Theoretical investigation on the stability, mechanical and thermal properties of the newly discovered MAB phase Cr₄AlB₄

1. Introduction

Recently, a new family of materials called MAB phases (M is a transition metal, A is Al or Si, and B is boron) has attracted great research interests [[1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]], due to the fact that they share many similarities with the well-known MAX phases (M_n+1AX_n, M is an early transition metal, A is a III_A-VI_A group element, X is carbon or nitrogen, n = 1-6) [[17], [18], [19], [20], [21]]. For example, they both crystallize in layered structure, which are usually formed by alternatively stacking of M-X (X = C, N, or B) layer and A element layer. The chemical bonds within the M-X layer are usually stiff, while bonding between the M-X layer and the A element layer is weak. The diverse chemical bonds make them display both metal-like and ceramic-like properties, including good electrical and thermal conductivities, moderate elastic modulus, good thermal shock and oxidation resistance, and tolerant to damage, etc. [[17], [18], [19], [20], [21]]. The unique combination of these properties makes them promising for applications in high and ultra-high temperatures. Moreover, it was found that etching the A element out from MAX phases can produce a new set of 2D materials, MXene [22], which stimulates enormous research interests into these materials. In analogous, possible 2D MBenes has also been reported by selective etching Al out from Cr₂AlB₂ [23].

Up to now, both experimental and theoretical investigations on MAB phases have been carried out [[1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]]. The new MAB compounds that have been investigated include (CrB₂)_nCrAl (n = 1-3) [1], MoAlB, WAlB [[2], [3], [4]], Fe₂AlB₂ [7,8], Y₅Si₂B₈ [24] and compounds with solid solutions (Fe_2-xMn_x)AlB₂ [5], (Mo_xW_1-x)AlB, (Mo_xCr_1-x)AlB [6]. Among these MAB phases, compounds in Cr-Al-B system are the most typical group which are formed by inserting one layer of Al into hard boride Cr-B nano-layers (Cr₂B₂, Cr₃B₄, Cr₄B₆). The crystal structure and microhardness of (CrB₂)_nCrAl (n = 1,2,3) have been investigated experimentally [1], and their chemical bonding characteristic, elastic properties and preferred failure models have been studied by first principles calculation based on density functional theory (DFT) [[9], [10], [11]]. Recently, we have synthesized a new MAB compound in Cr-Al-B system, Cr₄AlB₄. Its crystal structure and composition were determined by using a combination of X-ray diffraction and energy dispersive spectroscopy [25]. However, the properties of Cr₄AlB₄ have not been investigated yet. It is the scope of the present work to predict the stability, elastic properties and temperature dependent properties of Cr₄AlB₄ by first principles calculations based on DFT.

2. Calculation methods

First-principles calculations based on DFT were performed using the Cambridge Serial Total Energy Package (CASTEP) code [26] with the norm-conserving pseudopotential [27] and exchange-correlation described by generalized gradient approximation (GGA) based on the Perdew-Burke-Ernzerhof (PBE) scheme [28]. The plane wave basis set cutoff energy was set to be 700 eV, and k-points mesh with a separation of 0.04 Å^-1 according to Monkhorst-Pack method [29] was adopted in the Brillouin zone. The unit cells of MAB phases were first optimized under zero pressure by using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) [30] minimization scheme. The convergence criteria for optimizations were set as follows: the difference in total energy within 1 × 10^-6 eV/atom, the maximum ionic Hellmann-Feynman force within 0.002 eV/Å, the maximum ionic displacement within 1 × 10^-4 Å, and maximum stress difference within 0.02 GPa. These settings for calculation have been tested in our previous work [10,11], with which the relaxed crystal structures are consistent with experiments. Elastic constants were determined from a linear fit of the calculated stress as a function of strain. Both positive and negative strains were applied for each strain component, with a maximum strain value of 0.3%. Phonon dispersions and density of state (DOS) was calculated by using supercell method with the cutoff radius defining the supercell being 5.0 Å.

