#  Order–Disorder Competitive Cooperation in Equiatomic 3d–Transition–Metal
Quaternary Alloys: Phase Stability and Electronic Structure

Hiroshi Mizuseki  Korea Institute of Science and Technology (KIST), Seoul
02792, Republic of Korea,  Ryoji Sahara  National Institute for Materials
Science (NIMS), Tsukuba 305-0047, Japan,  Kenta Hongo∗ Research Center for
Advanced Computing Infrastructure, JAIST, Asahidai 1-1, Nomi, Ishikawa
923-1292, Japan

###### Abstract

We use high-throughput first-principles sampling to investigate competitive
factors that determine the crystal structure of high-entropy alloys (HEAs) and
the energetics dependence of the stable phase on the atomic configuration of
fully ordered L12, D022, and random solid solution (RSS) phases of equiatomic
quaternary alloys comprising four of the six constituent elements (Cr, Mn, Fe,
Co, Ni, and Cu). Considering the configurational entropy, we demonstrate that
valence electron concentration (VEC) and temperature are crucial to determine
the phase stability of HEAs at finite temperatures, wherein the ordered phases
are energetically more favorable than RSS phases. Some D022 phases with high
VEC are energetically more stable than L12 phases, though both phases are
metastable. Further, we explore magnetic configurations to identify the origin
of the enthalpy term. The calculations reveal that ordered phases comprising
antiferromagnetic atoms surrounded by ferromagnetic atoms are energetically
stable. The quantitative structure–property relationship is also discussed.

##  I Introduction

High-entropy alloys (HEAs) HEAbook ; 2020GEO  have been gaining increasing
attention because of their novel material properties that cater to a wide
range of applications. Although HEAs were originally applied to structural
materials that required both strength and ductility, 2016VAR ; 2017VAR ;
2019WEI ; 2020MAR ; 2021HAN  their recent applications have been extended to
other functional materials such as catalysis, 2021SUN  superconductors,
2021MIZ  radiation resistance materials 2019EL- , and magnetic materials.
2015KOZ ; 2021MA  These extensions of their applications represent a new
direction of materials design based on the fundamental and key concept of
“high-entropy.” Along with this new direction, the comprehensive understanding
of quantitative structure–property relationship (QSPR) is crucial for
exploiting their potential applications. However, its most fundamental
information—microscopic/atomistic crystal structures of HEAs—remains unknown
2021SAV ; this implies that the atomistic characterization of HEA structures
is challenging for both “experiments” and “theories/computations,” in terms of
“spatial resolutions” 2018YAO  and “computational costs for vast amounts of
possible atomic configurations,” 2018LED ; 2021UTI  respectively.

It is considered that HEAs have random solid solutions (RSSs) instead of
ordered phases. Although disordered face-/body-centered cubic (FCC/BCC) phases
(A1/A2) have been reported as single HEA phases, 2020CHA  ordered FCC/BCC
(L12/B2) and hexagonal close-packed (HCP, A3) have also been found. 2014ZHA ;
2017MIR ; 2017ROG  Although there are a few examples of ordered phases,
perfect randomness in some types of HEAs has been excluded in several studies.
For example, short range order (SRO) in HEAs has been reported 2018DIN ;
2019LIb ; 2020ZHAb ; however, the existence of long-range order (LRO) is
rarely discussed and remains a controversial topic. 2012LUC ; 2015NIU  The
most famous ordered phase in HEAs is CrFeCoNi, with the L12 structure, 2015NIU
wherein the antiferromagnetic Cr atoms occupy the ordered lattice sites
surrounded by randomly distributed ferromagnetic Ni, Fe, and Co atoms. This
can be interpreted as a spin-driven stabilization of the atomic configuration,
which indicates that enthalpy enhancement by magnetization cooperates partial
randomness and then overcomes the entropy because of perfect randomness. This
was theoretically predicted 2014MID ; 2015NIU ; 2017FUK  and verified
experimentally only for the “surface” using scanning transmission electron
microscopy (STEM) 2015NIU ; experimental verifications for the “bulk” have not
been feasible because the constituent atoms are adjoining elements in the
periodic table and have similar chemical properties and comparable radii,
which makes them indistinguishable when using spectroscopic methods. 2018YAO
Thus, the question arises if the spin-driven stabilization is unique for
CrFeCoNi or if it generally holds for other quaternary alloys. To address this
issue from a microscopic perspective, instead of experiments, first-principles
approaches are the most relevant for identifying atomistic structures of HEAs
and elucidating their QSPR 2018CHO ; 2019RIC ; 2019YIN ; 2021WIT , though the
previous studies dealt with only single composites.

We systematically focus on 3d-transition-metal-based quaternary alloys with
Cr, Mn, Fe, Co, Ni, and Cu as the constituent elements; a total of
$15~{}(=\binom{6}{4}=:\frac{6!}{4!2!})$ different types of composites are
considered (TABLE 1). These elements are most commonly used magnetic HEA
constituents 2017MIR , and all quaternary alloy combinations that results from
them can be speculated to have the FCC phase 2017DIA . Therefore, they are
expected to be disordered/RSSs as their atomic sizes and the nature of
chemical bonds formed by them are similar. Our first-principles study revealed
that differences in their magnetic properties with respect to CrFeCoNi lead to
different magnetic orderings: our target phases were not restricted to only
RSSs and L12 ordered phases; they extended to the D022 ordered phases as well
(Figure 1), motivated by well-known studies on order–disorder phase
competition in ordered alloys 1983BIE ; 1996CAB ; 2018STO ; 2019HE ; 2019ZHO ;
2019ZHA . The L12 and D022 phases are ordered crystal structures for 25:75
at.% FCC alloys. As prototype systems for ordered phases, their phase
competition has been well investigated for conventional alloys in terms of
valence electron concentration (VEC). 1983BIE ; 1996CAB  For example, the L12
phase of intermetallic compounds is observed for VEC $\geq$ 7.5. However, to
the best of our knowledge, there is no previously reported study on their
counterpart for HEAs; in HEAs, VEC values are known to correlate well with
structure types (FCC/BCC structures prefer higher/lower VEC values). 2011GUO1
; 2011GUO2 ; 2015TIA ; 2017LEO  Thus, we analyze the RSS–L12–D022
thermodynamical competition in the 3d-transition-metal quaternary system in
terms of VEC.

