Computational characterization of the structural and mechanical properties of AlxCoCrFeNiTi1−x high entropy alloys

AlCoCrFeNiTi is one kind of high entropy alloys with potential applications in aerospace, electronics and machinery manufacturing, etc. Its microstructure, thermodynamic and mechanical properties vary with Al and Ti contents. The disordered structures of AlxCoCrFeNiTi1−x (x = 0–1.0) alloys were generated and screened with the special quasi-random method. First-principles calculations with the Perdew–Burke–Ernzerhof functional and projector-augmented wave potential were carried out to further identify the structures and assess their thermodynamic and mechanical properties. The measured lattice constants and observed phase transition from bcc to fcc with Ti addition and Al reduction were reproduced in the calculations. The predicted transition point is at the Al content of about 8 at%. The heat capacity of the bcc and fcc structures exhibit similar temperature dependence, matching well with the empirical Dulong-Petit Model and the quantum Debye Model. The elastic constants and elastic moduli vary with the phase structures and compositions. The studied AlxCoCrFeNiTi1−x alloys are predicted to possess good comprehensive mechanical performances, especially for x = 0.8 and 0.6 for the bcc structures and x = 0.5 for the fcc structures. The addition/reduction of Al atoms from the systems alters the electron localization/delocalization that has considerable influence on the interatomic interaction strength in the alloy systems.

The SQS approach simulates the disordered state of a limited supercell by optimizing the atomic distribution and minimizing the correlation function. To generate the initial structures, the named mcsqs algorithm in the ATAT package [58,59] was employed. The supercell size for the SQS screening is usually dependent on three conditions: ensuring chemical disorder as good as possible, matching experimental composition, and computationally efficient [49,50]. Screening the best SQS structure from the numerous candidates of a sixcomponent alloy system is a computationally demanding job. One can hardly figure 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 after a long period of time (>100 h). The screening process was repeated at least ten times for each composition. Each run we obtained one structure with a reasonably low correlation factor, which was defined in mcsqs for measuring the randomness of atomic distribution. Consequently, a family of more than ten candidate structures was generated for a given composition. Seven families of candidate structures were then generated for the seven compositions of Al x CoCrFeNiTi 1−x (x=1.0, 0.8, 0.6, 0.5, 0.4, 0.2 and 0.0) with 20,25,25,20,25,25 and 20 atoms per cell. The detailed positional information is presented in table S1 in the supplementary material is available online at stacks.iop.org/MRX/6/096519/mmedia. The above procedures were applied to the bcc and fcc lattices respectively. All these candidate structures were sent to next step for further screening at the density functional theory (DFT) level. The one with the lowest energy for each composition was used to evaluate their thermodynamic and mechanical properties.
Density functional theory (DFT) calculations under the generalized gradient approximation (GGA) of the Perdew-Burke-Ernzerhof (PBE) parametrization [60] and projector-augmented wave (PAW) potential [61], which were implemented in the VASP package [62], were employed to relax the candidate structures. The total energies were converged to 10 -7 eV, and the forces on each atom were relaxed to less than 10 -5 eV/Å for consecutive structures. Both cell parameters and internal coordinates were relaxed in the calculations. The cutoff of energy for the plane wave expansion was set to 600 eV. The k points of 11×11×11 were used to integrate the Brillouin region. All calculations were performed with spin-polarization to account for the magnetic properties of the studied alloys. It has been shown [45,46,48,63] that these settings produce reliable results for HEA systems.
where E total is the energy of the SQS structure optimized at the first-principles level, N is atom number of SQS cell and x is the number of atoms of element i in the 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 were carried out for Al x CoCrFeNiTi 1−x alloys which based on the analysis of the total energies of properly strained states of the material [56,64,65]. Although atoms are on the fcc (or bcc) lattice, the chemical distribution in small SQS cells may lead to an anisotropic environment and scattering elastic constants. To overcome this problem, an averaging scheme [50] was proposed to obtain the C 11 , C 12 , and C 44 parameters for the small SQS cubic structures: C 11 =(c 11 +c 22 +c 33 )/3, C 12 =(c 12 +c 23 +c 13 )/3, and C 44 =(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 [56,66]. The bulk modulus (B) is evaluated with The shear modulus (G) is given by the upper and lower (G R ) bounds, which are evaluated with = - G is estimated as (G V +G R )/2, according to Hill averaging method [56,67]. Young's modulus (E) is derived from B and G by: Finally, first-principles phonon method was applied to predict the phonon density of states (PHDOSs), vibrational entropy (ΔS vib ) and heat capacity at constant volume (C v ). Table 1 presents the lattice constants and VEC of the studied structures. The calculated lattice constants are in good agreement with available experiments [37,41] for both the bcc and fcc structures. The fcc lattices are larger than the bcc lattices, which is common in HEAs with the same compositions. For the bcc phase, the calculated lattice constant increases from 2.852 Å to 2.879 Å with decreasing Al content, resulting from the aggravating lattice distortion caused by Ti addition [37]. The fcc phase exhibits no clear composition dependence on Ti addition and Al reduction. As an indicator of phase structures, VEC is often used to predict that a solid solution adopts fcc or bcc structure. HEAs usually adopt bcc phase for VEC<6.87, fcc phase for VEC>8, and fcc-bcc mixture for 6.87<VEC<8 [3]. The VEC of Al x CoCrFeNiTi 1−x alloys is about 7.2-7.4, suggesting that the alloys may have both bcc and fcc phases. In fact, both the bcc and fcc phases of the alloys with various Al and Ti contents have been identified experimentally [5,[36][37][38][39][40][41].

