Adsorption and Diffusion of Sodium on Graphene with Grain Boundaries

Effects of grain boundaries (GBs) in graphene on adsorption and diffusion of sodium were investigated using first principle calculations. Results showed that the presence of GBs in graphene enhanced the adsorption of sodium, with their adsorption energies in the range of -1.32~-0.79 eV, which were lower than the value of -0.67 eV for sodium adsorbed on pristine graphene. The diffusion energy barriers were in the range of 0.09 to 0.35 eV when sodium was diffused along GBs of graphene, whereas they were decreased when sodium was gradually diffused into the GBs. Results showed that graphene with GBs had a larger energy storage capacity for sodium than the pristine one, indicating that it can be used as a good anode material for sodium ion batteries. Keyword: Graphene; Grain boundaries; Adsorption and diffusion; Diffusion batteries; first principle calculations

It was also found that a monolayer of MoS2 with defects showed better adsorption and diffusion properties of lithium atoms compared to those of its perfect counterpart [31].
Due to the limited lithium resources, LIBs are insufficient for the increasing demands of energy storage [32,33]. As sodium is located below lithium in the periodic table and these two elements show similar chemical properties in many aspects, sodium ion batteries (NIBs) have recently received much interest to be used as a low cost alternative to LIBs [34]. Various 2D materials have been investigated to be used as the electrode materials for NIBs, and some of them showed better electrochemical properties as the electrodes for NIBs than those for LIBs [3, 6,35,36]. For example, the diffusion energy barriers for lithium and sodium atoms on silicene were found to be 0.21 and 0.12 eV, respectively [5]. The diffusion energy barriers for sodium atom diffusing along zigzag and armchair directions of monolayer black phosphorene were 0.04 and 0.38 eV, respectively [35,36], whereas the corresponding diffusion energy barriers for the lithium atom to diffuse along zigzag and armchair directions were 0.08 and 0.68 eV, respectively [35,36]. Using these 2D materials as the electrodes, NIBs showed faster charging/discharging rates than those using LIBs [37].
Inspired by the findings that point defects or GBs in the graphene can enhance the adsorption and diffusion of lithium atoms [25,26], there were previous studies to understand the effects of these defects in graphene on the adsorption and diffusion of sodium atoms.
Graphene with vacancies could strengthen the Na-C interactions [38]. Using graphene as the anode for NIBs, the energy capacities were found to be 1450 and 1071 mAh/g for the graphene with divancancy and Stone-Wales defects, respectively [22,39]. The GBs can induce defect states close to the Fermi level of graphene [40,41] and can interact with sodium atoms, therefore, the adsorption and diffusion of sodium atoms could be tuned by GBs in graphene.
As far as we know, there are no reports to investigate the effects of GBs in graphene on adsorption and diffusion of sodium. In the present work, using a density functional theory (DFT), we systematically studied the adsorption and diffusion of sodium on graphene with two types of commonly observed GBs, i.e., zig-zag and armchair ones [26,42]. The adsorption and diffusion behaviors of sodium atoms were also compared with those of lithium ones.

