First-principles study of the mechanical properties of M₂CT₂ (M=Ti, Zr, Hf; T=O, F, OH) MXenes

First-principles study of the mechanical properties of M 2 CT 2 (M=Ti, Zr, Hf; T=O, F, OH) MXenes 1 2 first-author 1 3 corresp Corresponding author. E-mail address: rencuilan@sinap.ac.cn (Cui-Lan Ren) Corresponding author. E-mail address: rencuilan@sinap.ac.cn (Cui-Lan Ren) rencuilan@sinap.ac.cn 4 corresp E-mail address: wcy58462006@126.com (Chang-Ying Wang) E-mail address: wcy58462006@126.com (Chang-Ying Wang) wcy58462006@126.com 1 2 1 1 3 1 5 6 corresp E-mail address: huaiping@sinap.ac.cn (Ping Huai) E-mail address: huaiping@sinap.ac.cn (Ping Huai) huaiping@sinap.ac.cn Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China University of Chinese Academy of Sciences, Beijing 100049, China University of Chinese Academy of Sciences, Beijing 100049, China Key Laboratory of Interfacial Physics and Technology, Chinese Academy of Sciences, Shanghai 201800, China Key Laboratory of Interfacial Physics and Technology, Chinese Academy of Sciences, Shanghai 201800, China Changzhou Institute of Technology, Changzhou 213032, China Changzhou Institute of Technology, Changzhou 213032, China Center of Shanghai Light Source, Shanghai Advanced Research Institute, Shanghai 201204, China Center of Shanghai Light Source, Shanghai Advanced Research Institute, Shanghai 201204, China School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China Two-dimensional (2D) transition metal carbides known as MXenes belong to a new branch of 2D material family and their fundamental properties vary with their compositions and surface functionalizations. In this study, the structural and ideal mechanical properties of M 2 C-type MXenes and their functionalized M 2 CT 2 MXenes (M=Ti, Zr, Hf; T=O, F, OH) were systematically examined via first-principles methods. The stress–strain (SS) curves of the MXenes under homogenous biaxial and uniaxial tension are identified, and the fundamental quantities (e.g., Young’s modulus, in-plane stiffness, and Poisson’s ratio) are addressed. With significantly higher strength and extended critical strains, the M 2 CO 2 MXenes exhibit optimal flexibility when compared with that of M 2 C, M 2 CF 2 , and M 2 C(OH) 2 . Additionally, Hf 2 CT 2 exhibits optimal tensile performance under uniaxial or biaxial tension when compared to that of Ti 2 CT 2 and Zr 2 CT 2 . The Young’s modulus, in-plane stiffness, and Poisson’s ratio of MXenes with different surface functionalization increase in a sequence corresponding to OH < F < O. Furthermore, the effects of vacancy on the mechanical properties of MXenes are further explored and indicate that vacancy can significantly weaken the tensile properties of MXenes that are considered. Moreover, vacancy also results in a certain anisotropy of stress along armchair and zigzag directions even under the biaxial tension condition. MXenes Mechanical properties Vacancy First-principles study 1. Introduction 1 MXenes constitute a new group of two-dimensional (2D) transition metal carbides, nitrides, and carbonitrides with the chemical formula (M n +1 X n T x , n = 1, 2, 3) and have been successfully synthesized since 2011 [ 1 1 ]. Specifically, MXenes can be fabricated by selectively etching the A layer from the MAX phases (M n +1 AX n , n = 1, 2, 3) materials via hydrofluoride acid (HF) [ 1 1 - 3 3 1-3 ]. The "M" denotes early transition metal, "A" stands for element in III A or IV A group, and "X" denotes carbon or nitrogen. Given the HF-etching preparation method, the MXene surfaces are generally covered with terminal groups "T x " such as oxygen (O), fluorine (F), and hydroxyl (OH) [ 4 4 , 5 5 ]. Additionally, MXenes exhibit excellent thermodynamic stability, superior mechanical strength, and flexibility [ 1 1 ]. Furthermore, they exhibit long-term stability (no / low activation with their low-Z chemical composition), large specific surface areas, high ion-exchange capacity, and acid resistance [ 6 6 ]. The aforementioned unique properties enable the use of MXenes in various applications such as supercapacitors[ 7 7 ], lithium ion batteries[ 8 8 ] catalysis[ 9 9 - 10 10 9-10 ], and spent fuel treatment[ 11 11 - 12 12 11-12 ]. It is known that the spent fuel treatment becomes a challenging environmental concern due to the contamination of actinides (e.g., U and metallic fission products Am, Cs, Sr, and I). Adsorption and separation constitute regular and efficient approaches to clear radioactive contamination. Moreover, 2D MXenes with the aforementioned abundant active sites and high specific surface area are proposed as effective adsorbents of the radioactive metallic contaminations [ 11 11 , 12 12 ]. For example, Wang et al. [ 11 11 ] reported the utility of Ti 2 CT x nanosheets in enhanced separation of U(VI) ion (the uranium