# Deformation and Failure of MXene Nanosheets

^{1}

^{2}

^{*}

## Abstract

**:**

_{3}C

_{2}T

_{z}monolayer flake and to validate the material model. For the evaluation of the adhesive strength of the free-standing Ti

_{3}C

_{2}T

_{z}-based film, the model comprised single-layered MXene nanosheets with a specific number of individual flakes, and the reverse engineering method with a curve fitting approach was used. The interlaminar shear strength, in-plane stiffness, and shear energy release rate of MXene film were predicted using this approach. The results of the sensitivity analysis showed that interlaminar shear strength and in-plane stiffness have the largest influence on the mechanical behavior of MXene film under tension, while the shear energy release rate mainly affects the interlaminar damage properties of nanosheets.

## 1. Introduction

_{2}CT

_{z}, Ti

_{3}C

_{2}T

_{z}) have the unique characteristics of both groups. Due to the combination of the electrical conductivity of transition metal carbides and the hydrophilicity of hydroxyl or oxygen-terminated surfaces, these MXenes behave as “conductive clays” [2].

_{3}C

_{2}T

_{z}, was obtained at 0.33 ± 0.03 TPa [8]. According to classical molecular dynamics simulation [7], which does not take into account various material defects, the modulus was higher and equal to 0.502 TPa.

## 2. Modeling Methods

#### 2.1. FE Model of Nanoindentation

_{3}C

_{2}T

_{z}MXene monolayer flake for analysis of the nanoindentation process in LS-Dyna software and, using the force vs. deflection curve, validate the FE model and material characteristics. The FE model (Figure 1) was developed according to the experimental data presented by Lipatov et al. [8], where it was considered that the MXene flake has isotropic properties, and therefore, the membrane can be parametrized using Young’s modulus, E and Poisson’s ratio, v.

_{3}C

_{2}T

_{z}monolayer flake is an important parameter as it influences the results of the nanoindentation experiments. Using AFM for the determination of the thickness of monolayers of 2D materials has some limitations. The Ti

_{3}C

_{2}T

_{z}MXene flake thicknesses obtained by AFM can differ significantly [8,14], and this directly affects the determined value of Young’s modulus. MXene flakes with a thickness of 0.98 nm were modeled [1,2,11]. The mechanical properties of MXenes and the SiO

_{2}support ring used in the model are based on the analysis of the literature data. The indenter was a diamond crystal with a modulus of 1 TPa. The mechanical properties of the materials used in the nanoindentation simulation are presented in Table 1.

_{3}C

_{2}T

_{z}MXene flake was modeled with shell elements using a linear elastic material model. The size of the shell elements was 10 nm, while the center of the monolayer (contact zone with nanoindenter) was decreased to as little as 1.25 nm. The nanoindenter was defined as an elastic solid sphere (diameter 14 nm). The bottom of the MXene flakes was supported on the ring surface of SiO

_{2}, which was fully fixed. Two pinball-type contacts (AUTOMATIC_NODES_TO_SURFACE) were used: surface of SiO

_{2}—bottom flake, and surface of the nanoindenter—top flake. Two simulations were performed with initial pretension and without. As it was an explicit analysis, the initial pretension initiated the oscillation of the flake; therefore, the *CONTROL_DYNAMIC_RELAXATION must be activated.

^{3}relationship with a coefficient of E

^{2D}, which dominates at large loads. For comparison, only the second part of Equation (1) was used; therefore, the prestretching was not taken into account during the simulation.

#### 2.2. FE Model of Pure MXene Film

_{3}C

_{2}T

_{z}and Ti

_{2}CT

_{z}, with a SiO

_{2}-coated Si spherical tip is one of the recent studies of adhesion properties [17].

_{3}C

_{2}T

_{z}-based films [4] were used. It was assumed that the single-layered nanosheet has a square form of 1 μm length [22,23,24,25], consisting of 18 Ti

_{3}C

_{2}T

_{z}individual flakes and an average thickness of 20 nm [25]. In total, 164 layers of single-layered nanosheets per film of 3.3 μm thickness were used in experimental testing [4]. As the overlapping of nanosheets has an essential influence on the strength of the interlayer surface, a 2D analysis of randomly placed rectangle nanosheets was performed using materials modeling software, Digimat. For the overlapping analysis, MXene nanosheets with dimensions of 1000 × 20 nm were chosen. The overall thickness of MXene film formed from these nanosheets was 3.3 μm, as it was in testing [4] (Figure 2a). The results show that the average overlapping of nanosheets is 20% (Figure 2b). This overlap value was used to create the FE model.

