Dissipation Mechanisms and Superlubricity in Solid Lubrication by Wet-Transferred Solution-Processed Graphene Flakes: Implications for Micro Electromechanical Devices

Solution-processed few-layer graphene flakes, dispensed to rotating and sliding contacts via liquid dispersions, are gaining increasing attention as friction modifiers to achieve low friction and wear at technologically relevant interfaces. Vanishing friction states, i.e., superlubricity, have been documented for nearly-ideal nanoscale contacts lubricated by individual graphene flakes. However, there is no clear understanding if superlubricity might persist for larger and morphologically disordered contacts, as those typically obtained by incorporating wet-transferred solution-processed flakes into realistic microscale contact junctions. In this study, we address the friction performance of solution-processed graphene flakes by means of colloidal probe atomic force microscopy. We use a state-of-the-art additive-free aqueous dispersion to coat micrometric silica beads, which are then sled under ambient conditions against prototypical material substrates, namely, graphite and the transition metal dichalcogenides (TMDs) MoS2 and WS2. High resolution microscopy proves that the random assembly of the wet-transferred flakes over the silica probes results into an inhomogeneous coating, formed by graphene patches that control contact mechanics through tens-of-nanometers tall protrusions. Atomic-scale friction force spectroscopy reveals that dissipation proceeds via stick–slip instabilities. Load-controlled transitions from dissipative stick–slip to superlubric continuous sliding may occur for the graphene–graphite homojunctions, whereas single- and multiple-slips dissipative dynamics characterizes the graphene–TMD heterojunctions. Systematic numerical simulations demonstrate that the thermally activated single-asperity Prandtl–Tomlinson model comprehensively describes friction experiments involving different graphene-coated colloidal probes, material substrates, and sliding regimes. Our work establishes experimental procedures and key concepts that enable mesoscale superlubricity by wet-transferred liquid-processed graphene flakes. Together with the rise of scalable material printing techniques, our findings support the use of such nanomaterials to approach superlubricity in micro electromechanical systems.


S1. Preparation of colloidal probes and force calibration
We assembled colloidal probes by gluing silica spheres of mean nominal diameter 25.24 (MicroParticles GmbH SiO 2 -R-SC93; standard deviation ) to rectangular Si cantilevers with 0.75 molten Shell Epikote resin (heating temperature ), using a custom-built micromanipulation ~140°C stage coupled with an optical microscope. 1 The nominal elastic constant of each cantilever (MikroMasch HQ:NSC35/tipless, NSC12/tipless) was measured by Sader's method 2 and was in the The applied normal force was estimated as , where is the variation of the = ∆ ∆ photodiode current induced by the cantilever vertical deflection and the normal force calibration factor . Here, is the effective spring constant of the colloidal probe 3 and is the = * * vertical deflection sensitivity of the AFM optical lever system. The lateral force calibration factor , required to estimate the lateral force , was determined by means of a diamagnetic = ∆ levitation spring system. 4 For calibration of normal forces, the sensitivity (in nm/nA) was obtained from the slope of normal deflection vs displacement curves acquired over a rigid Si wafer. To obtain the effective spring constant of the colloidal probe , we first evaluated the normal spring constant of the * bare silicon cantilevers (i.e. before attaching the silica bead) using Sader's method. 2 Next, we estimated the spring constant , where is the distance of the glued bead with = ( / ) 3 respect to the base of the lever ( Figure S1). Finally, following Edwards et al. 3 , the effective spring constant of the colloidal probe was * obtained as: * ≡ properties of the tipless cantilever and on the bead position along the long axis of the cantilever. Figure S1. Side view of the colloidal probe during AFM experiments. The angle of , formed by the cantilever's 20°l ong axis with the plane of the sample surface, is imposed by the AFM head design. This angle is taken into account for the calibration of normal forces as well as for the exact location of the contact region in SEM imaging.     Figure S6, right column). Once the process is finished WSXM can calculate the area of (h th ) the "drought islands". The threshold value was varied for each topography in small steps of below the h th 0.1 -0.2nm maximum surface height , to find out the threshold separating the regime of single-asperity contact ( h max h * th S-10 , only one island in the "flooded topography") from a multi-asperity contact ( , two or more 0 < h th ≤ h * th h th > h * th islands in the "flooded" topography). The area of the topographically-highest nanoasperity was estimated as ( . We conventionally assumed that values that are more than 1nm apart from ~30 and a tentative curvature radius for the nanoasperity). In such specific situations the area of the GPa = 50 -300nm topographically-highest nanoasperity was assumed to be . We obtained that is for (h max -1nm)~2 × 10 2 nm 2 all the beads. Arrows in the original AFM topographies (left column) and "flooded topographies" (right column) highlight the location of the single "drought island". S-11

