Engineering the Mechanics and Thermodynamics of Ti₃AlC₂, Hf₃AlC₂, Hf₃GaC₂, (ZrHf)₃AlC₂, and (ZrHf)₄AlN₃ MAX Phases via the Ab Initio Method

Abstract

When combined with ceramics, ternary carbides, nitrides, and borides form a class of materials known as MAX phases. These materials exhibit a multilayer hexagonal structure and are very strong, damage tolerant, and thermally stable. Further, they have a low thermal expansion and exhibit outstanding resistance to corrosion and oxidation. However, despite the numerous MAX phases that have been identified, the search for better MAX phases is ongoing, including the recently discovered Zr₃InC₂ and Hf₃InC₂. The properties of MAX phases are still being tailored in order to lower their ductility. This study investigated Ti₃AlC₂ alloyed with nitrogen, gallium, hafnium, and zirconium with the aim of achieving better mechanical and thermal performances. Density functional theory within Quantum Espresso module was used in the computations. The Perdew–Burke–Ernzerhof generalised gradient approximation functionals were utilised. (ZrHf)₄AlN₃ exhibited an enhanced bulk and Young’s moduli, entropy, specific heat, and melting temperature. The best thermal conductivity was observed in the case of (ZrHf)₃AlC₂. Further, Ti₃AlC₂ exhibited the highest shear modulus, Debye temperature, and electrical conductivity. These samples can thus form part of the group of MAX phases that are used in areas wherein the above properties are crucial. These include structural components in aerospace and automotive engineering applications, turbine blades, and heat exchanges. However, the samples need to be synthesised and their properties require verification.

1. Introduction

${\mathrm{M}}_{\mathrm{n}+1}\mathrm{A}{\mathrm{X}}_{\mathrm{n}}$ is the generic formula for the ternary carbides, nitrides, and borides that make up MAX phases (where X can be either carbon, nitrogen, or boron; A can be any element contained in groups 13–16 in the periodic table; and M is an early transition metal [1,2,3]). Because of their special combination of ceramic and metallic properties, these materials have a number of useful applications. The alternating hexagonal structure of the layers of the M and AX components is a characteristic that makes the MAX phases stand out [4,5].

MAX phases have good damage tolerance, the ability to bend plastically, high strength and stiffness that are comparable to ceramics, and machinability with current and standard techniques [6], in addition to having a low thermal expansion and dimensional stability under thermal cycling, stability at high temperatures with high melting temperatures, and excellent thermal conductivity for effective heat dissipation. Additionally, MAX phases can tolerate abrupt temperature changes without breaking, and are conductive (much like metals). Additionally, they have exceptional corrosion resistance in a variety of settings, and outstanding oxidation resistance in high-temperature applications [7,8,9].

Titanium aluminium carbide (Ti₃AlC₂) is a typical example of a MAX phase because of its strong thermal conductivity, good oxidation resistance, machinability, and stability at high temperatures, which makes it appropriate for thermal management systems, heat exchangers, and high-temperature coatings. It has been investigated computationally using the Cambridge Serial Total Energy Package (CASTEP), where its mechanical properties such as bulk modulus of 162.1 GPa, shear modulus of 140.2 GPa, and Poisson ratio of 1.16 have been obtained [10]. However, the study found out that the material is both brittle and anisotropic, which can limit its application in high-temperature components in the aerospace industry, and in electrical contacts and thermal interface materials in the electronic industry [11]. Ti₂AlC provides a blend of ceramic and metallic properties, despite having a distinct stoichiometry, and it is commonly used in structural components, high-temperature applications, and protective coatings. Further, chromium aluminium carbide (Cr₂AlC) exhibits excellent oxidation and corrosion resistance, good thermal and electrical conductivity, and is used in corrosive environments, protective coatings, and as a structural material in harsh environmental conditions. Niobium aluminium carbide (Nb₂AlC) is appropriate for structural, protective coating, and high-temperature applications, owing to its high melting temperature, better heat conduction, and resistance to oxidation [12,13,14,15]. In addition, the 314 boron-based (Zr₃CdB₄) MAX phase has recently been investigated and found to be ductile [3], which is a highly desirable property in structural, high-temperature, and biomedical applications.

