Figure 1

(Color online) Grain boundary sample, one out of 48, made by merging two graphene edges with the same chirality, but different orientation and random translation along the boundary; our samples are hence characterized by chirality. In this figure chiral indices are (11,6), chiral angle $\theta =20.4°$. Dashed vertical lines stand for periodicity across the horizontal direction.Reuse & Permissions

Figure 2

(Color online) Energies of the free edges of graphene as a function of chiral angle; zigzag edge has $\theta =0°$ and armchair $\theta =30°$. Free zigzag edge segments are metastable and prefer local reconstruction (two adjacent hexagons reconstruct into a pentagon and a heptagon); these so-called reczag edges are shown with gray connected symbols. The reczag segments are, however, important only for free edges and irrelevant for grain boundaries studied here.Reuse & Permissions

Figure 3

(Color online) Trends in energy density for graphene grain boundaries as a function of the chiral angle. The zero-energy means pristine graphene, and is achieved only for zigzag $\left(\theta =0°\right)$ and armchair $\left(\theta =30°\right)$ chiral angles.Reuse & Permissions

Figure 4

(Color online) (a) Relative number of polygons in grain boundaries as a function of chiral angle. The largest lattice mismatch, near chiral angle $\sim 15°$, is manifested by the appearance of mainly pentagons and heptagons, but also squares, octagons, and nonagons. In (b) each dot represents one polygon in all the grain boundary polygons in all of our 48 samples. Abscissa shows the area of that polygon (determined by triangulation) and ordinate shows the polygon energy cost per atom, relative to an atom in pristine graphene.Reuse & Permissions

Figure 5

(Color online) Inflection angles after optimizations for all grain boundary samples; $0°$ means planar structure (occurring only with pristine graphene, $\theta =0°$ and $\theta =30°$). The error bars remind of technical ambiguities in angle determination. The main trend here is that zigzag edges cause both large and small inflection, whereas armchair edges cause only small inflections. Inset: inflection angle illustrated.Reuse & Permissions

Figure 6

(Color online) Characterizing reactivity in grain boundaries. (a) Grain boundary sample with (9,5) chirality, including various polygons. (b) Characterizing chemical reactivity from the DFTB electronic structure directly: we calculated hydrogen adsorption energy for selected 350 atoms as adsorption sites and plot for each atom adsorption energy and ${ϵ}_i$. Given the correlation between the adsorption energy and ${ϵ}_i$, we access reactivity directly from the electronic structure, without additional calculations. The adsorption site numbers refer to panel a; atom ①, having three $\sim 120°$ angles has a small adsorption energy, atom ⑤ has a dangling bond, and other atoms have strained environment. (c) Using the correlation established in panels a and b, we calculate the averaged density of dangling bonds per unit length along grain boundary for all samples, as a function of $\theta$.Reuse & Permissions

Figure 7

(Color online) PVDOS as a function of the wave number and the chiral angle. PVDOS for each sample (given $\theta$) was calculated by solving vibrational spectrum from the dynamical matrix and projecting the vibration eigenmodes on atoms within the grain boundary; the contour plot is then gathered and smoothed from all the 48 samples. Intensity increases from blue to red (dark to bright).Reuse & Permissions