## 1 Introduction  
Computer simulations are used to describe various phenomena that occur during the crystallisation process. The development of knowledge and technology, which has been progressing for several decades, allows for the continuous improvement of the tools used, especially at the level of numerical calculations. More and more modern computational methods are used, which yield more satisfactory results and, more importantly, are very often consistent with the actual behaviour of the simulated materials [1]. In recent years, numerical models have been used to study nucleation and grain growth processes during the crystallisation of metallic glasses [2]. [PERSON] [3] proposed a division into macroscopic models (first generation) and micro-/macro-models (second generation). First-generation models are based on the relationship between the proportion of the locally solidified phase and temperature. Second-generation models are characterised by the inclusion of the laws of nucleation and grain growth in the calculations, which is typical for modelling at the micro-level [4]. The first work related to modelling the nucleation and growth of grains in alloys with an amorphous structure (called metallic glasses) was carried out by [PERSON] [5]. He derived equations that combine classical theories of nucleation and growth and those taken from the Kolmogorov-Johnson-Mehl-Avrami theory. This allowed for the determination of time-temperature transition diagrams and the prediction of critical cooling rates for various metallic glasses. [PERSON] [6] modelled crystallisation in the Ni-Si-B system. In this case, the author combined the thermodynamic model by [PERSON] and [PERSON] [7] with the previously mentioned [PERSON] model. [PERSON] [8], in turn, used the Gibbs free energy for his model. He calculated the crystallisation kinetics for the Fe\({}_{40}\)Ni\({}_{40}\)P\({}_{14}\)B\({}_{6}\) metallic glass by measuring the difference in specific heat between the liquid and crystalline phases. [PERSON] et al. [9] used the [PERSON] and CALPHAD kinetics databases, which allowed for the prediction of the possibility of obtaining metallic glasses of Cu-Zr and Cu-Zr-Ti. All the works cited above, as well as others [10; 11; 12; 13], assume polymorphic conditions, which means an equal chemical composition of the particle and matrix phases. Crystallisation then depends on the rate of attachment of atoms at the phase boundary, while the coupling related to grain growth and transport through diffusion to the grain-matrix phase boundary is omitted. [PERSON] et al. [14] and [PERSON] et al. [15] showed that polymorphic crystallisation is quite rare in metallic glasses. A model that fully shows the kinetics of the crystallisation of metallic glasses should describe both the nucleation of crystals and the growth of grains controlled by the diffusion and dissolution of particles. Such a model should show the phase transition under both isothermal and non-isothermal conditions. It is possible to design a numerical simulation model that, using appropriate models of the nucleation and growth division in metallic glasses, shows the evolution of the crystal size distribution over time. The modelling of the nucleation, growth, and dissolution of crystals for the amorphous Al-Cu-Zr alloy was shown by [PERSON] et al. [2]. In this model, the authors used two theories: the classical nucleation theory and the diffusion-controlled multicomponent growth developed by [PERSON] [16]. This method was then implemented using the numerical scheme of [PERSON] [17], which was designed by [PERSON] and [PERSON] [18]. In a recent publication, it was noted that the concentration gradient causes the movement of atoms, which are excited by high temperatures, causing a largely diffusive phase transformation [19; 20]. Due to the increase in temperature, secretory reactions occur. If they are formed from a supersaturated solid solution, three important processes occur [17; 21; 22]: nucleation, growth, and thickening.