_{3}C

_{2}T

_{z}MXene nanosheet was modeled with shell elements. The Young’s modulus of the MXene nanosheet was set at 333 GPa; Poisson’s ratio—0.227; the size of shell elements—5 nm. The nodes on the left-side edges had a fixed 4 degrees-of-freedom, allowing free contraction in the x-direction and rotation about the z-axis. For the right-side nodes, the displacement vs. time u(t) was applied with a speed of 1 mm/ms. The interface between the single-layered nanosheets was modeled using tiebreak contact, *CONTACT_AUTOMATIC_ONE_WAY_SURFACE_TO_SURFACE_TIEBREAK. The discrete crack model with power-law damage, which works with offset shell elements (option = 11), shown in Figure 4, was chosen. The parameters needed to describe tiebreak contact are presented in Table 2.

_{3}C

_{2}T

_{z}-based film depends on the nanosheet interface shear strength,

## 3. Results

#### 3.1. FE Simulation of Nanoindentation

#### 3.2. FE Simulation of Pure MXene Film

_{3}C

_{2}T

_{z}film strength, stiffness, and failure strain when the overlapping is increased.

## 4. Conclusions

_{3}C

_{2}T

_{z}MXene monolayer flake and validate the material model. Sensitivity analysis of Young’s modulus and the flake thickness was performed. The obtained force versus deflection curves showed lower results than the experimental ones. To achieve similar results to the experimental ones, the thickness of the MXene flake should be increased from 0.98 up to 1.1 nm, or the Young’s modulus should be increased from 333 up to 380 GPa.

_{3}C

_{2}T

_{z}-based films. The interlaminar shear strength, stiffness, and energy release rate were selected as the main variables; these variables affect the tensile strength of the assembled free-standing Ti

_{3}C

_{2}T

_{z}-based film. The experimental tensile test data has been used for the curve fitting approach. A simulation was performed using three and nineteen layers of single-layered nanosheets with 20% overlap. The best fitted FE curve was obtained using an interlaminar shear strength ${\tau}_{\mathrm{interl}}=2.2\mathrm{MPa}$, in-plane stiffness ${E}_{\mathrm{interl}}=0.26\frac{\mathrm{GPa}}{\mathsf{\mu}\mathrm{m}}$, and shear energy release rate ${G}_{IIc}=3.8\times {10}^{-2}\frac{\mathrm{J}}{{\mathrm{m}}^{2}}$. The results of the sensitivity analysis showed that the largest influences on the behavior of tensioned film of free-standing MXene nanosheets are interlaminar shear strength and in-plane stiffness, while the shear energy release rate mainly affects the interlaminar damage properties of nanosheets. The simulation results based on identified material parameters showed an increase in the free-standing Ti