Computational details
All the DFT calculations were performed with SIESTA (Spanish Initiative for Electronic Simulations with Thousands of Atoms) codes [43]. We chose a local density approximation (LDA) for exchange-correction functions parameterized by the Ceperley-Alder (CA) [44].
The valence electron wave functions were expanded using a double-ζ basis set plus polarization functions. The cut-off energy was set to be 180 Ry for the calculation of self-consistent Hamiltonian matrix. Larger cut-off energy of 250 Ry was also tested. The results are listed in Table S1, which shows that the adsorption energies of Li and Na on pristine graphene are same with that calculated with cut-off energy of 180 Ry. The pseudopotential generation was performed using the ATOM program within the SIESTA package, and electronic configurations for pseudopotentials generation and cut-off radii are listed in Table S2. The pseudopotentials were successfully used to calculate lattice constants of ionic compounds of Li2O and Na2O (see the supporting information).
A 6×6 hexagonal supercell was used to model pristine graphene, and a 3×3×1 k-point mesh including the Г-center was used to sample the Brillouin zones. Larger hexagonal supercells were also tested, and the results show that this does not apparently affect the adsorption energies (see Table S3). The simulation models used in this work were two parallel and equally spaced GBs in a rectangular simulation supercell in order to satisfy periodic boundary conditions. A vacuum thickness of 30 Å above graphene sheet was used to avoid the influence of the interlayer. Two types of GBs, i.e. zig-zag and armchair ones [40,45] shown in Figs. 1a and 1b, were considered. The initial structures of graphene with GBs were obtained by relaxing the positions of carbon atoms along with the lattice parameters until the force on each atom was less than 0.02 eV/Å. The Brillouin zones were sampled using a 2×4×1 k-point mesh. Results listed in Table S4 showed show that using a denser k-point mesh with 4×6×1 and a larger cut-off energy of 250 Ry achieved similar results, indicating that the 2×4×1 k-point mesh and cut-off energy of 180 Ry we used was good enough. The zig-zag GBs were composed of pentagon/heptagon/hexagon (5-7-6), whereas the armchair ones were composed of pentagons/heptagons/pentagons/heptagons/hexagon (5-7-5-7-6). There are generally three possible adsorption sites for a single sodium atom on graphene considering the hexagonal symmetry of graphene: the hollow site (H) at the center of a hexagon; the bridge site (B) at the midpoint of a carbon-carbon bond; and the top site (T) directly above a carbon atom. All these sites are illustrated in Fig. 1a. Previous analysis showed that the hollow site (H) is normally the most stable adsorption position for lithium and sodium atoms [25,28,38,39]. Our calculation results also showed that sodium prefers to occupy the H site. To investigate the stability of sodium/lithium atoms on graphene with GBs, the adsorption of sodium/lithium atoms was calculated based on equation (1): and GBs E are the total energies of the supercell with and without adsorption of number of n sodium/lithium atoms, respectively. The adsorption energy can be calculated using an isolated Li/Na atom or bulk metal as the reference states. Na/Li E is the energy of an isolated Na/Li atom or half of the energy body center cubic Li/Na bulk metal.
The basis set superposition error (BSSE) [46] induced by the artificial shortening of distances and strengthening of the interactions was corrected by applying the counterpoise corrections [47] using "ghost" atoms. The BSSE was calculated to be -0.15 eV (for Na) and -0.17 eV (for Li). The BSSE was considered for all the adsorption energies in the whole work. According to equation (1), a larger negative value of adsorption energy indicates a more favorable exothermic reaction occurring between graphene and adopted atoms.
Several methods, such as Nudged Elastic Band (NEB) method, dimer method, and constrained method, can be used to determine the diffusion energy barriers of atoms in the condensed matters [48]. The NEB method is often considered as the most efficient one, whereas when the final state is unknown, the dimer method was often used [48]. The constrained method is the simplest and the most intuitive one. Because there could be a large computational effort needed when using the NEB method for the parallel computation, we used the constrained method and compared with the results from the NEB method with several cases, and results indicate that the constrained method is reliable to be used to calculate the diffusion behavior of sodium atoms on graphene. With the constrained method, one degree of freedom of the sodium/lithium atom was fixed, whereas all the other n-1 degrees of freedom were allowed to relax, i.e. the energy of the system was minimized in an n-1 dimensional hyperplane.
The sodium/lithium atom was diffused from one stable H site to a nearest one by passing through the B site on graphene. The sodium/lithium atom was pushed along the H-B-H path but constrained in the direction along the path. It was free to move in the directions perpendicular to the path for sodium/lithium atom, and all other atoms are allowed to move freely.

Adsorption of sodium/lithium on graphene with GBs
The calculated adsorption energies for sodium atoms adsorbed at H, B and T sites on pristine graphene are -0.67, -0.49 and -0.48 eV, respectively. The sodium atom was found to stay preferably at H site, which agrees with the previous reported result [38,49]. The distance between sodium atom and graphene is 2.08 Å, and the bond length of the Na-C atoms is 2.52 Å.
Several adsorption sites for sodium on graphene with zig-zag oriented GB were calculated. The results are shown in Fig. S1, and the calculated adsorption energies are listed in Table S5. The sodium atom was found to stay preferably at H site. Therefore, we will focus on the H sites in the following part of the paper. The adsorption energies of sodium atoms on graphene with zig-zag and armchair-oriented GBs, obtained using an isolated sodium atom as the reference state are shown in Figs atoms, which consists with the previously reported results [25,26]. The adsorption energies of sodium/lithium on graphene with GBs using the bulk metal as a reference state are shown in Fig. S2, which reveals the same trend with that using isolated atoms as a reference state. 9 Results also reveal that sodium atom shows the same adsorption behavior with those from LDA and generalized gradient approximation (GGA) calculations expect for the absolute values (see Fig. S3 in the supporting information). We can conclude that the calculations using LDA do not apparently affect the conclusion of this work.

