Effect of Ta addition on the structural, thermodynamic and mechanical properties of CoCrFeNi high entropy alloys

Ta addition has considerable effects on the microstructures and mechanical performances of CoCrFeNi alloy systems. Structure search with the special quasirandom structure method and structure identification with first-principles calculations were carried out to investigate the structural, thermodynamic and mechanical properties of CoCrFeNiTax (x = 0.0–1.0) high-entropy alloys in the fcc and bcc lattice frameworks. The predicted lattice parameters of identified structures are in agreement with available experiments. Phase transition between the fcc and bcc lattices was predicted for the lowest-energy structures with increasing Ta content. The predicted temperature dependence of specific heat capacity for the identified structures matches well with the Dulong–Petit, Kepp and Debye Models. Both vibration and configuration entropy contribute to the stabilization of alloy systems, while the latter is about 2–3 times greater than the former. The elastic constants and moduli vary with composition and phase structure. Ta atoms have preference to some atoms like Ni, and form relatively strong bonds with adjacent atoms. The introduction of Ta promotes the electron localization and favors the formation of mixed structures.


Introduction
High-entropy alloys (HEAs) have attracted extensive research attention in the past decade, most of which was aimed at developing a series of alloy systems with novel properties. 1-8 A number of HEAs with unique microstructures, 9,10 predominant mechanical, 11,12 great thermal stability, 13 complex magnetic behavior 14 and brilliant properties of corrosion resistance 6 have been reported. Compared with conventional alloys, HEAs have high entropy of mixing that promote the formation of solid solution phase and intermetallic compounds in simple crystal structures. Three kinds structures, face-centered cubic (fcc), 15 body-centered cubic (bcc) 15 and hexagonal close-packed (hcp) crystallographic structures, 16 have so far been characterized for HEAs.
In recent years, HEAs with broad atomic compositions were investigated based on CoCrFeNi systems. [17][18][19][20][21] Elements like Al, Mo, Ti and Si were successfully added into the CoCrFeNi system to assess their alloying inuences on the structures and properties of the multi-component alloys. [22][23][24][25][26][27][28] Wang et al. 29 studied the transition from paramagnetism to superparamagnetism in the amorphous phase for the CoCrFeNiCuTi x alloys. Liu et al. 30 concluded that the CoCrFeNiMo 0.3 HEA exhibits a tensile strength as high as 1.2 GPa and a good ductility of $19%. Zhang et al. 31 found that the as-cast structure of Al x CoCrFeNiTi tends to be a single bcc phase. Ma et al. 32 reported that the compressive yielding strength and Vickers hardness of AlCoCrFeNb x Ni systems have signicant variations with the addition of Nb. On the computational side, Feng et al. 33 investigated the effects of Mn and Al additions on the structural stability and magnetic properties of FeCoNi-based alloys by means of density functional theory (DFT) calculations at Perdew, Burke, and Ernzerhof (PBE) level. 34 The alloy structures change from fcc to bcc with the increase of Mn and Al content for FeCoNi(MnAl) x alloys. First-principles electronic structure calculations at the PBE level were performed by Zaddach et al. 35 to determine the elastic constants and lattice parameters of NiFeCrCoMn alloys, which are in agreement with the mechanical testing and microstructure characterization. Tian et al. 36 predicted that NiCoFeCrAl x HEAs have excellent micromechanical properties, and strong metallic and ductility characters when x ¼ 1 through ab initio calculations using the PBE functional. Those computational studies have proven that computations are an effective approach to identify the microstructures and properties of alloy systems.
