Configurational Entropy in Multicomponent Alloys: Matrix Formulation from Ab Initio Based Hamiltonian and Application to the FCC Cr-Fe-Mn-Ni System

Abstract

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 25Fe 25Mn 25Ni 25 and Cr 18Fe 27Mn 27Ni 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).

Chemical short-range order in derivative Cr–Ta–Ti–V–W high entropy alloys from the first-principles thermodynamic study

“…The y ij n values can be obtained by the matrix inversion from eqn (5) following ref. 37. The SRO parameters can be calculated from the point and pair correlation functions.…”

“…30 The focus on configurational entropy frequently ignored essential contributions from the enthalpy of mixing to the phase stability. 37,38 However, the configurational entropy can change with temperature due to small changes in short-range atomic ordering or by chemical partitioning between different alloy elements. 17,[39][40][41][42][43][44][45][46][47][48] The fundamental hypothesis also assumes that the maximum configurational entropy is achieved at high temperature or in the liquid state.…”

“…We employ a combination of methods that allows investigating the phase stability at finite temperature for different compositions, namely density-functional theory (DFT), cluster expansion (CE) and canonical Monte-Carlo (MC) simulations. 37,[41][42][43]45,68 In a variance with other conventional methods such as special quasirandom structure (SQS) approaches or coherent potential approximation (CPA), which are more relevant for disordered substitutional alloys, our hybrid combination of ab-initio based CE Hamiltonian with MC simulations allows to investigate the dependence of configurational entropy in multi-component alloys as a function of temperature and composition and to integrate it into the free energy calculations by properly considering the contribution from the enthalpy of mixing. More importantly, by using statistical mechanics simulations, this approach, which uses many-body interactions, is able to clarify the important role of chemical short-range effect on the transformation of stable and ordered phases at low temperature into a fully disordered configuration at high temperature and therefore allows to predict the order-disorder transition temperature (ODTT) for different alloy compositions.…”

A Review of Multi-Scale Computational Modeling Tools for Predicting Structures and Properties of Multi-Principal Element Alloys

“…For instance, coherent potential approximation (CPA) to study the chemical and magnetic disorders, special quasi-random structure (SQS) as one of the most successful models known for binary and ternary alloys, and coarse-grain cluster expansion (CE) to investigate short-range order effects in Monte-Carlo simulations [55]. In two recent studies, Fernandez-Caballero et al [84,85] investigated the configurational entropy as a functional temperature, and calculated the multi-body ordering probabilities and short range ordering (SRO) for Cr-Fe-Mn-Ni and MNbTaVW MPE systems, by developing a hybrid combinations of effective cluster interactions (ECIs) derived from DFT and Semi-Canonical Monte Carlo Simulations.…”

New Advances in High-Entropy Alloys

“…Traditionally, configuration entropy was computed by the empirical formula. In Fernández-Caballero’s work [ 4 ], a new formalism based on a hybrid combination of the Cluster Expansion 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. Experimentally, Haas et al [ 5 ] used differential scanning calorimetry to determine the thermal entropy and compared it to the configurational entropy.…”