Interactions of polymers with reduced graphene oxide: van der Waals binding energies of benzene on graphene with defects

Abstract

The interaction of benzene molecules with various defects in graphene is studied using density functional theory enhanced by two different recent dispersion corrections. Both provide the same qualitative picture: the binding strength of benzene to the various defects is governed by steric hindrance. Our first principles calculations in combination with a simple model predict reduced stabilities of polymer–graphene nanocomposites made of reduced graphene oxides depending on the defect density. Above ∼15% defect coverage the interaction is lowered to roughly one third as compared to pristine graphene.

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Graphene, the thinnest truly two-dimensional material known to mankind, has shown physical properties (such as Young's modulus, yield strength, thermal and electrical conductivities) superior to most of the current materials, and consequently graphene is expected to have great potential for future applications in various technologies.¹ At the moment, the progress in the area of graphene synthesis falls far behind the large-scale production rates required for serious industrial applications. Bottom-up fabrication procedures for epitaxial graphene are still in their infancy. Devices made from such epitaxial graphene are still too expensive to compete with corresponding products from semiconductor assembly lines. Accordingly, major attention is paid toward different, scalable chemical routes, especially ones that extract graphene sheets from bulk graphite either directly (for instance by exfoliation with non-covalently bound molecules^2–4) or from oxidation–reduction intermediates. Due to their scalability these production methods provide currently the highest yields of graphene. For instance, current lab-scale oxidation–reduction routes⁵ can synthesize up to 100 g of reduced graphene oxide (RGO) per week. RGO is obtained by reducing chemically oxidized graphene (GO) sheets that have been exfoliated from graphite oxide by expanding gases formed from intercalated chemical impurities. The resulting graphene with defects has been studied intensively by considering structural defects,⁶ and oxygen and hydroxyl groups^7,8 including reduction pathways from GO to RGO.⁹ Although different structural models for GO have been reported,¹⁰ the actual structure of its reduced form RGO is largely unknown. The most popular structural model of GO¹¹ suggests the dominance of hydroxyl and epoxy groups on the surface of GO, while limiting the carboxylic groups (if any) to the sheet edges. The structure of RGO can be expected to be highly diverse consisting of different regions with varying content of oxidized graphene and graphene with defects. Indeed, a large number of defects and additional hydroxyl, epoxy and related chemical groups have been experimentally observed.¹²

Graphene without defects is chemically inert and can be regarded as a large stable macro-molecule. Its interaction with other stable molecules is mediated by non-covalent bonding via van der Waals interactions. This non-covalent bonding not only stabilizes natural graphite, but also enables the use of functionalized graphene flakes as carriers for water-insoluble drugs¹³ or creates nanocomposites between graphene and polymers¹⁴ that can be further bound to proteins.¹⁵ While the interaction between small flat molecules and the perfect graphite surface or ideal graphene sheets has been studied in the past,¹⁶ interactions with functionalized graphenes (such as RGO) are still largely unexplored.

In this article, we want to provide a first contribution to close this gap. Our goal is to explore the effect of defects and oxygen containing functional groups on the van der Waals interaction between an aromatic molecule and graphene. Our molecule of choice as the adsorbent is the smallest aromatic molecule: benzene. Benzene has a simple planar geometry that is not subject to any structural conformations (quite unlike the case with linear or even saturated cyclo-hydrocarbons). Moreover, it has the same continuous π bond cloud, a hallmark of graphene, so that the so-called “π–π interactions” prevail. Aromatic rings are common constituents of many widely-used polymer species (including polystyrene (PS), polyethylene terephthalate (PET), polybutylene terephthalate (PBT), acrylonitrile butadiene styrene (ABS), polyepoxide (Epoxy), and polycarbonate), which emphasizes the suitability of using benzene in this study to display the role that aryl functional groups play in the stability of RGO–polymer composites.