We performed high-throughput first-principles sampling from four types of L12
and D022 phases to understand the RSS–L12–D022 competition and their QSPR, and
from one type of RSS phase for 15 different composites. We found that
composites with lower VEC values tend to have ordered phases over wide
temperature ranges; however, for higher VEC values, RSSs emerge as the
temperature increases. In addition, we discuss the distribution of the
magnetic moment at the first nearest neighbor site and the constituent
elements in our HEAs. We believe that phase stability is realized by
order–disorder “competitive cooperation” rather than by the “commonly
conceived” order–disorder competition: the former is caused by the cooperation
between a “magnetically ordered atomic configuration composed of one element
gaining enthalpy” and “fully disordered/random configuration composed of the
other three elements gaining entropy,” which we propose in this study; the
latter is the realization of either “fully ordered configuration” or “fully
random configuration.” Their difference is not sufficiently appreciated in the
HEA community.

The paper is organized as follows: Computational details in the present study
are given in “METHODS”. In “RESULTS”, we start with our main finding
“Order–Disorder Competitive Cooperation”, followed by “Energetics” and
“Magnetic ordering”, which can be explained by “Quantitative
Structure–Property Relationship”. “SUMMARY AND FUTURE PROSPECTS” summarize our
findings and their significance towards further investigation in HEAs.

##  II Methods

###  II.1 HEA structural models

HEAs have been regarded as solid solutions, wherein every site in the crystal
structure is randomly occupied by the constituent elements. However, it is
difficult to reproduce such a random configuration with a small simulation
cell size. Therefore, finding an appropriate method to mimic the configuration
of HEAs under a limited periodic condition is crucial for theoretical
investigations. This problem can be solved by evaluating the correlation
functions of the atomic configurations in HEAs. For instance, Zunger et al.
proposed the concept of special quasirandom structures (SQSs) 1990ZUN , which
are supercell approximations for a disordered system optimized to mimic the
random local atomic environment. Therefore, the SQS approach is widely applied
to HEA research because it provides a good approximation for the RSS phase.
However, it is difficult to discuss the energetics and phase stability of HEAs
in detail using only one configuration obtained from the SQS model. It has
previously been reported that applying several SQS configurations of HEAs
leads to non-negligible energy fluctuations (0.01–0.02 eV/atom). HEAbook ;
2018NIU  We performed high-throughput first-principles sampling to provide
insights regarding the phase stability of HEAs with high accuracy and
reliability for overcoming the disadvantage of using a single configuration in
the SQS approach.

Two-, three-, and four-body clusters were selected for a correlation function,
with the Warren–Cowley short-range order (SRO) parameter evaluated by
homogeneous and heterogeneous elements 1960COW ; 1965COW . The range of the
SRO parameters was restricted to the fourth nearest-neighbor site for two-body
clusters and to the third nearest-neighbor site for three- and four-body
clusters to reduce the computational time. For the RSS phase, the
configurations of four elements were explored to realize a better correlation
function using the Metropolis algorithm 1953MET  with a wide range of
fictitious temperatures. For both the ordered and RSS phases, four elements
were considered for evaluating the correlation function. In the L12 ordered
phase, one element was fixed at the cubic corner (CC) sites in the FCC
lattice, while the other three elements in the quaternary system occupied the
face-centered sites to realize an ideal L12 ordered phase with a good
correlation function. The D022 phase was also constructed in a similar manner.
A wide range of occupancy rates was used to elucidate the influence of the
number of first nearest-neighbor sites on the formation energy. Accordingly, a
large number of atomic configurations were randomly generated for RSS
structures (Fig. 6). Our structural models of the quaternary alloys were
described by $2~{}\\!\times~{}\\!2~{}\\!\times~{}\\!2$ supercells with 32
sites or $4~{}\\!\times~{}\\!4~{}\\!\times~{}\\!4$ supercells with 256 sites,
depending on the computed quantities; the equiatomic system comprises 8 atoms
in the $2~{}\\!\times~{}\\!2~{}\\!\times~{}\\!2$ supercell and 64 atoms in the
$4~{}\\!\times~{}\\!4~{}\\!\times~{}\\!4$ supercell for each element. The
crystal structures were drawn using the VESTA software 2011MOM .

###  II.2 High-throughput first-principles sampling

The free energy of formation (Figure 4) and magnetic moment (Figure 5) for
each phase were obtained based on 100 configurations using a
$2~{}\\!\times~{}\\!2~{}\\!\times~{}\\!2$ supercell. Figure S-2 shows the
formation free energies obtained using 100 configurations. The dependences of
the formation energy on the occupancy rate (Figure 6) and the bond-length
distributions (Figure 7) for the CrFeCoNi quaternary alloy were obtained based
on 1,500 and 100 configurations, respectively, using a
$4~{}\\!\times~{}\\!4~{}\\!\times~{}\\!4$ supercell.

We carried out high-throughput spin-polarized first-principles simulations
based on DFT 1965KOH  using the Vienna ab initio simulation package (VASP) by
inputing all generated HEA models above 1996KRE1 ; 1996KRE2 . The PBEsol
functional 2008PER  was selected for the electronic structure calculations and
geometry relaxation. Projector-augmented wave (PAW) 1994BLO ; 1999KRE
potentials were used to consider the interactions between the ion cores and
valence electrons. The Brillouin zone was described by a set of k-points in a
$3~{}\\!\times~{}\\!3~{}\\!\times~{}\\!3$ grid mesh using the Monkhorst–Pack
method. 1976MON  For structural optimization, the energy convergence criterion
for electronic iterations was set at $10^{-2}$ eV/Å. The lattice parameter was
set at 3.495 Å during structural optimization to reduce the calculation time.
As an initial condition, Fe, Co, Ni, and Cu were set as ferromagnetic, whereas
Cr was set as antiferromagnetic. For systems without Cr, Mn was set as
antiferromagnetic; otherwise, Mn was set as ferromagnetic as an initial
condition because its magnetism is affected by Cr.

