Avrami-Evans equation

The first attempts to describe the conversion degree of melt into growing domains concerned the radial growth from nuclei distributed randomly in a material [1-6]. In spite of different reasoning, they led to the equation, best known as the Avrami-Evans equation, which is widely used to analyze the isothermal spherulitic crystallization, which is the crystallization at constant temperature. 

By use of the Poison distribution, Avrami derived the famous Avrami phenomenological equation to treat a kinetic process (Avrami 1939, 1940, 1941). Kolmogorov first discussed the formulation of this equation (Kolmogorov 1937). Johnson and Mehl also made similar derivation independently (Johnson and Mehl 1939). Evans proposed a very concise derivation as introduced below (Evans 1945). 

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. 

Isothermal crystallization of a polymer is frequently characterized by the induction time and the crystallization half-time. The crystallization induction time is the time that elapses from the moment when the desired crystallization temperature is reached to the onset of crystallization, characterized by the formation of the first nuclei. The crystallization half-time is the time when relative crystallinity reaches 0.5. More detailed analyses of the isothermal crystallization are usually based on the Avrami-Evans theory. Equation (7.10) yields ... 

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. 

Pantani et al. [11] gave an extensive review on available models to predict and characterize the morphology of injection-molded parts. The authors themselves proposed a model to predict the morphology of injection-molded iPP, in which flow kinematics are computed using a lubrication approximation. Polymorphism was accounted for, using the Avrami-Evans-Nakamura equation to describe the crystallization kinetics of the mesomorphic phase, while the evolution of the a phase was modeled using Kolmogorov s model [122]. 

The equation developed by Evans for DSC measurements based on the Avrami equation is ... 

Under the aforementioned hypotheses, as two different crystalline phases are formed (a and mesomorphic), at least two kinetic processes take place simultaneously. The simplest model is a parallel of two kinetic processes non-interacting and competing for the available molten material. The kinetic equation adopted here for both processes is the non-isothermal formulation by Nakamura et al. (Nakamura et al., 1973, Nakamura et al., 1972) of the Kolmogoroff Avrami and Evans model (Avrami, 1939,1940,1941, Evans, 1945). 

The Avrami equationhas been extended to various crystallization models by computer simulation of the process and using a random probe to estimate the degree of overlap between adjacent crystallites. Essentially, the basic concept used was that of Evans in his use of Poisson s solution of the expansion of raindrops on the surface of a pond. Originally the model was limited to expansion of symmetrical entities, such as spheres in three dimensions, circles in two dimensions, and rods in one, for which n = 2,2, and 1, respectively. This has been verified by computer simulation of these systems. However, the method can be extended to consider other systems, more characteristic of crystallizing systems. The effect of (a) mixed nucleation, ib) volume shrinkage, (c) variable density of crystallinity without a crystallite, and (random nucleation were considered. AH these models approximated to the Avrami equation except for (c), which produced markedly fractional but different n values from 3, 2, or I. The value varied according to the time dependence chosen for the density. It was concluded that this was a powerful technique to assess viability of various models chosen to account for the observed value of the exponent, n. 

The A vrami Equation The original derivations by Avrami (73-75) have been simplified by Evans (81) and put into polymer context by Meares (82) and Hay (83). In the following, it is helpful to imagine raindrops falling in a puddle. These drops produce expanding circles of waves that intersect and cover the whole surface. The drops may fall sporadically or all at once. In either... 

The general Avrami equation is applicable to any type of crystallization. It is not restricted to polymers. It describes the time evolution of the overall crystallinity. The pioneer work was conducted during the 1930s and 1940s by Evans, Kolmogoroff, Johnson and Mehl, and Avrami. Wunderlich (1978) concludes that without the parallel knowledge of the microscopic, independently proven mechanism, the macroscopic, experimentally derived Avrami equation and the Avrami parameters are only a convenient means to represent empirical data of crystallization. However, interest in the Avrami equation has been... 

Equation (5.45) represents the main theoretical formula resulting from the Kolmogoroff s treatment [5.12], Later it has been obtained in practically the same form by Johnson and Mehl [5.13], Avrami [5.14-5.16] and Evans [5.17]. 

Equation (9.36) is identical to Eq. (9.29) obtained by the Avrami analysis for the same types of nucleation and growth. Calculation of three-dimensional growth by this method yields Eq. (9.28). The Evans method can also be applied to other growth geometries with the corresponding Avrami expressions resulting. 

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