3.1. Structural properties of

The MAX phases have a hexagonal symmetry in space group P63/mmc with Wyckoff positions. The structure of 312 MAX phases consists of hexagonal layers stacked in the repeated sequence A–M(1)–X–M(2)–X–M(1). There are two non-equivalent M atoms denoting M(1) and M(2) with Wyckoff positions of 4f (1/3, 2/3, and 2a (0,0, 0), respectively. The A plane atoms are located in 2b (0, 0, 1/4) and X atoms in 4f (1/3, 2/3, .^[24] Analogously, the isomorphous structure of 211 MAX phases consists of a repeated atomic layer sequence of A–M–X–M. The octahedra, similar to those formed in respective MX binary carbides, are connected with each other by shared edges. The main difference in structure between 211 and 312 phases lies in the number of M layers in between every two A layers.

We perform a thorough research with 12 MAX phases (Ti₃SiC₂, Ti₃AlC₂, Ti₃SiN₂, Ti₃AlN₂, Cr₂AlC, Cr₂GeC, Cr₂GaC, Nb₂AlC, Ta₂AlC, Ti₂AlC, Ti₂AlN, and V₂AlC), most of which have been previously investigated experimentally. We first calculate the lattice constants of these MAX phases to validate both the pseudopotentials and the method used in this simulation. The calculated structural parameters are shown in Table 1. The calculated lattice parameters of a, c values from experiments and previous calculations are also included for comparison. The values in a/b–Err (%) and c–Err (%) denote the percentage differences among lattice parameters between the calculated values and experimental data, respectively. Some of the blanks in this table represent that they have not been synthesized experimentally yet or they have not been calculated with the DFT method before. As we can see, the lattice parameters in our calculation are similar to the experimental data, while the slight differences between different theoretical calculations can be reasonably attributed to the GGA function used. The results validate our method.

Table 1.

Table 1.

Table 1. The lattice parameters (in unit of Å) for Ti3SiC2, Ti3AlC2, Ti3SiN2, Ti3AlN2, Cr2AlC, Cr2GeC, Ti2AlC, Ti2AlN, and V2AlC; –Err (%) and c–Err (%) denote the largest percentage differences among lattice parameters between the calculated values and experimental data. . Composition a/b c a/b–Err (%) c–Err (%) Ti3SiC2 this work 3.075 17.708 0.26 0.209 DFT[6] 3.077 17.706 0.32 0.198 EXP[25] 3.067 17.671 Ti3AlC2 this work 3.082 18.641 0.032 0.833 DFT[6] 3.081 18.594 0 0.578 EXP[25] 3.081 18.487 Ti3SiN2 this work 2.983 17.826 DFT[26] 2.99 17.84 EXP Ti3AlN2 this work 3.002 18.463 DFT[26] 3.00 18.49 EXP Cr2AlC this work 2.844 12.685 –0.663 –1.0 DFT[22] 2.842 12.687 –0.733 –0.991 EXP[25] 2.863 12.814 Cr2GeC this work 2.938 12.024 –0.406 0 DFT [22] 2.925 12.024 –0.847 0 EXP 2.95 12.024 Cr2GaC this work 2.87 12.54 –0.347 0.555 DFT EXP[27] 2.88 12.61 Ti2AlC this work 3.068 13.731 0.557 0.689 DFT[28] 3.04 13.60 –0.36 –0.271 EXP[25] 3.051 13.637 Ti2AlN this work 2.993 13.614 0.133 0 DFT[29] 2.99 13.61 0.033 0.03 EXP[22] 2.989 13.614 V2AlC this work 2.905 13.118 0.239 0.167 DFT[27] 2.91 13.13 0.068 0.0761 EXP[24] 2.912 13.140

Table 1. The lattice parameters (in unit of Å) for Ti3SiC2, Ti3AlC2, Ti3SiN2, Ti3AlN2, Cr2AlC, Cr2GeC, Ti2AlC, Ti2AlN, and V2AlC; –Err (%) and c–Err (%) denote the largest percentage differences among lattice parameters between the calculated values and experimental data. .

