Minimal graphene thickness for wear protection of diamond

We show by means of molecular dynamics simulations that graphene is an excellent coating for diamond. The transformation of diamond to amorphous carbon while sliding under pressure can be prevented by having at least two graphene layers between the diamond slabs, making this combination of materials suitable for new coatings and micro- and nanoelectromechanical devices. Grain boundaries, vacancies and adatoms on the diamond surface do not change this picture whereas reactive adsorbates between the graphene layers may have detrimental effects. Our findings can be explained by the properties of layered materials where the weak interlayer bonding evolves to a strong interlayer repulsion under pressure.

We show by means of molecular dynamics simulations that graphene is an excellent coating for diamond. The transformation of diamond to amorphous carbon while sliding under pressure can be prevented by having at least two graphene layers between the diamond slabs, making this combination of materials suitable for new coatings and micro-and nanoelectromechanical devices. Grain boundaries, vacancies and adatoms on the diamond surface do not change this picture whereas reactive adsorbates between the graphene layers may have detrimental effects. Our findings can be explained by the properties of layered materials where the weak interlayer bonding evolves to a strong interlayer repulsion under pressure.
Control of friction and wear is one of the key challenges for the design of micro-and nanoelectromechanical systems (MEMS/NEMS). There is an ongoing quest to make these devices reliable, robust and able to resist demanding environments, under high stress and with sliding surfaces in contact. For these devices, lubrication has to be based on dry solid coatings rather than on liquids to avoid undesirable effects, associated with viscosity 1 , squeeze out 2 and stiction 3 .
The present MEMS/NEMS technology is based on silicon 4,5 , but its poor mechanical, chemical and tribological properties make alternatives desirable and actively sought after 6 . In particular, at the nanoscale wear is a limiting factor as it drastically shortens their lifetime 3 . Diamond is such an alternative material in view of its hardness and chemical inertness. Making perfectly crystalline diamond is difficult, but nanocrystalline diamond (grain sizes of 10-200 nm) shares many of its properties and is attainable by CVD 4 .
Although diamond is very hard, it is not resistant to wear and it can be polished. The polishing rate has been shown to depend on the surface orientation and sliding direction 7 . The amorphous layer which develops at the sliding interface is easily removed leading to wear of the surface. This amorphous phase, with many bonds at the interface, leads to a high friction coefficient. Fortunately, lowering of the friction coefficient after some time, also called running-in, is observed for sliding amorphous carbon 8,9 . The microscopic mechanisms for this behavior are still a matter of debate. Molecular dynamics based on a modified version of the empirical potential REBO, reports the formation of a graphene-like layer 10 during sliding under pressure that would inhibit the further growth of an amorphous layer at the interface. Recent ab-initio calculations, instead, attribute the reduction of friction after the initial phase (running in) to passivation of the dangling bonds by water or, preferably, by hydrogen [11][12][13] . For the latter, a minimum humidity or hydrogen gas pressure is necessary and the contact pressure needs to be below a critical value 11 for passivation. These results suggest that operation in vacuum or high-pressure environments would be difficult.
An approach to reduce wear is to look for suitable coatings, effective at the nanoscale. Moreover, it is desirable to have a very thin coating. Graphene is a nat-ural candidate for this purpose in view of its exceptional mechanical properties 14 . The frictional properties of (few-layer) graphene have been recently intensively studied 15,16 showing a lowering of friction with decreasing number of layers. Coating of sliding steel surfaces with few layer graphene has been demonstrated to reduce drastically the friction and wear during sliding 17 . On a smaller scale, coating an AFM probe with graphene also improved resistance to wear 18 . Moreover it has been recently shown that graphene withstands without damage much higher loads than diamond-like carbon 19 , making graphene suitable for high-pressure conditions.
Here we suggest combining the properties of diamond and graphene to form a hard but smoothly sliding structure to enable new MEMS/NEMS technologies.
We perform atomistic simulations to describe the wear of diamond surfaces during sliding under pressure when the surfaces are either bare or separated by one or two layers of graphene. We find that at least two layers of graphene form a contact that drastically reduces friction and wear.
The interatomic interactions are given by the reactive empirical potential LCBOP 20 as implemented in the molecular dynamics code LAMMPS 21 . This bond-order potential can accurately describe different phases of carbon 22 , the transformations between them and the elastic constants of diamond and graphite. It can also describe the interaction of single carbon atoms with the diamond surfaces and graphene. Since single atoms are very reactive we use them to represent the effect of reactive species and impurities.
