## 2 Materials and Methods
### Sample Surface Inspection and Tribological Testing  
Figure 1a shows the topography of the five samples after preparation and the corresponding value of the 3D areal surface roughness as per ISO 25178. Figure 1b illustrates the appearance of the sample surface under the optical microscope. The significant increment of roughness measured for the sample coated by directly grown graphene is presumably due to the curious globular-like structure observed on the surface of the sample during FESEM analysis (Figure 2). Some details are provided on this aspect in the following sections. The samples were inspected with FESEM (Zeiss, Jena) with an ultra-high-resolution Inlens EsB detector to image the carbon nano-coatings; images are grouped in Figures 2 and 3. Except for direct growth graphene, a transparent grainy structure was perceived on the surface of the sample through which one can see the roughness pattern of the substrate surface. It corresponds to the ultra-thin layer of sp\({}^{2}\)-bonded carbon atoms. The transferred graphene coating appeared as a uniform patchwork with patches of similar colour, and it looked like the bilayer graphene structure reported by [PERSON] et al. on copper [53]. The patches are single-crystalline graphene islands grown at nucleation sites on the catalyst foil. As for self-assembled graphene coatings, the enlarged images suggested that the recombination of GNPs was effective in covering the surface of the sample, but the pattern of dark and light patches hints that the carbon layer is uneven, as expected, with regions of locally increased thickness due to folds (see also the inset of Figure 3) or superpositions of many flakes. This graphene structure appeared very similar to the eight-layer graphene reported by [PERSON] et al. [53]. Figure 1: (**a**) Surface morphology of the coated samples before the tribological tests. (**b**) Inspection under the optical microscope of the sample surface before the tribological test. Interestingly, the surface of the samples with direct growth graphene features a pattern of platelets or grains which resembles the morphology of multiplayer CVD graphene on Ni substrate reported by [PERSON] et al. [27]. This led the authors to envisage a growth mode as ‘islands’ or agglomerations of platelets of different thickness. Figure 2: SEM imaging of transferred and directly grown graphene with magnification \(500\times\), \(2.5\) kX and \(25\) kX. The surface of the direct growth graphene appeared covered by a structure of spherical nodules protruding from the surface. The authors found no other evidence of CVD graphene on copper with a similar surface texture in the scientific literature. A possible explanation of this phenomenon could be that this peculiar globular structure is made of relatively hard amorphous carbon nodules. [PERSON] et al. suggested that amorphous carbon can originate on the surface of copper foils during the CVD process if an insufficient flow of precursor gas is supplied and, thus, an insufficient amount of C solubilizes into the substrate [54]. The concentration of the carbon precursor is an important influencing factor in triggering the graphene nucleation, irrespective of the reactor temperature (800-1000 \({}^{\circ}\)C) and the nature of the catalyst metal. This argumentation led the authors to raise the hypothesis that the globular structure visible under FESEM inspection is some form of agglomerated amorphous carbon grown in place of the pristine graphene layer at the nucleation sites. The reasons for this phenomenon are yet unclear. The flow rate or the partial pressure of CH\({}_{4}\) inside the reactor might have been insufficient to promote the supersaturation of C atoms on the surface of a thicker copper substrate compared to the case of deposition on Cu foils. At the same time, the state-of-the-art understanding of the deposition process evidences that carbon solubility decreases with cooling down, and carbon precipitation triggers the surface crystallization of the graphene lattice [55].