Crystallization Avrami equation

The overall nucleation and crystalhzation rates of PLA tmder heterogeneous conditions are relatively higher than in homogenous conditions. The nucleation and crystallization rates of propylene-ethylene copolymer are increased tmder isothermal conditions. Addition of nucleating agent accelerates crystallization. Avrami equation is in popular use in the analysis of isothermal crystallization kinetics of polymers ... 

Table 4.3 Summary of Exponents in the Avrami Equation for Different Crystallization Mechanisms...

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. 

The crystallization of poly(ethylene terephthalate) at different temperatures after prior fusion at 294 C has been observed to follow the Avrami equation with the following parameters applying at the indicated temperatures ... 

The Avrami—Erofe ev equation, eqn. (6), has been successfully used in kinetic analyses of many solid phase decomposition reactions examples are given in Chaps. 4 and 5. For no substance, however, has this expression been more comprehensively applied than in the decomposition of ammonium perchlorate. The value of n for the low temperature reaction of large crystals [268] is reduced at a 0.2 from 4 to 3, corresponding to the completion of nucleation. More recently, the same rate process has been the subject of a particularly detailed and rigorous re-analysis by Jacobs and Ng [452] who used a computer to optimize curve fitting. The main reaction (0.01 < a < 1.0) was well described by the exact Avrami equation, eqn. (4), and kinetic interpretation also included an examination of the rates of development and of multiplication of nuclei during the induction period (a < 0.01). The complete kinetic expressions required to describe quantitatively the overall reaction required a total of ten parameters. 

The overall isothermal crystallization kinetics of polymers can be described by the Avrami equation [88-90] ... 

Solid PET feedstock for the SSP process is semicrystalline, and the crystalline fraction increases during the course of the SSP reaction. The crystallinity of the polymer influences the reaction rates, as well as the diffusivity of the low-molecular-weight compounds. The crystallization rate is often described by the Avrami equation for auto-accelerating reactions (1 — Xc) = cxp(—kc/"), with xc being the mass fraction crystallinity, kc the crystallization rate constant and n a function of nucleation growth and type. 

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 3 The Avrami equation addresses the problem that crystals growing from different nuclei can overlap. Accordingly, the equation is sometimes called the overlap equation . 

One approach is to roughly estimate how the degree of crystallization would vary with time by making main simplifications in treating solidification, leading to the Avrami equation. 

Figure 4-13 Degree of crystallization based on the Avrami equation.

Kinetics of crystallization. Trick (79) has reported a dilatometric study of the bulk crystallization of PTHF. The rates he observed for a polymer of Mw = 130,000 (Polymer A) are shown in Fig. 27. He also found that a lower molecular weight polymer (Polymer B, Mn = 6760) crystallized to a higher degree of crystallinity, whereas the introduction of comonomer units (Polymer C) decreased the degree of crystallinity (Fig. 28). From attempts to fit the Avrami Equation to the experimental data in the early stages of crystallization, a tentative value of n = 3 was... 

The key to modelling the crystallization process is the derivation a kinetic equation for a(t,T). It is possible to find different versions of this equation, including the classical Avrami equation, which allows adequate fitting of the experimental data. However, this equation is not convenient for solving processing problems. This is explained by the need to use a kinetic equation for non-isothermal conditions, which leads to a cumbersome system of interrelated differential and integral equations. The problem with the Avrami equation is that it was derived for isothermal conditions and... 

This is a first-order approximation, only valid for the first stage of crystallization (otherwise AV would increase to infinity). As soon as growing crystals touch each other, their growth is limited. A better approximation is given by the Avrami equation ... 

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