An Analytical Model for Adsorption and Diffusion of Atoms / Ions on Graphene Surface

Theoretical investigations are made on adsorption and diffusion of atoms/ions on graphene surface based on an analytical continuous model. An atom/ion interacts with every carbon atom of graphene through a pairwise potential which can be approximated by the Lennard-Jones (L-J) potential. Using the Fourier expansion of the interaction potential, the total interaction energy between the adsorption atom/ion and a monolayer graphene is derived. The energy-distance relationships in the normal and lateral directions for varied atoms/ions, including gold atom (Au), platinum atom (Pt), manganese ion (Mn), sodium ion (Na), and lithium-ion (Li), on monolayer graphene surface are analyzed. The equilibrium position and binding energy of the atoms/ions at three particular adsorption sites (hollow, bridge, and top) are calculated, and the adsorption stability is discussed. The results show that H-site is the most stable adsorption site, which is in agreement with the results of other literatures. What is more, the periodic interaction energy and interaction forces of lithium-ion diffusing along specific paths on graphene surface are also obtained and analyzed. The minimum energy barrier for diffusion is calculated. The possible applications of present study include drug delivery system (DDS), atomic scale friction, rechargeable lithium-ion graphene battery, and energy storage in carbon materials.


Introduction
In recent decade, graphene has been paid much attention as an ideal candidate for the development of high-sensitivity sensors [1][2][3], drug delivery system (DDS) [4][5][6][7], rechargeable lithium-ion graphene battery [8][9][10], lithium storage in carbon materials [11][12][13][14][15], and so on, due to its unique characteristics, such as the thinnest known 2-dimensional nanomaterial, the lowest resistivity, and extreme mechanical strength.The characteristics of graphene have been widely investigated in the field of biology, chemistry, and physics.However, there is still a common unsolved major scientific question for its applications.It is to develop an analytical model to quantitatively describe the adsorption and diffusion of atoms/ions on graphene surface, which can be further applied for the interaction between nanoparticles or atomic force microscopy (AFM) tip and graphene surface.
For this problem, lots of theoretical and experimental researches have been made, and some interesting research results are achieved.In atomic and nanoscale, general theoretical research methods include density functional theory (DFT), first principle, and molecular dynamics (MD) simulation.A systematic density functional study was performed on the adsorption of Cu, Ag, and Au adatoms on pristine graphene, especially accounting for van der Waals (vdW) interactions by the vdW-DFT and Perdew, Burke, and Ernzerhof plus long-range dispersion correction (PBE + D2) methods [16].According to the differences of the total energies at the three adsorption sites, they found that the diffusion of Cu and Au took place along carboncarbon bonds, while the Ag adatoms could diffuse almost unrestrictedly on graphene sheet.Using density functional theory, theoretical calculations were also carried out for the diffusion of Si and C atoms on a monolayer graphene surface and between the multilayer graphene layers [17].The results showed that the diffusion barrier for a Si atom on monolayer graphene surface was extremely small, and the diffusion of Si atom between bilayer graphene was also almost barrier-free.However, the diffusion of a C atom on monolayer graphene was far more difficult than that of the Si atom.By the same methods, the adsorption of Li atom on graphene was investigated [9].Three different adsorption sites were taken into consideration.According to their researches, the hollow site was found to be the most stable, followed by the bridge and top sites.Moreover for Li atom, the minimum diffusion energy path where Li atom migrates from a hollow site to a neighboring hollow site should be a path through the bridge site between them.What is more, the adsorption, diffusion, and desorption of hydrogen on graphene were studied by DFT method [18].They concluded that the atomic hydrogen adsorbate produced by bombarding the surface with an atomic beam was not possible under equilibrium conditions, and below room temperature diffusion was frozen out and a beam-deposited atomic hydrogen adsorbate could be maintained, though completely out of equilibrium.
Apart from density function theory method, MD simulation methods have also been widely used in the study of graphene interacting with atom/ion.The diffusion processes of the lithium-ion on a fluorinated graphene (F-graphene) surface were investigated by means of a direct molecular orbital-molecular dynamics (MO-MD) method [19].It was found that the diffusion coefficient of lithium-ion on Fgraphene surface was close to that of normal graphene (Hgraphene), but the thermal behavior on the F-graphene surface was much different from that on the H-graphene surface.The direction of lithium-ion diffusion could be controlled by fluorinated substitution (F-substitution) on the graphene sheet around room temperature.But at higher temperatures, the lithium-ion diffused freely on surface and edge region of F-graphene.The interaction of Mg atom with graphene was also researched by using MD simulations [20].It was found that the Mg atom vibrated strongly on the graphene surface; even at high temperature (1000 K), the diffusion of Mg is difficult.
There are relatively few experimental researches on adsorption and diffusion of atoms/ions on graphene surface.The viability of utilizing functionalized graphene oxide with good solubility and stability, biocompatibility, and targetspecificity was demonstrated as a drug nanocarrier for controlled loading and folate receptor-targeted drug delivery of anticancer drugs [21].It revealed that mastering the stability of equilibrium for graphene was vital to investigate a new drug delivery system for cancer therapy.The possibility of higher lithium storage capacity was explored by controlling layered structures of graphene nanosheet materials [12].It enlightened that the spacing equilibrium distance of atom/ion on and between graphene sheets is critical for the future large-scale, high-capacity lithium-ion batteries.The diffusion of a cobalt bis-terpyridine, Co(tpy) 2 -containing tripodal compound (1•2PF6), noncovalently adsorbed on the surface of a single-layer graphene (SLG), was studied by scanning electrochemical microscopy [22].It was proved that surface diffusion is the main mechanism for the observed decrease in the electrochemical response due to radial diffusion of the adsorbed molecules outward from the microspots onto the unfunctionalized areas of the SLG surface.
In order to quantitatively understand atomic adsorption and diffusion on graphene surface from the perspective of atomic scale, an analytical continuous model is put forward in this work based on the Fourier expansion and Lennard-Jones potential.With the theoretical model, equilibrium position, binding energy, and adsorption stability of the atoms/ions (Au, Pt, Mn 2+ , Na 1+ , and Li 1+ ) on monolayer graphene surface are calculated and analyzed.The different metal atoms/ions selected in this work possess many application prospects.For example, Mn 2+ is an ideal candidate for drug delivery and Li 1+ is a common element for high-energy density rechargeable alkali batteries.The dependence of diffusion energy and diffusion forces on the path is illustrated.

