Design of high-entropy alloys with a single solid-solution phase: Average properties vs. their variances

Accepted Manuscript Design of high-entropy alloys with a single solid-solution phase: Average properties vs. their variances Yiming Tan, Jinshan Li, Shaowu Tang, Jun Wang, Hongchao Kou PII:

S0925-8388(18)30259-7

DOI:

10.1016/j.jallcom.2018.01.252

Reference:

JALCOM 44716

To appear in:

Journal of Alloys and Compounds

Received Date: 30 November 2017 Accepted Date: 17 January 2018

Please cite this article as: Y. Tan, J. Li, S. Tang, J. Wang, H. Kou, Design of high-entropy alloys with a single solid-solution phase: Average properties vs. their variances, Journal of Alloys and Compounds (2018), doi: 10.1016/j.jallcom.2018.01.252. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Design of high-entropy alloys with a single solid-solution phase: Average properties vs. their variances Yiming Tan, Jinshan Li*, Shaowu Tang, Jun Wang, Hongchao Kou

Xi’an, Shaanxi 710072, P.R. China Abstract

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State Key Laboratory of Solidification Processing, Northwestern Polytechnical University,

The empirical thermo-physical parameters for designing of high-entropy alloys (HEAs) with a

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single solid-solution phase (SSP) are based on the average properties of constituent elements, significant derivation from which might make them out of work. In the current work, both the

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important roles of average properties and their variances were studied. After an introduction of the variance of dimensionless enthalpy of mixing to characterize the chemical bond mismatch and a comparative study between two typical collections of HEAs to show the validity of previous designing rules, it was found that all the previous parameters to some extent fail to predict the formation of HEAs with a single SSP whereas the variance of dimensionless enthalpy of mixing is

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able to separate the regions of HEAs with a single SSP, with multi-phases and with an amorphous phase. Application to three typical HEA systems proved further that the lower value of the variance of dimensionless enthalpy of mixing corresponds to the higher possibility to form HEAs

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with a single SSP. The current work provides some necessary conditions for designing of HEAs, based on which preparation of potential HEAs with a single SSP could be accelerated. High-entropy

alloys;

Atom

size

difference;

Chemical

bond

mismatch;

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Keywords:

Thermodynamics; Designing rules.

* Corresponding author E-mail address: [email protected] (Jinshan Li) 1

ACCEPTED MANUSCRIPT 1. Introduction High-entropy alloys (HEAs) were reported independently by Cantor et al. [1] and Yeh et al. [2] in 2004 and from then on become a new approach for designing of alloys [3-5]. They make our attention change from the conventional alloys with only one principal element such as iron-based,

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copper-based, aluminum-based and nickel-based alloys to the alloys with multiple elements in equimolar or near-equimolar ratios [2], i.e., from the corner of phase-diagrams to the center where a huge number of alloys are potentially amenable for preparation. The change of configurational entropy for the formation of an ideal solid-solution phase (SSP) from n components is [2]: n

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∆Sconfig = − R ∑ ci ln ci . i =1

(1)

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Here R is the gas constant and ci is the molar fraction of component i . ∆Sconfig is maximum max = R ln n . In this sense, it is quite possible when the components are in equimolar ratios, i.e. ∆Sconfig

that a SSP with many principal elements could be more stable than other phases, e.g., the intermetallic compounds, being different to our understanding of the concentrated binary and ternary alloys in which many intermetallic compounds can be found. Since the HEAs with a single

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SSP could have superb properties (e.g. the CoCrFeNiMn HEA with a single face-centered cubic (FCC) SSP has an exceptional damage tolerance and its mechanical properties can be improved further at the cryogenic temperatures [6,7], the COCrFeNiPdx HEAs with a single FCC SSP

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exhibit a second order magnetic phase transition whose critical temperature is tunable from 100 K to well above the room temperature and can be used as potential commercially viable magnetic

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refrigerants [8,9] etc.), their designing has been paid much attention to [10-47]. From a very recent review of Gao et al. [10], HEAs can be designed using three typical methodologies, i.e., empirical thermo-physical parameters, the CALPHAD (calculation of phase diagrams) method and the first-principles density functional theory (DFT) calculations combined with hybrid Monte Carlo/Molecular Dynamics (MC/MD) simulations. Regarding that the CALPHAD method as an extrapolation of binary and ternary systems to high order systems [11-13] is not always satisfied and both the CALPHAD method and the DFT calculations combined with hybrid MC/MD simulations [14-19] need a large number of computing workload, the empirical thermo-physical parameters as an easy way to design of HEAs were studied 2

ACCEPTED MANUSCRIPT intensively [20-47]. The empirical parameters were generally proposed according to the Hume-Rothery rules [48] that were used to describe the stability of a SSP in binary alloys and the thermodynamic model of Takeuchi and Inoue [49-51] that was used to find new compositions of bulk metallic glasses

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(BMGs) with multi-components. From the Hume-Rothery rules [48], the stability of a SSP is controlled by the atomic size difference, the electronegativity difference and the electron concentration. Assuming that the difference between the Gibbs free energy of solid phase and liquid phase is proportional to the Gibbs free energy of mixing from n components, i.e., ∆Gmix = ∆H mix − T ∆S mix ,

SC

(2)

the thermodynamic model of Takeuchi and Inoue [49-51] indicates that a SSP is stable when the

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entropy of mixing is able to overcome the driving forces for ordering and segregation. Here ∆H mix and ∆Smix are the mixing of enthalpy and entropy, respectively.

Despite of their great success [20-47], the empirical parameters are generally based on the average thermo-physical properties of constituent elements, significant derivation from which actually may make the empirical design rules out of work. For example, in the work of Guo et al.

