E. V. Makoveeva, I. E. Koroznikova, A. E. Glebova, A. A. Ivanov, M. A. Nikishina, L. V. Toropova, D. V. Alexandrov
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Abstract—A theory is developed to describe the crystal nucleation and growth at the initial and intermediate stages of bulk solidification of melts and solutions in metastable liquids. An integro-differential model of phase transformation is formulated; it consists of a first-order kinetic equation for a crystal size distribution function, a balance equation for liquid supercooling (supersaturation), and boundary and initial conditions. The mathematical model of the process takes into account the effects of nonstationary growth of each individual crystallite (it takes into account the nonstationary temperature field (impurity concentration) around an evolving spherical particle). The mathematical model is formulated for arbitrary crystal nucleation kinetics (for calculations, the Meirs and Weber–Volmer–Frenkel–Zel’dovich kinetics are considered). A complete analytical solution to the integro-differential model of bulk solidification is obtained using the saddle point method for calculating a Laplace-type integral. In a parametric form, we found a particle radius distribution function, liquid supercooling/supersaturation, time, the total number of particles in a liquid, and their average size (modified time is a parameter). A fundamental solution and three correction coefficients for it are determined using the saddle point method. The analytical solution is shown to converge quickly: for calculations, it is sufficient to use the first three contributions to it. The supercooling of the melt (supersaturation of solution) decreases with time due to the latent heat of phase transformation released by growing crystals. During this process, the particle radius distribution function is limited to the maximum crystal size and eventually shifts toward larger crystal sizes as a result of the nucleation of new particles and the growth of existing particles. The developed theory determines the initial state of melts and solutions at the final stage of phase transformation.
期刊介绍：
Russian Metallurgy (Metally) publishes results of original experimental and theoretical research in the form of reviews and regular articles devoted to topical problems of metallurgy, physical metallurgy, and treatment of ferrous, nonferrous, rare, and other metals and alloys, intermetallic compounds, and metallic composite materials. The journal focuses on physicochemical properties of metallurgical materials (ores, slags, matters, and melts of metals and alloys); physicochemical processes (thermodynamics and kinetics of pyrometallurgical, hydrometallurgical, electrochemical, and other processes); theoretical metallurgy; metal forming; thermoplastic and thermochemical treatment; computation and experimental determination of phase diagrams and thermokinetic diagrams; mechanisms and kinetics of phase transitions in metallic materials; relations between the chemical composition, phase and structural states of materials and their physicochemical and service properties; interaction between metallic materials and external media; and effects of radiation on these materials.
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