The unique properties exhibited by MAX phases render them perfect for various applications, particularly where both metallic and ceramic properties are desired. Their development continues to expand with promising new materials with tailored properties for specific applications. Recently, two novel indium-incorporating MAX phases (Zr₃InC₂ and Hf₃InC₂) were successfully prepared by Zhang et al. [16] using spark plasma sintering. However, ensuring that the indium does not evaporate during the high-temperature synthesis is challenging because of its low melting temperature. Ti₃InC₂ has excellent resistivity at low ion beam fluences [17]. Further, Hf–In–C ternary compounds exhibit a good tolerance to ions at high ion beam fluences [18].

While MAX phases exhibit appropriate properties for various applications, they have a number of limitations, including being less resistant to oxidation compared to traditional ceramics (such as alumina or silicon carbide) at extremely high temperatures; an inferior fracture toughness compared to metals, which limits their ability to withstand high-stress applications where toughness is critical; and the high-temperature and controlled atmospheres required for their manufacture, making the manufacturing process complex and costly [19]. As the search for more MAX phases with better mechanical and thermal properties continues, this study carried out a computational investigation of the mechanics and thermodynamics of Ti₃AlC₂ alloyed with gallium (Ga), hafnium (Hf), Zr, and nitrogen (N), with the aim of tailoring the mechanical and thermal properties of the alloys to discover new MAX phases before they are manufactured. Five samples (Ti₃AlC₂ and four novel ones) were investigated. The mechanical characterisation included the elastic constants and Vickers hardness, whereas the thermal characterisation entailed the Debye temperature, melting temperature, entropy, specific heat capacity at a constant volume, and electrical and thermal conductivities.

2. Materials and Methods

The computations were performed using the first-principles density functional theory [20], as implemented in the Quantum Espresso module [21]. The original crystallographic information file, which included a hexagonal crystal of the $\mathrm{P}63/\mathrm{m}\mathrm{m}\mathrm{c}$ space group and cell parameters $\mathrm{a}=\mathrm{b}=3.072\mathrm{\AA }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{c}=18.73\mathrm{\AA }$ [22] was retrieved from the crystallographic open database (Figure 1). Thereafter, alloying was performed on the cell by replacing the Al atoms with Ga, and replacing the titanium (Ti) with Hf, Zr, and N. The replacement of the atoms was carried out using Burai, a graphical user interface for Quantum Espresso. Certain samples were co-alloyed. The resulting samples are shown in Figure 2.

Based on the Perdew–Burke–Ernzerhof generalised gradient approximation (PBE–GGA), the PBE–GGA system [23] was used to determine the electronic exchange and correlation energies. The plane-wave expansion was performed by varying the kinetic energy cut-off (ecut) from 10 Ry to 90 Ry in steps of 10 Ry, and convergence was observed at the ecut value of 50 Ry. For integration across the first Brillouin zone, the mesh was varied from 2 × 2 × 2 to 9 × 9 × 9 in steps of 1 × 1 × 1, and the convergence was observed at the 5 × 5 × 1 mesh for the self-consistent function. For the non-consistent function calculations, a denser mesh of 15 × 15 × 3 was utilised in accordance with the Monkhorst–Pack technique [24]. The lattice parameters $\mathrm{a}$ and $\mathrm{b}$ were varied from 2.765 to 3.379 Å in steps of 0.0409 Å (10% below and above the reference value of 3.072 Å, respectively). Following the same procedure, parameter $\mathrm{c}$ was varied from 16.857 to 20.603 Å in steps of 0.2497 Å. Atomic coordinates were configured using the Broyden–Fletcher–Goldfarb–Shanno algorithm [25]. The convergence thresholds were ×10⁻⁴ Ry/atom for the self-consistent field tolerance, and 3 × 10⁷ Pa for the stress.

