Limitations of Structural Superlubricity: Chemical Bonds versus

Limitations of Structural Superlubricity: Chemical Bonds versus Contact Size Dirk Dietzel,† Ján Brndiar,‡ Ivan Štich,‡,§,⊥ and André Schirmeisen*,† †

Institute for Applied Physics, Justus-Liebig-Universität, 35392 Gießen, Germany CCMS, Institute of Physics, Slovak Academy of Sciences, 845 11 Bratislava, Slovakia § Institute of Informatics, Slovak Academy of Sciences, 845 07 Bratislava, Slovakia ⊥ Department of Natural Sciences, University of Ss. Cyril and Methodius, 917 01 Trnava, Slovakia ‡

S Supporting Information *

ABSTRACT: Structural superlubricity describes the state of virtually frictionless sliding if two atomically flat interfaces are incommensurate, that is, they share no common periodicity. Despite the exciting prospects of this low friction phenomenon, there are physical limitations to the existence of this state. Theory predicts that the contact size is one fundamental limit, where the critical size threshold mainly depends on the interplay between lateral contact compliance and interface interaction energies. Here we provide experimental evidence for this size threshold by measuring the sliding friction force of differently sized antimony particles on MoS2. We find that superlubric sliding with the characteristic linear decrease of shear stress with contact size prevails for small particles with contact areas below 15 000 nm2. Larger particles, however, show a transition toward constant shear stress behavior. In contrast, Sb particles on graphite show superlubricity over the whole size range. Ab initio simulations reveal that the chemical interaction energies for Sb/ MoS2 are much stronger than for Sb/HOPG and can therefore explain the different friction properties as well as the critical size thresholds. These limitations must be considered when designing low friction contacts based on structural superlubricity concepts. KEYWORDS: nanotribology, structural superlubricity, AFM manipulation of nanoparticles, incommensurate interfaces, shear stress

S

result, interfaces suitable for structural superlubricity have typically been found only under UHV conditions4,5,13,14 or by shearing of layered materials with low diffusion barriers for surface atoms.6,8,9,11,19 Only recently, superlubricity has also been found for other systems like gold nanoparticles on HOPG under ambient conditions12 or in tribometer experiments between graphene coated surfaces.10 Second, relaxations at the interface can suppress structural superlubricity. One manifestation is the Aubry transition, where an incommensurate interface, which would otherwise exhibit superlubric behavior, is forced into registry under a load and shows friction.20 Alternatively, if the adhesion between the atoms of the slider and the substrate becomes stronger than the cohesion, instabilities can arise when the slider or substrate can no longer be considered as rigid. In this case, elastic deformations can appear to accommodate the pinning forces related to strong interatomic interactions between slider and

tructural superlubricity is a recently discovered phenomenon in the field of nanotribology that describes the state of ultralow friction. More precisely, this effect relies on an atomic lattice mismatch between two sliding surfaces that prevents interlocking between atomically flat surfaces. The basic framework of structural superlubricity has been introduced by Hirano et al.,1−3 and up to now it has been confirmed by a continuously growing number of experiments both on the nano- and on the mesoscale.4−14 Since most realistic contacts will consist of dissimilar materials and also have misaligned orientation, structural superlubricity should be the standard case. However, despite all recent experiments that could be linked to structural superlubricity, its occurrence seems to be more exotic than widespread. This is due to two effects. First, it is necessary to have clean and atomically flat interfaces to ensure the collective behavior of the interface atoms. Especially, the omnipresent interface contamination can cause the breakdown of structural superlubricity15−18 since mobile dirt molecules at the interface allow an effective interlocking between the two dissimilar surfaces.15,17 As a © 2017 American Chemical Society

