Simulation setup

We consider rectangular Ti_n+1C_n nanosheets with n = 1, 2, 3 and lateral size 20×17 nm in a Cartesian coordinate box with periodic boundary conditions in the xy direction and free boundary condition in the z direction. An example typical configuration of the system is shown in Fig. 1 (all MD snapshots in this paper were produced using Visual Molecular Dynamics software^[Reference Humphrey, Dalke and Schulten^27]). Previously developed in-house code for parallel computing was employed to calculate the interatomic forces between atoms in the samples during heating. Detailed description of the developed approach and corresponding interatomic potentials can be found in Refs. Reference Borysiuk, Mochalin and Gogotsi^15, Reference Borysiuk, Mochalin and Gogotsi¹⁷ and Reference Oymak and Erkoc^28, Reference Zhou, Wadley, Johnson, Larson, Tabat, Cerezo, Petford-Long, Smith, Clifton, Martens and Kelly²⁹, respectively.

Figure 1.

Typical configuration of Ti₂C nanosheet at T = 300 K used in this study.

Heating of the Ti_n+1C_n nanosheets was simulated by increasing temperature using Berendsen thermostat,^[Reference Berendsen, Postma, Vangunsteren, Dinola and Haak^30] according to the scheme described in Ref. Reference Alarifi, Atiş, Özdoğan, Hu, Yavuz and Zhou2. Temperature of the samples was increased from the initial value of 300 K in 50 K increments. Similar to Ref. Reference Alarifi, Atiş, Özdoğan, Hu, Yavuz and Zhou2, at each particular temperature the system was simulated for 100,000 MD steps ( ≈ 20 ps), during which all data required for calculation of the structural parameters were recorded. Then the temperature was increased to the next value and the procedure repeated. Around the thermal degradation point the temperature increment was reduced to 10 K for a better accuracy.

To characterize MXene stability and thermal degradation we calculated the Lindeman indexes. Lindemann index is a commonly used parameter in MD simulations of melting processes. It indicates changes in crystal structure of the sample due to thermal disorder. The local Lindemann index q _i of an atom i is defined as^[Reference Zhang, Stocks and Zhong^21]:

(1) $$q_i = \displaystyle{1 \over {N-1}}\sum\limits_{\,j\ne i} {\displaystyle{{\sqrt {\langle{r_{ij}^2} \rangle -{\langle{r_{ij}} \rangle} ^2}} \over {\langle{r_{ij}} \rangle}}}, $$

where r _ij is a distance between atoms i and j, and angle brackets indicate time average at constant temperature. Lindemann index q of a sample consisting of N atoms can be estimated by averaging over all atoms in the system.

The melting point of the sample can be determined from the critical value of the Lindeman index q _c. The range of values q _c ≈ 0.1–0.15 is typically used as a melting criterion for bulk solid structures. However, in recent studies of nanomaterials, a significantly lower value q _c ≈ 0.03 was proposed for determination of the melting points of nanosized samples, e.g., carbon nanotubes.^[Reference Zhang, Stocks and Zhong^21] Moreover, in MD simulations of melting of various nanostructures presented in the literature, the melting points were observed at values close to q _c ≈ 0.03 (see Table 1 and related references), thus it seems reasonable to use this value as an indicator of thermally induced degradation for Ti_n+1C_n nanosheets. Besides the actual value of the Lindemann index, its temperature dependence also provides information about melting of the sample. It is known that at low temperatures the Lindemann index is small and slowly increases in nearly linear manner with the temperature. At the onset of melting, q is expected to switch to nonlinear increase with temperature. Thus, examining the magnitude and critical behavior of the Lindeman index obtained from Eq. (1), it is possible to estimate the melting point of the sample. To further elucidate structural changes, temperature dependences of the potential energy and RDF below and above the thermal degradation temperature were also calculated.

Table I.

Approximate values of Lindemann indexes for various nanostructures near their melting points

Results and discussion

The results of MD simulations are presented in Figs. 2–5. Figure 2 presents the temperature dependence of Lindeman indexes (Fig. 2a) and averaged potential energy (Fig. 2b) for Ti₂C, Ti₃C₂, and Ti₃C₄ samples, calculated from atomistic configurations. As follows from the figure, Lindeman index q exceeds the critical value q _c = 0.03 at approximately 1050, 1500, and 1700 K for Ti₂C, Ti₃C₂, and Ti₃C₄ respectively. These temperatures correspond to abrupt changes in q(T) dependencies from slow to very fast growth, which further confirms the thermally induced degradation processes. It is worth noting that the corresponding degradation temperatures increase with the thickness of the MXene samples. Temperature dependencies of the average potential energy calculated around the degradation points (Fig. 2b) also exhibit behavior typical for the melting process with the regions of slow and rapid growth prior to phase transition that results in slowdown of the potential energy growth. Similar trends in potential energy versus temperature have been observed in various MD simulations of the melting processes of other nanostructures.^[Reference Alarifi, Atiş, Özdoğan, Hu, Yavuz and Zhou^2, Reference Yang, Yang and Xu^20, Reference Zhang, Stocks and Zhong^21]

Figure 2.

