Payen

R Dessapt

L Lajaunie

J J Calvino

P Deniard

I Trenque

C Payen

Nanocrystalline BaCo 3 (VO 4 ) 2 (OH) 2 with a kagome lattice of Co(II) ions: synthesis, crystal structure and magnetic properties

Nanocrystalline BaCo 3 (VO 4 ) 2 (OH) 2 with a kagome lattice of Co( ii ) ions: synthesis, crystal structure and magnetic properties

Introduction

Materials based on 3d transition metal (TM) ions with triangular, honeycomb or kagome lattices are interesting candidates for studying geometric frustration and spin liquids, 1,2 2D ferromagnetism, 3 layered multiferroics 4 or even Kitaev interactions. 5 Recent technological opportunities have also arisen in the field of spintronic. 6 While fundamental research ideally requires single-crystals, considering potential applications implies other material forms, including nanostructuration, 7 and many initially studies of new materials are undertaken on polycrystalline samples. Therefore, synthetizing and studying new materials with 2D lattices of TM ions in various forms (bulk, nano and thin film) is of great interest. In particular, high-spin d 7 materials are currently actively investigated as new quantum magnets with spin-orbit coupling (SOC). 5,8,[START_REF] Zhong | Proc. Natl. Acad. Sci[END_REF] In this context, we targeted the Co 2+ orthovanadate BaCo 3 (VO 4 ) 2 (OH) 2 because its magnetic properties remain unexplored. [START_REF] Dordević | [END_REF] Its layered rhombohedral crystal structure (R-3m space group) is similar but not identical to those of the copper mineral vesignieite BaCu 3 (VO 4 ) 2 (OH) 2 (monoclinic, C2/m space group), 11,12 which is a well-known spin-1/2 kagome antiferromagnet, and of the spin-1 nickel compound BaNi 3 (VO 4 ) 2 (OH) 2 (monoclinic, C2/m space group), 13 which exhibits a glassy magnetic transition at low temperature. Very small single-crystals of BaCo 3 (VO 4 ) 2 (OH) 2 were already obtained with moderate yield (ca 35%), together with an uninvestigated powder, by hydrothermal reaction. [START_REF] Dordević | [END_REF] Their crystal structure comprises regular 2D kagome planes of edgesharing CoO 6 octahedra. However, the size and the quality of these single-crystals did not permit measurements of the physical properties. Here, we present a new soft-chemistry route to quantitatively obtain phase-pure samples of BaCo 3 (VO 4 ) 2 (OH) 2 consisting of highly crystalline quasi-spherical nanoparticles (NPs). BaCo 3 (VO 4 ) 2 (OH) 2 NPs have been characterized using powder X-ray diffraction (PXRD), FT-IR and FT-Raman spectroscopies, energy-dispersive X-ray spectroscopy (EDS), aberration-corrected high-resolution scanning transmission electron microscopy (HR-STEM) including image simulations, and magnetisation measurements.

Results and discussion

BaCo 3 (VO 4 ) 2 (OH) 2 NPs were prepared in a reproducible manner by using a simple coprecipitation method in ambient pressure. An aqueous solution (4 mL) of barium acetate Ba(CH 3 COO) 2 (0.520 g -2 mmol) was added slowly to an aqueous solution (32 mL) of Co(NO 3 ) 2 6H 2 O (0.872 g -3 mmol) and ammonium metavanadate NH 4 VO 3 (0.234 g -2 mmol) at 90°C under vigorous magnetic stirring, leading to immediate precipitation of a light brown solid. The pH was adjusted to 6 by Please do not adjust margins Please do not adjust margins adding manually NH 3 6 M, and the slurry was stirred at 90°C for 3 hours. After filtration, the powder was washed with water and ethanol and dried in air at room temperature (96 % yield in Co). Noticeably, the as-prepared powder exhibits a high specific surface area of 61 m 2 /g.

The purity and crystallinity of the sample were confirmed by PXRD analysis at room temperature, and the Rietveld refinement was performed using previously published structural model. [START_REF] Dordević | [END_REF] Fig. 1a shows the quality of the refinement. The final reliability factors under the R-3m space group (a = 5.9085(4) Å, c = 21.099(2) Å) are R wp = 2.19 and Gof = 1.09. The computed crystallite sizes are 22.2(3) nm and 18.7(4) nm perpendicular and parallel to [001], respectively. Structural parameters are gathered in Table S1, ESI †. All refined parameters compare well with those previously obtained from single-crystals. The structure is built upon {Co 3 (VO 4 ) 2 (OH) 2 } 2-layers stacked along the c-axis and separated by 7.03 Å due to the presence of large Ba 2+ ions between the layers (Fig. 1b). These layers are made up of edge-shared CoO 4 (OH) 2 octahedra in which the Co 2+ ions are coordinated to four equivalent equatorial oxo ligands (Co-O2 = 2.176(4) Å) and two shorter axial hydroxide ones (Co-O3 = 1.9407(6) Å). The Co 2+ ions form a regular 2D kagome lattice (Fig. 1c and Fig. S2, ESI †), and the distance between two adjacent Co 2+ ions is 2.954 Å. The orthovanadate VO 4 3-groups are located on both sides of the M 2+ site deficiency of the kagome framework (Fig. 1c). Each slightly-distorted VO 4 tetrahedron belongs to the C 3v point group and connects the Co 2+ ions via three equivalent O-atoms (V-O2 = 1.765(2) Å). The fourth V-O bond is shorter (V-O1 = 1.668(14) Å), and the O-atom is involved in strong hydrogen bonding interaction (O1O3 distance of 2.60 Å) with the hydroxide ligand of a neighboring layer (Fig. 1b).

BaCo 3 (VO 4 ) 2 (OH) 2 NPs were also characterized by FT-IR and FT-Raman spectroscopies. Attribution of the vibration bands have been done by comparison with vibrational data of natural mineral BaCu 3 (VO 4 ) 2 (OH) 2 samples, 14 and other orthovanadatecontaining materials. [15][16][17] The FT-IR spectrum of BaCo 3 (VO 4 ) 2 (OH) 2 NPs (Fig. S1a, ESI †) shows a weak band at 980 cm -1 and two intense ones at 908 and 756 cm -1 which are assigned to the stretching modes of VO 4 3-units with C 3v point group. 14 The band at 509 cm -1 may be assigned to a Co-O stretching mode. The two broad bands at 3425 and 2995 cm -1 characterize the OH stretching modes, and the weak bands at 1852, 1635 and 1384 cm -1 are in direct line with combination and overtones modes. 17 In the FT-Raman spectrum of BaCo 3 (VO 4 ) 2 (OH) 2 NPs (Fig. S1b, ESI †), the bands at 885 and 790 cm -1 could be assigned to the  1 symmetric and  3 antisymmetric stretching modes of VO 4 3-units with C 3v point group. 14 The band at 472 cm -1 could be attributed to the  4 antisymmetric bending mode, while the more intense one at 327 cm -1 and that at 397 cm -1 could correspond to the  2 symmetric bending modes. All NPs are quite homogeneous in shape and size, as shown by SEM analysis (Fig. S3, ESI †). Fig. 2 shows the HR-STEM HAADF analyses performed on the BaCo 3 (VO 4 ) 2 (OH) 2 NPs. They are quite regular in shape with a quasi-spherical morphology, and their diameter is between 9 and 25 nm. In agreement with PXRD results, this well evidences that each nanoparticle is a single crystallite. The high crystalline quality of the NPs can be clearly appreciated in the micrographs (Fig. 2a and2b) and in the corresponding FFT pattern (Fig. 2c). In addition, the FFT pattern was successfully ascribed to the crystal structure of BaCo 3 (VO 4 ) 2 (OH) 2 . The crystallographic structure of the nanoparticles was also confirmed by the excellent agreement between the experimental and simulated HR-STEM HAADF images (Fig. 2d and2e). It should be noted that image simulations and FFT indexations were performed by using the PXRD-refined data as input. The Ba/Co, Ba/V and Co/V elemental ratios were determined by STEM-EDS analyses (Fig. S4,ESI †) and are equal to 0.33, 0.46 and 1.40, respectively. These values are in good agreement with the expected ones (0.33, 0.50 and 1.50, respectively). These findings confirm the results obtained from PXRD analysis regarding the crystallographic structure.