The MAX phases can be considered as robust materials when subjected to ion irradiation. Nevertheless, according to observations different MAX phases behave differently under irradiation and the reason for this remains unclear. Motivated by the necessity to focus extensively on the evolution of the microstructure during the irradiation, we summarize some experimental data to illustrate the diversity of radiation tolerance, and the results are given in Table 2.

Table 2.

Table 2.

Table 2. Summarized experimental data concerning the radiation tolerances of the six materials: Ti3AlC2, Ti3SiC2, Ti2AlC, Ti2AlN, Cr2GeC, and Cr2AlC. Different experimental conditions and variables are shown below (RT denotes room temperature). . Incident ion MAX material Fluences/cm−2 Temperature/°C dpa Degree of amorphization 1-MeV Xe[7] Ti3AlC2 3.25×1014 RT no amorphization Ti3SiC2 3.25×1014 RT begin amorphous neutron( )[10] Ti2AlC 4.00×1020 360(±20) 0.1 decompose into TiC from the pristine 6.7% to 8.3% 4.00×1020 695(±25) 0.1 decompose into TiC from the pristine 6.7% to 4.3% Ti2AlN 4.00×1020 360(±20) 0.1 decompose into TiN from the pristine 3.23% to 11.6% 4.00×1020 695(±25) 0.1 decompose into TiN from the pristine 3.23% to 13% Ti3AlC2 4.00×1020 360(±20) 0.1 decompose into TiC from the pristine 1.9% to 52.6% 4.00×1020 695(±25) 0.1 decompose into TiC from the pristine 1.9% to 44.4% Ti3AlN2 (fine - grained) 4.00×1020 360(±20) 0.1 decompose into TiN from the pristine 20% to 22.7% 4.00×1020 695(±25) 0.1 decompose into TiN from the pristine 20% to 25% 1-MeV Au[30] Ti2AlC 1.00×1014 RT no amorphization 3.00×1014 RT no amorphization, begin to emerge stacking faults 1.00×1016 RT partial phase transition from hcp to fcc 2.00×1016 RT entirely transformed into the fcc phase 100-keV Ar[9] Ti2AlN 1.8×1013–1.3× 1015 RT up to 12 no amorphization 320-keV Xe[11] Cr2GeC 5.00×1013 RT 0.41 begin amorphous 1.00×1014 RT 0.82 fully amorphous Cr2AlC 1.00×1014 RT 0.62 no amorphization 3.50×1014 RT 2.2 begin amorphous 6.00×1014 RT 3.7 fully amorphous 1.00×1015 RT 6.2 fully amorphous

Table 2. Summarized experimental data concerning the radiation tolerances of the six materials: Ti3AlC2, Ti3SiC2, Ti2AlC, Ti2AlN, Cr2GeC, and Cr2AlC. Different experimental conditions and variables are shown below (RT denotes room temperature). .

Basically, all the macroscopic properties of materials originate from their electronic structure properties as well as the natures of the chemical bonding. Trachenko et al. have given a more general comprehension of the physics of radiation damage.^[14] A material with high ionic bonding is more resistant to amorphization than a material with covalent bonding. In general, short-range covalent contributions lead to highly directional bonds and a number of distinct potential energy minima which act as barriers to recovery of periodicity in the vicinity of the collision cascade, whereas long-range ionic contributions minimizes the barriers to recovery of atomic periodicity. What is more, the local recrystallization process is promoted in a material with high ionicity of bonding by ions attracting oppositely charged neighbors and making the “defect” structures that consist of neighboring atoms of the same charge energetically unfavorable, whereas this effect is absent in a covalent structure.^[13,15]