The structural stabilities of the alloy samples were investigated by calculating their formation energies $({\mathrm{H}}_{\mathrm{f}})$ according to the following relationship [26]:

$${\mathrm{H}}_{\mathrm{f}}^{\mathrm{M}\mathrm{A}\mathrm{X}}=\mathrm{E}\left(\mathrm{M}\mathrm{A}\mathrm{X}\right)-\mathrm{a}\left(\mathrm{E}\left(\mathrm{M}\right)\right)-\mathrm{b}\left(\mathrm{E}\left(\mathrm{A}\right)\right)-\mathrm{c}(\mathrm{E}\left(\mathrm{X}\right)),$$

(1)

where $\mathrm{a},\mathrm{b},\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{c}$ are the number of atoms of each atomic species and $M,A,\mathrm{a}\mathrm{n}\mathrm{d}X$ are the atomic species of the MAX phases. For example, in the formation energy of Hf₃AlC₂, we have $\mathrm{a}=3,\mathrm{b}=1,\mathrm{c}=2,\mathrm{M}=\mathrm{H}\mathrm{f},\mathrm{A}=\mathrm{A}\mathrm{l},\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{X}=\mathrm{C}$. Using the stress–strain method [27], the elastic stiffness constants were calculated by utilising strains of ±0.006 in steps of 0.003 to maintain the crystals’ linear regime. Using the elastic stiffness constants $({\mathrm{c}}_{11},{\mathrm{c}}_{12},{\mathrm{c}}_{13},{\mathrm{c}}_{33},\mathrm{a}\mathrm{n}\mathrm{d}{\mathrm{c}}_{44})$, the elastic constants were calculated as described by Zhao et al. [10]. The Vickers hardness was determined using the Tian model according to the following relationship [28]:

$${\mathrm{H}}_{\mathrm{V},\mathrm{T}\mathrm{i}\mathrm{a}\mathrm{n}}=0.92{\left({\displaystyle \frac{1}{\mathrm{n}}}\right)}^{1.137}{\mathrm{G}}^{0.708},$$

(2)

where $\mathrm{n}$ is the Pugh’s modulus ratio $(\mathrm{n}=\mathrm{B}/\mathrm{G})$, $\mathrm{B}$ is the bulk modulus, and $\mathrm{G}$ is the shear modulus.

The thermal properties (entropy and specific heat capacity at constant volume) were computed using the thermo_pw module [29] within 0–1200 K in steps of 2 K. The Debye temperature $({\mathrm{\theta }}_{\mathrm{D}})$ and the melting temperature $({\mathrm{T}}_{\mathrm{m}})$ were calculated from the relations below [30,31]:

$${\mathrm{\theta }}_{\mathrm{D}}={\displaystyle \frac{\mathrm{h}}{{\mathrm{k}}_{\mathrm{B}}}}{\left[{\displaystyle \frac{3\mathrm{n}}{4\mathrm{\pi }}}{\displaystyle \frac{{\mathrm{N}}_{\mathrm{A}}\mathrm{\rho }}{\mathrm{M}}}\right]}^{{\displaystyle \frac{1}{3}}}{\mathrm{v}}_{\mathrm{m}},$$

(3)

$${\mathrm{T}}_{\mathrm{m}}=354+{\displaystyle \frac{4.5(2{\mathrm{c}}_{11}+{\mathrm{c}}_{33})}{3}}\pm 300,$$

(4)

where $\mathrm{h}$ is the Planck’s constant, ${\mathrm{k}}_{\mathrm{B}}$ is the Boltzmann constant, $\mathrm{n}$ is the number of atoms in the unit cell, ${\mathrm{N}}_{\mathrm{A}}$ is the Avogadro’s number, and $\mathrm{M}$ is the molar mass of the crystal. Further, ${\mathrm{v}}_{\mathrm{m}}$ is the average acoustic velocity, and was obtained as:

$${\mathrm{v}}_{\mathrm{m}}={\left[{\displaystyle \frac{1}{3}}\left({\displaystyle \frac{1}{{\mathrm{v}}_{\mathrm{l}}^3}}+{\displaystyle \frac{2}{{\mathrm{v}}_{\mathrm{t}}^3}}\right)\right]}^{-{\displaystyle \frac{1}{3}}};{\mathrm{v}}_{\mathrm{l}}=\sqrt{{\displaystyle \frac{3\mathrm{B}+4\mathrm{G}}{3\mathrm{\rho }}}};{\mathrm{v}}_{\mathrm{t}}=\sqrt{{\displaystyle \frac{\mathrm{G}}{\mathrm{\rho }}}},$$

(5)

where ${\mathrm{v}}_{\mathrm{l}}$ is the longitudinal acoustic velocity, ${\mathrm{v}}_{\mathrm{t}}$ is the transverse acoustic velocity, and $\mathrm{\rho }$ is the density of the alloy. For the computation of the electrical and thermal conductivities, the BoltzTrap2 code [32] was employed at a constant temperature of 300 K (room temperature).