Then, the magnetic properties of the BaCo 3 (VO 4 ) 2 (OH) 2 NPs were studied using a commercial SQUID magnetometer. The temperature dependence of the inverse dc magnetic susceptibility, 1/(T), is presented in Fig. 3. Data display Curie-Weiss behaviour,  = C/(T-) above ≈ 150 K. Its fitting from 150 K to 300 K yields C ≈ 0.01432 cm 3 K mol -1 and  ≈ -22 K. Using the molar weight of bulk BaCo 3 (VO 4 ) 2 (OH) 2 , the fitted C value corresponds to an effective magnetic moment of 4.7 µ B /Co atom, consistent with high-spin Co 2+ (d 7 , three unpaired electrons) ions for which unquenched orbital angular momentum is present. [START_REF] Mabbs | Magnetism and Transition Metal[END_REF] For 3d TM ions having orbitally nondegenerate ground term, experimental magnetic moments are often close to the spin-only values and a negative  implies dominant antiferromagnetic (AF) interactions. However, for high-spin Co 2+ ions in a weak octahedral crystal field, the  value determined from high-temperature susceptibility data could differ from the Weiss temperature of the molecular field approximation due to orbital degeneracy and SOC. [START_REF] Mabbs | Magnetism and Transition Metal[END_REF] Below ≈ 100 K, a negative deviation of 1/ from the high-temperature Curie-Weiss behaviour is observed. In parallel, the (T).T product, shown in Fig. S5, ESI †, decreases less rapidly than required by the Curie-Weiss law when decreasing temperature in the range of 150-10 K. Similar deviation from the hightemperature Curie-Weiss behaviour was reported in other octahedral high-spin Co 2+ bulk oxides, and was ascribed to the splitting between a Kramers doublet ground state with effective spin J eff = ½ and lowest excited states. [START_REF] Rawl | [END_REF]20 As can be seen in Fig. 4, the dc magnetic susceptibility measured at a small applied field of 100 Oe keeps on rising down to 2 K with no difference between zero-field-cooled (ZFC) and fieldcooled (FC) traces. There is no maximum in the T versus T curve, too (Fig. S5,ESI †). These features, which have been observed for two independently prepared samples, would indicate the absence of magnetic ordering or spin freezing above 2K for a micrometer-sized sample. Below 10 K, both and the T product sharply increase with decreasing temperature. A log-log plot of (T) versus T shows that, in the vicinity of 10 K, there is a crossover from the sub-Curie hightemperature region to a weakly sample-dependent low temperature regime where  varies as    with α = 2.1-2.4 (Fig. S6,ESI †). Magnetic field dependences of magnetization M(H) measured at temperatures between 2 and 50 K are shown in Fig. 4. At 50 K, only the anticipated linear paramagnetic response is observed. Below 10 K, the isothermal magnetization rapidly increases to 4-5 T and then linearly increases. As shown in Fig. S7, ESI †, isothermal magnetization data below 10 K do not follow the Brillouin H/T scaling for noninteracting spins, in agreement with the absence of a plateau in the T vs T curve below 10 K. Fig. S8, ESI † shows that the M(H) data at 2 K are significantly different from Brillouin equations calculated for J = 3/2 with g = 2 and for an effective J = 1/2 with g = 4, which is a typical value observed in compounds containing high-spin Co 2+ ion in a distorted octahedral environment. 8,[START_REF] Rawl | [END_REF]21,22 Also, the value g = 4 is consistent with the theoretical expression for the g factor that can be calculated within the effective J eff = 1/2 approach. 23 The magnetization versus field variation is not saturated even at 2 K and 7 T. The maximum value of the magnetization at 2 K (40 emu/g) corresponds to 1.4 µ B /Coatom, which is lower than the saturation values observed in bulk oxides or orthovanadate compounds containing high-spin Co 2+ ions in octahedral environment. 8,[START_REF] Zhong | Proc. Natl. Acad. Sci[END_REF]21,22,24,25 The slope of the isothermal M(H) scan at high field (H > 4 T) and low temperature (T < 5 K) is at least four times higher than the Van Vleck paramagnetic susceptibility observed in high-spin Co 2+ compounds, 21,22 suggesting the presence of AF interactions This journal is © The Royal Society of Chemistry 20xx

Please do not adjust margins Please do not adjust margins and/or of magnetic anisotropy. Octahedral high-spin Co 2+ moments are often anisotropic due to spin-orbit coupling. The magnetic behaviour of BaCo 3 (VO 4 ) 2 (OH) 2 NPs cannot be fully explained at this initial stage. The fine-structure splitting of the lowest levels of the Co 2+ ion under the action of crystal field, spin orbit-interaction, and magnetic field is unknown. The crystal structure contains compressed CoO 4 (OH) 2 octahedra with four long equatorial Co-O2 distances and two short axial Co-O3 distances, but the actual local symmetry of the metal cation is lower than D 4h (all the O-Co-O angles differ from 90°). For a D 4h compression, the 4 T 1g (t 5 2g e 2 g , L = 1) state of the regular CoO 6 octahedra is split into a 4 E g (M L = ±1) ground term and 4 A 2g (M L = 0) term. When spin-orbit coupling is switched on, each term splits into Kramers doublets. 26 Another issue relates to the magnetic couplings between Co 2+ ions. Nearest-neighbour (NN) superexchange pathways within a kagome layer are depicted in Fig. S2, ESI †. There are two possible superexchange pathways involving either a µ 3 -O(H) or a µ 2 -O site with Co-O-Co angles of 99,13(4)° and 85,5(2)°, respectively. According to the Goodenough-Kanamori-Anderson (GKA) rules, [START_REF] Goodenough | Magnetism and the Chemical Bond[END_REF] close to 90° superexchange is ferromagnetic (F) and a switch from F to AF is expected for a significant deviation from 90° Co-O-Co geometry. Published neutron diffraction results for high-spin Co 2+ oxide phases evidenced F interactions for Co-O-Co angles less than about 100°. 24,[START_REF] Mentre | [END_REF] Recent theoretical studies for high-spin Co 2+ ion in 90° Co-O-Co bonding geometry with pseudo-spin J eff = 1/2 have suggested a more complex situation in which F anisotropic Kitaev interactions coexists with a Heisenberg coupling, that can be F or AF depending on the local electronic structure. 29,30 Further neighbours interactions could also play an important role, as suggested in BaCu 3 (VO 4 ) 2 (OH) 2 . 31 Finally, we turn to the effects of finite size on the magnetic properties. The computed crystallite sizes (Table S1, ESI †) correspond to crystallites consisting of 1.1×10 4 to 1.4×10 4 unit cells and 25 to 28 kagome layers stacked in the c direction. As mentioned before, the magnetic susceptibility above ≈ 10 K bears resemblance to those of many bulk oxide phases containing high-spin Co 2+ ions in sixfold environment as well as to those of six-coordinated high-spin Co 2+ complexes. The low temperature behaviour is however likely to be impacted by the nanostructuration. The marked increases in and T below 10 K could be due to finite size or surface effects. The rapid initial increase of the experimental M(H) curve at 2 K compared to Brillouin functions for J = 3/2 or 1/2 (Fig. S8,ESI †) is consistent with a superparamagnetic contribution that could originate from the smallest nanoparticles or from surface layers. We also noted that the first magnetic study of synthetic vesignieite BaCu 3 (VO 4 ) 2 (OH) 2 , which was performed on a powder with crystallite size on the order of a few nm, revealed neither a magnetic order or a spin glass transition down to 2 K. 11 Further data collected for polycrystalline samples with much shaper PXRD reflections, 32 showed the same behaviour as in the nanosized sample in the Curie-Weiss regime (T > 150K), but evidenced a magnetic ordering at T N = 9 K.