_{3}C

_{2}T

_{z}film strength, stiffness, and failure strain when the overlapping was increased by 50%.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Naguib, M.; Kurtoglu, M.; Presser, V.; Lu, J.; Niu, J.; Heon, M.; Hultman, L.; Gogotsi, Y.; Barsoum, M.W. Two-dimensional nanocrystals produced by exfoliation of Ti
_{3}AlC_{2}. Adv. Mater.**2011**, 23, 4248–4253. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Naguib, M.; Mochalin, V.N.; Barsoum, M.W.; Gogotsi, Y. 25th anniversary article: MXenes: A new family of two-dimensional materials. Adv. Mater.
**2014**, 26, 992–1005. [Google Scholar] [CrossRef] [PubMed] - Ronchi, R.M.; Arantes, J.T.; Santos, S.F. Synthesis, structure, properties and applications of MXenes: Current status and perspectives. Ceram. Int.
**2019**, 45, 18167–18188. [Google Scholar] [CrossRef] - Ling, Z.; Ren, C.E.; Zhao, M.-Q.; Yang, J.; Giammarco, J.M.; Qiu, J.; Barsoum, M.W.; Gogotsi, Y. Flexible and conductive MXene films and nanocomposites with high capacitance. Proc. Natl. Acad. Sci. USA
**2014**, 111, 16676–16681. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Anasori, B.; Lukatskaya, M.R.; Gogotsi, Y. 2D metal carbides and nitrides (MXenes) for energy storage. Nat. Rev. Mater.
**2017**, 2, 1609. [Google Scholar] [CrossRef] - Ghidiu, M.; Lukatskaya, M.; Zhao, M.; Gogotsi, Y.; Barsoum, M. Conductive two-dimensional titanium carbide ‘clay’ with high volumetric capacitance. Nature
**2015**, 516, 78–81. [Google Scholar] [CrossRef] - Borysiuk, V.N.; Mochalin, V.N.; Gogotsi, Y. Molecular dynamic study of the mechanical properties of two-dimensional titanium carbides Ti
_{n+1}C_{n}(MXenes). Nanotechnology**2015**, 26, 1–10. [Google Scholar] [CrossRef] [Green Version] - Lipatov, A.; Lu, H.; Alhabeb, M.; Anasori, B.; Gruverman, A.; Gogotsi, Y.; Sinitskii1, A. Elastic properties of 2D Ti
_{3}C_{2}T_{x}MXene monolayers and bilayers. Sci. Adv.**2018**, 4, 1–7. [Google Scholar] [CrossRef] [Green Version] - Borysiuk, V.N.; Mochalin, V.N.; Gogotsi, Y. Bending rigidity of two-dimensional titanium carbide (MXene) nanoribbons: A molecular dynamics study. Comput. Mater. Sci.
**2018**, 143, 418–424. [Google Scholar] [CrossRef] - Guo, Z.; Zhou, J.; Si, C.; Sun, Z. Flexible two-dimensional Ti
_{n}+ 1C_{n}(n = 1, 2 and 3) and their functionalised MXenes predicted by density functional theories. Phys. Chem.**2015**, 17, 15348–15354. [Google Scholar] - Zhao, M.Q.; Ren, C.E.; Ling, Z.; Lukatskaya, M.R.; Zhang, C.; Van Aken, K.L.; Barsoum, M.W.; Gogotsi, Y. Flexible MXene/carbon nanotube composite paper with high volumetric capacitance. Adv. Mater.
**2015**, 27, 339–345. [Google Scholar] [CrossRef] - Kilikevičius, S.; Kvietkaitė, S.; Žukienė, K.; Omastová, M.; Aniskevich, A.; Zeleniakienė, D. Numerical investigation of the mechanical properties of a novel hybrid polymer composite reinforced with graphene and MXene nanosheets. Comput. Mater. Sci.
**2020**, 174, 109497. [Google Scholar] [CrossRef] - Monastyreckis, G.; Mishnaevsky, L., Jr.; Hatter, C.B.; Aniskevich, A.; Gogotsi, Y.; Zeleniakiene, D. Micromechanical modeling of MXene-polymer composites. Carbon
**2020**, 162, 402–409. [Google Scholar] [CrossRef] - Shearer, C.J.; Slattery, A.D.; Stapleton, A.J.; Shapter, J.G.; Gibson, C.T. Accurate thickness measurement of graphene. Nanotechnology
**2016**, 27, 125704. [Google Scholar] [CrossRef] - Fu, Z.H.; Zhang, Q.F.; Legut, D.; Si, C.; Germann, T.C.; Lookman, T.; Du, S.Y.; Francisco, J.S.; Zhang, R.F. Stabilization and strengthening effects of functional groups in two-dimensional titanium carbide. Phys. Chem.
**2016**, 94, 104103. [Google Scholar] [CrossRef] - Jang, J.; Suhr, J.; Gibson, R.F. Combined numerical/experimental investigation of particle diameter and interphase effects on coefficient of thermal expansion and Young’s modulus of SiO
_{2}/epoxy nanocomposites. Polym. Compos.**2012**, 33, 1415–1423. [Google Scholar] [CrossRef] - Zhang, H.; Fu, Z.H.; Legut, D.; Germannd, T.C.; Zhang, R.F. Stacking stability and sliding mechanism in weakly bonded 2D transition metal carbides by van der Waals force. RSC Adv.
**2017**, 7, 55912–55919. [Google Scholar] [CrossRef] [Green Version] - Jiang, T.; Zhu, Y. Measuring graphene adhesion using atomic force microscopy with a microsphere tip. Nanoscale
**2015**, 7, 10760–10766. [Google Scholar] [CrossRef] [PubMed] - Burnham, N.A.; Dominguez, D.D.; Mowery, R.L.; Colton, R.J. Probing the surface forces of monolayer films with an atomic-force microscope. Phys. Rev. Lett.
**1990**, 64, 1931–1934. [Google Scholar] [CrossRef] [PubMed] - Cappella, B.; Kappl, M. Force measurements with the atomic force microscope: Technique, interpretation and applications. Surf. Sci. Rep.
**2005**, 59, 1–152. [Google Scholar] - Li, Y.; Huang, S.; Wei, C.; Wu, C.; Mochalin, V.N. Adhesion of two-dimensional titanium carbides (MXenes) and graphene to silicon. Nat. Commun.
**2019**, 10, 3014. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mashtalir, O.; Naguib, M.; Mochalin, V.N.; Dall’Agnese, Y.; Heon, M.; Barsoum, M.W.; Gogotsi, Y. Intercalation and delamination of layered carbides and carbonitrides. Nat. Commun.
**2013**, 4, 1–7. [Google Scholar] [CrossRef] [PubMed] - Jastrzębska, A.M.; Karwowska, E.; Wojciechowski, T.; Ziemkowska, W.; Rozmysłowska, A.; Chlubny, L.; Olszyna, A. The atomic structure of Ti
_{2}C and Ti_{3}C_{2}MXenes is responsible for their antibacterial activity toward E. coli bacteria. J. Mater. Eng. Perform.**2019**, 28, 1272–1277. [Google Scholar] [CrossRef] - Wang, H.; Zhang, J.; Wu, Y.; Huang, H.; Li, G.; Zhang, X.; Wang, Z. Applied Surface Science Surface modified MXene Ti
_{3}C_{2}multilayers by aryl diazonium salts leading to large-scale delamination. Appl. Surf. Sci.**2016**, 384, 287–293. [Google Scholar] [CrossRef] - Chen, L.; Shi, X.; Yu, N.; Zhang, X.; Du, X. Measurement and analysis of thermal conductivity of Ti
_{3}C_{2}T_{x}MXene films. Materials**2018**, 11, 1701. [Google Scholar] [CrossRef] [Green Version] - LS-DYNA Keyword User’s Manual; Livermore Software Technology Corporation (LSTC): Livermore, CA, USA, 2014; Volume II, pp. 1–1265.

