najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМОnajo-1035Research ArticleCHEMISTRY AND MATERIAL SCIENCEХИМИЯ И МАТЕРИАЛОВЕДЕНИЕMetastable clusters and aggregative nucleation mechanismAlmjashevaO. V.

St. Petersburg

almjasheva@mail.ruGusarovV. V.

St. Petersburg

St. Petersburg State Electrothechnical University “LETI”; Ioffe InstituteRussian FederationIoffe InstituteRussian Federation20141508202553405416Copyright © Almjasheva O.V., Gusarov V.V., 20252025Almjasheva O.V., Gusarov V.V.Almjasheva O.V., Gusarov V.V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/1035

Feasible nucleation in condensed media by the aggregation mechanism of small metastable crystalline clusters is demonstrated. The presence of the stable small clusters in the initial phases makes the homogeneous and heterogeneous nucleation processes more similar.

nucleationhomogeneous and heterogeneous nucleationclustersThis work was supported by Russian Foundation for Basic Research (grant No 13- 03-00888).ReferencesWilson C.T.R. Condensation of water vapour in the presence of dast-free air and other gases. Phil. Trans. R. Soc. Lond. A, 189, P. 256–307 (1897).Wilson C.T.R. Condensation of water vapour in the presence of dast-free air and other gases. Phil. Trans. R. Soc. Lond. A, 189, P. 256–307 (1897).Tammann G. Uber die abh¨angigkeit der zahl der kerne, welche sich in verschiedenen unterk¨uhlten ¨ fl¨ussigkeiten bilden, von der temperature. Zeit. f. Physik. Chemie, 25, P. 441 (1898).Tammann G. Uber die abh¨angigkeit der zahl der kerne, welche sich in verschiedenen unterk¨uhlten ¨ fl¨ussigkeiten bilden, von der temperature. Zeit. f. Physik. Chemie, 25, P. 441 (1898).Wilson C.T.R. On the condensation nuclei produced in gases by the action of Rontgen rays, uranium rays, ultra-violet light, and other agents. Phil. Trans. R. Soc. Lond. A, 192, P. 403–453 (1899).Wilson C.T.R. On the condensation nuclei produced in gases by the action of Rontgen rays, uranium rays, ultra-violet light, and other agents. Phil. Trans. R. Soc. Lond. A, 192, P. 403–453 (1899).Wilson C.T.R. On the comparative efficiency as condensation nuclei of positively and negatively charged ions. Phil. Trans. R. Soc. Lond. A, 93, P. 289–308 (1900).Wilson C.T.R. On the comparative efficiency as condensation nuclei of positively and negatively charged ions. Phil. Trans. R. Soc. Lond. A, 93, P. 289–308 (1900).Tammann G. Aggregatzust¨ande. Verlag von Leopold Voss, Leipzig, 237 p. (1922).Tammann G. Aggregatzust¨ande. Verlag von Leopold Voss, Leipzig, 237 p. (1922).Volmer M., Weber A. Keimbildung in ¨ubers¨attigten gebilden. Z. Phys. Chem., 119, P. 277 (1926).Volmer M., Weber A. Keimbildung in ¨ubers¨attigten gebilden. Z. Phys. Chem., 119, P. 277 (1926).Becker R., D¨oring W. Kinetische behandlung der keimbildung in ¨ubers¨attiigten d¨ampfen. Ann. Phys., 416(8), P. 719–752 (1935).Becker R., D¨oring W. Kinetische behandlung der keimbildung in ¨ubers¨attiigten d¨ampfen. Ann. Phys., 416(8), P. 719–752 (1935).Volmer M. Kinetik der Phasenbildung. Steinkopff, Dresden-Leipzig (1939).Volmer M. Kinetik der Phasenbildung. Steinkopff, Dresden-Leipzig (1939).Frenkel J. A general theory of heterophase fluctuations and pretransition phenomena. J. Chem. Phys, 7(7), P. 538–546 (1939).Frenkel J. A general theory of heterophase fluctuations and pretransition phenomena. J. Chem. Phys, 7(7), P. 538–546 (1939).Zel’dovch Ya. On the theory of new phase formation. Cavitation. ZhETF, 12(11/12), P. 525–538 (1942),(in Russian).Zel’dovch Ya. On the theory of new phase formation. Cavitation. ZhETF, 12(11/12), P. 525–538 (1942),(in Russian).Pound G.M., La Mer V.K. Kinetics of crystalline nucleus formation in supercooled liquid tin. J. Amer. Chem. Soc., 74, P. 2323 (1952).Pound G.M., La Mer V.K. Kinetics