Ozawa equation

The Ozawa equation of isothermal crystallization dynamics applied to non-isothermal crystallization assumes that the crystallization proceeds under a constant cooling rate, from the valid mathematical derivation of Evans [47], In... 

Kq(T) is called the cooling function. Accordingly, the Ozawa equation is derived as given by... 

With a change of cooling rate, the temperature region for polymer crystallization shifts. The Ozawa method may not be easy to provide enough data points exhibiting a good linear relationship. Liu and Mo proposed a combination of Ozawa equation... 

Differential scanning calorimetry was used to study the non-isothermal crystallization behavior of blends of poly(phenylene sulfide) (PPS) with the thermotropic liquid-crystalline copoly(ester amide) Vectra-B950 (VB) [126], The PPS crystallization temperature and the crystallization rate coefficient increased significantly when 2-50% VB was added. The Ozawa equation was shown to be valid for neat PPS as well as for the blends. The values of the Avrami exponents matched well against those determined previously using isothermal analysis, and they are independent of the concentration of VB. 

To characterize the spherulitic nucleation during nonisothermal crystallization, the Ozawa equation is applied, which could be obtained by integrating twice by parts the Avrami equation and assuming cooling at the constant rate, a. The slope of the plot ln -ln[l - a(T)] versus In(fl) equals two or three for instantaneous nucleation, three or four for nucleation prolonged in time, in two- and three-dimensional crystallization, respectively. The values from three to four, depending on temperature range were obtained for iPP from DSC nonisothermal crystallization [4],... 

Keywords entanglement, disentanglement, cross-hatching, lamellae, crystallization, nucleation, reptation, nucleation (crystallization) regimes, nucleation agents, nucleation rate, spherulitic growth rate, Avrami-equation, Ozawa-equation, isothermal crystallization, nonisothermal crystallization, secondary nucleation, supercooling. 

Table 3.8 Exponent of Ozawa equation (m) and crystallization mechanism... 

In the non-isothermal method, the values of the Avrami parameter, n, are determined from the Ozawa equation (Qelikbilek et al., 2011 Ozawa, 1971 Prasad Varma, 2005) ... 

There exist also different approaches for the interpretation of the activation energy in the literature, such as the modified Ozawa equation by Matusita et al. (Equation 59) (Matusita Sakka, 1981) ... 

The values of the Avrami parameter, n, were calculated from the linear fits to the experimental data based on the Ozawa equation (Equation 57), as shown in Fig. 23. The n value was determined as 1.14 for 0.90Te02-0.10W03 glasses. On the basis of the determination about the non-integer value of the Avrami parameter, in this study the n value was determined as 1, indicating the formation of surface ciystallization during the crystallization process (see Table 1). 

For further clarification, the activation energy of crystallization, E, is calculated for glasses by using the modified Ozawa equation ... 

The experimental data were analyzed with the Ozawa and Ziabicki theories. The Ozawa equation was satisfactorily used to describe the dynamic solidification of PBl. The value of the Avrami exponent, calculated with the Ozawa method, was close to 3, as shown in Table 5, in quite good agreement with the value obtained in isothermal conditions (see Section II.C.4). Conversely, the use of Ziabicki theory was not in good agreement with the experimental results it was found that the zero-order approximation did not describe the nonisothermal crystallization process of PBl, probably indicating that athermal nucleation is not negligible. 

In order to describe the non-isothermal crystallization process more effectively for comparison, Liu et al. [75] suggested a convenient procedure for characterizing non-isothermal crystallization kinetics by combining the Avrami and Ozawa equations based on the assumption that the degree of crystallinity was correlated to the cooling rate and crystallization time. Therefore, their relationship for non-isothermal crystallization can be derived by combining the equations (3) and (4) as follows ... 

Figure 12. Plots of log a versus log t from the combined Avrami and Ozawa equations at different relative degree of crystallinity for (a) PEN and (b) the PEN/CNT 0.5 nanocomposites.

Table 5. Values of b and F T) for the PEN and the PEN/CNT nanocomposite obtained from the combined Avrami and Ozawa equation...

S.4.2.2 Nonisothermal Crystallization The Avrami equation does not apply when we cool the melt from a higher to a lower temperature continuously, or in the nonisothermal crystallization process. We can also use the Ozawa equation or a similar equation [51] ... 

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 ... 

In the view of equations describing the nonisothermal crystallization in detail, kc has no physical meaning. Liu et al. [68] combined isothermal Avrami equation (Eq. 7.10) with the nonisothermal Ozawa equation into a single equation ... 

A plot of log V against log t should give a straight line with the intercept of log F T) = log[H(T)/k] and the slope of equal to -n/M. In fact, the Ozawa equation and the isothermal Avrami equation (Eq. 7.10) are derived from the same general Equation (7.6a) and Equation (7.6b) but with different assumptions in the Ozawa equation a constant cooling rate is assumed, while in the Avrami Equation (7.10) the growth rate G is a constant. Therefore, the combination of the equations proposed in References [68] and used by many authors has no justification and is erroneous. 

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