species generally exist in the form of uranyl (UO 2 2+ ) in practical environments) from aqueous solutions with a high capacity corresponding to 470 mg/g. Simultaneously, an enhanced adsorption capacity for the radionuclide is reached when compared to that of dry T 3 C 2 T x via rational control of the interlayer space of hydrated T 3 C 2 T x MXenes. It is also reported that with higher adsorption capacity, MXenes exhibit significant prospects in terms of the removal of U(VI) when compared with other nanomaterials such as graphene oxide and metal organic frameworks [ 12 12 ]. Currently, several MXenes including M 2 C (TiC, Nb 2 C, V 2 C), Ti 3 C 2 , Ta 4 C 2 , (Ti 0.5 , Nb 0.5 ) 2 C, Ti 3 CN, et al., have been successfully synthesized [ 13 13 - 16 16 13-16 ]. Moreover, some MXenes, such as Mg 2 C, are theoretically predicted by using CALYPSO software [ 17 17 ]. Among the diverse MXenes, titanium-containing (Ti-) MXenes correspond to the most extensively studied species. Given the HF etching preparation method, the MXene surfaces are typically covered with terminal groups such as O, F, and OH [ 4 4 , 5 5 ]. Recently, full Cl-terminated MXenes are synthesized via a molten chlorinated salt etching method [ 18 18 ]. Thus, given the copious family members of such materials, their fundamental properties vary with their chemical compositions and surface functionalization. Most MXenes exhibit metallic properties, which can be tuned via element doping, surface functionalization, defects, external electric field, and stress among others [ 19 19 - 21 21 19-21 ]. Additionally, the mechanical properties are extremely important characters that should be considered during their potential application as irradiation-tolerant materials or spent fuel separation media. When compared with three-dimensional (3D) bulk materials, it is important to identify in-plane stiffness constants, Young’s elastic modulus, and in-plane Poisson’s ratio for the 2D MXenes. It is reported that the elastic constants of Ti n +1 X n ( n = 1, 2, 3) are excellent ( > 500 GPa) [ 22 22 ] and that their Young’s modulus is in the range of 430–520 GPa [ 23 23 ]. Yorulmaz et al. [ 20 20 ] extensively investigated the dynamical and mechanical stability of M 2 C (M = Sc, Ti, Zr, Mo, and Hf) and their functionalized M 2 CT 2 (T = O and F) MXenes wherein in-plane stiffness constants are predicted in a range of 92–161 N/m. The MXenes are still considerably stiff due to their stronger M-C/N bonds although they are less stiff than other 2D materials, such as graphene, single-layer h-BN monolayer, and MoS 2 with stiffness constants corresponding to 340 N/m [ 24 24 ], 275.9 N/m [ 25 25 ], and 140 N/m [ 26 26 ], respectively. Furthermore, the stress–strain (SS) curves are an important measure to clarify the mechanical properties of MXenes. Chakraborty et al. [ 27 27 ] revealed that the critical strain (i.e., strain at which the stress reaches its maximal value) of Ti 2 C can sustain ~ 8% and ~ 13–14% under biaxial and uniaxial loads, respectively. The aforementioned values can be significantly improved up to more than 20% via surface oxygen functionalization. Notably, very few MXenes (Mg 2 C) exhibit negative Poisson’s ratios [ 28 28 ]. In a manner similar to the modification of electronic properties, their mechanical properties can also be adjusted via substitutional element d oping and defects [ 27 27 ]. Currently, many theoretical studies were performed [ 20 20 , 23 23 , 27 27 - 30 30 27-30 ] and focus on the mechanical properties of MXenes. However, there is a paucity of studies that examine the systematical development of mechanical properties of MXenes and especially effects of metal composition, functionalized groups, and effects of vacancy on their mechanical properties. It should be noted that the aforementioned strengths of MXenes correspond to ideal mechanical strengths that are defined as the maximum stress required to homogeneously deform its defect-free crystal lattices at zero temperature [ 31 31 , 32 32 ] and is normally significantly higher than the actual stress required to produce a mechanical deformation in defective lattices. However, the ideal mechanical strength is still a fundamental mechanical parameter that characterizes the inherent bonding in the crystal, which sets clear limits on the mechanical properties that materials can achieve [ 32 32 ]. It is reported that the ideal strengths of graphene correspond to 110 GPa at strain of 19.4% along the zigzag direction and 121 GPa at strain of 26.6% along the armchair direction [ 33 33 ]. By introducing inhomogeneous lattice fluctuations, Fu et al. [ 34 34 ] indicated that the mechanical strength of Ti 2 CO 2 and Mo 2 CO 2 can be significantly shrunk when comparing to their ideal strengths. Furthermore, given the well-controlled