of crystalline nucleus formation in supercooled liquid tin. J. Amer. Chem. Soc., 74, P. 2323 (1952).Hillert M. A. Theory of nucleation for solid metallic solutions. Sc. D. Thesis, MIT (1955).Hillert M. A. Theory of nucleation for solid metallic solutions. Sc. D. Thesis, MIT (1955).Cahn J.W., Hilliard J.E. Free energy of a nonuniform system. III. Nucleation in a two-component incompressible fluid. J. Chem. Phys., 31(3), P. 688–700 (1959).Cahn J.W., Hilliard J.E. Free energy of a nonuniform system. III. Nucleation in a two-component incompressible fluid. J. Chem. Phys., 31(3), P. 688–700 (1959).Courtney W.G. Remarks on homogeneous nucleation. J. Chem. Phys., 35(6), P. 2249 (1961).Courtney W.G. Remarks on homogeneous nucleation. J. Chem. Phys., 35(6), P. 2249 (1961).Lothe J., Pound G.M. Reconsideration of nucleation theory. J. Chem. Phys., 36(8), P. 2080–2085 (1962).Lothe J., Pound G.M. Reconsideration of nucleation theory. J. Chem. Phys., 36(8), P. 2080–2085 (1962).Reiss H., Katz J.L. Resolution of the translation-rotation paradox in the theory of irreversible condensation. J. Chem. Phys., 46(7), P. 2496–2499 (1962).Reiss H., Katz J.L. Resolution of the translation-rotation paradox in the theory of irreversible condensation. J. Chem. Phys., 46(7), P. 2496–2499 (1962).Flettcher N.H. The physics of rainclouds. University Press, Cambridge (1962).Flettcher N.H. The physics of rainclouds. University Press, Cambridge (1962).Reiss H., Katz J.L., Cohen E.R. Translation-rotation paradox in the theory of nucleation. J. Chem. Phys., 48(12), P. 5553–5560 (1968).Reiss H., Katz J.L., Cohen E.R. Translation-rotation paradox in the theory of nucleation. J. Chem. Phys., 48(12), P. 5553–5560 (1968).Lothe J. Concentration of clusters in nucleation and the classical phase integral. J. Chem. Phys., 48(4), P. 1849–1852 (1968).Lothe J. Concentration of clusters in nucleation and the classical phase integral. J. Chem. Phys., 48(4), P. 1849–1852 (1968).Abraham F.F. Re-examination of homogeneous nucleation theory: statistical thermodynamics aspects. J. Chem. Phys., 48(2), P. 732–740 (1968).Abraham F.F. Re-examination of homogeneous nucleation theory: statistical thermodynamics aspects. J. Chem. Phys., 48(2), P. 732–740 (1968).Lin J. Equilibrium distribution of droplets in the theory of nucleation. J. Chem. Phys., 48(9), P. 4128– 4130 (1968).Lin J. Equilibrium distribution of droplets in the theory of nucleation. J. Chem. Phys., 48(9), P. 4128– 4130 (1968).Stillinger F.H. Comment on the translation-rotation paradox in the theory of irreversible condensation. J. Chem. Phys., 48(3), P. 1430–1431 (1968).Stillinger F.H. Comment on the translation-rotation paradox in the theory of irreversible condensation. J. Chem. Phys., 48(3), P. 1430–1431 (1968).Kuni F.M., Rusanov A.I. The homogeneous nucleation theory and the fluctuation of the center of mass of a drop. Phys. Letters., 29A(6), P. 337–338 (1969).Kuni F.M., Rusanov A.I. The homogeneous nucleation theory and the fluctuation of the center of mass of a drop. Phys. Letters., 29A(6), P. 337–338 (1969).Wood G.R., Walton A.G. Homogeneous nucleation kinetics of ice from water. J. Appl. Phys., 41(7), P. 3027–3036 (1970).Wood G.R., Walton A.G. Homogeneous nucleation kinetics of ice from water. J. Appl. Phys., 41(7), P. 3027–3036 (1970).Reiss H. Treatment of droplike clusters by means of the classical phase integral in nucleation theory. J. Stat. Phys., 2(1), P. 83–104 (1970).Reiss H. Treatment of droplike clusters by means of the classical phase integral in nucleation theory. J. Stat. Phys., 2(1), P. 83–104 (1970).Blander M., Katz J.L. The thermodynamics of cluster formation in nucleation theory. J. Stat. Phys., 4(1), P. 55–59 (1972).Blander M., Katz J.L. The thermodynamics of cluster