Patent Project

Calculations

                                                                (#)[pic 1]

where:

 is the enthalpy of melting[pic 2]

 is the specific heat of silicon[pic 3]

 is the change in temperature[pic 4]

and  is the reference Temperature, which is 298 K[pic 5]

For this for the purpose of these calculations, it will be assumed that  [3], the silicon begins at the reference temperature 298 K, and the silicon is heated to a temperature of 1420°C or 1693 K. Using these assumtions:[pic 6]

[pic 7][pic 8][pic 9]

                                                                        (#)[pic 10]

where:        

 is Gibbs free energy[pic 11]

 is the latent heat of formation[pic 12]

 is the amount of undercooling[pic 13]

and  is the melting temperature.[pic 14]

For the purposes of this report, it is possible to assume that  , using the “Richard’s Rule” discussed in lecture notes. This simplifies equation (#) to[pic 15]

                                                                        (#)[pic 16]

This can then be used to find an approximation for the driving force behind this solidification process.

[pic 17][pic 18]

                        (1)[pic 19]

where:        

 is the concentration of impurities in the solid[pic 20]

 is the concentration of impurities in the liquid[pic 21]

 is the fraction of impure melt trapped in the solid metal during the crystallization process[pic 22]

 is the segregation coefficient determined by Czochralski crystallization of silicon[pic 23]

and  is the segregation coefficient determined by the new process[pic 24]

 has a value of 6.4*10-6, but in order to find , it is required to first calculate R using the next equation:[pic 25][pic 26]