Surface-Kinetics-Limited Ostwald Ripening of Spherical Precipitates at Grain Boundaries
Ostwald ripening of sufficiently large (usually macroscopic) precipitates is the late stage of the diffusion decomposition of a supersaturated solid solution, occurring through the formation of fluctuations and subsequent growth of centers (nuclei) of a new phase. The paper describes a theoretical study of the Ostwald ripening of spherical precipitates of a newly formed phase at the grain boundary of finite thickness with the diffusion of impurity atoms from the grain interior to the grain boundary considered. The precipitate growth is assumed to be limited by the kinetics of impurity atom imbedding into the precipitate rather than by the impurity atom diffusion inside the grain boundary. The speed of diffusion growth of spherical precipitate located on the grain boundary is found. A system of equations which describes surface-kinetics-limited growth of Oswald ripening of spherical precipitates on the grain boundary is formulated. This system consists of the equation of growth rate of the precipitate, the kinetic equation for the precipitates size distribution function which is normalized by the precipitates density, and the equation of the balance of matter in the system (the law of conservation of matter). The law of conservation of matter takes into account the atoms of impurities which are in solid solutions of the grain boundary and the body of the grain as well as in the precipitates which is the specifics of our problem. The asymptotic time dependences are found for the average and critical precipitate radius, supersaturation of solid solution of impurity atoms in the grain boundary, precipitate size distribution function, precipitate density, and for the factor of grain boundary filling with precipitates (the area covered by the precipitates per unit area of the grain boundary) and the total number of impurity atoms in precipitates. The factor of grain boundary filling with precipitates is a characteristic of the two-dimensional Ostwald ripening problem. A discussion of the limits of validity of obtained results is given.
Ya.I. Frenkel’, Введение в Теорию Металлов, 4-e изд. [Introduction to the Theory of Metals] (Nauka, Leningrad, 1972), 424 p. (in Russian)
V.V. Slezov and V.V. Sagalovich, Sov. Phys. Usp. 30(1), 23-45 (1987).
R.D. Vengrenovitch, Acta Metall. 30(6), 1079-1086 (1982).
K.V. Chuistov, Упорядочение и Распад в Пересыщенных Твердых Растворах [Ordering and Decomposition in Supersaturated Solid Solutions] (RIO IMF, Kiev, 1999), 216 p. (in Russian)
K.V. Chuistov, Старение Металлических Сплавов, 2-e изд. [Aging of Metallic Alloys] (Academperiodika, Kiev, 2003), 568 p. (in Russian)
W. Ostwald, Zs. Phys. Chem. 34, 495-503 (1900). (in German)
I.M. Lifshitz and V.V. Slyozov, J. Phys. Chem. Solids. 19(1-2), 35-50 (1961).
C. Wagner, Zs. Elektrochem. 65(7/8), 581-591 (1961). (in German)
S.K. Bhattacharyya and K.C. Russell, Metall. Trans. 3(8), 2195-2199 (1972).
S.A. Kukushkin and A.V. Osipov, Phys. Usp. 41(10), 983-1014 (1998).
A.V. Koropov and V.V. Sagalovich, Poverkhnost’. Fizika, Khimiya, Mekhanika. No. 6, 50-55 (1987) (in Russian).
A.V. Koropov and V.V. Sagalovich, Poverkhnost’. Fizika, Khimiya, Mekhanika. No. 5, 55-63 (1989) (in Russian).
A.V. Koropov and V.V. Sagalovich, Poverkhnost’. Fizika, Khimiya, Mekhanika. No. 2, 17-26 (1990) (in Russian).
A.V. Koropov, P.N. Ostapchuk and V.V. Slezov, Sov. Phys. Solid State. 33(10), 1602-1607 (1991).
R.D. Vengrenovich, B.V. Ivanskii and A.V. Moskaliuk, Physics and Chemistry of Solid State. 10(1), 19-30 (2009). (in Ukrainian)
A.A. Chernov, in: Modern Crystallography III. Crystal Growth, edited by M. Cardona et al. (Springer-Verlag, Heidelberg, 1984), pp. 1-297.
A.V. Koropov, S.A. Kukushkin and D.A. Grigor’ev, Tech. Phys. 44(7), 786-791 (1999).
A.M. Gusak and G.V. Lutsenko, Phil. Mag. 85(10), 1323-1331 (2005).
A.M. Gusak, G.V. Lutsenko and K.N. Tu, Acta Mater. 54(3), 785-791 (2006).
B.V. Ivanskii, R.D. Vengrenovich, V.I. Kryvetskyi and Yu.M. Kushnir, J. Nano- Electron. Phys. 9(2), 02025 (2017). (in Ukrainian)
V.V. Slezov, J. Phys. Chem. Solids. 39(4), 367-374 (1978).
