Avrami equation secondary crystallization

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. 

Crystallization of PET proceeds in two distinct steps [97], i.e. (1) a fast primary crystallization which can be described by the Avrami equation, and (2) a slow secondary crystallization which can be described by a rate being proportional to the crystallizable amorphous fraction dXc/dt = (Xmax — tc)kc, with Xmax being the maximum crystallinity (mass fraction) [98], Under SSP conditions, the primary crystallization lasts for a few minutes before it is replaced by secondary crystallization. The residence time of the polymer in the reactor is of the order of hours to days and therefore the second rate equation can be applied for modelling the SSP process. 

The isotherms obtained in dilatometric measurements of the crystallization rate could be fitted with an Avrami (3) type equation only by assuming the existence of a secondary crystallization process much slower than the rate of spherulitic growth observed microscopically, and by taking into account the experimentally determined form of the nucleation rate. The nucleation rate was found to be a first-order process. Assuming that the secondary crystalliza-... 

Figure 3.14. Comparison between a typical experimental crystallization isotherm (solid line) and the Avrami equation (Eq 3.17, broken line). The three regions 1, II and III correspond to primary, primary and secondary, and secondary crystallization, respectively. [Perez-Cardenas et al., 1991].

In reality, polymer crystaHizatiOTi is too complex to be described by a simple expression such as the Avrami equation. For example, the assumption in Avrami s expression that the volume does not change is inaccurate because the specimen tends to shrink during crystallization. In addition, secondary crystallization and crystal perfecting processes are not taken into account. 

Equation 3.21 is valid for a < [, and Eq. 3.22 for a > C- InstEad of two Avrami parameters, five parameters are required to describe the process. They have the following physical meaning k and n (the primary crystallization parameters) depend on crystallization temperature, nature of primary nucleation, and the fast growth the secondary crystallization parameters, k and n, depend on the conditions under which the slow crystallization of the remaining amorphous regions takes place and a fifth parameter, i, indicates the weight fractiOTi of material crystallized up to the moment the primary crystallization ends, t is the moment at which the third region starts (e.g., pure secondary crystallization). 

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 Avrami equation represents only the initial portions of polymer crystallization correctly. The spherulites grow outward with a constant radial growth rate until impingement takes place when they stop growth at the intersection, as illustrated in Figures 6.13 and 6.22. Then a secondary crystallization process is often observed after the initial spherulite growth in the amorphous interstices (85). 

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. 

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