Density Functional Theory Calculations. All the first-principles calculations were performed by the full-potential, all-electron DFT package FHI-aims using light default settings for the basis set and integration grids, the PBE or rSCAN exchange-correlation functional, and a dipole correction. Dispersion corrections were included by either the Tkatchenko-Scheffler (TS) method or via the MBD scheme . K-point sampling was done with a density exceeding 60/Å kpoints. The slab model for Cu(111) comprised 4 atomic layers with a (1x1) surface unit-cell, which then hosted 2 C atoms of a slightly strained Gr sheet (Gr on Cu(111) in Table ). The liquid Cu slab model has a comparable thickness. Electronic self-consistency converged energies to within 10 -6 eV, with the exception of large Gr-Cu(l) configurations computed with rSCAN, where a looser criterion of 10 -2 eV was used. Bulk fcc Cu and solid Cu(111) geometries were relaxed until residual forces fell below 1 meV/Å and 10 meV/Å, respectively.