Catalytic effect of graphene on the inversion of corannulene using a continuum approach with the Lennard-Jones potential

The catalytic effect of graphene on the corannulene bowl-to-bowl inversion is confirmed in this paper using a pair-wise dispersion interaction model. In particular, a continuum approach together with the Lennard-Jones potential are adopted to determine the interaction energy between corannulene and graphene. These results are consistent with previous quantum chemical studies, which showed that a graphene sheet reduces the barrier height for the bowl-to-bowl inversion in corannulene. However, the results presented here demonstrate, for the first time, that the catalytic activity of graphene can be reproduced using pair-wise dispersion interactions alone. This demonstrates the major role that pair-wise dispersion interactions play in the catalytic activity of graphene.


Introduction
Geodesic polyarenes are polycyclic aromatic hydrocarbons in which structural constraints result in a curved p-system. [1][2][3][4] Geodesic hydrocarbons exhibit unique chemical properties, such as large dipole moments and dynamic bowl-inversion behavior. [2][3][4][5][6][7][8][9] Corannulene (C 20 H 10 ) is a prototypical geodesic molecule in which a pentagon surrounded by ve hexagons results in a bowl-shaped structure. 10 Corannulene undergoes a rapid bowl-to-bowl inversion via a planar transition structure as illustrated in Fig. 1. 11 Catalysis of this bowl-to-bowl inversion has attracted considerable attention aer it was demonstrated that a cyclophane receptor catalyzes this process via induced-t catalysis. [12][13][14] It was later found that graphene (a planar twodimensional (2D) material composed of sp 2 -hybridized carbons) can also catalyze the bowl-to-bowl inversion in corannulene [15][16][17] as well as rotation and inversion reactions in related molecules. [18][19][20] It has also been demonstrated via extensive density functional theory (DFT) and ab initio calculations that these catalytic processes are driven by strong noncovalent interactions that typically exceed a hundred kJ mol −1 . [15][16][17][18][19][20][21][22][23][24] In the system where graphene is used as a catalyst for corannulene bowl-to-bowl inversion, we envisage that the favourable conformation of corannulene (either concave-up or concave-down bowl) can be determined from the structure that gives rise to the minimum interaction energy with the graphene. Since van der Waals forces dominate the interaction between corannulene and graphene, this paper adopts the Lennard-Jones potential to determine the interaction energy between the two molecules. Here, we assume that carbon atoms on graphene are evenly distributed on its surface so we can use continuum surface approximation to model graphene. For corannulene, its three possible conformations are considered, which are concave-up bowl, concave-down bowl and at circular structure. Two approaches to model corannulene-graphene interaction are used. The rst approach considers corannulene as a collection of 30 discrete atoms (20 carbon and 10 hydrogen atoms), and so the total energy is obtained by summing 30 pairwise interaction energies between each atom on corannulene and a graphene sheet. In the second approach, due to its geometry, we model corannulene as a collection of four circular rings (three carbon rings and one hydrogen ring) centred on the same axis (Fig. 2), and on each ring, atoms are assumed uniformly distributed. As a result, the total interaction energy can be obtained from summing four pairwise interaction energies between each ring and a graphene sheet. For each of corannulene conformations, we nd that both approaches give the same energy prole, which is also in agreement with molecular dynamics studies. These results conrm the catalytic effect of graphene on the ability to control the orientation of corannulene that minimises the interaction energy of the system.
In the following section, we give mathematical background for the two approaches to model corannulene-graphene interactions. Detailed calculation of the integrals involved are provided in Appendices A and B. Numerical results for the interaction energies are shown in Section 3 for the three conformations of corannulene. These results are also conrmed by molecular dynamics (MD) and density functional theory (DFT) simulations which their detailed set-ups are given in Appendices C and D, respectively. Finally, concluding remarks is provided at the end of Section 3.