In order to be unbiased by empirical parameters, we use density functional theory (DFT) for our calculations. The inclusion of van der Waals interactions is known to be a difficult task for state-of-the-art approximations used in DFT.^17,18 Functionals employing the generalized gradient approximation (GGAs) that are very successful in the description of chemical bonds completely fail to provide dispersion related attraction, whereas the lower level local density approximation (LDA) shows some, but unphysical contribution.¹⁹ For example, the inter-sheet interaction in graphite is nearly zero in GGA and roughly half of the experimental value²⁰ in LDA.²¹ Therefore, corrections were proposed that add the missing van der Waals contributions to the GGA energy E^GGA, i.e. to write the energy as^22,23

where the sum extends over all atom pairs AB, R_AB is the distance between atoms, C₆ is a constant and f is a cut-off function to avoid the singularity at R_AB = 0. This approach is attractive due to the good performance of the GGAs in chemical binding and the low computational cost of the correction. The first approaches were highly parameterized²⁴ and higher order terms (∼C₈/R_AB⁸) were also included.²⁵ Recently Tkatchenko and Scheffler proposed the TS09 approach using only one fitted parameter, the distance for switching on the correction, and all other parameters are obtained from ab initio calculations.²⁶ The C₆ coefficients are not constants anymore, but depend on the electron density and in this way on the chemical environment.

It has also been shown in the last few years that functionals including a non-local kernel for the description of the correlation energy can also reliably describe vdW interactions. The exchange–correlation energy E_xc[n] depending on the electron density n is written as E_xc[n] = E^GGA_x[n] + E^nl_c[n]²⁷ and hence obtained from the GGA exchange E^GGA_x and a non-local correlation expression¹⁷

E^nl_c[n] = ½∫d³r∫d³r′n(r)Φ(r, r′)n(r′) (2)

where the kernel Φ(r, r′) is approximated. Efficient implementations have been developed for the evaluation of the non-local integral in eqn (2).^28,29 We will use both approaches in the present work and find that they give some quantitative differences, but show good qualitative agreement.

The DFT calculations have been carried out using the projector-augmented wave (PAW) method³⁰ as implemented in grid-based open-source GPAW software.^31,32 The TS09 approach is a correction to the GGA devised by Perdew, Burke, and Ernzerhof (PBE).³³ For the vdW-DF approach of eqn (2), we use the revPBE GGA exchange and include LDA correlation as a local and a plasmon-pole approximation for the non-local contribution of E_c (see ref. 17 for the exact definition). The smooth wave functions were represented on grids of 0.20 Å lattice spacing and this choice was carefully checked for convergence.

The structures were constructed using the Atomic Simulation Environment (ASE).³⁴ The graphene lattice was set up in the honeycomb geometry using the experimental C–C bond length of a = 1.42 Å. The simulation supercell was periodic in x and y directions, the directions of the graphene sheet. Zero boundary conditions were employed in the z-direction perpendicular to the sheet. The supercell contained 60 carbon atoms and has a rectangular shape of dimensions [9a, 5√3a, h_z], where the height h_z was adjusted such that at least 4 Å of vacuum are applied above and below all atoms in the cell. Testing the convergence with respect to the k-points revealed that the Γ-point was enough to achieve convergence of the binding energies within 15 meV in all configurations considered in this study. Therefore we report Γ-point energies exclusively.

There is a huge amount of possible defects in and on graphene which cannot be completely covered by an ab initio study. We therefore selected some typical examples that serve the goal of exploring the general trends. The chosen structures are reported in Fig. 1 and can be grouped into the following families: pristine graphene (labeled A), Stone–Waals defects (B), single vacancies (C), double vacancies (D) and larger holes (E). Oxygen dopants are labeled F, oxygen on bridge positions (i.e. epoxy groups) G and hydroxyl groups H. Different conformations of the same family are distinguished by numbers and their structures are explicitly given in the ESI.† The defects were relaxed using PBE until the maximum force was within 0.05 eV Å⁻¹.

Fig. 1 The eight defect families considered in this work (see the text). The color code of the substrate atoms is C: grey, O: red and H: white. The benzene molecule deposited on top of each defect is shaded in yellow.

The geometry of the benzene molecule was imported from the G2 database³⁵ of small molecules and placed on top of each of the defects. We systematically increased the benzene height in the range from 0.5 Å to 6 Å and found only a single energy minimum in all structures considered. The binding energy (BE) is obtained at this minimum as

BE = E_surface + E_benzene − E_{surface+benzene} (3)

where E_surface, E_benzene, and E_surf+benzene are the energies of the surface, the benzene molecule and their compound, respectively.