formation in nucleation theory. J. Stat. Phys., 4(1), P. 55–59 (1972).Abraham F.F. Homogeneous nucleation theory. Academic Press, NY, (1974).Abraham F.F. Homogeneous nucleation theory. Academic Press, NY, (1974).Bendig L.L., Larson M.A. Nuclei generation from repetitive contacting. AIChE Symposium Series, 25, P. 57 (1976).Bendig L.L., Larson M.A. Nuclei generation from repetitive contacting. AIChE Symposium Series, 25, P. 57 (1976).Katz J.L. Nucleation theory without Maxwell Demons. J. Coll. Interface Sci., 61(2), P. 351–355 (1977).Katz J.L. Nucleation theory without Maxwell Demons. J. Coll. Interface Sci., 61(2), P. 351–355 (1977).Khambaty S., Larson M.A. Crystal regeneration and growth of small crystals in contact nucleation. I&EC Fundamentals, 17, P. 160 (1978).Khambaty S., Larson M.A. Crystal regeneration and growth of small crystals in contact nucleation. I&EC Fundamentals, 17, P. 160 (1978).Katz J.L., Donohue M.D. A kinetic approach to homogeneous nucleation theory. Adv. Chem. Phys., 40, P. 137 (1979)Katz J.L., Donohue M.D. A kinetic approach to homogeneous nucleation theory. Adv. Chem. Phys., 40, P. 137 (1979)Slezov V.V., Sagalovich V.V. Diffusive decomposition of solid solutions. Sov. Phys. Usp., 30(1), P. 23–45 (1987).Slezov V.V., Sagalovich V.V. Diffusive decomposition of solid solutions. Sov. Phys. Usp., 30(1), P. 23–45 (1987).Garnier J.P., Mirabel P., Rabeony H. Experimental results of homogeneous nucleation of supersaturated vapors, J. Chem. Phys., 79(4), P. 2097–2098 (1983).Garnier J.P., Mirabel P., Rabeony H. Experimental results of homogeneous nucleation of supersaturated vapors, J. Chem. Phys., 79(4), P. 2097–2098 (1983).Ruth V., Hirth J.P., Pound G.M. On the theory of homogeneous nucleation and spinodal decomposition in condensation from the vapor phase. J. Chem. Phys., 88(11), P. 7079 (1988).Ruth V., Hirth J.P., Pound G.M. On the theory of homogeneous nucleation and spinodal decomposition in condensation from the vapor phase. J. Chem. Phys., 88(11), P. 7079 (1988).Larson M.A. Solute clustering and secondary nucleation. Conference on Advances in Industrial Crystallization. 1990, College of Engineering Iowa State University, Preprint 91130.Larson M.A. Solute clustering and secondary nucleation. Conference on Advances in Industrial Crystallization. 1990, College of Engineering Iowa State University, Preprint 91130.Girshick S.L., Chiu C.P. Kinetic nucleation theory: A new expression for the rate of homogeneous nucleation from an ideal supersaturated vapor. J. Chem. Phys., 93(2), P. 1273–1278 (1990).Girshick S.L., Chiu C.P. Kinetic nucleation theory: A new expression for the rate of homogeneous nucleation from an ideal supersaturated vapor. J. Chem. Phys., 93(2), P. 1273–1278 (1990).Zeng X.C., Oxtoby D.W. Gas-liquid nucleation in Lennard-Jones fluids. J. Chem. Phys., 94(8), P. 4472– 4478 (1991).Zeng X.C., Oxtoby D.W. Gas-liquid nucleation in Lennard-Jones fluids. J. Chem. Phys., 94(8), P. 4472– 4478 (1991).Cahn R.W., Haasen P. Kramer E.J. Materials Science and technology, 5 (Phase transformation in materials), VCH (1991).Cahn R.W., Haasen P. Kramer E.J. Materials Science and technology, 5 (Phase transformation in materials), VCH (1991).Brener E.A., Marchenko V.I., Formation of nucleation centers in a crystal. JETP Lett., 56(7), P. 368–372 (1992).Brener E.A., Marchenko V.I., Formation of nucleation centers in a crystal. JETP Lett., 56(7), P. 368–372 (1992).Adachi M., Okuyama K., Seinfeld J.H. Experimental studies of ion-induced nucleation. J. Aerosol. Sci., 23(4), P. 327–337 (1992).Adachi M., Okuyama K., Seinfeld J.H. Experimental studies of ion-induced nucleation. J. Aerosol. Sci., 23(4), P. 327–337 (1992).Girshick S.L. Comment on: “Self-consistency correction to homogeneous nucleation