Strain‐Driven Superlubricity of Graphene/Graphene in Commensurate Contact

The occurrence of structural superlubricity (SSL) requires that two sliding surfaces be in incommensurate contact. However, the incommensurate contact between two sliding surfaces is fundamentally an instable state whose maintenance over time is extremely laborious. To circumvent this difficulty, it is proposed in the present work to change the paradigm of making appear superlubricity and keeping it over time. Two graphene layers in sliding commensurate contact, which are subjected to an isotropic in‐plane synchronous strain, are considered and studied. First, by DFT calculations, it is demonstrated that the synchronous strain‐driven superlubricity (SSDSL) takes place for some particular sliding paths or for all sliding paths, once the compressive strain prescribed reaches 15% or 35%. Next, the Prandtl‐Tomlinson (P‐T) model is used to explain how to modulate stick‐slip, continuous and frictionless slides by the strain. Finally, the SSDSL of two graphene layers in commensurate contact is justified in detail by the interfacial charge density transfer due to the strain. The results obtained by the present work open a new perspective of realizing superlubricity in a robust way.


Introduction
Friction and wear are two main modes of energy dissipation and component failure in various mechanical systems, [1] especially in micro-nano devices under strong confinement and www.advmatinterfaces.de first-principles computations, we evidence that: i) a compressive isotropic in-plane synchronous strain ε reduces the energy barriers and consequently diminishes the frictional force; ii) on contrary, a tensile strain ε increases the energy barriers and thus augments the frictional force; iii) when the synchronous compressive strain ε is accentuated gradually, the frictional force disappears for some particular sliding paths or for all sliding paths once ε exceeds 15% or 35%. Next, the Prandtl-Tomlinson (P-T) model is adopted to explain how prescribing an appropriate isotropic in-plane synchronous strain ε can modify the ratio k s /k e of the in-plane stiffness k s to the effective stiffness k e so as to modulate the different sliding patterns: multi-slip, single-slip, continuous (smooth) slip, and frictionless slip. Finally, the interfacial charge transfer induced by the strain is quantitatively investigated and shown to be the origin of SSDSL of two graphene layers in commensurate contact. These results support the feasibility of SSDSL of Gr/Gr layers even if in commensurate contact, providing a novel approach to robust superlubricity at the 2D interfaces.

The Commensurate Contact between Sliding Graphene Layers
We consider a sliding system obtained by stacking two graphene sheets (including 2 × 2 primitive cells) in commensurate contact, and its vertical and side views are shown in Figure 1. The vertical view illustrates the commensurate stacking of Gr/Gr sheets and the loading pattern of isotropic in-plane biaxial strain ε. [19] The strain applied to the superior layer is the same as the one to which the inferior layer is subjected. In this case, the sliding system is said to undergo the synchronous strain and any interfacial atomic mismatch caused by strain can be avoided. The rectangle marked by a dotted line in Figure 1a corresponds to the area used to display a potential energy surface (PES) and its variation ΔE in the following  Figure S1, Supporting Information), which schematizes the evolution of sliding energy barriers inhibited by the synchronous strain (Figure 1b,c). As for Gr/Gr, the Hollow and Top sites are, respectively, the most stable and unstable stacking configurations under various synchronous strains. [23] Therefore, the graphene sheets are kept in commensurate contact during the interlayer sliding even if the synchronous strain is applied.

