Skip to main content
SpringerLink
Log in

A stochastic model for solidification

I. The basic equations, their analysis and solution

  • Published:
Pramana Aims and scope Submit manuscript

Abstract

A 3-dimensional (2-space, 1-time) model relating the diffusion of heat and mass to the kinetic processes at the solid-liquid interface, using a stochastic approach is presented in this paper. This paper is divided in two parts. In the first part the basic set of equations describing solidification alongwith their analysis and solution are given. The process of solidification has a stochastic character and depends on the net probability of transfer of atoms from liquid to the solid phase. This has been modeled by a Markov process in which knowledge of the parameters at the initial time only is needed to evaluate the time evolution of the system. Solidification process is expressed in terms of four coupled equations, namely, the diffusion equations for heat and mass, the equations for concentration of the solid phase and for rate of growth of the solid-liquid interface. The position of the solid-liquid interface is represented with the help of a delta function and it is defined as the surface at which latent heat is evolved. A numerical method is used to solve the equations appearing in the model. In the second part the results i.e. the time evolution of the solid-liquid interface shape and its concentration, rate of growth and temperature are given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. F C Flemings, inSolidification processing (McGraw Hill Inc., New York, 1974)

    Google Scholar 

  2. W Kurz and D J Fisher, inFundamentals of solidification (Trans-Tech Publication, Switzerland, 1986)

    Google Scholar 

  3. P Strimbord, D R Nelson and M Roucheti,Phys. Rev. Lett. 47, 1297 (1981)

    Article  ADS  Google Scholar 

  4. P Ramachandrarao, G V S Shastri, L Pandey and A Sinha,Acta, Cryst. A47, 206 (1991)

    Google Scholar 

  5. D Domb and J L Lebowitz (eds.) inPhase transitions and critical phenomena. (Academic, London, 1983)

    Google Scholar 

  6. N Sounders and A P Miodownik,J. Mater. Res. 1, 1803 (1986)

    Google Scholar 

  7. F X Kelly and L H Ungar,Phys. Rev. B. 34, 1746 (1986)

    ADS  Google Scholar 

  8. Z Chvoj, Z Kozisek and J Sestak,Thermochim. Acta. 153, 349 (1989)

    Article  Google Scholar 

  9. D T Gillespie,J. Chem. Phys. 74, 661 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  10. D Kashchiev,Cryst. Res. Tech. 19, 1413 (1984)

    Article  Google Scholar 

  11. A K Ray, M Chalom and L K Peters,J. Chem. Phys. 85, 2161 (1986)

    Article  ADS  Google Scholar 

  12. P Gordon, inPrinciples of phase diagrams in materials (Mc-Graw Hill, 1968; Jersey, 1972)

  13. A K Jena and M C Chaturvedi, inPhase transformation in materials. (The Metals Society, Eaglewood Cliffs, New Jersey 1992)

    Google Scholar 

  14. A Bruce and D Wallace,Phys. Rev. Lett. 4, 457 (1982)

    Google Scholar 

  15. J S Langer and L A Turski,Phys. Rev. A8, 3230 (1973)

    ADS  Google Scholar 

  16. J S Langer and A J Turski,Phys. Rev. A22, 2189 (1980)

    ADS  Google Scholar 

  17. J S Langer and A J Swartz,Phys. Rev. A21, 948 (1980)

    ADS  Google Scholar 

  18. V P Skripov and A V Skripov,Usp. Fiz. Nauk. 128, 193 (1979)

    Google Scholar 

  19. V P Skripov and V P Koverda, inSpontaneous crystallization of undercooled liquids (Nauka, Moscow, 1984)

    Google Scholar 

  20. A D J Haymet and D W Oxtoby,J. Chem. Phys. 84, 1769 (1986)

    Article  ADS  Google Scholar 

  21. A D J Haymet,Prog. Solid State Chem. 17, 1 (1986)

    Article  Google Scholar 

  22. B B Mandelbrot, inThe fractal geometry of nature (Freeman, San Francisco, 1982)

    MATH  Google Scholar 

  23. D S Cannel and C Aubert, inFractal and non fractal patterns in physics, in growth and form edited by H E Stanley and N Ostrowski (Martinus Nijhoff, Dordrecht, 1986)

