jmak-1759577105

JMAK

The John-Mehl-Avrami-Kolmogorov model is a Solid State Phase Transformation(SSPT) model based on the Avrami equation for isothermal phase transition. Though the range of application of this model is generally contested, it is applied widely for its simplicity. (Fanfoni & Tomellini, 1998)

This model will be the benchmark for kMC-FEA and will be integrated into [[transient-thermal-1756651292|Single Element Transient Thermal with deal.II]] for comparison with Kinetic Monte Carlo method

Avrami Equation

The basic Avrami equation is as follows:

$$V(t)=1-\exp (-V_e(t))$$

where,

• $V$ is the fraction of transformed phase, and
• $V_e$ is the extended volume of transformed phase.

The extended volume of the transformed phase refers to the volume the phase would acquire if the overlap between the growing nuclei were disregarded. Once this extended volume is determined, the kinetics would be entirely known.

History of Avrami equation

The Avrami equation and its mathematical predecessors were independently derived by the Soviets and the western world.

Kolmogorov

Kolmogorov solved for the mathematical representation of grain growth and probability theory. The solution was general purpose and solves for constant and simultaneous grain growth and nucleation.

Johnson and Mehl

Johnson and Mehl unknown about Kolmogorov’s solution publish the same result but with a much more mathematically involved solution.

Avrami

Avrami provided a model for grain growth by describing the grain nucleations as mutually non exclusive events and the probability for nucleation as probability of $A$ given an infinite number of mutually non exclusive events $A_k$. The solution was in the form of a series but was thought to be too tough to handle for kinetics estimation. However, in his next paper, Avrami provided the solution to the infinite series as the Avrami equation.

Additionally Avrami introduces the concept of phantom grains. Phantom grains are grains which have been swallowed by other grains during growth but are still modelled to be growing themselves to provide the general solution to the series. These grains are fundamental to modelling the grain growth kinetics.

JMAK kinetics

Common terms

Volume ~ The volume of the $nD$ space.

Germ ~ A nucleation site

Nucleus(Grain) ~ Grown germ

Problem definition

1. The phase transformation occurs by nucleation and growth processes.
2. Germs are distributed randomly in the volume.
3. The critical size of the nucleus is $0$.

The overall problem can be defined as, what is the probability that a point of the space has not transformed at time $t$?

Derivation

Let us consider the case where all nucleations start at the same time and at $t->0^{+}$. This means that the nucleation rate is a Dirac delta function. Assuming all clusters are circular, the radius as a function of time $r(t)$ will result in a cluster area of $\pi r^2$. A point $C$ in space will not lie in a cluster if there are no germs within a distance $r(t)$ of the point. Hence the probability of $C$ not belonging to a cluster is that no germs lie in a circle of radius $r(t)$ centred at $C$.

This problem can be described with a Poisson point process. For a germ density of $N_0$, the probability $P$ as a function of $r(t)$ can be described as:

$$P(r)=\exp (-\pi N_0{r}^2)$$

Since this equation is written in a 2D space, the cluster area corresponds to volume in this space and the area $\pi r^2$ represents the excluded volume.

For the case where the nucleation is not simultaneous, the density of germs $n_i$ which start growing at a time $z_i$ will create a set of nucleation events $n_i$. Now let us isolate each item in the set into a different plane where the transformation takes place. The probability in a plane $P_i$ can be written as:

$$P_i=\exp (-\pi n_i{r}_i^2)$$

Since each plane is an independent space, the overall probability at point $C$ in the real space would be the product of probabilities across all planes.

$$P=\prod _i{P}_i=\prod _i\exp (-\pi n_i{r}_i^2)=\exp (-\pi \sum _i{n}_i{r}_i^2)$$

Hence, the excluded volume can be represented for non simultaneous nucleations in 2D space as:

$$V_e=-\pi \sum _i{n}_i{r}_i^2$$

At the continuum limit, $V_e$ for a set of nuclei starting growth at time $z$ at time $t$ becomes:

$$V_e={\int }_0^t\frac{dN}{dz}\pi (r(t,z){)}^{2}dz$$

Where $dN/dz$ is nucleation rates of the considered set of nuclei.

In this case of non simultaneous nucleation, a grain in any plane which is surrounded by another grain in a different plane wont contribute to the final transformed phase but it still needs to be considered with the rest of its plane. These grains are what are called phantom grains.

References

Fanfoni, M., & Tomellini, M. (1998). The Johnson-Mehl-Avrami-Kohnogorov model: A brief review. Il Nuovo Cimento D, 20(7), 1171–1182. https://doi.org/10.1007/BF03185527