theory”. J. Chem. Phys., 94(1), P. 826–828 (1991).Girshick S.L. Comment on: “Self-consistency correction to homogeneous nucleation theory”. J. Chem. Phys., 94(1), P. 826–828 (1991).Olemskoi A.I., Koplyk I.V. The theory of spatiotemporal evolution of nonequilibrium thermodynamic systems. Phys. Usp., 38(10), P. 1061–1097 (1995).Olemskoi A.I., Koplyk I.V. The theory of spatiotemporal evolution of nonequilibrium thermodynamic systems. Phys. Usp., 38(10), P. 1061–1097 (1995).Oxtoby D.W. Nucleation of first-order phase transitions. Acc. Chem. Res., 31(2), P. 91–97 (1998).Oxtoby D.W. Nucleation of first-order phase transitions. Acc. Chem. Res., 31(2), P. 91–97 (1998).Antonov N.M., Popov I.Yu., Gusarov V.V. Model of spinodal decomposition of phases under hyperbolic diffusion. Phys. Solid State., 41(5), P. 824–826 (1999).Antonov N.M., Popov I.Yu., Gusarov V.V. Model of spinodal decomposition of phases under hyperbolic diffusion. Phys. Solid State., 41(5), P. 824–826 (1999).Gorbachev Yu.E., Nikitin I.S. Evolution of cluster size distribution during nucleation with rapidly changing dynamic gas processes. Tech. Phys., 45(12), P. 1538–1548 (2000).Gorbachev Yu.E., Nikitin I.S. Evolution of cluster size distribution during nucleation with rapidly changing dynamic gas processes. Tech. Phys., 45(12), P. 1538–1548 (2000).Yau S.-T., Vekilov P.G. Quasi-planar nucleus structure in apoferritin crystallization. Nature, 406, P. 494–497 (2000).Yau S.-T., Vekilov P.G. Quasi-planar nucleus structure in apoferritin crystallization. Nature, 406, P. 494–497 (2000).Oxtoby D.W. Phase transitions: Catching crystals at birth. Nature, 406, P. 464–465 (2000).Oxtoby D.W. Phase transitions: Catching crystals at birth. Nature, 406, P. 464–465 (2000).Kuni F.M., Shchekin A.K., Grinin A.P. Theory of heterogeneous nucleation for vapor undergoing a gradual metastable state formation. Phys. Usp., 44(4), P. 331–370 (2001).Kuni F.M., Shchekin A.K., Grinin A.P. Theory of heterogeneous nucleation for vapor undergoing a gradual metastable state formation. Phys. Usp., 44(4), P. 331–370 (2001).Radhakrishnan R., Trout B.L. A new approach for studying nucleation phenomena using molecular simulations: application to CO2 hydrate clathrates. J. Chem. Phys., 117(4), P. 1786–1796 (2002).Radhakrishnan R., Trout B.L. A new approach for studying nucleation phenomena using molecular simulations: application to CO2 hydrate clathrates. J. Chem. Phys., 117(4), P. 1786–1796 (2002).Auer S., Frencel D. Numerical predication of absolute crystallization rates in hard-sphere colloids. J. Chem. Phys., 120(6), P. 3015–3029 (2004).Auer S., Frencel D. Numerical predication of absolute crystallization rates in hard-sphere colloids. J. Chem. Phys., 120(6), P. 3015–3029 (2004).Xu D., Johnson W.L. Geometric model for the critical-value problem of nucleation phenomena containing the size effect of nucleating agent. Phys. Rev. B., 72, P. 052101 (2005).Xu D., Johnson W.L. Geometric model for the critical-value problem of nucleation phenomena containing the size effect of nucleating agent. Phys. Rev. B., 72, P. 052101 (2005).Bushuev Yu.G., Davletbaeva S.V. Molecular dynamics simulation of the kinetics of nucleation of supercooled NaCl melt clusters. Russ. J. Phys. Chem. A, 83(4), P. 630–636 (2009).Bushuev Yu.G., Davletbaeva S.V. Molecular dynamics simulation of the kinetics of nucleation of supercooled NaCl melt clusters. Russ. J. Phys. Chem. A, 83(4), P. 630–636 (2009).Al’myashev O.V., Gusarov V.V. Features of the phase formation in the nanocomposites. Russ. J. Gen. Chem., 80(3), P. 385–390 (2010).Al’myashev O.V., Gusarov V.V. Features of the phase formation in the nanocomposites. Russ. J. Gen. Chem., 80(3), P. 385–390 (2010).Al’myasheva O.V., Gusarov V.V. Nucleation in media