Ta, with a high melting point, 37 was oen added to alloys to enhance their structure stability. Zheng et al. 38 investigated the effect of Ta addition on the stress rupture properties and microstructural stability of Ni-based alloys, verifying that the eutectic phase enhances the stress rupture. Stelmakh et al. 39 synthesized the Ta-W solid solution alloy photonic crystals as spectrally selective components for high-temperature energy conversion. Jiang et al. 37 reported the alloying effects of Ta on the microstructures and mechanical properties of CoCrFeNi alloy, nding a high fracture strength of 2.29 GPa for CoCrFeNiTa 0.4 . It has been proven experimentally that Ta addition has great inuence on the performances of alloys. However, such inuence has been scarcely investigated computationally, especially for the case of Ta-containing HEAs. Computational studies not only predict the microstructures of alloys lack of experimental characterization, but also suggest the correlation between the microstructures and the properties, which are helpful for the design of new alloys with target performances. This work aims to establish the relationship between the microstructures and the performances of CoCrFeNiTa x alloys. The microstructures of CoCrFeNiTa x alloys, including their interatomic bonding, g(r), ELF, and formation energy, etc., were computed and correlated with their performances, such as heat capacity, elastic constant and modulus.

Computational methods
Despite of the single solid solution phase, it is still a great challenge to study HEAs by rst-principles calculations because of their multiple-component and complex structures with numerous possible candidates. One strategy to solve this problem is to use the special quasirandom structure (SQS) 40 method to construct the most disordered HEA structures, which are then sent to subsequent rst-principles calculations for further identication. For example, Feng et al. 33 investigated structural stability of quaternary FeCoNiX (X ¼ Al, Mn) alloys using the model structures screened with the SQS method. Wang et al. 41 applied the rst-principles phonon method to predict the major phase separations for the refractory VNbMo-TaW HEAs which are based on the SQS structures.
In the studied CoCrFeNiTa x structures, the fraction of Ta varies between x ¼ 0.0 and x ¼ 1.0 with an interval of 0.2. It has been noted that lattice structures of HEAs are relevant to their average valence electron concentrations (VEC). 42 As an indicator of phase structures, VEC is oen used to predict that a solid solution adopts fcc or bcc structure. 6 HEAs usually adopt bcc phase for VEC < 6.87, fcc phase for VEC > 8, and fcc-bcc mixture for 6.87 < VEC < 8. The VEC of CoCrFeNiTa x alloys are listed in Table 1, which are about 7. 60-8.25, suggesting that the alloys may have both bcc and fcc phases.
The Alloy-Theoretic Automated Toolkit (ATAT) developed by van de Walle and coworkers 43,44 was used to generate the SQS structures of the quinary random solid solution CoCrFeNiTa x HEAs. Screening the best SQS structure from the numerous candidates of a ve-component alloy system is a challenging job. One can hardly gure out the SQS structures for these systems with completely random atomic distribution. In our computations, the screening process was interrupted when the best SQS structure did not change on the list aer a long period of time (>100 hours). The lattice vector and atomic positions of the acquired SQS models were given in Table S1 in the ESI. † Ten SQS runs were performed for each composition and all these ten SQS structures were collected for further identication at the rst-principles level.
A number of quantities were used to identify the structures of the alloys. Valence electron concentration is computed with where c i and e i are the concentration and number of valence electrons of atom i in the cell. To describe the disordered SQS structures, a parameter named total pair distribution function (PDF) is dened as 45-47 where N is the number of atoms in the simulation cell and r i (r) is the density of atoms in the shell. r 0 is the average density of atoms in the SQS models. To compare the atomic arrangements between the two atoms, the partial pair distribution function is also dened as where a and b represent the atomic species. The structural optimization and energy calculation were carried out for all the SQS screened structures using the DFT calculations with the Vienna Ab initio Simulation Package (VASP). 48 The projector augmented-wave (PAW) method and standard PBE exchange-correlation functional were employed. 34 The convergence criteria were set to 10 À8 eV in energy and 10 À3 eVÅ À1 in force in structure optimization. The cut-off energy was selected as 600 eV in the simulations and a Monkhorst-Pack k-point mesh of 19 Â 19 Â 19 was applied for Brillouin zone sampling. It has been shown that these settings 35 produce reliable results for HEA systems.