3. Results and Discussions

3.1. Stability and structural properties

Fig. 1 shows the crystal structures of Cr_2nAlB_2n (n = 1-3). The space group of Cr₂AlB₂ and Cr₆AlB₆ is Cmmm (No. 65), and the space group of Cr₄AlB₄ is Immm (No.71) [25]. Here, Cr₆AlB₆ is an assumed phase. The figure reveals that Cr_2nAlB_2n is built by alternatively stacking of Cr-B layers and Al layers, which is formed by inserting one layer of Al into every one layer, two layers or three layers of Cr₂B₂. For all of these crystals, the stacking direction is defined as the b direction, while the average direction of B-chain is defined as the a direction. The optimized lattice parameters and refined atom positions of Cr_2nAlB_2n are listed in Table 1, wherein experimental results of Cr₂AlB₂ and Cr₄AlB₄ are also listed for comparison [1,25]. The calculated parameters agree well with experiments, indicating the reliability of the calculations.

To evaluate the stability of Cr₄AlB₄, enthalpy changes of the following reactions were calculated:

2Cr₄AlB₄ + 4CrB → 2Cr₆AlB₆ (1)

2Cr₂AlB₂ + 4CrB → 2Cr₄AlB₄ (2)

2Cr₄AlB₄ + 2Al → 4Cr₂AlB₂ (3)

The calculated enthalpy changes of the three reactions are 0.11 eV, -0.37 eV and -0.54 eV, respectively. The results reveal that Cr₄AlB₄ is a relatively stable phase in Al lean condition, since Reaction (1) shows a positive enthalpy change and Reaction (2) shows a negative enthalpy change. It means that selectively inserting Al into CrB by every two layers of Cr₂B₂ instead of other periods is preferred when Al is scanty. Reaction (3) suggests that Cr₄AlB₄ can further react with Al to form Cr₂AlB₂ in Al rich condition. From Reactions (1), (2), (3), enthalpy changes of Reactions (4), (5), (6) are -0.91 eV, -1.28 eV, and -1.17 eV, respectively.

4CrB + 2Al → 2Cr₂AlB₂ (4)

8CrB + 2Al → 2Cr₄AlB₄ (5)

12CrB + 2Al → 2Cr₆AlB₆ (6)

It indicates that bonding of Al to Cr₂B₂ layers in Cr₄AlB₄ may be the strongest, which agrees well with the shortest Al-B bond length in Cr₄AlB₄ (2.214 Å) in comparison to 2.267 Å in Cr₂AlB₂ and 2.255 Å in Cr₆AlB₆. The stiff bonding of Al in Cr₄AlB₄ is also consistent with experimental result which shows that Cr₄AlB₄ is more resistant to corrosion than Cr₂AlB₂ in dilute hydrochloric acid solution.

3.2. Mechanical properties

For an orthorhombic crystal, there are nine independent elastic constants (c₁₁, c₂₂, c₃₃, c₄₄, c₅₅, c₆₆, c₁₂, c₁₃, c₂₃) [31,32], which should satisfy the Born-Huang criteria for mechanical stability [33,34]:

c₁₁>0,c₂₂>0,c₃₃>0,c₄₄>0,c₅₅>0,c₆₆>0
c₁₁+c₂₂+c₃₃+2(c₁₂+c₂₃+c₃₁)>0
c₁₁+c₂₂-2c₁₂>0
c₂₂+c₃₃-2c₂₃>0
c₃₃+c₁₁-2c₃₁>0 (7)

Elastic constants c_ij of Cr₄AlB₄ were calculated using stress-strain approach, where the computed values are listed in Table 2. Elastic constants of Cr₂AlB₂ and two typical MAX phases Ti₂AlC and Ti₃AlC₂ are also listed in Table 2 for comparison. It is clear that the Born-Huang criteria are satisfied for Cr₄AlB₄, indicating the stability of Cr₄AlB₄ under elastic perturbations. Phonon dispersion curves of Cr₄AlB₄ were also calculated, where no imaginary mode was found, conforming the stability of Cr₄AlB₄.

