Anti-perovskite carbides and nitrides A₃BX: A new family of damage tolerant ceramics

1. Introduction

Ceramics are hard and brittle while metals are soft and ductile, which are underpinned by the strong directional covalent bonds in the former [1,2] and non-directional metallic bonds in the latter [3]. In the last decades, novel structural ceramics are expected to combine the advantages of both ceramics and metals. The typical high temperature ceramics, binary carbides and nitrides, have high modulus but brittle nature [[4], [5], [6], [7], [8], [9]]. A strategy to overcome their brittle issue is incorporation of non-directional metallic bonds to form the structures with alternative strong and weak stacking layers, such as MAX phases (M is an early transition metal, X carbon or nitrogen and A A-group element) [[10], [11], [12], [13]] and the similar layer structure compounds including Cr₂AlB₂, Zr₂Al₃C₅ and Hf₃AlN [2,14,15], which are proved owing good machinability and damage tolerance as metals. However, their stiffness is greatly decreased as the result of the incorporation of weak unit layers synergistically [2,14,15].

The mechanical properties depend greatly on the crystal structure, compositions and chemical bonds of materials. Herein, a strategy different from MAX phases is proposed to merge the strong covalent bonds into the metallic box forming A₃BX (A and B are different metals, X is C or N) phases. A₃BX ceramics crystallize in an anti-perovskite structure, consisting of A₃B metallic frameworks (i.e., metallic box) and inserted corner shared XA₃ octahedra as illustrated in Fig. 1(a). The quasi-ductility is achieved through the easy shear deformed A₃B metallic box in A₃BX which extends to the whole structural space. At the same time, the XA₃ octahedra containing strong covalent bonds contribute to the stiffness and make these compounds highly strong. A comprehensive structure-property investigation is urgently called for to fill the gap of the understanding of this family of materials.

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Fig. 1.   (a) Crystal structure of A₃BX and (b) Workflow of automated calculations and criteria for data screening.

High-throughput calculations have shown profound influences on both automatically theoretical simulations and date analysis with the aid of density functional theory (HTC-DFT). Desirable properties can be designed through the new material structure construction with exposed mechanisms and the construction information can be recorded into the database for further considerations. For example, Liu et al. has successfully evaluated the mechanical and thermal properties of over 300 perovskite oxides and predicted six promising thermal insulator materials as novel thermal barrier coating materials [16]. Through screening oxygen-bearing materials, Aykol et al. identified a series of functional materials (Caln₂O₄ and Li₂MnO₃ etc.) having acid scavenging capabilities and electrochemical stabilities, which are pivotal for the applications of cathode coatings in Li-ion batteries [17]. In this work, the mechanical properties of 126 A₃BX phases (from the Materials Informatics Platform [18]) including 65 carbides and 61 nitrides are studied using HTC-DFT processed as illustrated in Fig. 1(b). Six hard and four soft damage tolerant ceramics are predicted. Co₃AlC and Ti₃TlN that are representatives as hard and soft ceramics, respectively, are studied in details to explore the chemical bonding characteristics and their high temperature performances. These results provide a comprehensive map of the mechanical properties of A₃BX ceramics as well as uncover the underlying mechanisms of the property variation. It is proposed that future damage tolerant ceramics can be designed through the idea of merging the strong covalent bonding unites into metallic boxes.

2. Calculation methods

The structural optimizations and mechanical property calculations were performed employing VASP.5.4.4 [19] in which the projector augmented wave method was adopted [20]. The generalized gradient approximation was adopted in the form of Perdew-Burke-Ernzerhoffor solids (GGA-PBEsol) [21]. The cut-off energy for plane-wave basis was 500 eV and the special k-points sampling integration over the Brillouin zone was employed using the Monkhorst Pack method with a 6 × 6×6 special k-points mesh [22]. Spin polarization was considered in this work. During the structural optimization, lattice constants and internal atomic coordinates were modified independently until the force on individual atoms was smaller than 0.005 eV/Å. The elastic constants were obtained through applying a set of homogeneous deformations to the unit cell and calculating the corresponding stress-strain relationship [23]. Three independent elastic constants, i.e., c₁₁, c₁₂ and c₄₄, are considered for the A₃BX with the cubic structure. Here, only one strain pattern with four strain amplitudes up to 0.4 % was applied as in our previous works [[24], [25], [26], [27], [28]]. The ideal shear and tensile stress-strain curves were calculated using the method proposed by Roundy et al., in which a series strains were applied accompanying the constrained structural relaxation as described in Refs. [29,30].

