# 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
[34] discovered that solid solution formation tends to occur within the range of \(-15\) kJ/mol \(\leq\Delta H_{am}\leq 5\) kJ/mol and \(1\%\leq\delta\leq 6\%\). Conversely, [PERSON] et al. [32] reported a narrower range of \(\Delta H_{am}\) values for HEA formation: \(-5\) kJ/mol \(\leq\Delta H_{am}\leq 5\) kJ/mol. In a subsequent study, [PERSON] and [PERSON] [30] identified the following simultaneous ranges for HEA formation: (i) \(-22\) kJ/mol \(\leq\Delta H_{am}\leq 7\) kJ/mol, (ii) \(0\leq\delta\leq 8.5\), and (iii) \(11\) J/Kmol \(\leq\Delta S_{am}\leq 19.5\) J/Kmol. Positive values of the mixing enthalpy, \(\Delta H_{am}\), promote phase separation, whereas large negative values favor the formation of intermetallic compounds [35]. Electronegativity difference (\(\Delta\chi\)) is also discussed as a main parameter in deciding the structure type in alloys and was determined to be lower than 6% for HEA containing only solid solutions [36; 37]. Based on the same parameters and a set of different alloys, [PERSON] et al. [30] suggested that a high value of \(VEC\) (>8.0) stabilizes the FCC phase, while a low value (<6.87) stabilizes the BCC phase. A value between these limits leads to the formation of a mixture of FCC and BCC phases [35]. Conversely, the valence electron concentration (\(VEC\)) can differentiate phases when dealing with a limited number of alloys within a specific alloy family. For example, in cast alloys such as Al\({}_{x}\)CoCrCuFeNi and Al\({}_{x}\)CoCrFeNi\({}_{2}\) (where \(0\leq x\leq 2\)) [32], \(VEC\) can distinguish between BCC and FCC phases. Similarly, \(VEC\) can predict the formation of the \(\sigma\) phase in heat-treated alloys containing Cr and Fe along with Al, Co, Mn, Ni, Ti, and/or V [38]. However, these correlations become less accurate as more elements are introduced. For instance, the same VEC range can predict both BCC + FCC phase formation in [31] and \(\sigma\) phase formation in another context [38]. The addition of Mn is assumed to make these predictions uncertain [31]. Most empirical approaches for predicting solid solution phases or intermetallic compounds in HEAs use parameters such as \(\delta\) and \(\Delta H_{am}\) or \(\Omega\)[36; 39; 40; 41]. The atomic size difference and \(\Delta H_{am}\) are established empirical criteria for identifying amorphous alloys (AM) [40]. These parameters help distinguish between solid solution phases and amorphous phases in HEAs, although intermetallic compounds can overlap with both categories [34; 39]. In an attempt to distinguish between solid solution domains and intermetallic compounds, the parameter \(\Omega\) combines \(\Delta H_{am}\), \(\Delta S_{am}\), and \(T_{m}\) and was first used by [PERSON] and [PERSON] [39]. By calculating both parameters, \(\delta\) and \(\Omega\), the relationship between them was established (Figure 3), and a new rule for solid solution formation in high-entropy multicomponent alloys with high entropy was proposed. Figure 3: Relationship between \(\delta\) and \(\Omega\) parameters for multi-element alloys [39]. Figure 2: The relationship between \(\delta\) (in this case \(\delta\) is amplified 100 times for simplicity) and \(\Delta H_{mix}\) in some HEA alloys (S: indicates the alloy containing only solid solutions; C: indicates the alloy containing intermetallic compounds) S1–S8; 9: CrFeCoNiAlCu\({}_{0.25}\), 10: VCuFeCoNi, 11: Al\({}_{0.5}\)CrFeCoNi, 12: Ti\({}_{2}\)CrCuFeCoNi, 13: AlTiVYZ, 14: ZrTiVCuNiBe [33]. Thus, when the parameter \(\Omega>1.1\) and the parameter \(\delta<3.6\%\), exclusively solid solutions are formed. On the other hand, when \(\Omega\) and \(\delta\) fall within the ranges \(1.1<\Omega<10\) and \(3.6\%<\delta<6.6\%\), both solid solutions and intermetallic compounds are formed, whereas if \(\Omega>10\), only solid solutions are formed [39].