###  II.3 Formation energy and formation free energy

To discuss phase stability, the formation energy $E_{f}(\rm{HEA})$ is defined
as

$E_{f}({\rm{HEA}})=E({\rm{HEA}})-\sum_{i}x_{i}E(X_{i}).$ (1)

Here, $E(\rm{HEA})$ denotes the total energy per atom of the HEA with the
alloying elements; $X_{i}$ and $x_{i}$ represent the fractions of the alloying
elements; and $E(X_{i})$ denote the energies per atom of the alloying elements
$X_{i}$ in their ground state structures, i.e., BCC Cr, $I\bar{4}3m$ Mn, BCC
Fe, HCP Co, FCC Ni, and FCC Cu, which are estimated by referring to the Open
Quantum Materials Database 2013SAA ; 2015KIR . A negative value of the
formation energy implies that the HEA is stable, whereas a positive value
implies that it is less stable than the ground states of the pure alloying
elements.

Temperature is one of the most important factors that governs the phase
stability of materials because most materials are used in a finite temperature
range instead of being used at absolute zero. 2017ZHO . Here, the Helmholtz
free energy is defined as

$\displaystyle F(V,T)$ $\displaystyle=$ $\displaystyle
E_{\mathrm{el}}(V)+E_{\mathrm{vib}}(V,T)$ (2) $\displaystyle-$ $\displaystyle
TS_{\mathrm{ele}}(V,T)-TS_{\mathrm{vib}}(V,T)-TS_{\mathrm{config}}(V,T),$

where $E_{\mathrm{el}}(V)$ denotes the internal energy, and the phonon
vibration contribution includes the lattice vibration energy, i.e.,
$E_{\mathrm{vib}}(V,T)$, and lattice vibration entropy, i.e.,
$S_{\mathrm{vib}}(V,T)$. Further, $S_{\mathrm{config}}(V,T)$ represents the
atomic configurational entropy, and the last term $S_{\mathrm{ele}}(V,T)$
denotes the electronic-scale entropy contribution, which includes thermal
excitation and spin polarization. In this study, contributions from the
vibrational and electronic entropy terms are not included owing to the
limitations of the calculations. 2019NIU  Further, the vibrational
contributions for both the ordered and RSS phases were estimated and found to
be comparable 1998WAL . The contributions of the electronic and magnetic terms
are smaller than that of the configurational term 2015MA .
$S_{\mathrm{config}}$ in the configurational entropy term is given by
$S_{\mathrm{config}}=R\sum_{i}c_{i}\ln c_{i}$, where $c_{i}$ denotes the ratio
between the number of atoms of a disordered component and the total number of
disordered atoms, and $R$ represents the gas constant. In this study, we
introduced an equiatomic quaternary system and calculated
$S_{\mathrm{config}}=R\ln(3)\times 0.75=0.824R$ for the ordered phases and
$S_{\mathrm{config}}=R\ln(4)=1.386R$ for the RSS phase. In other words, the
RSS phase has a greater configurational entropy contribution. Therefore, the
free energy of formation is calculated by

$F_{f}({\rm{HEA}})=E({\rm{HEA}})-TS_{\mathrm{config}}.$ (3)

##  III Results

###  III.1 Order–Disorder Competitive Cooperation

One of the main objectives of this study is to investigate whether an ordered
phase (OP0) is more stabilized than the corresponding RSS for each studied
quaternary alloy. That is, we intended to determine the “crossover”
temperature ($T_{c}$) at which the phase transition between the OP0 and RSS
occurs for each composite of the quaternary alloy. Therefore, it is important
to consider the configurational entropy term to discuss phase stability in
HEAs in the finite temperature region. It is imperative to quantitatively
analyze the influences of VEC and temperature on the stable HEA crystal
structures for investigating the order–disorder competitive cooperation and
phase stability. This is because the enthalpy term defined at the absolute
zero temperature correlates with VEC, whereas the entropy effect is
temperature dependent; the most stable phase is determined by free energy,
i.e., the sum of the enthalpy and entropy terms. Since the direct elucidation
of QSPR is difficult, material features (or descriptors) such as VEC 2011GUO1
; 2011GUO2  are “inserted” between “structure” and “property”; then, two step
analyses are conducted based on “structure–feature” and “feature–property”
relationships. Guo et al. 2011GUO1 ; 2011GUO2  reported several HEA systems
wherein the single FCC phase was stable in the high VEC region (VEC $>8.0$),
the single BCC phase was stable in the low VEC region (VEC $<6.87$), and the
mixed FCC and BCC phases appeared in the intermediate region
($6.87<\rm{VEC}<8.0$). BCC and FCC enhance the strength and ductility,
respectively HEAbook . The average VEC values of magnetic HEAs are greater
than 7.5, and therefore, they possess the FCC structure. Hence, this study
does not focus on the lattice type of magnetic HEAs, but on their magnetic
orderings. We investigate the suitability of VEC to identify the orderings,
which would be the first step toward successful designing HEAs.

TABLE 1 lists the quaternary alloy composites investigated in this study
(Composite), average VECs (VECave), average melting temperature of the four
constituent elements in the alloy ($\bar{T}_{m}$), ordered phase with the
lowest energy at 0 K (OP0), temperature at which stability is attained for
each phase ($T_{s}(X)$ ($X=$ L12, D022, RSS)), “crossover” temperature between
the ordered phase OP0 and RSS ($T_{c}$), and the corresponding free energy of
formation at $T_{c}$ ($F_{\mathrm{c}}$). The the most stable solid phase
(MSSP) of each composite below $\bar{T}_{m}$ and the corresponding temperature
range ($\Delta T_{s}$) are also listed in the table. The free energy of
formation considering the configurational entropy contribution is elucidated
in Eq. (2). To estimate estimating the free energy of formation, we can
consider the ordered phases as a 75 at.% equiatomic ternary alloy for the
configurational entropy term because one of the elements is completely located
at a specific site (Fig. 1 (a) or (b)).

