# Structural Particularities, Prediction, and Synthesis Methods in High-Entropy Alloys
## 2 Structural Particularities of HEAs, Prediction and Estimation Concepts
### Prediction and Estimation Concepts
#### 2.3.1 Gibbs Law  
For a more efficient prediction of the number of phases in a multicomponent alloy, the phase rule proposed by [PERSON] in the 1870s can be applied [25]. [PERSON] rule relates to the number of phases, \(P\), in a multicomponent system at thermodynamic equilibrium, in connection with the degrees of freedom, \(F\) (the number of variables that can be independently adjusted without disturbing the equilibrium), and the number of components, \(N\). At constant pressure, [PERSON] phase rule is expressed by the equation:  
\[P=N-F+1. \tag{1}\]  
Given that phases defined by composition have (\(N-1\)) degrees of freedom, and phases defined by temperature have 1 degree of freedom, the maximum number of degrees of freedom \(F\) is \(N\). Conversely, the minimum number of degrees of freedom is 0, which corresponds to a point on a phase diagram where the number of phases is at its maximum, (\(N+1\)). Any deviation in composition or temperature reduces the number of equilibrium phases. The minimum number of phases is 1 when the degrees of freedom equal the number of components in the system (i.e., \(F=N\)). Gibbs' phase rule dictates the limits on the number of possible phases and determines the degrees of freedom for a given number of phases and components. However, it does not specify the exact number of phases present in a particular alloy or system at specific temperature and pressure conditions. In a system with \(N\) components, the number of phases can range from a minimum of 1 to a maximum of (\(N+1\)) without contradicting Gibbs' theorem [12]. Some researchers argue that the phase rule does not apply to High-Entropy Alloys (HEAs) [26].The fact that HEAs display fewer phases than the maximum possible is advantageous for validating the entropy effect [23; 27; 28]. Binary phase diagrams offer a clear method for examining the evolution and stability of a system, such as the mixture of alloy components under specific external conditions. In binary systems, the maximum number of phases is achieved when the degrees of freedom are zero, i.e., \(F=0\)[12]. The Hume-Rothery rules represent the oldest guidelines for obtaining solid solution alloys [29]. These rules indicate that solid solution formation is most likely in alloys where the constituent elements have comparable atomic sizes, crystal structures, electronegativities, and valencies. To predict solid solution phase formation in complex alloys, the HEA community has established new relationships involving atomic size difference (\(\delta\)), relative electronegativity (\(\Delta\chi\)), and the average valence electron concentration (\(VEC\)) [30; 31]. Thermodynamic considerations are reflected through the mixing enthalpy \(\Delta H_{am}\) and a parameter \(\Omega\) which incorporates \(\Delta H_{am}\), the mixing entropy \(\Delta S_{am}\), and the melting temperature \(T_{m}\)[32]. The formulas for these parameters are as follows:  
\[\delta=100*\sqrt{\sum c_{i}\Big{(}1-\frac{r_{i}}{\overline{\tau}}\Big{)}^{2}}, \tag{2}\]  
\[\Delta\chi=100*\sqrt{\sum c_{i}\Big{(}1-\frac{\chi_{i}}{\overline{\chi}}\Big{)}^ {2}}, \tag{3}\]  
\[VEC=\sum c_{i}VEC_{I}, \tag{4}\]  
\[\Delta H_{am}=\sum 4c_{i}c_{j}*\Delta H_{ij}, \tag{5}\]  
\[\Omega=\left(\sum c_{i}T_{m,i}\right)\left(\Delta S_{mix}\right)/\left|\Delta H _{am}\right|, \tag{6}\]  
where \(c_{i}\), \(c_{j}\)--the mole fractions for elements i and j;  
\(r_{i}\)--the atomic radius of each element;  
\(\chi_{i}\)--electronegativity of each component element;  
\(VEC_{i}\)--valence electron concentration for each element i;  
\(\Delta H_{ij}\)--binary enthalpy;  
\(T_{m,i}\)--melting point of element i;  
\(\overline{\tau}\)--average atomic radius calculated with the formula:  
\[\overline{\tau}=\sum c_{i}r_{i}; \tag{7}\]  
\(\overline{\chi}\)--average electronegativity calculated with the formula:  
\[\overline{\chi}=\sum c_{i}\chi_{i}; \tag{8}\]  
Studies in the specialized literature [33] have revealed correlations between the phases formed and parameters such as \(\delta\) or \(VEC\) (Figure 2). [PERSON] et al.