Johnson-Mehl rate 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. 

The Eq. (12) is the same as the one obtained for the case of random nucleation sites by Johnson-Mehl. This equation implies that the increase in the volume of new phases is caused by nucleation and growth. On the other hand, Eq. (13) does not include nucleation rate and it imphes that the nucleation sites are covered by new phases and the increase in the volume is dependent only on the growth of new phases. This situation is referred to as site saturation [73]. 

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 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... 

Overall Rate of Transformation Johnson-Mehl and Avrami Equations... 

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. 

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 ... 

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... 

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

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

The theoretical basis for use of the DSC or DTA for study of crystallization rates was developed independently by Johnson and Mehl and by Avrami. The volume fraction of a sample crystallized, x, as a function of time, t, is expressed in terms of the nucleation rate per unit volume, /, and the crystal growth rate, u, via the equation ... 

---