in which nanoparticles of another phase are distributed. Dokl. Phys. Chem., 424(2), P. 43–45 (2009).Al’myasheva O.V., Gusarov V.V. Nucleation in media in which nanoparticles of another phase are distributed. Dokl. Phys. Chem., 424(2), P. 43–45 (2009).Fedoseev V.B., Fedoseeva E.N. Size effects during phase transformations in stratifying systems. Russ. J. Phys. Chem. A, 88(3), P. 436–441 (2014).Fedoseev V.B., Fedoseeva E.N. Size effects during phase transformations in stratifying systems. Russ. J. Phys. Chem. A, 88(3), P. 436–441 (2014).Fedoseev V.B., Fedoseeva E.N. State supersaturated solution in systems of limited size. JETP Lett., 97(7), P. 408 (2013).Fedoseev V.B., Fedoseeva E.N. State supersaturated solution in systems of limited size. JETP Lett., 97(7), P. 408 (2013).Ebeling W., Engel A., Feistel R. Physik der Evolutionsprozesse. Akademie-Verlag, Berlin, 371 p (1990).Ebeling W., Engel A., Feistel R. Physik der Evolutionsprozesse. Akademie-Verlag, Berlin, 371 p (1990).Prigogine I. From Being to Becoming. Nauka, Moscow (1985). (in Russian)Prigogine I. From Being to Becoming. Nauka, Moscow (1985). (in Russian)Klimontovich Yu.L. Statistical theory of open systems V.1. Yanus, Moscow (1995). (in Russian)Klimontovich Yu.L. Statistical theory of open systems V.1. Yanus, Moscow (1995). (in Russian)Klimontovich Yu.L. Relative ordering criteria in open systems. Phys. Usp., 39(11), P. 1169–1179 (1996).Klimontovich Yu.L. Relative ordering criteria in open systems. Phys. Usp., 39(11), P. 1169–1179 (1996).Rusanov A.I. Thermodynamics of surface phenomena. Leningrad University, Leningrad (1960). (in Russia)Rusanov A.I. Thermodynamics of surface phenomena. Leningrad University, Leningrad (1960). (in Russia)Rusanov A.I. Phase equilibria and surface phenomena. Khimiya, Leningrad (1967). (in Russian); German edition; Rusanov A.I. Phasengleichewichte und Grenzflchenerscheinungen. Akademie-Verlag, Berlin (1978).Rusanov A.I. Phase equilibria and surface phenomena. Khimiya, Leningrad (1967). (in Russian); German edition; Rusanov A.I. Phasengleichewichte und Grenzflchenerscheinungen. Akademie-Verlag, Berlin (1978).Rusanov A.I. Thermodynamic fundamentals of mechanochemistry. Nauka, Saint-Petersburg, 221 p. (2006). (in Russian)Rusanov A.I. Thermodynamic fundamentals of mechanochemistry. Nauka, Saint-Petersburg, 221 p. (2006). (in Russian)Grinfeld M.A. Methods of continuum mechanics in theory of phase transformations. Nauka, Moscow (1990). (in Russian)Grinfeld M.A. Methods of continuum mechanics in theory of phase transformations. Nauka, Moscow (1990). (in Russian)Gusarov V.V. Statics and dynamics of systems based on polycrystalline refractory oxides. Sc. D. Thesis, St. Petersburg State Technological Institute (Technical University), St. Petersburg (1996). (in Russian)Gusarov V.V. Statics and dynamics of systems based on polycrystalline refractory oxides. Sc. D. Thesis, St. Petersburg State Technological Institute (Technical University), St. Petersburg (1996). (in Russian)Lomanova N.A., Gusarov V.V. Influence of synthesis temperature on BiFeO3 nanoparticles formation. Nanosystems: physics, chemistry, mathematics, 4(5), P. 696–705 (2013).Lomanova N.A., Gusarov V.V. Influence of synthesis temperature on BiFeO3 nanoparticles formation. Nanosystems: physics, chemistry, mathematics, 4(5), P. 696–705 (2013).Lomanova N.A., Gusarov V.V. Effect of surface melting on the formation and growth of nanocrystals in the Bi2O3-Fe2O3 system. Russ. J. Gen. Chem., 83(12), P. 2251–2253 (2013).Lomanova N.A., Gusarov V.V. Effect of surface melting on the formation and growth of nanocrystals in the Bi2O3-Fe2O3 system. Russ. J. Gen. Chem., 83(12), P. 2251–2253 (2013).Lomanova N.A., Gusarov V.V. Effect of the phase composition of the starting mixture