Avrami model

A kinetic model for single-phase polymerizations— that is, reactions where because of the similarity of structure the polymer grows as a solid-state solution in the monomer crystal without phase separation—has been proposed by Baughman [294] to explain the experimental behavior observed in the temperature- or light-induced polymerization of substimted diacetylenes R—C=C—C=C—R. The basic feature of the model is that the rate constant for nucleation is assumed to depend on the fraction of converted monomer x(f) and is not constant like it is assumed in the Avrami model discussed above. The rate of the solid-state polymerization is given by... 

Applying the Avrami model to the analysis of the isothermal crystallization of interesterified and noninteresterified 20%SSS/80%000 at 30°C, 40 C and 50 C, many differences can be observed (Table 17.3). At 30°C and 40°C growth would be described as rodlike with instantaneous nucleation for both interesterified and noninteresterified samples. Also, for the noninteresterified system at 50°C spherulitic growth with instantaneous nucleation takes place. The half-time of nucleation... 

It was found that is a function of temperature but the model was found to give a better fit than analytical expressions like the Avrami model or the modified Gompertz model (Kloek, Walstra and Van Vliet 2000). The main advantage of this model is that as it is formulated as a differential equation, it can be used to predict isothermal as well as dynamic crystallization. However, this model does not consider the polymorphism of the material which is a critical point in the crystallization of cocoa butter. Another contribution is the model of Fessas et al. (Fessas, Signorelli and Schiraldi 2005) which considers all the transitions possible between each... 

In microscale models the explicit chain nature has generally been integrated out completely. Polymers are often described by variants of models, which were primarily developed for small molecular weight materials. Examples include the Avrami model of crystallization,- and the director model for liquid crystal polymer texture. Polymeric characteristics appear via the values of certain constants, i.e. different Frank elastic constant for liquid crystal polymers rather than via explicit chain simulations. While models such as the liquid crystal director model are based on continuum theory, they typically capture spatiotemporal interactions, which demand modelling on a very fine scale to capture the essential effects. It is not always clearly defined over which range of scales this approach can be applied. 

E)uring crystallization of most compounds, including TAG, the formation of nuclei and their growth are events that occur nearly simultaneously at different rates, since there is a continuous variation of the conditions that produce crystallization. The Avrami model takes into account the formation of nuclei and their growth. However, is important to indicate that the Avrami model does not provide information regarding the size or the polymorph state of the crystals. 

The Avrami model (19,20) states that in a given system under isothermal conditions at a temperature lower than V. the degree of crystallinity or fractional crystallization (70 as a liinction of time (t) (Fig. 11) is described by Equation 5. Although the theory behind this model was developed for perfect crystalline bodies like most polymers, the Avrami model has been used to describe TAG crystallization in simple and complex models (5,9,13,21,22). Thus, the classical Avrami sigmoidal behavior from an F and crystallization time plot is also observed in TAG crystallization in vegetable oils. This crystallization behavior consists of an induction period for crystallization, followed by an increase of the F value associated with the acceleration in the rate of volume or mass production of crystals, and finally a metastable crystallization plateau is reached (Fig. 11). 

Fig. 13. Coexistence, hysteresis and kinetics, (a) Schematic of interface-controlled growth of phase ft into phase a (b) superposition of spectra and thermal hysteresis (c) time evolution of volume fraction of phase ft Avrami model (Eq. (14)) for different exponents n.

It was often found that, contrary to the theoretical prediction, the value of n is non-integer [Avrami, 1939]. The Avrami model is based on several assumptions, such as constancy in shape of the growing crystal, constant rate of radial growth, lack of induction time, uniqueness of the nucleation mode, complete crystallinity of the sample, random distribution of nuclei, constant value of radial density, primary nucleation process (no secondary... 

The Avrami model was originally derived for the study of kinetics of crystallization and growth of a simple metal system, and further extended to the crystallization of polymer. Avrami assumes the nuclei develop upon cooling of polymer and the number of spherical crystals increases linearly with time at a constant growth rate in free volume. The Avratni equation is given as follow ... 

K " and n can be extracted from the intercept and the slope of Avrami plot, lg[-ln(l-.A0] versus lg(f-f ), respectively. The prime requirement of Avrami model is the ability of spherulites of a polymer to grow in a free space. Besides, Avrami equation is usually only valid at low degree of conversion, where impingement of polymer spherulites is yet to take place. The rate of crystallization of polymer can also be characterized by reciprocal half-time (/ 5). The use of Avrami model permits the understanding on the kinetics of isothermal crystallization as well as non-isothermal crystallizatioa However, in this chapter the discussion of the kinetics of crystallization is limited to isothermal conditions. 

TABLE 10 The Avrami exponents and the rate constants after Avrami model for isothermal crystallization kinetics of PET, PBT, and PTT (Adapted from Dangseeyun and co workers (2004)... 

The linearized version of the modified Avrami model is slightly different from Eq. (9.56). 

The TTT equation for the modified Avrami model (Eq. (9.58)) is slightly different. 

The kinetics of crystallization has been simulated by many models, including the Avrami model (House 2007). The rate of conversion from amorphous to crystalline states can be measured by using thermal analysis (differential scanning calorimetry, DSC) and/or X-ray diffraction. The rate of conversion from amorphous to crystalline form depends on a number of factors. The process occurs in two steps, nucleation and growth (Mullin 2001), which are affected by various factors and occur at different rates. Specifically, for crystallization to occur, a seed or nucleus must form, on which subsequent growth will occur. Thus, the rate of nucleation is of primary interest. By analogy with Arrhenius-type processes, the nucleation rate can be written as... 

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