Johnson-Mehl-Avrami-Kolmogorov

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. 

The hydrogen absorption/desorption kinetics are usually analyzed by applying the JMAK (Johnson-Mehl-Avrami-Kolmogorov) theory of phase transformations, which is based on nucleation and growth events [166-168] where a is the fraction transformed at time t or alternatively for hydrides the fraction absorbed... 

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 kinetics of the transformation process of the old phase to the new phase depends on the nucleation process, the dimensionality and mechanism of growth. Despite the interplay of a number of mechanisms, the time variation of the fractional volume of the new phase ft in phase a, o(t) = Vp(t)/V (Vis the total volume) can be usually described by a single simple formula, the Johnson-Mehl-Avrami-Kolmogorov model183 186... 

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... 

The common method for modeling both isothermal and non-isothermal crystallization kinetics from amorphous solids is the Johnson-Mehl-Avrami -Kolmogorov... 

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 ... 

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... 

After switching from fast cooling to isothermal conditions at time zero, the measured heat flow rate exponentially approaches a constant value (-10.3 mW) with a time constant of about 3 seconds for this DSC. The observed crystallization peak is often symmetric, and then the time of the peak maximum (nunimum) is a measure of crystallization half time. Integration of the peak yields the enthalpy change, which can be transformed into relative crystallinity by dividing by the limiting value at infinite time. To obtain development of absolute crystallinity (mass fraction) the curve has to be divided by the enthalpy difference between crystal and liquid at the crystallization temperature, which is available from ATHAS-DB [124], The commonly applied Kolmogorov-Johnson-Mehl-Avrami (KJMA) model for the kinetic analysis of isothermal crystallization data is based on volume fractions. Therefore, the mass fraction crystallinity, Wc, as always obtained from DSC, should be transformed into volume crystallinity. 

The theory of the kinetics of concurrent nucleation and growth reactions has a rich history that includes work by Kolmogorov [1], Johnson and Mehl [2], Avrami [3-5], Jackson [6], and Cahn [7]. Cahn s time-cone method for treating a class of these problems is the most general of these, with the most transparent assumptions, and is presented here. The method of Johnson, Mehl, and Avrami is covered in Section 4 of Christian s text [8]. 

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... 

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). 

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. 

Many mathematical models have bees advanced relating nucleation and nuclei growth rates to the overall kinetics of phase transformation, such as Johnson and Mehl [427], Avrami [428], Yerofyeyev [429], Kolmogorov [430] as well as Jacobs-Tompkins [431] or Mampel [432] and were agreeably suinmarized elsewhere [1,3,413,144, 421,422,423,426,43 I ]. 

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