on the formation of the layered perovskite-like compound Bi7Fe3Ti3O21. Russ. J. Inorg. Chem., 55(10), P. 1541–1545 (2010).Lomanova N.A., Gusarov V.V. Effect of the phase composition of the starting mixture on the formation of the layered perovskite-like compound Bi7Fe3Ti3O21. Russ. J. Inorg. Chem., 55(10), P. 1541–1545 (2010).Pozhidaeva O.V., Korytkova E.N., Romanov D.P., Gusarov V.V. Formation of ZrO2 nanocrystals in hydrothermal media of various chemical compositions. Russ. J. Gen. Chem., 72(6), P. 849–853 (2002).Pozhidaeva O.V., Korytkova E.N., Romanov D.P., Gusarov V.V. Formation of ZrO2 nanocrystals in hydrothermal media of various chemical compositions. Russ. J. Gen. Chem., 72(6), P. 849–853 (2002).Pozhidaeva O.V., Korytkova E.N., Drozdova I.A., Gusarov V.V. Phase state and particle size of ultradispersed zirconium dioxide as influenced by conditions of hydrothermal synthesis. Russ. J. Gen. Chem., 69(8), P. 1219–1222 (1999).Pozhidaeva O.V., Korytkova E.N., Drozdova I.A., Gusarov V.V. Phase state and particle size of ultradispersed zirconium dioxide as influenced by conditions of hydrothermal synthesis. Russ. J. Gen. Chem., 69(8), P. 1219–1222 (1999).Rao C. N. R., Raveau B. Transition metal oxides: structure, properties, and synthesis of ceramic oxides. Wiley, NY, 392 p. (1998).Rao C. N. R., Raveau B. Transition metal oxides: structure, properties, and synthesis of ceramic oxides. Wiley, NY, 392 p. (1998).Svensson G. HREM Studies of intergrowth between NbO and perovskite in the Ba-, Sr and K-Nb-O systems. Microsc. Microanal. Microstruct., 1(5-6), P. 343–356 (1990).Svensson G. HREM Studies of intergrowth between NbO and perovskite in the Ba-, Sr and K-Nb-O systems. Microsc. Microanal. Microstruct., 1(5-6), P. 343–356 (1990).Magn´eli A. On heterogeneous crystalline compounds and the phasoid concept. Microsc. Microanal. Microstruct., 1(5-6), P. 299–302 (1990).Magn´eli A. On heterogeneous crystalline compounds and the phasoid concept. Microsc. Microanal. Microstruct., 1(5-6), P. 299–302 (1990).Bernuy-Lopez C., Pelloquin D., Raveau B., Allix M., Claridgea J.B., Rosseinskya M. J., Wangc P., Blelochc A. Phasoid intergrowth between the double perovskite Sr2MgMoO6 and the n=2 R-P phase Sr3Mo2O7. Solid State Ionics, 181(19-20), P. 889–893 (2010).Bernuy-Lopez C., Pelloquin D., Raveau B., Allix M., Claridgea J.B., Rosseinskya M. J., Wangc P., Blelochc A. Phasoid intergrowth between the double perovskite Sr2MgMoO6 and the n=2 R-P phase Sr3Mo2O7. Solid State Ionics, 181(19-20), P. 889–893 (2010).Rusanov A.I., Shchukin E.D. Rebinder P.A. On the theory of dispersion. I. Thermodynamics of monodisperse systems. Colloid J. USSR, 30(5), P. 428 (1968). (in Russiun)Rusanov A.I., Shchukin E.D. Rebinder P.A. On the theory of dispersion. I. Thermodynamics of monodisperse systems. Colloid J. USSR, 30(5), P. 428 (1968). (in Russiun)Rusanov A.I., Kuni F.M., Shchukin E.D. Rebinder P.A. On the theory of dispersion. 3. Dispersion in liquid medium. Colloid J. USSR, 30(5), P. 561 (1968).(in Russian)Rusanov A.I., Kuni F.M., Shchukin E.D. Rebinder P.A. On the theory of dispersion. 3. Dispersion in liquid medium. Colloid J. USSR, 30(5), P. 561 (1968).(in Russian)Rebinder P.A., Shchukin E.D. Surface phenomena in solids during the course of their deformation and failure. Sov. Phys. Usp., 15(5), P. 533 (1973).Rebinder P.A., Shchukin E.D. Surface phenomena in solids during the course of their deformation and failure. Sov. Phys. Usp., 15(5), P. 533 (1973).Shchukin E.D., Pertsov A.V. Thermodynamic criterion of spontaneous dispersion. Colloid and interface science series. V. 1. Colloid stability: The role of surface forces. Part 1. Wiley, NY (2007). P. 23.Shchukin E.D., Pertsov A.V. Thermodynamic criterion of spontaneous dispersion. Colloid