Structural stability
The total pair distribution function (PDF) is often used to analyze atomistic structures of amorphous systems. It measures the number of atoms around a give atom as a function of distance (r). The details of g(r) evaluation is given in the supplementary material. Figure 1 shows the total PDF of the fourteen studied Al x CoCrFeNiTi 1−x structures. There are several sharp peaks at r<7 Å, which implies ordered structures adopted by the atoms at short distances. The g(r) approaches to 1 when r>7 Å, indicating disordered structures in the systems at long distances. The disordering of solid solutions is well addressed in the structures identified by the SQS-DFT method.  show the dependence of the formation energies ΔE f per atom on the fraction of the alloying element Al. One may note that some structures have a tendency to decompose and transform into their neighbors. The Al x CoCrFeNiTi 1−x alloys may adopt bcc or fcc lattice, or their mixture, depending on the Al and Ti contents. For both the fcc and bcc phases, their ΔE f are negative for all the calculated x values. This means that the studied alloys are stable at the studied concentrations. The relative stability of the alloys with a given composition is measured with their energy difference (ΔE) between the bcc and fcc phases, i.e., ΔE=(E bcc -E fcc )/N, as shown in figure 2(c). A positive ΔE implies a stable fcc structure, and vice versa. ΔE changes from negative to positive when the Al content decreases from 1.0 to 0.0. The bcc structure is more stable than the corresponding fcc one for x=0.5-1.0, and the fcc structure becomes more stable for x=0.0-0.4. Our calculations predict a phase transition from bcc to fcc with Ti addition and Al reduction. Several experiments have focused on the microstructures of AlCoCrFeNi systems, confirming that the alloys form bcc solid solutions at high Al concentration, and are fcc dominant at low Al concentration. Liu et al [35] found the Al x CoCrFeNi alloys form fcc solid solutions when x=0.15 and 0.4. Zhou et al [5] reported that the AlCoCrFeNiTi x alloys are composed mainly of bcc solid solution regardless of varying Ti content of x=0, 0.5, 1 and 1.5. A single bcc phase was also identified for NiCoCrFeAl 3 by Ji et al [68]. Butler and Weaver [69] found that the dominant structure of the low Al concentration HEAs is fcc, while the high Al concentration HEAs are bcc dominant, and the structural transition point is at ∼15 at%. Wang et al [70] found that Al x CoCrFeNi alloys form fcc structures for x=0.0-0.3, bcc structures for x=0.9-1.2, and mixed structures of fcc and bcc for x=0.5-0.7. Jiang et al [37] reported that Al x CoCrFeNiTi 1−x alloys form bcc solid solution when x=1.0, mixed bcc and fcc when x=0.8 and 0.5. A decrease of bcc volume and an increase of fcc volume were observed when x changes from 0.8 to 0.5. Moreover, the fcc or bcc content varies with temperature, annealing process, etc [70,71]. Our calculations reproduced the phase transition from bcc to fcc for the Al x CoCrFeNiTi 1−x alloys when Al content decreases. The predicted transition point is at about x=0.4, corresponding to Al concentration of about 8 at%.

Thermodynamic properties
The phonon density of states (PHDOSs) of the identified bcc and fcc structures were given in Fig. S1 in the supplementary material. No imaginary frequencies were noted for all the structures, verifying that the studied structures were local minima on the potential energy surfaces. In addition, the PHDOS was used to calculate heat capacity, C v , as a function of temperature, as presented in figure 3. C v varies with temperature in a similar way for the fcc and bcc alloys regardless of their compositions and structures. It increases rapidly at low temperature (<150 K). The increase slows down between 150-400 K, and the C v approaches to 25 J K −1 ·mol above 400 K.
