Modelling the interaction of graphene oxide using an atomistic-continuum model

Abstract

In this paper, we construct a continuum model for graphene oxide based upon the Lerf–Klinowski structure to investigate the interaction forces between sheets of graphene oxide. We use the Lennard-Jones potential and coulombic potential to determine the total potential energy between sheets of graphene oxide. We analytically calculate the interaction forces within the system using sums of hypergeometric functions. Our model is then modified to investigate different levels of hydration and oxidation within the system. Our investigations are reconstructed using the LAMMPS molecular dynamics simulator and we find that the analytical solution quickly and effectively calculates results that match well against our simulation data and values taken from literature.

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I. Introduction

Graphene science is a rapidly growing field with a plethora of potential applications including composites, energy storage and conversion, nanoelectronics, transparent conducting films and multifunctional 3D structures.^1–6 The chemical exfoliation of graphite is the most practical method for the production of graphene. In this method, a graphene dispersion is obtained through the oxidation of natural or synthetic graphite to give graphite oxide that can be exfoliated in water or organic solvents to generate single sheets of graphene oxide (GO).^7–10 This dispersed GO can then be utilised to fabricate devices directly or via composite formation, followed by reduction to graphene.^11,12 In this regard, an understanding of the interactions between GO sheets offers a significant advantage to extend our ability to readily process and fabricate them into a wide variety of structures using a number of different 3D fabrication techniques.¹³ The oxidation process causes the disruption of sp² networks in the structure of GO. As a result of this, GO has little thermal or electronic conductive properties.¹⁰ However, GO is hydrophilic and can be easily dispersed in water and many organic solvents. This high solubility suggests a use in medical applications, notably in drug delivery.¹⁴

When graphene is oxidised to form GO, various functional groups and deformations are introduced into the graphene basal plane. Some of the bonds between carbon atoms are stretched and distorted by the presence of the functional groups which displace the attached carbon atoms slightly out of the plane forming an sp³ hybrid carbon structure.¹⁵ Graphene oxide (GO) consists of a basal plane of sp² carbon with oxidated functional groups decorating both sides of the structure.¹⁰ The exact chemical structure of GO has long been the subject of debate and no definitive model exists.¹⁰ Originally, GO models were constructed as a periodic, regular lattice structure. The Hofmann model consists of epoxy groups spread over the basal plane with a general formula of C₂O. This model was later modified by Ruess who included hydroxyl groups into the structure to account for the hydrogen content of GO, as well as modifying the hybridisation of the basal structure to an sp³ system. The Scholz–Boehm model removed the epoxide groups.

In 1998, Lerf et al. constructed a model, as shown in Fig. 1, where the main functional groups attached to the basal plane consist of epoxy (C–O–C) and hydroxyl (C–OH) groups with carboxyl and carbonyl groups attached to the edges of the carbon structure.^16,17 Unlike earlier structures, the Lerf–Klinowski model is non-periodic. This model of GO is nonstoichiometric and predominantly amorphous due to distortions from the high fraction of sp³ C–O bonds.¹⁵ The Lerf–Klinowski model has become the most widely accepted model for GO in recent times and further nuclear magnetic resonance (NMR) experimental results support the model.^10,18,19 In constructing the model, Lerf et al. performed several investigations on the hydration properties of GO. This paper aims to replicate these experimental results using analytical techniques.

Fig. 1 The Lerf–Klinowski model.¹⁰

In this paper, we investigate the interaction forces between sheets of GO. We adopt the Lerf–Klinowski structural model to construct a surface representation of GO using a series of continuous and flat disk surfaces. The interaction energy (E) is defined as the total energy induced between our molecules by van der Waals and electrostatic forces. All other forces are considered trivial. The van der Waals force is calculated using the Lennard-Jones potential function and the electrostatic force using the Coulomb potential. We successfully analytically calculate the interaction forces within the system using the sums of the hypergeometric functions. These equations are then applied to sheets of GO under different conditions of both oxidation and hydration. The composition of the GO sheets is varied so we are able to view the change in the energy given different levels of oxidation. We also simulate hydration by inserting monolayers of water between sheets of GO. We find that the analytical solution quickly and effectively calculates results that match well against molecular dynamic simulation data and values taken from literature.