Theoretical Model
A monolayer graphene is assumed to be perfect, flat, and infinite compared to atomic scale (as shown in Figure 1(a)).Due to its periodic honeycomb lattice structure, a twodimensional lattice vector parallel to graphene can be defined [23][24][25]: where  1 and  2 are integers and a 1 and a 2 are the unit lattice vectors in the plane (as shown in Figure 1(a)).It is evident to know that the translation of a single adsorption atom by l will take it from one position above a surface lattice cell into an equivalent position above another different lattice cell (as shown in Figure 1(a)).Therefore, the total interaction potential (r) takes the form of where r is the position of the adsorption atom with respect to the graphene surface.
Thus, the periodic nature of the total interaction potential (r) is a Fourier series: where  is perpendicular distance between the adsorption atom and graphene surface,  is the two-dimensional translation vector, and g is the multiples of the reciprocal lattice vectors b 1 and b 2 : where  1 and  2 are integers and b 1 and b 2 are defined by Equation (5) indicates that b 1 and b 2 are perpendicular to a 2 , a 1 and of length csc / 1 , csc / 2 , respectively, where  is the angle between a 1 and b 1 .  () in ( 3) can be expressed in the following forms: where  is the area of unit cell,   (,  + l + m  ) is the partial interaction potential between the single atom and the carbon atom in graphene plane,  indicates the th atom in a unit cell, and the position of the atom in the plane is given by l + m  and t =  + l + m  .
Commonly, the average interaction potential per atom/ion on graphene is defined by the total energy of an atom/ion, a sheet of graphene, and the graphene with atom/ion adsorption, which consider the effects of temperature, chemical bond, charge transfer, van der Waals interaction, and so forth [9,10,26,27].In this paper, for the convenience of discussion, the interaction potential is only defined by van der Waals interaction.Even so, the model in this paper is also suitable for taking into account the effects of other factors mentioned above.Two empirical potentials commonly used are the L-J potential and the Morse potential.By referring the reader for detail of the Morse potential and its applications [28,29], this paper adopts the form of 6-12 L-J potential to determine the van der Waals interaction potential between the adsorption atom and the th carbon atom in the graphene, where r  , , and  denote the distance between the adsorption atom and the th carbon atom in the graphene, the potential well depth, and the distance when the interaction potential is equal to zero, respectively.Then, where  = 2 is the number of the carbon atoms in the unit cell and  2 and  5 are the modified Bessel function of the second kind.Substitute (8) into (3), and then the total interaction potential (r) can be derived as where  is the magnitude of g, As shown in Figure 1(a), when  is chosen to be zero at a hollow point, and the unit cell which contains two carbon atoms is marked by a rhombus, then m  for the two atoms in the unit cell indicated in Figure 1(a) are located at (1/3, 1/3), (2/3, 2/3), where  is the distance between the centers of two nearest hexagons.Summation over m  gives a value of cos[(2/3)( 1 +  2 )], so (9) can be further written as Because (, ) in ( 11) decreases rapidly with the increase of , and the -dependent terms are important only over a short range of distance  [23,25], there are approximately simplified expressions of |g| = 4/ √ 3 and cos[(2/3)( 1 +  2 )] = −1/2.As shown in Figure 1(a),  and  coordinates are marked;  = x + y can be expressed in terms of a 1 and a 2 with Thus, the total energy between an adsorption atom/ion and graphene can be derived as If we introduce the following dimensionless variables  * = / 2 ,  * = /,  * = ,  * = /,  * = /,  * = /, and  * = /, then the dimensionless form of total energy can be written as where dimensionless parameters , , and  have the following forms: Furthermore, let F * = −∇ * (r * ), and then the normal and lateral interaction forces between the atom/ion and graphene can be obtained in the Cartesian coordinate system as shown in Figure 1(a), In the same way, the lateral interaction force between the atom/ion and graphene in the polar coordinate system can be derived, where  * and  are dimensionless polar coordinates and  * coincides with the  * -axis when the angle  = 0.

