Avrami equation primary crystallization

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. 

Note In isothermal crystallization, primary crystallization is often described by the Avrami equation. 

The primary crystallization process is characterized by three parameters. These are the rate of radial growth of the spherulite, G, the time constant for nucleation, t , and the time constant for the primary crystallization process, Tc, which is determined from the Avrami equation. All three parameters seem to depend on the stereoregularity of the polymer, but the nucleation rate seems to depend most strongly. 

We saw in Section 4.1.1 that the increase of crystallinity with time at constant crystallization temperature follows the Avrami equation (eq. (4.1)). Since in earlier investigations (Santa Cruz etal, 1991) the increase in the H of PET during primary crystallization was shown to be directly proportional to the increase of crystallinity, one may expect that the increase of H will follow a similar relationship as a function of time ... 

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].

The time dependence of the overall (primary) crystallization is described by the Avrami equation, which was originally derived for the crystallization of metals. The crystallinity is expressed as the volume fraction of the crystalline material in the total sample. For the derivation, it is assumed that each nucleus leads to an entity (e.g., a rod, disk, or sphere). After an infinitely long time the whole sample is filled with these shapes. The crystallinity of the sample is then 4>oo. This, however, is the crystallinity of a single entity which is assumed not to change during crystallization. At the time r, the fraction of the sample volume filled by these entities is /randomly distributed nuclei the probability p that a point does not lie in any one entity is proportional to this fraction, i.e.,... 

The spheruhte growth may be initiated by either heterogeneous nucleation by foreign particles or homogeneous spontaneous nucleation. The growth rate of spherulites from the polymer melt increases as the temperature decreases. It reaches a maximum in the temperature range of 140-150°C, and then decreases upon further decrease of temperature. The time dependency of the primary crystallization can be described by the known Avrami equation [282]. 

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). 

The overall crystallization process includes two steps, primary nucleation followed by crystal growth. The process can be well described by the Avrami equation (63,64) as shown ... 

The ciystallization curve shown in Figure 2 was obtained in a SAXS/WAXS/DSC experiment from iPP [23] and shows the classic features of primary crystallization. The detailed molewlar structure of the polymer, the specific nature of the nucleation processes and the degree of under-coolteg, determines the magnitude of the lamellar thickness and the degree of crystallinity within the lamellar stadcs. The crystallization kinetics are analyzed using the Avrami model [24], expressed in terms of the equation... 

Crystallization kinetic data, obtained by differential scanning calorimetry (DSC) and dilatometry, can be analysed if the primary data are transformed to volume crystallinity as a function of crystallization time. The constants in the Avrami equation are obtained by taking the double logarithm of eq. (8.36) ... 

FIGURE 3.20 Data for the primary isothermal crystallization of PET at different temperatures analyzed on the basis of the Avrami equation where X/Xp.w represents mjmg. Deviation from linearity indicates a varying value of n with extent of crystallization. (Data from Lu, X. R, and J. N. Hay, Polymer, 42, 9423, 2001. With permission.)... 

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