II. Modelling approach

Using the continuum approach for modelling nanostructures as demonstrated by Cox et al.²⁰ we construct our model of GO from a series of continuous sheets. Firstly, we detail the structure of the functional groups used to construct the model, then we determine the interaction energy between two sheets of the GO model based on the van der Waals force, represented by the 6–12 Lennard-Jones potential, and the electrostatic potential, represented using the coulombic point-charge.

A. Structure of functional groups

We use the Lerf–Klinowski model to represent our GO structure. As discussed earlier, the hexagonal lattice of carbon atoms comprising the base of the GO structure is distorted but still largely intact. Consequently, we ignore these distortions in our model of GO and assume the carbon atoms are structured like an unaffected graphene sheet.

The energies of a GO sheet are greatly reduced when epoxy and hydroxyl groups are gathered together in one area.²¹ However despite this, the arrangement of the functional groups on the GO sheet still largely depends on the method and conditions involved during synthesis.¹⁸ We follow studies by Mkhoyan et al. which suggest the functional groups form a uniformly random distribution on both sides of the sheet.¹⁵

We use the structure of the epoxy and hydroxyl groups from the DFT study performed by Yan et al.²¹ to derive the positions of the oxygen and hydrogen atoms relative to the carbon plane. An epoxy functional group is constructed of a single oxygen atom bonded to two neighbouring carbon atoms. The C–O bond length at relaxation is given to be 1.44 Å and the two attached carbons C–C bond length is stretched to 1.51 Å. Further distortions include the carbon atoms moving out of the plane by 0.34 Å. From this information, we deduce that the oxygen atoms lie at a perpendicular distance of (1.23 + 0.34)/2 = 1.57 Å from the carbon basal plane. A hydroxyl functional group is constructed of a OH group bonded to a carbon atom. The C–O bond length at relaxation is given to be 1.44 Å, the O–H bond length is given as 0.98 Å and the C–O–H bond angle is 107.9°. The attached carbon atom is distorted out of the plane by 0.37 Å. For simplicity, we assume the hydrogen and oxygen atoms lie in the same plane perpendicular to the basal plane. Then, we find the hydrogen and oxygen atoms lie at perpendicular distance of 1.37 + 0.37 = 1.74 Å from the basal plane. Taking the average of the C–O distance in the hydroxyl and the height of the epoxy group, we get a value of (1.57 + 1.74)/2 = 1.66 Å for the averaged height of an oxygen/hydrogen atom from the basal plane.

B. Continuum approach

We represent a single sheet of GO using a construction of three distinct, continuous, flat surfaces, each lying perpendicular to the same axis. The central surface represents a sheet of graphene, while the upper and lower surfaces represent the attached functional groups as illustrated in Fig. 2. The basal plane of carbon is assumed to be fundamentally identical to sp² graphene. The surfaces representing the functional groups are modelled as two surfaces of combined oxygen/hydrogen, representing the attached epoxy (C–O–C) and hydroxyl (C–OH) elements, positioned both above and below the carbon sheet and separated by a distance of 1.66 Å. We assume that in this continuous approximation, the atoms are distributed uniformly on each surface.

Fig. 2 A continuum model for a single GO sheet represented by a surface of graphene and two surfaces of mixed hydrogen and oxygen atoms.

The continuum approach of modelling discrete atomic structures as continuous surfaces has been used previously for calculating various van der Waals interactions between carbon nanostructures.²⁰ We aim to use this model to reduce the computational costs involved with the modelling interactions of GO structures that would be necessary when using a molecular dynamics (MD) simulation or any other discrete approach. We determine the interaction energy between two atoms on distinct molecules using the Lennard-Jones potential function which incorporates both the van der Waals attraction force and the Pauli repulsion force between the molecules. The 6–12 Lennard-Jones potential function is written in the form, ϕ(ρ) = −A/ρ⁶ + B/ρ¹², where ρ is the distance between the two atoms and A and B are the appropriate attractive and repulsive Lennard-Jones constants. The presence of the oxygenated groups introduces a charged potential between the two sheets so electrostatic forces must also be considered. This electrostatic potential between two discrete, charged atoms is calculated using the Coulomb potential which is defined as, q₁q₂/4πε₀ρ, where q₁ and q₂ are the partial atomic charges on each atom, ε₀ is the permittivity constant and ρ is the distance between the two atoms.