Table 1:  Temperature dependence of the most stable structures for 15 cases of equiatomic quaternary systems (column name: “Composite”) investigated in the present study. VECave corresponds to the average valence electron concentration of the composite, where Cr, Mn, Fe, Co, Ni, and Cu have VEC values of 6, 7, 8, 9, 10, and 11, respectively. $\bar{T}_{m}$ represents the average melting temperature of the four elements that comprise the composite. OP0 represents the ordered phase possessing the lowest energy at 0 K. For example, Cr-L12 indicates the ordered phase wherein the ordered lattice positions are occupied by Cr. (NB: This does not necessarily mean the ordered phase has a lower energy than the RSS.) $T_{s}(X)$ ($X$ = L12, D022, RSS) indicates the temperature at which phase $X$ is stabilized, i.e., its free energy of formation becomes zero. $T_{c}$ represents a “crossover” temperature between the ordered phase OP0 and RSS. $F_{c}$ (eV/atom) denotes the free energy of formation at $T_{c}$. $\Delta T_{s}$ represents the temperature range up to $T_{m}$, wherein either one of the ordered phases or the RSS phase is realized as the most stable solid phase (MSSP). All temperatures are listed in Kelvin (K).  |  |  |  | $T_{s}$ |  |  |
---|---|---|---|---|---|---|---
Composite(#) | VECave | $\bar{T}_{m}$ | OP0 | L12 | D022 | RSS | $T_{c}$ | $F_{\mathrm{c}}$ | MSSP ($\Delta T_{s}$ )
CrMnFeCo(1) | 7.50 | 1820 | Cr-L12 | 380 | 683 | 638 | 1016 | -0.045 | L12 (380 $\sim$ 1016)
|  |  |  |  |  |  |  |  | RSS (1016 $\sim$ $\bar{T}_{m}$)
CrMnFeNi(2) | 7.75 | 1810 | Cr-L12 | 448 | 737 | 657 | 963 | -0.037 | L12 (448 $\sim$ 963)
|  |  |  |  |  |  |  |  | RSS (963 $\sim$ $\bar{T}_{m}$)
CrMnCoNi(3) | 8.00 | 1799 | Cr-L12 | 282 | 553 | 574 | 1003 | -0.051 | L12 (282 $\sim$ 1003)
|  |  |  |  |  |  |  |  | RSS (1003 $\sim$ $\bar{T}_{m}$)
CrMnFeCu(4) | 8.00 | 1717 | Cr-L12 | 2594 | 2755 | 1820 | 687 | 0.135 | unstable
CrFeCoNi(5) | 8.25 | 1872 | Cr-L12 | 0 | 127 | 494 | 1333 | -0.100 | L12 (0 $\sim$ 1333)
|  |  |  |  |  |  |  |  | RSS (1333 $\sim$ $\bar{T}_{m}$)
CrMnCoCu(6) | 8.25 | 1706 | Cr-L12 | 2251 | 2425 | 1649 | 768 | 0.105 | RSS (1649 $\sim$ $\bar{T}_{m}$)
CrFeCoCu(7) | 8.50 | 1779 | Cr-L12 | 1913 | 1917 | 1469 | 819 | 0.078 | RSS (1469 $\sim$ $\bar{T}_{m}$)
CrMnNiCu(8) | 8.50 | 1696 | Cr-L12 | 1881 | 1950 | 1398 | 691 | 0.085 | RSS (1398 $\sim$ $\bar{T}_{m}$)
MnFeCoNi(9) | 8.50 | 1706 | Mn-L12 | 0 | 128 | 333 | 970 | -0.076 | L12 (0 $\sim$ 970)
|  |  |  |  |  |  |  |  | RSS (970 $\sim$ $\bar{T}_{m}$)
CrFeNiCu(10) | 8.75 | 1769 | Cr-D022 | 1786 | 1546 | 1223 | 750 | 0.074 | RSS (1223 $\sim$ $\bar{T}_{m}$)
MnFeCoCu(11) | 8.75 | 1614 | Mn-L12 | 1597 | 1665 | 1218 | 663 | 0.066 | RSS (1218 $\sim$ $\bar{T}_{m}$)
CrCoNiCu(12) | 9.00 | 1758 | Cr-D022 | 1711 | 1594 | 1117 | 417 | 0.092 | RSS (1117 $\sim$ $\bar{T}_{m}$)
MnFeNiCu(13) | 9.00 | 1604 | Mn-D022 | 1395 | 1205 | 956 | 592 | 0.057 | RSS (956 $\sim$ $\bar{T}_{m}$)
MnCoNiCu(14) | 9.25 | 1593 | Mn-D022 | 1497 | 1204 | 981 | 653 | 0.060 | RSS (981 $\sim$ $\bar{T}_{m}$)
FeCoNiCu(15) | 9.50 | 1666 | Fe-L12 | 421 | 474 | 469 | 539 | -0.008 | L12 (421 $\sim$539)
|  |  |  |  |  |  |  |  | RSS (539 $\sim$ $\bar{T}_{m}$)
Figure 1:  Crystal structures of (a) L12 (Cu3Au-type, space group
$Pm\bar{3}m$, no. 221), (b) D022 (Al3Ti-type, space group $I4/mmm$, no. 139),
and (c) RSS for quaternary alloys. Labels indicate the first, second, third,
and fourth nearest neighbor sites from the red site located at the cubic
corner.

Consequently, one element of the ordered phases can be ignored for the number
of the elements considering the configurational entropy term. Alternatively,
the RSS phases need to be treated as a quaternary alloy. Therefore, the
contribution of the configurational entropy term for the RSS phase is larger
than that for the ordered phases. Accordingly, the slope of the free energy of
formation for the RSS phase is steeper than that for the ordered phases. The
ordered phases consequently appear gradually as the temperature increases, and
this is followed by the appearance of the RSS phases at higher temperatures.
The above discussion presupposes thermal equilibrium. Note that ordered phases
that thermally equilibrate at higher temperatures can appear even at ambient
temperatures by non-equilibrium processing, i.e., rapid solidification and
cooling. 2013YEH

Figure 2 shows the free energy of formation of the CrMnCoNi quaternary alloy
as a function of temperature. The temperature dependence of the free energy of
formation of the other alloys, including the CrMnCoNi alloy, is shown in Fig.
S-1 (supplementary information). For the both ordered phases, Cr-L12 and
Cr-D022 are the most stable phase of each ordered phase (Fig. 4 (3)).
Therefore, we compare the free energies of formation of the Cr-L12 and Cr-D022
ordered phases and the RSS phase up to $\bar{T}_{m}$(1799 K). The temperature
dependence can be seen as follows:

  1. 1.

Within the low-temperature region (up to 282 K), all ordered and RSS phases
exhibit a positive free energy of formation, and they are less stable than the
corresponding ground state of each pure element.

  2. 2.

At 282–1003 K, Cr-L12 shows a negative free energy of formation, which is
lower than those of the Cr-D022 and RSS phases. Therefore, Cr-L12 is the most
stable phase in this temperature region.