The results in Table 2 reveal that the elastic constants of Cr₄AlB₄ display similar trends to those of MAX phases. Firstly, the elastic constants that represent stiffness against principle (tensile/compression) strains (c₁₁, c₂₂, c₃₃) are much higher than those against shear deformation strains (c₄₄, c₅₅, c₆₆). Secondly, tensile/compression resistance along the layer stacking direction (c₂₂ for Cr₂AlB₂ and Cr₄AlB₄, c₃₃ for Ti₂AlC and Ti₃AlC₂) are low, and shear resistance in the stacking plane (c₄₄ and c₆₆ for Cr₂AlB₂ and Cr₄AlB₄, c₄₄ and c₅₅ for Ti₂AlC and Ti₃AlC₂) are low. Thirdly, c_ij values increase with the thickness of M-X (X = C, N, B) layers, which indicates that the easy deformation is resulted from weak bonding between A layer and M-X layer. The bulk modulus B, shear modulus G, Young's modulus E and Poisson's ratio ν of polycrystalline Cr₄AlB₄ can be evaluated by applying the Voigt-Reuss-Hill approximations [[35], [36], [37]]. The values of B, G, E and ν of polycrystalline Cr₄AlB₄ are 247 GPa, 189 GPa, 452 GPa, 0.196, respectively. The Pugh’s shear to bulk modulus ratio G/B of Cr₄AlB₄ is 0.764. Though the Pugh’s ratio G/B is larger than 0.571 (the criterion to distinguish ductile and brittle materials), it is comparable to experimentally proven damage tolerant MAX phases. In combination with its layered structure, Cr₄AlB₄ can be judged as a damage tolerant MAB phase.

For a comprehensive understanding of elastic properties, elastic anisotropy was analyzed. In principle, for a given direction n (n is a unit vector), the Young’s modulus can be calculated from

1/E_n = s^t_ijkln_in_jn_kn_l (8)

where the Einstein’s summation for repeated indices is employed. In analogous, for a given shear system with shear plane normal n and shear direction b (both n and b are unit vectors), the shear modulus G_nb can be evaluated as

1/G_nb = s^t_ijklb_in_jb_kn_l (9)

where s^t_ijkl is the component of the fourth-rank compliance tensor s^t. The components of s^t_ijkl can be derived from its matrix notation s_ij, which is the inverse of the elastic stiffness matrix c_ij, referring to Nye’s book [38] for more details. Fig. 2a shows the surface of directional dependent Young’s modulus. Fig. 2b and c show the directional dependent Young’s modulus and shear modulus on (100), (010) and (001) plane, respectively. From the figures, it is clearly seen the anisotropic nature of elastic properties. The maximum Young’s modulus E_max in (100) plane is 444 GPa at 0°, while the minimum Young’s modulus E_min in (100) plane is 425 GPa at 56°. E_max in (010) plane is 503 GPa at 90°, while E_min in (010) plane is 429 GPa at 31°. E_max in (001) plane is 491 GPa at 0°, while E_min in (001) plane is 430 GPa at 55°. Herein, 503 GPa in (100) plane and 425 GPa in (010) plane are respectively the maximum and minimum Young’s modulus of Cr₄AlB₄ over all directions. The maximum and minimum shear modulus in (100), (010) and (001) plane are 219 GPa and 173 GPa, 176 GPa and 173 GPa, 219 GPa and 176 GPa, respectively.

It is well known that the distinctive unusual mechanical properties of MAX phases are resulted from the weak bonding between the A element layer and the M-X layer, which leads to preferred cleavage along and/or shear in the A element plane to dissipate deformation energy. It is therefore essential to determine whether Cr₄AlB₄ prefers to cleavage along and/or shear in (010) plane. Stress-strain relations under (100), (010) and (001) tensile deformations and (001)[100], (010)[100] and (010)[001] shear deformations are simulated and the results are shown in Fig. 3. Ideal tensile strengths normal to (100), (010) and (001) plane are 28.5 GPa, 21.9 GPa and 24.2 GPa, respectively. Ideal shear strengths under (001)[100], (010)[100] and (010)[001] shear are 39.1 GPa, 21.0 GPa and 20.4 GPa, respectively. The simulations conform that Cr₄AlB₄ prefers to cleavage along and/or shear in the A element plane, which is similar to other MAB phases, including Cr₂AlB₂, Cr₃AlB₄, Cr₄AlB₆ [10,11] and MoAlB, WAlB [2,12]. It indicates that Cr₄AlB₄, like most other MAB phases, displays similar mechanical properties as those of MAX phase.

