Theoretical Prediction and Experimental Synthesis of Zr₃AC₂ (A = Cd, Sb) Phases

Abstract

MAX phases have great research value and application prospects, but it is challenging to synthesize the MAX phases containing Cd and Sb for the time being. In this paper, we confirmed the existence of the 312 MAX phases of Zr₃CdC₂ and Zr₃SbC₂, both from theoretical calculations and experimental synthesis. The Zr₃AC₂ (A = Cd, Sb) phase was predicted by the first-principles calculations, and the two MAX phases were confirmed to meet the requests of thermal, thermodynamic, and mechanical stabilities using formation energy, phonon dispersion, and the Born–Huang criteria. Their theoretical mechanical properties were also systematically investigated. It was found that the elastic moduli of Zr₃CdC₂ and Zr₃SbC₂ were 162.8 GPa and 164.3 GPa, respectively. Then, differences in the mechanical properties of Zr₃AC₂ (A = Cd, In, Sn, and Sb) were explained using bond layouts and charge transfers. The low theoretical Vickers hardness of the Zr₃CdC₂ (5.4 GPa) and Zr₃SbC₂ (4.3 GPa) phases exhibited excellent machinability. Subsequently, through spark plasma sintering, composites containing Zr₃CdC₂ and Zr₃SbC₂ phases were successfully synthesized at the temperatures of 850 °C and 1300 °C, respectively. The optimal molar ratio of Zr:Cd/Sb:C was determined as 3:1.5:1.5. SEM and the EDS results analysis confirmed the typical layered microstructure of Zr₃CdC₂ and Zr₃SbC₂ grains.

1. Introduction

Nowotny discovered the H-phases in 1967, and Barsoum defined this kind of material as the M_n+1AX_n phase (MAX phase) until 2000 [1,2,3,4]. M_n+1AX_n phases are a family of hexagonal, ternary layered compounds. Generally, M-site elements are early transition metals, A-site elements mainly come from ⅢA-ⅤA elements, and X-site elements are C, N, or B. All MAX phases have a hexagonal crystal structure, and their space group belongs to P6₃/mmc. The X atom is located at the center of the octahedron composed of M atoms, and the entire crystal structure is composed of densely packed M₆X octahedron layers and A layers, which are alternated periodically. According to the difference in the number of M layers between two A layers, MAX phases can be generally divided into 211 (n = 1), 312 (n = 2), and 413 (n = 3) phases [5,6,7,8,9,10,11,12,13]. The MAX phase contains covalent, ionic, and metallic bonds and has both ceramic and metallic properties [14]. The strong covalent and ionic bonds between MX atoms contribute to the thermodynamic and mechanical stability of the MAX phase [15]. The weak covalent bonds and metal bonds exist between MA atoms, and metal bonds exist between MM atoms. The special nanolayered crystal structure endows them with a high elastic modulus, high strength, and excellent machinability, as well as high electrical conductivity and high thermal conductivity. Therefore, MAX phases are promising to be high-temperature structural materials, electrode brush materials, chemical resistant materials, high-temperature heating materials, etc. [16,17].

Nowadays, MAX phases are becoming popular research materials due to their excellent properties and wide range of potential applications [18]. In recent years, more and more 312 MAX phases have been discovered and studied. For instance, 312 MAX phases containing In and Sn as A-position elements have been successfully synthesized recently. Lapauw et al. successfully synthesized Zr₃SnC₂ by spark plasma sintering with Fe, Co, or Ni additives in 2017 and proposed the mechanism for the synthesis of Zr₃SnC₂ [19,20]. Most recently, an indium-containing MAX phase of Zr₃InC₂ was also successfully sintered by spark plasma sintering [21]. In the periodic table of elements, Cd and Sb are in the same period of In and Sn, and these four elements are located next to each other. However, perhaps due to the challenging synthesis of MAX phases in which the A elements are Cd and Sb, we found that previous research reports on MAX phases containing Cd and Sb were uncommon. For example, Ti₂CdC was successfully sintered at 750 °C by Nowotny in 1964 [22]. The crystal structure, chemical bonding, and elastic properties of Ti₂CdC were investigated by first-principles calculations, and this cadmium-containing MAX phase was predicted to have potential applications as electric friction materials [23,24]. Therefore, theoretical prediction and experimental synthesis of Zr₃AC₂ (A = Cd, Sb) MAX phases is a necessary and meaningful endeavor. To the best of our knowledge, this article is the first report on Zr₃CdC₂ and Zr₃SbC₂ phases. The discovery of these two MAX phases may provide ideas for the synthesis of high-purity Zr₃CdC₂ and Zr₃SbC₂ phases and new options in the field of electrically and thermally conductive materials.