  3. 3.

Beyond 1003 K, the RSS phase shows a lower free energy of formation than
Cr-L12 and becomes a stable phase in the high-temperature region.

Meanwhile, the Cr-D022 phase does not appear as the most stable phase in the
entire temperature range. Consequently, in the case of the CrMnCoNi quaternary
alloy (VEC${}_{\mathrm{ave}}=8.00$), the L12 crystal structure appears as an
ordered phase in a specific temperature region.

Figure 2:  Free energy of formation of CrMnCoNi quaternary alloy as a function
of temperature up to its melting temperature (1799 K). Blue, red, and green
lines indicate the RSS, Cr-D022, and Cr-L12 ordered phases, respectively.

Further, as shown in Fig. S-1 and TABLE 1, the CrMnFeCo (1;
VEC${}_{\mathrm{ave}}=7.5$), CrMnFeNi (2; VEC${}_{\mathrm{ave}}=7.75$),
CrFeCoNi (5; VEC${}_{\mathrm{ave}}=8.25$), MnFeCoNi (9;
VEC${}_{\mathrm{ave}}=8.5$), and FeCoNiCu (15; VEC${}_{\mathrm{ave}}=9.5$)
quaternary alloys exhibit the same phase transition between the L12 ordered
phase and RSS phase at a specific temperature. However, the CrFeCoNi (5;
VEC${}_{\mathrm{ave}}=8.25$) and MnFeCoNi (9; VEC${}_{\mathrm{ave}}=8.5$)
quaternary alloys exhibited the L12 structure at 0 K. Furthermore, the
CrMnCoCu (6; VEC${}_{\mathrm{ave}}=$ 8.25), CrFeCoCu (7;
VEC${}_{\mathrm{ave}}=$ 8.5), CrMnNiCu (8; VEC${}_{\mathrm{ave}}=$ 8.5),
CrFeNiCu (10; VEC${}_{\mathrm{ave}}=$ 8.75), MnFeCoCu (11;
VEC${}_{\mathrm{ave}}=$ 8.75), CrCoNiCu (12; VEC${}_{\mathrm{ave}}=$ 9.0),
MnFeNiCu (13; VEC${}_{\mathrm{ave}}=$ 9.0), and MnCoNiCu (14;
VEC${}_{\mathrm{ave}}=$ 9.25), quaternary alloys show the RSS phase only below
the melting temperature because of the small difference in the free energy of
formation at 0 K. (Fig. S1 (6)-(8), and (9)-(14)) The CrMnFeCu (4;
VEC${}_{\mathrm{ave}}=$ 8.0) quaternary alloy is unstable in the entire
temperature range.

The results of this study are summarized as follows: The alloys without Cu as
a constituent prefer the L12 ordered phases for small VEC values ($\leq
8.25$). The Mn-based L12 of MnFeCoNi (9) and the Fe-based L12 of FeCoNiCu (15)
are also stable under a specific temperature. The free energy of formation of
the D022 ordered phases are lower than L12 ordered phases for large VEC values
($\geq 8.50$), except for MnFeCoCu (11) and FeCoNiCu (15). However, D022
ordered phases do not appear as the most stable structure, because there is no
condition that the free energy of formation is negative and smaller than the
RSS in the entire temperature range. In contrast, alloys with large VEC values
and Cu as a constituent exhibit the RSS phases. Finally, we may conclude that
neither entropy nor enthalpy is dominant in stabilizing HEAs, but both
cooperatively stabilize their structures (see Figure 3) depending on
temperature and other factors such as VEC.

Figure 3: Phase stability of 3d-transition metal high-entropy alloys (HEAs) is
realized by an order–-disorder “competitive cooperation,” rather than a usual
order–-disorder competition. The former was newly proposed in this study,
while the latter has already been accepted in the HEA community. 2015NIU

###  III.2 Energetics

A clear understanding of the energetics of each phase is important to
determine the free energy of formation at a finite temperature for discussing
the phase stability of HEAs. Figure 4 shows the average free energies of
formation of 100 different configurations, with the standard deviation at 1000
K for all 15 quaternary alloys, each of which has eight types of ordered
phases and the RSS phase. Ordered phases with the lowest energy at 0 K (OP0)
for each quaternary alloy are listed in Table 1. For all quaternary alloys
(Fig. S-1), the free energy of formation of at least one ordered phase is
lower than that of the RSS phase at 0 K. Interestingly, the D022 ordered
phases exhibit a lower free energy of formation than the L12 ordered phases in
the high VEC region in some alloys. However, because the free energy of
formation of D022 ordered phases is not negative below the RSS value, the D022
phases are not stable phase in the entire temperature range.

Figure 4:  Free energy of formation at 1000 K for 15 cases of quaternary
alloys listed in Table 1. Error bars indicate the standard deviations (SD) for
each phase. The original raw data for each phase are shown in Fig. S-2
(Supplementary Information).

As shown in Fig. 4, CrFeNiCu (10; VEC${}_{\mathrm{ave}}=8.75$), CrCoNiCu (12;
VEC${}_{\mathrm{ave}}=9.0$), MnFeNiCu (13; VEC${}_{\mathrm{ave}}=9.0$), and
MnCoNiCu (14; VEC${}_{\mathrm{ave}}=9.25$) exhibit D022 phases with lower
energies than that of the L12 phases. These alloys demonstrate that the free
energy of L12 increases remarkably, as a result, these free energies of
formation show a positive value at 1000K. This result means that these ordered
structures are thermodynamically unstable, however, we compare the free
energies of L12 and D022 to get insight of a competitive cooperation. Similar
competitive behavior between the phase stability of L12 and D022 for a wide
range of VECs was previously reported for Ni3V, Pd3V, and Pt3V, which are
fundamental binary alloys, in contrast with HEAs 1996CAB . The VEC at which
the crossover of the energy difference between the L12 and D022 phases occurs
in this study (VEC${}_{\mathrm{ave}}=8.5$) is comparable to the value reported
for the previously studied binary system 1996CAB . One key result obtained
from Fig. 4 is that the systems comprising Cu have high free energies of
formation. Moreover, quaternary alloys in which Cu forms the ordered phase
entail higher free energies of formation of the ordered phase than those of
the RSS phase. We attribute the high energies of formation associated with Cu
to the positive formation energies exhibited by the least energetic structures
of the binary compounds consisting of Cu and 3d transition metals (Cr, Mn, Fe,
Co, and Ni) 2015TRO , 2021KUC , 2005TAK . The present results are consistent
with previous studies because as the number of bonding pairs between Cu and
other 3d transition metals increased in the Cu-ordered phases (cf. Table 2).

