Skip to main content
SpringerLink
Log in

A transient distribution of particle assemblies at the concluding stage of phase transformations

  • Original Paper
  • Published:
Journal of Materials Science Aims and scope Submit manuscript

Abstract

There are a number of experimental and theoretical papers testifying that the large-time behavior of the particle-size distribution function heavily depends on the initial conditions at the final stages of phase transformation processes. However, still now there is no theoretically derived distribution confirming these conclusions. The present paper is concerned with a new theoretical approach verifying the fact that the large-time distribution can actually be dependent on the details of the initial data. The concluding stage of evolution of a particulate ensemble as a result of coalescence and coagulation processes is considered. The transient kinetic and balance equations for the particle-size distribution function are modified into a single nonlinear equation for arbitrary collision frequency factors. This integro-differential equation is solved analytically in the limit of large times. The obtained particle-size distribution function depends on the initial condition and describes the concluding stages of transient phase transformation processes. The area of applicability of the present asymptotic solution for the large-time particle-size distribution is discussed. The obtained distribution function is in agreement with experiments.

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

Figure 1

Similar content being viewed by others

References

  1. Mullin JW (2001) Crystallization. Butterworth, Oxford, p 181–215

    Book  Google Scholar 

  2. Slezov VV (2009) Kinetics of first-order phase transitions. Wiley VCH, Weinheim, 415 pages

  3. Kelton KF, Greer AL (2010) Nucleation in condensed matter: applications in materials and biology. Elsevier, Amsterdam

    Google Scholar 

  4. Dubrovskii VG (2014) Nucleation theory and growth of nanostructures. Springer, Berlin, 603 pages

  5. Alexandrov DV (2014) On the theory of transient nucleation at the intermediate stage of phase transitions. Phys Lett A 378:1501–1504

    Article  Google Scholar 

  6. Alexandrov DV (2014) Nucleation and crystal growth in binary systems. J Phys A Math Theor 47:125102

    Article  Google Scholar 

  7. Alexandrov DV (2016) Mathematical modelling of nucleation and growth of crystals with buoyancy effects. Phil Mag Lett 96:132–141

    Article  Google Scholar 

  8. Lifshitz IM, Slezov VV (1959) Kinetics of diffusive decomposition of supersaturated solid solutions. Sov Phys JETP 35:331–339

    Google Scholar 

  9. Lifshitz IM, Slyozov VV (1961) The kinetics of precipitation from supersaturated solid solutions. J Phys Chem Solids 19:35–50

    Article  Google Scholar 

  10. Wagner C (1961) Theorie der Alterung von Niderschlägen durch Umlösen (Ostwald Reifung). Z Electrochem 65:581–591

    Google Scholar 

  11. Slezov VV (1978) Formation of the universal distribution function in the dimension space for new-phase particles in the diffusive decomposition of the supersaturated solid solution. J. Phys Chem Solids 39:367–374

    Article  Google Scholar 

  12. Slezov VV, Sagalovich VV (1987) Diffusive decomposition of solid solutions. Sov Phys Usp 30:23–45

    Article  Google Scholar 

  13. Sagui C, O’Gorman DS, Grant M (1998) Nucleation, growth and coarsening in phase-separating systems. Scanning Microsc. 12:3–8

    Google Scholar 

  14. Ratke L, Voorhees P (2002) Growth and coarsening: Ostwald ripening in material processing. Springer, New York, p 127–165

    Google Scholar 

  15. Baluffi RW, Allen SM, Carter WC (2005) Kinetics of materials. Wiley, Hoboken, p 459–542

    Book  Google Scholar 

  16. Xiao X, Liu G, Hu B, Wang J, Ma W (2013) Coarsening behavior for M23C6 carbide in 12 % Cr-reduced activation ferrite/martensite steel: experimental study combined with DICTRA simulation. J Mater Sci 48:5410–5419. doi:10.1007/s10853-013-7334-5

    Article  Google Scholar 

  17. Sordi VL, Feliciano CA, Ferrante M (2015) The influence of deformation by equal-channel angular pressing on the ageing response and precipitate fracturing: case of the AlAg alloy. J Mater Sci 50:138–143. doi:10.1007/s10853-014-8573-9

    Article  Google Scholar 

  18. Marder M (1987) Correlations and Ostwald ripening. Phys Rev A 36:858–874

    Article  Google Scholar 

  19. Yao JH, Elder KR, Guo H, Grant M (1992) Theory and simulation of Ostwald ripening. Phys Rev B 47:14110–14125

    Article  Google Scholar 

  20. Dubrovskii VG, Kazansky MA, Nazarenko MV, Adzhemyan LT (2011) Numerical analysis of Ostwald ripening in two-dimensional systems. J Chem Phys 134:094507

    Article  Google Scholar 

  21. Alexandrov DV (2015) On the theory of Ostwald ripening: formation of the universal distribution. J Phys A Math Theor 48:035103

    Article  Google Scholar 

  22. Alexandrov DV (2015) Relaxation dynamics of the phase transformation process at its ripening stage. J Phys A Math Theor 48:245101

    Article  Google Scholar 

  23. Alexandrov DV (2016) On the theory of Ostwald ripening in the presence of different mass transfer mechanisms. J Phys Chem Solids 91:48–54

    Article  Google Scholar 

  24. Alexandrov DV (2017) Kinetics of diffusive decomposition in the case of several mass transfer mechanisms. J Cryst Growth 457:11–18

