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):
α = L k T E ⋅ η v {\displaystyle \alpha ={\frac {L}{kT_{\mathrm {E} }}}\cdot {\frac {\eta }{v}}}
where L {\displaystyle L} is the latent heat of fusion; k {\displaystyle k} is the Boltzmann constant; T E {\displaystyle T_{\mathrm {E} }} is the freezing temperature at equilibrium; η {\displaystyle \eta } is the number of nearest neighbours an atom has in the interface plane; and v {\displaystyle v} is the number of nearest neighbours in the bulk solid. As L T E = Δ S f {\displaystyle {\frac {L}{T_{\mathrm {E} }}}=\Delta S_{f}} , where Δ S f {\displaystyle \Delta S_{f}} is the molar entropy of fusion of the material,
α = Δ S f k ⋅ η v {\displaystyle \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):
α = Δ S f R g ⋅ η 1 Z {\displaystyle \alpha ={\frac {\Delta S_{f}}{R_{\mathrm {g} }}}\cdot {\frac {\eta _{1}}{Z}}}
where R g {\displaystyle R_{\mathrm {g} }} is the universal gas constant. η 1 Z {\displaystyle {\frac {\eta _{1}}{Z}}} is similar to previous, always 1 4 < η 1 Z {\displaystyle {\frac {1}{4}}<{\frac {\eta _{1}}{Z}}} < 1.
References