The total non-bonded interaction energy (E) is determined by summing all of the interactions between every pair of atoms. That is,

where i and j iterate over all the atoms on each molecule respectively, q_i and q_j are the charges of the i-th and j-th atoms respectively and ρ_ij is the distance between the i-th and j-th atoms. However these calculations become quite computationally costly when dealing with large numbers of atoms. Using continuum methods as in ref. 20, our atoms are assumed to be evenly distributed over the surface. Thus, we calculate the energy between any two surfaces as,where η₁ and η₂ are the uniform surface densities of the atoms on the surfaces S₁ and S₂ respectively, where q_S1 and q_S2 are the average charges of the surfaces S₁ and S₂ respectively and ρ is the distance between the two surfaces.

The total interaction energy between two sheets of GO is the sum of all surface interactions. These interactions consist of the interaction between the two graphene surfaces (E_G–G), the total interaction between a graphene surface and the oxygen/hydrogen surface on different GO sheets (E_G–OH) and the total interactions between each pair of oxygen/hydrogen surfaces on different GO sheets (E_OH–OH). We define E = E(z) so we calculate the variation in interaction energy as the distance between the two sheets varies along the z-axis as shown in Fig. 3. Summing all our interactions gives us

E(z) = E_G–G(z) + 2E_G–OH(z + δ) + 2E_G–OH(z − δ) + 2E_OH–OH(z) + E_OH–OH(z + 2δ) + E_OH–OH(z − 2δ),

where z is the distance between the carbon planes on the two GO sheets and δ is the distance between the carbon plane and the oxygen/hydrogen planes.

Fig. 3 Interaction between two GO sheets. Here, z is the distance between the sheets and δ is the distance between the C and OH layers.

The various constants used in our model are now defined. Firstly, the Lennard-Jones constants are taken from the Universal Force Field (UFF) atomic potentials²² and are shown in Table 1. In the cases where we must determine the Lennard-Jones constants for a pair of different atoms, we use the Lorentz–Berthelot mixing rules defined as ε_xy = (ε_xε_y)^1/2 and σ_xy = (σ_x + σ_y)/2.

Table 1 Lennard-Jones constants

	C	O	H
ε (eV)	0.0045	0.0026	0.0019
σ (Å)	3.851	3.500	2.886
A (eV Å^−6)	29.45	9.49	2.19
B (eV Å^−12)	48 033.03	8718.39	631.65

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The point charges required to measure the coulombic potential are taken from the atomic point charge model calculated by Stauffer et al. for the electrostatic potential (ESP) of GO from ab initio calculations.²³ We use the simple ESP charges defined in ref. 23 and shown in Table 2, which also includes the ESP of water which is necessary for later use in Section D. We use the permittivity constant for water ε₀ = 1.3245 × 10⁻⁷ F m⁻¹.

Table 2 Electrostatic potential of charged groups on graphene oxide

	Atom	ESP
Water	O	−0.56e
	H	+0.28e
Epoxy	O	−0.24e
	C	+0.12e
Hydroxyl	O	−0.38e
	C	+0.12e
	H	+0.26e

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The density of our carbon graphene surface is taken to be η = 0.3812 Å⁻².²⁰ We then take the density of the oxygen/hydrogen surfaces from the C/O ratio of the GO sheet which vary given the oxidation of the sheet. Note that the comparative density of the oxygen and hydrogen atoms must be halved as they are distributed over two surfaces.

Recall from eqn (1) that the interaction energy between two surfaces S₁ and S₂ is expressed as

This surface integral is now applied to the circular sheets of the GO model using polar coordinates.

An arbitrary point on each circular surface is represented in polar coordinates as (r₂ cos θ₂, r₂ sin θ₂, Z) and (r₁ cos θ₁, r₁ sin θ₁, 0). The distance between two arbitrary points is simply ρ² = (r₂ cos θ₂ − r₁ cos θ₁)² + (r₂ sin θ₂ − r₁ sin θ₁)² + Z². Due to the angular symmetry of the circular sheets, we fix θ₁ = 0. Then, taking θ = θ₁ to simplify our notation, we have

ρ² = (r₂ − r₁ cos θ)² + r₁² sin² θ + Z²

= r₁² + r₂² − 2r₁r₂ cos θ + Z².

As θ₂ is fixed to be constant, eqn (2) becomes

We simplify this integral by using hypergeometric functions, the details of which are explained in Appendix A. The final result for the interaction energy is

where
here F(a, b; c; z) defines an ordinary hypergeometric function and (x)_n represents the rising pochhammer symbol.

