Ashivni Shekhawat*^{abc},
Colin Ophus^{d} and
Robert O. Ritchie^{ac}
^{a}Department of Materials Science and Engineering, University of California Berkeley, USA. E-mail: shekhawat.ashivni@gmail.com
^{b}Miller Institute for Basic Research in Science, University of California Berkeley, USA
^{c}Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, USA
^{d}National Center for Electron Microscopy, Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, USA

Received
23rd March 2016
, Accepted 27th April 2016

First published on 29th April 2016

The grain boundary (GB) energy is a quantity of fundamental importance for understanding several key properties of graphene. Here we present a comprehensive theoretical and numerical study of the entire space of symmetric and asymmetric graphene GBs. We have simulated over 79000 graphene GBs to explore the configuration space of GBs in graphene. We use a generalized Read–Shockley theory and the Frank–Bilby relation to develop analytical expressions for the GB energy as a function of the misorientation angle and the line angle, and elucidate the salient structural features of the low energy GB configurations.

Thus, it is clear that in order to understand the properties of polycrystalline graphene, it is necessary to characterize graphene GBs. Perhaps the most important property of a GB is its excess energy (per unit length). The excess GB energy, or simply the GB energy, has direct influence on the grain morphology,^{32,33} and thus influences all grain morphology dependent properties including strength and transport. It is not feasible to measure the GB energy directly in an experiment. Instead, GB energy in crystalline materials is typically inferred from the equilibrium structure of triple junctions (interface of three GBs).^{34–36} Such equilibrium junctions satisfy the Herring equations,^{34} which can be used to deduce the GB energy if a statistically large amount of experimental data is available.^{36} No such experimental study has been performed for graphene. Indeed, we (with co-authors) published the first statistically large dataset of high-resolution transmission electron microscopy (HRTEM) observation of graphene GBs only recently.^{37} Neither us, nor any other group has reported a statistically relevant number of experimental observations of triple junctions in graphene. On the other hand, computer simulations can be used to directly measure the GB energy without resorting to the indirect method of triple junctions.^{32} Although there have been several numerical studies of graphene GBs, most of these studies have focused on a few special configurations or symmetric GBs, and have not explored the entire configuration space of graphene GBs.^{38–48} Here we present a comprehensive numerical and theoretical study of the energy of a very large set of graphene GBs. We have used molecular dynamics (MD) simulations to measure the energy of about 79000 grain boundary configurations, corresponding to 4122 unique (θ_{M},θ_{L}) points, and spanning the entire space of all possible graphene grain boundaries.

Traditionally, the energy of low angle grain boundaries in bulk materials is understood in terms of Read and Shockley's theory.^{49–51} This theory was developed for bulk materials, and is not directly applicable to two dimensional membranes. This is due to the fact that the individual dislocation cores in a two dimensional membrane can buckle out of plane – a mode of relaxation that is not available in their three dimensional counterparts. By buckling out of plane, the dislocation can trade in-plane strain for out of plane bending, thereby lowering its energy considerably.^{38,52,53} We use a generalized Read–Shockley theory for two dimensional membranes, and combine it with the Frank–Bilby^{50,54–57} equation to elucidate the structure of the energy function of all possible graphene GBs. Further, we develop a theoretical understanding of the salient structural features of graphene GBs. Our results should be applicable to a large class of 2D materials, and will lead to a better understanding of fundamental processes such as grain growth, transport, and strength in these materials.

δn_{1}e_{1} + δn_{2}R_{60°}e_{1} + δn_{3}R_{120°}e_{1} = (∂n^{0}/∂θ^{0}_{M})δθ_{M} + (∂n^{0}/∂θ^{0}_{L})δθ_{L},
| (1) |

(2) |

Fig. 3a shows the numerically measured energy function for symmetric GBs, i.e., γ(θ_{M},0). It is well known that the GB energy has cusps at special high symmetry (low Σ CSL, where CSL stands for the ‘Coincident Site Lattice’, and Σ denotes the ratio of the volume of the unit cell of the CSL to that of the regular lattice, see ref. 50 for a detailed discussion of CSL) boundaries,^{50,64–67} and these can be seen clearly in Fig. 3a and 4a. There are two prominent cusps for graphene:^{39} first at the Σ_{7}(θ_{M} = 21.78°, θ_{L} = 0°) GB, and the second at Σ_{13}(θ_{M} = 32.2°, θ_{L} = 0°) GB. Apart from these, there are the obvious families of cusp singularities at θ_{M} = 0°, 60°. The Σ_{7,13} GBs are strong local energy minima of the GB energy. These minima arise due to favorable interactions between the dislocation cores. For intermediate values of misorientation (15° ≲ θ_{M} ≲ 45°), the density of required dislocations is high, and the individual cores cannot be well separated. We note that for isolated (1,0) as well as (1,0) + (0,1) cores, there is compression at the tip of the leading pentagonal ring and dilation at the tail of the trailing heptagonal ring (seen by the relative shortening and stretching of the bonds, most clearly visible in Fig. 2a and b).^{23} This local straining leads to significant out of plane buckling near the dislocation, as seen in Fig. 3d.^{38} However, as the dislocation density increases with increasing θ_{M}, and two (1,0) or (1,0) + (0,1) approach each other, their strain fields cancel, and there is a reduction in the elastic energy of the system. This cancellation of strain fields can be inferred from Fig. 3b and c, where it can be seen that the Σ_{7,13} GBs have almost no out of plane buckling, because the strain fields cancel out very effectively in these GBs with tightly arranged dislocations. On the other hand, at higher θ_{M} (note that the dislocation density peaks at θ_{M} = 30°), the increased density of the dislocations leads to higher energy per-unit-length. Thus, there is a competition between the energy increase due to higher dislocation density, and energy decrease due to dislocation interaction. It can be seen that initially the GB energy increases with θ_{M}, thus the energy increase dominates over the energy reduction. However, the reduction becomes significant, and the net energy starts to decrease at about θ_{M} = 18°. This reduction in energy reaches a first optimum for the Σ_{7} = (θ_{M} = 21.78°) CSL GB (Fig. 3b and e) where all (1,0) dislocation pairs are aligned, and there is a separation of exactly 1 carbon–carbon bond between them. At this optimal configuration there is significant reduction in the elastic energy, resulting in the first cusp in the GB energy (Fig. 3a). Increasing the misorientation θ_{M} further initially leads to an increase in the GB energy. This is due to the fact that a higher dislocation density pushes dislocations closer; however, geometrically it is still favorable to have all dislocations aligned in the same direction. Thus, creating a (1,0) + (0,1) pair incurs an energy penalty. However, with further increase in θ_{M}, it becomes progressively more favorable for the individual dislocations to stagger and merge to form (1,0) + (0,1) cores. This process leads to a reduction in energy starting at about θ_{M} = 24°. An optimal configuration is reached at the Σ_{13} = (θ_{M} = 32.2°) CSL GB where all (1,0) + (0,1) line up perfectly (Fig. 3c and f), and leads to a large reduction in the elastic energy, resulting in the second, deeper cusp in the GB energy. On increasing θ_{M} further, the net dislocation density decreases and the (1,0) + (0,1) cores separate, ultimately resulting in the behavior for large misorientations discussed previously.

