Kolmogorov-Avrami equation

As it is known [3, 33], crystallisation process kinetics can be described with the aid of the Kolmogorov-Avrami equation ... 

The Kolmogorov-Avrami equation for the case when the crystallisation parameter is the stress in a uniaxially stretched sample can be presented as follows [39] ... 

Comparison of Equation 4.26 with the Kolmogorov-Avrami equation in Relationship 4.18 [41,42] allows determination of the function x t). The values of K are accepted according to the data of paper [24]. 

Eet us consider crystallisation mechanism changes for HDPE/EP nanocomposites in comparison with the initial HDPE, which define the degree of reduction in crystallinity with an increase in c p. As it is known [8], the crystallisation kinetics of polymers is often described with the aid of the Kolmogorov-Avrami equation, obtained for low-molecular substances (Equation 4.19), in which the exponent n value can be changed within the range of 1-4 [9]. 

The rate of transformation of a metastable solid (parent) phase (A) to form a more stable solid (product) phase (B) is usually modeled using the Avrami equation (Avrami, 1939, 1940), which is also known as the Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation. This equation is based on a model that assmnes that the transformation involves the nucleation of the product phase followed by its growth imtil the parent phase is replaced by the... 

It follows that the so-called empirical kinetic model function can be generally described by all-purpose, three-exponent relation, first introduced by (and often named after the authors as) Sestdk and Berggren (SB) equation [480], h(q) = (/ ( - a) [-In (1 - a)f AX is practically applicable as either form, SB equation, oT (1 - a) , and/or modified Johnson, Mehl, Avrami, Yerofeev and Kolmogorov (JMAYK) equation, (1 - a) [-In (1 - a)f (related to its original form, - ln(l - a) = (krtf, through the exponentsp and r,. Q.,p (1 - 1/r ). 

The Avrami equation, also referred to as the Kolmogorov-Johnson-Mehl-Avrami equation [34-37,40], can be considered to be one of the possible solutions of Equation 11.3, and in its simplest form it can be expressed as [31,34—39,41]... 

This relationship is known as the Avrami equation (also known as the Kolmogorov-Johnson-Mehl-Avrami or KJMA equation). Half of the material will be transformed when fcf" = 0.693. Only 1% of the transformation has occurred when fcf" = 0.01 and the transformation is 99% complete when fcf = 4.6. The transformation time is generally specified as the time required for half of the material to be transformed, which is given by... 

The authors [10] demonstrated that the Kolmogorov-Avrami exponent n increased linearly with increase in the fractal dimension of a chain part between clusters, characterising the extent of molecular mobility for a polymer. The value of can be calculated according to Equation 4.15 and parameters and dp which are necessary for calculation of are connected with one another by Relationship... 

L. E. Levine, K. Lakshmi Narayan, K. F. Kelton. Finite size corrections for the Johnson-Mehl-Avrami-Kolmogorov equation. J Mater Res 72 124, 1997. 

Karty et al. [21] pointed out that the value of the reaction order r and the dependence of k on pressure and temperature in the JMAK (Johnson-Mehl-Avrami-Kolmogorov) equation (Sect. 1.4.1.2), and perhaps on other variables such as particle size, are what define the rate-limiting process. Table 2.3 shows the summary of the dependence of p on growth dimensionality, rate-limiting process, and nucleation behavior as reported by Karty et al. [21]. 

The temperature T is the kinetic temperature of vitrification. The liquid cooled down to temperatures below T transforms into a polycluster amorphous solid even at stationary temperature. Since the crystallization and clusterization are the competing processes, it is of importance to know how the volume fraction and the cluster size distribution depend on the cooling rate. The answer to this question has been obtained by integrating the Avrami-Kolmogorov equations [6.81] generalized to the case of simultaneous formation of two new phases. The consideration of these results is beyond the scope of the present communication and will be published elsewhere [6.82]. 

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). 

To study the phase transformation, which involves nudeation and growth, many methods are developed. Most of the methods deprend on the transformation rate equation given by Kolmogorov, Johnson, Mehl and Avrami (Lesz Szewieczek, 2005 Szewieczek Lesz, 2005 Szewieczek Lesz, 2004 Jones et al, 1986 Minic Adnadevic, 2008), pxrpularly known as KJMA equation, basically derived from experiments carried out under isothermal conditions. The KJMA rate equation is given by... 

On the other hand, kinetic data on first-order transformations are often obtained by isothermal analysis. The isothermal crystallization kinetics of the amorphous phase can be usually analyzed in terms of the generalized theory of the well-known Kolmogorov-Johnson-Mehl-Avrami (JMA) equation (Christian, 2002) for a phase transition ... 

Weinberg (1992a), Weinberg et al. (1997), and Zanotto (1997), reported in detail on transformation kinetics via nucleation and crystal growth. The standard theory of this type of phase transformation kinetics was developed by Johnson and Mehl and Avrami and Kolmogorov (see Weinberg et al., 1997). Therefore, this theory is called the JMAK theory. The JMAK equation (Eq. 1-6) is universal and applicable to glass-ceramics. 

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]. 

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