Based on the model provided by Trachenko, the relations between radiation resistance and the bonding trend of these MAX phases can be further analyzed by the calculated total electron density of states (TDOS) as well as their partial density of states (PDOS) as demonstrated in Figs. 1 and 3. The DOS at Fermi level ( determines the electronic conductivity characteristics and all of the MAX phases we calculated are finite at the Fermi energy level, which indicates the metallic properties of MAX phases. What is more, there is a high degree of hybridization of M orbitals and X orbitals, suggesting a strong covalent interaction between M–X in the material. The position of the in TDOS reveals the stability of the system while a local minimum represents relative stablility. As for Ti₃AlC₂, Ti₃AlN₂, Cr₂AlC, Ti₂AlC, V₂AlC, Ta₂AlC, Nb₂AlC, Cr₂GaC, and Ti₂AlN, the TDOS around the Fermi level is at a local minimum in contrast to that of Ti₃SiC2, Ti₃SiN₂, and Cr₂GeC. The conduction bands above are composed mainly of the M atom states. The dose below contains mainly 3s and 2s states contributed by A and X atoms, respectively. The increase in N ( tends to coincide with the increase in valence electrons of the corresponding element.^[24]

Figure 1.

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	Figure 1. (color online) Calculated energy-dependent densities of states and partial densities of states for (a) Cr2AlC, (b) V2AlC, and (c) Ti2AlC, where the green line represents DOS at Fermi level.

To illustrate the influence of transition metal M in MAX phase in terms of its radiation resistance observed in experiment, we choose Ti₂AlC, Cr₂AlC, and V₂AlC for investigating the relations between their radiation resistance and bonding trend. The DOS and PDOS of them are presented in Fig. 3. There is no obvious difference in PDOS of the C atom among these three MAX phases, but a distinct difference can be seen from the hybridizations of M_d–Al_p in Cr₂AlC, V₂AlC, and Ti₂AlC. The hybridization of Ti–Al in Ti₂AlC is closer to E_F than that of V–Al in V₂AlC, while hybridization of Cr–Al in Cr₂AlC is farthest to E_F. This difference possibly originates from the fact that the Ti atom has one less charge electron than V, and the V atom has one less charge electron than Cr. The weaker bonding of Ti–Al behaves more like an ionic bond which allows the damaged structure to reestablish coherence with the crystalline lattice more easily, whereas the structural reconstruction with stronger bondings of V–Al and Cr–Al should require additional energy cost.

Regarding the bonding trend between M and A, the various resistances to amorphization of Ti₂AlC, V₂AlC, and Cr₂AlC under the same irradiation condition can be easily explained. The weak interaction of M–A in MAX phase promotes irradiated defects recombination, which contributes to preserving their crystalline structures. So the substitutions of Cr by V and V with Ti can increase the resistance of radiation, leading to the conclusion that the crystal structure of the Ti-based phase is more resistant to amorphization of radiation than that of the V-based phase, and so is the case of V-based phases to Cr-based phases.

The charge density contours can also be used to explain the differences in bonding characters. We expand our ideas to the M–A bonding character in the context of the calculated charge density. It is suggested that the high radiation damage tolerance exhibited by many ceramic M–X compounds and the rapid recovery from damage which is inherent in metallic M–A compounds are both present in MAX phases. Figure 2 shows the charge density contours on the (1120) plane for (a) Cr₂AlC, (b) V₂AlC, and (c) Ti₂AlC. One can clearly observe the polarization of the electronic charge around the M atoms which points toward the C atoms in the center. The charge around the C atom is non-spherical, consistent with the existence of covalent bonds between M and C. The charge densities around X atoms are obviously greater than those around A atoms, demonstrating the stronger covalent bonding between M and X. The coupling between M–C and M–Al shows a declining trend for each of Cr₂AlC, V₂AlC, and Ti₂AlC, illustrating that a strong covalent bonding is formed between Cr–C, whereas the bonding around Ti turns more ionic. The stronger bondings of M–X and M–A in Cr-based phases than in Ti-based phases exhibit that they have lower radiation tolerances. The charge density contours we calculated are consistent with the DOS we simulated and also with the experiment results.

