grain morphology development during isothermal treatments for several nucleation regimes. In particular, thermal nucleation, preexisting nuclei, and the combination of both nucleation mechanisms are analyzed. For the first two cases, the universal grain-size distribution is obtained. The high accuracy of the model is stated from its comparison to analytical predictions. Finally, the validity of the Kolmogorov-Johnson-Mehl-Avrami model SSSR, is verified for all the cases studied

Fractionated crystallization and fractionated melting of confined PEO microdomains in PB-b-PEO and PE-b-PEO diblock copolymers

The confined crystallization of poly(ethylene oxide) (PEO) in predominantly spherical microdomains formed by several diblock copolymers was studied and compared. Two polybutadiene-b-poly(ethylene oxide) diblock copolymers were prepared by sequential anionic polymerization (with approximately 90 and 80 wt % polybutadiene (PB)). These were compared to equivalent samples after catalytic hydrogenation that produced double crystalline polyethylene-b-poly(ethylene oxide) diblock copolymers. Both systems are segregated into microdomains as indicated by small-angle X-ray scattering (SAXS) experiments performed in the melt and at lower temperatures. However, the PB-b-PEO systems exhibited a higher degree of order in the melt. A predominantly spherical morphology of PEO in a PB or a PE matrix was observed by both SAXS and transmission electron microscopy, although a possibly mixed morphology (spheres and cylinders) was formed when the PEO composition was close to the cylinder-sphere domain transitional composition as indicated by SAXS. Differential scanning calorimetry experiments showed that a fractionated crystallization process for the PEO occurred in all samples, indicating that the PE cannot nucleate PEO in these diblock copolymers. A novel result was the observation of a subsequent fractionated melting that reflected the crystallization process. Sequential isothermal crystallization experiments allowed us to thermally separate at least three different crystallization and melting peaks for the PEO microdomains. The lowest melting point fraction was the most important in terms of quantity and corresponded to the crystallization of isolated PEO spheres (or cylinders) that were either superficially or homogeneously nucleated. This was confirmed by Avrami index values of approximately 1. The isothermal crystallization results indicate that the PE matrix restricts the crystallization of the covalently bonded PEO to a higher degree compared to PB.

Crystallization in poly(L-lactide)-b-poly(epsilon-caprolactone) double crystalline diblock copolymers: a study using X-ray scattering, differential scanning calorimetry, and polarized optical microscopy

The crystallization of well-defined poly(L-lactide)-b-poly(epsilon-caprolactone) diblock copolymers, PLLA-b-PCL, was investigated by time-resolved X-ray techniques, polarized optical microscopy (POM), and differential scanning calorimetry (DSC). Two compositions were studied that contained 44 and 60 wt % poly(L-lactide), PLLA (they are referred to as (L44C5614)-C-11 and (L60C409)-C-12, respectively, with the molecular weight of each block in kg/mol as superscript). The copolymers were found to be initially miscible in the melt according to small-angle X-ray scattering measurements (SAXS). Their thermal behavior was also indicative of samples whose crystallization proceeds from a mixed melt. Sequential isothermal crystallization from the melt at 100 degreesC (for 30 min) and then at 30 degreesC (for 15 min) was measured. At 100 degreesC only the PLLA block is capable of crystallization, and its crystallization kinetics was followed by both WAXS and DSC; comparable results were obtained that indicated an instantaneous nucleation with three-dimensional superstructures (Avrami index of approximately 3). The spherulitic nature of the superstructure was confirmed by POM. When the temperature was decreased to 30 degreesC, the PCL block was able to crystallize within the PLLA negative spherulites (with an Avrami index of 2, as opposed to 3 in homo-PCL), and its crystallization rate was much slower than an equivalent homo-PCL. Time-resolved SAXS experiments in (L60C409)-C-12 revealed an initial melt mixed morphology at 165 degreesC that upon cooling transformed into a transient microphase-separated lamellar structure prior to crystallization at 100 degreesC.

Melt structure and its transformation by sequential crystallization of the two blocks within poly(L-lactide)-block-poly(epsilon-caprolactone) double crystalline diblock copolymers

Sequential crystallization of poly(L-lactide) (PLLA) followed by poly(epsilon-caprolactone) (PCL) in double crystalline PLLA-b-PCL diblock copolymers is studied by differential scanning calorimetry (DSC), polarized optical microscopy (POM), wide-angle X-ray scattering (WAXS) and small-angle X-ray scattering (SAXS). Three samples with different compositions are studied. The sample with the shortest PLLA block (32 wt.-% PLLA) crystallizes from a homogeneous melt, the other two (with 44 and 60% PLLA) from microphase separated structures. The microphase structure of the melt is changed as PLLA crystallizes at 122 degrees C (a temperature at which the PCL block is molten) forming spherulites regardless of composition, even with 32% PLLA. SAXS indicates that a lamellar structure with a different periodicity than that obtained in the melt forms (for melt segregated samples). Where PCL is the majority block, PCL crystallization at 42 degrees C following PLLA crystallization leads to rearrangement of the lamellar structure, as observed by SAXS, possibly due to local melting at the interphases between domains. POM results showed that PCL crystallizes within previously formed PLLA spherulites. WAXS data indicate that the PLLA unit cell is modified by crystallization of PCL, at least for the two majority PCL samples. The PCL minority sample did not crystallize at 42 degrees C (well below the PCL homopolymer crystallization temperature), pointing to the influence of pre-crystallization of PLLA on PCL crystallization, although it did crystallize at lower temperature. Crystallization kinetics were examined by DSC and WAXS, with good agreement in general. The crystallization rate of PLLA decreased with increase in PCL content in the copolymers. The crystallization rate of PCL decreased with increasing PLLA content. The Avrami exponents were in general depressed for both components in the block copolymers compared to the parent homopolymers. Polarized optical micrographs during isothermal crystalli zation of (a) homo-PLLA, (b) homo-PCL, (c) and (d) block copolymer after 30 min at 122 degrees C and after 15 min at 42 degrees C.