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A complete analytical solution of unsteady coagulation equations and transition between the intermediate and concluding stages of a phase transformation

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Abstract

In this paper, a complete analytical solution to the unsteady kinetic equation of particle coagulation in a metastable liquid is constructed with allowance for an arbitrary initial particle-volume distribution function. Based on the analytical solution found, a theory of the transition of the phase transformation process from the intermediate stage to the coagulation stage is constructed. For this purpose, an exact analytical solution describing the intermediate stage of a phase transformation is used. It is shown that the maximum of the particle-volume distribution function decreases, and its right branch (“tail”) grows with increasing time during the coagulation stage.

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References

  1. W. Kurz, D.J. Fisher, Fundamentals of Solidification (Trans Tech Publications, Aedermannsdorf, 1989)

    Google Scholar 

  2. D. Herlach, P. Galenko, D. Holland-Moritz, Metastable Solids from Undercooled Melts (Elsevier, Amsterdam, 2007)

    Google Scholar 

  3. V.V. Slezov, Kinetics of First-Order Phase Transitions (Wiley-VCH, Weinheim, 2009)

    Book  Google Scholar 

  4. P.K. Galenko, D.V. Alexandrov, Philos. Trans. R. Soc. A 376, 20170210 (2018)

    Article  ADS  Google Scholar 

  5. D.V. Alexandrov, A.Y. Zubarev, Philos. Trans. R. Soc. A 377, 20180353 (2019)

    Article  ADS  Google Scholar 

  6. D.V. Alexandrov, A.Y. Zubarev, Philos. Trans. R. Soc. A 378, 20200002 (2020)

    Article  ADS  Google Scholar 

  7. P. Pelcé, Dynamics of Curved Fronts (Academic Press, Boston, 1988)

    MATH  Google Scholar 

  8. R. Trivedi, W. Kurz, Int. Mater. Rev. 39, 49 (1994)

    Article  Google Scholar 

  9. P.K. Galenko, V.A. Zhuravlev, Physics of Dendrites (World Scientific, Singapore, 1994)

    Google Scholar 

  10. W. Kurz, D.J. Fisher, R. Trivedi, Int. Mater. Rev. 64, 311 (2019)

    Article  Google Scholar 

  11. D.V. Alexandrov, P.K. Galenko, L.V. Toropova, Philos. Trans. R. Soc. A 376, 20170215 (2018)

    Article  ADS  Google Scholar 

  12. P.K. Galenko, D.V. Alexandrov, E.A. Titova, Philos. Trans. R. Soc. A 376, 20170218 (2018)

    Article  ADS  Google Scholar 

  13. D.V. Alexandrov, P.K. Galenko, Philos. Trans. R. Soc. A 378, 20190243 (2020)

    Article  ADS  Google Scholar 

  14. A.C. Zettlemoyer, Nucleation (Dekker, New York, 1969)

    Google Scholar 

  15. V.P. Skripov, Metastable Liquids (Wiley, New York, 1974)

    Google Scholar 

  16. K.F. Kelton, A.L. Greer, Nucleation in Condensed Matter: Applications in Materials and Biology (Elsevier, Amsterdam, 2010)