J.A. Marqusee and J. Ross, J. Chem. Phys. 80(1), 536-543 (1984).
M. Marder, Phys. Rev. A. 36(2), 858-874 (1987).
P.G. Cheremskoi, V.V. Slezov and V.I. Betekhtin, Поры в Твердом Теле [Pores in Solids] (Energoatomizdat, Moscow, 1990), 376 p. (in Russian)
B. Giron, B. Meerson and P.V. Sasorov, Phys. Rev. E. 58(4), 4213-4216 (1998).
V.V. Slezov and J. Schmelzer, Phys. Rev. E. 65, 031506 (2002).
V.V. Slezov, J. Colloid Interface Sci. 255(2), 274-292 (2002).
P. Guyot, L. Lae and C. Sigli, in: Thermodynamics, Microstructures and Plasticity, edited by A. Finel et. al. (Kluwer Academic Publishers, Dordrecht, 2003), pp. 107-121.
A. Onuki, Phase Transition Dynamics (Cambridge, Cambridge University Press, 2004), 724 p.
A.S. Shirinian and M.P. Kudyn, Metallofiz. Noveishie Tekhnol. 29(11), 1537-1553 (2007). (in Russian)
A.S. Shirinian and M.P. Kudyn, Ukr. J. Phys. 53(1), 50-60 (2008). (in Ukrainian)
R.D. Vengrenovich, B.V. Ivanskii and M.O. Stasyk, Metallofiz. Noveishie Tekhnol. 32(8), 1085-1104 (2010). (in Ukrainian)
H. Kreye, Zs. Metallkunde. 61(2), 108-112 (1970). (in German)
A.J. Ardell, Acta Metall. 20(4), 601-609 (1972).
R.D. Vengrenovich, Yu.V. Gudyma and S.V. Yarema, Phys. Met. Metallogr. 91(3), 228-232 (2001).
R.D. Vengrenovich, A.V. Moskalyuk and S.V. Yarema, Phys. Solid State. 49(1), 11-17 (2007).
H.O.K. Kirchner, Metall. Trans. 2(10), 2861-2864 (1971).
J.W. Martin, R.D. Doherty and B. Cantor, Stability of Microstructure in Metallic Systems, 2nd ed. (Cambridge University Press, Cambridge, 1997), 426 p.
O.V. Koropov, J. Nano- Electron. Phys. 6(1), 01025 (2014). (in Ukrainian)
O.V. Koropov, in: П’ятнадцята Міжнародна Наукова Конференція ім. акад. Михайла Кравчука [Fifteenth International Scientific Mykhailo Kravchuk Conference] (NTUU “KPI”, Kyiv, 2014), 1, pp. 162-167. (in Ukrainian)
V.N. Voyevodin and I.M. Neklyudov, Эволюция Структурно-Фазового Состояния и Радиационная Стойкость Конструкционных Материалов [Evolution of the Structure Phase State and Radiation Resistance of Structural Materials] (Nauk. Dumka, Kiev, 2006), 376 p. (in Russian)
B.S. Bokstein, Ch.V. Kopezky and L.S. Shvindlerman, Термодинамика и Кинетика Границ Зерен в Металлах [Thermodynamics and Kinetics of Grain Boundaries in Metals] (Metallurgiya, Moscow, 1986), 224 p. (in Russian)
I. Kaur, Y. Mishin and W. Gust, Fundamentals of Grain and Interphase Boundary Diffusion, 3rd rev. and enl. ed. (John Wiley & Sons Ltd., New York, 1995), 512 p.
B.S. Bokstein and A.B. Yaroslavtsev, Диффузия Атомов и Ионов в Твердых Телах [Diffusion of Atoms and Ions in Solids] (MISIS, Moscow, 2005), 362 p. (in Russian)
A.V. Koropov, Visnyk of Sumy State University. Seriya: Fizyka, Matematyka, Mekhanika. 9(93), 49-62 (2006). (in Russian)
A.V. Koropov, Phys. Solid State. 50(11), 2184-2189 (2008).
A.V. Koropov, J. Surf. Investig. X-Ray, Synchrotron and Neutron Techniques. 5(4), 780-786 (2011).
A.V. Koropov, J. Nano- Electron. Phys. 2(4), 31-46 (2010).
A.V. Koropov, Tech. Phys. 56(12), 1781-1786 (2011).
F. Olver, in: Справочник по специальным функциям с формулами графиками и математическими таблицами [Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables] (Transl. from Engl.), edited by M. Abramowitz and I.A. Stegun (Nauka, Moscow, 1979), pp. 177-253. (in Russian)
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