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.^[1][2]

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

${\displaystyle \alpha ={\frac {L}{kT_{\mathrm {E} }}}\cdot {\frac {\eta }{v}}}$

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

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

${\displaystyle \alpha ={\frac {\Delta S_{f}}{k}}\cdot {\frac {\eta }{v}}}$ ^[3]

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

${\displaystyle \alpha ={\frac {\Delta S_{f}}{R_{\mathrm {g} }}}\cdot {\frac {\eta _{1}}{Z}}}$

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