DFT Prediction of Structural and Physical Properties of Cr₃AlC₂ Under Pressure

Abstract

This work explores the physical properties of the MAX-phase material Cr₃AlC₂ through the application of density functional theory (DFT). The refined lattice parameters were determined through the minimization of the total energy. In order to explore the electronic properties and bonding features, we carried out computations on the band structure and charge density distribution. The calculated elastic constants (C_ij) validated the mechanical stability of Cr₃AlC₂. To assess the material’s ductility or brittleness, we calculated Pugh’s ratio, Poisson’s ratio, and Cauchy pressure. The hardness was determined. This study examined the anisotropic behavior of Cr₃AlC₂ using directional analyses of its elastic properties and by computing relevant anisotropy indicators. We examined several key properties of Cr₃AlC₂, including the Grüneisen parameter, acoustic characteristics, Debye temperature, thermal conductivity, melting point, heat capacity, Helmholtz free energy, entropy, and internal energy. Phonon dispersion spectra were analyzed to assess the dynamic stability of Cr₃AlC₂.

1. Introduction

Cr₃AlC₂, which belongs to the renowned MAX-phase family, exhibits unique attributes that combine the advantages of both metals and ceramics. The material is distinguished by its reduced density, superior resistance to oxidation, notable ductility, efficient electrical conductivity, self-lubricating properties, and favorable machinability [1,2,3,4,5]. These exceptional characteristics make Ti₃AlC₂ a material of significant technological interest, with great potential for applications at high temperatures. As a result, both Ti₃AlC₂ and its structurally analogous compound Ti₃SiC₂ have been extensively investigated [6]. These results are emphasized in a recent review article [7], focusing on layered, electrically conductive Ti₂AlC and Ti₃AlC₂, as well as in another study by Barsoum [8]. Much of the prior research has focused on comprehending and modeling M₂AX materials [9,10,11,12]. For M₃AX₂ phases, only a limited number of compounds have been examined, with Ti₃SiC₂ and Ti₃AlC₂ being notable exceptions [13]. However, as mentioned earlier, many M₃AX₂ compounds have recently been synthesized. For example, Ta₃AlC₂ was synthesized from metallic melts [14], whereas Ti₃SnC₂ was discovered by Dubois and colleagues [15]. Thus, a theoretical study of M₃AX₂ phases is both pertinent and essential for gaining insights into their characteristics. To date, it is commonly acknowledged that Ti₃AlC₂ and Ti₂AlC exhibit superior oxidation resistance compared to Ti₃SiC₂ [16,17]. This exceptional oxidation resistance is believed to result from the development of a protective, dense Al₂O₃ surface layer. Given the assumption that other M₃AlC₂ compounds might also exhibit oxidation resistance similarly to Ti₃AlC₂, studying them would prove beneficial. To the best of our knowledge, apart from the significant case of Ti₃AlC₂, the properties of M₃AlC₂ compounds have not been thoroughly explored. The purpose of this study is to fill this research gap. The overall characteristics of the crystal structure, electronic properties, and elastic behavior of Cr₃AlC₂ were thoroughly examined.

2. Theoretical Methods

The characteristics of Cr₃AlC₂ were investigated by employing the DFT method, utilizing the Vienna ab initio simulation code [18]. The electron–nucleus interaction was modeled using pseudopotentials created through the projector-augmented wave (PAW) approach [19]. The exchange–correlation energy was calculated based on the generalized gradient approximation (GGA) in conjunction with the Perdew–Burke–Ernzerhof (PBE) functional [20]. Convergence tests indicated that a cutoff energy of 480 eV and k-point mesh of 8 × 8 × 10 were appropriate for achieving accurate results. An energy convergence criterion of 10⁻⁴ eV/cell was used, together with a force convergence threshold of 0.001 eV/Å for minimizing Hellmann–Feynman forces. Furthermore, the dynamic stability of Cr₃AlC₂ was assessed through phonon frequency analysis. Phonon computations were performed utilizing a 2 × 1 × 2 supercell, employing the Phonopy package [21,22].