The computed results indicate that the contribution of each atom to C v is basically the same at the same temperature. At a sufficiently low temperature (near 0 K) under which the total heat capacity is mainly contributed by free electrons for conductors, C v approaches to 0 and is proportional to T. With increasing temperature, the contribution from phonon vibrations increases. For temperature below 400 K, both phonon vibrations and free electrons have considerable contribution to C v . At high temperature, contributions from free electrons becomes neglectable compared to those from phonon vibrations. C v is then dominated by phonon vibration contribution, which is approximately a constant of 25 J K −1 ·mol at a temperature well above the namely Debye temperature. Our calculations match well with the empirical Dulong-Petit Model [72] and the quantum Debye Model [73].
The calculated vibration entropy ΔS vib and configurational entropy ΔS con at 300 K of the studied alloys are listed in table 2. ΔS con is evaluated from the number of configurations in the systems, as given in the supplementary material. Both ΔS con and ΔS vib values are positive, implying that the formation of the alloys is a process of entropy increment. The magnitudes of ΔS con of Al x CoCrFeNiTi 1−x are in line with the quantitative criterion for solid solutions formation, ΔS con >13.38 J K −1 ·mol [3]. A large entropy change in both  configuration and vibrational entropy promotes the extent of confusion in alloys and the reduction of Gibbs free energy, favoring the random distribution of different elements in crystal lattice [74]. The ΔS con reaches maximum at x=0.5, which provides the largest number of configurations in the systems. ΔS vib was computed based on the phonon frequencies at the first-principles level. As shown in figure 4, ΔS vib increases with temperature, implying that its contribution to the HEA stability increases at high temperature. Moreover, ΔS vib varies with composition. For both phases, the structures of x=0.4 have the largest ΔS vib values. The differences in ΔS vib for the structures with different compositions are small at low temperature, increase with temperature, and become almost unchanged for temperature over 300-1000 K, reaching at about 3.3 J K −1 ·mol. At 300 K, the magnitudes of ΔS vib are much smaller than those of ΔS con , as presented in table 2. Although ΔS vib becomes greater at high temperature, its contribution to total entropy is still smaller than the ΔS con counterpart. Moreover, the ΔS vib values in fcc and bcc have similar composition dependence, indicating that ΔS vib is sensitive to composition rather than phase structure. Table 3 presents the averaged elastic constants, C 11 , C 12 and C 44 , which are defined above, of the fcc and bcc structures. The computed elastic constants for all the studied structures satisfy the dynamical stability conditions of the fcc and bcc lattice structures, i.e., C 44 >0, C 11 >|C 12 | and C 11 +2C 12 >0 [75]. 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=0.6 and C 44 of x=0.2, differ remarkably in the two phases. In the same phase, fcc or bcc, the elastic constants are different either for the structures with different compositions. For example, C 11 of the bcc structures varies between 251 and 336 GPa for x=0-1. Therefore, the elastic constants of Al x CoCrFeNiTi 1−x alloys vary with their phase structures and compositions. Their Cauchy pressure (Γ) and Zener ratio (A z ) are also given in table 3. Positive Γ is a characteristic of ductile alloys, while negative Γ is a definitive signature of brittle alloys [76]. All the studied structures have positive Γ, indicating that they are ductile regardless of phase structures and Al/Ti contents. A z is used to predict the elastic anisotropy of materials. A z =1 represents completely elastic isotropy, and its deviation from 1 measures the degree of elastic anisotropy [52,77]. The predicted A z values of all the structures are well above 1, verifying the anisotropic distribution of atoms in the lattice framework, as noted in the SQS structures in table S1. Using 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 Al x CoCrFeNiTi 1−x alloys, as shown in  composition. For most of the structures, the introduction of Al and Ti promotes their mechanical properties. As discussed above, the alloys prefer the bcc lattice to the fcc one at high Al content. For the bcc structures, B, E and G increase first and then decrease with the Ti addition and Al reduction. The turnover occurs at about x=0.6 and the alloys of x=0.8 and 0.6 were predicted to possess good mechanical performances. Our predictions basically comply with previous experiments [5,37]. For example, Zhou et al [5] observed that the AlCoCrFeNiTi x ( x=0, 0.5, 1.0 and 1.5) alloys system have good comprehensive mechanical properties, especially for x=0.5. Jiang et al [37] found that among Al x FeCoCrNiTi 1−x (x=1.0, 0.8 and 0.5) alloys the composition of x=0.8 exhibits good comprehensive mechanical properties. Room-temperature mechanical properties have been reported for CoCrFeNiTiAl x (x=0, 0.5, 1.0, 1.5 and 2.0) alloys among which good performances were found at x=1.0 [40]. Wang et al [41] also found that AlCoCrFeNiTi 0.5 in which Al content is about 18 at% exhibits the highest Vickers hardness. For the studied fcc structures, which are favored at low Al content, their computed G, E and B values increase with Al addition and Ti reduction at low Al content, and reach the largest values at about x=0.5. For both the bcc and fcc structures, their Pugh ratios (B/G) is larger than the critical value 1.75, indicating that the Table 3. Calculated elastic constants (C 11 , C 12 and C 44 ), Zener ratio (A z ) a , and Cauchy pressure (Γ) b of Al x CoCrFeNiTi 1−x alloys. All quantities but A z are in GPa. a A z =2C 44 /(C 11 -C 12 ). b Γ=C 12 -C 44 .  