The lowest-energy structures screened at the rst-principles level for all the studied compositions were further veried with phonon spectroscopy calculations. 49,50 No imaginary frequencies were found for these structures. The phonon density of states and the heat capacity at constant volume (C v ) and the vibration entropy DS vib were then computed. The formation energy, DE f , is evaluated with where E total is the energy of the SQS structure optimized at the rstprinciples level, N is atom number of SQS supercell and x is atom number of element i in SQS structure. E i is the energy of element i, which was obtained from the computations on its most stable phase at the same level. The elastic properties of a crystal is calculated using the basic elastic stress-strain relationship: where s i , 3 j , and C ij are the elastic stress, strain and tensor in the Voigt notation, respectively. C ij can be derived from this relation by performing six nite distortions of the lattice. Although the atoms are on an fcc (or bcc) lattice, the chemical species distribution in small SQS cells may lead to an anisotropic environment and scattering elastic constants. To overcome this problem, an averaging method 51 was employed to acquire the C 11 , C 12 and C 44 for cubic structures: C 11 ¼ (c 11 + c 22 + c 33 )/3, C 12 ¼ (c 12 + c 23 + c 13 )/3, and C 44 ¼ (c 44 + c 55 + c 66 )/3 in which c ii , c ij are computed elastic constants. The mechanical properties can be obtained with the elastic constants and the Voigt Reuss (V-R) average method. 52 The bulk modulus (B) is evaluated with while the shear modulus (G) is obtained as a mean of the upper (G V ) and lower (G R ) bounds given by 51,53,54 Young's modulus (E) can be derived from the B and G:

Results and discussion
The computed lattice parameters of the studied structures are presented in Table 1. The XRD measured parameters are also given for comparison. 35,37,55 Good agreement between our calculations and measurements were reached. The deviations from the measurements for fcc structures are about 2.0%, 1.4%, and 0.1%, for x ¼ 0, 0.2 and 0.4, respectively. Moreover, the lattice parameters increase with Ta content for both bcc and fcc structures. Ta has big atomic size, its addition usually leads to lattice expansion, which has been observed in other experiments. [56][57][58] Total PDF, g(r), measures the number of atoms around given atoms as a function of distance (r). Fig. 1 shows the total PDF of the studied CoCrFeNiTa x structures. There are several sharp peaks at r < 8.0Å, which implies ordered structures adopted by the atoms at short distances. The g(r) approaches to 1 when r > 8.0Å, indicating disordered structures in the system at long distance. g(r) approaches to 1 when r increases because the density of atoms in a large concentric layer is close to that of averaged density of atoms of the whole system. The partial PDF, which measures the number of specic atoms around a given atom as a function of r. The partial PDFs of all pairs in CoCrFeNiTa x are given in Fig. S1 to S12 in the ESI, † which highlight the relative locations of all atoms. Basically, the rst peaks of g(r) occur at about 2.6Å for all the structures, reecting the fact that these atoms have similar bondlengths with each other. However, the atomic environments are to some extent different from each other. For example, Co atoms have a tendency to coordinate with each other and with Ni atoms in the fcc structure of x ¼ 0.4 (Fig. S5 †), while Cr atoms tend to coordinate with Ta and Ni atoms. Moreover, such tendencies are different in the corresponding bcc structures (Fig. S6 †) in which Co atoms prefer Ta and Cr atoms, and Fe atoms prefer Cr and Ta atoms. The preferences of interatomic coordinations also change with Ta content. For example, Co atoms prefer Fe, Ta, Co, Ta, Co and Ni atoms, respectively, in the fcc structures when x changes from 0 to 1. Ta atoms show some preference in their coordinations. The highest peaks of x ¼ 0.2-1.0 are for Ni, Ni, Co, Ni and Cr atoms in the fcc structures, and Ni, Ni, Ni, Fe and Cr atoms in the bcc structures, respectively. These preferences reect that the diverse microstructures of CoCrFeNiTa x alloys. Fig. 2a Fig. 2c. A positive DE implies a stable fcc structure, and vice versa. DE changes when the Ta content increases from 0.0 to 1.0. The bcc structure is more stable than the corresponding fcc one for x ¼ 0.2, 0.4, and 1.0, the fcc structure becomes more stable for x ¼ 0.0, 0.6, and 0.8. Our calculations predict that the CoCrFeNiTa x alloys may adopt bcc or fcc lattice, or their mixture, depending on the Ta contents. However, one can note that absolute values of DE f are rather small for all these structures. Jiang et al. 37 found that with increasing Ta content, the microstructure changes from an initial fcc solid solution to a mixed structure in CoCrFeNiTa x consisting of bcc and other phases. One should note that the SQS structures represent the most randomly distributed conguration of atoms in the bcc or fcc lattice at a given composition. These structures were generated within a reasonable period of computing time. Other congurations with similar random atomic distributions may also exist. The characterization of all these possible structures are extremely challenging for multi-component alloys, and we used only one of the most possible structures to explore the thermodynamic and mechanical properties of CoCrFeNiTa x alloys.