In this work, the new 312 MAX phases of Zr₃AC₂ (A = Cd, Sb) were predicted to be stable by using the first-principles calculations, and their basic mechanical properties were systematically calculated. Furthermore, by adopting spark plasma sintering, the Zr₃CdC₂ and Zr₃SbC₂ phases were successfully synthesized, and their laminar microstructures were well determined.

2. Experimental Procedures

2.1. First-Principles Calculation

In this work, all the calculations were performed by the density functional theory (DFT) calculations in a framework of generalized gradient approximation with the projector augmented-wave method implemented in the Vienna ab initio simulation package (VASP.5.4.4, Hafner Group, University of Vienna, Vienna, Austria), where the Perdew–Burke–Ernzerhof (PBE) functional has been adopted for exchange-correlation potential [25,26]. The convergence criteria for energy and force were set to 10⁻⁶ eV and 0.01 eV/Å, respectively. The plane-wave basis set cutoff was 450 eV after the convergent test, and the special k-points sampling integration over the Brillouin zone was employed using the Monkhorst–Pack method with 12 × 2 × 2 spatial k-points mesh. The elastic constants, mechanical moduli, and phonon dispersion were calculated through the finite displacement method. In order to obtain an accurate acoustic vibration mode, we first optimized the lattice structure of the MAX phase. Secondly, the mechanical constants were calculated using density functional perturbation theory (DFPT). Finally, lattice vibration and thermodynamic properties were extracted from the VASP output by the Phonopy post-processing 2.17.1 software. Because MAX phases had a layered structure, the van der Waals force was involved through the Grimme-D3 scheme to correctly address the interactions between layers. This method can more accurately describe the weak interaction between molecules by introducing additional van der Waals terms [27]. Additionally, the crystal structures of the Zr₃AC₂ (A = Cd, Sb) phases were visualized by using the VESTA-gtk3 software [28].

2.2. Synthesis of Zr₃CdC₂ and Zr₃SbC₂ Phases

Commercial powders of zirconium (99.5% purity, 500 mesh; Eno High-Tech Material Development Co., Ltd., Qinhuangdao, China), cadmium (99.5% purity, 300 mesh; Eno High-Tech Material Development Co., Ltd., China), antimony (99.9% purity, 500 mesh; Eno High-Tech Material Development Co., Ltd., China), and graphite (99.95%, 500 mesh; Eno High-Tech Material Development Co., Ltd., China) were used as the initial materials. To successfully synthesize the Zr₃CdC₂ and Zr₃SbC₂ phases, different sintering temperatures and molar ratios were designed. The powders were weighed using an electric balance with an accuracy of 10⁻⁴ g and mixed in a plastic bottle for 12 h at 50 rpm. Then, the mixture was placed into a graphite die with a diameter of 20 mm. The sintering process was carried out in a spark plasma sintering furnace (SPS-20T-10, Shanghai Chenhua Technology Co., Ltd., Shanghai, China). The heating rate was 50 °C/min, and the applied pressure was 30 MPa. After sintering, the samples were cooled down to ambient temperature with the furnace. The surface contaminations of samples were removed by a diamond grinding wheel.

2.3. Composition and Microstructure Characterization

X-ray diffraction (XRD) (Ultima IV, Rigaku, Japan) with a Cu-Kα radiation source in the 2θ range of 5–75° was used for the composition analyses of the Zr₃CdC₂ and Zr₃SbC₂ phases. The theoretically calculated X-ray diffraction patterns of Zr₃CdC₂ and Zr₃SbC₂ were determined by the Materials Studio 8.0 software (Accelrys Software Inc., San Diego, CA, USA) using the optimized lattice parameters of Zr₃CdC₂ and Zr₃SbC₂ as the input. The microstructure features and elemental compositions of the fracture surfaces of sintered samples were monitored using a scanning electron microscope (SEM) (Apreo 2C, Thermo Fisher Scientific, Brno, Czech Republic) equipped with an energy dispersive spectroscope (EDS) (Oxford ultim Max 65, Oxford Ins., Oxford, UK).

