Superlubricity of molybdenum disulfide subjected to large compressive strains

The friction between a molybdenum disulphide (MoS2) nanoflake and a MoS2 substrate was analyzed using a modified Tomlinson model based on atomistic force fields. The calculations performed in the study suggest that large deformations in the substrate can induce a dramatic decrease in the friction between the nanoflake and the substrate to produce the so-called superlubricity. The coefficient of friction decreases by 1–4 orders of magnitude when a high strain exceeding 0.1 is applied. This friction reduction is strongly anisotropic. For example, the reduction is most pronounced in the compressive regime when the nanoflake slides along the zigzag crystalline direction of the substrate. In other sliding directions, the coefficient of friction will reduce to its lowest value either when a high tensile strain is applied along the zigzag direction or when a high compressive strain is applied along the armchair direction. This anisotropy is correlated with the atomic configurations of MoS2.


Introduction
In mechanical systems, friction is unavoidable. Friction increases energy consumption and wear, and reduces the life of device components [1,2]. These problems caused by friction are particularly important in nanometer-sized devices and machines because of the high surface-to-volume ratio of nanostructures. Two-dimensional (2D) materials [3][4][5], such as graphene [6] and molybdenum disulfide (MoS 2 ) [7,8] can considerably reduce the wear and friction in mechanical systems, owing to their chemical inertness [9][10][11][12], high strength [13], and peculiar structure [14]. For example, Kawai et al. discovered ultra-low friction between graphene nanoribbons and an Au surface [15]. Li et al. found that the coefficient of kinetic friction falls below 10 -4 when a single-layer MoS 2 flake slides on a MoS 2 surface [16]. Such ultralow friction state induced by an incommensurate interface registry is called structural superlubricity [17][18][19][20].
However, achieving robust superlubricity using 2D materials is challenging. Filippov et al. [21] demonstrated the instability of superlubricity in graphene caused by the reorientation of contacting layers at the interface. Bonelli et al. [22] reported that the rotation of graphene flakes can change the incommensurate registry of the interface, causing high friction. To achieve steady superlubricity, Leven et al. [23] proposed the use of a graphene-boron nitride (BN) for stabilizing the incommensurate registry of the lattice using the registry index concept. Wang et al. predicted, using molecular dynamics simulations, that stable superlubricity can be achieved by stretching the graphene substrate on which the graphene flake slides [24]. It was shown that robust superlubricity can be achieved in 2D nanostructures via strain engineering.
Lin et al. [25] reported that the friction between a graphene flake and a graphene substrate is almost insensitive to small compressive strains. The effect of high strains is unclear, and thus, requires exploration because nanostructures are known to sustain larger deformations than their bulk counterparts [26][27][28][29]. In this study, we used Tomlinson-type simulations [30][31][32][33][34] based on atomistic force fields to study the friction between a MoS 2 flake and a MoS 2 substrate. The substrate was subjected to large deformations, causing superlubric friction.

