  Fullerene-based sandwiches have emerged as new candidates for potential
applications of two-dimensional nanomaterials in electronics or energy storage.
Recently, experimentalists have observed the evolution of boundaries for
fullerene clusters sandwiched by two graphene layers, while vacuum space with
typical dimension of 30 {\AA} was found within the fullerene layer. Because the
pattern of the fullerene cluster impacts the physical properties of the
sandwiches, it is important to understand the mechanisms for their structural
transformations. In the present work, we find that the
graphene/fullerene/graphene sandwich structure transforms among three
configurations, depending on the fullerene to graphene area ratio. Molecular
dynamics simulations show that there are two critical values for the area
ratio. The fullerene pattern transforms from circular to rectangular at the
first critical area ratio of 1/{\pi}. The critical value of 1/{\pi} is
successfully derived by comparing the geometrical perimeter of the circular and
rectangular shapes for the fullerene cluster without any physical parameters.
At the second critical area ratio, the graphene layers surrounding the
fullerene cluster become separated due to the competition between the bending
energy and the cohesive energy. Based on the analytic model, the vacuum space
is predicted to be 34 {\AA}, which agrees quite well with the experimental
result. These findings provide fundamental insights into the mechanisms driving
structural transformation of fullerene-based sandwiches, which will enable the
selection of appropriate structures and materials to guide the design of future
electronic and energy storage applications.