Interaction energy between corannulene and graphene
Due to its simple form, the Lennard-Jones potential is commonly employed to determine the interaction energy between two non-bonded atoms, which is given by where r ij is the distance between atoms i and j, A ij and B ij are the attractive and repulsive constants, respectively. We note that A ij = 23 ij s ij 6 and B ij = 3 ij s ij 12 where 3 ij ¼ ffiffiffiffiffiffi ffi 3 i 3 j p is the energy well depth and s ij = (s i + s j )/2 is the van der Waals diameter. In this paper, the van der Waals parameters for carbon (C) and hydrogen (H) are taken from Rappe et al. 25 where 3 C = 0.4393 kJ mol −1 , s C = 3.8510 Å, 3 H = 0.1841 kJ mol −1 and s H = 2.8860 Å. Thus, the Lennard-Jones constants A ij and B ij can be evaluated as given in Table 1.
In a fully discrete approach, the total interaction energy between two non-bonded molecules can be obtained by summing the pairwise potential energy (1) between atom i on the rst molecule and atom j on the second molecule, which is given by where r ij is the distance between atoms i and j. Another method to model the interaction between two nonbonded molecules is known as a continuum approach. This approach assumes that atoms on each interacting molecule are uniformly distributed over its entire surface of the molecule. Thus, the double summation in (2) can be replaced by two surface integrals, namely where now r denotes the distance between typical surface elements dS 1 and dS 2 on the rst and second molecules, respectively. The constants h 1 and h 2 are the mean surface atomic densities of the two molecules. Note that r is the distance of the closet ring to graphene sheet, and the distances between each ring in corannulene are given by  Table 1 The attractive and repulsive constants (A ij and B ij ) for carboncarbon and carbon-hydrogen interactions The advantage of using (3) over (2) is the reduction in computational time, especially for large molecules. However, for the integrals in (3) to be traceable to yield analytical expressions, regular shape structures are generally assumed for the interacting molecules. Accordingly, this approach has been commonly adopted to determine the interaction energy involving carbon nanostructures, such as nanotubes, fullerenes, graphene, graphite and nanocones. 26 In the interest of modelling an irregularly shaped molecule interacting with a regular shaped structure, an alternative hybrid discrete-continuum approach is introduced, which is given by where h is the surface density of atoms on the regular shaped molecule, r i is the distance between a typical surface element dS on the continuous molecule and atom i on the molecule which is modelled as discrete.
In this paper, we use (4) to determine the non-bonded interaction energy between a graphene sheet and a corannulene. Three conformations of corannulene are considered which are depicted in Fig. 2. We model graphene sheet as a continuum at surface lying on the xy-plane. For corannulene, we rst assume a fully discrete structure with 30 atoms (10 hydrogen atoms (blue) and 20 carbon atoms (black)) as shown in Fig. 2 (Section 2.1). In Section 2.2, we consider corannulene as a structure comprising four continuous rings, where each ring is arranged as shown in Fig. 2.

Discrete model of corannulene
Here, we model a corannulene as a collection of 30 discrete atoms. The coordinates of a corannulene in all three conformations can be found from Karton. 15 Mathematically, we represent an atom on a corannulene as a typical point with coordinates (x, y, r) as shown Fig. 3. Each atom then interacts with a at graphene surface on which a typical point has coordinates (p, q, 0). Since there are 20 carbon and 10 hydrogen atoms on the corannulene, using (4) the total energy becomes where r i is the vertical distance of atom i from graphene sheet, h g is the atomic density of graphene sheet (h g = 0.3812 Å −2 ) and I n (r) (n = 3, 6) is dened by We note that the derivation of I n (r) is given in Appendix A.

Ring model of corannulene
Since the positions of atoms on a corannulene are as shown in Fig. 2, we assume that these atoms are located on rings R 1 to R 4 (see Fig. 2). We note that rings R 1 and R 2 each consists of ve carbon atoms, ring R 3 consists of ten carbon atoms and ring R 4 consists of ten hydrogen atoms. We also note that of these rings, R 1 involves chemically bonded carbons (i.e., the central pentagon ring of corannulene), whereas R 2 , R 3 , and R 4 involve non-bonded atoms. These rings are chosen since they are coplanar in the equilibrium (bowl-shaped) and transition state (at) structures of corannulene. Fig. 2 shows the mathematical representations of the equilibrium and transition state structures of corannulene along with the quantum chemically optimized structures. Further, physical parameters of each ring are given in Table 2.
Mathematically, the problem reduces to nding the interaction energy between a ring and a graphene sheet as shown in Fig. 4 and by using (4) we can obtain the total interaction energy as where h j denotes the atomic density of the ring R j , A j = A C-C and B j = B C-C when j = 1, 2 and 3 and A j = A C-H and B j = B C-H when j = 4. The integral J n (r) (n = 3 and 6) is dened by where its derivation is given in Appendix B. As shown in Fig. 2, r in (7) represents the vertical distance from the graphene sheet to the closet ring of corannulene. For the concave-up bowl, r is the distance from graphene sheet to ring R 1 and the distances  shaped corannulene, all rings are concentric and have the same vertical distance r from the graphene sheet.
In the next section, we plot the interaction energies for the three conformations of corannulene interacting with a graphene sheet. The results are also benchmarked with molecular dynamics simulations.