fabrication processes, the measured mechanical properties of graphene or MoS 2 can gradually approach their theoretically-predicted mechanical values [ 33 33 , 35 35 - 37 37 35-37 ]. Herein, a comprehensive first-principles study was conducted to obtain the ideal mechanical properties of M 2 C (M = Ti, Zr, Hf) and functionalized M 2 CT 2 (M = Ti, Zr, Hf; T = O, F, OH) MXenes. With the aid of systematical analyses on the stress–strain curves under homogenous tension in biaxial and uniaxial directions as well as the corresponding elastic parameters, tendency of mechanical properties among different metallic compositions and chemical group terminated systems were further identified. Furthermore, the effect of vacancy on the mechanical properties of MXenes is also established. 2. Computational details 1 All the first-principles calculations in the study were performed via the density functional theory (DFT) implemented in the VASP (Vienna ab-initio Simulation Package) package [ 38 38 , 39 39 ]. The electronic interactions were described by the projector-augmented wave (PAW) method [ 40 40 ]. The exchange-correlation energy was treated via the Perdew-Burke-Ernzerhof (PBE) functional of the generalized gradient approximation (GGA) [ 41 41 ]. For the plane-wave basis, the energy cutoff was set as 520 eV to ensure good convergence. The hexagonal unit cells of MXenes were selected to evaluate the fundamental structural properties, and two-formula-unit orthorhombic supercells were used to investigate the mechanical properties. The corresponding k -point meshes were set as 18 × 18 × 1 and 8 × 8 × 1 with the Monkhorst–Pack scheme for the two types of models, respectively. The repeating MXene slabs were separated by a vacuum space exceeding 15 Å in c axis to eliminate their interactions, and the periodic boundary conditions were set along the zigzag and armchair directions. The atomic structures of the MXenes were fully relaxed with the convergence criterions of total energies and Hellmann-Feynman forces less than 10 -7 eV and 0.001 eV/Å, respectively. The SS relations of the 2D MXenes under both biaxial and uniaxial tension were calculated to identify their intrinsic mechanical response to the external strain/stress. The biaxial tension was introduced via increasing the in-plane lattice parameters in both zigzag and armchair directions. With respect to the uniaxial tension, the strain was applied by increasing the lattice parameters in the zigzag or armchair direction. For each tensile test, the strain applied in the model increased with an interval of 0.01 strain until the fracture or deformation occurs. For each point, the supercells were relaxed in its orthogonal direction until all the components of the Hellman–Feynman stresses were less than 0.2 GPa to decrease the effect of the stress tensor in the orthogonal direction on the axial stress. Moreover, the box size in the orthogonal direction of the monolayer structure h is ambiguous under the framework of first-principles calculations. The equivalent stress was obtained via scaling the supercell stress with a factor corresponding to h/d in the c direction where h denotes the box lattice in c direction and d denotes the actual thickness of MXenes. This method was successfully employed in the previous calculations for the SS curves of 2D materials [ 27 27 ]. 3. Results and discussion 1 3.1 Structural properties and SS curves 2 As shown in Fig. 1a Fig. 1a , the 2D monolayer M 2 C MXenes are fabricated from the bulk M 2 AC (A = Si/Al) MAX phase materials by selectively removing the A layer (mostly aluminum) [ 13 13 ]. The unit cell of M 2 C crystal is a hexagonal lattice with carbon layer sandwiched between two M layers as identified by the black lines shown in Fig.1 b Fig.1 b . In order to perform the tension tests, the two-formula-unit orthorhombic supercells of M 2 C crystal as denoted by the dashed red lines are tailored from the pristine hexagonal M 2 C supercells. The first nearest-neighbor (1 NN) M-C bond direction of the orthorhombic cell is identified as the zigzag boundary while the second nearest-neighbor (2 NN) M-C bond direction is identified as the armchair boundary. Furthermore, the experimentally as-synthesized MXenes are typically chemically terminated with O, F, OH groups [ 13 13 ]. Specifically, the M 2 C and functionalized M 2 CT 2 MXenes (as shown in Fig. 1 c Fig. 1 c ) are investigated for their mechanical properties. The adsorbed sites for the functionalized groups are selected to be chemi-bonded at the hollow site under which there is no carbon atom, and these sites are identified as the most energetic favorable sites in previous studies [ 21 21 ]. The fundamental structural properties of MXenes investigated in the study are initially investigated by fully relaxing their hexagonal unit