G. M. Voldman, Independent Expert, e-mail: gvoldman@bk.ru, Moscow, Russia

The current practice is to use the Young – Laplace, Kelvin (Thomson) and Ostwald – Freundlich equations to describe the relationship between the properties of disperse particles (surface pressure, equilibrium steam pressure, concentration of saturated solution) defined by the surface energy and the particle size. These equations are based on the solutions of the derivative dS/dV calculated by Young and Laplace for a curved liquid surface: dS/dV = (1/r₁ + 1/r₂), where r₁ and r₂ — principal radii of curvature, and for a spherical drop (r₁ = r₂ = r): dS/dV = 2/r. These solutions are not applicable to flat surfaces (r₁ = r₂ = ∞) and, correspondingly, to solid particles confined by such flat surfaces. Besides, one cannot allow for the shape of particles when using the equivalent sphere radius as a parameter of these equations. That’s why the author proposed a general approach to describing the properties of disperse particles defined by the surface energy. In contrast to the Young – Laplace, Kelvin (Thomson) and Ostwald-Freundlich equations, the proposed method can be used to analyse the properties of solid particles of various shapes and structures. The method uses the same initial expressions as those that are used for deriving the above mentioned equations. These expressions describe the conditions of the corresponding interphase equilibria. However, the solutions found for the derivatives dE_∞/dV = σ(dS/dV), which are a part of these expressions, depend on the nature and shape of particular particles. This method allows to come up with a more detailed description of the processes that take place in disperse solid-phase systems. Thus, it takes into account the changing size of the particles but also their changing shape and the possibility of mass transfer from course particles to smaller ones at the initial stage.

1. Novikov I. I. Heat treatment of metals: Theory. Moscow : Metallurgiya, 1986. 480 p.
2. Utkin N. I. Production of non-ferrous metals. Moscow : Intermet Inzhiniring, 2000. 442 p.
3. Levinsky Yu. V., Lebedev M. P. Sintering of metallic powders: Theoretical basics. Moscow : Nauchny Mir, 2014. 372 p.
4. Adamson A. Physical chemistry of surfaces. Moscow : Mir, 1979. 568 p.
5. Frolov Yu. G. A course in colloid chemistry. Surface phenomena and disperse systems. Moscow : Khimiya, 1988. 464 p.
6. Landau L. D., Lifshits E. M. Theoretical physics. Vol. 2. Field theory. Moscow : Nauka, 1988. pp. 381–440.
7. Landau L. D., Lifshits E. M. Theoretical physics. Vol. V. Statistical physics. P. 1. Moscow : Nauka, 1976. pp. 561–583.
8. Landau L. D., Lifshits E. M. Theoretical physics. Vol. VI. Hydrodynamics. Moscow : Nauka, 1986. Ch. VII. pp. 333–349.
9. Bazhal I. G., Kurilenko O. D. Ostwald ripening in disperse systems. Kiev : Naukova dumka, 1975. 216 p.
10. Lifshits E. M., Slezov V. V. On the kinetics of diffusional decay of supersaturated solid solutions. Journal of Experimental and Theoretical Physics. 1958. V. 35, Iss. 2 (8). pp. 479–492.
11. Bokshtein B. S. Diffusion in metals. Moscow : Metallurgiya, 1978. 248 p.
12. Geguzin Ya. E. Physics of sintering. Moscow : Nauka, 1984. 312 p.
13. Shaskolskaya M. P. Crystallography. Moscow : Vysshaya Shkola, 1984. 376 p.
14. Voldman G. M., Zelikman A. N. Theory of hydrometallurgical processes. Moscow : Metallurgiya, 1993. 400 p.