About Kolmogorov's statistical theory of phase transformation

Many mathematical models of solidification use the following equation for the estimation of solid volume fS=1−exp(−n1τn2), where τ is time, n1 and n2 are constants. The fundamentals of the statistical theory of metal solidification were developed in classic papers written by Kolmogorov [A.N. Kolmogorov, Bull. Acad. Sci. USSR. Ser. Math., 3 (1937) 355-359 (in Russian)], Johnson and Mehl [W.A. Johnson, R.F. Mehl, Trans. Metall. Soc. AIME 135 (1939) 416-442], Avrami [M. Avrami, J. Chem. Phys. 7 (1939) 1103-1112; M. Avrami, J. Chem. Phys. 8 (1940) 212-224; M. Avrami, J. Chem. Phys. 9 (1941) 177-184]. The above equation in contemporary literature is named after these authors (K-J-M-A), but this equation is only a particular solution of Kolmogorov's general solution fS = 1−exp(−Ω), where the function Ω is the so-called total extended volume (per unit volume) of the growing grains, if their overlapping is neglected. Kolmogorov's general solution for the Ω-function is an accurate development only in the case of geometric similarity of the growing grains, their uniform distribution and equal orientation in space. Moreover, for an arbitrary moment and direction all grains must have the same absolute value of the growth velocity vector. If the conditions stated above are neglected, the above equations can give overestimated results. The source of this overestimation is the, so-called, screening effect. The presented solution takes into account the screening effect for the calculation of Ω-function. This solution expands the scope of the Kolmogorov's statistical theory of transformation. The proposed solution takes into consideration the growth of the grains of any shape, assuming their uniform distribution in space.