Fig. 3 (a) The energy of all simulated symmetric GBs in units of eV Å^{−1}. The filled black circles show the simulation data, while the solid line is a fit to eqn (3). The filled red circles show the magnitude of the fitting error, which is smaller than 0.06 eV Å^{−1} everywhere. (b and c) show the Σ_{7,13} GBs, and (d) shows a symmetric GB with θ_{M} = 10°, colored by the net out of plane displacement in unit of Å. (e)–(g) show the same GBs colored by excess energy per-atom in units of eV. |

Fig. 4 (a) Energy of all simulated symmetric and asymmetric GBs in units of eV Å^{−1}. The surface shows data from simulation, while the grid is a fit to eqn (4). (b) The error in fit shown in units of eV Å^{−1}. (c) and (d) The basis functions, |h_{1}(θ_{M},θ_{L},θ_{M}^{c}) + h_{2}(θ_{M},θ_{L},θ_{M}^{c})|, |h_{1}(θ_{M},θ_{L},θ_{M}^{c})| + |h_{2}(θ_{M},θ_{L},θ_{M}^{c})| used to fit the cusp singularity at θ_{M}^{c} = 21.78° in eqn (4). |

Having understood the most salient features of the GB structure, we now turn our attention to the GB energy. A simple analysis of eqn (2) shows that for 2D materials the GB energy function has a absolute value (| |) type singularity at the cusps (ESI Section S2†). Thus, we propose the following functional form for the energy of the symmetric GBs:

(3) |

(4) |

Our model for GB energy is based on a ‘small perturbation’ approach; however, all the evidence presented so far provides only indirect validation of the perturbation idea. The perturbation model can be supported by inspecting GBs in the vicinity of the high symmetry Σ_{7,13} boundaries. We consider both symmetric and asymmetric perturbations, as shown in Fig. 5. This figure (plots a–d) shows that GBs in the vicinity of the Σ_{7} GB have basically the same structure as Σ_{7}, plus an occasional extra (or missing) dislocation, as the case might be. The asymmetric perturbation ((δθ_{L} ≠ 0)) sometimes results in a faceted boundary (Fig. 5d and h), with the facet locally following the high symmetry GB. The kinks joining the facets are composed of extra dislocations that are not present in the high symmetry GB. The same observations are true for the GBs vicinal to the Σ_{13} GB. It is remarkable that the GB generation algorithm is able to capture all the features expected from well annealed graphene GBs.

We have found that all of the approximately 79000 lowest-energy configuration GBs that we have simulated consist of only pentagon–heptagon pairs, and the usual hexagonal rings. No other geometric configurations were observed for the lowest energy boundaries. For example, the 5-8-5 configurations that have been previously observed experimentally at grain boundaries^{28,68,69} were not found in our structures. From an energy point of view, the 5-8-5 defects are vacancy defects and thus should be precluded from the ground state structures. However, non-equilibrium structures, such as the 5-8-5 defects, can indeed be captured by our algorithm if the Hamiltonian (eqn (2) in ref. 37) is not driven to its minima (the convergence criteria could be suitably relaxed, or Metropolis sampling could be performed at a suitably defined “temperature”). Further, the absence of such defects from our GBs is consistent with the fact that, to the best of our knowledge, such defects have not been observed in free-standing graphene films. Rather, they have been observed either in films on substrate or in free-standing graphene films after electron beam irradiation has modified their structure.^{70} The ubiquity of pentagon–heptagon pairs in graphene grain boundaries is consistent with our HRTEM study of 176 boundaries,^{37} the majority of which did not contain any rings of more than 7 or less than 5 carbon atoms. Further, the GB generation algorithm is able to capture faceting where appropriate.

To conclude, the main contribution of this work is to develop a fundamental understanding of the structure and energy of the entire space of graphene GBs. We have developed analytical expressions for GB energy as functions of the misorientation and line angle that can be readily used in future calculations of grain growth or other GB related phenomena.^{32} We hope that our analysis will pave the way for a deeper understanding of GB interfaces in graphene and other 2D materials.

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6ra07584c |

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