Anisotropic Elastic and Thermal Properties of M₂InX (M = Ti, Zr and X = C, N) Phases: A First-Principles Calculation

Abstract

First-principles calculations were used to estimate the anisotropic elastic and thermal properties of Ti₂lnX (X = C, N) and Zr₂lnX (X = C, N) M₂AX phases. The crystals’ elastic properties were computed using the Voigt-Reuss-Hill approximation. Firstly, the material’s elastic anisotropy was explored, and its mechanical stability was assessed. According to the findings, Ti₂lnC, Ti₂lnN, Zr₂lnC, and Zr₂lnN are all brittle materials. Secondly, the elasticity of Ti₂lnX (X = C, N) and Zr₂lnX (X = C, N) M₂AX phase are anisotropic, and the elasticity of Ti₂lnX (X = C, N) and Zr₂lnX (X = C, N) systems are different; the order of anisotropy is Ti₂lnN > Ti₂lnC, Zr₂lnN > Zr₂lnC. Finally, the elastic constants and moduli were used to determine the Debye temperature and sound velocity. Ti₂lnC has the maximum Debye temperature and sound velocity, and Zr₂lnN had the lowest Debye temperature and sound velocity. At the same time, Ti₂lnC had the highest thermal conductivity.

1. Introduction

The ternary layered compound MAX phase has the common characteristics of ceramics and metals, and has become a research hotspot in the field of structural ceramics for more than 20 years. High damage tolerance and high fracture toughness are essential characteristics that distinguish it from traditional ceramics [1,2,3,4,5,6]. This type of material has a similar nano-layered crystal structure (space group P6₃/mmc), which is named the “M_n+1AX_n” phase, referred to as MAX phase, where M is a transition metal element, A is a IIIA or IVA group element, and X is C or N, (n = 1~3) [7,8,9]. MAX phase compounds can be regarded as the formation of a layer of main group atoms inserted into the binary carbon/nitride lattice. Characterization shows the extraordinary properties of MAX phase materials. They have many excellent properties, such as high damage tolerance, high fracture toughness, low hardness, machinability, thermal shock resistance, high damping, high stiffness, and good electrical and thermal conductivity [10,11], etc. Among them, 211-type MAX phase compounds also have similar properties. This extraordinary performance quickly attracted widespread attention in the field of ceramics.

At present, Z. J. Yang et al. have carried out many theoretical studies on the novel hysteresis behavior of Zr₂InC based on plane-wave pseudopotential (PW-PP) density functional theory (DFT) calculations, and the mechanical and electronic properties of Zr₂InC under pressure were investigated using first principles [12]. A.D. Bortolozo et al. investigated the Ti₂InN phase by X-ray diffraction and magnetic and resistivity measurements, and the results clearly showed that Ti₂InN is the first nitride superconductor belonging to the M_n+1AX_n family [13]. Sun et al. investigated the relationship between chemical bonds and the elastic properties of M₂AN (M = Ti, Zr, Hf, V, Nb, Ta, Cr, Mo and W, A = Al, Ga and Ge) by calculation [14]. The study showed that with the increase of valence electron concentration, the bulk modulus of M₂AN increases by a factor of 1.8, and the coupling between the MN layer and the A layer weakens. Despite these studies, theoretical studies of the M₂lnX phase are not sufficient. Previously, MAX carbides have received the greatest attention, while MAX nitrides have received less. In general, the nitride phase and the carbide phase are very similar. Therefore, first-principles calculations are employed to investigate the structure, elastic anisotropy, and thermodynamic properties of Ti₂lnX (X = C, N) and Zr₂lnX (X = C, N) M₂AX phase ceramics, providing a comprehensive complement to previously unexplored areas.

2. Methods

In this work, M₂InX (M = Ti, Zr and X = C, N) MAX phases were analyzed using the Cambridge Sequential Total Energy Package (CASTEP) [15,16] of Density Functional Theory (DFT) [17]. The interactions between electrons and ionic nuclei were calculated using on-the-fly generation (OTFG), ultrasoft pseudopotentials (USPPs), and the Perdew–Wang generalized-gradient approximation (PW91) [18,19,20,21] method in generalized gradient approximation (GGA) was utilized to model exchange correlation potential. In this work, the geometric optimization tolerance was set to 5 × 10⁻⁶ eV/atom of total energy difference, the maximum ionic Hellmann–Feynman force was 0.01 eV/atom, the maximum ion displacement was 5 × 10⁻⁴, and the maximum stress was 0.02 GPa; the M₂InX phases plan-wave cutoff energy was set to 450 eV and the k-point in the first irreducible Brillouin zone was selected as 10 × 10 × 2. In this calculation, we conducted relevant tests on 1 × 1 × 1, 1 × 1 × 2, 1 × 2 × 2, and 2 × 2 × 2 supercells, and found that when using 1 × 1 × 2, 1 × 2× 2, 2 × 2 × 2 supercells, the resulting energy change was about 1 meV/atom. The total number of atoms in 1 × 1 × 1, 1 × 1 × 2, 1 × 2 × 2, and 2 × 2 × 2 supercells were 8, 16, 32, and 64, respectively. Considering the computational cost and time, this study finally decided to use a 1 × 1 × 2 supercell for the calculation.

3. Results and Discussion

3.1. Single-Crystal Structural Properties

Firstly, the most stable hexagonal M₂InX (M = Ti, Zr and X = C, N) MAX phase crystal structure was determined, as shown in Figure 1. Fully relaxing atomic locations and lattice parameters yielded a crystal structure in equilibrium.