###  III.3 Magnetic ordering

Figure 5 illustrates the distribution of magnetic moments as a function of the
average magnetic moment at the first nearest-neighbor sites in the most
energetically stable ordered phase for 15 quaternary alloys. In Fig. 5, we
observe certain common trends: “Cr” and “Mn without Cr” show antiferromagnetic
properties, “Mn with Cr”, “Fe”, and “Co” show ferromagnetic properties, and
“Mn with a ferromagnetic moment” shows values that can be compared to those of
Fe. In contrast, Mn with antiferromagnetic properties behave similar to Cr.
Therefore, the magnetic nature of Mn depends on the presence or absence of Cr.
Ni and Cu exhibit nonmagnetic (paramagnetic) behavior in all systems. Although
the magnetic moments are mostly similar, the magnetic moment distributions of
some alloys show that the average magnetic moments at the first nearest-
neighbor sites are discrete; for example, at Cr and Fe in CrFeNiCu
(VEC${}_{\mathrm{ave}}=$8.75), as shown in Fig. 4 (10). Such a discrete
distribution can be attributed to the number of ferromagnetic or
antiferromagnetic atoms located at the first nearest-neighbor sites of the FCC
lattice (Fig. 2(b) in 2015NIU ). For example, the antiferromagnetic atoms in
the ordered phases are surrounded by four ferromagnetic atoms on average
because the other three elements are occupied at the 12 first nearest-neighbor
sites.

Figure 5:  Distribution of magnetic moments as a function of the average
magnetic moment at the first nearest-neighbor site in the most energetically
stable ordered phase for the 15 quaternary alloys listed in Table 1 as OP0.
Red, orange, green, light blue, dark blue, and purple indicate Cr, Mn, Fe, Co,
Ni, and Cu, respectively. $m$ and $m_{\mathrm{ave~{}\\!\\!NN}}$ represent the
magnetic moment for each atom and the average magnetic moment of the 12 first
nearest-neighbor sites, respectively.

Niu et al. 2015NIU  previously evaluated the magnetic moment of each element
and the average magnetic moment of the 12 first-nearest neighbor sites for the
RSS and Cr-L12 ordered phases of the CrFeCoNi quaternary alloy
(VEC${}_{\mathrm{ave}}=8.25$), and they are the same as that of the composite
shown in Fig. 4 (5). The results obtained in this study demonstrate that Fe
and Co exhibit ferromagnetic properties, whereas Cr exhibits anti-
ferromagnetic properties. Since the results obtained are consistent with those
reported previously 2015NIU , the validity of the calculation methods
described herein can be confirmed. Moreover, Niu et al. reported a discrete
distribution of the average magnetic moments at the first nearest-neighbor
sites in the Cr-L12 ordered phase. They revealed that the magnetism of Cr in
the CrFeCoNi quaternary alloy is strongly influenced by the magnetism of the
first nearest-neighbor sites 2015NIU . Zuo et al. 2017ZUO  evaluated the
magnetic moments of the individual atoms of the FCC MnFeCoNi quaternary alloy
and FCC CrMnFeCoNi quinary alloy and reported that Fe exhibits a higher
magnetic moment than Co, Ni exhibits a near-zero magnetic moment, and Cr
exhibits antiferromagnetism. In addition, Mn atoms cause a wider distribution
in the magnetic moments of atoms and exhibit both ferromagnetism and
antiferromagnetism in the RSS phase. 2017ZUO

Ghazisaeidi et al. reported a similar magnetism distribution in the hexagonal
close packed (HCP) and FCC phases of CrCoNi, FeCoNi, MnFeNi, MnCoNi, and
CrMnFeCoNi. 2018NIU  The magnetism of Cr and Mn atoms ranges from
ferromagnetism to antiferromagnetism in the RSS phase. Fedorov et al. 2020FED
and Schönfeld et al. 2019SCH  out that the magnetic properties of the
constituent elements are key factors governing phase stability and the
formation of ordered phases in 3d-transition-metal-based alloys.

###  III.4 Quantitative Structure–Property Relationship

The L12 and D022 phases have typical ordered crystal structures in the 25:75
(at.%) FCC binary alloys. For example, the L12 structure can be observed in
many alloys with a 3:1 composition, such as Cu3Au, Ni3Al, and TiPt3;
similarly, the D022 structure can be observed in trialuminides Al3 (Sc, Ti, V)
and Ni3V. Although the appearance of the L12 and D022 phases in FCC binary
alloys has been discussed previously, only a few studies have focused on the
configuration of the neighboring sites in multiprincipal-element alloys.
2015NIU  The number of first, second, third, and fourth nearest-neighbor sites
in the L12 and D022 phases are 12, 6, 24, and 12, respectively (Table 2). The
25 at.% element in the L12 and D022 phases is not located at the first nearest
neighbor sites, and thus, they can be isolated from each other and surrounded
by the 75 at.% elements. For instance, Au atoms in the L12 phase of Cu3Au are
not located at their first nearest-neighbor sites and are surrounded by 12 Cu
atoms.