Using second-order elastic coefficients c_ij and the compliance tensor s_ij (s_ij=c_ij^-1), all polycrystalline moduli (K, G and E) are calculated through Voigt (K_V and G_V)-Reuss (K_R and G_R)-Hill (K_H and G_H) equations as following [[31], [32], [33]]:

$K_{V}=\frac{c_{11}+2c_{12}}{3}$ (1)

$ G_{V}=\frac{c_{11}-c_{12}+3c_{44}}{5}$ (2)

$ K_{R}=\frac{1}{3s_{11}+6s_{12}}$ (3)

$ G_{R}=\frac{5}{4(s_{11}-s_{12})+3s_{44}}$ (4)

$ K_{ H }=\frac{K_{R}+K_{V}}{2}$ (5)

$ G_{H}=\frac{G_{R}+G_{V}}{2}$ (6)

Then, the Young’s modulus E and the Poisson’s ratio ν are acquired from K_H and G_H as [34]:

$E=\frac{9K_{H}G_{H}}{3K_{H}+G_{H}}$ (7)

$v=\frac{3K_{H}-2G_{H}}{2(3K_{H}+G_{H})}$ (8)

The vibrational free energies were calculated using PHONOPY code, in which the Parlinski-Li-Kawazoe method within the quasi-harmonic approximation (QHA) was adopted [35]. After the structure optimization, each atomic position was slightly displaced and the Hellmann-Feynman forces acting on the atoms in the supercells were evaluated to construct dynamical matrix to estimate the corresponding phonon frequencies. Once the phonon density of states g(v) was known, the vibrational contribution to the Helmholtz free energy at volume V and temperature T could be obtained by the relationship of:

$F_{vib}(T)=k_{B}T\int^{∞}_{0}dvg(v)ln[2sinh(hv/(2k_{B}T))]$ (9)

where g(v)dv is proportional to the number of phonons with frequency between v and v + dv, and k_B and h are the Boltzmann constant and the Planck constant, respectively. The total Helmholtz free energy F(V,T) was obtained by adding the vibrational free energy to the electronic energy. In this work, the F-V curve was obtained by fitting to the Vinet equation of states (EOS) [36]. The total Gibbs free energies G(p,T) and the equilibrium lattice parameters at a given T were immediately obtained from the F-V curves. Then, the temperature dependent expansion coefficients, heat capacities and mechanical properties were calculated.

3. Results and discussions

3.1. The crystal structure

The A₃BX phases crystalize in the cubic structure with the space group of Pm-3 m [37]. The A atoms locate at (0, 0.5, 0.5), B atoms at (0, 0, 0), and X atoms at (0.5, 0.5, 0.5), respectively [38]. This structure can be described using the lattice constant only due to its high symmetry. All calculated lattice constants (a) are collected in Table S1 in Supplementary 1 (Supporting information). The comparison between the predicted lattice constants and available experimental values from ICSD [39,40] is shown in Fig. 2. The deviation is less than 4 %, validating the reliability of our calculations.

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Fig. 2.   Comparison of calculated and experimental lattice constants [35,36].

