The graphene–substrate interaction energy can be determined by summing up all van der Waals forces between the graphene carbon atoms and the substrate atoms. The van der Waals force between a graphene–substrate atomic pair of distance

*r* can be characterized by a Lennard–Jones pair potential,

*V*
_{LJ}(

*r*) = 4ɛ(σ

^{12} /

*r*
^{12} − σ

^{6}/

*r*
^{6}), where

is the equilibrium distance of the atomic pair and ɛ is the bonding energy at the equilibrium distance. The number of atoms over an area d

*S* on the graphene and a volume d

*V*
_{s} in the substrate are ρ

_{c} d

*S* and ρ

_{s} d

*V*
_{
s
}, respectively, where ρ

_{c} is the homogenized carbon atom area density of graphene that is related to the equilibrium carbon–carbon bond length

*l* by

and ρ

_{s} is the molecular density of substrate that can be derived from the molecular mass and mass density of substrate. The interaction energy, denoted by

*E*
_{int}, between a graphene of area

*S* and a substrate of volume

*V*
_{s} is then given by

Since Lennard–Jones potential decays rapidly beyond equilibrium atomic pair distance, *E*
_{int} can be estimated by adding up the van der Waals forces between each graphene carbon atom and the substrate portion within a cut-off distance from this carbon atom. If the cut-off distance is large enough, such an estimate of interaction energy converges to the theoretical value of *E*
_{int}. In all simulations reported in this paper, a cut-off distance of 3 nm was used and shown to lead to variations in the estimated value of *E*
_{int} less than 1%.

We have developed a Monte Carlo numerical scheme to compute the multiple integrals in Eq.

1, as summarized below [

25]. For the

*i* th graphene carbon atom,

*n* random locations are generated in the substrate portion within the cut-off distance from this carbon atom. The interaction energy between this carbon atom and the substrate is estimated by

where

*r*
_{
ij
} is the distance between the

*i* th graphene carbon atom and the

*j* th random substrate location. Equation

2 is evaluated at

*m* equally spaced locations over the graphene of area

*S*. The graphene–substrate interaction energy over this area can then be estimated by

As *n* and *m* become large enough, Eq. 3 converges to the theoretical value of *E*
_{int}. In all simulations in this paper, *n* = 10^{6}, *m* = 400.

The strain energy in the graphene–substrate system results from the corrugating deformation of the graphene and the interaction-induced deformation of the substrate. When an ultrathin monolayer graphene partially conforms to a rigid substrate (e.g., SiO

_{2}), the substrate deformation due to the weak graphene–substrate interaction is expected to be negligible. Also, when the graphene spontaneously follows the substrate surface under weak interaction (imagine a fabric naturally conforming to a rough surface) and is not subject to any mechanical constraints (e.g., pinning [

26]), the in-plane stretching of the graphene is also expected to be negligible. Under the above assumptions, the strain energy in the graphene–substrate system can be reasonably estimated by the graphene strain energy due to out-of-plane bending, denoted by

*E*
_{g}. Effect of the above assumptions on results is to be further elaborated later in this paper. Denoting the out-of-plane displacement of the graphene by

*w*
_{g}(

*x* *y*), the graphene strain energy over an area

*S* can be given by

where *D* and *ν* are the bending rigidity and the Poisson’s ratio of graphene, respectively.

The out-of-plane herringbone corrugations of the substrate surface (Fig.

1a) and the out-of-plane corrugations of the graphene regulated by such a substrate surface are described by

respectively, where

*A*
_{s} and

*A*
_{g} are the amplitudes of the substrate surface corrugations and the graphene corrugations, respectively; for both the graphene and the substrate,

*λ*
_{
x
} is the wavelength of the out-of-plane corrugations,

*λ*
_{
y
} and

*A*
_{
y
} are the wavelength and the amplitude of in-plane jogs, respectively; and

*h* is the distance between the middle planes of the graphene and the substrate surface. Given the symmetry of the herringbone pattern, we only need to consider a graphene segment over an area of

*λ*
_{
x
}/2 by

*λ*
_{
y
}/2, and its interaction with the substrate. By substituting Eq.

5 into Eq.

4, the strain energy of such a graphene segment is given by

As shown in Eq. 6, for a given substrate surface corrugation (i.e., *A*
_{s}, *A*
_{
y
}, *λ*
_{
x
}
*,* and *λ*
_{
y
}), *E*
_{g} increases monotonically as *A*
_{g} increases. On the other hand, the graphene–substrate interaction energy, *E*
_{int}, minimizes at finite values of *A*
_{
g
} and *h*, due to the nature of van der Waals interaction. As a result, there exists a minimum of (*E*
_{g} + *E*
_{int}) where *A*
_{
g
} and *h* reach their equilibrium values. The energy minimization was carried out by running a customized code on a high performance computation cluster. In all computations, *D* = 1.41 eV,*l* = 0.142 nm, ρ_{s} = 2.20 × 10^{28}/m^{3},*σ* = 0.353 nm and *A*
_{s} = 0.5 nm, which are representative of a graphene-on-SiO_{2} structure [27, 28]. Various values of *ɛ* *λ*
_{
x
}
*,* *λ*
_{
y
}, and *A*
_{
y
} were used to study the effects of interfacial bonding energy and substrate surface roughness on the regulated graphene morphology.