3. Results and Discussion

3.1. Stability Calculation of Zr₃CdC₂ and Zr₃SbC₂ Phases

The crystal structures of experimentally synthesized Zr₃SnC₂ and Zr₃InC₂ are both P6₃/mmc [19,21]. Since Cd, In, Sn, and Sb belong to the same period in the periodic table of the elements, it was predicted that the properties and crystal structures of Zr₃CdC₂ and Zr₃SbC₂ are similar to those of Zr₃SnC₂ and Zr₃InC₂, and therefore, they were modeled and calculated according to the P6₃/mmc space group. Figure 1 shows the crystal structures of two new 312 MAX phases of Zr₃CdC₂ and Zr₃SbC₂. The optimized lattice parameters of Zr₃AC₂ (A = Cd, In, Sn, Sb) were calculated and are listed in

Table 1. Calculated lattice parameters (Å) and formation energies (eV/atom) of Zr₃AC₂ (A = Cd, In, Sn, and Sb) phases.

Phase	a (Å)	c (Å)	ΔH (eV/atom)
Zr3CdC2	3.319	20.393	−1.104
Zr3InC2	3.337 3.352 [21]	20.251 20.252 [21]	−1.220
Zr3SnC2	3.348 3.359 [19]	19.870 19.876 [19]	−1.282
Zr3SbC2	3.367	19.413	−1.306

. For comparison, the experimental parameters of two synthesized 312 phases of Zr₃InC₂ and Zr₃SnC₂ are also listed. It is clearly determined that when the M and X sites are identical, the planar lattice parameter a shows a monotonic increase from 3.319 to 3.367 Å with the increasing order of the atomic number of A-site elements, while the vertical lattice parameter c shows a decrease simultaneously from 20.393 to 19.413 Å, indicating that the bonds within the (0001) plane become weaker and the interlayer interactions become stronger when the A-site elements go from Cd to In to Sn and further to Sb. Additionally, it is known that if the crystals of Zr₃CdC₂ and Zr₃SbC₂ are stable, they must meet the requests of thermal, thermodynamic, and mechanical stabilities. Therefore, the thermal stability was quantized by the formation energy:

$$\Delta H=\frac{H_{crystal}-l{H}_M-m{H}_A-n{H}_X}{l+m+n}$$

(1)

where M, A, and X are composite elements; l, m, n are the number of each element in the unit cell; H_crystal is the enthalpy of the unit cell; and H_M, H_A, and H_X are the corresponding enthalpies of the composite atom in the MAX phases of Zr₃CdC₂ and Zr₃SbC₂. The calculated formation energies (ΔH) of the Zr₃CdC₂ and Zr₃SbC₂ phases were −1.104 and −1.306 eV/atom, respectively. Therefore, both the formation energies of the Zr₃CdC₂ and Zr₃SbC₂ phases are negative, corresponding to the thermal stability.

Furthermore, the thermodynamic stability of the Zr₃CdC₂ and Zr₃SbC₂ phases was investigated by calculating the phonon dispersion, as shown in Figure 2. The phonon dispersion curve can reflect the dynamic stability of materials to a certain extent. In the first-principles calculation, high symmetry points were selected in the Brillouin zone according to the crystal structure to calculate the mechanical constants. The calculation results showed that there was no negative frequency (imaginary frequency) in the Brillouin zone, which means that the dynamic stability of materials is studied under standard pressure. The minor softness around the Γ and I points are deemed to be caused by the numerical error in calculations but not by the solid-state physics theory. It is observed that Zr₃SbC₂ has a smaller maximum frequency than Zr₃CdC₂, which means that Zr₃CdC₂ has a higher melting point, i.e., the highest frequency mode is more difficult to activate for Zr₃CdC₂ [29]. The calculated melting point of Zr₃CdC₂ (1548.344 K) is higher than that of Zr₃SbC₂ (1451.880 K). In addition, it is known that the Debye temperature is connected to the lattice vibration, thermal expansion coefficient, specific heat, and melting point of the crystal. As the maximum frequency mode of vibration, the calculated T_D using elastic modulus is specified as one of the standard approaches [30,31]. The calculated Debye temperatures of Zr₃CdC₂ and Zr₃SbC₂ are 392.7 K and 386.2 K, respectively, lower than those of Zr₃InC₂ (488.9 K) and Zr₃SnC₂ (493.7 K).