10

 10

Fig. 1

 1 Fig. 1 (a) Rietveld refinement for BaCo3(VO4)2(OH)2 NPs showing observed, calculated and difference patterns with the total pattern as inset. The high-level noisy pattern is due to cobalt fluorescence. Structure of BaCo3(VO4)2(OH)2 viewed along the c-axis (b) and perpendicular to the c-axis (c). Blue octahedra: CoO6, yellow tetrahedra: VO4, orange sphere: oxygen, pink sphere: barium, white sphere: hydrogen. Hydrogen bonds are displayed as grey dotted lines.

Fig. 2

 2 Fig. 2 Large view (a) and detail (b) HR-STEM HAADF micrographs of BaCo3(VO4)2(OH)2 NPs. (c) FFT pattern acquired on the nanoparticle shown in (b). It has been successfully indexed with the XRD-refined BaCo3(VO4)2(OH)2 structure seen along the [241] zone axis. (d) HR-STEM HAADF micrograph of a nanoparticle which is seen along the [-141] zone axis. (e) Superimposition of the experimental and simulated HR-STEM HAADF images. The experimental image is an enlargement of the

3

 3 journal is © The Royal Society of Chemistry 20xx J . N a m e ., 2013, 00, 1-3 | Please do not adjust margins Please do not adjust margins nanoparticle shown in (d). The simulated image is delimited by a red dashed line.

Fig. 3

 3 Fig. 3 Inverse magnetic susceptibility 1/ versus temperature T for BaCo3(VO4)2(OH)2 NPs. Data were acquired at a dc applied field of 5 kOe. The solid line corresponds to a Curie-Weiss behaviour, 1/ = (T-)/C, obtained from a fit to the data in the temperature range of 150-300 K.

Fig. 4

 4 Fig. 4 Magnetic susceptibility  versus temperature for BaCo3(VO4)2(OH)2 NPs at low temperatures. Data were collected at dc applied magnetic fields of 100 Oe under ZFC (open circles) and FC conditions (plus signs). Inset: magnetic field dependence of magnetisation measured at 2, 3, 4, 5, 10, 20, and 50 K.

  .T product. See DOI: 10.1039/x0xx00000x This journal is © The Royal Society of Chemistry 20xx

	a. Université de Nantes, CNRS, Institut des Matériaux Jean Rouxel, IMN, F-44000
	Nantes, France. E-mail: remi.dessapt@cnrs-imn.fr, christophe.payen@cnrs-imn.fr
	b. Departamento de Ciencia de los Materiales e Ingeniería Metalúrgica y Química
	Inorgánica, Facultad de Ciencias, Universidad de Cádiz, Campus Río San Pedro
	S/N, Puerto Real, 11510, Cádiz, Spain. E-mail: luc.lajaunie@uca.es
	c. Instituto Universitario de Investigación de Microscopía Electrónica y Materiales
	(IMEYMAT), Facultad de Ciencias, Universidad de Cádiz, Campus Río San Pedro
	S/N, Puerto Real 11510, Cádiz, Spain. E-mail: luc.lajaunie@uca.es
	Electronic Supplementary Information (ESI) available: Experimental section with
	structural parameters, FT-IR and FT-Raman spectra, Fragment of kagome layer, SEM
	image, EDS spectrum,

Acknowledgement

Nicolas Stéphant is gratefully acknowledged for performing SEM measurements. This work was supported in part by the French Agence Nationale de la Recherche Grant ANR-18-CE30-022. The authors acknowledge the use of TEM instrumentation provided by the National Facility ELECMI ICTS ("Division de Microscopia Electronica", Universidad de Cadiz, DME-UCA). LL acknowledges funding from the Andalusian regional government (FEDER-UCA-18-106613), the European Union's Horizon 2020 research and innovation program (grant agreement 823717 -ESTEEM3) and the Spanish Ministerio de Economia y Competitividad (PID2019-107578GA-I00).

Conclusions

We have successfully elaborated a new simple synthetic route under ambient pressure to quantitatively produce morphologically homogeneous and highly crystalline NPs (9-25 nm) of the cobalt orthovanadate BaCo 3 (VO 4 ) 2 (OH) 2 . Its layeredkagome structure is topologically related to those of BaCu 3 (VO 4 ) 2 (OH) 2 11,12 and BaNi 3 (VO 4 ) 2 (OH) 2 . 13 In the case of BaCo 3 (VO 4 ) 2 (OH) 2 , the nanostructuration does not affect the crystal structure even at the atomic scale. Considering the magnetic properties, the effective moment indicates a high-spin state with an orbital contribution for the Co 2+ ions. Remarkably, the susceptibility increases with no ZFC-FC difference as the temperature decreases down to 2 K. At low temperature, the magnetic behaviour is most likely affected by finite-size effects. Further studies will be needed to better understand the magnetic properties of BaCo 3 (VO 4 ) 2 (OH) 2 , especially considering that high-spin Co 2+ compounds are currently very attractive candidates in the search for new quantum magnets with spinorbit coupling. 33 In particular, it would be useful to develop new synthetic routes to elaborate large micrometer-sized powder samples that would permit to investigate the bulk magnetic properties. It could also be interesting to prepare NPs with smaller crystallite size in order to reveal a superparamagnetic behavior with ZFC/FC irreversibility and blocking temperature above 2 K. For potential applications in spintronics, for instance, low-dimensional magnetic systems will have to be scaled down the nanometer regime where, eventually, superparamagnetism and thermal excitations will reduce the stability of the bulk magnetic state.

Conflicts of interest

There are no conflicts to declare.


James G Mchugh

Pavlos Mouratidis

Anthony Impellizzeri

Kenny Jolley

Dogan Erbahar

Chris P Ewels

Prismatic Edge Dislocations in Graphite

Keywords: DFT, prismatic edge, dislocation, graphite, Klein, zigzag, armchair, Bernal

Dislocations are a central concept in materials science, which dictate the plastic deformation and damage evolution in materials. Layered materials such as graphite admit two general types of interlayer dislocations: basal and prismatic dislocations, of which prismatic dislocations have been relatively less studied. Using density functional theory (DFT) calculations, we have examined different prismatic core structures in graphite and evaluated their structure, energetics and mobility. We find close energetic interplay between bonded and "free-standing" core structures in both zigzag and armchair directions, with a reconstructed stable zigzag core identified. We explore grain boundaries and prismatic dislocation pile-up, identifying metastable structures which may be important in energy storage. The role of interlayer stacking in core structure, dislocation glide and climb is also considered in-depth. Our calculations suggest that the prismatic dislocation core is stable up to high temperatures of approximately 1500K in bulk graphite. Above this temperature, the breaking of bonds in the dislocation core can facilitate climb, grain-boundary motion, and the annealing of damage through prismatic dislocation glide.

Introduction

Many of the most important properties of materials are determined not by the perfect bulk structure, but through the properties of defects such as dislocations. Dislocations represent one of the most fundamental concepts in materials science and, most importantly, they are the principle carriers of plastic deformation in crystalline materials [START_REF]The direct study by electron microscopy of crystal lattices and their imperfections[END_REF][START_REF] Hirsch | LXVIII. Direct observations of the arrangement and motion of dislocations in aluminium[END_REF][START_REF] Bollmann | Interference Effects in the Electron Microscopy of Thin Crystal Foils[END_REF]. Dislocations are one-dimensional line defects which accommodate strain through lattice deformation within a core region, and their dynamics largely accounts for the plastic deformation of materials under conditions of high temperature and under irradiation.