Table 1. The lattice parameters, cohesive energy E_c (in eV/atom), and formation enthalpy ∆H_f (in eV/atom) of M₂InX (M = Ti, Zr and X = C, N) MAX phases.

	Lattice Parameters (Å)		V (Å3)	Ec	∆Hf	Ref.
	a	c
Ti2lnC	3.15	14.18	121.67	−7.81	−0.665	This work
	3.14	14.17	120.74		−0.669	Exp. [22]
Ti2lnN	3.10	14.05	116.76	−8.02	−0.667	This work
				−8.01		Exp. [13]
Zr2lnC	3.36	15.04	147.47	−8.30	−0.690	This work
	3.35	15.04	146.62			Theo. [12]
	3.349	14.91	144.8			Theo. [12]
	3.358	15.09	147.437			Theo. [12]
Zr2AlC					−0.40	Theo. [24]
Zr2BiC					−4.17	Theo. [24]
Zr2lnN	3.31	14.92	141.25	−8.51	−0.751	This work
				−8.12		Theo. [23]

displays the computed lattice parameters a and c, as well as their volumes and formation enthalpies. The calculated structural parameters of the M₂InX phase are in agreement with other experimental results [12,22]. This demonstrates the dependability and precision of the simulations presented in this study, and provided us with the confidence to continue investigating the characteristics of Ti₂lnX (X = C, N) and Zr₂lnX (X = C, N) ceramics. Ti₂lnX (X = C, N) and Zr₂lnX (X = C, N) crystallize in the P6₃/mmc-space group with the Wyckoff positions of 4f (1/3, 2/3, Z_Zr/Ti) for Zr/Ti atoms, 2d (1/3, 2/3, 3/4) for In atoms, and 2a (0, 0, 0) for N/C atoms, and the unit cell was made up of 8 atoms, including 2X, 4M, and 2ln atoms. The atoms were aligned along the c-axis in the order X-ln-M-X, as shown in Figure 1.

The computed lattice parameters a and c, their volumes, the formation enthalpy, and cohesive energy are listed in Table 1. As we all know, thermodynamic stability can be understood to some extent as the concept of minimal energy, which states the lower the energy, the greater the system’s thermodynamic stability. As a result, thermodynamic stability is described using quantities such as formation enthalpy ΔH_f, and cohesive energy E_c. When the formation enthalpy and cohesive energy are both negative, the structure is stable. Therefore, the greater the negative value, the greater the substance’s stability. The calculation is as follows:

$$E_c\left({\mathrm{M}}_2\mathrm{lnX}\right)=\frac{1}{8}\left[E\left({\mathrm{M}}_2\mathrm{lnX}\right){-4E}_{\mathit{iso}}\left(\mathrm{M}\right){-2E}_{\mathit{iso}}\left(\ln \right){-2E}_{\mathit{iso}}\left(\mathrm{X}\right)\right]$$

(1)

$$\mathsf{\Delta }{H}_f\left({\mathrm{M}}_2\mathrm{lnX}\right)=\frac{1}{8}\left[E\left({\mathrm{M}}_2\mathrm{lnX}\right){-4E}_{\mathit{bulk}}\left(M\right){-2E}_{\mathit{bulk}}\left(\ln \right){-2E}_{\mathit{bulk}}\left(X\right)\right]$$

(2)

Here, E is the total energy of M₂lnX. E_bulk(M), E_bulk(ln) and E_bulk(X) represent the energies of single M, ln, and X atoms in a stable state, respectively. E_iso(M), E_iso(ln), and E_iso(X) are the energies of isolated M, ln, and X atoms, respectively.

Table 1 and Figure 2 show the computed cohesive energy E_c and formation enthalpy ΔH_f of Ti₂lnC, Ti₂lnN, Zr₂lnC, and Zr₂lnN, as well as other experimental data for related compounds [12,13,22–24]. The predicted E_c and ΔH_f values for the compounds in Table 1 are negative, indicating that Ti₂lnC, Ti₂lnN, Zr₂lnC, and Zr₂lnN are thermodynamically stable and have stronger bond strengths. Furthermore, the sequence of E_c and ΔH_f values is Ti₂lnC > Ti₂lnN and Zr₂lnC > Zr₂lnN. Therefore, the order of thermodynamic stability is Ti₂lnN > Ti₂lnC and Zr₂lnN > Zr₂lnC. The comprehensive comparison shows that Zr₂lnN and Ti₂lnN have the highest thermodynamic stability. If Zr₂lnN and Ti₂lnN continue to be compared, it can be seen that Zr₂lnN has the highest thermodynamic stability, which is shown in Figure 2.

3.2. Elastic Properties

This work is based on the stress-strain approach of Hooke’s law to analyze mechanical stability, and the elastic constants were determined by applying normal stress, shear stress, and six different deformations.

Table 2. The elastic constants C_ij (in GPa) of M₂InX (M = Ti, Zr and X = C, N) MAX phases.

