Avrami formulation

The influence of heterogeneous nucleation on the crystallization kinetics is examined in Fig. 9.12 in terms of the Avrami formulation. In this figure the... 

The isotherms illustrated in Fig. 9.12 only represent one type of nuclei activation. If, for example, all of the potential nuclei are activated at r = 0, equations of the form of Eqs. (9.28) to (9.31) result. There is obviously a variety of activation processes that can be postulated, each of which will result in its own unique isotherm. Hence, for a given growth geometry there can be many differently shaped isotherms that have n values that not only differ but apparently vary with the time course of the transformation. Hence, it should not be unexpected when considering the complete transformation to observe decreasing values of the Avrami exponent with time. This observation is not caused by a deficiency in the general Avrami formulation (Eqs. 9.26 and 9.27) but rather by the varying nucleation rate with time. 

Another multi-stage process that has been suggested focuses attention on nucle-ation rates.5(61) Two distinctly different steady-state nucleation rates, M and N2, are assumed. The rates change at time t = f. Thus at / < c, N2 - 0, and the derived Avrami formulation will apply. However, at time t > c the steady-state nucleation rate will be given by... 

When applying this concept to the Avrami formulation the phantom nuclei have to be taken into account. They will now be located in both the untransformable as well as the transformed regions. The analysis proceeds as previously with either Eq. (9.26) or (9.27). Neither the fraction transformed nor untransformable appear explicity in either of these equations. Hence, in order to introduce the influence of the untransformable fraction on the extent of the transformation a decrease in N(r) with the level of crystallinity, or time, needs to be postulated. Several efforts have been made to resolve the problem in this manner.(63-66) However, they all involve postulating an arbitrary retardation in either the nucleation rate, the hneal growth rate, or in both. There is no physical or molecular basis for the functions that have been proposed. The normalization procedure that led to Eq. (9.42) was an effort to account for the fractions of untransformable material in the kinetics. The formalism of nucleation and growth has also been applied to the development of the stable crystalline state from a metastable one, rather than from the pure melt.(66a) Formally, the analysis is qualitatively similar to the two-stage series process that was discussed previously.(54,55)... 

Avramis, V. and Panosyan, E. 2005. Pharmacokinetic/pharmacodynamic relationships of asparaginase formulations - the past, the present and recommendations for the future. Clinical Pharmacokinetics 44(4), 367-393. 

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... 

Dibildox-Alvarado, E., and J.F. Toro-Vazquez, Evaluation of Trijtalmitin Crystallization in Sesame Oil Through a Modified Avrami Equation, Ibid. 75 73-76 (1998). deMan, J.M., and L. deMan, Palm Oil as a Component for High- Quality Margarine and Shortening Formulations, Mai. Oil Sci. Tech. 4 56-60 (1995). 

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). 

By use of the Poison distribution, Avrami derived the famous Avrami phenomenological equation to treat a kinetic process (Avrami 1939, 1940, 1941). Kolmogorov first discussed the formulation of this equation (Kolmogorov 1937). Johnson and Mehl also made similar derivation independently (Johnson and Mehl 1939). Evans proposed a very concise derivation as introduced below (Evans 1945). 

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. 

To study kinetic parameters of nonisothermal crystallization processes, several methods have been developed and the majority of the proposed formulations are based on the Avrami equation, which was developed for isothermal crystallization conditions. 

The Kolmogoroff-Avrami formalism can be used without restrictions in the case of 3D, 2D and ID crystallization in three-, two- and onedimensional space, respectively [5.19]. However, the method cannot be applied directly to the nucleation, growth and coalescence of 3D clusters on a plane substrate. The reason is that such clusters caimot grow in the direction perpendicular to the substrate and therefore the spread of the 3D deposit is not random in space [5.53]. Since the formulation of a rigorous theoretical model encounters principle difficulties, here we do not consider this complex case of mass electrocrystaUization. However, theoretical treatment of the nucleation, growth and overlap of circular cones, hemispheres and three-dimensional clusters with more complex geometrical forms can be found in [5.29, 5.53-5.61],... 

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