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):

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

where $L$ is the latent heat of fusion; $k$ is Boltzmann’s 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 ${\frac {L}{T_{\mathrm {E} }}}=\Delta S_{f}$, where $\Delta S_{f}$ is the molar entropy of fusion of the material,

$\alpha ={\frac {\Delta S_{f}}{k}}\cdot {\frac {\eta }{v}}$

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

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

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.