    Google Scholar 

  24. B Caroli, C. Caroli and B J Roulet,J. Cryst. Growth 66, 575 (1984)

    Article  Google Scholar 

  25. T A Cherepanova,J. Cryst. Growth 59, 371 (1980)

    Google Scholar 

  26. T A Cherepanova,Phys. Status. Solidi A58, 469 (1980)

    Google Scholar 

  27. T A Cherepanova,J. Cryst. Growth 52, 319 (1981)

    Article  Google Scholar 

  28. L Pandey and P Ramachandrarao,Acta. Metall. 35, 10, 2549 (1987)

    Google Scholar 

  29. R Trivedi, P. Magnin and W Kurz,Acta Metall. 35, 971 (1987)

    Article  Google Scholar 

  30. G S Reddy and J A Sekhar,J. Mater. Sci. 21, 3535 (1985)

    Article  Google Scholar 

  31. Z Chvoj,Czech. J. Phys. B37, 1256 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  32. Z Chvoj,Cryst. Res. Tech. 21, 8, 1003 (1986)

    Article  ADS  Google Scholar 

  33. Z Chvoj,Czech. J. Phys. B37, 607 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  34. Z Chvoj,Czech. J. Phys. B33, 961 (1983)

    Article  ADS  Google Scholar 

  35. Z Chvoj,Czech. J. Phys. B33, 1060 (1983)

    Article  ADS  Google Scholar 

  36. Z Chvoj,Czech. J. Phys. B34, 548 (1984)

    Article  ADS  Google Scholar 

  37. Z Chvoj,Czech. J. Phys. B36, 863 (1986)

    Article  ADS  Google Scholar 

  38. Z Chvoj,Czech. J. Phys. B37, 1340 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  39. Z Chvoj,Czech. J. Phys. B40, 473 (1990)

    Article  ADS  Google Scholar 

  40. Z Chvoj,Czech. J. Phys. B40, 483 (1990)

    Article  ADS  Google Scholar 

  41. S Dass, G Johri, L Pandey and P Ramachandrarao, inII Annual General Meeting of MRSI held at NPL, New Delhi, February 9–10 (1991)

  42. C W Gardiner, inHandbook of stochastic methods (Springer-Verlag, Berlin, 1985)

    Google Scholar 

  43. C N R Rao and K J Rao, inPhase transitions in solids (Mc-Graw Hill, Chatham, 1978)

    Google Scholar 

  44. J B Scarborough, inNumerical mathematical analysis (Oxford and IBH Publishing Co., New Delhi, 1976)

    Google Scholar 

  45. M Krizek and P Neittaanmaki, inFinite element approximation of variational problems and applications (Longman Scientific with John Wiley Inc., New York, 1990)

    MATH  Google Scholar 

  46. B Chalmers, inPrinciples of Solidification (John Wiley and Sons, New York, 1967) p. 7

    Google Scholar 

  47. R E Reed-Hill, inPhysical metallurgy principles (EWP, New Delhi, 1975) p. 535

    Google Scholar 

  48. R C Weast (ed.)CRC Handbook of Chemistry and Physics, (CRC Press Inc., 1988)

Download references

Author information

Authors and Affiliations

  1. Govt. M.H. College of Science and Home Science, 482 002, Jabalpur, India

    Shobha Dass, Gautam Johri & Lakshman Pandey

  2. Department of Post Graduate Studies and Research in Physics, Rani Durgavati University, 482 001, Jabalpur, India

    Gautam Johri & Lakshman Pandey

Authors
  1. Shobha Dass
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gautam Johri
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lakshman Pandey
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dass, S., Johri, G. & Pandey, L. A stochastic model for solidification. Pramana - J. Phys 47, 447–470 (1996). https://doi.org/10.1007/BF02847540

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02847540

Keywords

PACS No

Search

Navigation