Received: March 31, 2017 Accepted: July 11, 2017 Published: July 17, 2017 7642

DOI: 10.1021/acsnano.7b02240 ACS Nano 2017, 11, 7642−7647

Article

www.acsnano.org

Article

ACS Nano substrate.21−28 Since such effects are based on intrinsic material parameters, they can constitute a much more fundamental obstacle on the road to ultralow friction. Although previous studies have pointed out, that interface interactions are often too small to induce pinning for incommensurate interfaces,29 especially two recent studies have highlighted the importance of system size to promote the breakdown of structural superlubricity. A critical length preventing structural superlubricity was for instance found for a 1D chain of atoms on a periodic surface potential using an edge-driven Frenkel-Kontorova model.25 Recently, atomic scale calculations by Sharp et al.27 have shown that a critical size for the occurrence of structural superlubricity also exists for crystalline 2D structures sliding on a substrate. In this case, friction depends sensitively on the ratio between the particle’s shear modulus and the interaction forces between particle and substrate atoms. Experimentally, a first indication of interface relaxation effects was found for Sb particles sliding on HOPG, which showed logarithmic contact aging, albeit still in the superlubric regime.30,31 In this paper, we now try to shed more light on the complex interface processes and especially the breakdown of structural superlubricity by contrasting these previous results to new measurements obtained for Sb particles sliding on MoS2. In many ways, HOPG and MoS2 are similar. Both materials are well-known lubricants and can easily be cleaved to yield clean and atomically flat interfaces on which Sbnanoparticles can be grown.32,33 On the basis of structural considerations alone, we would expect both material systems to exhibit structural superlubricity. Nonetheless, the two material systems behave distinctively different. As we will show later on, the shear-stress found for Sb particles sliding on MoS2 is almost constant over a wide range of contact sizes. This is in strong contrast to the HOPG substrate, where the shear-stress is continuously decreasing with contact area,5 which represents the unique fingerprint of structural superlubricity.34,35 The difference between the two substrates is related to different atomic bonding mechanisms at the interface. More specifically, our ab initio calculations will reveal particularly strong interatomic forces between Sb and the MoS2-substrate, which is the key parameter to understand the mechanism resulting in the breakdown of superlubricity.

Figure 1. Example of nanoparticle manipulation. (a) Ensemble of particles prior to manipulation. (b) Control image after nanoparticle pushing, which verifies the successful manipulation process. (c) Friction trace recorded while pushing the nanoparticle from the side. For x < 0, the tip approaches the nanoparticle and hits it at about x = 0. The following peak in the lateral force signal as described by ΔFlateral,static can be interpreted as static friction, whereas the subsequent level of constant lateral force corresponding to ΔFlateral,sliding represents sliding friction.

x < 0 corresponds to the friction between tip and substrate, and can be used as reference level, since tip/substrate friction is universally present throughout the whole tip trajectory. Once the tip reaches the nanoparticle (x = 0), a significant buildup of the lateral force is observed until the static friction of the nanoparticle is overcome for ΔFlateral,static. Once the particle starts sliding, a steep decline of the lateral force signal can be observed leading to an almost constant level, which corresponds to the combined sliding friction of tip and nanoparticle. Friction between nanoparticle and substrate can hence be calculated as the difference between this level and the lateral force signal measured before the tip reaches the particle, that is, ΔFlateral,sliding. This approach of pushing the nanoparticles from the side might potentially introduce shear stress inhomogeneities at the interface. However, previous experiments using different nanoparticle manipulation approaches showed no influence of the shear stress distribution on friction.36 As a first approximation, such effects have thus not been considered explicitly. For each nanoparticle, the average sliding friction level Fsliding and also the corresponding static friction peak Fstatic (i.e., the friction level at the onset of sliding after an extended rest time) have been determined. Afterward the sliding shear stress τ = Fsliding/Acontact and the ratio between static and sliding friction Fstatic/Fsliding have been calculated from these values for each particle. Figure 2 shows both (a) the shear stress and (b) Fstatic/ Fsliding as a function of contact area. For comparison, also the shear stress for a set of Sb particles sliding on HOPG has been added to Figure 2a. This set of data is based on previous friction measurements under UHV conditions5 and reveals a continuously decreasing shear stress in accordance to the concept of structural superlubricity. More precisely, on the HOPG substrate the power law describing friction of amorphous particles5 Fsliding,HOPG ∝ Acontact0.5 transforms into a continuously decreasing shear stress τHOPG ∝ Acontact−0.5. A distinctively different behavior is observed for the Sb particles sliding on MoS2 in Figure 2a (red markers). Here, a decrease of shear stress with contact area is found only for small and intermediate particle sizes (Acontact < 15 000 nm2). For