    Google Scholar 

  17. D.V. Alexandrov, A.P. Malygin, J. Phys. A: Math. Theor. 46, 455101 (2013)

    Article  ADS  Google Scholar 

  18. R.N. Hills, D.E. Loper, P.H. Roberts, Q., J. Appl. Maths. 36, 505 (1983)

  19. A.C. Fowler, IMA, J. Appl. Math. 35, 159 (1985)

  20. V.T. Borisov, Theory of Two-Phase Zone of a Metal Ingot (Metallurgiya Publishing House, Moscow, 1987)

    Google Scholar 

  21. S. Martin, P. Kauffman, J. Fluid Mech. 64, 507 (1974)

    Article  ADS  Google Scholar 

  22. D.L. Aseev, D.V. Alexandrov, Phys.-Dokl. 51, 291 (2006)

    Article  ADS  Google Scholar 

  23. D.V. Alexandrov, A.A. Ivanov, I.V. Alexandrova, J. Cryst. Growth 532, 125420 (2020)

    Article  Google Scholar 

  24. D.V. Alexandrov, I.A. Bashkirtseva, L.B. Ryashko, Philos. Trans. R. Soc. A 376, 20170216 (2018)

    Article  ADS  Google Scholar 

  25. I.G. Nizovtseva, D.V. Alexandrov, Philos. Trans. R. Soc. A 378, 20190248 (2020)

  26. Yu.A. Buyevich, V.V. Mansurov, J. Cryst. Growth 104, 861 (1990)

    Article  ADS  Google Scholar 

  27. D.V. Alexandrov, I.G. Nizovtseva, I.V. Alexandrova, Int. J. Heat Mass Trans. 128, 46 (2019)

    Article  Google Scholar 

  28. Y.A. Buyevich, Y.M. Goldobin, G.P. Yasnikov, Int. J. Heat Mass Trans. 37, 3003 (1994)

    Article  Google Scholar 

  29. Y.A. Buyevich, D.V. Alexandrov, I.O.P. Conf, Ser. Mater. Sci. Eng. 192, 012001 (2017)

    Google Scholar 

  30. V.V. Slezov, V.V. Sagalovich, Sov. Phys. Usp. 30, 23 (1987)

    Article  ADS  Google Scholar 

  31. M. Conti, B. Meerson, A. Peleg, P.V. Sasorov, Phys. Rev. E 65, 046117 (2002)

    Article  ADS  Google Scholar 

  32. D.V. Alexandrov, J. Phys. Chem. Solids 91, 48 (2016)

    Article  ADS  Google Scholar 

  33. I.V. Alexandrova, A.A. Ivanov, D.V. Alexandrov, J. Cryst. Growth 532, 125456 (2020)

    Article  Google Scholar 

  34. S.K. Friedlander, Smoke, Dust and Haze: Fundamentals of Aerosol Behaviour (Wiley-Interscience, New York, 1977)

    Google Scholar 

  35. J.R. Hunt, J. Fluid Mech. 122, 169 (1982)

    Article  ADS  Google Scholar 

  36. A.V. Alyab’eva, Yu.A. Buyevich, V.V. Mansurov, J. Phys. II France 4, 951 (1994)

    Article  Google Scholar 

  37. T. Goudon, F. Lagoutière, L.M. Tine, Math. Models Methods Appl. Sci. 23, 1177 (2013)

    Article  MathSciNet  Google Scholar 

  38. D.V. Alexandrov, J. Phys.: Condens. Matter 28, 135102 (2016)

    Google Scholar 

  39. D.V. Alexandrov, A.A. Ivanov, I.V. Alexandrova, J. Phys. A: Math. Theor. 52, 015101 (2019)

    Article  ADS  Google Scholar 

  40. D.V. Alexandrov, A.A. Ivanov, I.V. Alexandrova, Eur. Phys. J. Special Topics 228, 161 (2019)

    Article  ADS  Google Scholar 

  41. B. Liu, Y. Wang, M. Zhang, H. Zhang, Polymers 8, 55 (2016)

    Article  Google Scholar 

  42. V.A. Ditkin, A.P. Prudnikov, Integral Transforms and Operational Calculus (Pergamon Press, Oxford, 1965)

    MATH  Google Scholar 

  43. A.M. Cohen, Numerical Methods for Laplace Transform Inversion (Springer, New York, 2007)

    MATH  Google Scholar 

  44. D.V. Alexandrov, A.P. Malygin, Model. Simul. Mater. Sci. Eng. 22, 015003 (2014)

    Article  ADS  Google Scholar 

  45. I.V. Alexandrova, D.V. Alexandrov, Philos. Trans. R. Soc. A 378, 20190245 (2020)

    Article  ADS  Google Scholar 

  46. D.V. Alexandrov, Chem. Eng. Sci. 117, 156 (2014)

    Article  Google Scholar 

  47. E.V. Makoveeva, D.V. Alexandrov, Philos. Trans. R. Soc. A 376, 20170327 (2018)

    Article  ADS  Google Scholar 

  48. E.V. Makoveeva, D.V. Alexandrov, Philos. Trans. R. Soc. A 377, 20180210 (2019)

    Article  ADS  Google Scholar 

  49. E.V. Makoveeva, D.V. Alexandrov, Philos. Mag. Lett. 98, 199 (2018)

