Avrami model isotherms

Applying the Avrami model to the analysis of the isothermal crystallization of interesterified and noninteresterified 20%SSS/80%000 at 30°C, 40 C and 50 C, many differences can be observed (Table 17.3). At 30°C and 40°C growth would be described as rodlike with instantaneous nucleation for both interesterified and noninteresterified samples. Also, for the noninteresterified system at 50°C spherulitic growth with instantaneous nucleation takes place. The half-time of nucleation... 

It was found that is a function of temperature but the model was found to give a better fit than analytical expressions like the Avrami model or the modified Gompertz model (Kloek, Walstra and Van Vliet 2000). The main advantage of this model is that as it is formulated as a differential equation, it can be used to predict isothermal as well as dynamic crystallization. However, this model does not consider the polymorphism of the material which is a critical point in the crystallization of cocoa butter. Another contribution is the model of Fessas et al. (Fessas, Signorelli and Schiraldi 2005) which considers all the transitions possible between each... 

The Avrami model (19,20) states that in a given system under isothermal conditions at a temperature lower than V. the degree of crystallinity or fractional crystallization (70 as a liinction of time (t) (Fig. 11) is described by Equation 5. Although the theory behind this model was developed for perfect crystalline bodies like most polymers, the Avrami model has been used to describe TAG crystallization in simple and complex models (5,9,13,21,22). Thus, the classical Avrami sigmoidal behavior from an F and crystallization time plot is also observed in TAG crystallization in vegetable oils. This crystallization behavior consists of an induction period for crystallization, followed by an increase of the F value associated with the acceleration in the rate of volume or mass production of crystals, and finally a metastable crystallization plateau is reached (Fig. 11). 

K " and n can be extracted from the intercept and the slope of Avrami plot, lg[-ln(l-.A0] versus lg(f-f ), respectively. The prime requirement of Avrami model is the ability of spherulites of a polymer to grow in a free space. Besides, Avrami equation is usually only valid at low degree of conversion, where impingement of polymer spherulites is yet to take place. The rate of crystallization of polymer can also be characterized by reciprocal half-time (/ 5). The use of Avrami model permits the understanding on the kinetics of isothermal crystallization as well as non-isothermal crystallizatioa However, in this chapter the discussion of the kinetics of crystallization is limited to isothermal conditions. 

TABLE 10 The Avrami exponents and the rate constants after Avrami model for isothermal crystallization kinetics of PET, PBT, and PTT (Adapted from Dangseeyun and co workers (2004)... 

Ozawa extended the Avrami model to quantify polymer crystallization kinetics using noniso-thermal data [289]. It was reasoned that nonisothermal crystallization amounted to infinitesimal short crystallization times at isothermal conditions, given a crystallization temperature T [290]. This analysis led to the following equation ... 

The isothermal overall crystallization kinetics of polymer blends from the melt can be analyzed on the basis of the Avrami model [38, 39] ... 

In quite another approach to the problem it has been postulated that two distinctly different Avrami type crystallizations are operative during the transformation. These processes can occur in either series or parallel with one another. They are based on the derived Avrami expression, Eq. (9.31a). In the series type the first step is termed the primary crystallization, the other the secondary one. However, it is not made clear in many applications where in the model isotherm shown in Eig. 9.18 one process stops and the other begins. The primary process that is initiated at time t = 0 is given by (54,55)... 

The key to modelling the crystallization process is the derivation a kinetic equation for a(t,T). It is possible to find different versions of this equation, including the classical Avrami equation, which allows adequate fitting of the experimental data. However, this equation is not convenient for solving processing problems. This is explained by the need to use a kinetic equation for non-isothermal conditions, which leads to a cumbersome system of interrelated differential and integral equations. The problem with the Avrami equation is that it was derived for isothermal conditions and... 

An example of an intensively studied set of polymorphs whose decompositions are of great theoretical and practical importance (see Chapter 12) is CaCOj which may exist (in order of decreasing thermodynamic stability) as calcite, aragonite or vaterite [18]. Vaterite can be prepared by precipitation from aqueous solutions under carefully controlled conditions. A DTA curve for the vaterite calcite transition is shown in Figure 2.3. The transition is exothermic AH = -34.3 J g ) with onset at 704 K. Isothermal extent of conversion against time curves were described [18] by the Johnson, Mehl, Avrami, Erofeev model (see Chapter 3) with n = 2. The measured Arrhenius parameters were F, = 210 kJ mol and A = 1.15x10 min. The decomposition of vaterite and its concurrent transformation to calcite under various conditions were compared [18] with the decomposition of calcite xmder the same conditions (see Chapter 12). 

Thermal characteristics of encapsulates are important, especially for aroma active compounds, from the viewpoint of their release in thermally processed food. In one of the recent publications, the thermal release of vanillin encapsulated in Carnauba wax microcapsules was studied by isothermal thermogravimetric analysis at a temperature range of 170°C-210°C. Kinetic studies revealed that the release is not a single-step reaction but a complex kinetic process that can satisfactorily be described by the Avrami-Erofe ev kinetic model A3. More importantly, thermal release of vanillin encapsulated into Carnauba wax proceeded with an activation energy lower than 40 kJ moL, indicating that the Carnauba wax microcapsules release vanillin relatively easily and thus suggesting that the Carnauba wax can be suitably used as a carrier for aromas especially in the food industry. 

Under the aforementioned hypotheses, as two different crystalline phases are formed (a and mesomorphic), at least two kinetic processes take place simultaneously. The simplest model is a parallel of two kinetic processes non-interacting and competing for the available molten material. The kinetic equation adopted here for both processes is the non-isothermal formulation by Nakamura et al. (Nakamura et al., 1973, Nakamura et al., 1972) of the Kolmogoroff Avrami and Evans model (Avrami, 1939,1940,1941, Evans, 1945). 

This is in agreement to the results of rate corrstant obtained from Avrami isothermal crystallization model. Meanwhile, the artalysis of activation energy of isothermal crystallization of PHAs has been investigated from different approach. It has been proposed that the plot of In V versus (RT) according to... 

The common method for modeling both isothermal and non-isothermal crystallization kinetics from amorphous solids is the Johnson-Mehl-Avrami -Kolmogorov... 

Crystallisation processes in PEEK have been the subject of many academic papers [10-13]. However, the crystallisation of PEEK generally matches the classic behaviour of other polymers. The effect of time is described by Avrami kinetics ( 3) and secondary crystallisation occurs after the spherulites have impinged. This secondary crystallisation results from an increase in the crystallinity within the spherulites and is probably related to the existence of the low-temperature melting peaks (LTMP) described later. A number of non-isothermal crystallisation models have been developed. 

After switching from fast cooling to isothermal conditions at time zero, the measured heat flow rate exponentially approaches a constant value (-10.3 mW) with a time constant of about 3 seconds for this DSC. The observed crystallization peak is often symmetric, and then the time of the peak maximum (nunimum) is a measure of crystallization half time. Integration of the peak yields the enthalpy change, which can be transformed into relative crystallinity by dividing by the limiting value at infinite time. To obtain development of absolute crystallinity (mass fraction) the curve has to be divided by the enthalpy difference between crystal and liquid at the crystallization temperature, which is available from ATHAS-DB [124], The commonly applied Kolmogorov-Johnson-Mehl-Avrami (KJMA) model for the kinetic analysis of isothermal crystallization data is based on volume fractions. Therefore, the mass fraction crystallinity, Wc, as always obtained from DSC, should be transformed into volume crystallinity. 

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