On some limitations of the Johnson–Mehl–Avrami–Kolmogorov equation

Abstract

Some limitations of the Johnson–Mehl–Avrami–Kolmogorov (JMAK) equation used widely for describing kinetics of phase transformation are demonstrated using probabilistic analysis and Monte Carlo simulations. The JMAK equation predicts correctly the real transformed fraction only if the number of the growing nuclei in the controlled volume is large. If the number of growing nuclei is small, the transformed fraction predicted by the JMAK equation deviates significantly from the real transformed fraction, no matter how large the volume of the controlled volume is. As an alternative, another equation should be applied, which for any number of the growing nuclei predicts correctly the true amount of transformed fraction.

Introduction

The question concerning kinetics of a phase transformation characterised by constant rates of nucleation and radial growth is essentially geometrical and was first treated by Kolmogorov [1], Johnson and Mehl [2] and Avrami [3], [4], [5]. Assuming that the growth rate is constant in all directions and that nuclei “β” grow as regular spheres in the matrix “α” (Fig. 1), the equation

$$\text{ξ}{}_{\mathrm{\beta }}\text{(τ)=1−}\text{exp}\text{−I}\text{π}\text{3}\text{k}{}^3\text{τ}{}^4$$

has been derived for the time dependence of the transformed volume fraction ξ_β(τ) [1], [2], [3], [4], [5]. In equation (1) “I” is the nucleation rate (number of nuclei per unit volume per second), “k” (m/s) is the radial growth rate and τ (s) is the time since the start of the transformation. The two-dimensional analogue of equation (1) is:

$$\text{ξ}{}_{\mathrm{\beta }}\text{(τ)=1−}\text{exp}\text{−I}\text{π}\text{3}\text{k}{}^2\text{τ}{}^3$$

which gives the time dependence of the transformed areal fraction during growth of circular nuclei with constant radial rate “k”, which nucleate with a nucleation rate I (number of nuclei per unit area per second). Kinetics of one-dimensional growth is described by

$$\text{ξ}{}_{\mathrm{\beta }}\text{(τ)=1−}\text{exp}\text{(−Ikτ}{}^2\text{)}$$

where the nucleation rate I has a dimension “number of nuclei per unit length per second”.

Some drawbacks of equation (1) have been analysed in earlier work [6], where an exact kinetics equation was derived. According to this equation, the volume fraction of the transformed zone during phase transformation of type nucleation and growth [7] with a constant radial growth rate is given by

$$\text{ξ}{}_{\mathrm{\beta }}\text{(τ)=1−}\text{exp}\text{∫}\text{0}\text{τ}\text{ln}\text{[1−ψ(υ)]I(τ−υ)dυ}$$

where υ is a dummy integration variable; I(τ−υ) is the number of spherical nuclei of volume v(υ)=(4/3)πk³υ³ and volume ratio ψ(υ)=(4/3)πk³υ³/V, (0≤υ≤τ) which have nucleated in the elementary time interval (τ−υ,τ−υ+dτ) in the controlled volume V, (for unit volume $\text{V=1}\text{m}{}^3\text{ψ(υ)=(4/3)πk}{}^3\text{υ}{}^3\text{)}$). If the nucleation rate I is constant, equation (4) becomes

$$\text{ξ}{}_{\mathrm{\beta }}\text{(τ)=1−}\text{exp}\text{I}\text{∫}\text{0}\text{τ}\text{ln}\text{[1−ψ(υ)]dυ}$$

The two- and one-dimensional analogues of equation (5) have the same functional form. The only differences are the expressions giving the volume ratio ψ(τ) of the growing nuclei and the dimension of the nucleation rate I. For two-dimensional nucleation and growth, the areal ratio is $\text{ψ(υ)=πk}{}^2\text{υ}{}^2\text{/S,}\text{(0≤υ≤τ),}\text{(ψ(υ)=πk}{}^2\text{υ}{}^2$ for unit area $\text{S=1}\text{m}{}^2\text{)}$). In case of one-dimensional growth, the lineal ratio in equation (5) is $\text{ψ(υ)=2kυ/L,}\text{(0≤υ≤τ),}\text{(ψ(υ)=2kυ}$ for unit length $\text{L=1}\text{m}$).