and interface science series. V. 1. Colloid stability: The role of surface forces. Part 1. Wiley, NY (2007). P. 23.Samsonov V.M., Murav’ev S.D., Bazulev A.N. Surface characteristics, structure and stability of nanosized particles. Russ. J. Phys. Chem. A, 74(11), P. 1791–1795 (2000).Samsonov V.M., Murav’ev S.D., Bazulev A.N. Surface characteristics, structure and stability of nanosized particles. Russ. J. Phys. Chem. A, 74(11), P. 1791–1795 (2000).http://www.chem.msu.su/rus/history/Rebinder/17.htmlhttp://www.chem.msu.su/rus/history/Rebinder/17.htmlRebinder P.A. Formation and aggregative stability of disperse systems. Colloid J. USSR, 20(5), P. 493– 502 (1958).Rebinder P.A. Formation and aggregative stability of disperse systems. Colloid J. USSR, 20(5), P. 493– 502 (1958).Shchukin E.D., Rebinder P.A. The formation of new surfaces at deformation and destruction of a rigid body in a surface-active medium. Koll. Zh., 20(5), P. 645–654 (1958).Shchukin E.D., Rebinder P.A. The formation of new surfaces at deformation and destruction of a rigid body in a surface-active medium. Koll. Zh., 20(5), P. 645–654 (1958).Kligman F.I., Rusanov A.I. On the thermodynamic equilibrium states of disperse systems with solid particles. Koll. Zh., 39(1), P. 44–47 (1977). (in Russian)Kligman F.I., Rusanov A.I. On the thermodynamic equilibrium states of disperse systems with solid particles. Koll. Zh., 39(1), P. 44–47 (1977). (in Russian)Kuni F.M., Rusanov A.I. Statistical theory of aggregative equilibrium. Theor. Math. Phys, 2(2), P. 192– 206 (1970).Kuni F.M., Rusanov A.I. Statistical theory of aggregative equilibrium. Theor. Math. Phys, 2(2), P. 192– 206 (1970).Rusanov A.I. On thermodynamic conditions of spontaneous dispergation of solids. Vestn. Leningr. Univ., 10, P. 38–49 (1983). (in Russian)Rusanov A.I. On thermodynamic conditions of spontaneous dispergation of solids. Vestn. Leningr. Univ., 10, P. 38–49 (1983). (in Russian)Rusanov A.I. Thermodynamics of solid surface. Surf. Sci. Rep., 23, P. 173–247 (1996).Rusanov A.I. Thermodynamics of solid surface. Surf. Sci. Rep., 23, P. 173–247 (1996).Ubbelode A. R. The Molten State of Matter. London University, London, 384 p. (1978).Ubbelode A. R. The Molten State of Matter. London University, London, 384 p. (1978).Denisov V.M., Belousova N.V., Istomin S.A., Bahvalov S.G., Pastukhov E.A. Structure and properties of molten oxides. Ural Division of RAS, Ekateriburg (1999). (in Russian)Denisov V.M., Belousova N.V., Istomin S.A., Bahvalov S.G., Pastukhov E.A. Structure and properties of molten oxides. Ural Division of RAS, Ekateriburg (1999). (in Russian)Vatolin N.A., Pastukhov E.A. Diffraction studies of high-temperature melts. Nauka, Moscow, 188 p. (1980). (in Russian)Vatolin N.A., Pastukhov E.A. Diffraction studies of high-temperature melts. Nauka, Moscow, 188 p. (1980). (in Russian)Vatolin N.A., Kern E.M., Lisin V.L. X-ray diffraction study of the structure of silicate melts. Structure and physical-chemical properties of metal and oxide melts. Sverdlovsk, P. 38–56 (1986). (in Russian)Vatolin N.A., Kern E.M., Lisin V.L. X-ray diffraction study of the structure of silicate melts. Structure and physical-chemical properties of metal and oxide melts. Sverdlovsk, P. 38–56 (1986). (in Russian)Polukhin V.A. Nanostructure and precursor modeling. Ural Division of RAS, Ekateriburg, 208 p. (1999). (in Russian)Polukhin V.A. Nanostructure and precursor modeling. Ural Division of RAS, Ekateriburg, 208 p. (1999). (in Russian)Tolman R.C. The effect of droplet size on surface tension. J. Chem. Phys., 17(3), P. 333 (1949).Tolman R.C. The effect of droplet size on surface tension. J. Chem. Phys., 17(3), P. 333 (1949).Bykov T.V., Shchekin A.K. Surface tension, Tolman length, and effective rigidity