	C11	C33	C44	C66	C12	C13	Ref.
Ti2lnC	286	244	90	112	64	53	This work
	285	243	83	111	64	52	Theo. [22]
Ti2lnN	249	228	87	87	54	99	This work
Zr2lnC	256	241	82	96	64	96	This work
	258	237	83		64	88	Theo. [12]
Zr2lnN	242	225	89	83	57	88	This work
	237	199	76		55	82	Theo. [23]

shows the calculated elastic constants C_ij and elastic compliance constants S_ij of Ti₂lnC, Ti₂lnN, Zr₂lnC, and Zr₂lnN. Only five elastic constants (C₁₁, C₁₂, C₁₃, C₃₃, C₄₄) are independent for hexagonal crystals [25]. The mechanical stability criteria for hexagonal crystals are given by the following equation [26,27,28]:

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

(3)

Combining the elastic constants calculated in Table 2 and Figure 3, it is not difficult to see that these constants satisfy all the above conditions, proving that Ti₂lnC, Ti₂lnN, Zr₂lnC, and Zr₂lnN are mechanically stable. Usually, the elastic constant represents some important physical meaning, for example, C₁₁ and C₃₃ correspond to the linear compression resistance of the a and c axes, respectively. Table 2 reflects that the C₃₃ value of M₂lnX is significantly smaller than the C₁₁ value, indicating that M₂lnX has a high compressibility along the c-axis. To begin with, for the Ti₂lnX (X = C, N) system, Ti₂lnC has the highest C₁₁ value (286 GPa) and C₃₃ value (244 GPa), while Ti₂lnN has the lowest C₁₁ value (229 GPa) and C₃₃ value (228 GPa), suggesting that Ti₂lnC is the least compressible along the a-axes and c-axes, while Ti₂lnN is the most compressible along the a-axes and c-axes. This is consistent with Ti₂lnC having the strongest chemical bond and Ti₂lnN having the weakest chemical bond [13,22]. It can also be seen that for the system of Zr₂lnX (X = C, N), Zr₂lnC is the most incompressible along the a- and c-axes, while Zr₂lnN is the most compressible along the a- and c-axes. From the comparison of the above two systems, it can be understood that Ti₂lnC and Zr₂lnC are the most incompressible of the two systems, respectively. If the values of C₁₁ and C₃₃ are continuously compared in the M₂lnC (M = Ti, Zr) system, it can be seen that Ti₂lnC > Zr₂lnC, and therefore the compressibility of Ti₂lnC along the a-axis and c-axis is the smallest, which also confirms that Ti₂lnC has a great elasticity constant.

It is well-known that C₁₂, C₄₄, and C₆₆ are related to shear modulus, and larger values of C₁₂, C₄₄, and C₆₆ correspond to larger shear modulus. For Ti₂lnX (X = C, N) the order of C₁₂, C₄₄, and C₆₆ is Ti₂lnC > Ti₂lnN, while for Zr₂lnX (X = C, N) the order of C₁₂, C₄₄, and C₆₆ is Zr₂lnC > Zr₂lnN. Therefore, Ti₂lnC and Zr₂lnC should have the highest shear moduli. Similarly, by comparing the values of C₁₂, C₄₄, and C₆₆ in the M₂lnC (M = Ti, Zr) system, it can be seen that Ti₂lnC > Zr₂lnC, which proves that Ti₂lnC has the highest shear modulus.

3.3. Elastic Moduli

In general, using elastic modulus as a parameter to measure the mechanical properties of polycrystals is more convincing than using the elastic constant. In practice, because most materials are polycrystalline, the applied elastic modulus can more accurately describe the anisotropy of the material than the elastic constant. The Reuss-Hill-Voigt approximation is generally used to obtain the elastic modulus for M₂InX phase, which contains Young’s modulus E, bulk modulus B, and shear modulus G [24,29,30]. The following are the specific expressions of B_H, E_H, and G_H:

$$G_H=\frac{\left(G_V+G_R\right)}{2}$$

(4)

$$B_H=\frac{\left(B_R+B_V\right)}{2}$$

(5)

$$E=\frac{9G_{\mathrm{H}}{B}_{\mathrm{H}}}{\left(3B_{\mathrm{H}}+G_{\mathrm{H}}\right)}$$

(6)

Bulk modulus can be specifically divided into lower bulk modulus (B_V) and upper bulk modulus (B_R). Likewise, the shear modulus has a lower shear modulus (G_V) and an upper shear modulus (G_R).

Table 3. Calculated bulk modulus B (in GPa), shear modulus G (in Gpa), Poisson’s ratio ν, and Young’s modulus E (in GPa) of M₂InX (M = Ti, Zr and X = C, N) MAX phases.

	BV	BR	BH	GV	GR	GH	BH/GH	E	ν	Ref.
Ti2lnC	128.2	127.2	127.7	101.4	100.5	100.9	1.26	239.5	0.187	This work
Ti2lnN
Zr2lnC
Zr2lnN
			127					232.04	0.176	Theo. [31]
Ti2lnN	123.4	122.7	123.1	78.6	77.8	82.7	1.5	198.1	0.226	This work
Ti2lnN
Zr2lnC
Zr2lnN
			121.78							Theo. [13]
Zr2lnC	137.4	137.3	137.4	89.5	88.5	89	1.54	216.1	0.217	This work
Ti2lnN
Zr2lnC
Zr2lnN
			126.03							Exp. [12]
Zr2lnN	134.8	134.8	134.8	82.9	82.2	82.5	1.63	205.7	0.245	This work
Ti2lnN
Zr2lnC
Zr2lnN
			134			79		196		Theo. [23]

shows the estimated and reference data for the elastic modulus M₂lnX compound [12,13,23,31]. The results of this computation are obviously close to the findings stated in the reference. Bulk moduli are frequently used to define a material’s compressibility under hydrostatic pressure. A higher bulk modulus indicates less compressibility. To put it another way, the higher the bulk modulus, the stiffer the material. It can be seen from Table 3 that for Ti₂lnX (X = C, N), the B_H value of Ti₂lnC is greater than that of Ti₂lnN, indicating that Ti₂lnC has higher incompressibility. For Zr₂lnX (X = C, N), Zr₂lnC has higher incompressibility. It is well-known that the larger bulk modulus in compounds originates from the strong hybridization between orbitals. Therefore, Ti₂lnC and Zr₂lnC should have the strongest orbital hybridization.

