Deviations from Avrami Equation

Deviations from the Avrami equation are frequently encountered in the long time limit of the data. This is generally attributed to secondary nucleation occurring at irregularities on the surface of crystals formed earlier. 

We observe that quasi-isotherms shift downwards when the temperature T increases, it occurs as a consequence of slowing down of rate of ciystalhzation. One may state, isotherms and quasi-isotherms show deviations from Avrami at high conversions whereas, non-isotherms with s = const display deviations from Equation (34) at low conversions. The comparison of relevant parameters for isotherms and quasi-isotherms reveal that rates are lower and Avrami exponents are higher for isotherms as compared to quasi-isotherms. An example is given in Table 4. 

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. 

The Avrami exponent ( ) depends on nucleation type, the geometry of crystal growth and the kinetics of crystal growth (see Chapter 8). The kinetics at low degrees of conversion usually follows the Avrami equation but deviates from the linear trend in the plot... 

Eder and Wlochowicz [192] crystallized PE at constant cooling rates ranging from 0.5 to 10°C/min. Their experimental data did not conform to the theoretical treatment developed by Ozawa [177]. The authors attributed the deviation from the equation to factors such as secondary crystallization (for polyethylene it may be greater than 40% of the total [140]), dependence of the lamellar thickness on crystallization temperature, and occurrence of different mechanisms of nucleation. However, it is worth commenting that the occurrence of different kinds of nucleation would not affect the validity of Ozawa s equation, but only the value of the Avrami exponent. 

Another distinguishing aspect of copolymer crystallization is that with increasing counit content, the crystallization isotherms deviate from the Avrami relation, given in Equation 11.6 [79], at progressively lower extents of crystallization [69,76]... 

Equation (9.47) is compared with the derived Avrami, Eq. (9.31a), in Fig. 9.21. Here the extent of the transformation is plotted against the log time for integral values of the exponent n. There are only small differences between the two relations, particularly in the usual range of polymer crystalhzation. Analysis of typical kinetic data indicates that deviations from either theory occur at about the same crystallinity level. 

The Avrami exponent n is evaluated by the curve fitting procedure that has been described. It only applies to that portion of the isotherm that fits Eq. (9.3 la). However, there are situations where a subjective decision has to be made. These are cases where, although a significant portion of the transformation can be fitted by n = 3, about half the transformation is also satisfied by n = 4. The problem is whether n = 4 represents the actual mechanism with deviations from the Avrami equation ensuing. 

A similar plot for poly(butylene naphthalene 2,6-dicarboxylate) is given in Fig. 9.25.(96) In contrast to the previous figure, these plots are only linear at the lower portion and curvature is observed at the higher levels of crystallinity. These results indicate that in this case the crystallization has not been limited to Region I. Curvature in the Ozawa type plot has also been observed with poly(aryl ether ether ketone) (40), poly(aryl ether ether ketone ketone) (97) and poly(aryl ether ether sulfide).(97a) Curvature and deviation from the theory will be observed, if crystallization occurs beyond Region I, because the derived Avrami equation is no longer vaUd. 

FIGURE 3.20 Data for the primary isothermal crystallization of PET at different temperatures analyzed on the basis of the Avrami equation where X/Xp.w represents mjmg. Deviation from linearity indicates a varying value of n with extent of crystallization. (Data from Lu, X. R, and J. N. Hay, Polymer, 42, 9423, 2001. With permission.)... 

The abovementioned three equations are equivalent to the perspective of simulating the flavor release from spray-dried powder. All of the parameters - namely n in Avrami s equation, p in the KWW-equation, and a Gaussian distribution of AG with the standard deviation a in Eq. 6.9 - can be understood as a consequence of the activation energy distribution of the release rate. 

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