On the other hand, the crystals of Zr₃CdC₂ and Zr₃SbC₂ are also mechanically stable and are confirmed to meet the Born–Huang criteria (Equation (2)), where C_ij is presenting the elastic constants and is listed in

Table 2. Calculated elastic constants of C_ij of Zr₃AC₂ (A = Cd, In, Sn, and Sb) phases.

Phase	C11 (GPa)	C12 (GPa)	C33 (GPa)	C13 (GPa)	C44 (GPa)
Zr3CdC2	283.717	94.155	228.795	67.108	33.936
Zr3InC2	310.372	81.969	248.679	68.846	84.965
Zr3SnC2	286.005	109.654	290.242	89.750	118.166
Zr3SbC2	239.549	108.369	252.822	126.183	62.190

[32]:

$${\mathrm{C}}_{11}>\left|{\mathrm{C}}_{12}\right|,2{\mathrm{C}}_{13}^2<{\mathrm{C}}_{33}({\mathrm{C}}_{11}+{\mathrm{C}}_{12}),{\mathrm{C}}_{44}>0$$

(2)

It is seen that the pure shear elastic constants of C₄₄ are lower than the unidirectional elastic constants of C₁₁ and C₃₃. This result means that the shear deformation is easier to occur in comparison with linear compression along the crystallographic a- and c-axes. The unequal values of C₁₁, C₃₃, and C₄₄ (C₁₁ ≠ C₃₃ ≠ C₄₄) imply different atomic arrangements and hence, different bonding strengths along the a-axis, c-axis, and shear planes. The combination of C₁₂ and C₁₃ leads to functional stress along the crystallographic a-axis when a uniaxial strain exists in both the b- and c-axes. The low values of these constants imply that the Zr₃CdC₂ and Zr₃SbC₂ phases will accept shear deformation along the b- and c-axes when adequate stress is applied to the a-axis of the crystals.

3.2. Theoretical Mechanical Properties of Zr₃CdC₂ and Zr₃SbC₂ Phases

In order to clearly understand the mechanical properties of the Zr₃CdC₂ and Zr₃SbC₂ phases, the bulk modulus (B), shear modulus (G), Young’s modulus (E), Pugh’s ratio, and Vickers hardness (H) were calculated based on the equations (S is the elastic softness constant) [33,34,35,36,37,38,39,40,41]:

$$B_V=\left(2\right(C_{11}+2C_{12}+4C_{13}+C_{33})/9$$

(3)

$$B_R=1/(2S_{11}+2S_{12}+4S_{13}+S_{33})$$

(4)

$$G_V=(3.5C_{11}-2.5C_{12}-2C_{13}+C_{33}+6C_{44})/15$$

(5)

$$G_R=15/(8S_{11}-10S_{12}-8S_{13}+4S_{33}+6S_{44})$$

(6)

$$B=(B_V+B_R)/2$$

(7)

$$G=(G_V+G_R)/2$$

(8)

$$H_V^G=0.1769G-2.899$$

(9)

The calculated bulk moduli, shear moduli, and Vickers hardnesses of the Zr₃CdC₂ and Zr₃SbC₂ phases are listed in

Table 3. Calculated mechanical moduli of B, E, G, Pugh’s ratio of B/G, Vickers hardness of H_V, melting point of T_m, and Debye temperatures of T_D of Zr₃AC₂ (A = Cd, In, Sn, and Sb) phases.