constant in the surface layer of a drop with a large radius of curvature. Inorg. Mater., 35(6), P. 641–644 (1999).Bykov T.V., Shchekin A.K. Surface tension, Tolman length, and effective rigidity constant in the surface layer of a drop with a large radius of curvature. Inorg. Mater., 35(6), P. 641–644 (1999).Magomedov M. N. Dependence of the surface energy on the size and shape of a nanocrystal. Phys. Solid State, 46(5), P. 954–968 (2004).Magomedov M. N. Dependence of the surface energy on the size and shape of a nanocrystal. Phys. Solid State, 46(5), P. 954–968 (2004).Rekhviashvili S.Sh., Kishtikova E.V. On the size dependence of the surface tension. Techn. Phys., 56(1), P. 143–146 (2011).Rekhviashvili S.Sh., Kishtikova E.V. On the size dependence of the surface tension. Techn. Phys., 56(1), P. 143–146 (2011).Sharikov F.Yu., Almjasheva O.V., Gusarov V.V. Thermal analysis of formation of ZrO2 nanoparticles under hydrothermal conditions. Russ. J. Inorg. Chem., 51(10), P. 1538–1542 (2006).Sharikov F.Yu., Almjasheva O.V., Gusarov V.V. Thermal analysis of formation of ZrO2 nanoparticles under hydrothermal conditions. Russ. J. Inorg. Chem., 51(10), P. 1538–1542 (2006).Ostwald W. Ober die vermeintliche Isomerie des roten und gelben Quecksilberoxyds und die Oberflchenspannung fester. Z. Phys. Chem. Stoechiom. Verwandtschaftsl, 34, P. 495–503 (1900).Ostwald W. Ober die vermeintliche Isomerie des roten und gelben Quecksilberoxyds und die Oberflchenspannung fester. Z. Phys. Chem. Stoechiom. Verwandtschaftsl, 34, P. 495–503 (1900).Penn R.L., Banfield J.F. Morphology development and crystal growth in nanocrystalline aggregates under hydrothermal conditions: insights from titania. Geochimica et Cosmochimica Acta, 63(10), P. 1549– 1557 (1999).Penn R.L., Banfield J.F. Morphology development and crystal growth in nanocrystalline aggregates under hydrothermal conditions: insights from titania. Geochimica et Cosmochimica Acta, 63(10), P. 1549– 1557 (1999).Banfield J.F., Veblen D.R. Conversion of perovskite to anatase and TiO2 (B): A TEM study and the use of fundamental building blocks for understanding relationships among the TiO2 minerals. Am. Mineralogist, 77, P. 545–557 (1992).Banfield J.F., Veblen D.R. Conversion of perovskite to anatase and TiO2 (B): A TEM study and the use of fundamental building blocks for understanding relationships among the TiO2 minerals. Am. Mineralogist, 77, P. 545–557 (1992).Penn R.L., Banfield J.F. Oriented attachment and growth, twinning, polytypism, and formation of metastable phases: Insights from nanocrystalline TiO2. Am. Mineralogist, 83, P. 1077–1082 (1998).Penn R.L., Banfield J.F. Oriented attachment and growth, twinning, polytypism, and formation of metastable phases: Insights from nanocrystalline TiO2. Am. Mineralogist, 83, P. 1077–1082 (1998).Penn R.L., Banfield J.F. Imperfect Oriented Attachment: Dislocation Generation in Defect-Free Nanocrystals. Science, 281(5379), P. 969–971 (1998).Penn R.L., Banfield J.F. Imperfect Oriented Attachment: Dislocation Generation in Defect-Free Nanocrystals. Science, 281(5379), P. 969–971 (1998).Mitsuhashi T., Ichihara M., Tatsuke U. Characterization and Stabilization of Metastable Tetragonal ZrO2. J. Am. Ceram. Soc., 57(2), P. 97–101 (1974).Mitsuhashi T., Ichihara M., Tatsuke U. Characterization and Stabilization of Metastable Tetragonal ZrO2. J. Am. Ceram. Soc., 57(2), P. 97–101 (1974).Chaim R., Heuer A.H., Brandon D.G. Phase equilibration in ZrO2-Y2O3 alloys by liquid film migration. J. Am. Ceram. Soc., 69(3), P. 243–248 (1986).Chaim R., Heuer A.H., Brandon D.G. Phase equilibration in ZrO2-Y2O3 alloys by liquid film migration. J. Am. Ceram. Soc., 69(3), P. 243–248 (1986).

The authors declare that there are no conflicts of interest present.