cells. Calculated lattice parameters, total energies of the optimized unit cells, bond lengths of M-C and M-T, and monolayer thickness of M 2 C and their functionalized M 2 CT 2 (M = Ti, Zr, Hf; T = O, F, OH) MXenes are listed in Table 1 Table 1 . Among the MXenes considered in the study, the Ti 2 CT 2 systems are the most examined species as mentioned previously. The calculated lattice parameters and total energies of Ti 2 C and Ti 2 CT 2 MXenes are compared with the available experimental studies and DFT calculations via PBE method in previous studies, and they are found to be in good agreement with each other [ 20 20 - 22 22 20-22 , 29 29 , 42 42 ]. a Ref. 20 b Ref. 22 c Ref. 29 d Ref. 42 e Ref. 21 1 1   1 1 Lattice parameters (Å) 1 1 E (eV) 1 1 d M-C (Å) 1 1 d M-T (Å) 1 1 Thickness d (Å) 1 1 Ti 2 C  1 1 3.040 (3.039 a , 3.01 b , 3.040 c , 3.040 d ) 1 1 -24.675  1 1 2.101  1 1 —  1 1 2.308(2.29 b , 2.31 c )  1 1 Zr 2 C 1 1 3.280 1 1 -26.206 1 1 2.283 1 1 — 1 1 2.551 1 1 Hf 2 C 1 1 3.211 1 1 -29.170 1 1 2.246 1 1 — 1 1 2.538 1 1 Ti 2 CO 2   1 1 3.035 (3.035 a , 3.034 e , 3.04 c ) 1 1 -44.460(-44.459 e ) 1 1 2.186  1 1 1.979  1 1 4.456 (4.45 c )  1 1 Ti 2 CF 2 1 1 3.060 (3.060 a , 3.04 c ) 1 1 -39.360 1 1 2.104 1 1 2.161 1 1 4.777 (4.80 c ) 1 1 Ti 2 C(OH) 2 1 1 3.076 (3.04 c ) 1 1 -51.033 1 1 2.115 1 1 2.183 1 1 6.789 (6.78 c ) 1 1 Zr 2 CO 2 1 1 3.319 1 1 -46.994 1 1 2.375 1 1 2.131 1 1 4.670 1 1 Zr 2 CF 2 1 1 3.311 1 1 -41.221 1 1 2.278 1 1 2.323 1 1 5.116 1 1 Zr 2 C(OH) 2 1 1 3.319 1 1 -52.823 1 1 2.286 1 1 2.350 1 1 7.155 1 1 Hf 2 CO 2 1 1 3.265 1 1 -50.770 1 1 2.333 1 1 2.103 1 1 4.613 1 1 Hf 2 CF 2 1 1 3.263 1 1 -44.082 1 1 2.232 1 1 2.310 1 1 5.067 1 1 Hf 2 C(OH) 2 1 1 3.274 1 1 -55.899 1 1 2.243 1 1 2.318 1 1 7.052 The SS curves are extremely important measures to evaluate mechanical properties of a material. Generally, the SS curves are drawn via recording the amount of strain and corresponding stress during their tensile deformation. The curves reveal many essential mechanical properties including Young’s modulus, Poisson’s ratio, and critical strains. Here, the SS curves of M 2 C and their functionalized M 2 CT 2 MXenes under biaxial and uniaxial (along zigzag or armchair direction) tensile strains are calculated and shown in Fig. 2 Fig. 2 based on the methods described in the computational details section. This indicates that the SS curves of MXenes begin with a quasi-linear region at small strains and is followed by a plastic region at larger strains. The critical strain here is defined as the strain value at which the stress reaches its maximal value. Young’s modulus under biaxial and uniaxial tensions can be obtained by the linear fit on the initial slope of the corresponding SS curves. In the SS curves of pristine Ti 2 C MXene shown in Fig. 2 a Fig. 2 a , ideal tensile strength and critical strain of pristine Ti 2 C under the in-plane strain are calculated as 26.5 GPa and 7%, respectively, and they are significantly lower than that under the uniaxial strain with strengths and critical strains corresponding to 38-41GPa and 12%, respectively. The ideal tensile strengths and critical strains of Zr 2 C/Hf 2 C exhibit similar trends as those of Ti 2 C(see Fig. 2 b-c Fig. 2 b-c ). This indicates that the functionalized M 2 CT 2 systems reach maximum ideal strengths when they are subjected to 18–25% biaxial strains (Fig. S2 a, d, g). In a manner relative to the non-terminated M 2 C MXenes, the increased critical strains for the three functionalized M 2 CT 2 prove that the surface functionalized groups enhance the toughness of pristine M 2 C MXene, which allow them to sustain larger strains without plastic deformation. The results also indicate that the biaxial SS curves for all the chemical functionalized MXenes (the black lines) are exactly drawn above the uniaxial SS curves (see the blue and red lines in Fig. 2 d-l Fig. 2 d-l ) and indicate their relatively better mechanical properties. When the chemical functionalized MXenes are uniformly stretched in a uniaxial direction, the ideal strength along the zigzag direction exceed that along the armchair direction. It is observed that this type of behavior of Ti 2 CO 2 is consistent with the previously calculated SS relations for Ti 2 CO 2 and the Ti 2 (C 0.5 B 0.5 )O 2 systems [ 27 27 , 34 34 ]. The critical strains for F/OH terminated M 2 CT 2 along the armchair direction are lower than those along the zigzag direction and indicate that fracture or deformation can occur at lower stages when applying strain along the armchair direction. With respect to a more intuitive comparison of the mechanical properties among different metal compositions and different chemical terminated groups, the SS curves of MXenes with the sequence of metal compositions (M = Ti, Zr, Hf) and functionalized groups (T = O, F, OH) are redrawn and shown in Fig. S1 and Fig. S2, respectively. As shown in Fig. S1, the tensile properties of pristine Hf 2 C and Hf 2 CT 2 (T = O, F, OH) system are the most optimal among the aforementioned three systems irrespective of the loads applied in uniaxial or biaxial directions. The Ti 2 CT 2 and Zr 2 CT 2 systems exhibit similar tensile behavior when stretched in the uniaxial direction and tensile properties of Ti 2 CT 2 systems exceed those of Zr 2 CT 2 when biaxial tension is involved. It should be noted that for non-functional MXenes, the tensile properties exhibited in the SS curves vary significantly with different metal-containing systems (see Fig. S1 a-c). This indicates that the ideal tensile strength of Hf 2 C is 50% and 33% higher than Ti 2 C and Zr 2 C under biaxial tension, respectively, and 28–32% improvements are observed for that along the uniaxial direction. For the functionalized M 2 CT 2 , similar tensile properties of MXenes among the different transition metal compositions can arise from the similarity in the chemical properties of Ti, Zr, and Hf elements. The SS curves of MXenes with different functionalized groups are plotted in Fig. S2. It is clearly visible that the tensile properties of MXenes with different surface functionalization increase in a sequence corresponding to OH < F < O. With respect to the biaxial tensile tests (see Fig. S2 a, d, g), the biaxial tensile strengths and critical strains for all the chemical functionalized M 2 CT 2 MXenes are improved when compared to that of M 2 C (as also see Fig. 2 d-l Fig. 2 d-l ). The biaxial tensile critical strains of all M 2 CT 2 exceed that of graphene (~15%) [ 43 43 , 44 44 ]. Furthermore, for all the chemical functionalized MXenes, the O-terminal systems exhibit the maximum biaxial ideal strength (61-68GPa) at a lower biaxial strain of ~19%, when compared with that of F-terminal MXenes (42-47GPa; 20-22%) and OH-terminal MXenes (32–35 GPa; 23–25%). For the uniaxial tensile tests along the armchair (see Fig. S2 b, e, h) and zigzag (see Fig. S2 c, f, i) directions, it is also observed in the elastic parts (at smaller strains) of the SS curves that the O-terminated MXenes exhibit similar elastic properties to the non-terminated ones, which significantly exceed that of -F and -OH terminated MXenes. The strong interaction between the transition metal element and oxygen terminated groups aid in improving the toughness of M 2 C and further slow the collapse of M 2 CO 2 MXenes. This type of strong bonding arises from the hybrid state interaction between M-d and C-p orbitals. The aforementioned results indicate that the M 2 C and M 2 CO 2 MXenes can exhibit good mechanical properties, such as elastic constants and Young’s Modulus, and this is subsequently discussed. Furthermore, few uniaxial SS curves for the uniaxial tension of M 2 CO 2 appear to increase with large strain > 20%. The M-C and M-O bond lengths are also recorded during uniaxial tension to identify the atomic structure evolution. As shown in Fig. S3 and Table S1, we consider the uniaxial SS of Ti 2 CO 2 as an example. There are two types of Ti-C bonds wherein one is relatively more robust while another is significantly elongated during the tension test. The bond length elongates to 3.02 Å, which is close to 138% when compared to that without strain for tension along the armchair direction, indicating that the atomic structure is subject to serious lattice distortion. Therefore, the uniaxial SS curves are plotted with the same strain range of biaxial tension. 3.2 In-plane mechanical properties 2 By comparing with the bulk 3D materials, it extremely important to identify the in-plane elastic properties for these types of 2D materials. Young’s Modulus was calculated to identify the material's response for in-plane biaxial strain. The value of Young's modulus corresponds to the rigidity of materials, which is obtained by linear fitting of the elastic deformation stage of the stress–strain curve [ 23 23 , 45 45 ] as follows: <math display="block" id="M1"><mrow><mi>E</mi><mo>=</mo><mfrac><mi>σ</mi><mi>ε</mi></mfrac></mrow></math> where σ and ε denote stress and strain in the elastic deformation stage, respectively. With increases in Young's modulus, the material is more resistant to deformation. In the study, the Young’s modulus calculated from the stress–strain curves under biaxial tension are mainly discussed. The properties for uniaxial tension exhibit a similar trend via comparing the initial slope of relative SS curves as shown in Fig. 2 Fig. 2 . Specifically, the strain ranges of less than ~ 6% and ~ 8% are selected as the elastic region of the M 2 C and M 2 CT 2 MXenes, respectively, as shown in Fig. 2 Fig. 2 . As the calculated Young’s modulus shown in Fig. 3(a) Fig. 3(a) , non-terminated M 2 C MXenes