S8. Nonlinear force vs distance curves measured by means of the pristine colloidal probes
In a previous study 5 we discussed the origin of the adhesion for pristine colloidal probes in contact with HOPG. We showed that the long-range van der Waals (vdW) force gives only a modest contribution to . Assuming that the colloidal bead roughness limits the distance of closest approach to HOPG to about , we estimated a vdW attraction of (see Equation A12  ~1nm~209nN in the Supplementary Information of 5 ); this is much smaller that the adhesion measured in experiments. On the other side, using the theoretical model by Farshchi-Tabrizi et al., 6 we calculated that the capillary force of a water meniscus condensed around the bead-HOPG contact amounts to at the ambient relative humidity (see Equation A10 in the ~2.6μN RH = 60% Supplementary Information of 5 ). Therefore we concluded that the (total) estimated adhesion is dominated by the capillary force and amounts to , which is comparable with measured in ~2.8μN experiments. Accordingly, the unloading branch of the force vs distance curves should mainly reflect the kinetics of rupture of the capillary bridge at the bead-HOPG contact.
We found nonlinear unloading curves for all the pristine probes in contact with HOPG, albeit the magnitude of the effect varied from probe to probe. Nonlinearity took place in the tensile region a few hundreds of nm before the jump-off-contact. We believe that the nonlinearity derives < 0 from the superposition of distinct contributions.
Intrinsic contributions from cantilever tilt and contact mechanics. Qualitatively, the nonlinear behavior depicted in Figure 5(a) agrees with the picture that on retracting the probe from the surface a capillary neck is built, then it shrinks and at some point it breaks. In fact it is known that similar phenomena can deviate the unloading branch from linearity, and they can even transform the sharp jump-of-contact detachment into a smoother slide-off-contact event. 7 In our experimental set-up, the cantilever is tilted by 20° with respect to the sample plane so that the tip end of the lever typically displaces (over the sample surface) in the cantilever long-axis direction whenever the contact load is varied through a elongation/retraction of the AFM scanner. This implies that ∆ the contact spot between the colloidal bead and the substrate is not stationary in the sample plane but it slides horizontally forward/backward by a distance during both loading/unloading branches. 8 For a typical scanner displacement one gets . Therefore, both the horizontal translation of the contact spot (imposing a shear force) and the nm vertical increase of the probe-surface gap (imposing a tensile force) contribute in principle to the S-12 kinetics of the capillary rupture, hence to the shape of the unloading curve nearby the jump-offcontact event. Another strictly related issue (certainly affecting the unloading branch) is the frictionally-driven torque of the colloidal probe. In fact, due to the horizontal translation of the contact spot during a force vs distance curve, longitudinal friction forces come into play producing a torque that opposes the normal bending moment. 9 In our experiments we often found -as an evidence of the sizeable role of the frictionally-driven torque -the manifestation of a peculiar hysteresis between the forward and backward branches of the force vs displacement curves (e.g. see on the finite size and shape of the laser spot reflected by the backside of the cantilever into the PSPD active area. An analytical expression for the PSPD response that takes into account the finite size effects of an ideally-round spot is: where is the laser spot radius, is the cantilever deflection and is the optical level amplification factor. When the unloading branch is measured on an ideally-rigid and adhesive substrate and one gets the PSPD linear scaling for whereas for . A solution to mitigate the extrinsic PSPD contribution is to use (∆ )→∆ max N ∆ → / elastically stiffer cantilevers in order to reduce their deflection signal at the jump-of-contact, ∆ and thus make the optical level system always operate in the PSPD linear regime. However we avoided to use stiffer cantilevers, as this would have considerably decreased the sensitivity to lateral forces required to explore the superlubric response of the graphene-coated probes. In practice, the extrinsic contribution of the PSPD implies that the measured jump-of-contact amplitude underestimates the true adhesion force experienced by the pristine probes on HOPG.
We note that the above discussion has no impact on the phenomenon of the 'adhesion breakdown' depicted in Figure 4(k) and Figure 5(a),(b), as this holds regardless of the intrinsic/extrinsic nature of the nonlinearity. We also underline that AFM force spectroscopy experiments involving the S-13 graphene-coated probes are not affected by the PSPD nonlinearity (as the maximum contact force amounts to , see Figure 6(d)). ~700 ≪ | max | Since the nonlinearity of the unloading branch reflects multiple contributions besides contact mechanics effects, we avoided to transform experimental curves into curves for the case of the pristine colloidal probes. Figure S9. Variation of the long-ranged behavior of two spectroscopic curves (top) acquired with the very same graphene-coated colloidal probe on HOPG. Variation is triggered by the release of a loosely attached FLG flake from the coated probe to HOPG (occurred while sliding on HOPG in between the two spectroscopic experiments). S-17

S12. Role of damping in stick-slip dynamics
To better support our claim that the downward flexion of curves ( Figure 6i) is * ≡ / probably due to the underdamped frictional dynamics, we carried out simulations at varying Langevin damping parameter . To isolate the effect of damping and make results as clear as possible, we carried out the simulations at zero temperature T=0K. The lateral spring constant was fixed at . Figure S12 reports the normalized friction force as a function of the = 10N/m Tomlinson parameter , for various values of the damping , along with the theoretical limit by Gnecco et al. 10 in the zero-temperature single-slip overdamped case. For large , the simulated curve follows the theoretical prediction (e.g. , purple triangles in Figure S12(a)). As = 0.137ns -1 decreases, the curve develops the downward trend discussed in the main text, as expected.

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Moreover, these underdamped zero-temperature curves show sharp steps as a function of , marked in Figure S12(a) for . These steps correspond to the transition from single to double, = 0.048ns -1 triple, etc. slips. This is clearly shown in Figure S12(b) for : the force traces at = 0.048ns -1 different substrate potential (hence evolve from single to multiple stick-slips. Hence, for 0 ) smaller Langevin dissipation, inertia allows the transition from single to multiple stick-slip to occur at a smaller value of , while for very large dissipation, only single slips are allowed as predicted by Gnecco et al. 10 . This mechanism underpins the downward trend as a function of observed in experimental data, validating our assumption of underpinned dynamics.