Furthermore, the stiffness of a material can be defined by its Young’s modulus; the greater the Young’s modulus, the greater the stiffness of the material. Table 3 shows that Ti₂lnC and Zr₂lnC have the maximum stiffness in the Ti₂lnX (X = C, N) and Zr₂lnX (X = C, N) systems, respectively. Ti₂lnC has the highest stiffness in the M₂lnC (M = Ti, Zr) system, because its Young’s modulus is greater than that of Zr₂lnC. Additionally, the shear modulus can be used to forecast the hardness of a material. In general, the shear modulus characterizes the degree to which the shape of the material is affected by the shear force acting on it. The larger the shear modulus, the greater the corresponding deformation resistance. Therefore, the greater the shear modulus, the higher the hardness and strength of the material. From the conclusion drawn from the elastic constants in the previous section, it can be seen that Ti₂lnC and Zr₂lnC have the highest shear moduli, so the hardness of both Ti₂lnC and Zr₂lnC is highest. Further comparison of the M₂lnC (M = Ti, Zr) system shows that Ti₂lnC has the largest shear modulus, so the deformation resistance of Ti₂lnC is the largest. Finally, Figure 4a shows the comparison between B, G, and E. Generally, the plastic properties of materials can be characterized by B/G. Judging whether the material is ductile or brittle depends on the value of B/G. If B/G > 1.75, the material is plastic; if the contrary, the material is brittle. In addition, it is straightforward to see from the ratios in Table 3 that all four materials are brittle, as these values are all less than the critical value (1.75). In addition, Poisson’s ratio (ν) can also be used to measure the brittleness and toughness of materials. When ν < 1/3, it is a brittle material; in addition, the material exhibits toughness. Table 3 shows that the Poisson’s ratio of all four materials is less than 0.33. Likewise, Figure 4b can also prove that all four materials are brittle.

3.4. Anisotropy in Elastic Moduli

Elastic anisotropy is commonly responsible for the formation and development of microcracks in materials. As a result, the elastic anisotropy of solids must be discussed, and this anisotropy is best defined by three indicators: the anisotropy index A^U [32], the percentage of compression anisotropy A_comp, and the percentage of shear anisotropy A_shear [33,34,35]. The calculation formula is as follows:

$$A^{\mathrm{U}}=5\frac{G_V}{G_R}+\frac{B_V}{B_R}-6$$

(7)

$$A_{\mathrm{comp}}=\frac{B_V-B_R}{B_V+B_R}\times 100\%$$

(8)

$$A_{\mathrm{shear}}=\frac{G_V-G_R}{G_V+G_R}\times 100\%$$

(9)

$$A_1=\frac{4C_{44}}{C_{11}{+C}_{33}{-2C}_{13}}$$

(10)

$$A_2=\frac{4C_{55}}{C_{22}{+C}_{33}{-2C}_{23}}$$

(11)

$$A_3=\frac{4C_{66}}{C_{11}{+C}_{22}{-2C}_{12}}$$

(12)

Table 4. Calculated elastic anisotropic indexes (A^U, A_comp, A_shear, A₁, A₂, A₃) of M₂InX (M = Ti, Zr and X = C, N) of MAX phases.

	AU	Acomp (%)	Ashear (%)	A1	A2	A3
Ti2lnC Ti2lnN Zr2lnC Zr2lnN	0.053	0.004	0.005	0.851	0.851	1.000
Ti2lnN Ti2lnN Zr2lnC Zr2lnN	0.059	0.003	0.006	1.622	1.622	1.000
Zr2lnC Ti2lnN Zr2lnC Zr2lnN	0.034	0.001	0.004	1.072	1.072	1.000
Zr2lnN Ti2lnN Zr2lnC Zr2lnN	0.043	0.001	0.006	1.225	1.225	1.000

shows the anisotropy indices of the obtained ceramics of M₂InX phases. Figure 5 depicts the variation of A^U with A_shear. From Table 4, it can be seen that for Ti₂lnX (X = C, N), the A^U value of Ti₂lnC (0.063) is smaller than that of Ti₂lnN (0.042). For Zr₂lnX (X = C, N), the A^U value of Zr₂lnC is smaller than that of Zr₂lnN. Because the A^U value is proportional to the elastic anisotropy, the order of elastic anisotropy is Ti₂lnN > Ti₂lnC and Zr₂lnN > Zr₂lnC. It shows that Ti₂lnC and Zr₂lnC have better performance and lower possibility of microcracks.

