  Controllable synthesis of defect-free graphene is crucial for applications
since the properties of graphene are highly sensitive to any deviations from
the crystalline lattice. We focus here on the emerging use of liquid Cu
catalysts, which has high potential for fast and efficient industrial-scale
production of high-quality graphene. The interface between graphene and liquid
Cu is studied using force field and ab initio molecular dynamics, revealing a
complete or partial embedding of finite-sized flakes. By analyzing flakes of
different sizes we find that the size-dependence of the embedding can be
rationalized based on the energy cost of embedding versus bending the graphene
flake. The embedding itself is driven by the formation of covalent bonds
between the under-coordinated edge C atoms and the liquid Cu surface, which is
accompanied by a significant charge transfer. In contrast, the central flake
atoms are located around or slightly above 3 {\AA} from the liquid Cu surface
and exhibit weak vdW-bonding and much lower charge transfer. The structural and
electronic properties of the embedded state revealed in our work provides the
atomic-scale information needed to develop effective models to explain the
special growth observed in experiments where various interesting phenomena such
as flake self-assembly and rotational alignment, high growth speeds and low
defect densities in the final graphene product have been observed.