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[23] aiming to describe phase selection between the SSP and the amorphous phase, it was suggested that phase selection is essentially determined by the amount of atomic pairs with high negative enthalpy of mixing if the SSP is topologically un-favored. Sohn et al. [52] prepared a

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series of noble metal HEAs and found that all their HEAs follow the design rule of Yang and Zhang [29] but only two of them exhibit a one-phased structure. For the AuPdAgPt HEA, the

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ijmax ijmin highest positive enthalpy of mixing ∆H mix and the highest negative enthalpy of mixing ∆H mix

are 4 kJ mol-1 and -7 kJ mol-1. Addition of Cu to the AuPdAgPt HEA forms a second FCC phase because the pairs of Pt-Cu and Pd-Cu have higher negative enthalpies of mixing (i.e. ijmin ij ∆H mix = −14 kJ mol-1). Further addition of Ni that has a positive ∆H mix with Ag and Au to ij cancel the effect of large negative ∆H mix from the Pt-Cu and Pd-Cu pairs results in a two-phased

structure in the AuPdAgPtCuNi HEA, whereas, introduction of pairs of moderately positive ij by substituting Au and Ag with Rh and Ir leads to a one-phased structure in the ∆H mix

PdPtRhIrCuNi HEA. Similarly, the CoCrFeMnNiPdx (x=0.2-2 at.%) eutectic HEAs follows all the 3

ACCEPTED MANUSCRIPT current design rules of HEAs with a single SSP but exhibits a two-phased structure [53,54]. The ijmax ijmin highest positive enthalpy of mixing ∆H mix and the highest negative enthalpy of mixing ∆H mix

are 2 kJ mol-1 for the Cr-Mn pair and -23 kJ mol-1 for the Mn-Pd pair. It is also the high enthalpy of mixing of the Mn-Pd pair that forms the dual FCC and MnxPdy phases. All the work [23,52-54]

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shows that mere consideration of average properties of constituent elements is not sufficient to the design of HEAs with a SSP.

In the current work, not only the average properties of constituent elements but also their variances were adopted to design of HEAs with a SSP. First, the thermo-physical parameters in the

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previous work are summarized (Sec. 2). Similar to the variance of atomic radius [20] or atomic radiuses in pairs [33] for the atomic size difference, the variance of dimensionless enthalpy of

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mixing was introduced here to characterize the chemical bond mismatch. After a comparative study between the collections of HEAs by Gao et al. [10] and Ye et al. [44], it was found that all the previous parameters to some extent fail to predict the formation of HEAs with a single SSP, whereas, the current parameter, i.e., the variance of dimensionless enthalpy of mixing, could separate qualitatively the regions of HEAs with a single SSP, with multi-phases and with an

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amorphous phase (Sec. 3). The current work shows that both the average properties and the individual properties might be un-favored for designing of HEAs (Sec. 4.1) and both the noble metal HEAs of Sohn et al. [52] and the CoCrFeMnNiPdx (x=0.2-2 at.%) eutectic HEAs [53,54]

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can be predicted successfully by the variances of atomic radius and dimensionless enthalpy of mixing (Sec. 4.2). Finally, the current work is summarized in Sec. 5.

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2. Thermo-physical parameters 2.1. The ∆H mix ~ δ r diagram

n

For a HEA with n components, the average atomic radius is r = ∑ ci ri with ri the atomic radius i =1

of component i . The variance of atomic radius or the comprehensive effect of atomic size difference can be given as [20]:

δr =

2

 r ci  1 − i  . ∑  r i =1 n

(3)

The chemical compatibility among all the components is described by the mixing of enthalpy [20]: 4

ACCEPTED MANUSCRIPT n

n

n

n

ij ∆H mix = ∑∑ Ωij ci c j = ∑∑ 4 H mix ci c j . i =1 j > i

(4)

i =1 j > i

A plot of the ∆H mix ~ δ r diagram allows Zhang et al. [20] to conclude that SSPs are formed when the atomic size difference is relative small so that the atoms of different components have similar probability to occupy the lattice sites and ∆H mix is not so negative that intermetallic compounds

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cannot be formed. After calculating ∆H mix and δ r for HEAs in equimolar ratios for 73 elements, Takeuchi et al. [21] found that the ∆H mix ~ δ r diagram is useful for finding out HEAs with a SSP as well as high entropy BMGs. Guo et al. [22,23] analyzed δ r , ∆H mix , ∆Smix , ∆χ

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(electronegativity) and VEC (valence electron concentration) and found that the ∆H mix ~ δ r

2.2. The Ω ~ δ r diagram

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diagram alone is able to describe phase selection between the SSPs and the amorphous phase1.

Regarding that the negative ∆H mix makes different components combine to form intermetallic compounds while the positive ∆H mix makes mix of different components difficult and leads to segregation in solid, the absolute value of ∆H mix was taken to describe the resistance for the

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formation of a single SSP [29]. Meanwhile, the high value of ∆Smix can lower significantly ∆Gmix and reduces considerably the tendency of ordering and segregation. In other words, the

competition between T ∆S mix and ∆H mix

determines the formation of SSPs, according to which

Tm ∆Smix , ∆H mix

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Ω=

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Yang and Zhang [29] proposed a parameter Ω , i.e., (5)

where the melting temperature of n-component alloy Tm is calculated by the rule of mixture: n

Tm = ∑ ci Tmi ,

(6)

i =1

with Tmi the melting temperature of component i . In the case that ∆Smix is given by the change of ideal configurational entropy, i.e. Eq. (1), a plot of the δ r ~ Ω relation allows them to propose

1

Even though VEC is not a good parameter to design of HEAs with a SSP [22], it is useful to describe the

stability of FCC and BCC phases in HEAs [24], to design of intrinsically ductile refractory HEAs [25], to control the formation of sigma phase in HEAs [26,27] and to design of HEAs with high hardness [28]. 5

ACCEPTED MANUSCRIPT the designing rules of HEAs with a SSP, i.e. Ω ≥ 1.1 and δ r ≤ 6.6% 2. The Ω ~ δ r diagram has been widely used for designing of not only HEA with a single SSP [31] but also HEAs with an amorphous phase [32]. 2.3. Geometrical parameters