With these settings we calculate a binding energy of 591 meV using TS09 and 429 meV using vdW-DF for the interaction of the benzene molecule with pristine graphene in the standard AB stacking configuration (Fig. 1, sub-graph A). These values are in good agreement with the literature.³⁶ We have also checked for our unit cell that the TS09 dispersion attraction between benzene molecules in two neighboring cells is less than 1 meV. Our simulations show a gradual deterioration of the binding if benzene is tilted with respect to the pristine graphene surface, reaching 40% (50%) weaker TS09 (vdW-DF) binding when the molecule is perpendicular to the surface. Therefore, the parallel alignment provides the strongest binding among all valid geometries. This finding can be related to graphite's extended structure in contrast to the case of benzene dimers (and solid benzene) where the T-shaped and slipped-parallel configurations are very near in energy.³⁷

Next we consider the binding of benzene to graphene with defects. The stability of the considered defects after benzene adsorption is questionable and deserves further consideration. Therefore we have performed full structural optimization of the compound structure (defect plus benzene). When the benzene molecule is adsorbed on a given substrate, the substrate slightly bends in response. This bending breaks the symmetry and guides the substrate to a more energetically stable state (Fig. 2). We found that such relaxation was only important for two defect families: B (Stone–Wales) and C (single vacancies). The symmetry broken optimized configurations after removing the benzene molecule are shown in Fig. 2. This bending leads to a TS09 energy gain of 0.56 eV in B and 0.22 eV in C relative to the flat configuration. The vdW-DF predicts higher energetic gains, 1.01 eV for B and 0.33 eV for C, respectively. The difference between the functionals can be attributed to the different optimized C–C bond lengths, 1.423 Å in TS09 and 1.429 Å in vdW-DF. Therefore, the structures can further relax in vdW-DF than in TS09 when departing from the experimental bond length of 1.42 Å in the flat configuration.

Fig. 2 Ground state structures of graphene with a Stone–Wales (B) and a single vacancy (C) defect. Colors represent atomic heights in Å relative to the carbon at the lower left corner. The TS09 results are reported, but the vdW-DF relaxed structures are very similar.

One might wonder whether the binding of benzene to graphene with defects is purely van der Waals (i.e. mediated by mutually induced dipoles) or could the interaction of static dipoles of functional groups containing oxygen with an induced dipole on the benzene significantly contribute to the binding. The latter mechanism is already adequately covered by local and semi-local functionals. In order to clarify the nature of the binding we also performed GGA calculations without van der Waals corrections. In this case, the binding energy of the benzene to graphene with defects vanished, which shows that static dipole contributions are not sufficient to explain binding.³⁶

The main results of our study of the binding energy of graphene with defects are summarized in Fig. 3a (the explicit binding energies can be found in Table S2, ESI†), where the case of pristine graphene is highlighted. The BE of vdW-DF is systematically lower than the value obtained in TS09, but both clearly show the same energetic ordering of the different defect types and are in good qualitative agreement with each other. The most severe qualitative difference is observed for support B as a consequence of its deformed structure.

Fig. 3 (a) Overview of the interaction energies of benzene with pure graphene (A1) and a group of its derivatives with defects, (b) the corresponding equilibrium adsorption height between the benzene molecule and the surface.

We can distinguish roughly between two groups of defects: the first group contains in-plane defects (represented by families A to F) and shows BE comparable to binding energy on pristine graphene. The second group contains defects that have spatial extension outside the plane, represented by the families G and H and have reduced interaction energy as compared to pristine graphene (only ∼32% in TS09 and ∼40% in vdW-DF). The strong decrease in binding energy is understandable as the protrusion of the defect hinders better van der Waals interactions with the rest of the surface. The exceptions from this rule are G3, G4, H3 and H4 where the benzene molecule is below the graphene plane, and hence is not sterically hindered by the protrusion.