exhibit the highest Young’s modulus while surface functionalization decreases their values. Moreover, the Young's modulus of M 2 CT 2 with different functionalized groups is in a descending order of O > F > OH. Among different functionalized states, the Young's modulus of Zr 2 CT 2 is lower than that of the Ti 2 CT 2 and Hf 2 CT 2 as shown in Fig. 3(b) Fig. 3(b) . In-plane stiffness C and Poisson’s ratio ν are other two important constants to characterize the properties of reduced dimensional MXenes. In the case of 2D materials, given the reduced dimensionality of these materials and ambiguity of the thickness of a monolayer structure h , it is more important to define the in-plane stiffness C as opposed to the classical 3D Young’s modulus as a measure of the mechanical property, as extensively used in literature for different 2D materials [ 46 46 ]. The in-plane stiffness constant C is calculated as follows [ 28 28 , 46 46 ]: <math display="block" id="M2"><mrow><mi>C</mi><mo>=</mo><mtext> </mtext><mfrac><mn>1</mn><mrow><msub><mi>S</mi><mn>0</mn></msub></mrow></mfrac><mfrac><mrow><msup><mo>∂</mo><mn>2</mn></msup><msub><mi>E</mi><mi>S</mi></msub></mrow><mrow><mo>∂</mo><msup><mi>ε</mi><mn>2</mn></msup></mrow></mfrac></mrow></math> where S 0 denotes the equilibrium in-plane area of the MXenes, ε denotes biaxial strain while E s denotes the energy difference between the MXenes in their equilibrium and tensile states, and this is denoted as strain energy. The Poisson’s ratio of a 2D material is defined as the ratio of the contractile strain along one direction to the tensile strain along another direction, and it is positive for most materials [ 47 47 , 48 48 ]. It is expressed as follows: <math display="block" id="M3"><mrow><mi>v</mi><mo>=</mo><mo>−</mo><mfrac><mrow><msub><mi>ε</mi><mn>1</mn></msub></mrow><mrow><msub><mi>ε</mi><mn>2</mn></msub></mrow></mfrac></mrow></math> where <math id="M4"><mrow><msub><mi>ε</mi><mn>1</mn></msub></mrow></math> and <math id="M5"><mrow><msub><mi>ε</mi><mn>2</mn></msub></mrow></math> denote the strains in two orthogonal directions when performing the uniaxial tensile test. The calculated in-plane stiffness ( C ), Young’s modulus ( E ), ideal strength (σ), critical biaxial strain, and Poisson’s ratio ( ν ) for the M 2 C and their functionalized M 2 CT 2 (M = Ti, Zr, Hf; T = O, F, OH) MXenes are listed in Table 2 Table 2 . The calculated in-plane stiffness constants of M 2 C (M = Ti, Zr, Hf) MXenes are in the range of 125–154 N/m. These properties are significantly improved to be in the range of 268–294 N/m for M 2 CO 2 , and this significantly exceeds that for M 2 CF 2 and M 2 C(OH) 2 . The in-plane stiffness constants of MXenes are lower than that of graphene, single layer h -BN, and higher than MoS 2 as shown in Table 2 Table 2 [ 24 24 - 26 26 24-26 ]. a Ref. 20 b Ref. 27 c Ref. 23 d Ref. 24 e Ref. 25 f Ref. 33 1 1   1 1 C 1 1 E 1 1 σ 1 1 <math id="M6"><mrow><msubsup><mi>ε</mi><mrow><mtext>cri</mtext></mrow><mrow><mtext>bi</mtext></mrow></msubsup></mrow></math> 1 1 ν 1 1 Ti 2 C 1 1 563 / 130 (125 a , 142 b ) 1 1 634(577 b , 610 c )  1 1 26.720 1 1 7 (8 b ) 1 1 0.300 (0.366 b ) 1 1 Zr 2 C 1 1 521 / 133 (129 a ) 1 1 646 1 1 33.628 1 1 10 1 1 0.270 1 1 Hf 2 C 1 1 651 / 165 (154 a ) 1 1 771 1 1 44.316 1 1 10 1 1 0.263 1 1 Ti 2 CO 2 1 1 603 / 268 (238 a )  1 1 578 (570 b , 567 c )  1 1 67.946 1 1 20 (23 b ) 1 1 0.295 (0.303 b ) 1 1 Zr 2 CO 2 1 1 582 / 271 (240 a ) 1 1 548 1 1 61.181 1 1 19 1 1 0.298 1 1 Hf 2 CO 2 1 1 679 / 294 (271 a ) 1 1 594 1 1 67.968 1 1 18 1 1 0.250 1 1 Ti 2 CF 2 1 1 420 / 200 (159 a ) 1 1 393 1 1 47.358 1 1 20 1 1 0.300 1 1 Zr 2 CF 2 1 1 402 / 205 (141 a ) 1 1 346 1 1 42.261 1 1 20 1 1 0.294 1 1 Hf 2 CF 2 1 1 368 / 186 (157 a ) 1 1 371 1 1 46.096 1 1 22 1 1 0.270 1 1 Ti 2 C(OH) 2 1 1 289 / 195 1 1 267 1 1 35.417 1 1 23 1 1 0.228 1 1 Zr 2 C(OH) 2 1 1 266 / 190 1 1 245 1 1 32.013 1 1 23 1 1 0.228 1 1 Hf 2 C(OH) 2 1 1 294 / 207 1 1 273 1 1 34.964 1 1 25 1 1 0.220 1 1 Graphene 1 1 — / 340 d 1 1 h-BN 1 1 — / 275.9 e 1 1 MoS 2 1 1 — / 140 f 1 1   1 1   1 1   1 1   As mentioned in Sect. 3.1, the surface functionalization with O, F, and OH groups allow M 2 C MXenes to sustain extended strains. The critical strains for the non-terminated M 2 C are calculated in a range of 7–10%, and it increases to 18–20%, 20–22%, and 23–25% for M 2 CT 2 in the presence of surface terminal O, F, and OH groups, respectively. 