Table 2:  Comparison of the site occupancies of L12, D022, and RSS phases containing 25 at.% of ordered elements. For each neighbor site therein, the number of the ordered elements over the total number of the sites is given with its percentage in brackets. For the RSS phase, the occupancy rate for any site is 25 at.% on average (shown in brackets).  Neighbor site | L12 | D022 | RSS
---|---|---|---
First nearest | 0/12 $\langle 0\rangle$ | 0/12 $\langle 0\rangle$ | $\langle 25\rangle$
Second nearest | 6/6 $\langle 100\rangle$ | 4/6 $\langle 67\rangle$ | $\langle 25\rangle$
Third nearest | 0/24 $\langle 0\rangle$ | 8/24 $\langle 33\rangle$ | $\langle 25\rangle$
Fourth nearest | 12/12 $\langle 100\rangle$ | 4/12 $\langle 33\rangle$ | $\langle 25\rangle$

Figure 6 shows the relationship between the formation energy and occupancy
rate of Cr–Cr pairs at the first and second nearest-neighbor sites in the RSS
phase of the CrFeCoNi quaternary alloy. In the RSS phase, the rate of
occupancy by the same element can remain at 25 at.% on average at any nearest-
neighbor site. Meanwhile, the occupancy rates for the ordered phases depend on
the nearest-neighbor site, as listed in TABLE 2. For the first nearest-
neighbor sites (Fig. 6(a)), the calculated results reveal a lowering of the
energy with a decreasing occupancy rate, such that the energy approaches the
values corresponding to the L12 and D022 ordered phases on extrapolation of
the occupancy rate to zero. However, for the second nearest-neighbor sites
(Fig. 6(b)), an opposite trend is observed, i.e., the formation energies
decrease with increasing occupancy rate. The values for both ordered phases
can be found on the extrapolation line. These results imply that the
configurations at the first and second nearest-neighbor sites affect the
formation energy of the CrFeCoNi quaternary alloy. Furthermore, the lower
formation energies of the ordered phases can be attributed to their unique
occupancy rates.

Figure 6:  Dependence of the formation energies of ordered and RSS phases in
the CrFeCoNi quaternary alloy on the occupancy rate of the Cr–Cr pairs at the
(a) first and (b) second nearest-neighbor sites. The red dots represent the
RSS phase obtained using 1,500 samples, whereas the blue lines serve as
guidelines. The blue and green dots indicate the Cr-L12 and Cr-D022 ordered
phases, respectively.

Figures 7 (a) and (b) show the bond-length distributions in the RSS and Cr-L12
ordered phases, respectively. In Figure 7, the solid lines indicate
homogeneous element pairs, whereas the dotted and dashed lines indicate
heterogeneous element pairs. Many 3d transition metals, especially Cr, Fe, Co,
and Ni, exhibit comparable atomic radii. However, the CrFeCoNi quaternary
alloy shows a different bond-length distribution. These results indicate that
these element pairs have different interatomic distances. As shown in Figure
7(a), the Cr–Cr pairs in the RSS phase exhibit a broad bond-length
distribution and longer bond lengths than those corresponding to the first
nearest-neighbor sites, as estimated by the lattice constant. The bond lengths
of the homogeneous element pairs are in a decreasing order of Cr, Fe, Ni, and
Co. The same trend is observed for the Cr-L12 ordered phase in Fig. 7(b).
Notably, there are no Cr–Cr pairs in the Cr-L12 ordered phase because Cr atoms
occupy the sites constructing the framework of the L12 ordered phase, and they
are not located at the first nearest-neighbor sites (TABLE 2). Although the
structure optimization process is applied to all atoms in the Cr-L12 ordered
and RSS phases, the bond-length distributions of the Cr-L12 ordered phase are
narrower than those of the RSS phase.

Figure 7:  Relative frequency distribution of bond-lengths fitted with
Gaussian function for each element pair in the (a) RSS phase and (b) Cr-L12
ordered phase of the CrFeCoNi quaternary alloy after structure optimization.
The bond lengths in the legends indicate the average values obtained upon
fitting with the Gaussian function.

Let us consider the half widths at half maximum (HWHM) of the fitted Gaussian
functions. The HWHMs of Fe–Fe, Co–Co, and Ni–Ni in the L12 ordered phases are
64.5 %, 71.2 %, and 68.6 %, respectively, of the corresponding RSS phases. It
is presumed that the Cr occupancy at the second nearest-neighbor sites results
in a small bond-length distribution with regularity. Gao and Alman used ab
initio molecular dynamics (AIMD) simulations 2013GAO  and reported that the
Al1.3CrFeCoNiCu senary alloy exhibits a different distribution for each
element pair, which is similar to the results obtained in this study for
3d-transition-metal-based equiatomic quaternary alloys. However, the
HfNbTaTiZr quinary alloys show almost the same distribution for all its
constituent elements without a 3d transition metal. 2013GAO  In this study, we
conclude that the bond-length distributions of the described systems are
significantly different because each 3d transition metal has an individual
magnetic behavior.

##  IV Summary and Future Prospects

In this study, we performed first-principles calculations considering the
configurational entropy term to investigate the effect of VEC and temperature
on the stabilities of the two ordered phases: L12 and D022. Further, we drew a
comparison with the disordered atomic configurations in equiatomic quaternary
alloys comprising Cr, Mn, Fe, Co, Ni, and Cu. We found that the VEC is a
significant factor controlling phase stability in both ordered phases. In the
high-VEC region, the D022 ordered phase is energetically more stable than the
L12 ordered phase, though both phases are metastable. The magnetic moments
indicate that the anti-ferromagnetic Cr and Mn atoms are surrounded by the
ferromagnetic Fe, Co, and Ni atoms located at the first nearest-neighbor
sites. Such magnetic configurations make the ordered HEA phases energetically
stable. Since the constituent elements corresponding to 25 at.% in the L12 and
D022 ordered phases exhibit no bonding between the first nearest-neighbor
sites, the extent of the combination of anti-ferromagnetic and ferromagnetic
elements can be increased in the L12 and D022 phases. The bond-length
distribution results reveal that the Cr–Cr distances are longer and the
distributions are broader than those of other element pairs, which suggests
that the high formation energy of the RSS phase can be attributed to the
abnormal bond length between the anti-ferromagnetic atoms. Consequently, both
the ordered phases have enthalpy-driven energetically favorable structures
that are enthalpy-driven.

Since 3d-transition-metal quaternary systems are base alloys for their
extension to more multicomponent systems, a comprehensive understanding of
their structures would serve as a stepping stone toward efficient HEA design.
As shown for most 3d-transition-metal quaternary alloys, we should consider
that phase stability is realized by an order–-disorder “competitive
cooperation,” rather than a usual order–-disorder competition. The former was
newly proposed in this study, while the latter has already been accepted in
the HEA community. Most researchers in the HEA community naively consider that
the “fully disordered/random atomic configurations” of HEAs have higher
entropies, and therefore, they have higher phase stabilities. However, it is
important to recognize the cooperation between “a magnetically ordered atomic
configuration composed of one element” and “fully disordered/random
configurations composed of the other three elements,” which correspond to the
enthalpy and entropy terms, respectively, to understand the microscopic
structures of HEAs in detail.