**Figure 2.**Estimation of nanosheets overlapping: (

**a**) segment of the Digimat model of free-standing Ti

_{3}C

_{2}T

_{z}film, (

**b**) obtained overlapping length distribution by simulation of randomly placed nanosheets.

**Figure 3.**FE model for the simulation of MXene nanosheet interface shear strength with 20% overlapping length: (

**a**) 3-nanosheets-thick model, (

**b**) 19-nanosheets-thick model. Red and blue colors are used for contrast to better show the single-layered nanosheets.

**Figure 4.**Bilinear traction–separation and the mixed-mode traction–separation law [26].

**Figure 5.**Dependencies for FE modeling: (

**a**) displacement vs. time curve used for loading (red) and force vs. time for the curve fitting procedure (black); (

**b**) force vs. displacement curve recalculated from the experiments [4].

**Figure 7.**FE results of the nanoindentation simulation: (

**a**) velocity of the nanoindenter and the center point of the MXene monolayer flake, (

**b**) deflection of the nanoindenter and the center point of the MXene monolayer flake.

**Figure 8.**Sensitivity analysis results: (

**a**) thickness influence on the force vs. deflection curve when E = 333 GPa; (

**b**) Young’s modulus influence on the force vs. deflection curve when h = 0.98 nm.

**Figure 9.**Sensitivity analysis of free-standing three-layer MXene nanosheets: (

**a**) force vs. time curve sensitivity upon in-plane stiffness ${E}_{\mathrm{interl}}$;

**×**—experimental results obtained from [4] and curve fitting of MXenes, (

**b**) influence of the interlaminar shear strength and shear energy release rate as ${E}_{\mathrm{interl}}=0.26\frac{\mathrm{GPa}}{\mathsf{\mu}\mathrm{m}}$.

**Figure 10.**FE simulation results: (

**a**) Tiebreak contact gap development episode in cases of 20%/80% and 50%/50% overlap; (

**b**) tensile stress–strain curves of different thicknesses for free-standing MXene films; the experimental results are recalculated from the literature [4].

Material | Density, ρ, g/cm^{3} | Elastic Modulus, E, GPa | Poisson’s Ratio, ν | Tensile Strength, σ_{u}, GPa |
---|---|---|---|---|

Ti_{3}C_{2}T_{z} | 3.19 [8] | 333 [8] | 0.227 [15] | 17.3 [8] |

SiO_{2} [16] | 2.65 | 70.0 | 0.17 | - |

Diamond nanoindenter | 3.50 | 1000 | 0.20 | - |

Normal Failure Stress, nfls (T), MPa | Shear Failure Stress, sfls (S), Mpa | Normal Energy Release Rates, eraten (G_{IC}) mJ/m^{2} | Shear Energy Release Rates, erates (G_{IIC}) mJ/m^{2} | Ratio of Tangential Stiffness to Normal Stiffness, ct2cn | Normal Stiffness, Cn €, MPa/μm |
---|---|---|---|---|---|

2 ÷ 4 | 2 ÷ 4 | 30 ÷ 60 | 30 ÷ 60 | 1 | 200–350 |

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**MDPI and ACS Style**

Zeleniakiene, D.; Monastyreckis, G.; Aniskevich, A.; Griskevicius, P.
Deformation and Failure of MXene Nanosheets. *Materials* **2020**, *13*, 1253.
https://doi.org/10.3390/ma13051253

**AMA Style**

Zeleniakiene D, Monastyreckis G, Aniskevich A, Griskevicius P.
Deformation and Failure of MXene Nanosheets. *Materials*. 2020; 13(5):1253.
https://doi.org/10.3390/ma13051253

**Chicago/Turabian Style**

Zeleniakiene, Daiva, Gediminas Monastyreckis, Andrey Aniskevich, and Paulius Griskevicius.
2020. "Deformation and Failure of MXene Nanosheets" *Materials* 13, no. 5: 1253.
https://doi.org/10.3390/ma13051253