3. Results and Discussion

The equilibrium lattice parameter $\mathrm{a}$ was found at the minimum of the curve in Figure 3, which illustrates the change in the total energy as the unit cell volume changed for all the samples. Similarly, the equilibrium values for parameter $\mathrm{c}$ were determined.

Table 1. Structural properties of the computed samples. The cell parameters (a and c) are in m, and density is in kg/m³.

Sample	${\mathbf{a}}_{\mathbf{o}}(\times {10}^{-10})$	${\mathbf{c}}_{\mathbf{o}}(\times {10}^{-10})$	${\mathbf{c}}_{\mathbf{o}}/{\mathbf{a}}_{\mathbf{o}}$	$\mathbf{\rho }$	${\mathbf{E}}_{\mathbf{f}}\left(\mathbf{e}\mathbf{V}/\mathbf{a}\mathbf{t}\mathbf{o}\mathbf{m}\right)$
${\mathrm{H}\mathrm{f}}_3\mathrm{A}\mathrm{l}{\mathrm{C}}_2$	3.270 (3.332 [30])	19.659 (19.690 [30])	6.011 (5.909 [30])	10,699 (10,288 [30])	−4.825
${\mathrm{H}\mathrm{f}}_3\mathrm{G}\mathrm{a}{\mathrm{C}}_2$	3.279	19.256	5.872	11,652	−4.802
$\mathrm{T}{\mathrm{i}}_3\mathrm{A}\mathrm{l}{\mathrm{C}}_2$	3.077 (3.072 [33])	18.746 (18.547 [33])	6.092	4205 (4240 [33])	−4.874
${(\mathrm{Z}\mathrm{r}\mathrm{H}\mathrm{f})}_3\mathrm{A}\mathrm{l}{\mathrm{C}}_2$	3.319	19.951	6.011	6426	−5.431
${(\mathrm{Z}\mathrm{r}\mathrm{H}\mathrm{f})}_4\mathrm{A}\mathrm{l}{\mathrm{N}}_3$	3.190	24.857	7.792	10,546	−13.90

indicates that the cell parameters and densities of Ti₃AlC₂ and Hf₃AlC₂ were computed accurately and showed excellent agreement with the experimental values reported by Roknuzzaman et al. [30] and Vovk et al. [33]. Notably, the lattice parameters of all the alloyed samples were larger than those of Ti₃AlC₂, indicating cell expansion. This expansion can be attributed to the large alloying atoms (Ga replaced Al; Hf and Zr replaced Ti) introduced into the Ti₃AlC₂ lattice.

The density of the MAX phases facilitates the determination of their mechanical, thermal, electrical, chemical, structural, and functional properties. Typically, an increase in density enhances the properties of MAX phases, rendering them more suitable for demanding applications. This occurs as a result of an improved mechanical strength, enhanced fracture toughness, and electrical and thermal conductivities. As presented in Table 1, the densities of all four of the other samples were higher than that of Ti₃AlC₂, indicating that they could be better substitutes for demanding applications, such as high-temperature coatings, electrical contacts, thermal management materials, and structural components in harsh environments. Hf₃GaC₂ was the best material in this regard because it had the highest density. However, it is essential to balance density with other material properties to achieve the desired performance for specific applications.

The formation energy of a material is a critical parameter that significantly affects the properties and structural stability of the MAX phases. Because (ZrHf)₄AlN₃ had the lowest formation energy among all the samples (Table 1), it was the most structurally stable sample. In contrast, Ti₃GaC₂ was the least structurally stable, owing to its high formation energy and was thus more prone to decomposition into its constituent elements or other phases. Moreover, the lowest formation energy of (ZrHf)₄AlN₃ favours the formation of its pure phase, thereby reducing the likelihood of secondary phase formation or impurities.