In this study, the enthalpy terms that describe structural properties were
evaluated by first-principles calculations to obtain insight into an atomistic
origin of the energetics of the 3d-transition-element HEAs. Since the entropy
term is as important as the enthalpy term, the addition of a configurational
entropy term in the density functional theory (DFT) calculations is effective
for determining the crystal structure of HEAs. This enables us to estimate the
transition temperatures between the ordered and RSS phases, which is useful
for obtaining the relationship between structure and processing. It is already
widely recognized in the materials science field that the relationship among
structures, properties, and processing methods is a crucial factor in
realizing outstanding structural materials 2016XIO . For example, Kenel et al.
designed a 3d-transition-metal HEA using a 3D printing technique 2020HAN2
with oxide nanopowder, which exhibits prominent mechanical properties at
ambient and cryogenic temperatures 2019KEN . Futhermore, it is necessary to
predict an accurate phase diagram of HEA to realize QSPR by combining the
calculation of the phase diagram (CALPHAD) 1998SAU  and the first-principles
based thermodynamic assessment 2020HAN1 . We believe that the systematic
investigation described herein will be helpful for realizing such an
assessment and for predicting the microstructures of HEA by performing model
simulations such as “first-principles phase field model” without any
thermodynamic empirical parameters 2019BHA .

Finally, we note that our finding is applicable to “ceramics”, “oxides,” and
“carbide”, as well because the “high-entropy” concept has been adopted even in
them 2018ANA ; 2019SAR ; 2019HAR ; 2019YE ; 2020OSE ; 2020KAU ; 2021MCC ;
2021HOS . Furthermore, our proposed concept sheds light on the constituent
phases of even multiprincipal element alloys (MPEAs) 2015SEN ; 2019ZHE  and
multiphase compositionally complex alloys (CCAs) 2019MAN . Thus, our finding
can encourage experimentalists to investigate the microscopic identification
of their structures, which will help us elucidate their QSPR.

##  V Supplementary Information

###  V.1 Ab initio calculations

Figure S-1 shows the formation free energy as a function of temperature for 15
kinds of quaternary alloys, which are listed at Table I. The slope of the
formation free energy for RSS phase is steeper than that of the ordered
phases, because the entropy term of RSS is larger than that of the ordered
phases. On the other hand, the slope of both ordered phases are exact same.
Consequently, the comparison of the formation free energy of the ordered
phases is determined by formation energy at 0 K. Note that in the case of
CrFeCoCu (VEC${}_{\mathrm{ave}}=8.50$) alloy in (7) at Fig. S-1, the both
ordered phases have almost same formation free energy. Figure S-1 (14) and
Figure 2 show the formation free energy for MnCoNiCu alloy with different
vertical scale. In Fig. S-1, different scale is set for comparison with other
alloy systems.

Figure S-2 illustrates the distribution of the formation free energies of at
1,000 K for quaternary alloys using 100 samples. Figure 3 shows the average of
formation free energies with the standard deviation from the raw data of 100
samples in Fig. S-2. $X$-L12 and $X$-D022 ($X=$ Cr, Mn, Fe, Co, Ni, and Cu)
indicate the cubic corner (CC) sites of an face centered cubic (FCC) lattice
occupied by each element for L12 structure and the corner sites and body
centered sites of an FCC lattice when $c=2a$ for D022 structure, respectively.
a is a lattice constant of FCC lattice. RSS is random solid solution (cf.
Figure 1). Namely, 8 kinds of the ordered phases and RSS phase for each
composite can be compared in Fig. S-1. It becomes clear that the quaternary
alloys containing Cu show high formation free energy than those of other
alloys, as a result of comparison of 15 kinds of the alloys with the same
energy ranges on all vertical axes. A few ordered phases, such as Cr-L12
phase, show a narrow dispersion of formation free energy, while many ordered
phases show a same wide dispersion as RSS phases. Interestingly, $X$-L12 ($X=$
Cr and Mn) phases of CrMnNiCu (VEC${}_{\mathrm{ave}}=8.50$) in (8), CrFeNiCu
(VEC${}_{\mathrm{ave}}=8.75$) in (10), MnFeCoCu (VEC${}_{\mathrm{ave}}=8.75$)
in (11), CrCoNiCu (VEC${}_{\mathrm{ave}}=9.0$) in (12), MnFeNiCu
(VEC${}_{\mathrm{ave}}=9.0$) in (13), and MnCoNiCu
(VEC${}_{\mathrm{ave}}=9.25$) in (14) at Fig. S-2 show a scatter distribution.

Acknowledgements

This study was supported by the computational resources of the HPCI system
(Project ID: hp190014, hp190059, hp200040, hp210019), the JHPCN system
(Project ID: jh190019), Institute for Materials Research, Tohoku University
(Proposal No. 19S0501, 20S0505, 202012-SCKXX-0506), JAIST, and NIMS. H.M.
thanks the Korea Institute of Science and Technology (Grant Nos. 2Z05840, No.
2E31201, No. 2N62110) for providing the financial support for this research.
R.S. acknowledges the support from the Council for Science, Technology and
Innovation under the Cross-ministerial Strategic Innovation Promotion Program.
K.H. acknowledges the financial support from MEXT-KAKENHI (Grant Nos.
JP19K05029, JP19H05169) and the Air Force Office of Scientific Research (Award
Number: FA2386-20-1-4036).

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Table of Contents

Figure S-2: *

Phase stability of 3d-transition metal high-entropy alloys (HEAs) is realized
by an order–-disorder “competitive cooperation,” rather than a usual
order–-disorder competition. The former was newly proposed in this study,
while the latter has already been accepted in the HEA community.

(a)

(b)

(c)

(d)

(e)  Figure S–2:  Formation free energy as a function of temperature for 15
kinds of quaternary alloys, which are listed at Table I. Blue, red and green
lines indicate RSS, D022, and L12 ordered phases, respectively.

(f)

(g)

(h)

(i)

(j)  Figure S–2:  Distribution of the formation free energies at 1,000 K for
15 kinds of quaternary alloys, which are listed at Table I using 100 samples.
Red, orange, green, cyan, blue and purple indicate Cr, Mn, Fe, Co, Ni, and Cu,
respectively. Dark and light colors indicate X-L12, X-D022 phases,
respectively. Black indicates RSS phase.