Phase	B (GPa)	G (GPa)	E (GPa)	B/G	HV (GPa)	Tm (K)	TD (K)
Zr3CdC2	137.942	62.477	162.846	2.208	5.448	1548.344	392.7
Zr3InC2	144.437	95.251	242.262	1.455	15.842	1658.135	488.9
Zr3SnC2	160.068	102.306	253.013	1.565	14.625	1647.378	493.7
Zr3SbC2	159.913	61.810	164.265	2.587	4.309	1451.880	386.2

, compared to those of the Zr₃InC₂ and Zr₃SnC₂ phases. Also, the changing tendencies of the bulk moduli, shear moduli, Young’s moduli, and Vickers hardness of the Zr₃AC₂ (A = Cd, In, Sn, and Sb) phases are shown in Figure 3. It is seen that the bulk modulus shows a monotonic increase from 137.942 to 160.068 GPa when the A-site element goes from Cd to In and finally to Sn (Figure 3a). For the Zr₃SnC₂ and Zr₃SbC₂ phases, they have larger bulk moduli of 160.068 and 159.913 GPa, respectively. Whereas for the shear modulus, the Zr₃SbC₂ and Zr₃CdC₂ phases have lower values of 61.810 and 62.477 GPa than those of Zr₃InC₂ and Zr₃SnC₂ of 95.251 and 102.306 GPa, respectively. According to the calculated Pugh’s ratio, Zr₃CdC₂ (2.208) and Zr₃SbC₂ (2.587) are the ductile phases (B/G > 1.75), while Zr₃InC₂ (1.455) and Zr₃SnC₂ (1.565) belong to the brittle phases. As a result, the Zr₃InC₂ (15.842 GPa) and Zr₃SnC₂ (14.625 GPa) phases have higher Vickers hardnesses than those of Zr₃CdC₂ (5.488 GPa) and Zr₃SbC₂ (4.309 GPa), as shown in Figure 3b.

Additionally, the bond length, bond population, and Bader charge (ΔQ) of the Zr₃CdC₂ and Zr₃SbC₂ phases were calculated, as listed in

Table 4. The calculated bond length, bond population, and Bader charge (ΔQ) of Zr₃AC₂ (A = Cd, In, Sn, and Sb) phases.

Phase	Zr1-C Length (Å)	Zr1-C Population	Zr2-C Length (Å)	Zr2-C Population	ΔQZr1	ΔQZr2	ΔQC
Zr3CdC2	2.377	0.85	2.242	1.24	1.591	1.135	−1.692
Zr3InC2	2.383	0.87	2.245	1.24	1.592	1.111	−1.698
Zr3SnC2	2.380	0.87	2.260	1.23	1.578	0.948	−1.577
Zr3SbC2	2.384	0.88	2.268	1.24	1.642	1.016	−1.700

. Here, Zr1 is an atom far away from the atomic layer A, and Zr2 is an atom near the atomic layer A. It can be seen that the comprehensive bond populations of A-site atoms In (0.87 and 1.24) and Sn (0.87 and 1.23) are all larger than those containing Cd atoms (0.85 and 1.24), indicating that the covalent interaction between Zr and In/Sn elements is strong. Because Zr₃InC₂ and Zr₃SnC₂ crystals have stronger covalent bonds inside, they show a higher elastic modulus and Vickers hardness. Whereas, for the Zr₃SbC₂ phase, though its interatomic population number is larger (0.88 and 1.24), its interatomic bond length (2.384 Å and 2.268 Å) is larger than those of Zr₃InC₂ (2.383 Å and 2.245 Å) and Zr₃SnC₂ (2.380 Å and 2.260 Å). Generally speaking, the longer the bond length, the lower the bond energy and the lower the Vickers hardness. Additionally, the total electron gain/loss of electrons in the crystals of Zr₃CdC₂ (1.692 and 2.726) and Zr₃SbC₂ (1.700 and 2.658) is greater than those of Zr₃InC₂ (1.698 and 2.703) and Zr₃SnC₂ (1.577 and 2.526), which indicates that their ionic properties are stronger and their covalent properties are weaker; that is, their Vickers hardness values are lower than those of Zr₃InC₂ and Zr₃SnC₂. The relatively high atomic charge transfer indicates the main ionic bond characteristics. The number of charges transferred by Zr1 is higher than that transferred by Zr2, which indicates that its covalence is weak. Interestingly, this conclusion is self-consistent with the long bond length and weak bond energy of Zr1-C [42].