We find some structures (G3, G4, F1, F2, B3, B4, H3, and H4) which show a slightly enhanced adhesion (below 10%), relative to graphene itself. In G3, G4, H3 and H4 the benzene is deposited on the concave surface of an epoxy/hydroxyl-induced graphitic bulge. The enhanced interaction with the bulge can be explained by an improved coordination of the benzene by the surrounding bulging carbon atoms resulting in an enhanced van der Waals interaction. Similar effects can be seen near other curved graphitic structures. Fullerenes, for instance, bind better to the insides of carbon nanotubes (to the walls or inside the caps) than to their exterior.³⁸ Another example is a curved graphene nanoribbon, where benzene binds stronger on the concave than on the convex side (see ESI†). Such curvatures, whether local or extended, have increasing importance as recent experimental results suggest that the flat state is not the ground state of graphene in graphene-loaded polymer matrices,³⁹ where graphene is unstable with respect to folding and rolling.

Fig. 3b shows the adsorption height (AH) at the minimal energy. The AH is defined as the vertical distance of benzene to the carbon at the border of the unit cell, i.e. at the lower left corner as depicted in Fig. 2. Also here TS09 and vdW-DF show the same trend that is generally caused by geometrical constraints, with a systematically larger AH within vdW-DF. Interestingly, the difference between the predictions of the two functionals decreases with increasing AH.

In order to quantitatively estimate the influence of the defect density on the effective benzene–graphene binding, we use the epoxy ad-group as a representative case (configuration group G). Fig. 4 shows the binding energies when the benzene molecule is moved parallel to the substrate at constant height h across the epoxy group. At the height of maximum interaction over the epoxy oxygen (h = 4.9 Å in TS09 and 4.93 Å in vdW-DF) the binding energy is almost constant throughout the whole unit cell (200 meV for both TS09 and vdW-DF). Remarkably, the oxygen atom itself does not significantly add to the interaction with benzene. This is in line with our previous conclusion that reduced binding can be fully explained by steric hindrance: the protrusion formed by the oxygen atom prevents a closer approach of benzene towards the graphene surface resulting in larger R_AB in eqn (1) and consequently in weaker van der Waals binding. At a lower height (optimal for pristine graphene–benzene interaction, h = 3.58 Å for vdW-DF and h = 3.31 Å for TS09), the epoxy group suppresses the attraction over a range of roughly 6.5 Å. Therefore, assuming that each epoxy atom strongly reduces the benzene–graphene interaction in an area of A = π(6.5/2 Å)² ≈ 33 Ų, we estimate that a substantial decrease of van der Waals interaction can already be expected at 16% oxygen coverage (assuming that at 100% defect density the number of oxygen atoms is equal to the number of carbon atoms – pristine graphene contains 0.38 carbon atoms per Ų and we consider blocking of both sides of the sheet).

Fig. 4 Binding energy of benzene on substrate G depending on the lateral position at given constant height h. The epoxy-group oxygen atom is located at the origin and x denotes the position of benzene's center of mass on the path crossing the epoxy oxygen. The arrow indicates the effectively blocked area of 6.5 Å diameter.

Conclusions

In this article we report first insights into dispersion force mediated interactions between graphene with defects and benzene. According to two recent vdW functionals, the dispersion binding of graphene with defects is generally similar to the pristine case. While the two functionals predict different binding energies, they are in good qualitative agreement. The main influence of the defects on the van der Waals binding is given by geometric constraints. Defect related protrusions hinder optimal atom–atom interactions and result in reduced binding, while concave curvatures increase binding slightly. The similar binding strength between graphene with defects and pristine graphene suggests that the binding of graphene to polymers via dispersion forces is only marginally affected by defects provided defect densities are sufficiently low. This conjecture is supported by a simple model that predicts a substantial decrease of van der Waals binding for defect densities exceeding roughly 15%. Therefore, the stability of nanocomposites such as filler–polymer hybrids should not deteriorate significantly if pristine graphene is replaced by sufficiently reduced RGO as a filler, but would decrease considerably by using GO as a filler.

M.W. thanks P. Hyldgaard for useful discussions. The authors gratefully acknowledge financial support of this project by the German Federal Ministry of Education and Research (BMBF) within the “FUNgraphen” project (project 03X0111C). Calculations have been performed at NIC Jülich.

Notes and references

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Footnote

† Electronic supplementary information (ESI) available: Detailed comparison of the van der Waals energetics with literature values, as well as energetics and structures of all the configurations shown in Fig. 2. See DOI: 10.1039/c3cp53922a

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