Systematic investigation of phase stability of the magnetic fcc Fe-Cr-Mn-Ni system—promising candidate structural materials to replace conventional austenitic steels—has been performed using a combination of spin-polarized density-functional theory, cluster expansion, and Monte Carlo simulations. The developed model was able to reproduce all known ground states (GSs) in the studied system and to predict new ones with strongly negative formation enthalpy—ternary ${\mathrm{CrMnNi}}_2$ and quaternary ${\mathrm{FeCr}}_2{\mathrm{MnNi}}_4$. Investigation of phase stability was done at 0 K and finite temperatures in the whole concentration range and allowed us to observe the important role of Ni and Mn. Ni is the only element in the system that increases the order-disorder transition (ODT) temperature, which means that the fcc alloys with decreased concentration of Ni will form solid solutions at lower temperatures. Analysis of the effect of the addition of Mn to Fe-Cr-Ni alloy confirms a general trend of statistical correlation between the averaged magnitude of magnetic moments and volume per atom found from the predicted stable structures in the quaternary system and underlying subsystems. This linear magneto-volume relationship trend is, however, weaker in Fe-Cr-Mn-Ni alloys in comparison with those in the Fe-Cr-Ni system. Furthermore, Ni and Mn form the most stable GS—$\text{L}{1}_0$-MnNi, which has one of the strongest tendencies to segregate in fcc Fe-Cr-Mn-Ni alloys evidenced by the strength of Mn-Ni short-range ordering (SRO). Mn-Ni SRO significantly increases ODT temperature in the vicinity of $\text{L}{1}_0$-MnNi and to the equiatomic region. The ODT of ${\mathrm{Cr}}_{18}{\mathrm{Fe}}_{27}{\mathrm{Mn}}_{27}{\mathrm{Ni}}_{28}$ alloy is found to be $1290\pm 150$ K, which supports the experimental observation of the disordered solid solution structure in ${\mathrm{Cr}}_{18}{\mathrm{Fe}}_{27}{\mathrm{Mn}}_{27}{\mathrm{Ni}}_{28}$ alloy at higher temperatures.

Configuration entropy is believed to stabilize disordered solid solution phases in multicomponent systems at elevated temperatures over intermetallic compounds by lowering the Gibbs free energy. Traditionally, the increment of configuration entropy with temperature was computed by time-consuming thermodynamic integration methods. In this work, a new formalism based on a hybrid combination of the Cluster Expansion (CE) Hamiltonian and Monte Carlo simulations is developed to predict the configuration entropy as a function of temperature from multi-body cluster probability in a multi-component system with arbitrary average composition. The multi-body probabilities are worked out by explicit inversion and direct product of a matrix formulation within orthonomal sets of point functions in the clusters obtained from symmetry independent correlation functions. The matrix quantities are determined from semi canonical Monte Carlo simulations with Effective Cluster Interactions (ECIs) derived from Density Functional Theory (DFT) calculations. The formalism is applied to analyze the 4-body cluster probabilities for the quaternary system Cr-Fe-Mn-Ni as a function of temperature and alloy concentration. It is shown that, for two specific compositions (Cr${}_{25}$Fe${}_{25}$Mn${}_{25}$Ni${}_{25}$ and Cr${}_{18}$Fe${}_{27}$Mn${}_{27}$Ni${}_{28}$), the high value of probabilities for Cr-Fe-Fe-Fe and Mn-Mn-Ni-Ni are strongly correlated with the presence of the ordered phases L1${}_2$-CrFe${}_3$ and L1${}_0$-MnNi, respectively. These results are in an excellent agreement with predictions of these ground state structures by ab initio calculations. The general formalism is used to investigate the configuration entropy as a function of temperature and for 285 different alloy compositions. It is found that our matrix formulation of cluster probabilities provides an efficient tool to compute configuration entropy in multi-component alloys in a comparison with the result obtained by the thermodynamic integration method. At high temperatures, it is shown that many-body cluster correlations still play an important role in understanding the configuration entropy before reaching the solid solution limit of high-entroy alloys (HEAs).