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Wang et al. [33] tried to present a physical description of lattice distortions in HEAs. If the average atomic radius is taken to be the reference state, 1 − ri r

is the dimensionless

displacement of an atom of component i from an ideal atom. The total displacement is: n



r

i =1





α1 = ∑ ci  1 − i  . r

SC

(7)

Regarding that a single atom cannot reflect the local lattice distortion, a group of adjacent atoms

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should be used. In the case of atom pairs, the dimensionless displacement between an atom pair of atoms from component i , j and its counterpart pair is 1 − ( ri + rj ) 2r . The total displacement is: n



n

α 2 = ∑ ∑ ci c j  1 − 

i =1 j > i

ri + r j  . 2r 

(8)

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In a similar way, they defined a series of parameters to describe the average local distortion between the clusters of corresponding atoms and their counterpart of ideal lattice, i.e., α 3 , α 4 ,

α 5 and so on. It was found that α1 overestimates the local lattice distortion and α 2 can

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be defined as:

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represent other high order parameters. Similar to δ r , the variance of atomic radiuses in pairs can

δd =

 ri + rj  ci c j 1 −  , ∑∑ 2r  i =1 j > i  n

n

2

(9)

which is highly related to the intrinsic elastic strain energy in HEAs [33]. In another work of Wang et al. [34], the solid angles of atomic packing for the elements with the largest and smallest atomic sizes ( ωL , ωS ) were chosen to describe the atomic packing effect in 2

Poletti and Battezzati [30] classified and predicted the HEAs by both electronic (e.g. ∆χ , VEC and e/a

(itinerant electron concentration)) and thermodynamics parameters. The critical spinodal temperature TSC was calculated to proposed a new parameter µ = TSC Tm . Even though there is a tendency to have a single SSP at higher

µ with respect to two SSPs, the region of HEAs with a single SSP overlaps that of HEAs with multi-phases; see Fig. 6 in Ref. [30]. 6

ACCEPTED MANUSCRIPT HEAs and their ratio was proposed to be an indictor to reveal the atomic packing mismatch and topological instability, i.e., γ =

ωS  = 1− ωL  

( rs + r ) − r 2  2 ( rs + r )  2

 1 −  

( rL + r ) − r 2  . 2 ( rL + r )  2

(10)

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As a result, γ < 1.175 was suggested to be one of the necessary rules for designing of HEAs with a single SSP. Meanwhile, Ye et al. [35,36] proposed a geometrical model for intrinsic residual strain in HEAs. The root mean square residual strain ε RMS that measures the degree of fluctuation in the intrinsic residual strains from their mean value and relates highly to the elastic

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energy stored in HEAs was found to be able to predict the formation of BMGs and the transition from a single SSP to multi-SSPs. Regarding that a high value of ∆Smix prefers the formation

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SSPs while a high value of δ r promotes the formation of intermetallic compounds, a parameter Λ = ∆Smix δ r 2 was introduced and it was found that HEAs with a single SSP can be formed when Λ > 0.96 [37]. The Λ -parameter can be calculated from purely geometrical information such as

the configuration on a lattice and the atom radius.

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2.4. Mismatch entropy due to the atom size difference

All the geometrical parameters above mentioned, e.g., δ r , δ d and γ , are actually different expressions to characterize the atom size difference. Even though their high relationship with the

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strain energy was mentioned [33-37], the geometrical parameters are not a way to describe the atom size difference thermodynamically. After introducing the excess entropy of mixing that is a

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function of atomic size and atomic packing S E (i.e. mismatch entropy) to ∆Smix , i.e., ∆Smix = ∆Sconfig + S E , Takeuchi et al. [38] found that the ∆H mix ~ δ r diagram is similar to the ∆H mix ~ S E / k B diagram ( k B is the Boltzmann constant), indicating that δ r and S E are highly

correlated. According to the work of Mansoor et al. [55], S E can be given as: S E F − F id −2 −3 = − ln Z − ( 3 − 2ξ )(1 − ξ ) + 3 + ln (1 + ξ + ξ 2 − ξ 3 ) (1 − ξ )  ,   kB k BT

(11)

where F − F id 3 3 1  −1 −2 = − (1 − y1 + y2 + y3 ) + ( 3 y2 + 2 y3 )(1 − ξ ) + 1 − y1 + y2 − y3  (1 − ξ ) + ( y3 − 1) ln (1 − ξ ) , kBT 2 2 3 

(12) 7

ACCEPTED MANUSCRIPT −3 Z = (1 + ξ + ξ 2 ) − 3ξ ( y1 + y2ξ ) − ξ 3 y3  (1 − ξ ) ,

(13)

y1 = ∑ ∑ ∆ ij ( d i + d j )( d i d j )

(14)

n

n

−1/ 2

,

i =1 j > i

ξ ( di d j ) y2 = ∑∑ ∆ ij ∑ k dk i =1 j >i k =1 ξ n

−1/ 2

n

,

(15)

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n

3

 n  ξ  2/ 3  y3 =  ∑  i  ci1/ 3  ,  i =1  ξ  

+ dj )

1/ 2

∆ ij

i

j

ξ

n

n

i =1

i =1

i

2

(c c )

1/ 2

i

di d j

j

,

(17)

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(ξ ξ ) ( d =

(16)

1 6

(18)

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ξ = ∑ ξi = ∑ πρ ci d i 3 .

Here d i ( = 2ri ) is the atomic diameter of component i , ρ is the number density and ξ is the overall atomic packing fraction for a given ρ .

Noting from the Gibbs free energy of mixing Eq. (2) that if ∆Gmix is dominated by ∆Smix , S E ∆Sconfig << 1 − ∆H mix T ∆Sconfig , Ye et al. [39] proposed two

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∆H mix T ∆Smix << 1 , i.e.,

dimensionless thermodynamic parameters, i.e. SE ∆Sconfig and ∆H mix T ∆Sconfig , to design of HEAs with a single SSP. Furthermore, they defined the complementary entropy for the enthalpy of

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mixing, i.e., SH = ∆H mix Tm , and proposed a single-parameter thermodynamic rule for designing of HEAs [40]3, i.e., ∆Sconfig − S H

.