Figure 2.

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	Figure 2. (color online) Distributions of electron charge density on the (1120) plane for (a) Cr2AlC, (b) V2AlC, and (c) Ti2AlC, where charge density varies from 0.1 (blue) to 3.2 e/Å3 (dark red).

A similar method has also been used in Xiao et al.ʼs work,^[8] in which this model was validated. Our discussion also verifies the theory on the relationship between irradiation resistance and bonding type. To provide the further comprehension of the mechanism, much more atoms are tested.

An atom in MAX phases also plays a major role in radiation resistance.^[31] To elucidate the different behaviors of A atoms when subjected to the radiation of energetic ions, we focus on the bonding characters of two structures: Ti₃AlC₂ and Ti₃SiC₂. The TDOS and PDOS of them are shown in Fig. 3. Each of Ti₃SiC₂ and Ti₃AlC₂ has a high degree of hybridization between Ti 3d orbitals and C 2p orbitals, indicating a strong covalent interaction between Ti-C in these two materials. The hybridization of Ti–Al in Ti₃AlC₂ is much closer to E_F than that of Ti–Si in Ti₃SiC₂, revealing that the bonding between Ti–Al is weaker than Ti–Si.

Figure 3.

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	Figure 3. (color online) Calculated energy-dependent densities of states and partial densities of states for (a) Ti3AlC2 and (b) Ti3SiC2. The green line indicates DOS at Fermi level.

It can be further verified by the directional bonds between Ti-Si and Ti–Al in these two compounds as revealed by the electron density distribution plot demonstrated in Fig. 4. There are strong covalent bonds between Ti and C atoms, forming directional Ti–C bond chains. The density located in the Ti–Al region is much smaller than that in Ti–Si. The weaker bonding of Ti–Al behaves more like an ionic bond, which represents higher radiation tolerance, while the stronger bonding of Ti–Si shows lower tolerance. There are also some notable differences in bonding between A and X. It should be noted that there are some strong peaks at the same energy in the PDOS of some particular Si atoms and C atoms. A considerable overlap between C 2s, C 2p, Si 3s, and Si 3p orbitals shows a strong covalent bonding, while there is no significant overlap between the C 2s and C 2p orbitals with the Al 3s and Al 3p orbitals in Ti₃AlC₂. The weak overlap between Al and C denotes that minimal interaction is likely to exist between these two elements, primarily due to the difference in energy. The interaction between Si–C is much stronger than that between Al–C in these two MAX phases, leading to different recovery processes subjected to ion irradiation. Because of the notable overlap between Si and C, it is more likely to form nanoscale SiC^[32] rather than Ti₃AlC₂ in the recovery recrystallization process. The weaker bonding between Al and C in Ti₃AlC₂ is more likely to form point defects such as antisite defects and Frenkel pairs which increase the possibility to recombine into a crystalline structure. All the calculated results are consistent with our experimental data, validating the universality of our simulations.

Figure 4.

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	Figure 4. (color online) Distributions of electron charge density of (a) Ti3AlC2 and (b) Ti3SiC2, where charge density varies from 0.1 (blue) to 1.7 e/Å 3 (dark red).

This can also explain the greater resistance to amorphization of Cr₂AlC than Cr₂GeC under the same irradiation condition. The structural reconstruction with stronger bonding of Cr–Ge than Cr–Al should require additional energy cost, which leads to low radiation tolerance.