Graphite is an asymmetric, layered material, and bulk graphite consists of different stacking arrangements of individual graphene sheets. These layers differ from one another by an in-plane displacement along a C-C bond and can be labelled A, B and C depending on this relative shift. The asymmetry of graphite necessitates the existence of two types of interlayer dislocation: basal dislocations, where material is accommodated within a graphene sheet, creating regions of in-plane strain, and prismatic dislocations, where an additional partial graphene sheet is inserted into or removed from the bulk. In Bernal (AB-stacked) graphite [START_REF]The structure of graphite[END_REF], the (0001) basal plane is closed-packed and the closed-packed directions are the < > (or zigzag) directions, while the shortest lattice vectors are 1/3< >, see Fig. 1120 1120

dislocations readily dissociate into the corresponding partials, which have been the subject of extensive experimental and theoretical investigation. [START_REF] Amelinckx | Electron Optical Study of Basal Dislocations in Graphite[END_REF][START_REF] Eiichi Fujita | Observation of Lattice Defects in Graphite by Electron Microscopy, Part I[END_REF][START_REF] Delavignette | Dislocation patterns in graphite[END_REF] Prismatic edge dislocations have also been observed in graphite, using both STM [START_REF] Ouseph | Observation of Prismatic Dislocation Loops in Graphite by Scanning Tunneling Microscope[END_REF][START_REF] Atamny | Atomic resolution of defects in graphite studied by STM[END_REF] and TEM [START_REF] Thrower | Interstitial loops in graphite, their motion and their effect on elastic modulus[END_REF][START_REF] Baker | An electron microscope study of radiation damage in single crystal graphite[END_REF][START_REF] Bonfiglioli | Graphite Defects Observed by Electron Microscopy[END_REF][START_REF] Reynolds | The nucleation of radiation damage in graphite[END_REF] imaging, under both normal and damaged conditions [START_REF] Karthik | In situ transmission electron microscopy of electron-beam induced damage process in nuclear grade graphite[END_REF][START_REF] Krishna | An understanding of lattice strain, defects and disorder in nuclear graphite[END_REF]. In carbon nanotubes their diffusion can mediate the formation of carbon nanotube walls [START_REF] Huang | Self-Templated Growth of Carbon-Nanotube Walls at High Temperatures[END_REF]. They are readily formed under irradiation, where vacancies and interstitial atoms in excess of the equilibrium concentration can form prismatic dislocation loops [START_REF] Karthik | Neutron irradiation induced microstructural changes in NBG-18 and IG-110 nuclear graphites[END_REF][START_REF] Johns | Formation of carbon nanostructures in nuclear graphite under high-temperature in situ electron-irradiation[END_REF]. Yet, despite their importance to many fundamental material properties, prismatic dislocations have received relatively little theoretical attention. where the prismatic sheet has perfect "C" character,.

Fig. 2 schematically depicts a prismatic dipole configuration. A prismatic dislocation can be of either vacancy (2a) or interstitial (2b) character, where a graphene layer is correspondingly removed or inserted into graphite. The transition region between the extra or missing material and the perfect graphite lattice results in a pair of dislocation cores of opposite Burgers vector, which together comprise a dipole. The inserted layer of material experiences an energetic penalty, as there is a stacking fault between this layer and adjacent Bernal stacking layers. This is minimised when the additional layer has distinct (C) stacking to the adjacent A, B layers, leading to the low-energy rhombohedral ABC stacking fault [START_REF] Thrower | The burgers vector of dislocation loops in graphite[END_REF][START_REF] Heggie | Chapter 12. The Atomic-, Nano-, and Mesoscale Origins of Graphite's Response to Energetic Particles[END_REF]. While prismatic dislocations of interstitial character may automatically adopt this low-energy stacking, vacancy dislocations will initially result in unfavourable AA-stacking. This necessitates the in-plane shifting of material within a layer to restore preferable rhombohedral stacking, with the result that only layers adjacent to the stacking fault have distinct (AB, BC) stacking, which is restored to perfect Bernal lattice through passage of a basal dislocation, as shown in Fig. 2a.

The energetic and structural properties of different prismatic dislocation cores have been simulated in periodic supercells, using density functional theory (DFT) [START_REF] Suarez-Martinez | Dislocations of Burgers vector c/2 in graphite[END_REF], where it was found that it is possible for interlayer bonds to form in the core region. In principle however, there are many possible bond reconstructions along an edge or between the adjacent sheets as seen, for example, in graphene nanotube edges. [START_REF] Ivanovskaya | Low-Energy Termination of Graphene Edges via the Formation of Narrow Nanotubes[END_REF] In this work, we have undertaken a systematic study of the possible core configurations of the prismatic dislocation in graphite. In section 3.1, isolated prismatic cores are considered through explicit construction of graphene monolayer-bilayer junctions, which we find give a good approximation to the chemical structure of dislocation cores, while ignoring effects such as stacking & self strain. In this section, we focus particularly on the core reconstructions which are found to be most relevant in the bulk. In section 3.2, the structural and energetic properties of prismatic cores in the bulk are examined, both for isolated prismatic cores, and at regions of rotational disorder between graphite grains of different c-axis alignment, which correspond to different arrangements of dislocation arrays. In section 3.3, the dynamics of the bond-breaking process is considered, which is used to examine the stability of interlayer bonding, and dislocation motion through the mechanisms of dislocation glide and climb. Finally, in section 3.4 we compare simulated TEM images from our DFT structures to experiment.

Computational Methods

DFT calculations have been performed using the Quantum ESPRESSO ab-initio suite [START_REF] Giannozzi | QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials[END_REF][START_REF] Giannozzi | Advanced capabilities for materials modelling with Quantum ESPRESSO[END_REF]. We have used the generalized gradient approximation (GGA) as parametrized by Perdew, Burke, and Ernzerholf (PBE) [25]. Interlayer interactions have been incorporated into our calculations through Grimme's D2 dispersion correction, which is found to give good agreement with experimental structural and energetic parameters [26][27][28][29]. Efficient Vanderbilt ultrasoft pseudopotentials are employed to approximate the effect of core electrons [START_REF] Garrity | Pseudopotentials for high-throughput DFT calculations[END_REF][START_REF] Vanderbilt | Soft self-consistent pseudopotentials in a generalized eigenvalue formalism[END_REF][START_REF] Laasonen | Implementation of ultrasoft pseudopotentials in ab-initio molecular dynamics[END_REF].

The wavefunction basis is expanded as a series of plane-waves with a maximum cut-off of E cut = 40 Ry and a charge density cut-off of E ρ = 500 Ry, which are found to be in close agreement to more highly-converged calculations. A non-zero electron temperature of kT = 0.02 eV for electronic level occupation is applied to aid convergence, using a Gaussian smearing function. The Brillouin zone is sampled using the Monkhorst-Pack scheme, with a k-point grid density of . Structural 2π × 0. 05 Å -1 optimisations were performed with a convergence threshold of 0.01 eV energy difference between subsequent iterations, and until the residual component of all forces is less than 0.01 eV/Å. For cells containing vacuum, care has been taken to ensure that periodically-repeated images are well separated within the unit cell, and a separation of 20 Å is found to avoid any spurious self-interaction in the out of plane direction. All cells employed in our calculations employ an approximately equal length of material (~4 nm) in both monolayer and bilayer regions, so that we recover the interlayer separation of an isolated bilayer far from the cores. Reaction barriers are calculated using the climbing nudged elastic band (CI-NEB) method [START_REF] Sheppard | Optimization methods for finding minimum energy paths[END_REF][START_REF] Henkelman | A climbing image nudged elastic band method for finding saddle points and minimum energy paths[END_REF], where images are optimised so the force on each image orthogonal to the path reaches a value of less than 0.01 eV/Å. High resolution transmission electron microscopy (HRTEM) images are simulated using the QSTEM software package [START_REF] Koch | Determination of Core Structure Periodicity and Point Defect Density Along Dislocations[END_REF] at room temperature on a 200 kV electron beam with spherical aberration -0.06 mm, defocus set to the Scherzer value of +15 nm, chromatic aberration 1 mm, and convergence angle 24.5 mrad.

