Johnson-Mehl—Avrami equation

The function ((t) in Eq. 21.12 has a characteristic sigmoidal shape with a maximum rate of transformation at intermediate times. Examples are shown in Fig. 21.2. The d = 3 form of Eq. 21.12 is commonly known as the Johnson-Mehl-Avrami equation. 

Nucleation and Growth Johnson-Mehl-Avrami Equation... 

This is known as the Johnson-Mehl-Avrami equation.This equation reduces to Eq. (9.18) at small values of time. These assumptions are made in deriving this equation ... 

The kinetic evolution is usually represented by a sigmoid-type curve. Such a typical curve is given in Figure 9 for the case of displacement reaction, Ni + CuO Cu -F NiO realized in planetary ball mill [58]. Author of this overview analyzed these experimental results by one of the most frequently used kinetic model applied to various solid-state reactions, namely Johnson-Mehl-Avrami equation ... 

FIGURE 17.9 Fraction transformed for the mechanochemical reaction Ni + CuO —> Cu + NiO carried out in planetary ball mill as a function of milling time (experimental points taken from [58]) solid line is Johnson-Mehl-Avrami Equation (17.8). 

X 10 , 3.1 X 10 and 6.8 x 10 sec " for angular velocity of a supporting disc 240, 270, 300 and 330 rpm, respectively. However, although Johnson-Mehl-Avrami equation satisfactorily describes overall kinetics, it is hard to give any unambiguous physical interpretations of the derived values of Avrami exponent, n which varies from 2.14 to 3.57. 

The conversion can also be described by the Johnson-Mehl-Avrami equation ... 

Apart from empirical determinations of these transformation diagrams, methods of prediction based on nucleation theory and phenomenological growth theory using the Johnson-Mehl-Avrami equation have been devised to estimate TTT diagrams [1.83]. 

Isothermal investigation of crystallization in amorphous materials can be described by the Johnson-Mehl-Avrami equation as given in Equation 55 (Araujo Idalgo, 2009 Avrami, 1939,1940 Celikbilek et al., 2011 Prasad Varma, 2005). 

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]... 

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]. 

One of the earliest attempts to explicitly combine thermodynamics and kinetics in rapid solidification was by Saunders et al. (1985). They examined the equations derived by Davies (1976) and Uhlmann (1972) for predicting TTT diagrams. These were based on Johnson-Mehl-Avrami kinetics for predicting glass formation during rapid solidification where the ruling equation could be given as... 

The different combinations of nucleation, growth, and impingement processes give rise to the Johnson-Mehl-Avrami kinetic model [4], which results in the following equation... 

Equation (3.2) is often referred to as the Avrami-Erofeev (A-E) equation, or more fittingly, on account of the substantial contributions from other workers, especially Johnson and Mehl [26] in the field of metallurgy, as the Johnson-Mehl-Avrami-Erofeev-Kholmogorov (JMAEK) equation. The values of n obtained from kinetic... 

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 isothermal crystallization kinetics of amorphous materials is described by the Johnson-Mehl-Avrami QMA) equation. The Avrami exponent n for different temprerature ranges from 2.63 to 3.12 between 793 and 823 K (Fig. 29a), which indicates that the isothermal annealing was governed by diffusion-controlled three-dimensional growth for Nisa54Ti49.46... 

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 ... 

Critical cooling rates for glass formation can be obtained by Johnson-Mehl-Avrami isothermal transformation kinetics using the equation... 

The dynamics of the process in which nucleation and crystal growth occur together are expressed by the Johnson-Mehl-Avrami (JMA) equation. 

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 ). 

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... 

BaTiOj Kinetic Analysis. Utilization of the Hancock and Sharp modification of the Johnson-Mehl-Avrami analysis permits formation mechanism evaluation of an otherwise closed hydrothermal system(7,79). Through application of the following equation,... 

Equations (6.15) and (6.17) phenomenologically describe the overall growth kinetics after the initial nucleation took place and further nucleation is still occurring. Indeed, the sigmoidal form of the X(t) curve represents a wide variety of transformation reactions. Equation (6.13) is named after Johnson, Mehl, and Avrami [W. A. Johnson, R. E Mehl (1939) M. Avrami (1939)]. Let us finally mention two points. 1) Plotting Vin (1 -X) vs. t should give a straight line with slope km. 2) The time ty of the inflection point (d2X/dt2 = 0) on X(t) is suitable to derive either m or km, namely... 

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