# Development of a Partial Clustering Model of Alloy Viscosity
## 3 Results  
Verification of the proposed partial clustering model will be shown to calculate the dynamic viscosity of alloys on the experimental data of \(Cu\)-\(Sn\) alloy described in [31]. The authors of [31] presented the results of an experiment to determine the dynamic viscosity in the vicinity of the liquidus line for the \(Cu\)-\(Sn\) alloy on a high-frequency viscometer. A close correlation of the obtained equations for nine selected alloy samples was provided. It was of interest that the purpose of this experiment in [31] was to check a interpretation of superheating of the alloy above the liquidus line by the contrary principle. The repeated uniform heating of the alloy was set for the selected nine samples over the entire liquidus line every 20 degrees, starting from the line itself. As a result, viscosity polyterms close to the liquidus line were obtained with convergence and leveling at the increasing temperature and corresponding decrease in viscosity. However, the gradient of viscosity increase over the same temperature intervals turned out to be significantly different, although the superheating was equal. Therefore, the influence of such superheating is apparent. It has no relation to the equilibrium state of the thermodynamic system, which is independent of the path and the initial temperature of the alloy. It can be repeatedly heated and cooled and remain as the laws of thermodynamics dictate at a constant given temperature. Table 1 presents the results of these experiments using the Arrhenius equation. The equation includes a pre-exponential factor \(A\) (mPa\(\cdot\)s), activation energy of viscous flow \(E_{v}\) (\(\text{0/mol}\)) and heat of mixing \(\Delta_{mix}H\) (J/mol) [32]. Also, our calculations of viscosity polyterms using the Arrhenius equation for each intermediate numbered sample are presented. The liquidus temperature \(\text{T}_{\text{liq}}\), K obtained from our measurements on the state diagram of the \(Cu\)-\(Sn\) alloy in [31] is pointed out. It was found that, from the seemingly disparate data [31], a regular dependence of the activation energy of viscous flow on the alloy composition emerged. Remarkably, the same regularity was observed in the relationship between the total heat of chaos and the same alloy compositions of _Cu-Sn_. Moreover, it was found that the values of these regularities were strikingly similar in absolute magnitude. Based on this, Figure 1 is constructed. Figure 1 shows the dependence of the activation energy of the viscous flow on the composition of the _Cu-Sn_ alloy. In general, its shape repeats the shape of the viscosity change curve at the liquidus temperature, but it is much more closely correlated with the total barrier of randomization in the model (13). It should also be noted that when \(T=T_{m}=T_{liq}\) and \(RT_{liq}=RT_{m}\). The results of calculating the data using model (13) are shown in Figure 2 and Table 2. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{\(N\)} & \multicolumn{2}{c}{_Cu-Sn Alloy,_} & \multirow{2}{*}{\(A\), mPa-s} & \multirow{2}{*}{\(E_{\
u}\), J/mole} & \multirow{2}{*}{\(T_{liq}\), K} & \(\Delta_{mix}H_{\
u}\) & \(\Delta_{dt}H\) = \\  & & \(w\), \% & & & & \(\mathbf{J/mole}\) & \(RT_{liq}+\Delta_{mix}H\) \\ \hline
1 & _Cu_ & 1.769 & 10,833.7 & 1356 & 0 & 11,274 \\ \hline
2 & _Cu-_10\% _Sn_ & 1.219 & 14,116.8 & 1273 & 2300 & 12,884 \\ \hline
3 & _Cu-_25\% _Sn_ & 0.874 & 17,080.8 & 1104 & 3900 & 13,079 \\ \hline
4 & _Cu-_30.6\% _Sn_ & 1.187 & 13,373.8 & 1019 & 3950 & 12,422 \\ \hline
5 & _Cu-_38\% _Sn_ & 0.988 & 14,151.8 & 1004 & 3200 & 11,548 \\ \hline
6 & _Cu-_58.6\% _Sn_ & 1.436 & 6830.2 & 1004 & 900 & 9248 \\ \hline
7 & _Cu-_92.4\% _Sn_ & 1.036 & 3953.8 & 684 & 30 & 5717 \\ \hline
8 & _Cu-_99.3\% _Sn_ & 0.979 & 2963.0 & 490 & 10 & 4084 \\ \hline
9 & _Sn_ & 0.841 & 3488.8 & 505 & 0 & 4199 \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of the Arrhenius equation of the _Cu-Sn_ alloy. \(N\)—sample number of the alloy. \(T_{liq}\) according to our measurements from the diagram [31].