Corresponding TEM images simulated for different defocus values are presented in Supplementary Materials.

Results

Isolated Cores

We first consider isolated prismatic dipoles, which are constructed as monolayer-bilayer transitions. This gives a pair of dislocations (a dislocation dipole) with opposite Burgers vector and parallel dislocation lines (see Fig. 3). Core structures therefore involve linear "cutting" of a graphene sheet in the dislocation line direction, resulting in a row of dangling bonds. These edges can be stabilised either within a layer by self-passivation through rehybridisation and restructuring of dangling carbon bonds (a 'free-standing edge' structure, FSE shown in Fig. 3a), or by interlayer bonding to the adjacent graphene layer (an 'inter-layer bonded' structure, ILB shown in Fig. 3b). This second option removes dangling bonds from the free edge, at the energetic cost of disrupting the local resonant π-bonded structure of the sheet below. In both cases, there are two possible contributions to the Burger's vector: a c-axis component, which is automatically included through the transition from monolayer to bilayer, and an in-plane component which is parallel to the basal plane. For ILB cores, the latter may be included by varying the chemical structure within the core, while a simple translation is sufficient in the case of FSE cores. The relative stability of the different possible core structures is therefore not a-priori obvious and will depend on the local chemical bond structure in the dislocation core region. This approach allows evaluation of different edge configurations, while ignoring secondary effects such as stacking faults and elastic energy.

The formation energy of a prismatic dislocation core, E f , is given by the equation

(1) 𝐸 𝑓 = ( 𝐸 𝑐𝑒𝑙𝑙 -𝑛 𝑚𝑙 ϵ 𝑚𝑙 -𝑛 𝑏𝑙 ϵ 𝑙 )/2,
where is the total energy of the cell containing both defects, and are the number of carbon atoms 𝐸 𝑐𝑒𝑙𝑙 𝑛 𝑚𝑙 𝑛 𝑏𝑙 in the monolayer and bilayer regions respectively and are the calculated energy per carbon atom of ϵ 𝑚𝑙 ϵ 𝑏𝑙 isolated infinite monolayer and bilayer sheets, and we have divided by 2 due our use of dipole cells. The dislocation line energy, which is the formation energy of a dislocation core per unit length of dislocation, has also been calculated. This quantity determines the dislocation line tension which acts in the direction of the dislocation line and tends to shorten it, thus determining the energetic stability of a straight dislocation. [START_REF] Hirth | Theory of Dislocations[END_REF][START_REF] Schweizer | In situ manipulation and switching of dislocations in bilayer graphene[END_REF]

Free-standing edges

Before considering the effect of interlayer-bonding we first summarise the properties of monolayer FSE graphene edges. Graphene edges can lie along either the zigzag < > or armchair < > directions, or 1120 1010 a mixture of the two. Fig. 4 shows the edge terminations and associated notation used in this work. The free-standing zigzag (FSE ZZ) edge (Fig. 4a) consists of a line of energetically costly dangling bonds. It is highly unstable and a number of reconstructions are possible [START_REF] Koskinen | Self-Passivating Edge Reconstructions of Graphene[END_REF], which depend significantly on local hydrogen content [START_REF] Wassmann | Structure, Stability, Edge States, and Aromaticity of Graphene Ribbons[END_REF][START_REF] Rodrigues | Zigzag graphene nanoribbon edge reconstruction with Stone-Wales defects[END_REF]. At room temperature, for low hydrogen content, or under electron irradiation [START_REF] Koskinen | Self-Passivating Edge Reconstructions of Graphene[END_REF][START_REF] He | Temperature Dependence of the Reconstruction of Zigzag Edges in Graphene[END_REF] a bond rotation with a barrier of 0.4-0.7eV occurs [START_REF] Koskinen | Self-Passivating Edge Reconstructions of Graphene[END_REF][START_REF] Rodrigues | Zigzag graphene nanoribbon edge reconstruction with Stone-Wales defects[END_REF][START_REF] He | Temperature Dependence of the Reconstruction of Zigzag Edges in Graphene[END_REF], giving the more stable Dienes edge (FSE 57) (3b). This edge, constructed from pentagons and heptagons, is stabilised by distorted rehybridisation to sp-triple bonding in the edge C 2 pairs. Both of these edges lie along the glide plane, however cleavage is also possible along the parallel shuffle plane (see Fig. 4a). The latter gives the highly unstable Klein edge, with individual single neighbor carbon atoms which can stabilize through pentagonal pairwise reconstruction of adjacent dangling bonds, resulting in the reconstructed (FSE RK) edge (3c).

Armchair edges are shown in Fig. 4d-g. The armchair edge spontaneously reconstructs within a graphene layer by rehybridising the external C 2 pair into a distorted ~1.24 Å triple bond [START_REF] Thrower | The burgers vector of dislocation loops in graphite[END_REF][START_REF] Heggie | Chapter 12. The Atomic-, Nano-, and Mesoscale Origins of Graphite's Response to Energetic Particles[END_REF]. Since the armchair shuffle and glide planes have identical crystallographic character, their energetic and structural properties are the same (Fig. 4d,e). We include here two other armchair edges, the AC-56 edge [START_REF] Koskinen | Self-Passivating Edge Reconstructions of Graphene[END_REF], where removal or adsorption of a carbon atom at the edge carbon dimer results in a pentagonal reconstruction (Fig. 4f), and the AC-677 edge, which features a bond rotation similar to the zigzag FSE 57 edge (Fig. 4g). [START_REF] Koskinen | Self-Passivating Edge Reconstructions of Graphene[END_REF] Fig We have performed structural relaxation of FSE structures with and without an underlying graphene substrate, the latter providing a baseline for comparison with the interlayer-bonded cores. Graphene edges are modelled using a finite-width nanoribbon, which is suspended over a perfect graphene sheet in the bilayer case, as shown in Fig 3b,d. Table 1 shows the edge formation energies, which have been calculated using Eq. ( 1), and the effect of interlayer stacking has been accounted for by using appropriate values of the bilayer energy . ϵ 𝑏𝑙 = ϵ 𝐴𝐴 /ϵ 𝐴𝐵 Our results for isolated edges confirm previously reported well-known trends [START_REF] Koskinen | Self-Passivating Edge Reconstructions of Graphene[END_REF][START_REF] Gao | Transition Metal Surface Passivation Induced Graphene Edge Reconstruction[END_REF][START_REF] Zobelli | A comparative study of density functional and density functional tight binding calculations of defects in graphene[END_REF]. The most stable edge structure in the < > direction is the unreconstructed armchair edge (AC FSE). While the 1010 FSE ZZ edge is ~0.3 eV/Å more energetically costly than this, reconstruction into the FSE 57 edge lowers this energy to ~0.07 eV/Å more stable than the AC FSE edge, again in good agreement with previous work [START_REF] Koskinen | Self-Passivating Edge Reconstructions of Graphene[END_REF][START_REF] Gao | Transition Metal Surface Passivation Induced Graphene Edge Reconstruction[END_REF][START_REF] Zobelli | A comparative study of density functional and density functional tight binding calculations of defects in graphene[END_REF]. All structural properties such as bond lengths, and the energy differences between core configurations remain very similar in the presence or absence of a graphene substrate, regardless of the stacking configuration in the bilayer region.