Temperature dependencies of Lindeman index (a) and average potential energy (b) for 2D Ti₂C, Ti₃C₂, and Ti₃C₄.

Figure 3.

Top view of typical atomistic configurations of 2D Ti₂C, Ti₃C₂, and Ti₃C₄ during heating. Top and bottom panels show the samples with crystal (lower temperature) and amorphous (higher temperature) structures respectively. Corresponding temperatures of the samples (top and bottom snapshots) are 1000 and 1120 K for Ti₂C, 1450 and 1650 K for Ti₃C₂, and 1650 and 1710 K for Ti₃C₄.

Figure 4.

Top (top panel) and side (bottom panel) views of the MXene samples after complete thermal decomposition.

Figure 5.

Atomic RDF of the samples at initial and maximal simulated temperature (inner panels show detailed RDF at small distances) for Ti₂C (a), Ti₃C₂ (b), and Ti₃C₄ (c).

Snapshots of the systems at two different temperatures (right before the degradation starts and at the point when the degradation is clearly visible), indicating the changes in the atomic structure due to transition from crystal to amorphous configurations are shown in Fig. 3. The top panel represents clearly visible long-range ordered crystal structures of all three MXene samples at the temperatures below the onset of thermal degradation. Snapshots in the bottom panel, taken at higher temperatures, reveal local regions of the amorphous phases, indicating the thermal degradation process.

Top and side views of the atomistic configurations of the samples at maximum simulated temperatures are shown in Fig. 4. As can be seen from the figure, after complete MXene degradation most of the carbon atoms have been vaporized from the samples and move around freely in the simulation box. Titanium atoms tend to form continuous (due to periodic boundary conditions in the xy plane) rod-shaped nanostructures for Ti₂C and Ti₃C₂, while for Ti₄C₃ sample Ti atoms largely retain the original shape of a rectangular sheet without voids. This difference may be explained by a larger thickness of Ti₄C₃ sample. Another reason may be due to a greater number of the particles involved in simulations in case of Ti₃C₄, which need more time to re-arrange into a more stable configuration. The fact that carbon atoms do not form any particular structure after the structure degradation, whereas titanium atoms tend to arrange into a kind of a structure, is most likely related to the form of interatomic potentials for carbon and titanium used in our simulations and may be an artifact. Whereas titanium atoms, described by embedded atom method potential, exhibit typical metallic behavior forming nanoparticle-like amorphous structures, carbon atoms interact with titanium and with each other through simple Lennard-Jones potential and thus do not form any specific configurations.

It is important to note that a recent MD simulation with ReaxFF^[Reference Lotfi, Naguib, Yilmaz, Nanda and van Duin^26] showed that Ti and C atoms in Ti₃C₂ MXene rearrange into cubic titanium carbide TiC at 1500 K in vacuum. As we used simpler approach that does not capture chemical bonding, we only observed the thermal degradation of 2D Ti₃C₂ sample at approximately the same temperature of 1500 K instead of transition to TiC carbide. Thus our approach correctly reproduces the resistance of the 2D Ti₃C₂ MXene structure to heating until the point where the samples no longer maintain planar 2D configuration. At the same time, thermal degradation temperatures for Ti₂C and Ti₄C₃ MXenes obtained in this work are respectively lower (1050 K) and higher (1700 K) than the transition temperature of Ti₃C₂ into cubic TiC reported in Ref. Reference Lotfi, Naguib, Yilmaz, Nanda and van Duin26. This may be due to the inaccuracy of the used approach or due to the size effects, as Ti₂C and Ti₄C₃ are relatively thinner and thicker, correspondingly, than Ti₃C₂.

For quantitative characterization of the atomic configurations, we calculate RDF at initial and maximal simulated temperatures (Fig. 5). RDF for initial MXene samples (blue lines in the figure) have the shape typical for crystal structures with clearly visible narrow peaks related to equilibrium interatomic distances, indicative of the long-range ordering. Above the degradation point (orange lines in the figure), the RDF shows only short-range ordered structures. Thus, the RDF data confirm the amorphous structure of the samples at the temperatures above their thermal degradation points.

It is worth mentioning that the melting points of other 2D materials obtained from atomistic simulations have been reported at 4510 K (graphene)^[Reference Los, Zakharchenko, Katsnelson and Fasolino^22] and 3700 K (MoS₂)^[Reference Singh, Neek-Amal, Costamagna and Peeters^23] respectively. Thus all examined in this paper MXenes have much lower thermal stability compared to graphene and 2D MoS₂.

In general, to obtain more precise values of melting temperatures from MD simulations, it is critical to use a more accurate force field that reproduces breaking and formation of chemical bonds, depending on the external conditions. Other factors to take into consideration when comparing calculated thermal degradation temperatures to experimental ones are ideal vacuum conditions and the absence of surface functional groups, as well as defects in MXene models in our simulations, whereas experimental results (see e.g., Refs. Reference Li, Du, Huo, Wang and Cui18 and Reference Li, Wang, Sun, Zhang, Liu, Hu and Zhou19) are produced with non-ideal vacuum, often under argon or oxygen atmospheres and the real MXene samples are always terminated with surface functional groups and contain structural defects.