Avrami model polymers

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... 

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. 

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). 

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. 

It was often found that, contrary to the theoretical prediction, the value of n is noninteger (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 nucleation), and absence of overlap between the growing crystallization fronts. These assumptions are often not met in polymer (blend) crystallization. Also, erroneous determination of the zero time and an overestimation of the enthalpy of fusion of the polymer at a given time can lead to noninteger values for n (Grenier and Prud homme 1980). 

Ozawa extended the Avrami model to quantify polymer crystallization kinetics using noniso-thermal data [289]. It was reasoned that nonisothermal crystallization amounted to infinitesimal short crystallization times at isothermal conditions, given a crystallization temperature T [290]. This analysis led to the following equation ... 

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... 

Actually, it is very rare to obtain integer values of the Avrami exponent, which suggests the existence of parallel and/or competing processes of nucleation and growth of the crystalline zones. The treatment of experimental data using the Avrami model has thus less significance. Other, more complex models, which are better adapted to the case of polymers, were proposed—in particular, those that take into account the necessary disentanglement of the chains before crystallization. 

The isothermal overall crystallization kinetics of polymer blends from the melt can be analyzed on the basis of the Avrami model [38, 39] ... 

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. 

Crystallisation processes in PEEK have been the subject of many academic papers [10-13]. However, the crystallisation of PEEK generally matches the classic behaviour of other polymers. The effect of time is described by Avrami kinetics ( 3) and secondary crystallisation occurs after the spherulites have impinged. This secondary crystallisation results from an increase in the crystallinity within the spherulites and is probably related to the existence of the low-temperature melting peaks (LTMP) described later. A number of non-isothermal crystallisation models have been developed. 

It has to be emphasized that the classic Avrami and Evans equations, and consequently the Nakamura approach, were derived by assuming random positions of nuclei in a material therefore, they do not apply strictly when there is a correlation between positions of nucleation sites. Such nucleation of spherulites is accounted for in the model developed originally for fiber-reinforced polymers [53], described in Chapter 13. 

It was also demonstrated that in polymer composites, volume inhabited by embedded fibers inaccessible for crystallization and additional nucleation on internal interfaces [53,62,63], can markedly influence the overall crystallization kinetics, as described in Chapter 13. Similar problems might be encountered during crystallization in other polymer systems such as composites with particulate fillers and immiscible polymer blends. Under such conditions, the simplified Avrami equation (Eq. 7.10) does not apply and, as a consequence, the classic Avrami analysis may yield nonlinear plots and/ or noninteger n values. It must be emphasized that the problem cannot be solved by application of other, incorrect models, like that of Tobin, which are essentially based on the same assumptions as the Avrami-Evans theory but yield different equations due to incorrect reasoning. 

---