RESULTS AND DISCUSSION To experimentally assess the contact area dependence of friction for the Sb/MoS2 interface, we have performed nanomanipulation experiments for particles of varying size at room temperature in ultrahigh vacuum. For the nanoparticle experiments, we have chosen compact and mostly round nanoparticles, which are known to be amorphous.32,36 In principle, the concept of structural superlubricity is most apparent for crystalline particles. However, it is also applicable for amorphous particles5,15 with the added advantage that any influence of particle rotation can be neglected. To determine the interfacial friction between the nanoparticles and the substrate we have used the sharp tip of the AFM to push the nanoparticles across the substrate while simultaneously recording the lateral force signal related to the cantilever torsion.16,37 Figure 1 illustrates a typical nanomanipulation sequence. The lateral force signal recorded during the manipulation path is shown in Figure 1c. The initial friction level recorded for 7643


Article

ACS Nano

point it is not clear if there is a direct connection between the static and sliding ratio and the breakdown of structural superlubricity. However, this coincidence is an intriguing experimental observation, substantiating our above notion that the particle friction properties undergo a size dependent transition, which also seems to affect contact aging on long time scales. Coming back to the shape of the shear stress curve for the MoS2 substrate (accentuated dashed red lines in Figure 2a) we find a good qualitative overlap with the contact area dependence of the shear stress calculated by Sharp et al.27 In their model, the static friction (in this context meaning the peak forces observed during sliding) of particles moving on a substrate was calculated. These values can directly be compared to the experimental sliding friction data, which effectively averages over these peak forces.31 More specifically, friction was calculated for round 2D particles on a substrate of identical square lattice structure, while initial incommensurability was ensured by relative rotation between particle and substrate. They found that if a degree of compliance is introduced to the system (in this case by the 3D substrate), structural superlubricity (and thus a decrease in shear stress with contact area) only persists until a critical particle dimension is reached. This characteristic length scale of the system can be described by bcore = (d·G)/τmax, with d the lattice constant, G the shear modulus of the compliant component, and τmax representing the maximum lateral shear stress experienced by an atom on the surface potential. As long as the particle radius rparticle is smaller than bcore, the system can essentially be considered as fully rigid (and thus superlubric). Larger particle dimensions rparticle > bcore will allow the formation of dislocations at the sliding interface and the main friction contributions will originate from the motion of dislocations.27 As a consequence, the shear stress will change from a linear decrease with contact area to a constant value, independent of the contact area. In our case, the main factor that determines the size threshold is the ratio of the Sb particle’s shear modulus over the maximum shear stress of the particle−substrate interface, that is, either Sb/HOPG or Sb/MoS2. While the shear modulus of Sb of G = 20 GPa can be found in the literature,38 the particle−substrate interaction energies are unknown. Therefore, we performed DFT simulations to shed light on the atomic interface processes. Additionally, we have also performed a small number of simulations featuring “chemical defects” such as vacancies or Sb impurities; see SI for more details. Please note that those simulations are designed to capture the atomic level chemical/mechanical interaction and relaxation mechanisms at the interface. However, because of the small number of atoms in these simulations, we will always remain in the superlubric regime. In Figure 3, we show results of forces acting on an Sb atom in simulated sliding of Sb/MoS2 and Sb/ HOPG nanocontacts. It is immediately evident that the behavior of the two contacts is markedly different. While the Sb/MoS2 system exhibits quite substantial energy variations as a result of sliding transforming into forces of the order of 0.1 nN, in the Sb/HOPG contact both energies and forces during contact sliding are negligible. Perhaps surprisingly, in both cases it is the chemical component that dominates the forces, not the van der Waals, as might naively be assumed. We observe that sliding Sb atoms are able to explore the chemical corrugation of the underlying potential energy surface (PES), which is significantly larger in the Sb/MoS2 than in the