Results and Discussions
Equation (13) describes the interaction potential between a single adsorption atom/ion and graphene via the L-J potential and the first-order approximation of Fourier expansion.Obviously, it is dependent on the position of the adsorption atom/ion on graphene and the parameters of L-J potential which is varied for different atom/ion.In this work, several kinds of metal atoms and ions are selected, including Au, Pt, Mn 2+ , Na 1+ , and Li 1+ .If the L-J potential parameters for these atoms/ions are specified [30][31][32], the interaction potential of these atoms/ions can be obtained in terms of the position vector above the graphene surface.For a lithium-ion, the L-J potential parameters can be specified as  * = 1.005 and  * = 0.983.The variations of interaction potential between the lithium-ion and graphene are illustrated in Figure 2 with respect to the positions in the and -directions, respectively.It can be seen from Figure 2 that the potential curves are periodic in both directions, and there are three periodic equilibrium positions, corresponding to three particular adsorption sites of lithiumion on graphene surface.They are hollow (on top of a hexagon, H-), top (on top of a C-C bond, T-), and bridge (on top of a carbon atom, B-) sites (as shown in Figure 1(b)), respectively.At three adsorption sites, the equilibrium height and interaction potential at the equilibrium which can be called binding energy are different.Figure 3 shows the variations of interaction potential of the lithium-ion at three adsorption sites along the -direction.Although the variation trends are almost identical for the three sites, the equilibrium height and binding energy are different.The equilibrium height and binding energy at the H-site are the lowest, followed by those at the B-site, and those at the T-site are the highest.
Furthermore, via the equilibrium equations of lithiumion in the -direction,  *  ( * ,  * ) = 0, the equilibrium height  0 at three adsorption sites can be calculated, which is 0.241717 nm, 0.248800 nm, and 0.249502 nm, respectively.Compared with the results in the literature [30], there is a good agreement.The equilibrium height is very important for the design of lithium-ion battery based on graphene electrode materials, which can be used in predicting the minimum interlayer spacing 2 0 of two parallel graphene sheets so that the embedded Li 1+ undergoes no net force.According to our theoretical results, the minimum interlayer spacing for Li 1+ should be approximately 0.48-0.50nm.With the equilibrium heights at three adsorption sites, the binding energy  0 of lithium-ion can also be calculated.The minimum value of binding energy is −0.04059 eV at the H-site, subsequently −0.03746 eV at the B-site, and the maximum value is −0.03717 eV at the T-site.Thus, H-site is the most stable adsorption site, and T-site is the most unstable site.
Similarly, when the L-J potential parameters of other atoms/ions are given, the variations of interaction potential of these atoms/ions can be illustrated with respect to the positions in the -, -, and -directions, respectively (as shown in Figure 4).It can be found that the variation trends of interaction potential are almost the same in three directions for these atoms/ions.Three adsorption sites (H-, B-, and T-site) exist for every atom/ion on graphene surface.The equilibrium height and binding energy of these atoms/ions can be calculated and listed in Tables 1 and 2. Similar conclusions can be drawn that the order from low to high for both equilibrium height and binding energy is H-site, B-site, and T-site and H-site is the most stable adsorption site.In addition, the equilibrium height and binding energy of every atom/ion are different.Obviously, Pt atom adsorbed at the Hsite is the most stable among the atoms/ions selected in the work.
As the discussions above, the interaction potentials between adsorption atoms/ions and graphene are not constant but dependent on their positions on graphene surface.For different adsorptions sites, the energy for desorption of these atoms/ions is different.When an atom/ion migrates from an adsorption site to another, it must get over an energy barrier.The energy or force to drive the atom/ion is defined as   Figure 5: The variations of (a) interaction potential and (b) interaction force between a lithium-ion and graphene for different diffusion directions, 0 ∘ , 15 ∘ , 30 ∘ , and 60 ∘ angles from the -axis.
diffusion energy or diffusion force.Obviously, the diffusion energy and diffusion force are dependent on the diffusion path.
For a lithium-ion, the periodic interaction potentials along the and -directions are illustrated in Figure 2. The period is  in the -direction and √ 3 in the -direction.If its initial adsorption site is assumed to be H-site, in order to migrate along the -direction, a lithium-ion must get over an energy barrier between H-site and B-site, and the minimum diffusion energy is 0.00313 eV.In the same way, in order to migrate along the -direction, a lithium-ion must get over an energy barrier between H-site and T-site, and the minimum diffusion energy is 0.00342 eV.Similar results can also be obtained for other atoms/ions according to the periodic interaction potentials illustrated in Figure 4 and the binding energies listed in Table 2.
The initial adsorption site is still assumed to be H-site, and a lithium-ion is assumed to migrate along different directions, such as 0 ∘ , 15 ∘ , 30 ∘ , and 60 ∘ angles from the -axis.The dimensionless interaction potential and interaction force between the lithium-ion and graphene for different diffusion directions are illustrated in Figure 5.It can be seen from these figures that the periods of interaction potential and interaction force are varied for different diffusion directions, and the minimum diffusion energy and diffusion force are also varied; that is, the diffusion of atoms/ions on graphene surface is path-dependent.Simultaneously, it can be found that along the same lattice orientation, such as 0 ∘ and 60 ∘ angles, the variations of interaction potential and interaction force are nearly the same.These phenomena have been verified by the atomic scale friction experiments of AFM tip on graphene surface [33][34][35].
Furthermore, the diffusion from different initial positions along and -directions is investigated.The dimensionless interaction potential and interaction force of a lithiumion from different initial sites are illustrated along and -directions in Figures 6 and 7.In two directions, the periods of interaction potential and interaction force have path-independent period,  and √ 3, respectively.However, for different diffusion paths along or -direction, their amplitudes are varied; that is, the minimum diffusion energy and diffusion force are varied.