< > ZigZag Line energy (eV/Å) 1120

Isolated

Interlayer-bonded Zigzag Cores

ILB dislocation cores can be considered as the convergence of three graphene edges at the same line.

Here, ILB dislocation cores have been constructed using graphene edges which lie along different glide planes. This alters the chemical structure with the core, while allowing the incorporation of different in-plane Burger's vector components. In the zigzag case, for example, including a single FSE RK edge in the core will shift the bilayer to AB stacking in comparison to a dislocation core composed of three FSE ZZ edges, shown in Fig. 5.

Core cells have been labelled according to the three edge configurations within the core region, terminating the monolayer "ML" (white atoms in Fig. 5), "Top" and "Bottom" bilayer graphene sheets (red, blue atoms in Fig. 5), which we will refer to by the convention ML-Top-Bottom. Table 2 details the formation energies of the resulting ILB dislocation cores. For clarity, we also show the energy of the most favourable FSE configuration, and the difference between this and each ILB core. For all core configurations, the interlayer bonds bridging the two graphene sheets have sp 3 character due to their four-fold coordination. These sp 3 interlayer bonds are generally dilated compared to diamond, 1.53 Å or larger, due to additional strain in the core region. We will largely confine our discussion to those structures which preserve the hexagonal graphene lattice within the core. Two ILB configurations are energetically favourable or comparable to the FSE 57 edge (the most stable free-standing edge configuration), both of which lead to AA-stacking. The symmetric ZZ-ZZ-ZZ configuration (Fig. 6a) consists entirely of zigzag edges, giving perfect hexagonal graphene that is smoothly connected and shared between the monolayer and bilayer regions. While this core is slightly higher in energy (0.03 eV/Å) than the reconstructed FSE 57 edge, the energy difference is very small. For example, there is a similar energy difference between the FSE 57 and FSE AC edges, both of which are observed in TEM images [START_REF] He | Temperature Dependence of the Reconstruction of Zigzag Edges in Graphene[END_REF], and it is likely that some mixture of these two structures can co-exist within graphite. The ZZ-ZZ-ZZ edge structure can also bond into atoms on the opposite (β) sublattice rather than the same (α) sublattice by translating the bottom layer by a bond length, as shown in Fig. 6b. This gives AB stacking across the fault at the cost of a large increase in core formation energy. Henceforth, we will refer to these two configurations as the ZZ-ZZ-ZZ-α and ZZ-ZZ-ZZ-β structures, respectively. All of the other cores involve multiple RK edges meeting in the core region, as well as more energetically costly pentagonal rings, and are generally much higher in energy than the FSE 57 structure. An example of this is given by the RK-ZZ-RK core (Fig. 7b), which is the most stable AB-stacked core, and features a high density of pentagonal and heptagonal defects, as well as extensive local strain accommodation as indicated by the green bonds leading to a relatively large line energy. Some further images of these cores are available in the Supplementary Material, Fig. S1.

Zigzag cores

ML

Interlayer-bonded Armchair Cores

For armchair ILB configurations, there are only two distinct structures to consider, i.e. with or without a shift in one bilayer, giving either AA-or AB-stacking respectively (Table 2). The lowest energy configuration is the AA-stacked AC-AC-AC prismatic core (Fig. 8a) i.e. an AC-armchair edge bonding into the layer below, which is substantially lower in energy than the armchair free-standing edge, and has the lowest overall line energy of all cores considered in this work, again in good agreement with previous theoretical and experimental results [START_REF] Bonfiglioli | Graphite Defects Observed by Electron Microscopy[END_REF][START_REF] Thrower | The burgers vector of dislocation loops in graphite[END_REF][START_REF] Suarez-Martinez | Dislocations of Burgers vector c/2 in graphite[END_REF]. The AC-AC-AC core results in moderately more strain than the ZZ-ZZ-ZZ core, with C-C bonds along the core direction showing local double bond character. In contrast, the AB-stacked core is substantially less stable. While the zigzag direction can generate AB stacking through only one lateral translation, the armchair core requires both a lateral and a screw component. This results in material adjacent to the core shearing away from AB stacking in the equilibrium geometry, as seen in Fig. 8b. In summary, and in contrast to Ref [START_REF] Suarez-Martinez | Dislocations of Burgers vector c/2 in graphite[END_REF] we find that armchair-oriented dislocation cores are most stable when AA-stacked with interlayer bonding. The AB core is higher in energy, with FSE and ILB structures that are almost equal in energy. Finally, we note that our calculations suggest the possible importance of the local stacking environment to the type of reconstructions which are available. For example, while the ZZ-ZZ-ZZ-α ILB is almost equal in energy to the FSE-57 structure, this is not the case for AB-stacking, where all ILB configurations are much higher in energy. Detailed consideration of the stacking around a dislocation core will then be particularly important in the bulk of graphite. The lower formation energy of cores which give AA-stacking must be weighed against the additional energetic cost of faulted material adjacent to the core, which may restore AB-stacking (rhombohedral in bulk), at the energetic cost of a basal dislocation. As the latter energy is relatively low, at approximately 0.07 eV/Å [START_REF] Dai | Structure and energetics of interlayer dislocations in bilayer graphene[END_REF], it is therefore likely that the AA-stacked cores are preferred in reality. Incorporating this energy changes the relative stability of the cores and gives us a stability order for the most stable cores to be AC-AC-AC < FSE AC < RK-ZZ-ZZ < FSE 57 < ZZ-ZZ-ZZ-α, all with relatively close energies (within 0.06 eV/Å).

Bulk Graphite

Having discussed the structure and energetics of isolated cores, we now consider different possible bulk reconstructions. When performing simulations of dislocations in solids, due consideration must be given to the unit cell and boundary conditions employed. There are many inequivalent dislocation arrays which can be used to construct the type of periodic supercells required for DFT simulation [START_REF] Rodney | Ab initio modeling of dislocation core properties in metals and semiconductors[END_REF]. The two configurations of dislocation arrays used in this work are shown schematically in Fig. 9, corresponding to a quadrupolar array (Fig. 9a), where the Burgers vector alternates perpendicular to the line direction of the dislocation core, and a grain boundary configuration (Fig. 9b), where the dislocations separating regions of rotated graphite structure are all of the same sign (see also Fig. 9e-g). We have created quadrupole dislocation loops in bulk supercells through the insertion of extra material into an otherwise perfect graphite supercell (Fig. 9a) (equivalent to cleaving a perfect graphene sheet along the appropriate direction). The resultant orthorhombic cell contains a dislocation quadrupole, and through judicious use of appropriate boundary conditions, it is possible to construct an identical network using fewer atoms within a monoclinic cell as shown in Fig. 9c,d. [START_REF] Rodney | Ab initio modeling of dislocation core properties in metals and semiconductors[END_REF][START_REF] Lehto | Effects of Dislocation Interactions: Application to the Period-Doubled Core of the 90°Partial in Silicon[END_REF] All cells have been constructed such that the net Burgers vector within them is zero (some typical examples are shown in Fig. 9c,d), and the lowest energy configuration of every cell is therefore perfectly-stacked, Bernal AB graphite. The creation of a prismatic defect then requires some formation energy to nucleate. Due to cell size limitations and boundary conditions for the grain boundary cells, we have inserted prismatic cores into graphite cells which are initially AA-stacked. During optimization the graphene layers easily shear with respect to one another due to strain around the core, and the final configuration is close to perfect AB stacking in the non-defective regions away from the core. For this reason, we have calculated all formation energies with respect to the energy per atom in the AB stacking configuration. This also provides an absolute upper limit on the formation energy in bulk, as AB stacking is the global minimum of perfect graphite.

