# Effective Diffusion Distance

At large supersaturations, the growth rate of a precipitate is governed by the following equation.

$$\frac{dR}{dt} = \frac{D}{R} \frac{x_\infty^\alpha - x_R^\alpha}{\xi (x_R^\beta - x_R^\alpha)}$$

Where $\xi$ is the effective diffusion distance. This is given by

$$\xi = \frac{\Omega}{2 \lambda^2}$$

$$2\lambda^2 - 2\lambda^3 \sqrt{\pi} \exp{(\lambda^2)} \text{erfc}(\lambda) = \Omega$$

$$\Omega = \frac{x_\infty^\alpha - x_R^\alpha}{x_R^\beta - x_R^\alpha}$$

With $\Omega$ representing the supersaturation.

$\lambda$ must be solved numerically. However, there are analytical solutions for the following limits:

$$\lim_{\Omega \rightarrow 0} \lambda = \sqrt{\frac{\Omega}{2}}$$

$$\lim_{\Omega \rightarrow 1} \lambda = \sqrt{\frac{3}{2 (1 - \Omega)}}$$

Solving for $\xi$ in these limiting cases results in $\xi_{\Omega \rightarrow 0} = 1$ and $\xi_{\Omega \rightarrow 1} = 0$.

The effective diffusion distance is only used for binary systems. For multicomponent systems, the growth rate is calculated from the free energy curvature. This assumes small supersaturations and thus the effective diffusion distance is assumed to be 1.

### References #

1. Q. Chen, J. Jeppson and J. Agren, “Analytical treatment of diffusion during precipitate growth in multicomponent systems” Acta Materialia 56 (2008) p. 1890

2. T. Philippe and P. W. Voorhees, “Ostwald ripening in multicomponent alloys” Acta Materialia 61 (2013) p. 4237