3.2. Mechanical properties

The second order elastic constants are employed to describe the response of the single crystal material to the applied stress and they are also the basis for calculating the polycrystalline mechanical properties. The calculated elastic constants of 126 A₃BX compounds are collected in Table S2 in Supplementary 2. All the compounds have large tensile related c₁₁ and relative low shear related elastic constants of c₁₂ and c₄₄. The extremely low c₄₄ for a part of A₃BX compounds (Ti₃AlN, Mn₃ZnC, Ti₃TlN, etc.) benefits the shear deformation and the damage tolerance of ceramics. Furthermore, the studied A₃BX compounds are stable because all materials meet the Born-Huang structural stability criteria, i.e., c₁₁ > 0, c₁₁ > |c₁₂|, c₁₁+2c₁₂ > 0 and c₄₄ > 0 [38].

Polycrystalline moduli, including bulk modulus K, shear modulus G, and Young's modulus E, are always measured in experiments rather than the elastic constants c_ij. The predicted moduli are summarized in Table S2. It is shown that the predicted values are from 69.9 GPa to 469.2 GPa for Young’s moduli, from 39.4 GPa to 336.4 GPa for bulk moduli, and from 29.0 GPa to 187.2 GPa for shear moduli. In the case of Poisson’s ratios, their values are in the range of 0.16 to 0.39. Table S3 (Supplementary 3) compares the calculated bulk, shear and Young’s moduli for Ti₃AlC, Ti₃AlN and Ni₃MgC with the available data from literatures. Our calculated values are in good agreements with the other theoretical results. The agreement between calculated and experimental moduli of Ti₃AlC is good for K and G but fair for E. Based on both the theoretical outputs and the experimental results, we have constructed a reliable mechanical property database for A₃BX family, highlighting the tailorable mechanical properties.

The applications of the ceramics often suffer from their intrinsic brittleness and the difficulty in machining into complex shapes. Thus, it is of urgency to develop damage tolerant ceramics and gain a deep insight into their underlying mechanisms. Three indexes are generally accepted as reasonable criteria to describe the ductile-brittle nature of materials, including Cauchy's pressure, Pugh's ratio and the bond stiffness model [[41], [42], [43], [44], [45], [46], [47], [48], [49]]. For the Cauchy's pressure that is calculated by c₁₂-c₄₄, the negative value is typically for strong directional bonding and means brittle, while the positive value relates to certain characteristics being similar to metallic bonding and means quasi-ductile [[41], [42], [43]]. Pugh’s ratio that is defined as k = G/K, is proposed for metals at first and also makes a big achievement in ceramics recently [27,[44], [45], [46], [47]]. The boundary value is 0.571. If k < 0.571, the ceramic is quasi-ductile; otherwise, it is brittle. In order to overcome the failure of the Pugh’s ratio model on ternary layered ceramics, Bai et al. proposed the bond stiffness model instead, in which the ratio of bond stiffness of the weakest and strongest bonds is believed the key parameter [48,49]. The high damage tolerance and fracture toughness ceramics have the bond stiffness ratio being lower than 1/2, but above this value cracks are present in the Vickers’ indentation. This model gets big success in ternary layered ceramics but is not suitable for our current cubic structure materials. Therefore, the Pugh’s ratio and Cauchy’s pressure are adopted here to identify the potential damage tolerant ceramics. The Pugh’s ratio versus Cauchy’s pressure of 126 A₃BX is plotted in Fig. 3 where four regions are identified based on the above two criteria. All the compounds in area I meet the quasi-ductile requirements of both Cauchy's pressure and Pugh's ratio criteria, while the others are brittle. In experiments, Ti₃AlC is determined as a brittle ceramic [50], which is consistent with our conclusion as a result of its G/K = 0.68 and c₁₂ - c₄₄ = -24.94 GPa. Then, 49 A₃BX phases can be identified as damage tolerant ceramics.

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Fig. 3.   Pugh's ratio versus Cauchy's pressure of the A₃BX ceramics.

The high stiffness and/or mechanical moduli are also necessary for structural ceramics. In Fig. 4, the bulk, shear and Young’s moduli of 49 discovered damage tolerant A₃BX compounds are compared with the other well studied ceramics, including binary carbides/nitrides, MAX phases, and the other ternary layered ceramics. The A₃BX compounds have higher bulk moduli than MAX phases and the other ternary layered ceramics. Interestingly, the bulk moduli of certain A₃BX ceramics are similar to or even higher than binary carbides and nitrides. In the cases of shear and Young’s moduli, the values of A₃BX are lower than binary compounds, but comparable to MAX phases and the other ternary layered ceramics. The high bulk moduli and low shear moduli clearly uncover their synergistic effects of the high stiffness and the damage tolerance as expected.