The properties of MAX phases, such as their structural, mechanical, electrical, and chemical properties, are significantly affected by a decrease in the lattice constants as the pressure increases. Owing to their special blends of ceramic and metallic properties, MAX phases react to pressure in ways that can be helpful for various applications. As shown in Figure 4, (ZrHf)₄AlN₃ was the most resistant to pressure changes because a significant amount of pressure is required to compress it within the same volume range compared to the other samples. Therefore, they are expected to possess the greatest ability to withstand high pressures without significant deformation (high compressive strength, which is related to the bulk modulus). MAX phases that can maintain their structure under high pressures, such as (ZrHf)₄AlN₃ (as observed in this study), are desirable for applications requiring long-term stability under stress.

In the context of the MAX phases, the elastic stiffness constants ${\mathrm{c}}_{\mathrm{i}\mathrm{j}}$ are components of the stiffness tensor that describe the response of a material to mechanical stress. The calculated stiffness constants listed in

Table 2. Computed stiffness constants (in ×10⁹ Pa) of all the samples.

Sample	${\mathbf{c}}_{11}$	${\mathbf{c}}_{12}$	${\mathbf{c}}_{13}$	${\mathbf{c}}_{33}$	${\mathbf{c}}_{44}$
${\mathrm{H}\mathrm{f}}_3\mathrm{A}\mathrm{l}{\mathrm{C}}_2$	332.1 (347 [30])	75.9 (77 [30])	73.9 (80 [30])	284.7 (291 [30])	117.8 (127 [30])
${\mathrm{H}\mathrm{f}}_3\mathrm{G}\mathrm{a}{\mathrm{C}}_2$	340.7	92.9	116.9	280.6	109.9
$\mathrm{T}{\mathrm{i}}_3\mathrm{A}\mathrm{l}{\mathrm{C}}_2$	342.9 (356.8 [10])	69.8 (70.2 [10])	73.4 (69.2 [10])	285.2 (329.7 [10])	113.2 (139.8 [10])
${(\mathrm{Z}\mathrm{r}\mathrm{H}\mathrm{f})}_3\mathrm{A}\mathrm{l}{\mathrm{C}}_2$	307.1	82.6	128.4	248.3	94.7
${(\mathrm{Z}\mathrm{r}\mathrm{H}\mathrm{f})}_4\mathrm{A}\mathrm{l}{\mathrm{N}}_3$	337.1	68.2	107.3	298.3	143.5

were slightly lower than the values computed by Zhao et al. [10], which can be attributed to the CASTEP employed in the referenced work. The stiffness constant ${\mathrm{c}}_{11}$ is particularly significant because it represents the stiffness of the material along the principal crystallographic direction. A higher value of ${\mathrm{c}}_{11}$ compared with the other stiffness constants, as shown in Table 2, indicated that all the samples were stiffer in the direction of the crystallographic $\mathrm{a}$ axis. Thus, all the materials were more resistant to deformation along the $\mathrm{a}$ axis. This, in turn, highlighted the mechanical anisotropy of all the materials. The value for Ti₃AlC₂ was the highest, indicating that it was the stiffest along the $\mathrm{a}$ direction, whereas that of (ZrHf)₃AlC₂ was the least stiff.

The elastic stiffness constant ${\mathrm{c}}_{44}$ is particularly significant because it represents the resistance of the material to shear deformation. A higher value of ${\mathrm{c}}_{44}$ indicates a higher shear modulus, implying that the material is more resistant to shear stress. This is crucial for applications wherein the materials are subjected to torsional or shear forces. A high value of ${\mathrm{c}}_{44}$, as exhibited by (ZrHf)₄AlN₃ (Table 2), contributes to this by ensuring that the material retains its shear resistance even at elevated temperatures.

The mechanical properties of the samples are shown in

Table 3. Calculated mechanical properties of the samples. Μ is the Pugh’s ratio, while n = B/G. B, G, E, and H_V are in (×10⁹ Pa).

