Nonisothermal crystallization Avrami equation

So far the crystallization kinetics of HMFG have been less studied. Both continuous heating and isothermal experiments have been performed using DTA or DSC techniques. Bansal et al. (1983) studied the crystallization kinetics of a ZBL glass by means of isothermal and nonisothermal DSC heating above T. Under isothermal conditions, the volume fraction cc crystallized after a time t was found to follow the Avrami equation. 

The overall crystallization rate is used to follow the course of solidification of iPP. Differential scanning calorimetry (DSC), dilatometry, dynamic X-ray diffraction and light depolarization microscopy are then the most useful methods. The overall crystallization rate depends on the nucleation rate, 1(0 and the growth rate of spherulites, G(0. The probabilistic approach to the description of spherulite patterns provides a convenient tool for the description of the conversion of melt to spherulites. The conversion of melt to spherulites in the most general case of nonisothermal crystallization is described by the Avrami equation ... 

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. 

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. 

The nonisothermal crystallization kinetics of pure PP and 5% filled PP were studied using the Avrami equation corrected by the Jeziorny method. 

Ozawa extended the Avrami model to quantify polymer crystallization kinetics using noniso-thermal data [289]. It was reasoned that nonisothermal crystallization amounted to infinitesimal short crystallization times at isothermal conditions, given a crystallization temperature T [290]. This analysis led to the following equation ... 

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. 

Ziabicki proposed to analyze nonisothermal processes as a sequence of isothermal steps [170-172]. The proposed equation is a series expansion of the Avrami equation. In quasi-static conditions, provided that nucleation and growth of the crystals are governed by thermal mechanisms only, that their time dependence comes from a change in external conditions, and that the Avrami exponent is constant throughout the whole process, the nonisothermal crystallization kinetics can be expressed in terms of an observable half-time of crystallization, T]/2, a function of time, and of the external conditions applied. The following equation was derived for the dependence of the total volume of the growing crystal, E(t), with time ... 

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. 

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 ... 

These equations have been successfully used in practice to model crystallization data. Since equations have been derived for a number of Avrami models, this approach has the advantage that the model does not have to be known a priori, but instead can be chosen based on which equations fit the data (along with some physical insight). It is also possible to perform experiments at different temperatures or under nonisothermal conditions to facilitate further analyses such as obtaining growth activation energies, and the reader is referred to other works for detailed treatments (Khawam and Flanagan 2006). 

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