The mathematical model of the heterogeneous nucleation of a solid phase on a wetted spherical nanosubstrates is developed on the basis of the thermodynamic approach. According to the proposed model, new expressions for the nucleation energy and the nucleation rate of crystallization centers on spherical ultrafine seeds are obtained. The influence of size and capillary effects are taken into account. Numerical experiments were performed for an aluminum melt modified with silicon carbide nanoparticles. The results of calculations show that the wettability of the nanoparticle surface, as well as its size, has the greatest influence on the process of heterogeneous nucleation. Nanoseeds dispersion increase rises the nucleation rate of the solid, the wetting angle increase leads to an increase in the energy barrier, which impairs the nucleation process on large substrates to a bigger degree. The influence of size effects on heterogeneous nucleation on well-wetted spherical nanosubstrates (θ < 50) can be neglected in the case of only large seeds Rp > 50 nm. This study is of interest for the development of a mathematical model of heterogeneous crystallization, describing the processes of structure formation in metals and alloys, modified by refractory nanoparticles.

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