The Synchronous Strain-Driven Superlubricity (SSDSL) of Gr/Gr Sheets in Commensurate Contact
The computation and examination of PES allow us to study the interlayer sliding behavior of 2D structures. Indeed, the maximum amplitude ΔE max of the fluctuation of PES constitutes the sliding barrier and corresponds to the maximal energy that a friction process may dissipate. From a geometrical point of view, the maximum friction force is exactly equal to the maximum gradient of PES. Consequently, superlubricity takes place independently of sliding paths when and only when the gradient of PES is everywhere equal to zero. In short, superlubricity arises once ΔE max is null or, equivalently, PES is completely flat. The PES evolution of Gr/Gr system in function of the applied synchronous strain ε is shown in Figure 2a-d, where the contour lines of PE (potential energy) are given. In these figures, as usual, the red color represents high energy while but the blue color symbolizes low energy. Thus, a greater change in color corresponds to a larger PE difference. More details about the effect of the synchronous in-plane strain on the interlayer binding are given in Supporting Information 2.
It is important to note that the fluctuation of PES decreases with the increasing of compressive synchronous strain ε, as shown by Figure 2a-d. This implies that interlayer slip becomes more effortless upon exerting a compressive strain. From Figure 2b, we observe that particular sliding paths along which PE does not vary appear. In this case, we say that partial superlubricity occurs. Precisely, the appearance of partial superlubricity requires only a synchronous compressive strain ε ≥ 15% (Figure 2b). More importantly, once the compres- www.advmatinterfaces.de sive synchronous strain ε is equal to 35% (Figure 2d), ΔE max , the maximum fluctuation of PE, is almost null and thus the frictional force reduces to zero regardless of sliding direction ( Figure 2e). In this sense, we say that total superlubricity takes place. It is also important to remark that the fluctuation of PES augments with the increase of tensile synchronous strain ε ( Figure S3, Supporting Information). This means that the augmentation of a tensile synchronous strain makes the friction force larger (Figure 2e).
It should be emphasized that SSDSL described above is quite different from the strain-driven superlubricity reported by the previous works. [19][20][21] Indeed, in these works, the essence of superlubricity remains the SSL achieved by prescribing tensile or compressive strains so as to produce and/or maintain interfacial incommensurate contact. However, in the present work, superlubricity is enabled by imposing a synchronous compressive strain on two graphene layers in commensurate contact. In order to distinguish the strain-driven superlubricity reported by the present work from the strain-induced SSL, [19][20][21] it is named the synchronous strain-driven superlubricity (SSDSL).
The problem of maintaining a graphene flat while imposing an in-plane compressive strain to it is a challenging one from an experimental point of view. Nevertheless, one experimental solution for preventing the bending and corrugating of a graphene layer undergoing an in-plane compressive strain was suggested by Bousige et al. [17] According to this solution, a graphene layer bonded to a substrate is immersed in a compressive fluid in the experimental chamber of a diamond anvil cell (DAC). This cell allows the graphene-substrate system to be compressed together under a pressure of up to a few hundred of GPa. [24] According to the authors, the progressive compression of the fluid has two effects on the full system: i) the enhancement of adhesion between the graphene and the substrate due to the reduction of the graphene-substrate distance; ii) the biaxial compressive straining of the graphene layer and the reduction of the substrate volume.
Inspired by the experimental solution of Bousige et al., [17] we suggest to adapt it as shown in Figure S4 (Supporting Information): two graphene layers, each of which is bonded to a substrate, are first placed in sliding contact and then immersed www.advmatinterfaces.de together in a compressive fluid. In this way, the bending and corrugation of the two graphene layers should be avoided. Note that the reduction of the graphene-to-graphene distance under the effect of compression does not increase the interlayer interaction force, because the in-plane compressive strain leads to a decrease in the interlayer spacing due to the negative Poisson ratio of graphene ( Figure S2, Supporting Information). [25] In addition, the diamond anvil cell ( Figure S4, Supporting Information) can withstand pressure in the range of a few hundred GPa. [24] Meanwhile, the in-plane elastic strain of up to 25% for the graphene layer requires stress of ≈130 ± 10 GPa. [26] Thus, the in-plane compressive strain needed for SSDSL, at least partial SSDSL, namely 15% (Figure 2b), should be accomplished by this three-dimensional compression method. [17] However, the in-plane compressive strain should not be applied beyond the relevant experimental elastic limit strain, so the total SSDSL seems to be experimentally unrealistic.

The Kinetics Behavior of SSDSL
It is known that the friction between two sliding surfaces in commensurate contact varies with their relative displacement and exhibits the so-called stick-slip behavior. [27] There are two factors key to the stick-slip phenomenon: i) the adsorption/ desorption-like effect induced by the out-of-plane bending of a flexible 2D atomic layer, especially in contact with rough peaks; [28,29] ii) the intrinsic periodic potential determined by the atomic arrangement on the two sliding surfaces. [30] If there is no intervention of substrate bonding or pre-strain, [19,31] the adsorption/desorption-like effect is often the dominant element that brings the out-of-plane vibration of a 2D crystal lattice to form the phonon dissipation due to stick-slip. However, the case considered in the current work does not involve any single-or multiple-asperity contacts but two atomically flat graphene sheets, Hence it is worthless for the kinetics of SSDSL to evaluate the local pinning or out-of-plane bending. [28,32] In spite of this, the effects of synchronous strain on interplanar coupling and in-plane flexibility of graphene layers need to be inspected for SSDSL.
Indeed, even if the interference of out-of-plane deformation is eliminated, the stick-slip remains uncertain due to the existence of multiple slip systems. [33] Consequently, before reaching a superlubric state, the relative movement of two graphene layers experiences unstable multi-slip, stable single-slip, and continuous sliding. [6] To unveil the possible transition from stick-slip to frictionless slip under a compressive synchronous strain, the dynamic analyses based on the quasi-static P-T model are presented in Figure 3. In Figure 3a, the effective stiffness k e is estimated via where ΔE is the energy barrier of PES and Δy is the relative sliding distance along y-axis. In terms of energy dissipation, the maximum frictional force (F , see part 4. Experimental Section) is the most important for a stick-slip event, so the highest barrier path is selected to evaluate k e . Meanwhile, since the curvature of this path is not a constant but varies with the displacement Δy, the maximal value of k e , as outlined in Figure 3b, is selected to reflect its overall trend under the synchronous strain. Notably, k e decreases with the compressive The increase of the ratio η = k s /k e with compressive strain means the feebler interlayer coupling or/and the lower in-plane flexibility of Gr/Gr sheets, which is responsible for the transition from multiple-slip to frictionless slip.