Conclusions
By performing theoretical analyses of interaction potential and interaction force for a single atom/ion on graphene surface based on an analytical model, the mechanisms of adsorption and diffusion are revealed.The equilibrium height and binding energy on three particular adsorption sites are calculated, and the adsorption stability is discussed.The order from low to high for both equilibrium height and binding energy is H-site, B-site, and T-site, and Hsite is the most stable adsorption site.The variations of interaction potential and interaction force of lithium-ion diffusing along some specific directions on graphene surface are investigated, and it can be concluded that the diffusion of atoms/ions on graphene surface is path-dependent.The minimum diffusion energy and diffusion force along different diffusing path can be calculated by the analytical model.
The studies on the path-dependent diffusion of atoms/ions on graphene surface are beneficial to improve  the efficiency of drug delivery based on graphene, highperformance rechargeable lithium-ion batteries, and lithium storage in carbon materials, which can be further applied for the interaction between nanoparticles or AFM tip and graphene surface.Of course, there is a little inadequacy in present studies.For example, the graphene is assumed to be infinite, perfect, and flat; moreover, the influence of temperature is not taken into account.In fact, edge effect, temperature, defect and corrugation of graphene, and so forth can affect the adsorption and diffusion of atom/ion [30,[36][37][38][39][40].Therefore, these factors will be considered in our future works.

Figure 1 :
Figure 1: Schematics of the graphene where (a) the rhombus connecting the centers of four adjacent hexagonal is selected as a twodimensional primitive unit cell; (b) hexagon with three particular adsorption sites: hollow (on top of a hexagon, H-), top (on top of a C-C bond, T-), and bridge (on top of a carbon atom, B-).

Figure 2 :Figure 3 :
Figure 2: The variations of interaction potential between the lithium-ion and graphene in the (a) -direction and (b) -direction, respectively.

Figure 4 :
Figure 4: The variations of interaction potential of several kinds of metal atoms/ions along the (a) -direction, (b) -direction, and (c) -direction, respectively.

Figure 6 :
Figure 6: The variations of interaction potential between a lithium-ion and graphene from different initial sites along (a) -direction and (b) -direction.

Figure 7 :
Figure 7: The variations of interaction force between a lithium-ion and graphene from different initial sites along (a) -direction and (b) -direction.

Table 1 :
Equilibrium height  0 of atoms/ions on the three particular adsorption sites.

Table 2 :
Binding energy  0 of atoms/ions on the three particular adsorption sites.