Sample	$\mathbf{B}$	$\mathbf{G}$	$\mathbf{E}$	$\mathsf{\mu }$	$\mathbf{n}$	${\mathbf{H}}_{\mathbf{V}}$
${\mathrm{H}\mathrm{f}}_3\mathrm{A}\mathrm{l}{\mathrm{C}}_2$	150.7 (162 [30])	122.3 (127 [30])	288.9 (302 [30])	0.181 (0.189 [30])	1.232 (1.282 [30])	4.710 (4.9 [30])
${\mathrm{H}\mathrm{f}}_3\mathrm{G}\mathrm{a}{\mathrm{C}}_2$	156.2	119	284.7	0.196	1.313	4.461
$\mathrm{T}{\mathrm{i}}_3\mathrm{A}\mathrm{l}{\mathrm{C}}_2$	153.9 (162.1 [10])	122.8 (140.2 [10])	290.4 (326.4 [10])	0.185	1.253	4.631
${(\mathrm{Z}\mathrm{r}\mathrm{H}\mathrm{f})}_3\mathrm{A}\mathrm{l}{\mathrm{C}}_2$	144.4	103.4	250.4	0.211	1.400	4.096
${(\mathrm{Z}\mathrm{r}\mathrm{H}\mathrm{f})}_4\mathrm{A}\mathrm{l}{\mathrm{N}}_3$	183.6	118.7	293.4	0.234	1.547	4.061

and Figure 5. A material’s bulk modulus (B) indicates how resistant it is to homogenous compression. For MAX phases, a greater resistance to volume compression is indicated by a higher bulk modulus and shows that the MAX phases retain their volumes under external pressure because they are less compressible. This helps to maintain the materials’ structural integrity, guaranteeing that they will maintain their dimensions and form even under high mechanical pressures. With the highest bulk modulus (Table 3 and Figure 5a), the (ZrHf)₄AlN₃ sample was less likely to experience phase changes or structural instabilities under different pressure and temperature conditions. Furthermore, (ZrHf)₄AlN₃’s highest bulk modulus suggests that it is stiffer (which adds to its total rigidity and mechanical strength). As a result, it is more resilient to fracture and deformation.

A greater resistance to shear stress is indicated by a higher shear modulus; which is essential for preserving the material’s structural integrity (especially in applications like cutting, machining, and sliding contacts where shear forces are common). In this study, Ti₃AlC₃ produced the highest shear modulus (Table 3 and Figure 5b), suggesting that it can withstand sliding and abrasion forces without experiencing excessive deformation or surface damage, especially in areas like sliding contact surfaces, gears, cutting tool inserts for machining operations, and seals [34]. Additionally, the alloy with the lowest shear modulus was (ZrHf)₃AlC₂.

A higher Young’s modulus (E) indicates a greater stiffness, implying that the material requires more force to deform under the applied stress. This contributes to the overall mechanical strength and structural integrity of the material, rendering it more resistant to deformation, bending, and breakage. (ZrHf)₄AlN₃, which exhibited the highest Young’s modulus in this study (Table 3 and Figure 5c), could bear heavier loads without experiencing excessive deformation or failure. This property is crucial for the structural components used in aerospace and automotive engineering applications.

As ductility begins at 0.27 for the Poisson ratio and 1.75 for the Pugh’s ratio, as shown in Table 3 and Figure 5d,e for Poisson (µ) and Pugh’s (n) ratios, respectively, both indicate that all of the MAX phases simulated in this study were brittle. This was consistent with most MAX phases, with ‘A’ being the element that controls their ductility/brittleness. However, previous studies have shown that most MAX phases become ductile at high temperatures [35,36,37]. The brittleness of the samples obtained in this study is not suitable for most MAX phase applications because ductile MAX phases are easier to machine and form complex shapes without fracturing, which is important for manufacturing processes such as milling, turning, and drilling, where materials must be shaped accurately. Brittle MAX phases can be challenging to machine because of their propensity for fracturing rather than plastic deformation. This can result in tool wear, surface damage, and difficulty achieving the desired shapes or tolerances. Moreover, they may have limitations in forming processes, such as rolling, extrusion, and forging, as they are prone to cracking or fracturing under mechanical stress. This restricts their applications in industries that require complex-shaped components.

The resistance of a material to persistent deformation, particularly scratching or indentation, is measured based on its mechanical hardness. MAX phases with high hardness are more resistant to abrasion and wear, rendering them appropriate for applications involving sliding or abrasive contact, including cutting tools, wear-resistant coatings, and bearings. The Vickers hardness values computed in this study (Table 3 and Figure 5f) showed that Hf₃AlC₂ was the hardest. This was followed very closely by Ti₃AlC₂. The Vickers hardness values for (ZrHf)₃AlC₂ and (ZrHf)₄AlN₂ were lower than those of the rest. Hf₃AlC₂ and Ti₃AlC₂ are therefore suitable for use as cutting tool inserts, drills, and milling tools for machining hard materials such as metals, ceramics, and composites, where wear resistance is essential [38,39].