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strain but increases with the tensile strain (Figure 3b). This implies that the coupling strength difference between two Gr sheets can be canceled by a compressive stain. As shown in Figure 3c, the lateral in-plane stiffness k s is determined by where F x and F y are the forces and Δε x and Δε y are the strains along x and y, respectively. The slope of the curve is getting steeper with the synchronous compressive strain (Figure 3c), indicating that the in-plane stiffness of the graphene sheets augments with the compressive strain. As a result, the stronger in-plane stiffness under the compressive strain (Figure 3c) means the weaker electron-phonon coupling of the sliding systems, suggesting less dissipation. On the contrary, the tensile strain reduces the in-plane stiffness (Figure 3c), thus facilitating in-plane vibrations of carbon atoms of graphene sheets, viz., enhancing electron-phonon coupling. [34] According to the P-T model, [7,33] the ratio of k s to k e , namely η = k s /k e , can be used as a slip pattern indicator. Precisely, if the lateral stiffness k s is smaller than the effective one k e , i.e., η ≤ 1, the interlayer sliding is stick-slip, resulting in high energy dissipation. This case includes the single-slip sliding when η = 1 and the multi-slip sliding when η < 1. Once η > 1, the interlayer sliding becomes a continuous (smooth) slip with low energy dissipation; it can even get to a frictionless slip without energy dissipation when η ≫ 1. [33,35] There is no doubt that both the increase of k s (Figure 3c) and the decrease of k e (Figure 3b) under the effect of a synchronous compressive strain favor a lower energy dissipation of the interlayer sliding of the Gr/Gr system. Further, the ratio η = k s /k e is employed to analyze the synchronous strain-driven friction modulation in Figure 3d. As expected, the ratio η evolves from ≈0 to ≈10 4 as the synchronous strain is continuously applied from +30% (tensile) to -35% (compressive). Thus, from the views of according to the P-T model, [7,33,35] the sliding system undergoes a successive transition from multi-slip (η < 1), single-slip (η = 1), to continuous slip (η > 1) and ultimately enters frictionless slip (η ≫ 1). This supports the kinetic feasibility of adjusting the interlayer sliding of Gr/Gr sheets from a stick-slip state to a superlubric state upon applying a compressive strain.
Admittedly, only the path with maximum barrier is considered in above discussions. Given the slipping priority of the minimum barrier path (the red arrows in Figure 2b), there may be no need to compress the in-plane lattice to 65% for the frictionless slip (i.e., the total superlubricity) because k e is minified and η would be amplified along the minimum barrier path. In fact, this argument can be supported by Figure 1b, where the frictionless channel (i.e., partial superlubricity) has emerged under merely 15% compression along the red arrows (Namely, the isolated blue areas in Figure 2a under relaxed state have joined together). In addition, the thermal activation induced by the temperature experimentally, [36] is beneficial to the transition from stick-slip sliding to continuous (smooth) sliding. To sum up, the joint effects of slip path and activation temperature on the interlayer slipping of graphene sheets make that the synchronous compressive strain sufficient for SSDSL may be in practice much <15%.