    Article  ADS  Google Scholar 

  50. A.A. Ivanov, I.V. Alexandrova, D.V. Alexandrov, Philos. Trans. R. Soc. A 377, 20180215 (2019)

    Article  ADS  Google Scholar 

  51. D.V. Alexandrov, J. Phys. A: Math. Theor. 47, 125102 (2014)

    Article  ADS  Google Scholar 

  52. D.V. Alexandrov, Philos. Mag. Lett. 94, 786 (2014)

    Article  ADS  Google Scholar 

  53. D.V. Alexandrov, E.V. Makoveeva, Phys. Lett. A 384, 126259 (2020)

    Article  MathSciNet  Google Scholar 

  54. E.V. Makoveeva, D.V. Alexandrov, Eur. Phys. J. Special Topics 229, 2923 (2020)

    Article  ADS  Google Scholar 

  55. E.M. Lifshitz, L.P. Pitaevskii, Physical Kinetics (Pergamon, Oxford, 1981)

    Google Scholar 

  56. Yu.A. Buyevich, A.O. Ivanov, Phys. A 193, 221 (1993)

    Article  Google Scholar 

  57. I.V. Melikhov, T.. Ya.. Belousova, N.A. Rudnev, N.T. Bludov, Kristallografya 19, 1263 (1974)

    Google Scholar 

  58. J.W. Mullin, Crystallization (Butterworths, London, 1972)

  59. D.V. Alexandrov, Phys. Lett. A 378, 1501 (2014)

    Article  ADS  Google Scholar 

  60. D.V. Alexandrov, I.V. Alexandrova, Philos. Trans. R. Soc. A 378, 20190247 (2020)

    Article  ADS  Google Scholar 

  61. I.V. Alexandrova, D.V. Alexandrov, E.V. Makoveeva, Philos. Trans. R. Soc. A 379, 20200308 (2021)

    Article  ADS  Google Scholar 

  62. S. Simons, J. Phys. A: Math. Gen. 29, 1139 (1996)

    Article  MathSciNet  ADS  Google Scholar 

  63. D.V. Alexandrov, Int. J. Heat Mass Trans. 121, 884 (2018)

    Article  Google Scholar 

  64. E.D. McGrady, R.M. Ziff, Phys. Rev. Lett. 58, 892 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  65. R.M. Ziff, J. Phys. A: Math. Gen. 24, 2821 (1991)

    Article  ADS  Google Scholar 

  66. R.M. Ziff, J. Phys. A: Math. Gen. 25, 2569 (1992)

    Article  MathSciNet  ADS  Google Scholar 

  67. M.M.R. Williams, Aerosol Sci. Tech. 12, 538 (1990)

    Article  ADS  Google Scholar 

  68. D.V. Alexandrov, J. Phys. A: Math. Theor. 48, 245101 (2015)

    Article  ADS  Google Scholar 

  69. D.V. Alexandrov, J. Cryst. Growth 457, 11 (2017)

    Article  ADS  Google Scholar 

  70. D.V. Alexandrov, Commun. Theor. Phys. 68, 269 (2017)

    Article  ADS  Google Scholar 

  71. L. Foret, J.E. Dawson, R. Villaseñor, C. Collinet, A. Deutsch, L. Brusch, M. Zerial, Y. Kalaidzidis, F. Ju̇licher, Curr. Biol. 22, 1381 (2012)

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Acknowledgements

The theoretical part of this work is supported by the Russian Science Foundation (Grant no. 18-19-00008). Numerical calculations were performed due to the support from the Ministry of Science and Higher Education of the Russian Federation [project no. FEUZ-2020-0057].

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Authors and Affiliations

  1. Department of Theoretical and Mathematical Physics, Laboratory of Multi-Scale Mathematical Modeling, Ural Federal University, Lenin ave., 51, Ekaterinburg, 620000, Russian Federation

    Irina V. Alexandrova & Dmitri V. Alexandrov

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  1. Irina V. Alexandrova
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  2. Dmitri V. Alexandrov
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Correspondence to Dmitri V. Alexandrov.

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Alexandrova, I.V., Alexandrov, D.V. A complete analytical solution of unsteady coagulation equations and transition between the intermediate and concluding stages of a phase transformation. Eur. Phys. J. Spec. Top. 231, 1115–1121 (2022). https://doi.org/10.1140/epjs/s11734-022-00522-9

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