Advertisement

Acta Applicandae Mathematicae

, Volume 122, Issue 1, pp 473–483 | Cite as

Shock-Induced Phase Transitions in Systems of Hard Spheres with Attractive Interactions

  • Shigeru TaniguchiEmail author
  • Masaru Sugiyama
Article
  • 110 Downloads

Abstract

Shock-induced phase transitions are studied by adopting the recently-developed theoretical framework, which is applicable for shock waves in three phases (gas, liquid, and solid), based on the system of hard spheres with mutual attractive interactions. The Rankine-Hugoniot conditions derived from the system of Euler equations with caloric and thermal equations of state are studied, and the admissibility (stability) of a shock wave is analyzed. Two typical scenarios of the shock-induced phase transitions from gas phase to solid phase are found. A scenario of shock-induced phase transitions involving three phases simultaneously near the triple point is also found.

Keywords

Shock wave Shock-induced phase transition Hard-sphere system with attractive interaction Rankine-Hugoniot relations Stability of shock waves 

Mathematics Subject Classification

76L05 82C26 35L67 

Notes

Acknowledgements

The authors thank Prof. Tommaso Ruggeri and Dr. Andrea Mentrelli and Prof. Nanrong Zhao for valuable discussions.

References

  1. 1.
    Meier, G.E.A.: Liquefaction shock waves. In: van Dongen, M.E.H. (ed.) Multiphase Flows I. Shock Wave Science and Technology Reference Library, vol. 1, pp. 231–267. Springer, Berlin (2007), Chap. 7 Google Scholar
  2. 2.
    Matsuda, A., Kondo, K., Nakamura, K.G.: Nanosecond rapid freezing of liquid benzene under shock compression studied by time-resolved coherent anti-Stokes Raman spectroscopy. J. Chem. Phys. 124, 054501 (2006) (4 pp.) CrossRefGoogle Scholar
  3. 3.
    Yoo, C.S., Holmes, N.C., Ross, M., Webb, D.J., Pike, C.: Shock temperatures and melting of iron at earth core conditions. Phys. Rev. Lett. 70, 3931–3934 (1993) CrossRefGoogle Scholar
  4. 4.
    Belonoshko, A.B.: Atomistic simulation of shock wave-induced melting in argon. Science 275, 955–957 (1997) CrossRefGoogle Scholar
  5. 5.
    Kadau, K., Germann, T.C., Lomdahl, P.S., Holian, B.L.: Microscopic view of structural phase transitions induced by shock waves. Science 296, 1681–1684 (2002) CrossRefGoogle Scholar
  6. 6.
    Duvall, G.E., Graham, R.A.: Phase transitions under shock-wave loading. Rev. Mod. Phys. 49, 523–579 (1977) CrossRefGoogle Scholar
  7. 7.
    Thompson, P.A.: Liquid-vapor adiabatic phase changes and related phenomena. In: Kluwick, A. (ed.) Nonlinear Waves in Real Fluids, pp. 147–213. Springer, New York (1991), Chap. 6 Google Scholar
  8. 8.
    Morris, D.G.: An investigation of the shock-induced transformation of graphite to diamond. J. Appl. Phys. 51, 2059–2065 (1980) CrossRefGoogle Scholar
  9. 9.
    Taniguchi, S., Mentrelli, A., Zhao, N., Ruggeri, T., Sugiyama, M.: Shock-induced phase transition in systems of hard spheres with internal degrees of freedom. Phys. Rev. E 81, 066307 (2010) (13 pp.) CrossRefGoogle Scholar
  10. 10.
    Zhao, N., Sugiyama, M., Ruggeri, T.: Phase transition induced by a shock wave in hard-sphere and hard-disk systems. J. Chem. Phys. 129, 054506 (2008) (13 pp.) CrossRefGoogle Scholar
  11. 11.
    Zheng, Y., Zhao, N., Ruggeri, T., Sugiyama, M., Taniguchi, S.: Non-polytropic effect on shock-induced phase transitions in a hard-sphere system. Phys. Lett. A 374, 3315–3318 (2010) zbMATHCrossRefGoogle Scholar
  12. 12.
    Zhao, N., Mentrelli, A., Ruggeri, T., Sugiyama, M.: Admissible shock waves and shock-induced phase transitions in a van der Waals fluid. Phys. Fluids 23, 086101 (2011) (18 pp.) CrossRefGoogle Scholar
  13. 13.
    Taniguchi, S., Mentrelli, A., Ruggeri, T., Sugiyama, M., Zhao, N.: Prediction and simulation of compressive shocks with lower perturbed density for increasing shock strength in real gases. Phys. Rev. E 82, 036324 (2010) (5 pp.) CrossRefGoogle Scholar
  14. 14.
    Taniguchi, S., Zhao, N., Sugiyama, M.: Shock-induced phase transitions from gas phase to solid phase. J. Phys. Soc. Jpn. 80, 083401 (2011) (4 pp.) CrossRefGoogle Scholar
  15. 15.
    Longuet-Higgins, H.C., Widom, B.: A rigid sphere model for the melting of argon. Mol. Phys. 8, 549–556 (1964) CrossRefGoogle Scholar
  16. 16.
    Young, D.A., Alder, B.J.: Critical point of metals from the van der Waals model. Phys. Rev. A 3, 364–371 (1971) CrossRefGoogle Scholar
  17. 17.
    Münster, A.: Statistical Thermo-dynamics, vol. 2. Springer, Berlin (1974) Google Scholar
  18. 18.
    Barker, J.A., Henderson, D.: What is “liquid”? Understanding the states of matter. Rev. Mod. Phys. 48, 587–671 (1976) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Hansen, J.P., McDonald, J.R.: Theory of Simple Liquids. Academic Press, London (1986) Google Scholar
  20. 20.
    Woodcock, L.V.: Hard-sphere fluid equation of state. J. Chem. Soc. Faraday Trans. 272, 731–735 (1976) Google Scholar
  21. 21.
    Liu, T.-P.: The entropy condition and the admissibility of shocks. J. Math. Anal. Appl. 53, 78–88 (1976) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Liu, T.-P.: Admissible solutions of hyperbolic conservation laws. Mem. Am. Math. Soc. 30, No. 240 (1981) Google Scholar
  23. 23.
    Liu, T.-P., Ruggeri, T.: Entropy production and admissibility of shocks. Acta Math. Appl. Sin., Engl. Ser. 19, 1–12 (2003) MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lax, P.D.: Hyperbolic systems of conservation laws II. Commun. Pure Appl. Math. 10, 537–566 (1957) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Thompson, P.A., Kim, Y.-G.: Direct observation of shock splitting in a vapor-liquid system. Phys. Fluids 26, 3211–3215 (1983) CrossRefGoogle Scholar
  26. 26.
    Thompson, P.A., Chaves, H.G., Meier, G.E.A., Kim, Y.-G., Speckmann, H.-D.: Wave splitting in a fluid of large heat capacity. J. Fluid Mech. 185, 385–414 (1987) CrossRefGoogle Scholar
  27. 27.
    Cramer, M.S.: Shock splitting in single-phase gases. J. Fluid Mech. 199, 281–296 (1989) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Center for Social Contribution and CollaborationNagoya Institute of TechnologyNagoyaJapan
  2. 2.Graduate School of EngineeringNagoya Institute of TechnologyNagoyaJapan

Personalised recommendations