A transient distribution of particle assemblies at the concluding stage of phase transformations / Alexandrov D.V. // Journal of Materials Science. - 2017. - V. 52, l. 12. - P. 6987-6993.

ISSN:
00222461
Type:
Article
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. © 2017, Springer Science+Business Media New York.
Author keywords:
Index keywords:
Differential equations; Integrodifferential equations; Intermetallics; Light transmission; Nonlinear equations; Particle size; Particle size analysis; Phase transitions; Size distribution; Asymptotic
DOI:
10.1007/s10853-017-0931-y
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Affiliations Department of Theoretical and Mathematical Physics, Laboratory of Multi-Scale Mathematical Modeling, Ural Federal University, Ekaterinburg, Russian Federation
Funding Details 16-11-10095, RSF, Russian Science Foundation
References Mullin, J.W., (2001) Crystallization, pp. 181-215. , Butterworth, Oxford; Slezov, V.V., (2009) Kinetics of first-order phase transitions. Wiley VCH, Weinheim; Kelton, K.F., Greer, A.L., (2010) Nucleation in condensed matter: applications in materials and biology, , Elsevier, Amsterdam; Dubrovskii, V.G., (2014) Nucleation theory and growth of nanostructures. Springer, Berlin; Alexandrov, D.V., On the theory of transient nucleation at the intermediate stage of phase transitions (2014) Phys Lett A, 378, pp. 1501-1504; Alexandrov, D.V., Nucleation and crystal growth in binary systems (2014) J Phys A Math Theor, 47, p. 125102; Alexandrov, D.V., Mathematical modelling of nucleation and growth of crystals with buoyancy effects (2016) Phil Mag Lett, 96, pp. 132-141; Lifshitz, I.M., Slezov, V.V., Kinetics of diffusive decomposition of supersaturated solid solutions (1959) Sov Phys JETP, 35, pp. 331-339; Lifshitz, I.M., Slyozov, V.V., The kinetics of precipitation from supersaturated solid solutions (1961) J Phys Chem Solids, 19, pp. 35-50; Wagner, C., Theorie der Alterung von Niderschlägen durch Umlösen (Ostwald Reifung) (1961) Z Electrochem, 65, pp. 581-591; Slezov, V.V., Formation of the universal distribution function in the dimension space for new-phase particles in the diffusive decomposition of the supersaturated solid solution (1978) J. Phys Chem Solids, 39, pp. 367-374; Slezov, V.V., Sagalovich, V.V., Diffusive decomposition of solid solutions (1987) Sov Phys Usp, 30, pp. 23-45; Sagui, C., O’Gorman, D.S., Grant, M., Nucleation, growth and coarsening in phase-separating systems (1998) Scanning Microsc., 12, pp. 3-8; Ratke, L., Voorhees, P., (2002) Growth and coarsening: Ostwald ripening in material processing, pp. 127-165. , Springer, New York; Baluffi, R.W., Allen, S.M., Carter, W.C., (2005) Kinetics of materials, pp. 459-542. , Wiley, Hoboken; Xiao, X., Liu, G., Hu, B., Wang, J., Ma, W., Coarsening behavior for M23C6 carbide in 12 % Cr-reduced activation ferrite/martensite steel: experimental study combined with DICTRA simulation (2013) J Mater Sci, 48, pp. 5410-5419; Sordi, V.L., Feliciano, C.A., Ferrante, M., The influence of deformation by equal-channel angular pressing on the ageing response and precipitate fracturing: case of the AlAg alloy (2015) J Mater Sci, 50, pp. 138-143; Marder, M., Correlations and Ostwald ripening (1987) Phys Rev A, 36, pp. 858-874; Yao, J.H., Elder, K.R., Guo, H., Grant, M., Theory and simulation of Ostwald ripening (1992) Phys Rev B, 47, pp. 14110-14125; Dubrovskii, V.G., Kazansky, M.A., Nazarenko, M.V., Adzhemyan, L.T., Numerical analysis of Ostwald ripening in two-dimensional systems (2011) J Chem Phys, 134, p. 094507; Alexandrov, D.V., On the theory of Ostwald ripening: formation of the universal distribution (2015) J Phys A Math Theor, 48, p. 035103; Alexandrov, D.V., Relaxation dynamics of the phase transformation process at its ripening stage (2015) J Phys A Math Theor, 48, p. 245101; Alexandrov, D.V., On the theory of Ostwald ripening in the presence of different mass transfer mechanisms (2016) J Phys Chem Solids, 91, pp. 48-54; Alexandrov, D.V., Kinetics of diffusive decomposition in the case of several mass transfer mechanisms (2017) J Cryst Growth, 457, pp. 