A_comp is used to reflect the compressible anisotropy of the material. From the data in Table 4, it can be seen that the A_comp value of Ti₂lnC is the largest (0.004%), indicating that it has the strongest elastic anisotropy. In addition, it can be seen from A_shear that Ti₂lnN (0.006) and Zr₂lnN (0.006) have the highest shear elastic anisotropy. Meanwhile, from the A_shear value, Ti₂lnC (0.005) and Zr₂lnC (0.004) have the lowest anisotropy of shear modulus. A₁ denotes shear anisotropy in the (100) plane, A₂ shear anisotropy in the (010) plane, and A₃ shear anisotropy in the (001) plane. Table 4 shows the |A₁ − 1| values of Ti₂lnX (X = C, N) and Zr₂lnX (X = C, N). Ti₂lnN and Zr₂lnN have the highest absolute values, respectively, indicating that the shear anisotropy is high. On the other hand, Ti₂lnC and Zr₂lnC have the lowest shear anisotropy on the (100) and (010) planes, respectively. The shear anisotropy results for A₁ agree with those of A_shear. Therefore, the order of shear anisotropy is consistent with the order of elastic anisotropy. To sum up, the elastic anisotropy order of the M₂InX system is Ti₂InX > Zr₂InX (X = C, N) and M₂InN > M₂InC (M = Ti, Zr).

Crystal orientation, or the arrangement of crystal atoms in distinct crystal planes and orientations, can also influence elastic anisotropy. As a result, the elastic modulus anisotropy of the M₂InX phase is represented by the 3D surface structure using the following formula [36,37,38]:

$$\frac{1}{B}=\left(S_{11}{+S}_{12}{+S}_{13}\right){-(S}_{11}{+S}_{12}{-S}_{13}{-S}_{33}{)l}_3^2$$

(13)

$$\frac{1}{G}{=S}_{44}+\left[\left(S_{11}{-S}_{12}\right)-\frac{1}{2}{S}_{44}\right]\left({1-l}_3^2\right)+2\left(S_{11}{+S}_{33}{-2S}_{13}{-S}_{44}\right)l_3^2{(1-l}_3^2)$$

(14)

$$\frac{1}{E}{=S}_{11}{{(1-l}_3^2)}^2{+S}_{33}{l}_3^4{+(2S}_{13}{+S}_{44}{)l}_3^2{(1-l}_3^2)$$

(15)

In this case, l₃ represents the direction cosine.

Figure 6 and Figure 7 display the 3D surface structures of Ti₂lnC, Ti₂lnN, Zr₂lnC, and Zr₂lnN in terms of E, B, and G. Elastic isotropy may be described if the spherical 3D surface structure does not stray significantly from the sphere. Otherwise, it is anisotropic.

As can be seen from Figure 6 and Figure 7, for bulk modulus, the 3D views of Ti₂lnC and Zr₂lnC show significant compression along the c-axis, while the 3D views of Ti₂lnN and Zr₂lnN show significant compression along the a- and b-axes. Combined with the data analysis in

Table 5. Calculated uniaxial elastic moduli in the [100], [10] and [1] directions (in GPa) of M₂InX (M = Ti, Zr and X = C, N) MAX phases.

		Ti2lnC	Ti2lnN	Zr2lnC	Zr2lnN
E	[1]	304	205	211	174
	[10]	309	216	215	174
	[100]	309	216	215	180
	[100]/[1]	1.02	1.05	1.02	1.04
B	[1]	160	86	156	163
	[10]	172	163	160	170
	[100]	172	163	160	170
	[100]/[1]	1.08	1.90	1.03	1.04
G	[1]	122	118	112	80
	[10]	125	122	115	84
	[100]	125	122	115	84
	[100]/[1]	1.02	1.03	1.03	1.05

, the B[100]/B[1] values of Ti₂lnC, Zr₂lnC, Ti₂lnN, and Zr₂lnN have a large deviation from 1, which can also indicate that the materials are anisotropic.

When the shear moduli of Ti₂lnN and Zr₂lnN are compared to those of a spherical shape, it is obvious that their shapes are considerably different. As a result, Ti₂lnN and Zr₂lnN shear moduli are anisotropic. Ti₂lnN and Zr₂lnN shear moduli also are anisotropic, and the anisotropy order is Ti₂lnN > Ti₂lnC and Zr₂lnC > Zr₂lnN. Ti₂lnN has a substantially higher degree of shear anisotropy than Zr₂lnN. At the same time, this conclusion is compatible with the previously reported order of percent shear anisotropy. When compared to bulk and shear modulus, Young’s modulus E considers both bulk modulus B and shear modulus G. The 3D plots of the Young’s modulus of M₂InX (M = Ti, Zr and X = C, N) MAX phases are more compressible along the a- and b-axes than the c-axis, as shown in Figure 6 and Figure 7. Ti₂lnN is more anisotropic than Ti₂lnC, whereas Zr₂lnN is more anisotropic than Zr₂lnN.

The three-dimensional elastic modulus graph shows that Ti₂lnN and Zr₂lnN are elastically anisotropic, with Ti₂lnN having more elastic anisotropy than Zr₂lnN. This is consistent with the A^U value results presented above.

To further illustrate the elastic anisotropy of M₂InX (M = Ti, Zr and X = C, N) MAX phases, Figure 8 plots the 2D projections of the elastic modulus of the M₂InX system on crystal planes (001) and (100), respectively. Based on Figure 8, Table 5 shows the elastic modulus of the M₂InX phases. As can be shown in Table 5, the G[100]/G[1] values of Ti₂lnC and Ti₂lnN are 1.02 and 1.03, respectively, indicating that there is a divergence between G[100]/G[1] and 1, indicating that the material has a greater shear modulus and anisotropy. As a result, Ti₂lnN has the largest shear modulus anisotropy, followed by Ti₂lnC, showing that the order of anisotropy of G is Ti₂lnN > Ti₂lnC. Zr₂lnC and Zr₂lnN have G[100]/G[1] values of 1.03 and 1.05, respectively, showing that the anisotropic order of G is Zr₂lnN > Zr₂lnC. This is consistent with the results of A₁ above.