3.3. Temperature dependence of properties

MAB phases are regarded as candidate materials for high temperature applications. It is therefore essential to determine temperature dependences of their properties. Herein, temperature dependent properties of Cr₄AlB₄ were predicted by quasi-harmonic approach. In this approach, Helmholtz free energy F(V, T) can be approximated as the summation of static energy, phonon vibrational energy and thermal electron energy [[39], [40], [41]]:

$F(V,T)=E_{static}(V)+k_{B}T\int^{\infty}_{0}\frac{\hbar\omega}{2k_{B}T}+\ln(1-e^{-\frac{\hbar\omega}{k_{B}T}})g(\omega)\text{d}\omega+\int^{\infty}_{0}\varepsilon(f-f_{0})n(\varepsilon)\text{d}\varepsilon$ (10)

where V is the cell volume, T is temperature, k_B is Boltzmann constant, ħ is Planck constant, ω is oscillation circular frequency, g(ω) is phonon DOS, n(ε) is electron DOS, f and f₀ are Fermi-Dirac distribution at temperature T and 0 K, respectively. Using Eq. (10), for a given temperature T, the variation of Helmholtz free energy F with respect to cell volume V can be determined, which is fitted to the Burch-Murnaghan’s state equation [40] to find the minimum energy F and the corresponding cell volume V at temperature T. Then, variation of F, V, a, b and c as functions of T can be obtained, i.e. F(T), V(T), a(T), b(T) and c(T), and thermal expansion coefficients along a, b, c direction can be derived, which are shown in Fig. 4. It can be seen that thermal expansion along b direction is the largest, while thermal expansion along a direction is the lowest. The low thermal expansion along a direction results from the stiff B-B chain, while large thermal expansion along b direction is caused by the weak bonding between Al and Cr₂B₂ layers. The average linear thermal expansion coefficients between 500 and 1000 K along a, b and c directions are 7.5 × 10^-6 K^-1, 8.8 × 10^-6 K^-1, and 8.6 × 10^-6 K^-1, respectively. Therefore, the overall linear thermal expansion coefficient is 8.3 × 10^-6 K^-1.

Temperature dependent elastic constants can be determined by the following three steps [39]:

(i) calculate elastic stiffness constants as a function of cell volume c_ij(V) at 0 K;

(ii) determine temperature dependent isothermal elastic stiffness constants c_ij^T(V) according to c_ij(V) and V(T);

(iii) convert isothermal elastic stiffness constants c_ij^T(V) to isentropic elastic stiffness constants c_ij^S(V) based on the equation given by Davies [42]:

$c^{s}_{ij}(T)=c^{T}_{ij}(V)+\frac{TV\chi_{i}\chi_{j}}{C_{V}}$ (11)

where C_V is the heat capacity at constant volume. χ_i is evaluated as:

$\chi_{i}=-\sum^{6}_{j=1}\alpha_{j}c^{T}_{ij}(V)$ (12)

where α_j is linear thermal expiation coefficient along direction j, which can be determined from components of matrix Λ:

$\Lambda=R^{-1}(\frac{\partial R}{\partial V})_{T=0}\frac{\partial V}{\partial T}$ (13)

with α₁ = Λ₁₁, α₂ = Λ₂₂, α₃ = Λ₃₃, α₄ = Λ₂₃+Λ₃₂, α₅ = Λ₃₁+Λ₁₃, α₆ = Λ₁₂+Λ₂₁. In Eq. (13), R^-1 means the inverse of matrix R, and R is the matrix representation of a lattice unit cell with its row vectors being a, b and c of the lattice unit cell, respectively. For a lattice with orthorhombic symmetry, matrix Λ is in a simple diagonal form with Λ₁₁, Λ₂₂, Λ₃₃ respectively being linear thermal expansion coefficient along a, b, c, i.e. α_a, α_b, α_c, and 0 for non-diagonal components.

Using Eqs. (11), (12), (13), temperature dependent elastic constants c_ij^S of Cr₄AlB₄ can be evaluated. Then, bulk modulus B, shear modulus G, and Young’s modulus E of polycrystalline Cr₄AlB₄ can be deduced according to the Voigt-Reuss-Hill approximations [[35], [36], [37]]. The variations of c_ij^S and B, G, E of Cr₄AlB₄ with respect to temperature T are shown in Fig. 5a and b, respectively. The figures reveal that variations of elastic properties with respect to T are typical, which keep almost constant when T < 200 K, and then decrease linearly with the increase of T. To characterize the speed of decrease in elastic properties, slopes are fitted in the range of 500-1000 K for each elastic property. The decrease speeds of c₁₁, c₂₂, c₃₃, c₄₄, c₅₅, c₆₆, c₁₂, c₂₃, c₃₁ are -33.8, -37.4, -27.5, -14.9, -16.9, -12.4, -6.9, -9.6 and -11.2 MPa K^-1, respectively. The decrease speeds of B, G and E are -17.2, -13.5, and -32.1 MPa K^-1, respectively. The results reveal that decrease of c₂₂ is faster than c₁₁ and c₃₃, which can also be well seen in Fig. 5a. It is consistent with the result that thermal expansion along b direction is the largest, both of which result from the weak bonding between Al and Cr₂B₂ layers.