3. Results and Discussion

3.1. Structural Properties

As shown in Figure 1, Cr₃AlC₂ adopts a hexagonal crystal structure belonging to the P63/mmc (194) space group.

Table 1. The optimized lattice parameters a and c (Å), density ρ (g/cm³), and volume V (ų) of Cr₃AlC₂.

Pressure (GPa)		a	c	ρ	V
0	Present	2.8699	17.3922	5.5415	124.052
0	Ref. [23]	2.8508	17.272	5.66	121.56
10	Present	2.8260	17.1720	5.7881	118.767
20	Present	2.7896	16.9888	5.9269	115.986
30	Present	2.7572	16.8486	6.0464	113.693
40	Present	2.7287	16.7252	6.1557	111.694
50	Present	2.7040	16.6027	6.2567	109.872

presents the calculation results of this study as well as the relevant reference data [23]. The resulting values show reasonable consistency when contrasted with theoretical predictions. Notably, the difference between the computed lattice parameters and the theoretical values remains below 0.69%. Furthermore, Table 1 also provides the pressure-dependent lattice parameters, density, and volume.

3.2. Electronic Properties

In Figure 2, the band structure of Cr₃AlC₂ is presented. At E_F (Fermi energy level), the conduction and valence bands overlap, confirming the metallic nature of Cr₃AlC₂. Additionally, we calculated the density of states for Cr₃AlC₂ to analyze the bonding properties and contributions of various electronic states. The valence band is divided into two distinct sub-bands. (i) The lowest valence band spans from −15.0 to −10 eV for Cr₃AlC₂, where most of the states in this total density of states region are attributed to C-s orbitals, with minor contributions from Cr-s and Cr-p states. (ii) The upper valence band of Cr₃AlC₂ lies between −10 and 0.0 eV. It should be emphasized that the hybridization peak between Cr-d and C-p states appears in the lower-energy region, whereas the Cr-d peak is situated in the higher-energy range, close to the Fermi level (E_F) (Figure 3).

Figure 4 displays the charge density mapping (CDM) on the (101) plane, which provides insights into the bonding interactions among different atoms. With increasing pressure from 0 to 50 GPa, a significant enhancement in charge density is evident at specific locations. This elevated charge density enhances the bonding between Cr and C atoms, attributed to the overlapping of their electronic states. The CDM results indicate that the Cr-C bond exhibits greater strength compared to the Al-C bond.

3.3. Mechanical Properties

The elastic constants C_ij of Cr₃AlC₂ under various pressure conditions are presented in

Table 2. Calculated C_ij (GPa), elastic moduli (GPa), and E (GPa) of Cr₃AlC₂.

	Pressure (GPa)	C11	C12	C13	C33	C44	C66	BV	BR	B	GV	GR	G	E
Present	0	359.5	114.8	136.8	368.9	119.0	122.4	207.2	206.9	207.0	118.7	118.5	118.6	298.8
Theo. [23]	0	381	100	136	381	118	141			210			127	317
Present	10	408.9	134.9	169.4	415.9	148.9	137.0	242.3	241.8	242.1	137.6	136.4	137.0	345.8
Present	20	446.3	152	205.3	460.2	173.6	147.2	275.3	273.9	274.6	151.6	148.1	149.8	380.3
Present	30	484.7	172.7	238.4	491.7	195.3	156.0	306.7	305.0	305.8	163.4	157.0	160.2	409.2
Present	40	524.5	195.5	270.7	528.5	214.8	164.5	339.0	337.0	338.0	174.9	165.8	170.3	437.5
Present	50	554.4	214.9	299.0	565.0	233.2	169.7	366.6	363.8	365.2	184.6	172.7	178.6	460.8

. Based on the Born criteria, the mechanical stability requirements at equilibrium are influenced by different crystal symmetries [24]:

C₄₄ > 0; C₁₁ + C₁₂ − 2C₁₃²/C₃₃ > 0; and C₁₁ − C₁₂ > 0

(1)

As shown in Table 2, the calculated C_ij values for Cr₃AlC₂ align well with the corresponding theoretical values [23], suggesting that the current computational results are accurate. Based on the Voigt [25] and Reuss [26] methods, we calculated the values of bulk modulus (B) and shear modulus (G) using Hill’s approximation [27]. The computed values are summarized in Table 2, together with other previously reported results [23]. As shown in Table 2, the calculated B, G, and Young’s modulus (E) of Cr₃AlC₂ at 0 GPa are 207.0 GPa, 118.6 GPa, and 298.8 GPa, respectively, which align well with earlier findings.

Table 3. Calculated G/B, Poisson’s ratio v, C₁₃–C₄₄, and C₁₂–C₆₆ of Cr₃AlC₂.

Pressure (GPa)		G/B	v	C13–C44	C12–C66
0	Present	0.573	0.2595	17.8	−7.6
0	Theo. [23]	0.61	0.25
10	Present	0.566	0.2619	20.5	−2.1
20	Present	0.546	0.2692	31.7	4.8
30	Present	0.524	0.2770	43.1	16.7
40	Present	0.504	0.2843	55.9	31
50	Present	0.493	0.2897	65.8	45.2

presents Poisson’s ratio (v), G/B ratio, C₁₃–C₄₄, and C₁₂–C₆₆ values for Cr₃AlC₂ [28]. The mechanical behavior of brittleness or ductility can be evaluated using these parameters. A material is considered brittle if ν < 0.33, G/B > 0.57, C₁₃–C₄₄ < 0, and C₁₂–C₆₆ < 0; otherwise, it is classified as ductile. In Table 3, it is clear that Cr₃AlC₂ exhibits Poisson’s ratios (ν) below 0.33, G/B < 0.57, negative C₁₃–C₄₄ values, and negative C₁₂-C₆₆ values. These results indicate that Cr₃AlC₂ demonstrates brittle characteristics.

According to the literature [29,30,31], we can calculate various anisotropic parameters of the material such as three shear anisotropic factors, A₁, A₂, and A₃, percentage anisotropy in compressibility (A_B) and shear (A_G), and the universal anisotropic index A^U. As shown in

Table 4. Calculated elastic anisotropic factors of Cr₃AlC₂.

Pressure (GPa)	AU	AB (%)	AG (%)	A1	A2	A3
0	0.0064	0.0006	0.0001	1.0466	1.0466	1
10	0.0508	0.0044	0.0034	1.2255	1.2255	1
20	0.1206	0.0111	0.0040	1.4003	1.4003	1
30	0.2135	0.0203	0.0031	1.5637	1.5637	1
40	0.2817	0.0271	0.0017	1.6794	1.6794	1
50	0.3588	0.0338	0.0046	1.7890	1.7890	1

, the Cr₃AlC₂ under consideration exhibits anisotropic behavior since the values of A₁, A₂, and A₃ differ from unity. The non-zero values of A_B, A_G, and A^U (as shown in Table 4) confirm the anisotropic nature of Cr₃AlC₂.

To offer more distinct and thorough insight into the anisotropic properties of the investigated materials, we employed both two-dimensional (2D) and three-dimensional (3D) graphical representations of linear compressibility β, E, and G by ELATE software [32,33,34]. Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10 present the 2D and 3D visualizations produced. A perfect sphere represents complete isotropy in solids, while deviations from this spherical shape indicate varying degrees of anisotropy (Figure 5, Figure 6 and Figure 7). Figure 8 illustrates the anisotropy in β, where the xy-plane shows isotropic behavior, and the xz- and yz-planes show anisotropic behavior. E also demonstrates anisotropic behavior, as depicted in Figure 9. This figure reveals that E exhibits isotropic properties within the xy-plane but displays anisotropic characteristics in the xz- and yz-planes. The graphs indicate that for Cr₃AlC₂, the blue line achieves its peak values along the axes in both the xz- and yz-planes, while it reaches its lowest value at a 45-degree angle relative to these axes. The shear modulus presents two surfaces in both 2D and 3D representations, as shown in Figure 10. Additionally,

Table 5. The values of β_max/β_min, G_max/G_min, and E_max/E_min of Cr₃AlC₂ under various pressures.