3.3. Synthesis and Microstructure Characterization of Zr₃CdC₂ and Zr₃SbC₂ Phases

Initially, in an attempt to synthesize the Zr₃CdC₂ and Zr₃SbC₂ phases by spark plasma sintering, mixed powders with the molar ratio of Zr:Cd/Sb:C = 3:1.5:1.5 were sintered based on the consideration of easy evaporation of Cd/Sb and C deficiency in the P6₃/mmc hexagonal crystals. Considering the synthesis temperature of Ti₂CdC (750 °C [22]), Zr₃InC₂ (1400 °C [21]) and Zr₃SnC₂ (1200 °C [19]), the sintering temperatures of complexes were initially designed as 850 °C and 1300 °C, respectively. The examined X-ray diffraction (XRD) patterns acquired from the synthesized samples are displayed in Figure 4a and Figure 5a. In Figure 4a, in combination with the PDF cards, it is determined that a large amount of Zr, ZrC, ZrCd₂, and unknown phases exist in the cadmium-containing sample; while, in Figure 5a, it is judged that ZrC, Zr₂Sb₃, Zr₂Sb, and unknown phases are in the antimony-containing sample. Significantly, in comparison with the theoretical XRD patterns, as shown in Figure 4b and Figure 5b, the (002) and (004) diffraction peak positions of Zr₃CdC₂ and Zr₃SbC₂ have a similar degree of rightward shift. This might be due to the occurrence of crystal defects in the Zr₃CdC₂ and Zr₃SbC₂ phases, resulting in smaller lattice parameters. Anyhow, the (002) and (004) diffraction planes of Zr₃CdC₂ and Zr₃SbC₂ can still be presumed to be in greater accord with the theoretical results. Therefore, it is expected that the Zr₃CdC₂ and Zr₃SbC₂ phases have been synthesized. Furthermore, in order to enhance the purity of the samples, the mixture powders were sintered at different temperatures, and the different molar ratios of the mixture powders were designed and annealed at the optimal temperature for comparison. Figure S1 shows the XRD patterns of the cadmium-containing samples synthesized at 650–1050 °C with the molar ratio of Zr:Cd:C = 3:1.5:1.5. And Figure S2 shows the XRD patterns of the antimony-containing samples synthesized at 900–1400 °C with the molar ratio of Zr:Sb:C = 3:1.5:1.5. By comparing the XRD patterns in Figure S1, it can be found that only at 850 °C does the cadmium-containing sample have the least impurity phases, and the XRD pattern is closest to the theoretical result in Figure 4b. Therefore, 850 °C is considered to be the optimum sintering temperature for the Zr₃CdC₂ phase. Similarly, the optimum sintering temperature for synthesizing the Zr₃SbC₂ phase was determined as 1300 °C. Continuously, for the Zr₃CdC₂ phase, the mixed powders with the molar ratios of Zr:Cd:C = 3:(1.4–1.8):2 were sintered at 850 °C, and the examined XRD patterns are shown in Figure S3. It is seen that no obvious existence of the Zr₃CdC₂ phase could be determined. Then, the molar ratios of the mixture powders were modified to be 3:1.5:(1.5–2.25), and the samples were sintered at the optimum temperature of 850 °C. It is seen that there is no remarkable enhancement of purity of the Zr₃CdC₂ phase, as shown in Figure S4. It seems that the optimal molar ratio is 3:1.5:1.5 to synthesize the Zr₃CdC₂ phase. Similarly, for the Zr₃SbC₂ phase, the mixed powders with the molar ratios of Zr:Sb:C = 3:(1.5–1.95):2 were sintered at 1300 °C. By comparing the XRD patterns of samples sintered with the different molar ratios, the purity of the Zr₃SbC₂ phase was not obviously enhanced, as shown in Figure S5. Therefore, the optimized molar ratio of synthesizing the Zr₃SbC₂ phase is confirmed as Zr:Sb:C = 3:1.5:1.5. Owing to the existence of too many impurities in the sintered samples, it is difficult to refine the obtained XRD patterns by using the Rietveld method. Anyhow, in order to richen the information of the new phases of Zr₃CdC₂ and Zr₃SbC₂, the theoretical crystal constants, atomic positions, and the XRD data are given and listed in Tables S1–S3.

