# Development of a Partial Clustering Model of Alloy Viscosity
## 2 Materials and Methods
### Viscosity of Alloy and Its Partial Clustering Model
Therefore, the calculation of viscosity of the alloy can be subordinated to a proportion. If the fraction of clusters at the alloy temperature T is greater or less than the fraction of clusters for the pure components at the same temperature, then the viscosity of the alloy of this composition will be correspondingly greater or less than the known viscosity for the pure components. And this applies in all cases as \(P_{cl}=P_{cm}^{2}\) (12). The main idea in constructing a new model of viscosity is to look for its connection with clusters, not with temperature as a determining argument, not with the mechanics of motion and rest, not with the kinetics of viscous flow! The connection with clusters in the form of a certain material substrate is subject to the rigorous energy balance of its invisible existence. The quantitative model of viscosity of the \(A\)-\(B\) alloy is presented by a combination of the fractional content of clusters in the alloy and pure components of the alloy including the subordination of clusters to their probability distribution (10) as part of the concept of chaotic particles (3-6) based on the Boltzmann distribution (energy spectrum) (1):  
\[ \begin{array}{l}\eta_{T}=X_{A}\dot{\eta}_{T,A}\left[\frac{1-exp(-\Delta_{ch}H/ (RT))}{1-exp(-T_{m,A}/T)}\right]^{2}\\ +X_{B}\dot{\eta}_{T,B}\left[\frac{1-exp(-\Delta_{ch}H/(RT))}{1-exp(-T_{m,B}/T)} \right]^{2}\end{array}  \tag{13}\]  
where \(X_{A}\) and \(X_{B}\) are the partial shares of substances \(A\) and \(B\); \(\dot{\eta}_{T,A}\) and \(\dot{\eta}_{T,B}\) are the viscosity values of pure substances \(A\) and \(B\), mPa-s, at temperature \(T\geq T_{liq}\), K; \(\Delta_{ch}H=RT_{liq}\) + \(\Delta_{\rm mix}H+\Delta_{H}H+\ldots\) is a total thermal barrier to destruction (chaotization) of clusters when the alloy is heated in opposition to its increasing thermal content \(RT\), J\(\cdot\)mol\({}^{-1}\). In this model, alongside temperature, the heat of chaos emerges as a crucial factor, which includes the temperature and heat of the liquidus, heat of mixing, and other chaos barriers. The sum of these components is denoted as the generalized heat of chaos across the entire composition range of the alloy (0-100%, w, Sn). Moreover, a correlation was identified between the activation energy of viscous flow and the heat of chaos of the melt during heating. To use the model (13), it is necessary to know the liquidus temperature in the full range of the A-B alloy state diagram, preferably at the melting points of the formed compounds;to have experimental or approximated (according to Arrhenius or the cluster-associate model) temperature dependences of the viscosity of the pure components; to use the values of the heat of mixing \(\Delta_{\text{mix}}H\) and the formation \(\Delta_{\text{f}}H\) of \(A_{x}B_{y}\) intermetallic compounds as a function of temperature to account for the heat of chaotization \(\Delta_{\text{c}}H\). Literature analysis has shown the need to take into account various factors affecting the viscosity of melts [26; 27; 28; 29; 30; 31]. So, the authors in [26; 27; 28] do not take into account the presence of associates in the melt. Or the proposed models are developed only for a specific alloy system without possibility of further prediction [29]. The authors in [30] proposed a model to estimate the viscosity related to associates. But data on the thermodynamic characteristics of the alloys, the molar volume of the alloys and the activation energy of the associates are required to use this model. Our study aims to fill this gap. This paper aims to fill this gap in the theory of the liquid state and to develop a new model based on a fundamentally new approach including the energy component of matter. Previously, testing of the partial clustering model of viscosity on known data on viscosity of \(Cu\)-\(Al\) and \(Cu\)-\(Sn\) alloys showed the reliability of this model and its high demands on the volume and accuracy of the initial thermal and physical parameters, but the possibility of testing significantly reduces.