11-18; Lifshitz, E.M., Pitaevskii, L.P., (1981) Physical kinetics, pp. 427-447. , Pergamon Press, Oxford; Enomoto, Y., Okada, A., Effects of Brownian coagulation on droplet growth in a quenched fluid mixture (1990) J Phys Condens Matter, 2, pp. 4531-4535; Alyab’eva, A.V., Buyevich, Y.A., Mansurov, V.V., Evolution of a particulate assemblage due to coalescence combined with coagulation (1994) J Phys II France, 4, pp. 951-957; Baldan, A., Progress in Ostwald ripening theories and their applications to nickel-base superalloys Part I: Ostwald ripening theories (2002) J Mater Sci, 37, pp. 2171-2202; Alexandrov, D.V., Kinetics of particle coarsening with allowance for Ostwald ripening and coagulation (2016) J Phys Condens Matter, 28, p. 035102; Carr, J., Penrose, O., Asymptotic behaviour of solutions to a simplified Lifshitz–Slyozov equation (1998) Phys D, 124, pp. 166-176; Niethammer, B., Pego, R.L., Non-self-similar behavior in the LSW theory of Ostwald ripening (1999) J Stat Phys, 95, pp. 867-902; Snyder, V.A., Alkemper, J., Voorhees, P.W., Transient Ostwald ripening and the disagreement between steady-state coarsening theory and experiment (2001) Acta Mater, 49, pp. 699-709; Carr, J., Stability of self-similar solutions in a simplified LSW model (2006) Phys D, 222, pp. 73-79; Goudon, T., Lagoutière, F., Tine, L.M., Simulations of the Lifshitz–Slyozov equations: the role of coagulation terms in the asymptotic behavior (2013) Math Models Methods Appl Sci, 23, pp. 1177-1215; Herrmann, M., Laurençot, P., Niethammer, B., Self-similar solutions with fat tails for a coagulation equation with nonlocal drift (2009) C R Acad Sci Paris Ser I, 347, pp. 909-914; Laurençot, P., The Lifshitz–Slyozov–Wagner equation with conserved total volume (2002) SIAM J Math Anal, 34, pp. 257-272; Friedlander, S.K., (1977) Smoke, dust and haze. Wiley, New York; Hunt, J.R., Self-similar particle-size distributions during coagulation: theory and experimental verification (1982) J Fluid Mech, 122, pp. 169-185; Friedlander, S.K., Wang, C.S., The self-preserving particle size distribution for coagulation by Brownian motion (1966) J Colloid Interface Sci, 22, pp. 126-132; Sonntag, H., Strenge, K., (1987) Coagulation kinetics and structure formation, pp. 58-126. , Springer, Berlin; Smoluchowski, M., Versuch einer mathematischen theorie der koagulationskinetik kolloider lösungen (1917) Ann Phys Chem, 92, pp. 129-168; Wan, G., Sahm, P.R., Particle growth by coalescence and Ostwald ripening in rheocasting of Pb–Sn (1990) Acta Metall Mater, 38, pp. 2367-2372; Randolph, A., Larson, M., (1988) Theory of particulate processes. Academic Press, New York; Buyevich, Y.A., Alexandrov, D.V., Mansurov, V.V., (2001) Macrokinetics of crystallization, , Begell House, New York; Sagui, C., Grant, M., Theory of nucleation and growth during phase separation (1999) Phys Rev E, 59, pp. 4175-4187; De Smet, Y., Danino, D., Deriemaeker, L., Talmon, Y., Finsy, R., Ostwald ripening in the transient regime: a cryo-TEM study (2000) Langmuir, 16, pp. 961-967; Carrillo, J.A., Goudon, T., A numerical study on large-time asymptotics of the Lifshitz–Slyozov system (2004) J Sci Comput, 20, pp. 69-113; Alexandrov, D.V., Nizovtseva, I.G., Nucleation and particle growth with fluctuating rates at the intermediate stage of phase transitions in metastable systems (2014) Proc Roy Soc A, 470, p. 20130647; Alexandrov, D.V., Malygin, A.P., Nucleation kinetics and crystal growth with fluctuating rates at the intermediate stage of phase transitions (2014) Model Simul Mater Sci Eng, 22, p. 015003; Crowe, C.T., (2006) Multiphase flow handbook. Taylor & Francis, New York; Luo, X., Benichou, Y., Yu, S., Calculation of collision frequency function for aerosol particles in free molecule regime in presence of force fields (2013) Front Energy, 7, pp. 506-510
Correspondence Address Alexandrov, D.V.; Department of Theoretical and Mathematical Physics, Laboratory of Multi-Scale Mathematical Modeling, Ural Federal UniversityRussian Federation; email: dmitri.alexandrov@urfu.ru
Publisher Springer New York LLC
CODEN JMTSA
Language of Original Document English
Abbreviated Source Title J Mater Sci
Source Scopus