C. Effects of various oxidation levels on the interlayer distance between GO sheets

GO is a loosely defined material which can have varying structures and properties depending on the level of oxidation. Partial oxidation is considered to be thermodynamically favourable and chemically reduced forms of GO are often considered more useful in many applications.¹⁰ The degree of oxidation of GO varies depending on the synthesis conditions. The typical measure of the oxidation level is the ratio of carbon and oxygen atoms.²⁴ Generally, the range of oxidation varies from a C : O ratio of 4 : 1 to 2 : 1, although chemical reduction can give a ratio of 12 : 1 or higher.^24,25 Fully oxidised graphene oxide has around a 50 percent coverage of functional groups and an ideal composition of C₈O₂(OH)₂ or C₆O₁(OH)₂.²⁴

Different degrees of oxidation are simulated with the above GO model by varying the density of the oxygen/hydrogen layers. Here we show the interaction energy for three differently oxidised GO sheets (Fig. 4).

Fig. 4 The interaction energy between two graphene oxide sheets, shown at a C : O ratio of 2 : 1, 4 : 1 and 8 : 1.

We see that the functional groups are indeed instrumental for determining the interlayer distance and energy between GO sheets where the interlayer distance is the distance at which the interaction energy between the two layers is minimised. We also see clearly that the more oxidised a sheet of graphene oxide is, the further apart the interlayer distance and the stronger the binding energy between two sheets are. This interlayer distance matches well with that of dehydrated graphene oxide which has been experimentally measured at values from 5.9 to 6.7 Å.^26,27

As a comparison to the analytical result, a molecular dynamics simulation was run, the setup conditions of which are explained in Appendix C. A pair of periodic GO sheets were constructed and the system was then run until an equilibrium was reached. After the simulation was completed, as shown in Fig. 5, the averaged interlayer distance between the sheets was found as 4.7 Å. This is reasonably comparable to the value of 5.7 Å given by the equivalent analytical result. It is likely that the difference between the results is largely caused by the distribution of the functional groups on the surface.

Fig. 5 (a) A pair of 2 nm × 2 nm periodic sheets of GO shown after minimisation. (b) The same sheets displayed after a 1000 ps simulation. Both images are created using the Visual Molecular Dynamics (VMD) molecular visualization program.²⁸

D. Effects of hydration on the interlayer distance between GO sheets

GO is highly absorbent and tends to have some water molecules collected between the layers. We examine the effect of this intercalated water on the intermolecular distance by inserting a monolayer of water between the two GO sheets.

The oxygenated functional groups on the surface of GO cause it to have strongly hydrophilic properties.²⁵ Subsequently, it is very hard to remove all of the water from GO and intermellar water molecules are always present in the structure, even after prolonged drying.^29,30 Intercalated water has only marginal effects on the layer structure of GO sheets, however it causes the interlayer distance to increase to as high as 12 Å.^10,24

The hydrogen atoms in the water couple to the oxygen atoms in the functional groups via hydrogen bonds or electrostatic interactions.³¹ The electrostatic interactions between the oxygenated groups and water form an interfacial H-bond network, that is, the interaction between the layers is mediated by a network of hydrogen bonded water molecules.³² As GO becomes more hydrated, there is a gradual, linear increase in the interlayer distance. However, once the hydration of the GO sheets passes around 75 percent humidity, the interlayer distance between the sheets sharply increases in steplike transitions. This suggests the introduction of monolayers of water between the sheets.^27,31,33

We represent the intercalated water as a surface fixed directly in the centre of the two GO layers. The GO sheets are then allowed to interact as before, with the inclusion of additional interactions between each GO surface and the water surface.

E = E_GO + 2(E_G–W + E_OH–W).

We use the same values for the oxygen and hydrogen Lennard-Jones constants as we did for our GO sheet. We represent the water as a point on a flat surface. We approximate the area occupied by a molecule of water as the area of a circle with radius equal to the molecular distance of water at room temperature or 2.965 Å.³⁴ The density of water molecules on this surface is then equal to 1/(π × 2.965²) = 0.0362077 Å⁻². Consequently, the density of hydrogen is 0.024138 Å⁻² and the density of oxygen is 0.012069 Å⁻² (Fig. 6).