Figure 2. (a) Contact area dependence of the sliding shear stress for Sb particles sliding on MoS2 (red and blue spheres) and HOPG5 (black spheres). The HOPG data can well be fitted by a continuous decrease in shear stress (τHOPG ∝ Acontact−0.5). As indicated by the dashed lines, the shear stress for the MoS2 substrate is mostly constant and a trend of decreasing shear stress can only be recognized for Acontact < 15 000 nm2. A few data points (blue markers) have shown unusually high friction, probably related to defects or interface contamination. (b) Friction ratio Fstatic/Fsliding calculated for the nanoparticles on MoS2 represented by the red data points in panel a. For small contact areas (Acontact < 20 000 nm2), the ratio is always close to 1, and only for larger contact areas, Acontact > 20 000 nm2, the ratio increases significantly, a transition that is indicated by the dashed lines.

larger particles, a constant level of τMoS2 ≈ 1 MPa is maintained, thus indicating that structural superlubricity breaks down for Acontact > 15 000 nm2. Obviously, there is a transition from the superlubric to the constant shear stress regime, which depends on the particle size. Please note that a few nanoparticles do not follow the general trend (blue markers in Figure 2a) but instead show a significantly higher shear stress as indicated by the dashed blue line. Such high shear stresses can be induced by surface defects and thus constitute a different friction mechanism that needs to be treated separately; see Supporting Information (SI) for more details. Figure 2b shows the ratio between static and sliding friction for the Sb on MoS2 nanoparticles, where we also find a sharp transition. For smaller particle sizes (Acontact < 15 000 nm2), there is almost no difference in static and sliding friction and the ratio is approximately one for all particles measured. Only above this threshold large ratios of static vs sliding friction up to ∼5 can be found. This is the same size threshold as observed for the contact area dependence of the shear stress. At this 7644


Article

ACS Nano

A direct quantitative comparison of absolute values between experiment and simulations, however, is more difficult, since the experimental Sb particles are amorphous, while the simulated structures are crystalline. Also the effects of the supercell size in the simulation is unclear with respect to the size scaling. Lastly, the nanoparticles are of finite height (height range ∼20−50 nm), where the added stability can influence the dimension at which dislocations might occur.39 With this in mind, we still may estimate the critical particle radius for Sb on MoS2. With a maximum lateral force of Fmax ≈ 7.5 × 10−11 N from Figure 3, we calculate τmax = 290 MPa and find a critical radius of bcore = 30 nm. This length is in quite good agreement with Figure 2a, from which a critical particle radius of rparticle ≈ 50−80 nm can be inferred. So far our results unambiguously show that the interfacial forces between Sb and MoS2 are considerable higher than between Sb and HOPG. Interestingly, some of our experiments revealed a very direct indication for a particularly strong interface interaction on MoS2. This is illustrated in Figure 4a−c.

Figure 4. AFM-image (deflection signal) recorded for an ensemble of nanoparticles (a) before and (b, c) after the manipulation of the two particles indicated in panel a. The contact areas of the upper and lower particle are Acontact,upper = 19 500 nm2 and Acontact,lower = 28 000 nm2. After manipulation, both particles show distinct dislocation lines.

Figure 3. Calculated energy variation (top panel, energy of the first sliding position taken as zero) and x- and y-force components acting on one Sb atom for simulated sliding of Sb/MoS2 and Sb/ HOPG nanocontacts. Note that two almost identical variations are shown corresponding to periodicity of the substrate atoms. The insets (center panel) show the charge redistribution due to nanocontact formation (blue, charge accumulation; red, charge depletion). For results for other sliding directions and forces acting on other atoms, see SI.

Here, two particles have subsequently been pushed from left to right using the single vector manipulation approach as discussed for Figure 1. The images taken after each manipulation (Figure 4b and c) verify that the particles have indeed been translated, but more importantly they also document structural changes. Initially, both particles have been round and without any major defects, but directly after manipulation, a distinct dislocation is running across each particle. Since the onset of movement is the only major instability during the manipulation process, it is probably also the point at which the dislocation is formed. Please note that such effects have not been observed for all manipulations, but still ∼15−20% of particles showed discernible signs of damage. However, when analyzing the friction traces of particles with and without damage after manipulation, no significant differences were found, meaning that the friction values still fit well into the general trends as shown in Figure 2. This also means that the general stress levels induced by the AFM tip when trying to overcome the static friction must be comparable to the internal yield stress of the Sb particles. Therefore, large Sb particles on MoS2 cannot be considered as rigid when pushed by an AFM tip, but instead internal stress brings them close to the transition between elastic and plastic deformation. On HOPG, with its weaker interface interaction, similar effects have never been observed.