The average vibrational energy of the atoms in a solid substance is measured using the Debye temperature. This property is crucial for applications in high-temperature environments such as aerospace components and cutting tools [39]. Ti₃AlC₂ exhibited the highest Debye temperature, followed by (ZrHf)₃AlC₂ (

Table 4. Thermal properties of the samples. ${\mathrm{\theta }}_{\mathrm{D}}$ and ${\mathrm{T}}_{\mathrm{m}}$ are in K, S, and ${\mathrm{c}}_{\mathrm{V}}$ are in J/K, $\mathrm{\sigma }$ is in ${\mathrm{\Omega }}^{-1}{\mathrm{m}}^{-1}$, and k is in $\mathrm{W}/\mathrm{m}/\mathrm{K}$.

Sample	${\mathbf{\theta }}_{\mathbf{D}}$	${\mathbf{T}}_{\mathbf{m}}$	$\mathbf{S}$	${\mathbf{c}}_{\mathbf{V}}$	$\mathbf{\sigma }(\times {10}^6)$	$\mathbf{\kappa }$
${\mathrm{H}\mathrm{f}}_3\mathrm{A}\mathrm{l}{\mathrm{C}}_2$	448.3 (459.5 [30])	2077	145.6	135.0	1.84	12.61
${\mathrm{H}\mathrm{f}}_3\mathrm{G}\mathrm{a}{\mathrm{C}}_2$	426.2	2097	154.5	136.1	1.53	11.54
$\mathrm{T}{\mathrm{i}}_3\mathrm{A}\mathrm{l}{\mathrm{C}}_2$	757.3 (680 [40]), (764 [41])	2111 (2100 [42])	82.5	111.2 (115 [10])	2.34 (1.7 [43]), (2.9 [44])	16.22 (16 [43])
${(\mathrm{Z}\mathrm{r}\mathrm{H}\mathrm{f})}_3\mathrm{A}\mathrm{l}{\mathrm{C}}_2$	524.6	1947	126.9	129.5	2.22	17.46
${(\mathrm{Z}\mathrm{r}\mathrm{H}\mathrm{f})}_4\mathrm{A}\mathrm{l}{\mathrm{N}}_3$	462.4	2112	191.2	178.2	1.77	11.54

and Figure 6a). However, the melting temperature indicated that both Ti₃AlC₂ and (ZrHf)₄AlN₃ had almost equal values (Table 4 and Figure 6b). The elevated melting temperatures of Ti₃AlC₂ and (ZrHf)₄AlN₃ ensure that these two MAX phases can be (processed and) utilised in high-temperature applications without experiencing phase transformations or degradation.

The computed entropy of the samples used in this investigation is shown in Table 4 and Figure 6c, where Ti₃AlC₂ had the lowest value, while (ZrHf)₄AlN₃ had the highest. Even at high temperatures, a material with high entropy maintains a single-phase structure by preventing phase separation and stabilising its multiple phases. This stability is essential for preserving the integrity and functionality of MAX phases in demanding environments. Moreover, materials with high entropy, such as (ZrHf)₄AlN₃ obtained in this study, are more resistant to phase transformations at elevated temperatures. This makes them suitable for high-temperature applications like turbines and heat exchangers. A high entropy value also improves corrosion and oxidation resistance by forming a more stable and inert surface, which is particularly advantageous for MAX phases in harsh chemical environments, such as the chemical processing and petrochemical industries [45].

Figure 7a shows that, at low temperatures, the entropy of all samples approached zero. This observation aligns with the third law of thermodynamics and can be attributed to the significant influence of quantum effects at low temperatures, where the system’s entropy changes are governed by the discrete energy levels available to the particles. However, as the temperature increased, the entropy increased rapidly. This is because the samples behaved similarly to classical systems.

