# Development of a Partial Clustering Model of Alloy Viscosity
## 2 Materials and Methods
### Viscosity of Alloy and Its Partial Clustering Model  
The previously proposed and verified partial clustering model of viscosity of alloys [18] is based on the Boltzmann distribution (energy spectrum) and the concept of chaotic particles. It only includes the kinetic energy of the thermal motion in a thermodynamic system at the rest state. In this case, the viscosity is determined by a material substrate, clusters, as the lowest-energy unstable aggregations occurring during the melting from a solid phase. They are present in the liquid as random aggregations that transform into a crystal lattice on cooling. It ensures a harmonious combination of the solid and liquid states in subordination to the system-wide criteria of stability. The temperature dependence of the viscosity of a liquid alloy of pure components of A and B is called a partial clustering model because it obeys [PERSON]'s law. Thus, it is an equilibrium characteristic and depends on a molar fraction of components. A similar function is the liquidus line which separates the solid and liquid states of the alloy at the liquidus temperature \(T_{liq}\). This line, being an equilibrium line on state diagrams, displays the equilibrium deviation from the ideal linear distribution. This deviation is expressed as a line of complex configuration, containing extreme values and kinks. Such line is observed as the mole fraction of any of the components continuously changes. The analytical form of this dependence is unknown. However, certain regularities have been stated for viscosity isotherms [24] constructed on the same fractional distribution of components of A and B; namely, as the melt temperature increases over the same interval, the viscosity decreases, the isotherms smooth out, and they converge and become less similar to the liquidus line. Our research [18] has described this before using the example of a model dependence of viscosity for pure substances of A and B for discussion. The transition was realized from mapping models to experimental conditions. The fundamental Boltzmann distribution was used by us as the basis of responsibility for the deviation of viscosity from the ideal viscosity according to [PERSON]'s law to express the correlation of these deviations to the cluster content and to represent the quantitative form of viscosity \(\eta_{T}=f(T_{liq})\). The partial clustering model is consistent with the concept of viscosity as internal friction of liquid layers, in which clusters of different sizes create additional roughness of the friction surface. This model does not contradict the \"hole\" theory of viscosity due to the exchange presence and participation of vapor-mobile particles that transmit their impulses to formation of holes. The best similarity was found with the quasi-polycrystalline fluid theory [25], in which two forms of exchange appear as aggregations and single particles. The connection of the partial clustering model of viscosity with the state diagram of alloys is based on representation of the liquidus line as a thermal barrier \(RT_{liq}\) similar to the thermal barrier \(RT_{m}\) during the melting of pure substances. Other thermal barriers associated with the formation and destruction of unstable intermetallic \(A_{x}B_{y}\) aggregations are also taken into account. These aggregations are analogous to clusters for pure substances. An important role is played by consequences of mixing the alloy components, expressed as the total thermal barrier of \(RT_{ch}\) chaotization. This barrier is opposed to the thermal barrier \(RT\) when heating the melt to a temperature \(T\). The fraction of the intermetallic clusters is expressed through the proportion of crystalline particles, also the same for pure components, based on the same Boltzmann distribution, chaotic particles and cluster size distribution. The combined effect of clusters on viscosity of the alloy is determined by Raoul's law in the form of partial contributions from the pure components.