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Fig. 4.   (a) Bulk modulus, (b) Shear modulus and (c) Young's modulus versus Pugh's ratio for binary and ternary carbides/nitrides.

Based on the information as revealed in Fig. 4 and Table S2, sixteen A₃BX materials are found to have both high moduli (E > 320 GPa and K > 240 GPa) and good damage tolerance (G/K < 0.571). However, ten of them containing noble metals, such as Rh, Ru, and Ir, are not practical in applications because of their extra high cost. Therefore, six A₃BX phases (Mn₃NiN, Mn₃GaC, Mn₃GaN, Mn₃SnC, Cr₃SnN, Co₃AlC) are finally predicted as new promising damage tolerant ceramics which have comparable stiffness to binary materials as listed in Table 1. Furthermore, four A₃BX materials including Ti₃AlN, Mn₃CuN, Ti₃TlN, and Ni₃SnN with extra low Pugh's ratio (G/K < 0.3) may be machinable and can be used as the promising second soft phases for improving the damage tolerance of binary carbides and nitrides. Among these predicted compounds, the Co₃AlC and Ti₃TlN that are representatives of hard and soft damage tolerant ceramics, respectively, are selected to be studied in details, including the bonding characteristics that are related to the mechanical property difference, strength, thermal expansion coefficients, heat capacities, and temperature dependent elastic moduli as discussed in the following section.

3.3. Electronic structures and high temperature mechanical properties of Co₃AlC and Ti₃TlN

The excellent tailorable mechanical properties of A₃BX compounds can be traced to the chemical bond discrepancies in their specific structures. The projected density of states (PDOS) of Co₃AlC and Ti₃TlN are illustrated in Fig. 5. The Co-Al bonds are weaker than Co-C bonds because the Co-Al bonds locate in -4.9 ~ -2.5 eV and the Co-C bonds locate in -8.7 ~ -2.5 eV. So is for Ti₃TlN that shows the similar bonding environments, in which Ti-Tl bonds from -3.9 to -1.6 eV) are weaker than Ti-N bonds (from -6.7 to -2.2 eV). It should be noted that the energy level of the main Co-Al bond peak (-3.3 eV) totally overlaps with one Co-C bond peak (Fig. 5(a)), but the energy level of the main Ti-Tl bond peak (-2 eV) is higher than all Ti-N peaks (Fig. 5(b)), indicating the higher stiffness of Co₃AlC than Ti₃TlN according to the conclusion of Liao et al. [48,49,55,56]. For further understanding of their chemical bonding nature, the electron density difference maps on (100) and (110) planes of Co₃AlC and Ti₃TlN are displayed in Fig. 6(a) and (b), respectively. Charge accumulations are absent between Ti and Tl on the (100) plane in Ti₃TlN, which indicates the fully metallic bonds between Ti and Tl in the A₃B box. A clear accumulation can be observed between Co and Al in Co₃AlC, indicating the metallic-covalent Co-Al bonds in the A₃B box. In the view of the structural characteristics, the metallic A₃B box contributes to the easy shear deformation and should be the origin of the damage tolerance of the materials. Meanwhile, the more metallic the bond is, the better the damage tolerance and/or quasi-ductility of the materials is. In addition, the charge transfer is obvious and the electron segregation around the C and/or N is observed in both compounds as shown in the (110) plane, which indicates the covalent-ionic interactions to strengthen the materials. Namely, metallic or metallic-covalent bonds between A-B atoms determine their shear deformation resistance, while the strong covalent-ionic A-X bonds are responsible for the high stiffness.

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Fig. 5.   Projected electronic density of states of (a) Co₃AlC and (b) Ti₃TlN.