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Φ=

SE

(19)

A critical value of Φ * ≈ 20 was found, larger (smaller) than which corresponds to HEAs with a single SSP (multi-phase). In another work of Ye et al. [44], all the parameters, i.e., δ r , ∆H mix , SC , VEC , Φ and ε RMS were calculated and the predictions of Φ were found to be similar to that

of ε RMS . 3

It should be pointed out that the parameters Ω and Φ were both proposed from the idea that the entropy of mixing is able to override the enthalpy of mixing. Other thermodynamic principles, e.g. the minimum of Gibbs free energy of mixing corresponds to the compositions of stable alloy [41] and a single SSP is stable if its mixing of Gibbs free energy is smaller than all of the possible intermetallic compounds in the alloy system [42,43], were also used. 8

ACCEPTED MANUSCRIPT 2.5. Configurational entropy of non-ideal mixing The ideal configurational entropy ∆Sconfig in Eq. (1) actually comes from the ideal mixing rule, i.e. the Boltzmann equation which is a measure of the available microstates corresponding to the same macrostate [45]. Based on the Gibbs entropy, He et al. [45] derived a model to describe the

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configurational entropy of mixing by considering the possible correlations among the constituent elements due to various factors such as atomic size difference and chemical bond mismatch, which may disturb the potential energy of the system and thus reduce the configurational entropy of mixing. In the case that the number of the microstates for each macrostate remains on the same

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order of magnitude, one has:

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n  x x 1 + e− x  E ideal ∆Sconfig = ∆Sconfig + ∆Sconfig = − k B ∑ ci ln ci + k B 1 + − ln x + ln (1 − e − x ) − , 2 1 − e− x  i =1  2

(20)

where x = ∆ε kBT with ∆ε the range of energy fluctuation is the normalized energy fluctuation. For the energy fluctuation caused by atom size difference: KV . k BT

xe = 3 2ε RMS

(21)

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Here K is the average bulk modulus and V is the average atomic volume. For the energy fluctuation caused by chemical bond mismatch: ij − ∆H mix ) ∑∑ ci c j ( H mix n

i =1 j > i

2

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xc = 2

n

k BT

.

(22)

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Other kinds of energy fluctuation such as magnetism may also change the configurational entropy of mixing and can be also described by Eq. (20) [46] 4. It was found that the configurational entropy of mixing decreases more dramatically for HEAs with multi-phases and an amorphous ideal phase than HEAs with a single SSP [45]. If the ratio ∆Sconfig ∆Sconfig was calculated, HEAs with a

ideal <1. single SSP can be found when 0.85 < ∆Sconfig ∆Sconfig

2.6. Variance of dimensionless enthalpy of mixing As mentioned, the atomic size difference can be described by the variance of atomic radius δ r

4 It should be noted that Eq. (20) is valid for small perturbations [45]. Furthermore, the non-ideal formation of configurational entropy of mixing was found to be theoretically connected with the Hume-Rothery rules [47]. 9

ACCEPTED MANUSCRIPT (i.e., Eq. (3)) or the variance of atomic radiuses in pairs δ d (i.e., Eq. (7)). δ r is highly correlated to the mismatch entropy due to the atomic size difference S E in Eq. (11) [36] and δ d is directly related to the intrinsic elastic strain energy [33]. Regarding the importance of atomic size difference in designing of HEAs, the current work introduces here the variance of

∑∑ c c ( H n

δ H mix =

xc 4

2

i =1 j > i

=

T = Tm

n

i

j

ij mix

k B Tm

− ∆H mix )

2

,

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dimensionless enthalpy of mixing, i.e.,

(23)

SC

which is actually the square of xc 2 at T = Tm in Eq. (22), to describe the chemical bond mismatch. Regarding the fact that the negative enthalpy of mixing makes different components

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combine to form intermetallic compounds while the positive enthalpy of mixing makes mix of different components difficult and leads to segregation in solid, it might be of physical significance to re-define the variance of dimensionless enthalpy of mixing as: ij − 0) ∑∑ ci c j ( H mix n

0 δ H mix =

n

2

i =1 j > i

,

k BTm

(24)

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ij the enthalpy of mixing of component i , j In other words, it is the difference between H mix

and 0 but not the average enthalpy of mixing ∆H mix that is more appropriate to characterize the

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resistance for the formation of a single SSP. Similarly, the resistance for ordering and segregation can be characterized by:

∑∑ c c ( H n

i

j

ij + mix

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n

0+ δ H mix =

i =1 j > i

2

k BTm

∑∑ c c ( H n

0− δ H mix =

− 0)

n

i =1 j > i

i

j

k BTm

ij − mix

− 0)

.

(25)

,

(26)

2

The superscripts ‘+’ and ‘-’ stand for the positive and negative enthalpy of mixing, respectively. 3. Results In this section, some typical empirical parameters that describe solely the atom size difference (e.g., δ r , δ d , γ ), both the atom size difference and the chemical bond mismatch (e.g., Φ ) and 10

ACCEPTED MANUSCRIPT solely the chemical bond mismatch (e.g., ∆H mix , Ω , 5

δ H mix ,

0 , δ H mix

0+ and δ H mix

0− δ H mix

) are calculated. Two collections of HEAs, i.e., Ye et al. [44] including 33 HEAs with a single

FCC SSP, 7 HEAs with a single BCC SSP, 4 HEAs with a single hexagonal close packed (HCP) SSP, 37 HEAs with multi-phases and 9 HEAs with an amorphous phase, and Gao et al. [10]6

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including 16 alloys with a single FCC SSP, 32 alloys with a single BCC SSP, 4 alloys with a single HCP SSP, 216 alloys with multi-phases and 59 alloys with an amorphous phase, are chosen, through a comparative study between which the importance to consider their variances of the average properties of constituent elements is highlighted. The calculation results are summarized