Methods
In our simulations, a hexagonal MoS 2 flake composed of 81 atoms was made to slide atop an infinite MoS 2 substrate. As shown in Fig. 1, the flake was connected with three springs, which were used to monitor the motion of the flake. The deformation (either compressive or tensile) was applied to the substrate by changing the coordinates of the atoms for a Poisson ratio of 0.267 [35]. In accordance with the deformation, the periodic length and width of the simulation box were adjusted from L 0 and W 0 to L and W, respectively, for completely relaxing the pressure on the substrate along the x-and y-axes, respectively. The in-plane strain is given by  x = (L-L 0 )/L 0 or  y = (W-W 0 )/W 0 , depending on the direction of the applied deformation. A negative deformation represents a compressive strain while a positive one represents a tensile strain. The applied strain was below the previously reported elongation limit of the material by a value between -0.2 and 0.2 [36]. Buckling occurs when large compressive strains are applied on free-standing MoS 2 [37]. Although buckling was ignored in the simulation setup we used, Moiré template strain patterning methods can be used, if necessary, to reduce the buckling [38,39]. The sliding direction is represented by the angle between the direction of the movement of the MoS 2 flake and the x-axis (Fig. 1).
The interaction between the flake and the substrate was here considered at the atomistic level. It consisted of two parts: van der Waals (vdW) and electrostatic (elec) interactions. Because the two contacting bodies were both assumed to be rigid, the total interaction potential ccould be written as where i and j are the index of atoms in the flake and substrate, respectively. The Lennard-Jones force field [40] was as follows: with a potential well depth of 0.586, 2.800, and 13.860 meV and an equilibrium distance of 0.420, 0.367, and 0.313 nm for Mo-Mo, Mo-S, and S-S interactions, respectively [41]. The cutoff radius for the vdW potential was set to be 1.0 nm. The electrostatic potential is was calculated using the pairwise Coulomb function given below.
where C is the Coulomb constant. q = 0.76e and  [42,43]. The cutoff radius for the elec potential was set to be 3.0 nm. The total potential energy of the system included the total interaction potential and the elastic potential stored in the springs as indicated below.
where f R is the vector pointing to the center of the mass of the MoS2 flake and s R is that pointing to the root of the spring. The rigidity constants of the x-and y-springs were set to x k = y k = 55 N/m, based on experimentally reported values [44,45]. The Z-spring was used to maintain a constant normal load. In the Tomlinson model, stick-slip instability and the associated energy dissipation critically depend on the force constant of the loading springs. As shown in Fig. S1 in the Electronic Supplementary Material (ESM), the coefficient of friction decreases as the rigidity constant increases and becomes zero when the rigidity constant is large. The stick-slip behavior was active for the rigidity constant used.
In our simulation, the MoS 2 flake slid in discrete steps of 0.001 nm. Its position was spontaneously adjusted at each step by the springs for a nearby local energy minimum [31,44,46]. The friction force F was calculated as the average force acting on the flake against the sliding direction during the last 70% of the simulation steps (in a total sliding distance of 10 nm) [31]. To determine the influence of the initial position of the flake, we simulated 100 different initial positions for the flake. As a benchmark, we calculated the friction force as a function of the normal load without any applied strain, as shown in Fig. S2 in the ESM. The friction linearly increased with the load at a relatively large load, endorsing past experimental observations [47][48][49]. Figure 2 shows the friction force as a function of the sliding distance of the flake. A typical stick-slip behavior is observable even with high tensile strain ( Fig. 2(a)). This type of behavior has been observed in many nanoscale friction experiments [17,44]. However, the stick-slip motion appears to have been significantly hindered by the high compressive strain applied to the substrate. This is because, as Fig. 2(b) depicts, under a high compressive strain, the abrupt jumps in the curves of the friction force disappear, signifying the achievement of a stable superlubric state.