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Fig. 6.   Electron density differences of crystal planes (100) and (110) of (a) Co₃AlC and (b) Ti₃TlN.

The elastic parameters that are calculated under nearly equilibrium conditions can not totally evaluate the material deformation and/or strengths under large strain, as they associated with significant change of atomic bonding characteristics. A further study is implemented to reveal the ideal stress-strain relationships to understand the structural stability, strengths, hardness, and ductility of these ceramics. As shown in Fig. 7(a) and (b), the calculated tensile strengths of Ti₃TlN are 27.7 GPa, 28.3 GPa, and 36.4 GPa for [100], [110] and [111] directions, respectively, while those of Co₃AlC are 42.3 GPa, 39.5 GPa, and 39.7 GPa, respectively. These values are similar with the ideal strength of binary compounds, such as 33 GPa for TiC and 30 GPa for TiN, and also the ternary layer structural ceramics, such as 25 GPa for Ti₂AlC and 28 GPa for Ti₂AlN [30]. At the same time, the calculated shear strengths of Ti₃TlN are 17.8 GPa, 21.3 GPa, and 21.8 GPa for (001)[100], (001)[110] and (101)[111] directions, respectively, while those of Co₃AlC are 23.7 GPa, 25.2 GPa, and 27.1 GPa, respectively. Being different with the tensile deformation, these values are again similar with the values of binary compounds, such as 32 GPa for TiC and 28 GPa for TiN, but are much larger than the ternary layer structural ceramics, 11 GPa for Ti₂AlC and 14 GPa for Ti₂AlN [30]. Furthermore, for both Ti₃TlN and Co₃AlC, their shear strengths are lower than the tensile strengths, and the ratio of shear strength to tensile strength is in the range from 0.49 to 0.75. It uncovers that the shear deformation is preferred, and Ti₃TlN and Co₃AlC should be quasi-ductile. Namely, Co₃AlC and Ti₃TlN are damage tolerant materials similar to the ternary layer structural ceramics but still own high strength similar to binary compound, which is different with the ternary layer structural ceramics.

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Fig. 7.   Calculated stress-strain relation for (a) Ti₃TlN and (b) Co₃AlC.

As these ceramics potentially used in high temperatures, the temperature dependent thermal expansion coefficients (TECs), heat capacities and elastic moduli are important. In the theory framework, these properties can be calculated by combining the first-principles calculations and the QHA calculations. At first, the phonon dispersions of Ti₃TlN and Co₃AlC are calculated and shown in Fig. S1 in Supplementary 4. There are no imaginary modes in the phonon dispersion of these two compounds, indicating their dynamic stability. Secondly, their Helmholtz free energies at elevated temperatures are calculated through performing the phonon calculations at 18 volumes by varying a/a₀ from 0.988 to 1.07 for each compound. The volume corresponding to the minimum free energy is determined as the equilibrium volume at a certain temperature. Then, the TECs can be calculated by [35,57]:

$a=\frac{1}{a}(\frac{\partial a}{\partial T})$ (10)

The calculated TECs are shown in Fig. 8(a). From 300 K to 1200 K, the TECs of Ti₃TlN is in the range of 9.2-12.1 × 10^-6 K^-1 that is larger than those (7.3-9.8 × 10^-6 K^-1) of Co₃AlC, as the bonds in Ti₃TlN is weaker than those in Co₃AlC. Furthermore, the heat capacities of Ti₃TlN and Co₃AlC are also calculated. As they are both metal-like structure, the effect of the electron excitation must be included. Here, the free electron gas approximation is used, in which the electronic heat capacity is calculated using the following equation [58]:

$C^{el}_{v}=δT $ (11)

where δ = (1/3)π²N(E_F)(k_B)², k_B is the Boltzmann constant and N(E_F) is the density of states at the Fermi level. The total heat capacities that are the sum of the vibrational and electronic heat capacities are plotted for Ti₃TlN and Co₃AlC in the Fig. 8(b). It is found that their values are quite close.