Sb/HOPG nanocontact. This is corroborated by the chemical bonds due to contact formation, see Figure 3, which show very little chemical response in the Sb/HOPG nanocontact, while polar bonds are formed in the Sb/MoS2 nancontact. This directly provides a qualitative explanation of the observed different behavior for Sb on MoS2 versus HOPG. Already in the superlubric regime the much higher energy corrugation of the PES and corresponding forces on MoS2 are directly reflected in the higher shear stress. In the experiments, the relative ratio of the shear stress values on MoS2 versus HOPG is about 5, which is consistent with the simulations. Furthermore, on MoS2, the experiments showed that superlubric sliding breaks down at a certain particles size threshold of ∼15 000 nm2, while on HOPG, superlubricity prevailed up to the largest contact sizes that were investigated. As a first approximation, we used the above formula for the critical size threshold and found that a ratio in τmax between MoS2 and HOPG of a factor of 5 would translate into a factor of ∼25 for the critical contact area. This means that for HOPG we expect the critical size to be around 375 000 nm2, consistent with the experiments where superlubric sliding was observed on HOPG even for the largest particles with a contact area of 200 000 nm2 (Figure 1) but where the critical system size was not yet reached. 7645


Article

ACS Nano

pseudopotential45 for all species. van der Waals (vdW) interactions were included at the DFT-D2 level.46 All simulations were done at zero temperature by sliding the interface in static manner in directions of high symmetry with respect to the substrate and terminated after mapping out the first period. The simulations are similar in nature to those described elsewhere.17,18 Forces acting on individual atoms were calculated by displacing the atoms to counteract the relaxation process. More details can be found in the SI.

CONCLUSIONS To summarize, we show that the occurrence of incommensurate interface conditions does not necessarily lead to low interfacial friction as expected from the concept of structural superlubricity. Instead, the specific interface interaction can play a major role in determining the friction scenario. This becomes clear by comparing friction of sliding Sb-particles on two layered materials, HOPG and MOS2. Our ab initio simulations reveal that the interaction between Sb and MoS2 is considerably stronger than that between Sb and HOPG. While sliding on HOPG is fully consistent with the concept of structural superlubricity, Sb particles sliding on MoS2 show a transition from superlubricity to constant shear stress behavior above a certain particle size threshold. This can be described by current theories that identify movement of dislocations as the primary dissipation channel for particles beyond a critical size.27 In our case, the Sb/MoS2 interaction is dominated by strong chemical components, which facilitate the necessary formation of dislocations and in turn lead to the breakdown of structural superlubricity.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano.7b02240. Construction of periodic approximants to incommensurable interfaces; calculated results for other sliding directions; forces acting on other structurally distinct atoms; discussion of friction forces induced by chemical defects in MoS2 system (PDF)

AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

EXPERIMENTAL METHODS

ORCID

In our experiments, the antimony nanoparticles have been prepared by thermal evaporation onto the (1111) basal plane of naturally grown single crystalline MoS2. Typical evaporation times of ∼20 min resulted in the formation of nanoparticles of varying size and shape with compact and round nanoparticles, that is, those with amorphous structure,32,36 selected for further nanoparticle experiments. The growth process results in an overall flat structure of the particles. This ensures that particle motion is limited to sliding and the geometry well matches the theoretical systems. To avoid interface contamination due to prolonged storage of the sample systems within the UHVchamber,16 the data have been recorded in three separate runs. Each time sample preparation was done right before measurement, and for each run the lateral force sensitivity of the cantilever had been calibrated using the procedure suggested by Varenberg et al.40 The nanoparticles were pushed across the substrate with a sharp tip of the AFM while simultaneously recording the lateral force signal related to the cantilever torsion.16,37 Initially, the AFM was operated in dynamic mode (namely CE-Mode41), which allowed for unperturbed imaging of a group of particles (Figure 1a). From this group, a nanoparticle suitable for manipulation can be chosen. The AFM tip was then placed beside this nanoparticle before switching from dynamic to contact mode. Subsequently, the tip is moved along a defined manipulation vector path, Figure 1b, and the lateral force signal recorded, Figure 1c. After the manipulation process, we switch back to the dynamic mode and record a control image (Figure 1b), which verifies the successful manipulation of the nanoparticle and also the contact area Acontact between particle and substrate can be evaluated from the topography images. If applied to large nanoparticles, which are sticking more firmly to the substrate, this approach may fail. In this case, we have employed an alternative manipulation protocol,16,37,42 where continuous scanning results in multiple pushing attempts. Still, upon manipulation, the lateral force signal is fully equivalent to Figure 1c,12,16,37 and quantitative results obtained by this approach have been found to be highly reproducible.36,37 Altogether, friction has been measured for 35 Sb nanoparticles on MoS2. Theoretical Methods. In our calculations, the nanocontacts were approximated by infinite crystalline interfaces enclosed in periodic boundary conditions. The substrates were modeled by one/two fixed graphene/MoS2 sheets. The antimony slab consisted of two bilayers, the top layer kept fixed, with antimony lattice parameter slightly adjusted so as to form a periodic supercell. Density functional theory (DFT) modeling was performed with plane−wave pseudopotential VASP43 simulation package using PBE44 exchange correlation functional and a plane waves basis set at 500 eV with PAW

Dirk Dietzel: 0000-0001-6158-6971 Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS Financial support was provided by the German Research Foundation (Project DI917/5-1), in part by COST Action MP1303 and the Laboratory for Material Science of the Justus Liebig University Giessen, APVV-0759-15, VEGA 2/0162/15, and 2/0167/16, and by V4-Japan Joint Research Program on Advanced Materials (NaMSeN) projects. We also gratefully acknowledge use of the Hitachi SR16000/M1 supercomputer system at CCMS/IMR, Tohoku University, Japan. REFERENCES (1) Hirano, M.; Shinjo, K. Atomistic Locking and Friction. Phys. Rev. B: Condens. Matter Mater. Phys. 1990, 41, 11837. (2) Hirano, M.; Shinjo, K.; Kaneko, R.; Murata, Y. Anisotropy of Frictional Forces in Muscovite Mica. Phys. Rev. Lett. 1991, 67, 2642. (3) Hirano, M.; Shinjo, K. Superlubricity and Frictional Anisotropy. Wear 1993, 168, 121−125. (4) Dienwiebel, M.; Verhoeven, G.; Pradeep, N.; Frenken, J.; Heimberg, J.; Zandbergen, H. Superlubricity of Graphite. Phys. Rev. Lett. 2004, 92, 126101. (5) Dietzel, D.; Feldmann, M.; Schwarz, U. D.; Fuchs, H.; Schirmeisen, A. Scaling Laws of Structural Lubricity. Phys. Rev. Lett. 2013, 111, 235502. (6) Koren, E.; Lörtscher, E.; Rawlings, C.; Knoll, A. W.; Duerig, U. Adhesion and Friction in Mesoscopic Graphite Contacts. Science 2015, 348, 679−683. (7) Hirano, M.; Shinjo, K.; Kaneko, R.; Murata, Y. Observation of Superlubricity by Scanning Tunneling Microscopy. Phys. Rev. Lett. 1997, 78, 1448. (8) Liu, Y.; Grey, F.; Zheng, Q. The High-Speed Sliding Friction of Graphene and Novel Routes to Persistent Superlubricity. Sci. Rep. 2015, 4, 4875. (9) Liu, Z.; Yang, J.; Grey, F.; Liu, J. Z.; Liu, Y.; Wang, Y.; Yang, Y.; Cheng, Y.; Zheng, Q. Observation of Microscale Superlubricity in Graphite. Phys. Rev. Lett. 2012, 108, 205503. (10) Berman, D.; Deshmukh, S. A.; Sankaranarayanan, S. K. R. S.; Erdemir, A.; Sumant, A. V. Macroscale Superlubricity Enabled by Graphene Nanoscroll Formation. Science 2015, 348, 1118. 7646