3.3 Vacancy effect on the mechanical properties of MXenes 2 Defects and especially vacancy defects are unavoidable during the preparation or application of MXenes and can deteriorate material performance. On another hand, defects can tailor the properties of a material to a significant extent. Therefore, it is highly desirable to evaluate the vacancy effect on the mechanical properties of MXenes. The relative stability of vacancy is evaluated by the defect formation energy ( E f ), which is defined as follows [ 49 49 - 51 51 49-51 ]: <math display="block" id="M7"><mrow><msub><mi>E</mi><mtext>f</mtext></msub><mo>=</mo><msub><mi>E</mi><mrow><mtext>def</mtext></mrow></msub><mo>−</mo><msub><mi>E</mi><mrow><mtext>per</mtext></mrow></msub><mo>+</mo><msub><mi>μ</mi><mn>0</mn></msub><mo>,</mo></mrow></math> where E def denotes the total energy of the defective supercell, E per denotes the energy of the defect-free supercell, and μ 0 denotes the chemical potential of the vacancy species. Three possible types of vacancy defects can exist, namely metal vacancy ( V M ), carbon vacancy ( V C ), and functional group vacancy ( V T ) in MXenes. The titanium vacancies were experimentally obtained [ 52 52 ] while the carbon vacancies intrinsically exist in MAX phase materials [ 53 53 ]. We consider Ti 2 CO 2 MXene as an example, and the formation energy of typical vacancies in Ti 2 CO 2 MXene were systematically investigated as a function of vacancy concentrations [ 21 21 ]. The chemical potentials of Ti/Zr/Hf and C refer to the energy of their pure solids, and the chemical potential of O corresponds to the half energy of an O 2 molecule. From the point of vacancy formation energy, this indicates that the carbon vacancy exhibits the lowest formation energy for various concentration when compared to that of titanium and oxygen vacancies, indicating that carbon vacancies are most likely to be formed [ 54 54 ]. Conversely, the atomic structures of Ti 2 CO 2 MXene with titanium vacancy are quite unstable for high vacancy concentration, and it becomes stable in lower defect concentration supercells [ 52 52 ]. Specifically, in order to prevent the structure from losing stability due to excessive vacancy defect concentration, single carbon vacancy in the two-formula-unit orthorhombic supercells of M 2 CT 2 (i.e., the mostly stable type of defect in MXenes) is selected to identify the defect effect on the mechanical properties of functionalized MXenes. The carbon vacancy exhibits stability in a comparable small cell of MXene lattice, which also saves computational resources in terms of a systematical calculation of their mechanical properties. The calculated formation energies of carbon vacancy in M 2 CT 2 (M = Ti, Zr, Hf; T = O, F, OH) MXenes are listed in Table 3 Table 3 . It is reported that the carbon formation energy for MXenes depends on the carbon vacancy concentration. Specifically, the formation energy of carbon vacancy in Ti 2 CO 2 is calculated as 0.0015 eV, which is slightly smaller when compared to the predicted value in previous studies [ 21 21 , 54 54 ] due to the difference in the shape of super lattices. For all the MXenes with different terminated groups, the tendency for carbon formation energies increases in the following order corresponding to Ti, Zr, and Hf. While for the MXene with specific metal compositions, it exhibits an increasing trend similar to the terminated O, OH, and F groups. The carbon vacancy in the case of Ti 2 CO 2 MXene is identified with the lowest formation energy, indicating that carbon defect is most likely to be formed in all the cases that are considered. a Ref. 21 b Ref. 53 1 1 3 1 Vacancy formation energy (eV) 1 1 1 1 Ti 1 1 Zr 1 1 Hf 1 1 M 2 CO 2 1 1 0.0015 (0.385 a ) 1 1 1.5808 1 1 2.3617 1 1 M 2 CF 2 1 1 2.5718 (2.7 b ) 1 1 3.1134 1 1 3.5381 1 1 M 2 C(OH) 2 1 1 2.2903 1 1 2.8672 1 1 3.3730 The effect of carbon vacancy on the mechanical properties of M 2 CO 2 (M = Ti, Zr, Hf) systems are systematically investigated by exploring their SS curves as shown in Fig. 4 Fig. 4 . It is noted that the stress components along the armchair and zigzag directions for the biaxial tension are slightly different from each other, and we consider the smaller component of the stress obtained along the orthogonal directions. The results indicate that the mechanical properties of the materials are significantly weakened due to the effect of carbon vacancy. The in-plane strength and critical strain of Ti 2 CO 2 with single carbon vacancy for the biaxial tensile correspond to 34.1 GPa and 9%, respectively, which are approximately half that of defect-free structures. The values are also lower than the results (~ 44.4 GPa, 15%) calculated in previous studies by introducing lattice fluctuations [ 34 34 ]. Moreover, the strength of Zr 2 CO 2 and Hf 2 CO 2 also decreased by ~ 40%. The elastic parameters are also calculated as listed in Table 4 Table 4 , and Young’s modulus and in-plane stiffness of each system are also reduced by approximately 20–27% and 17–21%, respectively. The results indicate that the toughness of the material is significantly reduced, and the defective M 2 CO 2 may deform under lower strain than those without defects. Subsequently, we compare the tensile behavior of the M 2 CO 2 (M = Ti, Zr, Hf) MXene systems with carbon vacancy under biaxial tension ( Fig. 5(a) Fig. 5(a) ). After the carbon vacancy is introduced, the critical strain of Zr 2 CO 2 system is derived as ~ 10%, which indicates that the Zr 2 CO 2 can deform under ~ 10% biaxial strain while Ti 2 CO 2 and Hf 2 CO 2 can withstand up to 12% biaxial strain. As previously mentioned, Zr 2 CO 2 exhibits the lowest Young's modulus, and thus the material is most prone to deformation. 