SC

in Table S1 and S2 in the supplementary material. In what follows, the regions of HEAs with a single SSP, with multi-phases and with an amorphous phase in the diagrams are briefly denoted as

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the single-SSP-region, the multi-phase-region and the amorphous-phase-region, respectively. 3.1. Atom size difference

Fig. 1 shows the variance of atomic radiuses in pairs δ d vs. the variance of atomic radius δ r . For the collection of HEAs of Ye et al. [44], both δ r and δ d are able to distinguish the three

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regions, even though some of the single-SSP-region and the multi-phase-region coincide with each other; see Fig. 1a. It should be noted that the number of HEAs is not large enough to make δr

both

and

δd

change

continuously

from

the

multi-phase-region

to

the

amorphous-phase-region. This is not the case for the collection of HEAs of Gao et al. [10]

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including more reported HEAs. In this case, the multi-phase-region covers almost all of the single-SSP-region and the amorphous-phase-region; see Fig. 1b. For δ r , the single-SSP-region

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and the amorphous-phase-region are separated clearly, whereas for δ d , these two regions almost does not coincide with each other. A critical value of δ d * = 0.03 can be found, above (below) which is the amorphous-phase-region (the single-SSP-region). Fig. 2 shows the ratio of the solid angles of atomic packing for the elements with the largest and 5

It should be pointed out that it is not the variance of dimensionless enthalpy of mixing but its square root (i.e., δ H mix

,

0 δ H mix

,

0+ δ H mix

and

0− ) that is calculated to make the thermo-physical parameters for the δ H mix

atom size difference the chemical bond mismatch comparable with each other; see Eqs. (18) and (19). 6

It should be pointed out that the collection of HEAs of Gao et al. [10] includes typical binary and/or ternary

alloys in equimolar ratios with a single FCC SSP, a single BCC SSP, a single HCP SSP or multi-phases, considering which or not does not influence the conclusions of current work. 11

ACCEPTED MANUSCRIPT smallest atomic sizes γ vs. the variance of atomic radius δ r . Similar to δ r and δ d , γ is able to distinguish the three regions for the collection of HEAs of Ye et al. [44]; see Fig. 2a. Some of the single-SSP-region and the multi-phase-region coincide with each other. The gap between the multi-phase-region and the amorphous-phase-region is so small that a critical value of can be defined above (below) which is the amorphous-phase-region (the

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γ * =1.27

multi-phase-region and the single-SSP-region). As for the collection of HEAs of Gao et al. [10], the predictions of γ are the same as that of δ d except that the single-SSP-region and the

SC

amorphous-phase-region can be separated accurately. A critical value of γ * =1.27 can be found above (below) which is the amorphous-phase-region (the single-SSP-region). In contrast to Wang

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et al. [34] in which γ < 1.175 was found to be one of the necessary rules for designing of HEAs with a single SSP, the current work shows that HEAs with a single SSP can be formed if γ < 1.27 . 3.2. Atom size difference plus chemical bond mismatch

Fig. 3 shows the Φ parameter vs. the variance of atomic radius δ r 7. For the collection of HEAs of Ye et al. [44], the single-SSP-region and the multi-phase-region can be found when Φ > 14

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and 3.6 < Φ < 20 , respectively; see Fig. 3a. In other words, a critical value of Φ * = 20 can be defined above which forms the HEAs with a single SSP [44]. As the number of HEAs with an amorphous phase 9 is small and five of them are not shown in Fig. 3a due to the fact that Φ < 0 ,

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the amorphous-phase-region is not marked. As for the collection of HEAs of Gao et al. [10], the single-SSP-region locates at

Φ > 14

and the multi-phase-region covers all of the

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single-SSP-region and almost all of the amorphous-phase-region. Even though δ r , δ d , γ

and Φ can separate roughly the single-SSP-region, the

multi-phase-region and the amorphous-phase-region for the collection of HEAs of Ye et al. [44], they are out of work for the collection of HEAs of Gao et al. [10], e.g., the multi-phase-region covers almost all of the single-SSP-region and the amorphous-phase-region; see Figs. 1-3. In other words, all the previous empirical parameters that describe the atom size difference or both the atom size difference and the chemical bond mismatch fail to predict the stability of a SSP in HEAs. Anyway, δ d and γ are still very good parameters to separate the single-SSP-region and the 7

The Φ parameter is set to be an average value of the cases of ξ = 0.74 and ξ = 0.68 [39,44]. 12

ACCEPTED MANUSCRIPT amorphous-phase-region; see Figs. 1 and 2. 3.3. Chemical bond mismatch Fig. 4 is the ∆H mix ~ δ r diagram. For the collection of HEAs of Ye et al. [44], the three regions overlap with each other; see Fig. 4a. More than half of the single-SSP-region and the

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multi-phase-region are overlapped and more than half of the multi-phase-region is covered by the amorphous-phase-region. For the collection of HEAs of Gao et al. [10], the three regions also overlap with each other; see Fig. 4b. All of the single-SSP-region is covered by the multi-phase-region and about two thirds of the amorphous-phase-region overlaps with the

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multi-phase-region. The ∆H mix ~ δ r diagram therefore is not a good choice for designing of HEAs, even though it shows that a high negative mixing of enthalpy prefers the formation of

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amorphous-phase while a small mixing of enthalpy tends to form a single SSP.