Figure 2 Adsorption energies of sodium/lithium on graphene with (a)/(c) zigzag-and (b)/(d)
armchair-oriented GBs. The corresponding adsorption sites are presented in Fig. 1a and 1b, respectively.
With increasing the Na content, Na clusters may form. It is known that the formation of Na cluster would seriously reduce the charge/discharge capacity of the batteries. Also the formation of Na cluster will induce dendrite growth of sodium, which is harmful for the safety of NIBs. We have investigated more sodium atoms adsorbed on graphene and compared the binding energies and adsorption energies with those of small Nan clusters (see the supporting information). The results indicate that graphene with GBs can store more sodium atoms with a 10 larger Na/C ratio, and the existence of GBs could prevent formation of Na clusters on graphene.

Diffusion of Na/Li on graphene with GBs
The charging rate of NIBs depends on the diffusion behavior of sodium atoms on graphene. Smaller values of the diffusion energy barriers will lead to a faster charging rate.
The diffusion energy barriers of sodium and lithium atoms on the pristine graphene are 0.19 and 0.25 eV, respectively. The diffusion energy barrier of lithium atom is close to the value of 0.26 eV calculated using the NEB method [50].

Density of states
To calculate the density of states (DOS), dense k-point meshes with 21×21×1 and 15×21×1 were used to sample the Brillouin zones for pristine graphene and graphene with GBs. The DOS results of pristine graphene with and without adsorption of a sodium atom are shown in Figs. 5a and 5b, respectively. The Fermi level was set to be zero. The adsorption of a sodium atom does not apparently change the DOS of graphene except for an upshift of the Fermi energy level into the conduction band. Na 3s state (green line in Fig. 5b) is located at 0.36 eV, which is higher than the Fermi energy level, indicating that Na atom will donate its electron to graphene. Because of the existence of GBs in graphene, the Dirac point is destroyed and a localized band is formed near Fermi level [54], as the peaks marked with red  Fermi level is set to be zero.

Discussion
To be used as good anode materials for NIBs, the anode materials should have large values of exothermic reaction energy with sodium atoms so that anode materials have large storage capacities [57]. The presence of GBs enhances the adsorption of sodium atoms on graphene, indicating that a more favorable exothermic reaction occurs between graphene and sodium atom. The graphene with GBs should be better to store sodium atoms than those of pristine The dependence of the diffusivity on the diffusion energy barrier (EA ) obeys the Arrhenius equation [58]: in which kB is the Boltzmann constant and T is the temperature. Based on equation (2), a 60 meV increase (or decrease) in the diffusion energy barrier results in ten times of decrease (or increase) in the diffusivity. From Figs. 4a and 4b, the diffusion barriers are not affected when the sodium diffuses at sites far away from the GBs, whereas the diffusion energy barriers decrease from 0.19 eV to 0.13 eV as sodium atoms diffuse near the GBs. Based on this result, the diffusivity of the sodium atom on graphene with GBs can be enhanced for the fast charging process. The reverse diffusion barriers near GBs are larger than those of sodium diffusion on pristine graphene; however they are much smaller than the diffusion barriers of 0.55-0.59 eV for the Li diffusion in silicon [59,60]. Therefore, graphene with GBs is good to be used for discharge process of NIBs, and a good anode material for NIBs.
Above results also show that existence of GBs in graphene can prevent the formation of clustering of sodium atoms effectively, and thus inhibit dendrite growth of sodium, which is helpful to improve the safety of NIBs. GBs are likely to be introduced into graphene during chemical vapor deposition (CVD) growth process. Therefore, CVD growth can be used to fabricate low-cost, high-quality, and large area graphene as the anode for NIBs [61,62].

CONCLUSION
In summary, the adsorption and diffusion of sodium atom on graphene with GBs were systematically studied using the DFT calculations. The sodium atom prefers to be adsorbed at GBs with large adsorption energies. The diffusion energy barriers decrease with the decrease of distance from sodium atom to GBs as it diffuses along the path which is moving towards to GBs. The diffusion energy barriers are in the ranges of 0.09-0.33 and 0.09-0.35 eV as the sodium atom diffuses along both zigzag-and armchair-oriented GBs, respectively. Therefore, the existence of GBs in graphene can enhance the adsorption and diffusion of sodium atoms, indicating that graphene with GBs is a promising anode material for NIBs.

Acknowledgement:
This work was financially supported by the National Natural Science Foundation of China    Fig. 1a and 1b, respectively.