The E[100]/E[1] values of Ti₂lnC and Ti₂lnN are 1.02 and 1.05, respectively, as shown in Table 5. Zr₂lnC and Zr₂lnN have E[100]/E[1] values of 1.02 and 1.04, respectively. As a result, the anisotropic order of E is Ti₂lnN > Ti₂lnC, Zr₂lnN > Zr₂lnC. This is consistent with the A^U results.

The expected values of the bulk modulus along the [100] and [10] directions are significantly larger than those along the [1] direction. As shown in Table 5, the B[100]/B[1] values of Ti₂lnC and Ti₂lnN are 1.08 and 1.90, respectively. The B[100]/B[1] values of Zr₂lnC and Zr₂lnN are 1.03 and 1.04, respectively. It can be seen that Ti₂lnN has the largest anisotropy, which can be clearly perceived in combination with the 2D diagram, showing that the compression ratio along the c-axis is smaller than the a-axis and b-axis compression [39,40].

3.5. Debye Temperatures and Anisotropy of Sound Velocities

The Debye temperature (${\theta }_{\mathrm{D}}$) is a fundamental thermodynamic property of a solid. The Debye temperature is commonly used to characterize a material’s thermal properties, which may be computed using the single crystal elastic constant [41]:

$${\theta }_{\mathrm{D}}=\frac{h}{k_{\mathrm{B}}}{\left[\frac{3n}{4\mathsf{\pi }}\left(\frac{N_{\mathrm{A}}\rho }{\mathit{Mc}}\right)\right]}^{\frac{1}{3}}{v}_{\mathrm{m}}$$

(16)

N_A stands for Avogadro’s constant, k_B is Boltzmann’s constant, and h is Planck’s constant. the total number of atoms in the unit cell is represented by n. M stands for molecular weight, $\rho$ is density, and v_m represents the average sound velocity, which is determined by the longitudinal and transverse sound velocities, as shown in the following equation [42,43]:

$$v_l={\left[\left(B+\frac{4G}{3}\right)/\rho \right]}^{\frac{1}{2}}$$

(17)

$$v_t={\left(\frac{G}{\rho }\right)}^{\frac{1}{2}}$$

(18)

$$v_m={\left[\frac{1}{3}\left(\frac{2}{v_t^3}+\frac{1}{v_l^3}\right)\right]}^{-\frac{1}{3}}$$

(19)

Table 6. The density ρ, sound velocities (longitudinal ν_l, transverse ν_t, and mean ν_m), and Debye temperature θ_D of M₂InX (M = Ti, Zr and X = C, N) MAX phases.

TM5Al3C	ρ (g/cm3)	vl(km/s)	vt (km/s)	vm(km/s)	θD (K)	Ref.
Ti2lnC	6.079	6.738	4.271	4.697	564	This work
Ti2lnN	6.389	6.013	3.555	3.938	480	This work
Zr2lnC	6.949	6.066	3.573	3.960	446	This work
Zr2lnN	7.302	5.791	3.362	3.731	426	This work
		5.770	3.297		423	Theo. [23]

shows the calculated values of v_l, v_t, v_m, and θ_D for the compounds. In general, the greater the density, the slower the sound speed. Due to the higher densities of Ti₂lnN and Zr₂lnN (6.389 g/cm³ for Ti₂lnN and 7.302 g/cm³ for Zr₂lnN), their sound velocities are relatively slow in Table 6.

Figure 9a shows the variation in different sound velocities. Ti₂lnC has the highest Debye temperature due to its low density and high elastic modulus, and Zr₂lnN has the lowest Debye temperature due to its high density and low elastic modulus. As we all know, the Debye temperature is commonly employed to reflect the strength of chemical bonds and the hardness of materials. As a result, the higher the Debye temperature, the stronger the chemical bond and the harder the solid.

It can be shown that Ti₂lnC has the highest Debye temperature (564 K), suggesting that its chemical bond is the strongest, while Zr₂lnN has the lowest (426 K), showing that its chemical bond is the weakest. At the same time, it can be demonstrated that Ti₂lnC has the highest hardness and Zr₂lnN has the lowest. This confirms the previous section’s conclusion about the materials’ strength and hardness based on the elastic constant.

In addition, there is a certain proportional relationship between Debye temperature and thermal conductivity, so the higher the Debye temperature of the material, the greater the thermal conductivity, which is consistent with the literature [44,45,46]. Therefore, Ti₂lnC has the largest Debye temperature and thus exhibits the largest thermal conductivity.

The transverse sound velocity has two modes: v_t1 (first transverse mode) and v_t2 (second transverse mode). The directional sound velocity calculation results of M₂InX phases are shown in

Table 7. Anisotropic sound velocities (m/s) of M₂InX (M = Ti, Zr and X = C, N) MAX phases.