Pressure (GPa)	βmax/βmin	Gmax/Gmin	Emax/Emin
0	1.1565	1.08	1.042
10	1.2024	1.234	1.175
20	1.3580	1.421	1.302
30	1.4025	1.599	1.44
40	1.4434	1.727	1.536
50	1.5528	1.847	1.607

provides the β_max/β_min, G_max/G_min, and E_max/E_min values of β, E, and G for Cr₃AlC₂, confirming its significant anisotropic properties. As the pressure increases, the anisotropy also strengthens.

In this study, we computed both the Chen hardness (H_chen) and the Miao hardness (H_miao) [35,36].

$$H_{Chen}=2{[{({\displaystyle \frac{G}{B}})}^{2}G]}^{0.585}-3$$

(2)

$$H_{Miao}={\displaystyle \frac{(1-2v)E}{6(1+v)}}$$

(3)

Fine’s principle establishes the criteria for defining the melting point of a solid [37,38].

T_m = 354 + 4.5(2C₁₁ + C₃₃)/3

(4)

The thermal conductivity κ of the material is obtained from the Clarke model [39]:

$${\kappa }_{\min }=0.87k_{B}M{a}^{-{\displaystyle \frac{2}{3}}}{\rho }^{{\displaystyle \frac{1}{6}}}{E}^{{\displaystyle \frac{1}{2}}}$$

(5)

Here, k_B is Boltzmann’s constant, ρ is the density, M_a is the average mass per atom, and Young’s modulus (E).

Furthermore, based on a method in the literature, we were able to calculate the average sound velocity v_m, the longitudinal velocity (v_l), the transverse velocity (v_t), and the Debye temperature θ [40]. In addition, the Grüneisen γ_a can be described by v_l and v_t [41]:

$${\gamma }_a={\displaystyle \frac{3}{2}}({\displaystyle \frac{3v_l^2-4v_t^2}{v_l^2+2v_t^2}})$$

(6)

The changes in v_l, v_t, v_m, and θ under various pressures are summarized in

Table 6. The hardness (H_Chen and H_Miao) (GPa), wave velocity (v_l, v_t, v_m) (km/s), Debye temperature θ (K), Grüneisen constant γ_a, melting point T_m (K), and thermal conductivity k (W.m⁻¹ K⁻¹) of Cr₃AlC₂.

Pressure (GPa)	HChen	HMiao	vl	vt	vm	θ	γa	Tm	k
0	14.03	19.02	8.1173	4.6262	5.1417	702.4	1.5465	1985.9	1.8586
Ti3AlC2 0 [13]	9.2
Ti3SiC2 0 [13]	11.1
10	15.27	21.75	8.5666	4.8651	5.4088	749.7	1.5591	2204.6	2.0140
20	15.44	23.05	8.9460	5.0274	5.5941	781.6	1.5966	2383.2	2.1205
30	15.29	23.82	9.2684	5.1473	5.7330	806.3	1.6386	2545.7	2.2069
40	15.11	24.49	9.5810	5.2598	5.8635	829.6	1.6794	2720.3	2.2888
50	14.99	25.05	9.8199	5.3428	5.9601	847.9	1.7109	2864.7	2.3553

