Avrami equation modified

Finally, the Avrami equation modified by Jeziorny (II.I) was applied for studying the non-isothermal crystallization kinetics (Xu et al. 2005) ... 

Figure 4-14 Calculated F versus for bubble growth using the program of Proussevitch and Sahagian (1998) modified by Liu and Zhang (2000). F is the volume of the bubble versus the final equilibrium volume of the bubble. The calculated trend may he fit by the Avrami equation with an n value of 0.551.

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). 

Marangoni (8) has warned of the misuse of a modified form of the Avrami equation. 

The caution by Marangoni (8) on the misuse of a modified form of the Avrami equation is rightly justified within the context of the warning. However, if the equation is to be applied to the crystallization of oils and fats in order to elucidate information on the mechanisms of their nucleation and growth, the equation will have to be modified to suit the conditions prevailing in the system. 

Crystallization kinetics was studied using the modified Avrami equation. 

J. N. Hay, Application of the modified Avrami equation to polymer crystallization kinetics, Brit. Polym. J. 3,74 (1971). 

Transformation kinetics according to Ozawa Ozawa (2) proposed a method to predict crystallization of polymers under constant cooling or heating rate by modifying the Avrami equation ... 

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). 

Davidson et al. (2008) measured the rates of this transformation over the temperature range of 60 to 190°C and fit the results to the modified Avrami equation. The average value of n from their fits was 2.1(0.25). Fitting their reported rate constants to the Arrhenius equation gives an estimated value of of 32.4 kJ/mol with log A = 1.31. These values are substituted into Eq. (9.72) to create the TTT diagram shown in Figure 9.8. 

For non-isothermal crystallization, the process can be described by the modified Avrami equation in which the crystallization temperature is converted to time through heating or cooling rate,, by dividing crystallization temperature into infinity small [Equation (3)]. 

The crystallization rate can be described further by the reciprocal of half-time of crystallization. Jeziomy modified Avrami equation to accoimt for the effect of cooling... 

AppHcation of the modified Avrami equation (Eq. 29) gave values for n of about 4 (decreasing to 3.1 at 60 wt % PCL) these higher values were attributed... 

The nonisothermal crystallization kinetics of polymers can start with isothermal crystallization, and be corrected considering the characteristics of nonisothermal crystallization. The common DSC methods include Jeziomy, Ozawa, and MoZhishen methods. The nonisothermal crystallization kinetics of pure PP and 5% whisker-filled PP are compared using the Jeziorny modifying method of the Avrami equation. 

The Avrami equation should be modified when dealing with nonisothermal crystallization kinetics. 

In turn, the share of the transformed components participating in phase interaction can be expressed with the equation, which can be considered as a modified Avrami equation. 

Thus, with the help of Avrami equations or their modified analogs we can determine the optimal duration of the process to obtain the required result. It opens up the possibility of defining other parameters of the process and characteristics of nanostmctures obtained (by shape and sizes). 

Hyperbolic curves can also be described by the modified Avrami equation, but in this case n should set to 1. The resulting equation is an exponential growth curve. There also was a good fit of the model to the data over the same range of fractional crystallizations. Tables 5 and 6 show the values of and the correlation coefficients for samples 16, 15, and 14 (second hydrogenation process) and 8, 7, and 6 (first hydrogenation process), respectively. 

The rate of crystallization can now be assessed using the modified Avrami equation (Eq. 5.162) as... 

The Avrami equation is frequently applied directly to isothermal DSC data where the heat output of the crystallisation process is thought to describe exactly the crystallisation. It can also be applied in modified form to non-isothermal data such as the crystallisation of PET on cooling. It should be noted that there is a certain ambiguity in the exponent value. For example, a value of 3 can mean either 2D nucleation from nuclei appearing randomly in time, or 3D nucleation from pre-existing nuclei. 

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