The Ω ~ δ r diagram is shown in Fig. 5. For both the collections of HEAs of Ye et al. [44] and Gao et al. [10], the predictions of Ω are better than that of ∆H mix . There is no overlap between the single-SSP-region and the amorphous-phase-region and only less than two thirds of the

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single-SSP-region are covered by the multi-phase-region; see Fig. 5a. The overlap between the single-SSP-region and the amorphous-phase-region is not that serious but all of the single-SSP-region is still covered by the multi-phase-region; see Fig. 5b. A combination of Figs. 4b and 5b indicates that both ∆H mix and Ω are not able to distinguish the single-SSP-region

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and the amorphous-phase-region. In other words, the average properties of constituent elements alone cannot characterize accurately the chemical bond mismatch and its applicability depends

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highly on the collection of HEAs. Fig. 6 shows the square root of the variance of dimensionless enthalpy of mixing

δ H mix vs.

the variance of atomic radius δ r . One can see that besides the Al20Li20Mg10Sc20Ti30 HEA with a single HCP SSP prepared by mechanical alloying [56] in Fig. 6a,

δ H mix increases continuously

from the single-SSP-region to the multi-phase-region and then to the amorphous-phase-region for both the collections of HEAs of Ye et al. [44] and Gao et al. [10]. Even though the three regions overlap with each other, they can be separated roughly by

δ H mix as compared with parameters

13

ACCEPTED MANUSCRIPT δ r , δ d , γ , Φ , ∆H mix and Ω , indicating that the variance of dimensionless enthalpy of

mixing is a useful parameter for designing of HEAs. The square root of the variance of dimensionless enthalpy of mixing of atomic radius δ r is shown in Fig. 7. The prediction of

0 as a whole is similar to that δ H mix

δ H mix but a close look shows that for both the collections of HEAs of Ye et al. [44] and Gao

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of

0 vs. the variance δ H mix

et al. [10], less of the single-SSP-region is covered by the multi-phase-region and the 0 with 0 but not the enthalpy of mixing ∆H mix as the δ H mix

reference value predicts better than

δ H mix ; see Figs. 6 and 7. According to the collection of

HEAs

single-SSP-region,

et

al.

[10],

amorphous-phase-region

the

locate

at

the

multi-phase-region

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Gao

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amorphous-phase-region, indicating

0 0 < δ H mix < 1.63

,

and

0 0.54 < δ H mix < 2.32

the and

0 1.35 < δ H mix < 3.01 , respectively; see Fig. 7b. Therefore, one of the necessary condition for the

formation of HEAs with a single SSP is

0 δ H mix < 1.63 .

0+ δ H mix and

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Figs. 8 and 9 summarize the square root of the variance of dimensionless enthalpy of mixing 0− δ H mix vs. the variance of atomic radius δ r . For both the collections of HEAs of

Ye et al. [44] and Gao et al. [10], the formation of HEAs with a single SSP, with multi-phases and

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with an amorphous phase seems have nothing to do with (relates highly to) the chemical bond mismatch of atom pairs with positive (negative) enthalpy of mixing. Furthermore, the prediction 0− δ H mix is comparable with that of

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of

δ H mix but is not as good as that of

6,7 and 9. It should be pointed out that if

+ δ H mix and

0 δ H mix ; see Figs.

+ − − δ H mix as well as ∆H mix and ∆H mix

are adopted, similar results as Figs. 8 and 9 can be found, indicating the primary condition for the formation of a single SSP in HEAs is to avoid ordering but not segregation. In this sense, the thermodynamic principle that a single SSP is stable if their mixing of Gibbs free energy is smaller than all of the possible intermetallic compounds in the alloy system [42,43] might be of general meaning, even though it does not consider the effect of segregation in HEAs. 4. Discussion 14

ACCEPTED MANUSCRIPT 4.1. Variances of average properties vs. extremums of individual properties In principle, deviation from the average properties can be characterized by their variances as what have done in the current work and the maximum and/or minimum of the properties of constituent elements, i.e., extremums of individual properties. In Sec. 3, it has been shown that the variance of

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atomic radiuses in pairs δ d and the variance of atomic radius δ r cannot provides clear conditions for designing of HEAs with a single SSP but δ d can separate the single-SSP-region and the amorphous-phase-region with a critical value of δ d * = 0.03 (Fig. 1), whereas, the square root of the variances of enthalpy of mixing δ H mix ,

0 δ H mix

and

0− δ H mix

are able to

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distinguish roughly the three regions (Figs. 6, 7 and 9). Furthermore, γ as the ratio of the solid

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angles of atomic packing for the elements with the largest and smallest atomic sizes is actually a parameter characterize the deviation from the average properties by individual properties. As a result, a critical value of γ * =1.27 is found above (below) which is the amorphous-phase-region (the single-SSP-region), even it cannot distinguish the three regions. In other words, individual properties could also be a good choice to show the deviation from the average properties.

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In the case of chemical bond mismatch, Guo et al. [23] pointed out the more negative enthalpy of mixing between Zr and other elements, e.g., -41 kJ mol-1 for Zr-Co, -49 kJ mol-1 for Zr-Ni and -23 kJ mol-1 for Zr-Cu, makes the formation of Zr-containing intermetallic compounds and

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prevents the formation of a fully amorphous structure in the AlCoCrFeNiZr and CoCrCuFeNiZr HEAs. Similarly, the formation of a two-phased structure in the AuPdAgPtCu and AuPdAgPtCuNi

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HEA as mentioned in Sec. 1 is ascribed to the more negative enthalpy of mixing between Cu and ij Pd, Cu and Pt, i.e. ∆H mix = −14 kJ mol-1 [52]. The formation of CoCrFeMnNiPdx (x=0.2-2 at.%) ij eutectic HEAs is due to the Mn-Pd pair with the highest negative enthalpy of mixing ∆H mix = −23

kJ mol-1 [53,54]. All the work [23,52-54] shows that the extremums of individual properties as well as their amount play important roles in designing of HEAs. Fig. 10 (11) shows the contribution from the maximum (minimum) enthalpy of mixing ijmax ijmin ci c j ∆Hmix ( ci c j ∆H mix ) vs. the variance of atomic radius δ r . One can see that the formation of a

single SSP, multi-phases and an amorphous phase in HEAs seems have nothing to do with the

15

ACCEPTED MANUSCRIPT contribution from the maximum enthalpy of mixing and this is the case for both the collections of HEAs of Ye et al. [44] and Gao et al. [10]. As for the contribution from the minimum enthalpy of mixing, the HEAs with a single phase (an amorphous phase) trends to have small (large) negative ijmin ci c j ∆H mix , being inconsistent with the conclusions of Guo et al. [23]. Similarly, one can calculate

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ij the extremums of individual properties, e.g., ∆H mix , but one will find similar results as Figs. 10

and 11. In other words, the extremums of individual properties for and individual contributions from atom pairs to chemical bond mismatch could also be very important for designing of HEAs but corresponding applicable parameters should be proposed.