. The θ, v_l, v_t, and v_m at 0 GPa are determined to be 702.4 K, 7.1182 km/s, 3.8357 km/s, and 4.2820 km/s, respectively. The values for H_V, T_m, γ_a, and κ of Cr₃AlC₂ are provided in Table 6. The melting point of Cr₃AlC₂ is recorded as 1985.9 K under 0 GPa. If the maximum operational temperature is estimated as 0.8 T_m, then the maximum service temperature of Cr₃AlC₂ approaches approximately 1300 °C, suggesting that Cr₃AlC₂ possesses favorable thermal stability for practical applications. As shown in Table 6, when the pressure increases, H_V, T_m, γ_a, and k tend to rise. Furthermore, the hardness values for Cr₃AlC₂ obtained using Chen’s and Miao’s models at 0 GPa are 11.189 and 15.455 GPa, respectively. And our hardness is slightly greater than that of materials of the same type (Ti₃AlC₂ and Ti₃SiC₂) [13]. The hardness values calculated using the two methods are presented in Table 6. It has been observed that there is strong concurrence between them, suggesting the accuracy of the computation.

3.4. Dynamical Properties

To ensure that the dynamical stability is thoroughly examined, it becomes imperative to conduct an analysis of the phonon dispersion of Cr₃AlC₂, when subjected to elevated pressure. The phonon dispersion and phonon density of state for Cr₃AlC₂ were calculated at 0 GPa, 30 GPa, and 50 GPa, as depicted in Figure 11 and Figure 12. It is clear that Cr₃AlC₂ exhibits dynamical stability, as no imaginary phonon frequencies are detected in this crystal structure. In correlation with the representation of the phonon density of state illustrated in Figure 12, the frequency range of 0–15 THz primarily corresponds to Cr and Al vibrations, while the frequency range of 15–25 THz predominantly corresponds to C vibrations.

3.5. Thermal Properties

The thermodynamic potential functions of Cr₃AlC₂ were calculated based on the phonon density of states obtained using the quasi-harmonic approximation [42]. These functions include the internal energy E, Helmholtz free energy F, and entropy S. Figure 13 shows S and C_V as functions of temperature. Below 50 K, the S values are nearly zero, as shown in Figure 13a. Above 100 K, there is a noticeable nonlinear increase, a trend that is quite common and becomes increasingly evident with increasing temperature as natural processes evolve. Figure 13b depicts C_V for Cr₃AlC₂ across the temperature interval from 0 to 1000 K. At T < 300 K, C_V follows the Debye T³ power law, while at T > 800 K, it closely aligns with the Dulong–Petit model.

Table 7. The calculated F (KJ/mol) and E (KJ/mol) of Cr₃AlC₂ under various pressures.

Temperature/K	0 GPa		30 GPa		50 GPa
	F	E	F	E	F	E
300	43.31	114.81	62.43	122.38	72.42	126.92
400	15.85	139.34	39.03	145.54	50.96	149.30
500	−18.04	165.71	9.51	170.89	23.55	174.06
600	−57.36	193.14	−25.26	197.57	−9.00	200.30
700	−101.32	221.23	−64.57	225.09	−46.02	227.47
800	−149.34	249.75	−107.86	253.17	−86.99	255.28
900	−200.95	278.57	−154.69	281.63	−131.46	283.53
1000	−255.79	307.61	−204.71	310.38	−179.10	312.09

presents F and E at various temperatures and pressures. These thermodynamic results are expected to provide a valuable reference for the practical utilization of Cr₃AlC₂.

4. Conclusions

In this study, a thorough investigation of Cr₃AlC₂ was conducted based on DFT. The mechanical properties were examined in detail under varying pressure conditions. Additionally, the calculated phonon dispersion further validates the material’s dynamical stability. This compound demonstrates a reasonable level of mechanical anisotropy alongside sufficient hardness. All thermal properties exhibit distinctive behavior across the pressure range from 0 to 50 GPa. However, these properties (θ, γ_a, T_m, and k) gradually increase as pressure rises to 50 GPa. We believe that these results will encourage researchers to delve deeper into this material, both through theoretical modeling and experimental exploration.