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Fig. 8.   Calculated (a) TECs and (b) heat capacities of Ti₃TlN and Co₃AlC.

The temperature dependent mechanical properties are predicted using the method constructed by Wang et al. [59,60]. In experiments, the isentropic elastic constants and mechanical properties are measured and therefore the isothermal elastic constants (cijT) are converted to isentropic elastic constants cijS [61], as:

$c^{S}_{ij}(T)=c^{T}_{ij}(T)+\frac{TV\lambda_{i}\lambda{j}}{C_{V}}$(12)

$\lambda_{i}=-\sum_{j}a_{j}c^{T}_{ij}(T) (i,j=1,2,3)$ (13)

Then, the Eqs. (2)-(9) are used to calculate the K, G and E. As shown in Fig. 9(a) and (b), all the elastic constants and mechanical moduli decrease with the increased temperature. Until 1200 K, both these two materials still keep 80 % and/or higher ratio of their ground state moduli, indicating their promising application in high temperature environments. Furthermore, the residual elastic constants and moduli of Co₃AlC (89.5 % for c₁₁, 86.9 % for c₁₂, 93.5 % for c₄₄, 88.6 % for K, 92.2 % for G and 91.7 % for E) is larger than those of Ti₃TlN (82.0 % for c₁₁, 90.0 % for c₁₂, 89.0 % for c₄₄, 86.3 % for K, 81.1 % for G and 81.5 % for E), which is again originated from the stronger bonding of Co₃AlC.

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Fig. 9.   Temperature dependent elastic constants and mechanical moduli of (a) Co₃AlC and (b) Ti₃TlN.

4. Conclusions

In summary, the high-throughput calculation was employed to investigate the structure and properties of 126 A₃BX phases. The mechanical property database containing 126 A₃BX phases were constructed and 49 damage tolerant ceramics were identified based on both Cauchy’s pressure and Pugh’s ratio criteria. Then, six compounds (Mn₃NiN, Mn₃GaC, Mn₃GaN, Mn₃SnC, Cr₃SnN, Co₃AlC) were predicted as potential high temperature ceramics owning to the combination of both high stiffness (E > 320 GPa, K > 240 GPa) and damage tolerance (G/K < 0.571); meanwhile, four A₃BX phases (Ti₃AlN, Mn₃CuN, Ti₃TlN and Ni₃SnN) with the promising machinability (G/K < 0.3) were proposed as the alternative second soft phases for binary carbides/nitrides.

The detailed investigations on representative materials of Ti₃TlN and Co₃AlC were performed. The mechanical properties of A₃BX phases are strong correlated to the structure and bond characteristics, i.e., A-X bonds being stronger than A-B bonds. The quasi-ductility is attributed to the A₃B metallic box with the A-B metallic or metallic-covalent bonds, while the high stiffness comes from the strong A-X covalent-ionic bonds. Therefore, the A₃BX is identified as the damage tolerant ceramics. Meanwhile, their mechanical properties are tailorable through controlling the different A₃B metallic box and XA₃ octahedra. Furthermore, except for the good damage tolerance, Ti₃TlN and Co₃AlC also own comparable tensile and shear strengths as binary carbides and nitrides, indicating their high stiffness. Lastly, the good high temperature performance of Ti₃TlN and Co₃AlC is predicted.

The established database and structure-property relationship give the comprehensive knowledge of this A₃BX ternary ceramic family, which highlights their potential applications as damage tolerant ceramics. More importantly, the research outputs propose a promising strategy for ceramic design through engineering the crystal structure and/or chemical bonds to accelerate the development of the next generation ceramics owning both good damage tolerance and high stiffness.

Acknowledgements

This work was supported financially by the National Natural Science Foundation of China (No. 51602188) and the Program for Professor of Special Appointment (Eastern Scholar) by Shanghai Municipal Education Commission (No. TP2015040).

Appendix A. Supplementary data

Supplementary material related to this article can be found, inthe online version, at doi: https://doi.org/10.1016/j.jmst.2019.08.043.