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4.2. Application of statistical designing rules to specific HEA systems

The designing rules proposed from the statistical results of either the average properties or their

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variances are actually the necessary conditions but not the sufficient conditions. Taking the designing rules of Yang and Zhang [29] as an example, both the HEAs with a single SSP and with multi-phases can be formed if Ω ≥ 1.1 and δ r ≤ 6.6% are fulfilled; see Fig. 1 in Ref. [29]. If Ω and δ r are set to be smaller so that overlap between the single-SSP-region and the

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multi-phase-region is avoided, the designing rules in this case are the sufficient conditions which in fact may loss lots of potential HEAs with a single SSP. The statistical results therefore can only provide semi-quantitatively information on designing of HEAs. In what follows, three typical HEA systems are studied to show further the validity of the variance of dimensionless enthalpy of

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mixing as well as the problems in application of the statistical designing rules. To show the relative effects of enthalpy and entropy on the phase stability of HEAs, the

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CoCrFeNiMn HEA of Cantor et al. [1] was chosen to be the base alloy and the Co, Cr, Fe and Ni elements were replaced one at a time by the 3d and 4d transition element [57]. As a result, six HEAs, i.e., CoCrFeNiMn, CoCrFeCuMn, CoMoFeNiMn, CoVFeNiMn, TiCrFeNiMn and CoCrVNiMn were prepared and the as-cast ingots were annealed at 1123 K and 1273 K for 3 days. One can see form Table 1 that all of the six HEAs follow the designing rules of Yang and Zhang [29] but only one is the HEA with a single SSP. Similarly, only the CoVFeNiMn HEA with Φ = 5.01 violates the single-parameter thermodynamic rule of Ye et al. [40] (i.e., Φ > 20 ) and it

is not the CoCrFeCuMn HEA with the highest value of Φ = 39.49 that exhibits a one-phased 16

ACCEPTED MANUSCRIPT structure. As for the variance of dimensionless enthalpy of mixing, the minimum value of 0 δ H mix = 0.17 is achieved by both the CoCrFeNiMn and CoMoFeNiMn HEAs. The higher

variances of atomic radius and atomic radiuses in pairs of CoMoFeNiMn, i.e., δ r = 4.21 and

δ d = 0.02 , are the reason why it exhibits a two-phased structure.

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ij Regarding that the high negative enthalpy of mixing between Mn and Pd is ∆H mix = −23 kJ

mol-1, CoCrFeNiMnPdx (x=0.1-2) eutectic HEAs were designed by addition of Pd to the CoCrFeNiMn HEA with a single FCC phase [53]. One can see form Table 1 that all the five HEAs follow the designing rules of Yang and Zhang [29], but none of them is the HEA with a single SSP.

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Similarly, the CoCrFeMnNiPd0.2 ( Φ = 31.42 ), CoCrFeMnNiPd0.6 ( Φ = 31.42 ), CoCrFeMnNiPd

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( Φ = 31.42 ) follow the single-parameter thermodynamic rule of Ye et al. [40] but all of them are the HEAs with a eutectic structure. Among the CoCrFeNiMn and CoCrFeNiMnPdx (x=0.1-2) HEAs, the minimum values of the variances of atomic size difference and dimensionless enthalpy of mixing are achieved by the CoCrFeMnNi HEA and, δ r ,

0 increase continuously with δ H mix

the composition of Pd. As for the noble metal HEAs of Sohn et al. [52], all of them follow the

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designing rules of Yang and Zhang [29] and but only the AuPdAgPt and PdPtRhIrCuNi HEAs that follow the single-parameter thermodynamic rule of Ye et al. [40] exhibit a one-phased structure. Furthermore, their square roots of the variances of dimensionless enthalpy of mixing are much

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smaller than that of AuPdAgPtCu and AuPdAgPtCuNi HEAs with two-phased structure. One can see that the designing rules of Yang and Zhang [29] and Ye et al. [40] are necessary but

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not sufficient conditions, even though the latter was shown to have the ability to separate the three regions satisfactorily [40,44] which is however not the case for the current work with more collections of HEAs (Fig. 3). For all of the three specific HEA systems, the variance of dimensionless enthalpy of mixing proposed currently is a good parameter for designing of HEAs with a single SSP, i.e. the lower value of the variance of dimensionless enthalpy of mixing corresponds to the higher possibility to find a HEA with a single phase. 5. Conclusions In this work, the variance of dimensionless enthalpy of mixing was introduced to characterize the chemical bond mismatch and both the important roles of average properties and their variances 17

ACCEPTED MANUSCRIPT were studied. Our main conclusions are as follows. (1) Even though δ r , δ d , γ , Φ , ∆H mix and Ω , can separate roughly the three regions for the collection of HEAs of Ye et al. [44], they are out of work for the collection of HEAs of Gao et al. [10]. All the previous empirical parameters that describe solely the atom size difference, both

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the atom size difference and the chemical bond mismatch and solely the chemical bond mismatch fail to predict the stability of a SSP in HEAs.

(2) Compared with the parameters δ r , δ d , γ , Φ , ∆H mix and Ω , the square root of the 0 is able to separate roughly the three δ H mix

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variance of dimensionless enthalpy of mixing

regions of HEAs and thus is a useful parameter for designing of HEAs. One of the necessary

collection of HEAs of Gao et al. [10].

