Shape factors

Calculations of driving forces, interfacial compositions and growth rates were previously shown for spherical precipitates, in which the radius of curvature is constant across the surface. For non-spherical precipitates, the curvature varies.

While it is possible to model a single precipitate to investigate the thermodynamic and kinetic behavior across the surface, this is impractical for the KWN model. Rather, a correction factor is applied to the thermodynamic and kinetic terms.

In defining the precipitate shape, three terms will be used: the long axis ($l$), the short axis ($r$) and the equivalent spherical radius ($R$). Additionally, for ellipsoids, the eccentricity can be defined (which simplifies some of the following equations). The equivalent spherical radius is defined as the radius of a sphere that gives the same volume as the non-spherical precipitate. Defining the precipitate size by the equivalent radius minimizes the amount of changes needed to the KWN model (such as calculating volume fraction and composition).

$$ \alpha = \frac{l}{r} $$

$$ e = \sqrt{1 - \frac{1}{\alpha^2}} $$

The changes made to the growth rate and Gibbs-Thomson effect are then:

$$ \frac{dR}{dt} = f(\alpha) \frac{dR}{dt} \biggr\rvert_{sphere} $$

$$ \mu_A^\alpha = \mu_A^\beta + \left(g(\alpha) \frac{2\gamma}{R} + \Delta G_{el} \right) V_M^\beta $$


The equations for a spherical precipitate reduces the KWN model to its original equations.

$$ R_{eq} = r $$

$$ f(\alpha) = 1 $$

$$ g(\alpha) = 1 $$

Spherical precipitate shape

Spherical precipitate shape


$$ R_{eq} = r \sqrt[3]{\alpha} $$

$$ f(\alpha) = \frac{2 e \sqrt[3]{\alpha^2}}{\ln{(1+e)} - \ln{(1-e)}} $$

$$ g(\alpha) = \frac{1}{2 \sqrt[3]{\alpha^2}} \left(1 + \frac{\alpha}{e} \sin^{-1}{e} \right) $$

Needle-like precipitate shape

Needle-like precipitate shape


$$ R_{eq} = r \sqrt[3]{\alpha^2} $$

$$ f(\alpha) = \frac{e \sqrt[3]{\alpha^2}}{\pi/2 - \cos^{-1}{e}} $$

$$ g(\alpha) = \frac{1}{2 \sqrt[3]{\alpha^4}} \left(\alpha^2 + \frac{1}{2 e} \ln{ \left(\frac{1+e}{1-e} \right)} \right) $$

Plate-like precipitate shape

Plate-like precipitate shape


$$ R_{eq} = \sqrt[3]{\frac{3\alpha}{4\pi}} l $$

$$ f(\alpha) = 0.1 \exp{ \left(-0.091 (\alpha-1)\right)} + \frac{1.736 \sqrt{\alpha^2 - 1}}{\sqrt[3]{\alpha} \ln{ \left( 2\alpha^2 + 2\alpha \sqrt{\alpha^2 - 1} - 1 \right)}} $$

$$ g(\alpha) = \frac{2\alpha + 1}{2\pi} \left( \frac{4 \pi}{3 \alpha} \right)^{2/3} $$

Cuboidal precipitate shape

Cuboidal precipitate shape


  1. K. Wu, Q. Chen and P. Mason, “Simulation of precipitation kinetics with non-spherical particles” Journal of Phase Equilibria and Diffusion 39 (2018) p. 571

  2. B. Holmedal, E. Osmundsen and Q. Du, “Precipitation of non-spherical particles in aluminum alloys Part I: Generalization of the Kampmann-Wagner numerical model” Metallurgical and Materials Transactions A 47A (2016) p. 581