The improved radiation tolerance of Ti₂AlC compared with Ti₃AlC₂ observed in the experiments has drawn a great deal of attention in recent years. Rietveld analyses of the XRD spectra for Ti₂AlC and Ti₃AlC₂ revealed a drastic difference in irradiation tolerance between the two compounds in Tallman et al.ʼs work.^[10] By enhancing the inherent ability to accommodate radiation-induced defects in nanostructured composites, the A atomic plane is supposed to play a decisive role in improving the radiation tolerance of the material.^[31] In terms of the same chemical elements, other factors should be introduced to fully explain the improvements of radiation tolerance. We focuse on the increased ratio of the Al/TiC layer and the relative interlayer distance in lattice. The density of A atomic planes grows with the increase of the Al/TiC layer. The interlayer distance between Ti–C–Ti in Ti₂AlC is almost half that in Ti₃AlC₂, indicating that Ti₂AlC has more intercalated A atomic planes per MX layer than Ti₃AlC₂. From this perspective, we can explain the improvement of radiation tolerance from Ti₃AlC₂ to Ti₂AlC observed in experiments.

By making a comparison of DOS and charge density among different MAX phases, we find that crystal-structured Ti-based MAX phases are more resistant to amorphization than V/Cr-based MAX phases, and the phases are more resistant than phases. In conclusion, the substitution of Si by Al and Cr/V by Ti introduce extra valence electrons into each atom, improving the resistance to amorphization. The difference in dominating hybridization between C_n and N_n is not obvious, which means that they have similar resistances.

Though the irradiation tolerance of the MAX phase could be roughly estimated by analyzing its DOS and charge density map and so on, it is still difficult to compare more than three MAX phases. It is also difficult to obtain a conclusion by using the density of electron in a certain surface. It should depend on the three-dimensonal (3D) distribution of electrons. Therefore, we try to use the effective charge as a new tool and it is generally perceived to be more accurate than bonding characters presented by DOS overlap and visual observation of the charge density map.^[8,24]

The criterion in terms of charge is based on the good empirical correlation of resistance with the ionicity of the chemical bond, using Pauling or Phillips definitions.^[33] It should be noted that neither of the concepts of ‘ionicity’ and ‘covalency’ is clearly defined because both of them contribute considerably to a bond, making it impossible to draw a clear line between them. One of the reliable ways to determine the bonding character is to analyze the electron density maps obtained from either experimental diffraction methods or quantum-mechanical calculations. Unlike the Bader analysis method in our research, the Voronoi deformation density charges^[34] and Mulliken overlap population^[35,36] were calculated in Trachenko et al.ʼs work. In their report, they proposed that the increase of Q and decrease of M result in the increase of resistance of a complex compound.^[14] But it is not obvious to attribute the charges to one atom or the other in Mulliken overlap population, which is based on the plane wave basis. According to the above discussion, we choose the Bader analysis because it is intuitive to calculating the charge in individual atoms in molecules and crystals, and it also conforms to chemical experience. We have recently compiled the list of twelve materials to illustrate that their resistances can be generally explained by charge transfer as shown in Figs. 5 and 6. The bonding type and charge transfer values of these different 312 and 211 MAX phases are analyzed for allowing the comparison among them.

Figure 5.

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Figure 5. (color online) Charge transfer values of different atoms in different 211 MAX phases, illustrating the discussion about the resistance to amorphization. The red column denotes charge transfer of the M atom in M2AlC, the green column refers to charge transfer of the A atom in Cr2AC, while the blue column represents charge transfer of the X atom in Ti2Al X. A positive value shows that they lose electrons, whereas the negative value denotes that they gain electrons.

Figure 6.

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	Figure 6. (color online) Charge transfer values of different atoms in different 312 MAX phases, illustrating the discussion about the resistance to amorphization.

The amount of charge transfer can characterize the degree of ionicity in a certain MAX-phase compound. The bond behaves more like an ionic bond when the charge transfer value is a high positive or low negative value, allowing the damaged structure to reestablish coherence with the crystalline lattice more easily, whereas the low positive or high negative charge transfer behaves more like a covalent bond, which should require additional energy cost in structural reconstruction.^[37] In this case, we make an in-depth analysis on charge transfers of different M, A, and X atoms in MAX phases.