0 δ H mix < 1.63 according to the

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conditions for the formation of HEAs with a single SSP is

(3) γ is actually a parameter characterizing the deviation from the average properties by individual properties for the atomic size difference and a critical value of γ * =1.27 was found above (below) which is the amorphous-phase-region (the single-SSP-region). The extremums of

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individual properties for or individual contributions from atom pairs to chemical bond mismatch could be also very important for designing of HEAs. (4) The designing rules proposed from the statistical results of either the average properties or

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their variances are actually the necessary conditions but not the sufficient conditions. The current work shows that the primary condition for the formation of a single SSP in HEAs is to avoid

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ordering but not segregation and the lower value of the variance of dimensionless enthalpy of mixing corresponds to the higher possibility to form HEAs with a single SSP, based on which preparation of potential HEAs with a single SSP could be accelerated. Acknowledgements

This work was supported by the financial support from the National Nature Science Foundation of China (51571161 and 51774240), the Natural Science Basic Research Plan in Shaanxi Province of China (2016JQ5003), and the Program of Introducing Talents of Discipline to Universities (B08040). References 18

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23

ACCEPTED MANUSCRIPT Captions of Tables and Figures ijmax Table 1. Calculated parameters ∆S mix (J K-1 mol-1), ∆H mix (kJ mol-1), ∆H mix (kJ mol-1),

ijmin ∆H mix (kJ mol-1), Ω , Φ , δ r , δ d , γ ,

0 for three typical HEA systems. δ H mix

HEAs collected by Ye et al. [44] (a) and Gao et al. [10] (b).

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Fig. 1. The variance of atomic radiuses in pairs δ d vs. the variance of atomic radius δ r for

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Fig. 2. The γ parameter vs. the variance of atomic radius δ r for HEAs collected by Ye et al.

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[44] (a) and Gao et al. [10] (b).

Fig. 3. The Φ parameter vs. the variance of atomic radius δ r for HEAs collected by Ye et al. [44] (a) and Gao et al. [10] (b). It should be pointed out that Φ can be negative especially for HEAs with an amorphous phase and in this case it is not shown in Fig. 3.

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Fig. 4. The mixing of enthalpy ∆H mix vs. the variance of atomic radius δ r (i.e., the ∆H mix ~ δ r diagram) for HEAs collected by Ye et al. [44] (a) and Gao et al. [10] (b).

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Fig. 5. The Ω parameter vs. the variance of atomic radius δ r (i.e., the Ω ~ δ r diagram) for

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HEAs collected by Ye et al. [44] (a) and Gao et al. [10] (b).

Fig. 6. The square root of the variance of dimensionless enthalpy of mixing

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variance of atomic radius δ r for HEAs collected by Ye et al. [44] (a) and Gao et al. [10] (b).

Fig. 7. The square root of the variance of dimensionless enthalpy of mixing

0 vs. the δ H mix

variance of atomic radius δ r for HEAs collected by Ye et al. [44] (a) and Gao et al. [10] (b).

Fig. 8. The square root of the variance of dimensionless enthalpy of mixing

0+ vs. the δ H mix

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Fig. 9. The square root of the variance of dimensionless enthalpy of mixing

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ijmax Fig. 10. The contribution from the maximum enthalpy of mixing ci c j ∆Hmix vs. the variance of

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atomic radius δ r for HEAs collected by Ye et al. [44] (a) and Gao et al. [10] (b).

ijmin vs. the variance of Fig. 11. The contribution from the minimum enthalpy of mixing ci c j ∆H mix

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∆Smix

∆H mix

ijmax ∆H mix

ijmin ∆H mix

Ω

Φ

δr

δd

γ

CoCrFenNiMn CoCrFeCuMn CoMoFeNiMn CoVFeNiMn TiCrFeNiMn CoCrVNiMn CoCrFeMnNiPd0.2 CoCrFeMnNiPd0.6 CoCrFeMnNiPd CoCrFeMnNiPd1.6 CoCrFeMnNiPd2 AuPdAgPt AuPdAgPtCu AuPdAgPtCuNi PdPtRhIrCuNi

13.38 13.38 13.38 13.38 13.38 13.38 14.22 14.78 14.9 14.74 14.53 11.53 13.38 14.9 14.9

-4.16 4.16 -4.00 -13.28 -8.96 -9.12 -5.12 -6.61 -7.67 -8.71 -9.14 -2.00 -6.56 -2.22 -2.56

2 13 5 2 0 2 2 2 2 2 2 4 4 15 6

-8 -5 -8 -35 -18 -18 -23 -23 -23 -23 -23 -7 -14 -14 -14

5.76 5.52 6.50 1.84 2.69 2.73 4.98 4.01 3.49 3.05 2.86 9.28 3.18 10.64 11.57

36.7 39.49 23.08 5.01 25.61 27.59 31.41 24.55 20.75 17.48 16.11 75.07 17.87 16.19 33.58

3.27 3.15 4.21 6.59 3.44 3.33 3.59 4.02 4.29 4.53 4.62 2.23 4.38 5.57 3.88

0.01 0.01 0.02 0.03 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.01 0.02 0.02 0.02

1.10 1.10 1.11 1.20 1.10 1.09 1.12 1.12 1.12 1.12 1.12 1.05 1.14 1.17 1.12

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HEAs

0 δ H mix

0.17 0.28 0.17 0.55 0.34 0.34 0.23 0.30 0.34 0.38 0.39 0.19 0.36 0.37 0.21

Phase FCC [1,57] FCC1+FCC2 [57] FCC+µ [57] FCC+χ+? [57] FCC+σ [57] FCC+σ [57] FCC+Mn2Pd3 [53] FCC+Mn2Pd3 [53] FCC+Mn7Pd9 [53] FCC+Mn7Pd9 [53] FCC+Mn3Pd5 [53] FCC [52] FCC1+FCC2 [52] FCC1+FCC2 [52] FCC [52]

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1. The variance of dimensionless enthalpy of mixing was introduced. 2. All the thermo-physical parameters were summarized and analyzed. 3. The important roles of average properties and their variances were shown.

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4. The extremums of individual properties and their contributions were discussed.