Avrami approach

Figure 3.4 Geometric representations of (a) the Evans (a) and (b) the Avrami approach to the statistical formulation of the coverage.

Since Equation (3.5) was derived independent of any assumptions regarding the shape of the growth centers, it follows that = E for all growth forms. However, the use of the Avrami approach requires to write Si g t = Ee e. Thus, calculations involving Si g t will evidently require a more complex approach than the use of E directly, and this latter route will be the preference when formulating all forms of growth. 

Additional assumptions about the relation between nucleation and the growth rate as in the Nakamura isokinetic model [8, 9] can simplify the description of overall kinetics. In the Avrami approach [3-5], I>o nuclei present initially in a unit volume are activated according to the activation frequency q(t) ... 

Barium titanate powders were produced using either an amorphous hydrous Ti gel or anatase precursor in a barium hydroxide (Ba(OH)2) solution via a hydrothermal technique in order to discern the nucleation and formation mechanisms of BaTi03 as a function of Ti precursor characteristics. Isothermal reaction of the amorphous Ti hydrous gel and Ba(OH)2 suspension is believed to be limited by a phase boundary chemicd interaction. In contrast, the proposed BaTi03 formation mechanism from the anatase and Ba(OH>2 mixture entails a dissolution and recrystallization process. BaTi03 crystallite nucleation, studied using high resolution transmission electron microscopy, was observed at relatively low temperatures (38°C) in the amorphous hydrous Ti gel and Ba(OH)2 mixture. Additional solution conditions required to form phase pure crystallites include a C02-free environment, temperature >70°C and solution pH >13.4. Analysis of reaction kinetics at 75°C was performed using Hancock and Sharp s modification of the Johnson-Mehl-Avrami approach to compare observed microstructural evolution by transmission electron microscopy (7). 

The approach taken by Nakamura et al, in another theoretical development, is also based on the Avrami approach.(86,87) However, in this analysis the general Avrami expression is used. Therefore, the integral in Eq. (9.27) has to be evaluated. The evaluation of this integral has been simplified in the theory by assuming the isokinetie condition. This condition requires that the ratio G(T)IN(T) be a constant. With this requirement, the integral can be evaluated. The result is that the relative extent of the transformation can be expressed as (87,88)... 

Magnitudes of n have been empirically established for those kinetic expressions which have found most extensive application e.g. values of n for diffusion-limited equations are usually between 0.53 and 0.58, for the contracting area and volume relations are 1.08 and 1.04, respectively and for the Avrami—Erofe ev equation [eqn. (6)] are 2.00, 3.00 etc. The most significant problem in the use of this approach is in making an accurate allowance for any error in the measured induction period since variations in t [i.e. (f + f0)] can introduce large influences upon the initial shape of the plot. Care is needed in estimating the time required for the sample to reach reaction temperature, particularly in deceleratory reactions, and in considering the influences of an induction period and/or an initial preliminary reaction. 

The most commonly employed approach is that of Avrami and Erofe ev. This rate law has been derived in a munber of ways, indicating its general apphcabiUty and vaUdity. This equation takes the form... 

One approach is to roughly estimate how the degree of crystallization would vary with time by making main simplifications in treating solidification, leading to the Avrami equation. 

Takebe et al. [28] studied the effect of temperature and molecular weight on the crystallization rate of SPS by DSC. When SPS was melted at 300 °C, then rapidly cooled to the crystallization temperature, T, the evolution of crystallization showed a sigmoidal curve with reference to crystallization time (Figure 18.6). The crystallization rate becomes slower as T approaches close to melting point. When the crystallization rate of SPS is analyzed based on Avrami theory, the Avrami index, n, is equal to 3, which suggests that crystallization of SPS proceeds via three-dimensional heterogeneous growth [28,29]. 

Avrami Analysis The Avrami equation, a general approach for description of isothermal phase transformation kinetics originally developed for polymers (46), is often used for describing nucleation and crystal growth in fats. The Avrami equation is given as... 

In microscale models the explicit chain nature has generally been integrated out completely. Polymers are often described by variants of models, which were primarily developed for small molecular weight materials. Examples include the Avrami model of crystallization,- and the director model for liquid crystal polymer texture. Polymeric characteristics appear via the values of certain constants, i.e. different Frank elastic constant for liquid crystal polymers rather than via explicit chain simulations. While models such as the liquid crystal director model are based on continuum theory, they typically capture spatiotemporal interactions, which demand modelling on a very fine scale to capture the essential effects. It is not always clearly defined over which range of scales this approach can be applied. 

Erofeev s approach [5,8,22], with contributions fi om Kholmogorov and Bel kevich [23,24], was more general than that of Avrami and was based on the probability of the reaction step occurring in a particular time interval. The rate of reaction is ... 

Figure 6.28. Examples of crystallization isotherms satisfying Avrami s equation (Equation 6.30, with K obtained from Equation 6.31). Curve labels denote ti/2 and n. Changing ti/2 from 100 to 200 seconds at a constant n (3 in this example) results in the simple scaling of the crystallization isotherm along the time axis without any other change in its shape. Decreasing n at a constant ti/2 (100 seconds in this example) broadens the crystallization isotherm so that the induction time (the time lapsed before the onset of appreciable crystallization) becomes shorter but it takes a much longer time for crystallization to approach completion.

Calka, A. Radlinski, A. P. (1988). The local value of the Avrami exponent A new approach to devitrification of glassy metallic ribbons. Materials Science and Engineering, 97, 241-6. 

The Avrami-exponent m = 1.48 points to nucleation according to an exponential law coupled with surface diffusion-controlled growth (Eq. (34) and Eq. (35), solid line). The experimental results in panels (b) and (c) indicate that the spectroscopic and the electrochemical transients probe different interfacial properties of the dissolution process and illustrate the complementary information of both approaches (reproduced from Ref. [475]). 

From a technological point of view non-isothermal crystallization is important since many processing procedures are carried out under non-isothermal conditions. There are quite a number of approaches to non-isothermal crystallization. We restrict the discussion to non-isothermal conditions with constant cooling rate s. The Avrami equation serves as starting point for generahzations to non-isothermal conditions. We will discuss two approaches in detail that have been successfully applied. 

A series of non-isotherms with 5 = const is in that way characterized by an isothermal eqnivalent. There is a different law of conversion for each period of time. An example is given in Figure 10 for T=12 °C. Only a selected number of nonisotherms contributes to the Avrami-like relationship, from 5 = 7.5 to 30 K/min. As in isothermal Avrami plots (see Figure 2), deviations occnr at high conversions. The quasi-isothermal approach is depicted in Fignre 14 for non-isothermal ciystallization of PHBV. 

The crystallization of polymer in bulk as well as in solution is initiated by nucle-ation followed by growing of spheralites (Mandelkem, 2002). A common fundamental approach to study isothermal crystallization kinetics is the heuristic Avratni phase transition theory (Avrami, 1939 1940 1941). 

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... 

It is interesting to note that the deviations from the Avrami expression occur at a level that is given by the limit of applicability of the free growth approximation in Fig. 3.55. Similar data for a single molar mass, but at different temperatures, are shown in Fig. 3.101. All crystallizations seem to approach a common limit, but deviate at different temperatures from the Avrami equation. 

Accordingly, on a series of crystallinity versus time curves (see Fig. 10.31b), one can take the data points along the horizontal equal-crystallinity line, and then obtain the ratios of the Avrami indexes and the rate constants separately from the slope and the intercept of log(fl) versus log(f)- The experiments have verified that a better linear relationship can be obtained in comparing this approach to the conventional Ozawa method. 

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