Results and concluding remarks
Here, the interaction energy is determined as a function of r which is the closest distance between corannulene and graphene sheet. We obtain identical results for both discrete and continuous ring approaches, which are plotted as solid lines in Fig. 5 for the three conformations. The results from our model also agree with those of molecular dynamic simulations, which are plotted as square boxes in Fig. 5. From the gure, we can see that there is a preferred distance (r min ) for each conformation of corannulene that minimises the interaction energy of the system. The values of r min and the corresponding minimum energy are given in Table 3. The interaction energy between the planar corannulene transition structure and the graphene sheet amounts to 178.6 kJ mol −1 , whereas the interaction energy between the concave-up bowl and concave-down bowl and the graphene sheet amount to 129.7 kJ mol −1 and 167.3 kJ mol −1 , respectively. Thus the pair-wise dispersion interactions between the planar graphene sheet and the planar transition structure are stronger by 48.9 and 11.3 kJ mol −1 , respectively, than the concave-up and concave-down structures. This result is due to the closer proximity of the carbon atoms of corannulene and graphene in the planar transition structure than in the concaveup and concave-down structures.
The above results are signicant since they demonstrate that even in the absence of any explicit quantum chemical interactions, pair-wise dispersion interactions alone would result in a graphene sheet catalyzing the bowl-to-bowl inversion in corannulene. This result is consistent with previous dispersioncorrected, double-hybrid DFT calculations, which were obtained on the Gibbs free potential energy surface. 15,17 In order to compare on an even keel between the interaction energies obtained using our pair-wise dispersion model we need to calculate the DFT interaction energies on the electronic potential energy surface. For this purpose, we performed DFT calculations on the electronic potential energy surface using the PW6B95-D4 functional (see Appendix D for further details). At the PW6B95-D4/def2-TZVPP level of theory with basis set superposition error (BSSE) corrections, we obtain the following Fig. 4 Interaction between a ring of corannulene and a graphene sheet. The graphene sheet is assumed to lie on the plane z = 0 and a typical point on a ring of radius r situated at a distance r away from the graphene sheet is given by (r cos q, r sin q, r), where q˛[0, 2p).  interaction energies between corannulene and graphene 82.6 (concave-up bowl), 111.3 (planar TS), and 87.1 (concave-down bowl) kJ mol −1 . Using a larger quadruple-z basis set without BSSE corrections, namely at the PW6B95-D4/def2-QZVPP level of theory, we obtain similar interaction energies of 87.1 (concave-up bowl), 117.1 (planar TS), and 91.6 (concave-down bowl) kJ mol −1 . There is little to choose between the two levels of theory since both have different advantages and disadvantages in terms of basis-set completeness. However, the differences of 4.4-5.8 kJ mol −1 between both sets of results indicate that we are only a few kJ mol −1 away from the complete basis set limit. Here, we will focus on the results obtained with the larger def2-QZVPP basis set. The pair-wise dispersion model predicts much larger interaction energies of 129.7 (concave-up bowl), 178.6 (planar TS), and 167.3 (concave-down bowl) kJ mol −1 (Table 3). However, due to the systematic overestimation of the interaction energies for the concave-up, planar, and concave-down complexes, the catalytic enhancements predicted by the pair-wise dispersion model are in reasonable agreement with the PW6B95-D4/def2-QZVPP results.
In particular, the pair-wise dispersion model predicts catalytic enhancements of 48.9 and 11.3 kJ mol −1 for the forward and reverse directions, whereas the PW6B95-D4/def2-QZVPP level of theory results in catalytic enhancements of 30.0 and 25.5 kJ mol −1 for the forward and reverse directions. We note that the smaller interaction energies obtained in the DFT simulations are partly attributed to the use of a C 96 H 24 graphene nanoake model. We expect that using larger graphene nanoake models would result in larger interaction energies (for further details, see ref. 23). We also note that the pair-wise dispersion model and DFT interaction energies both suggest that the concave-down complex is energetically more stable on the electronic potential energy surface, albeit the DFT results suggest a smaller energy difference between the concave-up and concave-down complexes. Overall, these results demonstrate that pair-wise dispersion interactions play a major role in the catalytic activity of graphene.
Appendix A evaluation of integral I n in section 2.1 Here, we consider a typical point (x, y, r) as a location of an atom on corannulene interacting with a graphene sheet lying on the xy-plane on which its typical point has coordinates (p, q, 0). To determine the interaction energy between a single atom and a graphene sheet using the Lennard-Jones potential, we introduce the integral I n (r) which is given as where dS is the surface element of graphene sheet and d is a typical distance between atom on the corannulene and graphene sheet (Fig. 3) such that d 2 = (x − p) 2 + (y − q) 2 + r 2 .