1 1 V C -doped M 2 CO 2 1 1 C 1 1 E 1 1 σ 1 1 <math id="M8"><mrow><msubsup><mi>ε</mi><mrow><mtext>cri</mtext></mrow><mrow><mtext>bi</mtext></mrow></msubsup></mrow></math> 1 1 ν 1 1 Ti 2 CO 2 -V 1 1 479 / 210 1 1 420 1 1 34.108 1 1 9 1 1 0.300 1 1 Zr 2 CO 2 -V 1 1 478 / 221 1 1 428 1 1 37.085 1 1 10 1 1 0.330 1 1 Hf 2 CO 2 -V 1 1 533 / 244 1 1 475 1 1 40.721 1 1 12 1 1 0.300 For the biaxial tension of the defect-free MXenes, the derived stress components along both armchair and zigzag directions are identical to each other, and there exists anisotropy between the unidirectional tension along armchair and zigzag directions as shown in Fig. 2 Fig. 2 . For the defective MXenes, the stress components along the zigzag and armchair directions under biaxial tension are plotted as solid and dashed lines in Fig. 5 Fig. 5 , respectively. This indicates that the two stress components along the armchair and zigzag directions are different from each other after the carbon vacancy is introduced. The anisotropy is mainly attributed to the difference in the linear defect densities along armchair and zigzag directions. In order to further explore the underlying mechanism for the phenomenon, the bonding conditions between the adjacent carbon and surrounding titanium atoms are analyzed. As shown in Fig. 5(b) Fig. 5(b) , there are two types of bonds, namely "ba" and "bz", which are quasi-parallel to the armchair and zigzag directions, respectively. The average length of Ti-C bonds, denoted as "ba" and "bz", during biaxial tension are plotted in Fig. 5(c) Fig. 5(c) . When the biaxial strain increases, the bond length of "ba" and "bz" exhibits a trendency of separation. The length of Ti-C bond elongates in the zigzag direction although it shrinks in the orthogonal armchair direction with increases in strain. Thus, the stresses in the two orthogonal directions are inconsistent and exhibit anisotropic manners even for the biaxial tension loads that are applied on the in-plane surface. Furthermore, increases in stress applied to the model cause the bond lengths to exhibit a more serious separation during their elastic stages. Subsequently, atomic structures can be deformed and reconstructed in the following plastic stages. 4. Conclusion 1 The structural and mechanical properties of MXenes are crucial to their potential applications. In this study, the mechanical properties of M 2 CT 2 (M = Ti, Zr, Hf; T=O, F, OH) MXenes were systematically investigated via a first-principles method. The SS curves of MXenes under homogenous tension in their biaxial and uniaxial directions and elastic parameters including Young’s modulus, in-plane stiffness, and Poisson’s ratios were calculated and analyzed. The main findings are summarized as follows: First, ideal strength and critical strain of M 2 C were in a range of 26–44 GPa and 7–10%, respectively. With significantly higher ideal strength and extended critical strains, the M 2 CO 2 MXenes exhibited optimal flexibility when compared with that of M 2 C, M 2 CF 2 , and M 2 C(OH) 2 . The Young’s modulus, in-plane stiffness, and Poisson’s ratio of MXenes with different surface functionalization decrease in a sequence corresponding to O > F > OH. Second, for the M 2 CT 2 with different metal compositions, the Hf 2 CT 2 exhibited the best tensile performance under uniaxial and biaxial tension, and similarity in the tensile properties of MXenes among the different transition metal compositions can arise from the similarity in the chemical properties of Ti, Zr, and Hf elements. Furthermore, the anisotropy between biaxial and uniaxial tension was identified for each MXene. 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The authors acknowledge the TMSR Supercomputer Center, CAS and the Shanghai supercomputer center for providing the computing resources. 10.1007/s41365-019-0688-x The study was partially supported by the National Natural Science Foundation of China (Nos. 11605273 and 11847064), the Shanghai Municipal Science and Technology Commission (No.16ZR1443100). 2019-05-17 2019-07-01 2019-07-09 2019-10-30 2019-11-01 2021-04-11 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2019 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-019-0688-x http://dx.doi.org/10.1007/s41365-019-0688-x . 2019 Nuclear Science and Techniques 11 30