	[100]			[1]
	[100]vl	[10]vt1	[1]vt2	[1]vl	[100]vt1	[10]vt2
Ti2lnC	4289	6858	3846	6331	3846	3846
Ti2lnN	3906	5985	3683	5974	3683	3683
Zr2lnC	3715	6070	3431	5888	3431	3431
Zr2lnN	3564	5760	3295	5555	3295	3295

. Table 7 shows that the first transverse sound velocity ([10]v_t1) of the M₂InX system along the [100] direction is the greatest, followed by the first longitudinal sound velocity ([1]v_l). The directional sound velocity is well-known to be related to the elastic constants (C₁₁, C₃₃, C₄₄). As a result, Figure 9b shows the relationship between the elastic constants of sound velocity in different directions for the M₂InX phases. All sound speeds are obviously anisotropic in both Ti₂lnX (X = C, N) and Zr₂lnX (X = C, N) systems due to the shift in sound speed direction. It is also demonstrated that the link between sound speed and elastic constant is consistent. For example, the initial transverse sound velocity ([10]v_t1) along the [100] direction is related to C₁₁, for which the order is Ti₂lnC > Zr₂lnC > Ti₂lnN > Zr₂lnN. Similarly, the change trend of [1]v₁ matches that of C₃₃, and the change trend of [1]v_t2, [100]v_t1, and [10]v_t2 matches that of C₄₄. [100]v_l is associated with C₁₁-C₁₂. Therefore, Figure 9b plots the relationship between different sound velocities for M₂InX phases.

3.6. Thermal Properties

Generally speaking, the thermal properties of materials are usually characterized by thermal conductivity, heat capacity, and thermal expansion coefficient. Thermal conductivity is a measure of the thermal conductivity of a substance. The lattice thermal conductivity k_ph is one of the most important indicators to describe the thermal behavior of solids. Therefore, according to Slack’s model [47], the lattice thermal conductivity of M₂InX phases can be calculated with the following empirical formula:

$$k_{\mathit{ph}}=A\frac{M_{\mathit{av}}{\delta \theta }_D^3}{{\gamma }^2{\mathit{Tn}}^{2/3}}$$

(20)

Here, the volume of each atom is denoted by δ³. The average atomic mass of each atom is denoted by M_av, T is the temperature, and n is the number of atoms in the unit cell. γ is the Grüneisen parameter that can be obtained from Poisson’s ratio v, while A_γ is the component associated with γ, and the formulas are as follows [48]:

$$\gamma =\frac{3\left(1+v\right)}{2\left(2-3v\right)}$$

(21)

$$A_{\gamma }=\frac{5.720\times {10}^7\times 0.849}{2\times \left(1-\frac{0.154}{\gamma }+\frac{0.228}{{\gamma }^2}\right)}$$

(22)

The lattice thermal conductivities of Ti₂lnC, Ti₂lnN, Zr₂lnC, and Zr₂lnN at two temperatures (300 K and 1300 K) are shown in

Table 8. Calculated δ (in Å), M_av (in kg/mol), Grüneisen parameter γ, A_γ (×10⁻⁸), and lattice thermal conductivities k_ph (in W·m⁻¹·K⁻¹) of M₂InX (M = Ti, Zr and X = C, N) MAX phases.

	δ	Mav	γ	n	Aγ	kph (300 K)	kph (1300 K)	kmin
Ti2lnC	2.48	55.66	1.16	8.00	3.34	51.48	11.88	1.23
Ti2lnN	2.44	56.16	1.41	8.00	3.24	20.43	4.72	1.12
Zr2lnC	2.60	77.31	1.43	8.00	3.23	23.59	5.44	0.96
Zr2lnN	2.64	77.81	1.48	8.00	3.21	19.42	4.48	0.94

. The results show that most of the M₂AX phases have thermal conductivities ranging from 12 to 60 W·m⁻¹ [49]. The calculation result is within this range. The lattice thermal conductivities of Ti₂lnC, Ti₂lnN, Zr₂lnC, and Zr₂lnN at room temperature (300 K) are 51.48, 20.43, 23.59, and 19.42 W·m⁻¹·K⁻¹, respectively. Therefore, Ti₂lnC, Ti₂lnN, Zr₂lnC, and Zr₂lnN can serve as potential thermal conductive materials at room temperature. As shown in Figure 10, as the temperature increases, k_ph decreases rapidly and then tends to a limit value. In the temperature range of 300 K–1300 K, the order of k_ph is Ti₂lnC > Zr₂lnC > Ti₂lnN > Zr₂lnN.

The thermal conductivity is mainly derived from the lattice thermal conductivity at the ground state temperature. Therefore, the thermal conductivity reflection of ceramics can be characterized as the minimum thermal conductivity [50]. Consequently, the Clark model is used here to calculate the minimum thermal conductivity k_min of the M₂InX system, and the expression is as follows:

$$k_{min}=K_B{V}_m{\left(\frac{n\rho N_A}{M}\right)}^{\frac{2}{3}}$$

(23)

Table 8 shows the calculated minimum lattice thermal conductivities for the M₂InX system. It can be seen from Table 8 that the k_min of M₂InX are 1.23, 1.12, 0.96, and 0.94 W·m⁻¹·K⁻¹, respectively. The difference between the lattice thermal conductivity and θ_D is that the higher Debye temperature has a larger lattice thermal conductivity, so Ti₂lnC has the largest thermal conductivity, which corresponds to the highest Debye temperature (564 K) of Ti₂lnC. Furthermore, the calculated order of minimum thermal conductivity is Ti₂lnC > Ti₂lnN > Zr₂lnC > Zr₂lnN. As is known, M₂InX phases are not potential high-temperature thermal barrier coatings when compared with Ln₂Zr₂O₇ (1.2~1.4 W·M⁻¹·K⁻¹) [51]. The thermal conductivity of new ceramic materials is between 1.2 W·m⁻¹·K⁻¹–1.6 W·m⁻¹·K⁻¹. Among these compounds, the thermal conductivity of Ti₂lnC is within this range, so Ti₂lnC may become a potential insulating material [52].