3.4. Discussions

Ade and Hillebrecht [1] suggested that the general formula of MAB phases in Cr-Al-B system can be written as (CrB₂)_nCrAl (n = 1-3), which assumes that the Cr-B compound layer changes gradually from CrB to CrB₂ with its thickness increase. The discovery of Cr₄AlB₄ opens a new window to synthesize new MAB phases, which can be regarded as inserting Al layers into Cr-B compounds (CrB, Cr₃B₄, Cr₂B₃) with different periods, e.g. every one layer or two layers of Cr₂B₂. The other direction to develop new MAB phases is solid solution on either M site or A site or both. Chai et al. [5] and Okada et al. [6] have carried out the pioneer work on solid solution of MAB phases. Despite those diborides like compounds (CrB, Cr₃B₄, Cr₂B₃), MAB phases can also be derived from other borides, e.g. YB₄, where Y₅Si₂B₈ [24] was found crystallize in layered structure and exhibit similar properties as those of MAX phases. It means that the crystal structures of MAB phases are more diverse than those of MAX phases due to the diverse crystal structures of borides, and it is hopeful that more MAB phases can be discovered in the future.

The results predicted in previous sections reveal that the layered crystal structure and the weak bonding nature between Al and Cr₂B₂ layers dominate the properties. For example, c₂₂ decreases faster than c₁₁ and c₃₃ with respect to temperature; shear resistances in (010) plane are lower than in other planes; thermal expansion along [010] direction is larger than along other directions. Most importantly, failure is proven to preferentially cleavage long and/or shear in (010) plane, which gives rise to Cr₄AlB₄ the unusual mechanical properties in analogous to MAX phases, including readily machinable, thermal shock resistant, and damage tolerant. In combination with the fact that Cr, Al and B all can form dense oxides in oxygen rich environment to protect the material from further oxidation, it is suggested that Cr₄AlB₄ is a promising high temperature ceramic.

4. Conclusion

In the present work, the stability, mechanical properties, thermal expansions and temperature dependent elastic properties of a newly discovered MAB phase Cr₄AlB₄ were investigated by first principles calculations based on DFT. Enthalpy changes of chemical reactions indicate that Cr₄AlB₄ is more stable than Cr₂AlB₂ in Al lean condition, and can transform to Cr₂AlB₂ in Al rich condition. The full set of elastic constants c_ij of single crystalline Cr₄AlB₄ and bulk modulus B, shear modulus G, Young’s modulus E of polycrystalline Cr₄AlB₄ and their dependences on temperature were predicted. Ideal strengths under (100), (010) and (001) tensile deformations and (001)[100], (010)[100] and (010)[001] shear deformations were simulated to clarify the preferred failure model of Cr₄AlB₄. Anisotropic thermal expansions were predicted by quasi-harmonic model, where the values along a, b and c directions are 7.5 × 10^-6 K^-1, 8.8 × 10^-6 K^-1, and 8.6 × 10^-6 K^-1, respectively. The results reveal that the layered crystal structure and weak bonding nature between Al and Cr₂B₂ layers impact the properties significantly. For example, (1) elastic stiffness along [010] direction (the stacking direction of Al and Cr₂B₂ layers) is relatively low and decreases faster with respect to temperature than along other directions, (2) elastic stiffness is lower in (010) plane than in other planes, (3) failure is proven to preferentially cleavage long and/or shear in (010) plane, (4) thermal expansion along [010] direction is larger than along other directions. It is therefore suggested that the newly discovered MAB phase Cr₄AlB₄ may display similar properties in analogous to MAX phases, which warrants it a promising high temperature ceramic.

Acknowledgements

This work was supported by National Natural Science Foundation of China under Grant No. U1435206 and No. 51672064.

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