Regarding the charge transfer trend with 211 MAX phases, M elements have significant influences on the irradiation damage process. From Fig. 5, it can follow that the transition-metal atoms always lose charges. The calculations show that the charge transfers on M atoms decrease in the sequence of Ti₂AlC, V₂AlC, and Cr₂AlC. The charge transfer of Cr in Cr₂AlC is 0.5011, indicating that it loses 0.5011 electrons and V in V₂AlC loses 0.8692 electrons and Ti in Ti₂AlC loses 1.1903 electrons. The higher charge transfer value for early transition metal represents higher ionicity and reduced barriers to regain coherence with the crystalline lattice which makes them significantly more resistant to amorphization. Our predictions of the contrasting stability are also inconsistent with the experimental results. We also calculate the other two M atoms whose radiation tolerance has not been tested in experiment. Nb in Nb₂AlC and Ta in Ta₂AlC lose 0.9040 and 0.9699 electrons respectively. According to our calculated charge transfer of M atoms, we can make a prediction that the crystal-structured Ti-based MAX phase materials are more resistant to amorphization than other materials with the same A and X atoms.

The charge transfer of A atoms in MAX phases plays an important role in their resisting amorphization. The intercalated A atoms in the nanolaminated crystal promote the accommodation of disordering defects within their structures by enhancing the inherent ability to accommodate radiation-induced defects. So we further study the significant difference in radiation tolerance between Cr₂GeC, Cr₂GaC, and Cr₂AlC in 211 MAX phases. It is noted that Al in Cr₂AlC loses 0.3063 electrons, whereas Ga in Cr₂GaC obtains 0.2296 electrons and Ge in Cr₂GeC gains 0.3727 electrons. That is to say, the bonding around Ge in Cr₂GeC behaves as being more covalent than those of the bonding around Ga in Cr₂GaC and Al in Cr₂AlC, conducive to its chemistry forming a covalent network. As a result, an additional energy is required to break the strong covalent bond, inducing the radiation tolerance of Cr₂GeC to be lower than those of Cr₂GaC and Cr₂AlC. Though there are no experimental data about irradiation for Cr₂GaC, we can theoretically predict that it is not the most promising candidate for the nuclear system.

Finally, we focus on the effects of X atoms on radiation tolerance of 211 MAX phases. In Ti₂AlC, the charge transfer value of C is slightly lower than that of N in Ti₂AlN, while C obtains 1.7389 electrons and N gains 1.6 electrons. The similar charge transfer value manifests the basically consistent radiation tolerance of them and no quantitative description of their differences has been shown experimentally so far.

The mechanism of amorphization in terms of the bonding characters in these 211 MAX phases demonstrates that the substitution of Ge/V/Ta/Nb by Ti and Si/Ga by Al in phases induces higher radiation resistance because the bonds behave as being more ionic. That is to say, as the weight of the short-range covalent forces goes down and the weight of long-range ionic forces goes up, its resistance increases.

A meaningful comparison of the character of charge transfer electrons among 312 MAX phases can also be carried out. Our calculation about charge transfer contrasts the result from the readily amorphizable materials with resisting the radiation. It is illustrated that the Si in Ti₃SiC₂ and Ti₃SiN₂ obtain more electrons than Al in Ti₃AlC₂ and Ti₃AlN₂ as shown in Fig. 6. The increase of positive charge transfer value and decrease of negative charge transfer value improve the radiation tolerance of 312 MAX phases. Thus, the covalent bonding character of elemental Si in Ti₃SiC₂ and Ti₃SiN₂ lower the resistances to amorphization compared with the scenario of Al in Ti₃AlC₂ and Ti₃AlN₂.

Covalency and ionicity are different characterization ways in which the electron density can be distributed in a solid. Higher ionicity promotes the recombination of irradiated defects, contributing to preserving their crystalline structure. The difference in charge transfer between different atoms in MAX phases can explain the diversity of resistance to amorphization under the same irradiation condition, and also leads to a prior consideration of selecting the MAX phases that can highly resist the radiation for the applications in a future nuclear system.