Specific heat is closely related to entropy. As shown in Table 4 and Figure 6d, the specific heat of (ZrHf)₄AlN₃ was the highest, whereas that of Ti₃AlC₂ was the lowest. Moreover, the high specific heat capacity of (ZrHf)₄AlN₃ enhanced its ability to withstand high temperatures without compromising its mechanical properties. This renders it ideal for high-temperature applications such as turbine blades, heat exchangers, and other components in the aerospace and power generation industries [46]. Figure 7b shows that the specific heat of all the samples approached zero at lower temperatures. This is because only low-energy phonons (quantised vibrational modes) are excited at low temperatures. However, as the temperature increased, there was a corresponding increase in the specific heat, approaching the classical Dulong–Petit limit. A further increase in temperature above this limit resulted in significant anharmonic effects, and the specific heat capacity remained relatively constant.

Table 4 and Figure 6e and Figure 8a present the electrical conductivities of all samples, indicating that Ti₃AlC₂ had the best electrical conductivity, followed by (ZrHf)₃AlC₂. The high electrical conductivity of MAX phases arises from their unique crystal structure and the nature of bonding within these materials. MAX phases have a layered structure consisting of alternating layers of: M layers of transition metals like Ti, V, or Cr, A layers of elements from groups 13–16, such as Al or Si, and X layers of carbon or nitrogen atoms. [47]. Their ability to efficiently conduct electricity ensures minimal electrical energy loss and high performance. Both Ti₃AlC₂ and (ZrHf)₃AlC₂ can thus be utilised in these applications.

Table 4 and Figure 8b depict the thermal conductivities of all the MAX phases, which demonstrate that Ti₃AlC₂ has the maximum thermal conductivity (indicating improved performance), while Hf₃GaC₂ and (ZrHf)₄AlN₃ have the minimum values. The layered arrangement of the MAX phases influences thermal conductivity in that the covalent bonding in the carbide or nitride layers provides a stiff lattice, enabling efficient phonon transfer. Moreover, the metallic layers allow for free electrons to conduct heat efficiently, supplementing phonon-mediated heat transfer. Thus, Ti₃AlC₂ is perfect for use in heat sinks and thermal spreaders because of its high thermal conductivity, which allows it to dissipate heat effectively, and also in power electronics, graphics processing units, and central processing units [48].

4. Conclusions

The mechanical and thermal properties of Hf₃AlC₂, Hf₃GaC₂, Ti₃AlC₂, (ZrHf)₃AlC₂, and (ZrHf)₄AlN₃ were successfully calculated in this study. The calculated lattice parameters and densities of Ti₃AlC₂ and Hf₃AlC₂ were found to be in agreement with published values after an examination of their structural properties. The simulated samples’ higher densities than Ti₃AlC₂ suggest that they were better suited for demanding applications (such as thermal management materials and high-temperature coatings). Hf₃GaC₂ had a maximum density of 11,652 kg/m³. All of the samples were structurally stable, according to the calculated formation energies, with (ZrHf)₄AlN₃ having the highest stability at −13.9 eV/atom. (ZrHf)₄AlN₃ had the highest bulk modulus (of 183.6 × 10⁹ Pa) and Young’s modulus (of 193.4 × 10⁹ Pa), which indicates that it is the most rigid, resistant to deformation, and capable of supporting heavier loads without suffering from excessive deformation and fracture.

The highest shear modulus of 122.8 × 10⁹ Pa and Vickers hardness of 4.670 × 10⁹ Pa were exhibited by Ti₃AlC₂. These properties are ideal for withstanding sliding and abrasion forces, and Ti₃AlC₂ is therefore ideal for fabricating cutting tools and drills. While Ti₃AlC₂ exhibited the highest Debye temperature (of 757.3 K), both Ti₃AlC₂ and (ZrHf)₃AlC₂ showed the highest melting temperatures of 2111 K and 2112 K, respectively. (ZrHf)₄AlN₃ exhibited the best entropy and specific heat capacity of 191.2 J/K and 178.2 J/K, respectively, which imply enhanced stable multiple phases. Further, Ti₃AlC₂ recorded the highest electrical conductivity of 2.34 ${\mathrm{\Omega }}^{-1}{\mathrm{m}}^{-1}$, while the best thermal conductivity was observed in case of the (ZrHf)₃AlC₂ sample at 17.46 W/m/K. Thus, the new (ZrHf)₃AlC₂ and (ZrHf)₄AlN₃ MAX phases engineered in this study significantly contribute to the search for better MAX phases with enhanced mechanical and thermal properties. However, this study is purely computational. Thus, experiments should be performed on the modelled samples to verify the observed properties.