Phonon frequencies were calculated under the harmonic approximation for the lowest-energy bcc and fcc structures. The computed density of phonon states (PHDOS) are given in Fig. S13 in the ESI. † No imaginary frequency was noted in the PHDOS for all the structures, verifying that the identied structures were local minimum on the potential energy surfaces. In addition, the PHDOS were used to calculate heat capacity, C v , as a function of temperature, as presented in Fig. 3. The C v values appear to be quite similar for the studied alloys, increasing rapidly at low temperature (<150 K), increasing slowing at 150-400 K and approaching to 25 J K À1 mol À1 at high temperature. For T near to 0 K, the valence electrons contribute mainly to C v , making it proportional to gT, where g ¼ 10 À4 . Both phonons and electrons make contribute to the total C v when T < q D (Debye temperature). When T [ q D , C v ¼ 3Nk z 25 J K À1 mol À1 . The C v variations of CoCrFeNiTa x are consistent with the prediction of Dulong-Petit, Kepp and Debye Models. [59][60][61] For HEA systems entropy changes are usually dominant because of their rather small formation energies. In contrast to the small DE f values presented above, relatively large entropy contributions were noted in the CoCrFeNiTa x systems. The entropy change contains two parts, vibration (DS vib ) and conguration (DS con ). The DS vib values of CoCrFeNiTa x , which were computed based on the phonon frequencies at the rst-principles level, are given in Fig. 4. The DS vib exhibits similar variations with temperature, close to zero at low temperature and increasing gradually with temperature. The magnitudes of DS vib vary with Ta content. The differences are small at low temperature, increase with temperature between 0-250 K, but become almost unchanged at T > 300 K. The largest differences are about 2.5 J K À1 mol À1 for the fcc structures between x ¼ 0.0 and 0.6, and for the bcc structures between x ¼ 0.2 and 0.4. The congurational entropy DS con and DS vib at 300 K of the studied alloys are presented in Table 2. Both DS con and DS vib values are positive, implying that the formation of the alloys is a process of entropy increment. The magnitudes of DS con are about 2-3 times greater than those of DS vib , indicating that congurational entropy plays a dominant role in stabilizing the alloys. Although DS vib becomes greater at high temperature, its contribution to total entropy is still smaller than the DS con . A high entropy change promotes the extent of confusion in alloys and reduces the Gibbs free energy, favoring the random distribution of different elements in crystal lattice.
