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Alpha factor

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Factors found in materials science

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

  1. Bennema, P. (1993). "Morphology of crystals determined by alpha factors, roughening temperature, F faces and connected nets". Journal of Physics D: Applied Physics. 26 (8B): B1 – B6. doi:10.1088/0022-3727/26/8b/001.
  2. Uhlmann, Don; Fratello, Vincent (2022). "Kenneth A. Jackson (1930–2022)". National Academy of Engineering. Retrieved 2024-04-01.
  3. Gruzleski, John E. (1996). Microstructure Development During Metalcasting.
  4. Glicksman, Martin (2011). Principles of Solidification: An Introduction to Modern Casting and Crystal Growth Concepts.
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