# Structural Particularities, Prediction, and Synthesis Methods in High-Entropy Alloys
## 2 Structural Particularities of HEAs, Prediction and Estimation Concepts
### Prediction and Estimation Concepts
#### 2.3.2. CALPHAD Method
Typically, most databases are developed for alloy systems centered around a primary element. For instance, a thermodynamic database for the AlCoCrFeNi system was created by extrapolating data from binary and ternary systems across a broad range of compositions. The phase diagrams predicted using this database have been validated by experimental results [53]. Thermodynamic modeling based on the CALPHAD method makes significant contributions to the field of complex concentrated alloys. The reliability of CALPHAD calculations is assessed using the binary phase diagram fraction (fAB), which is defined as the ratio of the number of binary phase diagrams shared between the analyzed alloy and the thermodynamic database to the total number of binary systems for the alloy [55]. When fAB = 1, calculations tend to be the most accurate, whereas those with fAB \(\geq\) 0.6 can still offer reliable predictions regarding the number and types of phases. In studies using CALPHAD for equimolar alloys composed of 3 to 6 elements, the phases most frequently identified include FCC, M5 Si3 silicides, BCC, B2, and Laves (C15). Notably, for fAB = 1, the phases most commonly encountered are FCC, HCP, M5 Si3, B2, and BCC, independent of the specific fAB value. Alloys that meet all four Hume-Rothery rules are always predicted to form solid solutions, though not every alloy predicted by this method adheres to all four rules. For example, alloys that form solid solutions can have a \(\chi\) (electronegativity difference) of up to 30% and a \(VEC\) (valence electron concentration) of up to 55%. Among the Hume-Rothery conditions, the atomic size difference is the most important, ensuring that any alloy predicted to form solid solutions will have a \(\delta\) value of 15% or lower. When fAB = 1, there is a strong agreement between the calculated and observed phases, a pattern that holds consistently across the entire dataset [53; 60; 61]. The widely recognized \"high mixing entropy\" theory suggests that High-Entropy Alloys (HEAs) are likely to form single-phase solid solutions (SS) due to their high configurational entropy. However, numerous studies have demonstrated that many HEAs display multiple phases, and the solid solutions in some single-phase HEAs may become unstable after prolonged annealing. To investigate and better understand the phase formation and stability of HEAs, various methods have been developed and proposed. For example, [PERSON] and [PERSON] [39] suggested that the phase formation in HEAs is governed by a set of thermodynamic parameters, such as atomic size difference, mixing entropy, and mixing enthalpy. [PERSON] and colleagues [62] applied the method of fundamental principles to refractory HEAs to forecast their phase formation behavior across the temperature-composition range. [PERSON] and colleagues [57] applied phase diagram calculation methods (CALPHAD) to large quantities of HEAs, studying their phase formation behaviors and also developing several specialized commercial thermodynamic databases for HEAs. Other methods, such as machine learning, have also been applied to predict the phase formation behavior of HEAs [63]. In HEAs, the primary single-phase solid solution (SS) structures are BCC, FCC, or HCP. However, phases can also co-exist, such as BCC + HCP, BCC + FCC, FCC + HCP, and even combinations like BCC + FCC + HCP. Alongside these, intermetallic compound (IM) phases and amorphous phases may also form. The presence of intermetallic compounds in HEAs is generally undesirable because they often reduce ductility, negatively impacting mechanical performance and complicating processing. For this reason, achieving solid solution structures in HEAs is typically preferred [64]. Examples of HEA compositions with structures simulated by the CALPHAD method are presented in Table 2.  
The VEC rule was recently revised for the AlCoCrFeNi alloy system, setting the BCC phase within a VEC range of 5.7 to 7.2, and the FCC phase above a VEC of 8.4.