Results and discussion
To confirm this, we computed the coefficient of friction  and plotted it as a function of the applied strain (Fig. 3). As can be seen in Fig. 3, when no strain is applied,  = 0.23 for the given normal force. Under a small tensile or compressive strain,  starts to decrease, confirming the results of previous studies [24,25]. When the magnitude of the applied strain The effect of high tensile strain is however less significant than that of high compressive strain, which endorses the results of Wang et al. [24]. To understand the dramatic decrease of the coefficient of friction under large strains, the potential energy surface (PES), which is strongly correlated with the friction properties of the interface [50,51], was plotted in Fig. 4. The energy values shown were obtained by computing the total interaction potential when displacing the flake atop the substrate in discrete steps. As shown in Fig. 4, the PES has been significantly changed by the high strain applied to the substrate. The friction has to change accordingly because the principle of minimum energy makes the flake to "surf" waves on the PES along a path corresponding to the lowest energy corrugation. The energy corrugation along the x-axis reaches its minimum for  x = -0.175 (Fig. 4(c)), while the energy barriers in the sliding pathway reach the maximum when there is no applied strain (Fig. 4(a)).
The positive and negative strains cause different friction reductions. This anisotropy is correlated with the fact that a compressive strain will result in a shorter interatomic distance, enhancing the overlapping of the long-range interaction potential  energy of the atoms, which makes the peaks in the long-range interaction potential surface to be closer, as can be seen in Fig. 5(a). Hence, the PES will be "smoother"in the case of compressive strain. By contrast, tensile strain increases the energy surface corrugation with an increased interatomic distance, causing higher friction [24]. This effect can be seen by comparing the energy profile in Fig. 4(b) with that in Fig. 4(c).
The difference between the friction resulting from the straining of the substrate in the x-direction and the corresponding friction in the y-direction, which is another aspect of the anisotropy of the friction reduction effect, is correlated with the difference in the atom densities of MoS2 along different crystalline directions. As shown in Fig. 5(b), the atom density www.Springer.com/journal/40544 | Friction along the zigzag direction is higher than that along the armchair direction. Thus, when a strain is applied to the substrate, the induced lattice mismatch between the two layers at the interface can be higher for the strain along the zigzag direction. An enhanced lattice mismatch is known to cause higher friction [31].
The results mentioned are for the flake sliding along the x-axis. To consider a more general scenario, we simulated the sliding of the flake in different directions. The coefficient of friction was computed for different applied strains (Fig. 6). In all sliding directions, had the highest value for the pristine substrate under no strain. By contrast, generally had the lowest value when a tensile strain of 0.15 was applied along the x-axis, as shown in Fig. 6(a), or when a compressive strain of -0.15 was applied along the y-axis, as shown in Fig. 6(b). The sliding direction corresponding to the minimum changes with the applied strain. For example, when  = 0 or π for  x = -0.15, while  = π/6 or 5π/6 for  x = 0.05 or  x = -0.05, as shown in Fig. 6(a). For the strain applied in the y-direction, the sliding direction corresponding to the lowest friction is  = π/6 or 5π/6 for  y = -0.15, while it is  = π/3 or 2π/3 for  y = 0.15.
We also calculated the coefficient of friction for three different flake sizes (Fig. 7). As Fig. 7

Summary
We simulated the sliding of a rigid MoS 2 flake on a strained MoS 2 substrate using a modified Tomlinson model combined with atomistic force fields. The simulations indicate that a large deformation of the substrate can induce a dramatic decrease in the friction coefficient causing the so-called superlubricity. For example, the coefficient of friction decreased by 4 orders of magnitude when a high compressive strain of 0.2 was applied to a pristine substrate. This friction reduction effect is found to be highly anisotropic. In most sliding directions, the coefficient of friction is, in general, the lowest when a high tensile strain is applied along the x-direction, or with a high compressive strain applied along the y-direction. This anisotropic effect is correlated with the atomic configurations of MoS 2 as explained by the strain-induced change in the PES between the MoS 2 flake and the substrate. The simulations were expanded to represent more general scenarios by making the flake to slide in different directions. The sliding direction corresponding to the lowest coefficient of friction depended on the magnitude of the applied strain. Guangxi University (No. XTZ160532) are acknowledged.
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To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Shengcong WU. He received his bachelor degree in applied physics in 2017 from Jiangsu University of Science and Technology, Zhenjiang, China. Then, he was a master student in the Guangxi Key Laboratory for Relativistic Astrophysics at Guangxi University. He has recently obtained his master degree in Department of Physics at Guangxi University. His research interests include structural superlubricity of nanomaterials.
Zhao WANG. He received his Ph.D. degree in physics at the University of Besancon in France. He has been working at EMPA in Thun (Switzerland) and CEA in Grenoble (France) as a postdoc after graduation. He came back to China in 2011, for working at Xi'an Jiaotong University. He joined Guangxi University since 2016 as a professor. His research concentrates on molecular physics, as well as on mechanical and thermal transport properties of nanoscale interfaces.