studied Al x CoCrFeNiTi 1−x alloys are ductile. This is consistent with the above prediction indicated by Cauchy pressure and with previous observation [52,76,77]. Good comprehensive mechanical performances were predicted in our computations for these alloys, especially for x=0.6 for the bcc structures and x=0.5 for the fcc structures. To further correlate their performances with their microstructures, the local electron localization function (ELF) on the (100) facets for the bcc structures of x=1.0 and 0.6, and for the fcc structures of x=0.0 and 0.5 were given in figure 6. ELF is in general used to analyze interatomic interaction in crystal. ELF=0 and 1 corresponds to a completely delocalized state and a perfect localized state, respectively [43]. For the studied alloys, their ELF values are less than 0.5, representing a delocalized state of electrons. One notes in figure 6 that Al atom has a great tendency to localize electrons around it. In the bcc structure of x=1.0, the Al atom has the strongest electron localization. The strong localization is weakened in the bcc structure of x=0.6 in which two Al atoms sit together in presence of the neighboring Ti atom. In the fcc structure of x=0, electron delocalization is dominant in absence of Al atoms. The delocalization is also weakened in the fcc structure of x=0.5. The addition/reduction of Al atoms into/from the systems makes the electron localization/delocalization at an appropriate degree that differs from the x=0 or x=1 systems.

Mechanical properties
Finally, it should be mentioned that the above analysis is based on the SQS structures, which represent a possible distribution of atoms in the bcc or fcc lattice at a given composition. This distribution has the highly disordering atomic arrangement characterized within a reasonable period of computing time.

Conclusion
Stimulated by the good comprehensive mechanical performances of six-component AlCoCrFeNiTi alloys, computational characterizations on the microstructures, thermodynamic and mechanical properties of Al x CoCrFeNiTi 1−x (x=0-1.0) alloys were carried out by using the combined SQS and DFT approach. The disordered distributions of atoms in bcc and fcc lattices at a given composition were generated with SQS and then sent to further characterization at the DFT level with the PBE functional and PAW potential. First-principles phonon method was applied to predict the phonon density of states, vibrational entropy and heat capacity. All the identified structures were verified to be local minima on the potential energy surfaces and characterized by their disordered atomic distributions that are reflected by their pair distribution functions. The predicted lattice constants are in good agreement with the measurements. By computing their formation energies, the relative stability of the bcc and fcc structures was compared. The bcc structures are more stable at high Al content, while the fcc structures are mores table at low Al content. The observed phase transition from bcc to fcc with Al reduction was reproduced in our calculations. The predicted transition point is at the Al content of about 8 at%. The heat capacity and its temperature dependence over the range of 0-1000 K were evaluated based on the computed phonon density of states. All the studied structures exhibit similar temperature dependence regardless of their phase structures and compositions. The predicted temperature dependence is in agreement with the prediction from the Dulong-Petit model and the Debye model. The computed vibrational entropy increases with temperature, but is much smaller than its configurational counterpart. The elastic constants and elastic moduli vary with the phase structures and compositions. The Zener ratio and Cauchy pressure indicate that the studied structures are anisotropy and have good ductility. The computed B, E and G values, which are in agreement with previous available computations, imply that these alloys possess good comprehensive mechanical performances. Moreover, these moduli vary with composition and phase structure. Promoted mechanical properties were predicted at x=0.8 and 0.6 for the bcc structures and x=0.5 for the fcc structures. Electron localization function analysis revealed that the addition/reduction of Al atoms from the systems alters the degree of electron localization/delocalization and consequently alters the interatomic interaction in the alloy systems. Although only some representatives of Al x CoCrFeNiTi 1−x alloys were investigated in this work, our calculations shed light on the variations of microstructures, thermodynamic and mechanical properties of the with their compositions, and established the correlation of our computations with other experimental studies, which would be helpful for the design and preparation of Al x CoCrFeNiTi 1−x alloys with promoted performances.