3.7. Electronic Properties

The electronic properties (density of states) of M₂InX (M = Ti, Zr and X = C, N) MAX phases were studied to better understand chemical bonding and bond behaviors. Figure 11 depicts the M₂InX phases’ total density of states (TDOS) and partial density of states (PDOS). To begin, it is clear that DOS has a significant finite value at the Fermi level, indicating that these compounds exhibit metallic conductivity. Figure 11 shows that the total density of states (E_f) value of Ti₂InN and Zr₂lnN is greater than that of Ti₂InC and Zr₂lnC, indicating that Ti₂InN and Zr₂lnN are more conductive than Ti₂InC and Zr₂lnC. Secondly, the peak topologies and relative heights of the peaks around E_f in the TDOS plot are highly comparable, indicating the presence of similar chemical bonds in Zr₂AN. The time difference around the Fermi level is mostly made up of ln-p and M-d states. The time difference below the Fermi energy is caused mostly by the X-s, M-s, and M-p states, whereas the time difference above the Fermi energy is caused primarily by the ln-s and X-p states. As shown in Figure 11, the ln-p, M-d, and X-p states exhibit substantial hybridization, allowing M-C and M-N chemical bonds to form, resulting in the high elastic modulus of M₂InX. PDOS exhibits multiple hybridizations of the electronic states M, ln, and X. The valence band of M₂InX in Figure 11 displays substantial hybridization of the M-d and X-p states, as predicted for covalent compounds. The d-p hybrid state corresponding to the M-ln bond was discovered to be in a greater energy range than the M-X bond. As a result, the M-X d-p hybridization helps to maintain the crystal structure. Finally, it is demonstrated that M’s electronic charge density almost overlaps that of ln, indicating that the bonding between M and ln is quite weak. These findings are consistent with the observation that the biggest phase features very strong M-X bonds and very weak M-ln bonds.

As can be seen from

Table 9. Calculated Mulliken charge and bond population (BP) analysis of M₂InX (M = Ti, Zr and X = C, N) MAX phases.

	Atom	Charge Number					Charge	Bond	BP	Length(Å)
		s	p	d	f	Total
Ti2lnC	Ti	2.18	6.79	2.66	0.00	11.62	0.38	Ti-C	1.04	2.12
	In	1.11	1.94	9.97	0.00	13.02	−0.02
	C	1.46	3.27	0.00	0.00	4.73	−0.73
Ti2lnN	Ti	2.19	6.77	2.69	0.00	11.65	0.35	Ti-N	0.76	2.10
	In	1.05	1.97	9.97	0.00	12.99	0.01
	N	1.68	4.04	0.00	0.00	5.71	−0.71
Zr2lnC	Zr	2.28	6.63	2.68	0.00	11.59	0.41	Zr-C	1.06	2.30
	In	1.11	1.93	9.98	0.00	13.02	−0.02
	C	1.49	3.31	0.00	0.00	4.80	−0.80
Zr2lnN	Zr	2.30	6.63	2.72	0.00	11.65	0.35	Zr-N	0.69	2.27
	In	1.03	1.92	9.97	0.00	12.92	0.08
	N	1.70	4.07	0.00	0.00	5.77	−0.77

, the M atom loses electrons, while the ln and X atoms gain electrons. Among them, for the Ti₂InX system, the Ti-C bond has the largest BP value, indicating that the Ti-C bond has a strong chemical bond. Therefore, it is proved that Ti₂lnC has the strongest chemical bond and Ti₂lnN has the weakest chemical bond. For the Zr₂InX system, it can also be stated that Zr₂InC has the strongest chemical bond and Zr₂InN has the weakest chemical bond. The M and X atoms form a strongly directed M-X covalent bond originating from the hybrid M d-X p state. These results are also consistent with the finding that the largest phase typically has very strong M-X bonds and relatively weak M-A bonds.

4. Conclusions

In summary, this work uses first-principles calculations to estimate the anisotropic elastic and thermal properties of M₂InX (M = Ti, Zr and X = C, N) MAX phases. The structural properties and elastic constants of the obtained M₂InX phases are in agreement with the results reported in the literature, the obtained data are quite reliable, and the M₂InX phase’s mechanical stability has been studied. According to the elastic constants, Ti₂lnC and Zr₂lnC are more incompressible along the a- and b-axes, while Ti₂lnN and Zr₂lnN are more compressible along the a- and b-axes. For all M₂InX phase materials, shear modulus is a better measure of hardness than bulk modulus. Therefore, Ti₂lnC has high hardness and can be used to make superhard materials. The anisotropic elasticity of the M₂InX phase is Ti₂lnN > Ti₂lnC, Zr₂lnN > Zr₂lnC, based on A^U, A_comp, and A_shear values, 3D graphs, and 2D projection analysis. In addition, Ti₂lnC has the highest speed of sound (4.697) and Debye temperature (564 K), while Zr₂lnN has the lowest speed of sound (3.731) and Debye temperature (426 K). Meanwhile, Ti₂lnC may become a potential insulation material. The electronic properties of the M₂InX phase were investigated, and the presence of strong M-X d-p hybridization helps to maintain the crystal structure.