Johnson-Mehl equation

However, this is valid only for f 1. At longer times the fraction of the volume that is available for growth and nucleation is (1 — /). When this term is included, the Johnson-Mehl equation becomes... 

This equation, which relates the fraction transformed to the nucleation rate, the growth rate, and the time elapsed since the start of the transformation (at constant temperature), is known as the Johnson-Mehl equation. The fact that the exponential term depends on can be understood on the basis that growth is assumed to proceed spherically, and thus the volume transformed increases with the cube power of the linear growth rate. 

A simplified version of the Johnson-Mehl equation, known as the Avrami equation, is often employed. The Avrami equation is typically expressed as... 

The Johnson-Mehl equation describes the overall fraction of material transformed as a function of time [F (f)] assuming spherically growing nuclei and constant nucleation and growth rates ... 

The resulting equation relating the fraction transformed to nucleation rate, growth rate and time is called Johnson-Mehl equation. 

Or the Johnson-Mehl equation [72]. In these equations, the concept of extended volume ftaction is adopted. By using this concept, the hard impingement can be taken into consideration indirectly. The extended volume ftaction is the sum of the volume ftaction of all new phases without direct consideration of the hard impingement between new particles and is related to the actual volume fraction by... 

Larsen and Livesay (1980) chose SmCoj as a representative compound on which to study the intrinsic hydriding kinetics of compounds of the RCoj family. The compound LaNij and several pseudobinaries were studied by Belkbir et al. (1980, 1981). All these authors analysed their data in terms of the Johnson-Mehl equation (Johnson and Mehl, 1939 Avrami, 1940)... 

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... 

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... 

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... 

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... 

In order to integrate the differential equations (2)-(5) the additional relation is need because number density of the dimmers is determined from number density of monomers in case of steady state only. In general case, the time dependence of the number densities of all clusters can be found as solving of master equation of cluster dynamics. Now we have interest to the short description and we are looking for the additional approximation relation. The fluence dependence of the volume fraction of CEC is estimated by saturation tank) form " that is roughly equivalent to a Johnson Mehl-type function ... 

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 ... 

Equation 12.3 serves as the basis for the Johnson-Mehl-Avrami (JMA) 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. 

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