    Article  Google Scholar 

  25. Lifshitz EM, Pitaevskii LP (1981) Physical kinetics. Pergamon Press, Oxford, p 427–447

    Book  Google Scholar 

  26. Enomoto Y, Okada A (1990) Effects of Brownian coagulation on droplet growth in a quenched fluid mixture. J Phys Condens Matter 2:4531–4535

    Article  Google Scholar 

  27. Alyab’eva AV, Buyevich YuA, Mansurov VV (1994) Evolution of a particulate assemblage due to coalescence combined with coagulation. J Phys II France 4:951–957

    Article  Google Scholar 

  28. Baldan A (2002) Progress in Ostwald ripening theories and their applications to nickel-base superalloys Part I: Ostwald ripening theories. J Mater Sci 37:2171–2202 doi:10.1023/A:1015388912729

    Article  Google Scholar 

  29. Alexandrov DV (2016) Kinetics of particle coarsening with allowance for Ostwald ripening and coagulation. J Phys Condens Matter 28:035102

    Article  Google Scholar 

  30. Carr J, Penrose O (1998) Asymptotic behaviour of solutions to a simplified Lifshitz–Slyozov equation. Phys D 124:166–176

    Article  Google Scholar 

  31. Niethammer B, Pego RL (1999) Non-self-similar behavior in the LSW theory of Ostwald ripening. J Stat Phys 95:867–902

    Article  Google Scholar 

  32. Snyder VA, Alkemper J, Voorhees PW (2001) Transient Ostwald ripening and the disagreement between steady-state coarsening theory and experiment. Acta Mater 49:699–709

    Article  Google Scholar 

  33. Carr J (2006) Stability of self-similar solutions in a simplified LSW model. Phys D 222:73–79

    Article  Google Scholar 

  34. Goudon T, Lagoutière F, Tine LM (2013) Simulations of the Lifshitz–Slyozov equations: the role of coagulation terms in the asymptotic behavior. Math Models Methods Appl Sci 23:1177–1215

    Article  Google Scholar 

  35. Herrmann M, Laurençot P, Niethammer B (2009) Self-similar solutions with fat tails for a coagulation equation with nonlocal drift. C R Acad Sci Paris Ser I 347:909–914

    Article  Google Scholar 

  36. Laurençot P (2002) The Lifshitz–Slyozov–Wagner equation with conserved total volume. SIAM J Math Anal 34:257–272

    Article  Google Scholar 

  37. Friedlander SK (1977) Smoke, dust and haze. Wiley, New York, 317 pages

  38. Hunt JR (1982) Self-similar particle-size distributions during coagulation: theory and experimental verification. J Fluid Mech 122:169–185

    Article  Google Scholar 

  39. Friedlander SK, Wang CS (1966) The self-preserving particle size distribution for coagulation by Brownian motion. J Colloid Interface Sci 22:126–132

    Article  Google Scholar 

  40. Sonntag H, Strenge K (1987) Coagulation kinetics and structure formation. Springer, Berlin, pp 58–126

    Book  Google Scholar 

  41. Smoluchowski M (1917) Versuch einer mathematischen theorie der koagulationskinetik kolloider lösungen. Ann Phys Chem 92:129–168

    Google Scholar 

  42. Wan G, Sahm PR (1990) Particle growth by coalescence and Ostwald ripening in rheocasting of Pb–Sn. Acta Metall Mater 38:2367–2372

    Article  Google Scholar 

  43. Randolph A, Larson M (1988) Theory of particulate processes. Academic Press, New York, 386 pages

  44. Buyevich YuA, Alexandrov DV, Mansurov VV (2001) Macrokinetics of crystallization. Begell House, New York

    Google Scholar 

  45. Sagui C, Grant M (1999) Theory of nucleation and growth during phase separation. Phys Rev E 59:4175–4187

    Article  Google Scholar 

  46. De Smet Y, Danino D, Deriemaeker L, Talmon Y, Finsy R (2000) Ostwald ripening in the transient regime: a cryo-TEM study. Langmuir 16:961–967

    Article  Google Scholar 

  47. Carrillo JA, Goudon T (2004) A numerical study on large-time asymptotics of the Lifshitz–Slyozov system. J Sci Comput 20:69–113

    Article  Google Scholar 

  48. Alexandrov DV, Nizovtseva IG (2014) Nucleation and particle growth with fluctuating rates at the intermediate stage of phase transitions in metastable systems. Proc Roy Soc A 470:20130647

    Article  Google Scholar 

  49. Alexandrov DV, Malygin AP (2014) Nucleation kinetics and crystal growth with fluctuating rates at the intermediate stage of phase transitions. Model Simul Mater Sci Eng 22:015003

    Article  Google Scholar 

  50. Crowe CT (2006) Multiphase flow handbook. Taylor & Francis, New York, 1156 pages

  51. Luo X, Benichou Y, Yu S (2013) Calculation of collision frequency function for aerosol particles in free molecule regime in presence of force fields. Front Energy 7:506–510

    Article  Google Scholar 

Download references

Acknowledgement

This work was supported by the Russian Science Foundation (Grant No. 16-11-10095).

Author information

Authors and Affiliations

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

    D. V. Alexandrov

Authors
  1. D. V. Alexandrov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. V. Alexandrov.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alexandrov, D.V. A transient distribution of particle assemblies at the concluding stage of phase transformations. J Mater Sci 52, 6987–6993 (2017). https://doi.org/10.1007/s10853-017-0931-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10853-017-0931-y

Keywords

Search

Navigation