The computed averaged elastic constants C 11 , C 12 and C 44 , as well as the Cauchy pressure (C 12 -C 44 ) and the Zener ratio A z ¼ 2C 44 /(C 11 À C 12 ) of CoCrFeNiTa x are presented in Table 3. The dynamical stability conditions, i.e., C 44 > 0, C 11 > |C 12 | and C 11 + 2C 12 > 0, 62 are satised by the presented lowest-energy fcc and bcc structures. For the structures with the same compositions, their elastic constants are different for the fcc and bcc ones. Some components, for example, C 11 of x ¼ 0.2 and C 12 of x ¼ 0.8, differ remarkably between the two phases. In the same phase, either fcc or bcc, the elastic constants are also different for the structures with different compositions. For example, C 44 of the fcc structures varies between 66 and 160 GPa for x ¼ 0-1, while the C 44 of the bcc structures varies between 86 and 148 GPa. Therefore, the elastic constants of CoCrFeNiTa x alloys vary with their phase structures and compositions. Positive Cauchy pressure is featured by ductile alloys, while negative Cauchy pressure is a signature of brittle alloys. 63 The fcc structure of x ¼ 0 and the bcc structures of x ¼ 0, 0.4 and 0.8 are brittle, the rest of the structures are ductile. A z is used to predict the elastic anisotropy of materials. A z ¼ 1 represents completely elastic isotropy, and its deviation from 1 measures the degree of elastic anisotropy. 64 The predicted A z values of most structures are far from 1, verifying the anisotropic distribution of atoms in those lattice framework, as noted in the SQS structures presented in Table S1. † Based on the computed elastic constants, we further evaluated the shear modulus G, Young's modulus E, bulk modulus B, and Pugh ratio B/G of the CoCrFeNiTa x alloys, which are The microstructures, thermodynamic and mechanical properties of CoCrFeNiTa x alloys are related to their electronic structures. Analysis on their electronic structures may shed light on their composition-and structure-dependent variations that are presented above. Electron localization function (ELF) is oen used to analyze the interatomic interaction in alloy systems. ELF illustrates electron density among adjacent atoms that is used to classify the interatomic bonds and measure their  bonding strengths. 66 To highlight the effect of Ta addition on interatomic interaction, we calculated the ELF on the Tacontaining facets of the studied structures, as given in Fig. 6. In the gures ELF ¼ 0 and 1 corresponds to a completely delocalized state and a perfect localized state, respectively. The ELF values are about 0.20-0.30 in the regions around Ta atoms, indicating that they has a great tendency to localize electrons. The electron localization tends to be enhanced with increasing Ta content for both the fcc and bcc structures. Two structures of x ¼ 1.0 exhibit the largest ELF values around the Ta atoms. Although the enhanced localization strengthens the interatomic interaction between Ta and other atoms, it favors to form other phases. Similar interatomic interaction strengths are one of the features in HEA systems. The strong interaction between Ta and other atoms may disequilibrate the random distribution in alloys. Other experiments 37,67 have veried that the addition of Ta in CoCrFeNi tend to form a mixed structures.
The band structures of CoCrFeNiTa x are presented in Fig. S14 and S15 in the ESI. † All these alloys exhibit metallicity,

Conclusions
Using a combined SQS structure search and rst-principles calculation approach, we have investigated the microstructures, thermodynamic and mechanical properties of CoCrFeNiTa x (x ¼ 0.0-1.0) high entropy alloys. The SQS method was used to construct the disordered candidate structures, which were further veried with rst-principles calculations.
The identied structures have their lattice constants in good agreement well with the measures. The total and partial PDF reveal that the atoms have different environments from each other, varying with composition and phase structures. Ta atoms have preferences to coordinating with specic atoms. The C v variations with temperature are similar for the studied alloys, and are consistent with the prediction of Dulong-Petit, Kepp and Debye Models. Structure transition between the fcc and bcc structures was noted with Ta addition, but the formation energies are small for both phases. The magnitudes of DS con are about 2-3 times greater than their DS vib counterparts. It is congurational entropy that plays a crucial role in stabilizing the alloys. The computed elastic constants and moduli revealed that the mechanical properties of the CoCrFeNiTa x alloys vary with Ta content and phase structure, but the variations are relatively small. Using the computed moduli, the fcc structure of x ¼ 0 and the bcc structures of x ¼ 0, 0.4, and 0.8 were predicted to be brittle, and the rest of the structures ductile. The Ta addition alters the electron localization that has considerable inuence on the equilibrium among atoms in HEA systems because Ta atoms have relatively stronger interaction with some specic atoms in the systems. Our computations revealed the variations in microstructures, thermodynamic and mechanical properties of the CoCrFeNiTa alloys, which would be helpful for their design and preparation with target performances.

Conflicts of interest
There are no conicts to declare.