The Electron-Scale Mechanism of SSDSL
Despite the kinetic feasibility of SSDSL argued above, the physical mechanism underlying the strain-driven transition from stick-slip to continuous slip in commensurate contact remains to be clarified. The redistribution of interlayer charges conditions the relation between the relative slipping and interfacial coupling of two sliding surfaces in contact, [37] so the interlayer charge density and its variation during the interlayer sliding are quantified and displayed in Figure 4 ( Figure S5, Supporting Information).
First, the interlayer charge density increases with the compressive strain (left in Figure 4a,c), leading to stronger interlayer binding and in-plane bonding. [38] However, the gap between the interlayer electron densities of Top and Hollow sites gradually becomes smaller with the increase of the compressive strain (left in Figure 4a,c), resulting in less coupling strength difference. As shown by blue-grey marked zones, the overlapping of the z-axis projected curves accurately reveals the indistinctive interlayer interaction difference between Top and Hollow sites under the compressive strain (right in Figure 4a,c). For a tensile strain, the redistribution of interlayer charges is opposite to what happens in the case of a compressive strain ( Figure S6, Supporting Information). Therefore, SSDSL is due to the fluctuation of the interplanar charge density (Δρ inter− ) induced by compressive strain.
Using the quantified curves projected along the z-axis, the fluctuations of the interplanar charge density under various synchronous strains are summarized in Figure 4d. It is seen that the compression leads to a less Δρ inter− while the stretch gives rise to a bigger Δρ inter− . Further, there is a direct dependence of the sliding barrier on the Δρ inter− driven by biaxial strain. In view of these results, the mechanism underlying SSDSL can be summarized as follows: i) the distribution of the interlayer charge density can be tuned by the synchronous strain; ii) the effects of this charge redistribution are disparate on different sites; iii) there is a linear relation between ΔE max and Δρ inter− ; iv) as the synchronous compressive strain applied is equal to 35%, the interlayer coupling strength difference disappears and superlubricity occurs. In brief, SSDSL takes place once the strain-driven Δρ inter− becomes null.

Conclusion
To avoid the difficulties of maintaining over time the incommensurate contact conditions necessary for structural superlubricity (SSL), the present work has suggested and implemented a new strategy for superlubricity by considering and investigating, as a prototype, two graphene layers in sliding commensurate contact and undergoing an isotropic in-plane synchronous strain. It has been shown by DFT calculations that partial superlubricity or total superlubricity occurs once an applied compressive synchronous strain reaches 15% or 35%. This synchronous strain-driven superlubricity (SSDSL) has www.advmatinterfaces.de been further explained and justified first by the P-T model and then by the interfacial charge transfer induced by the strain. The strain engineering strategy proposed and illustrated in the present work suggests a new experimental way to obtain superlubricity lasting over time.

Experimental Section
The Simulation of Slipping: As shown in Figure 1b, the upper graphene layer slides along the x-axis while the bottom one was fixed, going through the path from Top, Bridge, Hollow, Saddle, Hollow, Bridge, back to Top site ( Figure S1, Supporting Information). An isotropic in-plane strain was applied simultaneously to the upper and bottom layers to ensure that it did not break the commensurability of the contact between the graphene layers. The interlayer spacing d was the distance between the centers of the top and bottom graphene sheets free from strain. The PES in Figure 2 was estimated by ΔE = E tot − E tot min , where E tot and E tot min were, respectively, the unstable (such as Top and Saddle sites) and most stable (Hollow site) stackings. Then, the friction force, [39] was calculated by f max F E nL = ∆ , where L is the distance between the two stacks used to compute ΔE max and n is the number of interfacial atoms. More computational details were presented in Supporting Information. First-Principles Calculations: In this study, the DFT calculations were employed to obtain the static energy and electron redistribution in function of the synchronous strain applied. The total energy is given by E tot = E PBE + E vdW , where E PBE is the DFT-GGA energy calculated by Perdew-Berke-Ernzerhof (PBE) form and E vdW represents the vdW interaction energy delivered by the modified DFT-D2 (Grimme) version. [40] To screen off the impacts from z-axis periodic lattices, a vacuum layer of 15 angstroms was added to the double graphene sheets along the z-axis direction. According to the convergence tests, the cutoff energy of ultrasoft pseudopotential and k-points of integral space were set as 650 eV and 15 × 15 × 1 points in Gr/Gr sheets, respectively. The precision of the self-consistent energy was selected as 0.5 × 10 −6 eV per atom between both electronic steps. The positions of carbon atoms in the strain-free Gr/Gr sheets were optimized until the maximum of the interaction (Hellmann-Feynman) force among all atoms was <0.01 eV Å −1 . Besides, the results in current work did not depend on the choices of dispersion interactions or exchange-correlation functions. Use had been made of VASP and CASTEP codes, [41] and the results obtained by these two codes were consistent.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.