Fig. 6 The interaction energy between two graphene oxide sheets with layers of water intercalated between the sheets.

We observe upon adding each successive layer of water between the sheets, the interlayer distance increases by around 3 Å. This correlates with experimental results which suggest that the insertion of a water monolayer causes a jump of an equivalent degree in the interlayer distance.^27,33 Fully hydrated GO has sample values measured up to 12 Å in ref. 27 which is similar to the result given by adding two monolayers of water to the model. After two layers of water have been intercalated between the pair of sheets, the properties of any additional water is indistinguishable from bulk water, i.e. the GO sheets are not simply hydrated but are considered to be fully separated.³¹

A molecular dynamics simulation of GO was repeated, now with a single layer of water molecules distributed between the sheets. The interlayer distance between the sheets after equilibrium is given as 7.7 Å after the simulation is completed as seen in Fig. 7.

Fig. 7 (a) A pair of 2 nm × 2 nm periodic sheets of GO separated with water molecules shown after minimisation. (b) The same sheets displayed after a 1000 ps simulation.

III. Conclusions

This paper considered an analytical model for calculating the interaction energy between sheets of graphene oxide. Our model is then modified to investigate different levels of hydration and oxidation within the system. Our investigations are reconstructed using the LAMMPS molecular dynamics simulator and we find that the analytical solution quickly and effectively calculates results that match well against our simulation data and values taken from the literature. The techniques used to derive the energy are able to be used for any disk–disk type molecular interaction. Knowledge of the interlayer distance between sheets of graphene oxide with different hydration and oxidation levels would be useful for tailoring the materials for applications such as a carrier for drug-delivery and energy storage.

IV. Appendix

A. Interaction energy between two disks

We have our interaction energy between two disks aswhere

In order to solve this, we start by taking the integral

As cos θ is an even function, we have

Let K₁ = r₁² + r₂² + z² and K₂ = 2r₁r₂. Then

Using the identity cos θ = 1 − 2 sin²(θ/2), we get

We simplify our function by taking ω = θ/2. Now

We convert this integral into the form of a hypergeometric function by taking t = sin² θ and .

We use the quadratic transformation to convert our function into the more convenient form

We take

Taking α = r₂² + z² to simplify our working, we get

We solve this integral using the method in Appendix B to get

Finally, we must solve

We solve this integral using the method in Appendix B to get

which can be substituted into eqn (3) to get our interaction energy between two disks as required.

B. Integration method required for deriving interaction energy

We want to solve an integral with the form

Let , then x = (αt)^1/2 and . Our equation becomes

Let and to force our limits to the interval [0, 1]. We get

Let t′ = 1 − s and dt′ = −ds to convert our integral into a form that is easier to work with

We now write our integral in the form of a hypergeometric function

C. Molecular dynamics simulation setup

Molecular dynamics (MD) simulations were performed using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) MD simulator. A pair of graphene oxide sheets were modelled in a 2 nm × 2 nm simulation box. This system was repeated periodically to represent an infinite sheet of graphene oxide and water molecules were inserted between the two sheets. The number of epoxy and hydroxyl groups was kept the same and the groups were distributed randomly over both sides of each sheet to give a C : O ratio of 4 : 1. The water was simulated using the extended simple point charge (SPC/E) water model. The SHAKE algorithm was applied to the water to reduce high frequency vibrations of the hydrogen bonding. The interactions between the graphene oxide molecules were modelled using the reactive forcefield ReaxFF which is known to be accurate in modelling hydrocarbon nanostructures.³⁵ The water and GO interactions were calculated by the Lennard-Jones 6–12 potential between the oxygen and carbon atoms using ε = 0.0937 kcal mol⁻¹ and σ = 3.19 Å.³² All coulombic forces were calculated using a particle–particle particle–mesh (PPPM) solver. The simulations were performed under the NVT canonical ensemble at 300 K. Temperature control was managed using the Nose–Hoover thermostat. The system energy was first minimized using the conjugate gradient (CG) algorithm and was then run at a timestep of 0.5 femtoseconds.

V. Acknowledgement

Funding from the Australian Research Council Centre of Excellence Scheme (Project Number CE 140100012) is gratefully acknowledged by RJ. The Australian National Fabrication Facility (ANFF) Materials Node is acknowledged by RJ for their provision of research facilities.

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