Application of equilibrium phase diagrams for calculation of segregation kinetics during two-component melt cooling

According to the equilibrium state diagrams, when the melt is cooled to a certain temperature below liquidus, compositions of liquid and solid phases are uniquely determined by corresponding curves in the diagram. However, it does not happen in reality. For equilibrium (which the diagram describes), it is necessary that the melt is maintained indefinitely at each temperature, or thermal conductivity of liquid and solid phases, and the diffusion coefficients of their components, are infinitely large. We made an attempt to find out how these processes occur in reality. In this work, we consider the growth of individual crystal during cooling of a two-component melt. Mathematical model is constructed based on the following. 1. The melt area with volume corresponding to one grain, the periphery of which is cooled according to a certain law, is considered. 2. At the initial instant of time, a crystal nucleus of a certain minimum size is in the liquid. 3. At the surface of crystal, compositions of liquid and solid phases correspond to equilibrium state diagram at a given temperature on its surface. 4. Changes in temperature and composition in liquid and solid phases occur according to the laws of heat conduction and diffusion, respectively. As the melt gets cold and the crystal grows, the liquid phase is enriched in one component and depleted in another, the solid phase – on the contrary. Since the diffusion coefficients of the components in the solid phase are small, the composition of the crystal does not have time to completely equalize its cross section. The model proposed in the work allows us to study this phenomenon, to calculate for each cooling mode how the composition of the crystal will vary over its cross section. The calculations have shown that the temperature equalization occurs almost instantly, and composition of the liquid phase equalizes much slower. Equalization of the solid phase composition does not occur in the foreseeable time. The results of the work will help to improve technology of generation of alloys with an optimal structure.

state diagram, local equilibrium, segregation, crystal growth, phase transition, mathematical model, diffusion, thermal conductivity

1. Tourret D., Gandin Ch. A generalized segregation model for concurrent dendritic, peritectic and eutectic solidification. Acta Materialia. 2009, vol. 57, no. 7, pp. 2066–2079.

2. Ferrandini P.L., Rios C.T., Dutra A.T., Jaime M.A., Mei P.R., Caram R. Solute segregation and microstructure of directionally solidified austenitic stainless steel. Materials Science and Engineering A. 2006, vol. 435-436, pp. 139–144.

3. Bellmann M.P., Meese E.A., Arnberg L. Impurity segregation in directional solidified multi-crystalline silicon. Journal of Crystal Growth. 2010, vol. 312, no. 21, pp. 3091–3095.

4. Steiner M.A., Garlea E., Agnew S.R. Modeling solute segregation during the solidification of g-phase U–Mo. Journal of Nuclear Materials. 2016, vol. 474, pp. 105–112.

5. Gong L., Chen B., Du Zh., Zhang M., Liu R., Liu K. Investigation of solidification and segregation characteristics of cast Ni-base superalloy K417G. Journal of Materials Science & Technology. 2018, vol. 34, no. 3, pp. 541–550.

6. Gao Zh., Jie W., Liu Yo, Luo H. Solidification modelling for coupling prediction of porosity and segregation. Acta Materialia. 2017, vol. 127, pp. 277–286.

7. Chatelain M., Botton V., Albaric M., Pelletier D., Cariteau B., Abdo D., Borrelli M. Mechanical stirring influence on solute segregation during plane front directional solidification. International Journal of Thermal Sciences. 2018, vol. 126, pp. 252–262.

8. Hou Z., Guo D., Cao J., Chang Yi. A method based on the centroid of segregation points: A Voronoi polygon application to solidification of alloys. Journal of Alloys and Compounds. 2018, vol. 762, pp. 508–519.

9. Lianga J., Zhaoa Zh., Tanga D., Yeb N., Yangc Sh. Improved microstructural homogeneity and mechanical property of medium manganese steel with Mn segregation banding by alternating lath. Materials Science & Engineering A. 2018, vol. 711, pp. 175–181.

10. Martinsen F.A. Purification of melt-spun metallurgical grade silicon micro-flakes through a multi-step segregation procedure. Journal of Crystal Growth. 2013, vol. 363, pp. 33–39.

11. Robson J.D. Analytical electron microscopy of grain boundary segregation: Application to Al–Zn–Mg–Cu (7xxx) alloys. Materials Characterization. 2019, vol. 154, pp. 325–334.

12. Li J., Guo Zh. Thermodynamic evaluation of segregation behaviors of metallic impurities in metallurgical grade silicon during Al–Si solvent refining process. Journal of Crystal Growth. 2014, vol. 394, pp. 18–23.

13. Drozin A.D. Rost mikrochastits produktov khimicheskikh reaktsii v zhidkom rastvore [Growth of chemical reaction products microparticles in a liquid solution]. Chelyabinsk: izd. YuUrGU, 2007, 57 p. (In Russ.).

14. Budak B.M., Gol’dman N.L., Uspenskii A.B. Difference scheme with front straightening for solving multi-front problems of the Stefan’s type. Doklady AN SSSR. 1966, vol. 167, no. 4, pp. 735–738. (In Russ.).

15. Tikhonov A.N., Samarskii A.A. Uravneniya matematicheskoi fiziki [Equations of mathematical physics]. Moscow: Nauka, 1972, 736 p. (In Russ.).

16. Samarskii A.A. Teoriya raznostnykh skhem [Theory of difference schemes]. Moscow: Nauka, 1977, 656 p. (In Russ.).

17. Samarskii A.A., Nikolaev E.S. Metody resheniya setochnykh uravnenii [Methods for grid equations solving]. Moscow: Nauka, 1978, 592 p. (In Russ.).

18. Lyakishev N.P. ed. Diagrammy sostoyaniya dvoinykh metallicheskikh sistem: Spravochnik [State diagrams of double metal systems: Reference book. Vol. 1]. Lyakishev N.P. ed. Мoscow: Mashinostroenie, 1996, 992 p. (In Russ.).

19. Zakharov A.M. Diagrammy sostoyaniya dvoinykh i troinykh sistem [State diagrams of binary and ternary systems]. Moscow: Metallurgiya, 1990, 250 p. (In Russ.).

20. Petrunin I.E. ed. Spravochnik po paike [Soldering reference]. Petrunin I.E. ed. Moscow: Mashinostroenie, 2003, 480 p. (In Russ.).

21. Fizicheskie velichiny. Spravochnik [Physical quantities. Reference book]. Grigor’ev I.S., Meilikhov E.Z. eds. Moscow: Energoatomizdat, 1991, 1250 p. (In Russ.).

22. Rabinovich V.A., Khavin Z.Ya. Kratkii khimicheskii spravochnik [Brief chemical reference]. Leningrad: Khimiya, 1978, 392 p. (In Russ.).

23. Khairulin R.A., Stankus S.V., Abdullaev R.N., Sklyarchuk V.M. The density and interdiffusion coefficients of bismuth-tin melts of eutectic and near-eutectic composition. High Temperature. 2010, vol. 48, no. 2, pp. 188–191.

24. Makhniy V.P., Protopopov E.V., Skripnik N.V. Mechanism of tin diffusion in ZnTe single crystals. Inorganic Materials. 2011, vol. 47, no. 9, pp. 945–946.