Article

ACS Nano

(34) Müser, M. H. Fundamentals of Friction and Wear; Springer, 2007; pp 177−199. (35) de Wijn, A. S. (In)commensurability, Scaling, and Multiplicity of Friction in Nanocrystals and Application to Gold Nanocrystals on Graphite. Phys. Rev. B: Condens. Matter Mater. Phys. 2012, 86, 085429. (36) Dietzel, D.; Feldmann, M.; Herding, C.; Schwarz, U. D.; Schirmeisen, A. Quantifying Pathways and Friction of Nanoparticles During Controlled Manipulation by Contact-Mode Atomic Force Microscopy. Tribol. Lett. 2010, 39, 273−281. (37) Dietzel, D.; Mönninghoff, T.; Jansen, L.; Fuchs, H.; Ritter, C.; Schwarz, U. D.; Schirmeisen, A. Interfacial Friction Obtained by Lateral Manipulation of Nanoparticles using Atomic Force Microscopy Techniques. J. Appl. Phys. 2007, 102, 084306. (38) Poole, C. P. Encyclopedic Dictionary of Condensed Matter Physics; Academic Press, 2004. (39) Guerra, R.; Tosatti, E.; Vanossi, A. Slider Thickness Promotes Lubricity: from 2D Islands to 3D Clusters. Nanoscale 2016, 8, 11108− 11113. (40) Varenberg, M.; Etsion, I.; Halperin, G. An Improved Wedge Calibration Method for Lateral Force in Atomic Force Microscopy. Rev. Sci. Instrum. 2003, 74, 3362. (41) Ueyama, H.; Sugawara, Y.; Morita, S. Stable Operation Mode for Dynamic Noncontact Atomic Force Microscopy. Appl. Phys. A: Mater. Sci. Process. 1998, 66, S295−S297. (42) Dietzel, D.; Schwarz, U. D.; Schirmeisen, A. Nanotribological Studies Using Nanoparticle Manipulation: Principles and Application to Structural Lubricity. Friction 2014, 2, 114−139. (43) Kresse, G.; Furthmüller, J. Efficiency of Ab-Initio Total Energy Calculations for Metals and Semiconductors using a Plane-Wave Basis Set. Comput. Mater. Sci. 1996, 6, 15. (44) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation made Simple. Phys. Rev. Lett. 1996, 77, 3865. (45) Blöchl, P. E. Projector augmented-wave method. Phys. Rev. B: Condens. Matter Mater. Phys. 1994, 50, 17953. (46) Grimme, S. Semiempirical GGA-Type Density Functional Constructed with a Long-Range Dispersion Correction. J. Comput. Chem. 2006, 27, 1787.

