Avrami equation classical

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... 

A similar derivation, which assumes that the transformation rate is controlled by the nucleation rate, gives an equation where t is raised to the fourth power. The classical Avrami equation generalizes these results by assuming that a is related to t by an equation with an arbitrary value for n. 

A modified version of the Avrami raises both k and t to the power of n, which gives k more conventional units of sec rather than the units of sec " that are required for the classical Avrami equation (Maffezzoli et al, 1995). 

It is often useful to express the conversion in terms of a characteristic time. The classical Avrami equation (9.55) can be solved to express t as a function of a. 

It was also demonstrated that in polymer composites, volume inhabited by embedded fibers inaccessible for crystallization and additional nucleation on internal interfaces [53,62,63], can markedly influence the overall crystallization kinetics, as described in Chapter 13. Similar problems might be encountered during crystallization in other polymer systems such as composites with particulate fillers and immiscible polymer blends. Under such conditions, the simplified Avrami equation (Eq. 7.10) does not apply and, as a consequence, the classic Avrami analysis may yield nonlinear plots and/ or noninteger n values. It must be emphasized that the problem cannot be solved by application of other, incorrect models, like that of Tobin, which are essentially based on the same assumptions as the Avrami-Evans theory but yield different equations due to incorrect reasoning. 

In the classical treatment, the crystallization kinetics, i.e., the crystallinity )-time (/) relationship, of the isothermal ciystallization process is described by the Avrami equation. ... 

The classic Avrami Equation (1) is used to study the isothermal bulk crystallization kinetics of PLA. 

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). 

The a-time curves were then fitted to the following classical Avrami-Erofe ev (AE) rate equation, which has been widely applied to the analysis of nucleation and crystallization kinetics... 

The crystalline phase typically grows as spherical aggregates called spherulites. However, other geometries such as disks or rods may be found with, as shown below, a consequent modification of the rate equation. M. Avrami [26] first derived these rate equations in the form used for polymer kinetics for the solidification of metals. The weight of the crystalline phase is calculated as a function of time at constant temperature. As will be described below, the temperature dependence of crystallization can be derived from classical nucleation theory. 

The form adopted in equation (12) for the time derivative of the expectancy reduces to the classical Avrami form, with a dimensionality index ni for the i > phase, if an isothermal experiment is considered. As for the dependence of the rate constant K, on temperature, the simplest expression that one can consider is a Gaussian shaped curve ... 

The TTT equation for the classical Avrami model is developed from Eq. (9.56). 

The ciystallization curve shown in Figure 2 was obtained in a SAXS/WAXS/DSC experiment from iPP [23] and shows the classic features of primary crystallization. The detailed molewlar structure of the polymer, the specific nature of the nucleation processes and the degree of under-coolteg, determines the magnitude of the lamellar thickness and the degree of crystallinity within the lamellar stadcs. The crystallization kinetics are analyzed using the Avrami model [24], expressed in terms of the equation... 

Therefore, a minor modification into the Avrami classical equation has to be introduced, in order to take into consideration the experimental induction time fg. Equation 11.4 can be then rewritten as... 

It has to be emphasized that the classic Avrami and Evans equations, and consequently the Nakamura approach, were derived by assuming random positions of nuclei in a material therefore, they do not apply strictly when there is a correlation between positions of nucleation sites. Such nucleation of spherulites is accounted for in the model developed originally for fiber-reinforced polymers [53], described in Chapter 13. 

Similarly, as in the case of the Avrami analysis of isothermal crystallization, the discrepancies between experimentally determined curves and predictions of the Ozawa equation originate mainly from oversimplified assumptions concerning the polymer crystallization. Those discrepancies inspired some authors to search for other equations enabling a better description and analysis of nonisothermal crystallization. For instance, the classic isothermal Avrami analysis based on Equation (7.5) with E expressed by Equation (7.10) was applied to nonisothermal crystallization [65, 66]. Such an approach has no theoretical justification. Even if a straight line Avrami plot is obtained, the parameters k and n are, at best, two adjustable parameters without a clear physical meaning. The Jeziomy method [67] deserves similar criticism. Jeziomy proposed using Equation (7.5) and Equation (7.10) and characterizing the process with the parameter kc defined as ... 

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