In Fig. 1 we show a sketch of our model. Our initial sample consists of two slabs of diamond with (100) surfaces, which are pressed against each other. The (100) surface has a square unit cell given by one face of the cubic lattice with lattice parameter 3.5668Å. Periodic boundary conditions are imposed in the in-plane x and y directions. Each diamond slab is made of 9 × 6 × 8 unit cells. This size is chosen to avoid strain and match the periodic boundary conditions when one or two graphene layers of 260 atoms each are placed between the diamond slabs, as shown in Fig. 1b  top rigid part can also move as a whole in the z direction under the influence of a constant force on each atom, which results in a pressure of 10 GPa. The temperature is controlled by a Langevin thermostat with damping constant γ −1 =0.1 ps applied to the 4 atomic layers adjacent to the top and bottom rigid layers. All simulations are performed at room temperature (300 K).
Randomly placed carbon atoms in the region between the two bare diamond slabs prevent cold welding, that is the joining of the two slabs 7 . When one or two graphene layers are present, these atoms allow bond formation between the graphene layer and the diamond surface as we discuss later.
It has been shown 7 that when two diamond slabs slide against each other, the crystalline structure at the interface is damaged, leading to an amorphous structure with a rate of amorphization which depends on the surface and on the sliding direction.
We consider (100) diamond surfaces sliding in the 100 direction, which is a fairly soft direction and find that the bare contact area transforms, as shown in Fig. 2a, into amorphous carbon with a ∼ 90 % sp 2 bonds. The precise percentage of bonding in disordered, liquid or amorphous, phases may depend on the used potential 9,23 . A single graphene layer between the two surfaces leads to the same result, namely the graphene layer is destroyed within tens of picoseconds and the contact area becomes amorphous. This situation changes dramatically for a bilayer graphene (Fig. 2c). Sliding occurs in this case preserving the structure of the diamond surface as well as that of the bilayer. The different behaviors are also visible in the velocity and temperature profiles along the height of the sample, shown in Fig. 3. While the samples which degrade to amorphous carbon show a gradual change in velocity, the sample with two layers of graphene shows a sharp transition where the two slabs slide over each other. In this case, the temperature remains constant at 300 K while for the amorphous contact area is raises to 600 K at the interface.
To understand the reason for the marked difference between one-and two-layer graphene coating of the diamond surface we have considered all the systems sketched in Fig. 4 that we have divided into those that do not present wear within the timescale of our simulations and those that do. We see that it is important to consider the possibile imperfections of the surfaces or the presence of adsorbates and reactive molecules. In fact for ideally planar, clean surfaces with either one or two graphene layers in between but no adsorbates, no wear occurs during sliding (see panel NW1,NW2). Adatoms in between the diamond surfaces and graphene lead to the formation of bonds as shown in Fig. 5, pulling graphene out of planarity. The consequences are very different for one or two layers. In fact, for one layer, once a bond is formed with the upper diamond surface, the deviations from planarity facilitate bonding of a neighboring atom with the lower diamond surface. Bonds with upper and lower diamond surface become sp 3 -like and propagate leading to an amorphous structure as in Fig. 2b. If instead there is a second layer of graphene, as in Fig. 5b, the bonding between the two graphene layers does not occur because it would require the two graphene layers to approach to distances below 2Å, which is prevented by the high energy barrier due to interlayer repulsion 20,24 . Wear occurs only when single carbon atoms are placed also between the graphene layers.
Next, we have considered less idealized systems by considering the most common defects in graphene, namely grain boundaries and vacancies. In the grain boundary shown in Fig. 6a the bonds form pentagons and heptagons which are more prone to rearrangement than the ideal hexagonal structure and the vacancies in Fig. 6b lead to unsaturated bonds. Also for these cases, we have found the same drastic difference between one and two layers. The only effect on the graphene layer with the grain boundary is a flattening of the curvature of the minimal energy structure without pressure 25 . For the sample with vacancies, we find that they remain intact and smooth when one percent or three percent of the atoms is missing. If we increase further the ratio of deleted atoms to five percent, the graphene layers degrade to amorphous carbon. That the graphene layers do not need to be perfect in order to inhibit wear is encouraging, since growth of perfect graphene is still a technological challenge. As a last test, we have increased the potential energy corrugation which is underestimated by LCBOP. Therefore, we repeated the simulation of diamond with two graphene layers, but with the interactions between atoms in different graphene layers described by a registrydependent potential 26 but did not find any qualitative difference. In summary, we have shown that two layers of graphene between diamond slabs may provide a strong wear-resistant layer. While clean diamond surfaces or separated by only one layer of graphene transform to an amorphous phase during sliding under pressure, two layers of graphene preserve their structure and protect the diamond from wear. This result holds also when the graphene layers present defects such as a grain boundaries or vacancies. We believe that our findings can be relevant for the development of fully carbon based MEMS/NEMS. We thank M. Patelkou for useful discussions. This work is part of the research program of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for Scientific Research (NWO).