Reconstructions and stacking in bulk graphite

In order to evaluate different structures, we initially compare the line energy of different core reconstructions in a monoclinic cell, for both zigzag and armchair directions with approximate dimensions of for both . Additional calculations using different boundary conditions have been performed 23 Å × 25 Å and are briefly reviewed in the Supplementary Material Section 2, the cell structure will not have a large impact on results for the cell sizes considered here. Note when comparing bulk and isolated structures, the "ML" and "Bottom" edges from the isolated case can be considered as belonging to the same single graphene layer, while the "Top" sheet contains the bonding-in prismatic sheet edge. The layers within the bulk are explicitly labelled to clarify this point in Fig. 10a, and this notation is used in labelling different core configurations. A number of structures are considered based on the isolated core results, including the FSE-57, ZZ-ZZ-ZZ in α and β configurations and RK edges with and without pentagonal reconstruction, and both AC-AC-AC and AC-AC-AC' cores, in all cases considering both ILB and FSE configurations.

Table 5 shows the line energy of the different prismatic cores calculated in this way. We note a net increase in the formation energy of all cores compared to the corresponding isolated structure, due to the additional elastic energy created by a lattice of prismatic dipoles in bulk graphite. The armchair cores remain the lowest-energy of all terminations examined in this work. Direct comparison between isolated and bulk cells suggests an approximate elastic energy of 0.384 eV/Å and 0.363 eV/Å for the zigzag and armchair cores, respectively. These are in quite close agreement with values calculated using elasticity theory for a superlattice of dipoles (0.45 eV/Å, 0.35 eV/Å for zigzag and armchair respectively) [START_REF] Suarez-Martinez | Dislocations of Burgers vector c/2 in graphite[END_REF]. Bond structure around the core is very similar between the isolated and bulk cores, shown in Fig. 10 for the three lowest energy ILB structures. In both crystallographic directions, we find relatively low energy ILB configurations. For the zigzag direction, we again find that the non-bonded FSE Dienes reconstructed edge is preferred over ZZ-ZZ-ZZ ILB with a moderate increase in the energy difference from 0.03eV/Å to 0.08 eV/Å. The α-ILB reconstruction remains energetically preferred over β-ILB, although the difference is significantly reduced to 0.18 eV/Å, which we partly attribute to reduction in the bond angle compared to the isolated case due to the surrounding layers. In the armchair core, we find that the ILB structure is the lowest energy overall which is in contrast to an earlier report for the armchair edge, where it was found that the free-standing armchair edge was more stable. [START_REF] Suarez-Martinez | Dislocations of Burgers vector c/2 in graphite[END_REF] This could be due to differences in cell construction, or due to computational approximations, such as the use of a Gaussian rather than plane-wave basis.

ZigZag Line energy (eV/Å)

ML

. ZZ-RK-ZZ cores, created through removal of a line of atoms in the "top" or bonding-in layer followed by different reconstructions (Fig. 11a-c), all have significantly higher energies compared to the ILB ZZ (Fig. 10 a,b) and Dienes reconstructed edges (Fig 11d). We have also considered the RK-ZZ-ZZ dislocation core (that is, an RK and ZZ edge within the same graphite layer). While this structure was more stable than the ZZ-ZZ-ZZ core for an isolated core, it would require the removal or addition of a row of carbon atoms within a layer in the bulk, in addition to a screw component which generates extensive additional strain which is not evident for the isolated cells due to lower dimensionality. In the bulk this gives a relatively distorted core structure with a high line energy of 1.89 eV/A, which is higher in energy than all examined reconstructions apart from the unreconstructed ZZ-RK-ZZ core (Fig. 11b).

Thus for the zigzag orientation we have an interesting situation, whereby the most stable zigzag oriented prismatic core is the ZZ-57-ZZ, non-bonded, with the dislocation line edge reconstructed through bond rotations to give a pentagon-heptagon edge sequence with triple-bond character at edge bonds (Fig. 11d). However close in energy is the ZZ-ZZ-ZZ interlayer bonded core, where an unreconstructed zigzag edge bonds directly into the sheet below. These two structures may potentially result in an interesting mechanical response that depends on annealing and restructuring at the core, and is discussed further below. Turning now to the armchair case, we consider both the purely hexagonal AC-AC-AC and the less stable 4-8 AC-AC-AC' reconstruction. Similarly to the difference between the α and β cores, we can create both structures in the bulk by shifting the adjacent flat sheet by a bond length, as indicated in Fig. 12 by arrows, showing how the breaking of bonds and shifting of layers creates these different cores. The FSE configurations have an almost identical line energy and structure, and are slightly less stable than the ILB structures. The shifted AC' core is also higher in energy than the AC core, due to the bonding strain from 4-8 defects, and is much closer in energy to its corresponding FSE structure than the shifted AC core.

Thus in the armchair case we find that bonding into the layer below is the most stable configuration, with relatively little energy difference when the layer debonds and becomes free-standing. In general the armchair oriented cores are slightly more stable than zig-zag oriented cores.

Grain boundary cells

We now consider asymmetric c-axis tilt grain boundaries. These consist of two graphite grains, one of which lies flat with , while the other has been tilted in the c-axis direction such that . This θ = 0 θ > 0

gives an array of graphene edges (prismatic dislocations) where the tilted grain meets the flat grain in the boundary region (see Fig. 9b). These cells have been created using appropriate boundary conditions where the atoms bordering one cell edge (i.e. the right-hand side) in one layer are continuously connected to the left-hand side atoms of the layer above (Fig. 13b). As the inter-grain tilt varies, the geometric incompatibility between tilted regions results in different densities of graphene edges along the grain boundary. This results in the tilted graphene sheets being "cut" with a specific crystallographic character at the grain boundary. Using simple geometric considerations, the separation (d) between carbon atoms along a prismatic edge at an asymmetric c-axis tilt grain boundary is .

𝑑 = 𝑐 sinθ

Graphite c-axis grain boundaries change the angle of the interlayer sp 3 bonds, and naturally create a mismatch in the lattice parameters between the flat and tilted grains. These two effects modify the formation energy of a prismatic array at the boundary. The DFT cells used in this work are necessarily small, and we stress that this could influence conditions of strain accommodation at the grain boundary. Nonetheless, we are able to observe some broad trends in our calculations. Fig. 13a shows the formation energy of a prismatic dislocation as a function of angle, for the armchair and zigzag core in both α and β-orientations. At large angles (close to 90°), the terminating edge bonds are close to each other and their separation is on the order of the interlayer separation. This allows the prismatic edges to form perfect graphitic bonds between adjacent edges, resulting in the sp 2 -capped configurations shown in Fig. 13c, which our calculations confirm are lower in energy than the corresponding sp 3 interlayer bonding. This type of looped closed-edge is readily observed at graphite crystallite boundaries in TEM experiments. [START_REF] Posligua | The closed-edge structure of graphite and the effect of electrostatic charging[END_REF][START_REF] Liu | Open and Closed Edges of Graphene Layers[END_REF][START_REF] Rotkin | Analysis of non-planar graphitic structures: from arched edge planes of graphite crystals to nanotubes[END_REF] At smaller angles the bond separation increases and the density of bonds along the boundary decreases, such that there are no adjacent uncompensated bonds for the edge atoms to form a perfect sp 2 termination. These edge structures then reduce their energy through interlayer bonding, so that the "edge capping" now effectively occurs via bonding into the adjacent layers, resulting in an array of prismatic dislocations. Interestingly, there is a marked change in the energy dependence for angles below ~15°, where we find a larger curvature in the flat layers immediately adjacent to the dislocation core (for additional images (see Supplementary Material Fig. S2). At these low tilt angles, the quadrupole core energies (i.e. dispersed dislocations) are lower than along a grain boundary. Typically, migration of dislocations towards a grain boundary can significantly lower defect formation energy, and our calculations indicate that this process may be inhibited for low-angle tilt grain boundaries which could lead to incoherent interfaces between graphite grains. Above 15°, grain boundary configurations are lower in energy than the quadrupoles and will tend to align, similar to dislocation cores in silicon [START_REF] Lehto | Effects of Dislocation Interactions: Application to the Period-Doubled Core of the 90°Partial in Silicon[END_REF] Notably, our calculations at an intermediate angle of θ ≈ 60°reveal a very low core energy, and no clear preference for either the α or β ILB cores, with tilted graphene edge atoms bonding into different atoms in the flat grain (see Supplementary Material Fig. S3). The sp 3 bond angle remains close to 90°at these angles, and the grain tilt arises from net curvature of the adjacent graphene sheet. We can understand this by noting that, in general, there is an energetic competition at play in the angled grain boundary interfaces. On the one hand, the sp 3 -bonded carbon atoms in the tilted grain will prefer a 90°contact angle, which drives the tilted planes towards attaching orthogonal to the flat sheet. However, a 90°match is not favoured when matching the interlayer spacing in the tilted grain to the in-plane periodicity of the flat grain. In practise, for large-angle grain boundaries such as the 60°, this results in a configuration where the layer spacing matches, with local curvature in the layers at the boundary towards a 90° contact angle (see Fig. S2).

