Nonisothermal crystallization nucleation rate

Parameter Estimation. The kinetic parameters of the model given above that must be estimated for model identification include kg, g, Eg, kb, b, Eb,j. Parameter estimation for this type of model is quite difficult because the parameters appear nonlinearly, the nucleation rate parameters enter only in the boundary condition, and availability of accurate data is limited. Certainly a model that describes the behavior of a nonisothermally operated crystallizer is needed if the temperature is to be manipulated, but there have been only a few studies of the effect of temperature on crystallization processes (Kelt and Larson 1977 Randolph and Cise 1972 Rousseau and Woo 1980). For isothermal crystallization, the terms involving Eg and Eh are absorbed into kg and kb, and only kg, g, kb, b, and i need to be estimated. 

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 ... 

The analysis of the above equations is often applied for obtaining the nucleation data from isothermal and nonisothermal crystallization experiments. Several simplifications of the equations are developed and used for isothermal crystallization (with instantaneous or spontaneous nucleations only) and nonisothermal processes with a constant cooling rate. It was found that the crystallization of iPP follows the dependence log [1 — a(f)] t where n is around three for relatively low supercoolings which indicates instantaneous character of primary nucleation. 

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],... 

Recently the statistical approach was developed [5] for the description of the kinetics of conversion of melt to spherulites and the kinetics of formation of spherulitic pattern during both isothermal and nonisothermal crystallizations. The final spherulitic pattern can also be described. The rates of formation of spherulitic interiors and boundaries (boundary lines, surfaces and points) as well as the their final amounts could be predicted if spherulite growth and nucleation rates are known. Applied to iPP crystallized during cooling with various rates, the approach allowed for the predictions of tendencies in the kinetics of formation of spherulitic structure and its final form. 

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. 

It must be mentioned that there are many reports on deviation of the Ozawa plots from linearity and also on M values that are different from those predicted theoretically. The analysis of nonisothermal crystallization encounters even more difficulties than that of isothermal crystallization. Additional problems result from the requirement to combine the results of several crystallization experiments performed at different cooling rates. The Ozawa theory is based on the assumption that the nucleation rate dependence on temperature f(T) is unaffected by a cooling rate. As a consequence, the validity of the Ozawa approach is limited to a narrow range of cooling rates that results in the crystallization in similar temperature intervals. Markedly different cooling rates cause variation of M, for instance as shown in Reference [64]. The same applies to other simplified approaches, for example, the Nakamura isokinetic model, in which nonisothermal crystallization is treated as a succession of isothermal processes. The results are reasonable as long as the nucleation process is not infiu-... 

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