In addition, in order to characterize the Zr₃CdC₂ and Zr₃SbC₂ grains in the sintered samples, the fracture surfaces of sintered bulks that underwent the shear loading were observed by using the scanning electron microscope, as shown in Figure 6 and Figure 7. Clearly, the grains with a typical nanolaminar character of MAX phases in the sintered bulks are the Zr₃CdC₂ and Zr₃SbC₂ phases (Figure 6a–f and Figure 7a–f), which are the same as other typical MAX phases of Ti₃AlC₂ and Ti₃SiC₂ [43,44]. Obvious bending and twisting phenomena were observed in these grains. Many equiaxed carbide grains are nearby. Additionally, the testing positions of the energy dispersive spectroscopy (EDS) analysis are labeled by white crosses in Figure 6 and Figure 7b,d,f, and the results are displayed in

Table 5. EDS analysis results of Zr₃CdC₂ grains at the test positions in Figure 6b,d,f.

Element	At.% in Figure 6b	At.% in Figure 6d	At.% in Figure 6f
Zr	75.67	75.31	75.38
Cd	24.33	24.69	24.62

and

Table 6. EDS analysis results of Zr₃SbC₂ grains at the test positions in Figure 7b,d,f.

Element	At.% in Figure 7b	At.% in Figure 7d	At.% in Figure 7f
Zr	74.48	74.10	75.74
Sb	25.52	25.90	24.26

. It is confirmed that the average element ratios are Zr:Cd = 3:0.976 for Zr₃CdC₂ grains and Zr:Sb = 3:1.012 for Zr₃SbC₂ grains, respectively, very close to the ratio of 3:1. Here, no data of the C element was collected, as the EDS analysis could not guarantee the accuracy of the C element.

Therefore, based on the above results, it is believed that the Zr₃CdC₂ and Zr₃SbC₂ phases could stably exist and be successfully synthesized. Combined with the above theoretical calculations, the pure Zr₃CdC₂ and Zr₃SbC₂ MAX phases are shown to have excellent machinability. However, due to the limitation of sintering, it is rather difficult to purify the Zr₃CdC₂ and Zr₃SbC₂ phases, and it is greatly worth it for us to explore new synthesis methods to improve the purity of the two MAX phases.

4. Conclusions

Two new ternary laminar MAX phases of Zr₃CdC₂ and Zr₃SbC₂ were predicted and synthesized. The obtained results are listed as follows:

• It was confirmed that the Zr₃CdC₂ and Zr₃SbC₂ phases belonged to the space group of P6₃/mmc with the hexagonal crystal structure. The calculated crystal parameters of Zr₃CdC₂ were a = 3.319 Å and c = 20.393 Å, and those of Zr₃SbC₂ were a = 3.367 Å and c = 19.413 Å. The calculated formation energies of Zr₃CdC₂ and Zr₃SbC₂ were −1.104 and −1.306 eV/atom, respectively, which confirm the thermal stability of the two phases. The absence of imaginary frequencies in the acoustic branches of the phonon band structure and the calculation results using the Born–Huang criterion confirmed the thermodynamic and mechanical stability of the two MAX phases. Additionally, the calculated Young’s moduli of Zr₃CdC₂ and Zr₃SbC₂ were 162.846 GPa and 164.265 GPa, respectively. The theoretical Vickers hardnesses of the Zr₃CdC₂ (5.448 GPa) and Zr₃SbC₂ (4.309 GPa) phases were low due to the weaker covalent bonds, exhibiting excellent potential machinability.
• Through spark plasma sintering, composites containing the Zr₃CdC₂ and Zr₃SbC₂ phases could be synthesized at temperatures of 850 °C and 1300 °C, respectively. The optimal molar ratio of Zr:Cd/Sb:C was determined as 3:1.5:1.5. Based on the SEM micrographs, the nanolayered characters of the Zr₃CdC₂ and Zr₃SbC₂ grains were determined. The average element ratios were Zr:Cd = 3:0.976 for Zr₃CdC₂ grains and Zr:Sb = 3:1.012 for Zr₃SbC₂ grains, respectively, very close to the ratio of 3:1.