Alpha factor

The α-factor is a dimensionless quantity used to predict the solid–liquid interface type of a material during solidification. It was introduced by physicist Kenneth A. Jackson in 1958. In his model, crystal growth with larger values of α is smooth, whereas crystals growing at smaller α (below the threshold value of 2) have rough surfaces.

Method

According to John E. Gruzleski in his book Microstructure Development During Metalcasting (1996):

$\alpha =\frac{L}{k{T}_{\mathrm{E}}}\cdot \frac{\eta }{v}$

where $L$ is the latent heat of fusion; $k$ is the Boltzmann constant; $T_{\mathrm{E}}$ is the freezing temperature at equilibrium; $\eta$ is the number of nearest neighbours an atom has in the interface plane; and $v$ is the number of nearest neighbours in the bulk solid.

As $\frac{L}{T_{\mathrm{E}}}=\mathrm{\Delta }{S}_f$, where $\mathrm{\Delta }{S}_f$ is the molar entropy of fusion of the material,

$\alpha =\frac{\mathrm{\Delta }{S}_f}{k}\cdot \frac{\eta }{v}$

According to Martin Glicksman in his book Principles of Solidification: An Introduction to Modern Casting and Crystal Growth Concepts (2011):

$\alpha =\frac{\mathrm{\Delta }{S}_f}{R_{\mathrm{g}}}\cdot \frac{{\eta }_1}{Z}$

where $R_{\mathrm{g}}$ is the universal gas constant. $\frac{{\eta }_1}{Z}$ is similar to previous, always $\frac{1}{4}<\frac{{\eta }_1}{Z}$ < 1.