In this paper, thermodynamical approach has been used to simulate the influence of shape on phase equilibria in the two-phase-region between liquidus and solidus temperatures in case of Si-Ge alloy nanoparticles. Volumes and shapes of considered nanoparticles have been described by their effective radii and fractal dimensions, the dependence of fractal dimensions on temperature has been obtained using a simple geometrical model. It has been shown that decreasing the volume of a nanoparticle and its fractal dimension (which corresponds to nanoparticles of a more complicated shape) leads to narrowing down the temperature range of the heterogeneous region and changes the phase transition temperatures and equilibrium compositions of co-existing phases. At different temperatures, the dependences of the composition of the liquid phase differ which is explained by implementing different mechanisms of reducing the surface energy.

In this paper, a dispersed system formed by an ensemble of particles of different volume has been modeled in the framework of a thermodynamical approach. Particle shape has been determined by its fractal dimension which correlates its volume and surface area. Using the methods of number theory and Hardy-Ramanujan-Rademacher formula, we have calculated the equilibrium size distributions for nanoparticles of different shape in an ensemble. Estimates of the average volume and fractal dimension of dispersed particles have been obtained based on distribution functions. The correlation between average geometrical characteristics of particles in the ensemble, thermodynamical conditions of the dispersed system and properties of its substance have also been revealed.