(11) Martin, J. M.; Donnet, C.; le Mogne, T.; Epicier, T. Superlubricity of Molybdenum Disulphide. Phys. Rev. B: Condens. Matter Mater. Phys. 1993, 48, 10583−10586. (12) Cihan, E.; Ipek, S.; Durgun, E.; Baykara, M. Z. Structural Lubricity under Ambient Conditions. Nat. Commun. 2016, 7, 12055. (13) Kawai, S.; Benassi, A.; Gnecco, E.; Söde, H.; Pawlak, R.; Feng, X.; Müllen, K.; Passerone, D.; Pignedoli, C. A.; Ruffieux, P.; Fasel, R.; Meyer, E. Superlubricity of Graphene Nanoribbons on Gold Surfaces. Science 2016, 351, 957−961. (14) Annett, J.; Cross, G. L. W. Self-Assembly of Graphene Ribbons by Spontaneous Self-Tearing and Peeling from a Substrate. Nature 2016, 535, 271−275. (15) Müser, M.; Wenning, L.; Robbins, M. Simple Microscopic Theory of Amontons’s Laws for Static Friction. Phys. Rev. Lett. 2001, 86, 1295−1298. (16) Dietzel, D.; Ritter, C.; Mönninghoff, T.; Fuchs, H.; Schirmeisen, A.; Schwarz, U. D. Frictional Duality Observed during Nanoparticle Sliding. Phys. Rev. Lett. 2008, 101, 125505. (17) Brndiar, J.; Turanský, R.; Štich, I. Simulation of Frictional Behavior of Sb Nanoparticles on HOPG: Frictional Duality and Biduality. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 84, 085449. (18) Brndiar, J.; Turanský, R.; Dietzel, D.; Schirmeisen, A.; Štich, I. Understanding Frictional Duality and Bi-Duality: Sb-Nanoparticles on HOPG. Nanotechnology 2011, 22, 085704. (19) Vu, C. C.; Zhang, S.; Urbakh, M.; Li, Q.; He, Q.-C.; Zheng, Q. Observation of Normal-Force-Independent Superlubricity in Mesoscopic Graphite Contacts. Phys. Rev. B: Condens. Matter Mater. Phys. 2016, 94, 081405. (20) Kim, W. K.; Falk, M. L. Atomic-Scale Simulations on the Sliding of Incommensurate Surfaces: The Breakdown of Superlubricity. Phys. Rev. B: Condens. Matter Mater. Phys. 2009, 80, 1. (21) Aubry, S.; Le Daeron, P. Y. The Discrete Frenkel-Kontorova Model and its Extensions: I. Exact Results For the Ground-States. Phys. D (Amsterdam, Neth.) 1983, 8, 381−422. (22) Sharma, S. R.; Bergersen, B.; Joos, B. Aubry Transition in a Finite Modulated Chain. Phys. Rev. B: Condens. Matter Mater. Phys. 1984, 29, 6335−6340. (23) Weiss, M.; Elmer, F.-J. Dry Friction in the Frenkel-KontorovaTomlinson Model: Static properties. Phys. Rev. B: Condens. Matter Mater. Phys. 1996, 53, 7539−7549. (24) Benassi, A.; Ma, M.; Urbakh, M.; Vanossi, A. The Breakdown of Superlubricity by Driving-Induced Commensurate Dislocations. Sci. Rep. 2015, 5, 16134. (25) Ma, M.; Benassi, A.; Vanossi, A.; Urbakh, M. Critical Length Limiting Superlow Friction. Phys. Rev. Lett. 2015, 114, 1. (26) Reguzzoni, M.; Ferrario, M.; Zapperi, S.; Righi, M. C. From the Cover: Onset of Frictional Slip by Domain Nucleation in Adsorbed Monolayers. Proc. Natl. Acad. Sci. U. S. A. 2010, 107, 1311−1316. (27) Sharp, T. A.; Pastewka, L.; Robbins, M. O. Elasticity Limits Structural Superlubricity in Large Contacts. Phys. Rev. B: Condens. Matter Mater. Phys. 2016, 93, 121402. (28) Bylinskii, A.; Gangloff, D.; Counts, I.; Vuletić, V. Observation of Aubry-Type Transition in Finite Atom Chains via Friction. Nat. Mater. 2016, 15, 717−721. (29) Müser, M. H. Dry Friction Between Flat Surfaces: Multistable Elasticity vs. Material Transfer and Plastic deformation. Tribol. Lett. 2001, 10, 15−22. (30) Feldmann, M.; Dietzel, D.; Fuchs, H.; Schirmeisen, A. Influence of Contact Aging on Nanoparticle Friction Kinetics. Phys. Rev. Lett. 2014, 112, 155503. (31) Feldmann, M.; Dietzel, D.; Tekiel, A.; Topple, J.; Grütter, P.; Schirmeisen, A. Universal Aging Mechanism for Static and Sliding Friction of Metallic Nanoparticles. Phys. Rev. Lett. 2016, 117, 025502. (32) Stegemann, B.; Ritter, C.; Kaiser, B.; Rademann, K. Crystallization of Antimony Nanoparticles: Pattern Formation and Fractal Growth. J. Phys. Chem. B 2004, 108, 14292−14297. (33) Stegemann, B.; Kaiser, B.; Rademann, K. Subsequent Layer Growth of Supported Nanoparticles by Deposition of Sb4 clusters onto MoS2(0001). New J. Phys. 2002, 4, 89. 7647


Limitations of Structural Superlubricity: Chemical Bonds versus

Recommend Documents