We also notice that these interlayer configurations can localise sp 2 bonds. At these angles, arrays of dislocation cores effectively create lines of internal surfaces, which show similar behaviour to an external reconstructed non-functionalised diamond surface, so-called "Pandey π-bonded chains". [START_REF] Walter | Geometry of the (2×1) reconstruction of diamond (111)[END_REF][START_REF] Pandey | New dimerized-chain model for the reconstruction of the diamond (111)-(2×1) surface[END_REF] This is exemplified by the two parallel zigzag chains, where the "external" atoms are sp 3 with only single bonds. This results in pairs of atoms with a spare p z orbital, which are completely isolated from all other sp 2 atoms in the lattice, so that they form a strong π-bond together, as shown in Supplementary Material Fig. S3, where they are joined by green bonds due to substantial bond compression.

Dislocation migration and thermal stability

Thus far the discussion has focussed on the thermodynamic stability of prismatic dislocation core structures.

Processes of bond formation, breaking and reconstruction within a core region are among the most fundamental properties of the prismatic edge dislocation core, and the kinetics of dislocation motion is a key factor in determining structure and properties. Notably, while certain structures may be thermodynamically favoured, kinetically they may not be easily accessible. Additionally, bond-breaking processes are paramount both in determining the relative thermal stability of ILB configurations and in determining dislocation mobility. Prismatic dislocation motion can occur along two inequivalent directions: between different basal planes, and therefore along the Burgers vector direction (i.e. in the c-axis direction) via dislocation glide, and along a basal plane or perpendicular to its Burgers vector, via dislocation climb.

Dislocation glide must proceed by bonding into a different layer, through the successive debonding and rebonding of the carbon atoms along the dislocation line in a process of double-kink formation followed by kink propagation, which necessitates the breaking of interlayer bonds within the core region.

To address these questions, CI-NEB calculations have been performed between the ILB (bonded) and FSE (broken) configurations of the bulk zigzag (α-ZZ) and armchair (AC) core configurations. Initially, we work with the same thin slab unit cells used in our energetic calculations, and so we are obliged to break all bonds along the dislocation core simultaneously. This means that our results do not incorporate the formation and propagation of kinks (in glide) and jogs (in climb) respectively, which are beyond the scope of the current study. These processes are likely to be important in realistic physical scenarios, and so these results should be taken purely as an indication of the most important physical processes mediating dislocation motion. We also consider the energetics of the bond breaking process in isolated cores, which have been created by extending bulk cores along the dislocation line.

Bond-breaking and glide

Fig. 14a shows the energy barrier vs transition coordinate of the debonding process for both the zigzag and armchair configurations and, in the inset, the initial and final structures of the NEB calculation.

For the zigzag core, the energy barrier to break an interlayer bond is 0.56 eV in our ribbon cells (0.56 eV/bond or 0.23 eV/Å), whereas the reverse rebonding process costs only 0.218 eV (0.109 eV/bond or 0.054 eV/Å). The armchair core demonstrates a similar, though less pronounced, asymmetry, with forward and reverse barriers of 1.478 eV (0.739 eV/bond or 0.346 eV/Å) and 0.676 (0.338 eV/bond or 0.158 eV/Å), respectively. This process remains endothermic and extensive in the number of interlayer bonds broken along the dislocation line. For n = 2 broken bonds along the armchair core and n=1 along the zigzag, we find that the core spontaneously reconstructs, and so this point has been omitted. c) Side and d) Plan view of broken bonds along the α-ZZ core.

On the basis of these first-principles energy barriers, we can estimate the probability that a single individual bond will fully break at a given temperature and infer the approximate temperature dependence of dislocation mobility. A minimal requirement for the formation of dislocation kinks will be the breaking of some fraction of the interlayer bonds along a dislocation line. We assume that the thermal activation of an individual interlayer bond is completely uncorrelated from that of adjacent bonds, and ignore conditions of in-plane and interlayer strain, and kink nucleation. We anticipate that strain effects will tend to increase the energy barrier while the nucleation and propagation of kinks can lower the required temperature since it is necessary for fewer bonds to break. Hence, the following calculation is a rough order of magnitude estimate only of the dislocation mobility.

The transition rates between the broken and unbroken configurations are evaluated using the Arrhenius equation, which is appropriate under the assumption that the timescales of bond-breaking and thermal processes are widely separated. The transition rate between any two configurations is given by ,

𝑟 𝑖→𝑗 = 𝐴 0 𝑒 - ∆𝐸 𝑖→𝑗 𝑘 𝐵 𝑇 (1)
where is the corresponding energy barrier between configurations i and j, is Boltzmann's constant, ∆𝐸 𝑖→𝑗 𝑘 𝐵 T is the temperature and is the attempt frequency, which is taken to have the value Hz. Given 𝐴 0 𝐴 0 = 10 13 a set of transition rates and under that assumption of thermal equilibrium, the probability that a single interlayer bond is in any particular configuration at a given temperature is ,

𝑝 𝑖 = Σ 𝑗 𝑟 𝑗→𝑖 Σ 𝑖,𝑗 𝑟 𝑗→𝑖 (2)
which we then evaluate using the respective DFT barriers heights. The resulting probability distribution (see

Supplementary Information Fig S5)

shows that at low temperatures (T<1500K) the probability that an individual bond is broken is essentially zero, implying a very low density of broken bonds along a dislocation line. Cores are highly thermodynamically stable and immobile up to relatively high temperatures (>1500K), above which there is an appreciable population of broken bonds at thermodynamic equilibrium.

Although approximate, this order of magnitude estimate compares reasonably well with experimental imaging of prismatic cores in carbon nanotubes, where spontaneous prismatic glide is not seen at temperatures below 2000K. [START_REF] Huang | Self-Templated Growth of Carbon-Nanotube Walls at High Temperatures[END_REF] We note that this discussion does not consider the ZZ-57-ZZ FSE core, which in principle is the most stable ZZ prismatic core. This cannot glide in bulk without first reconstructing, since otherwise the glide process would involve incorporation of a 57-reconstructed edge into a fully hexagonal layer, which is highly unstable. This edge may come into play, for example through constrained annealing, but in the temperature range 1500-2000K we expect the edge will be able to reconstruct and hence seems unlikely in general to impede prismatic dislocation glide.

In the above discussion we assume that all bonds along the dislocation core break and reform during motion simultaneously. In practice however dislocation migration is likely, as in other materials, to commence with double kink formation, followed by kink migration. To model this process requires symmetry breaking along the dislocation core. In general this requires the use of extremely large supercells which are beyond the scope of this current study. However, to provide some insight into this process, we have performed a set of structural relaxations of the isolated α-ZZ core for different numbers of broken interlayer bonds (Fig. 14b). These calculations have been performed at the Γ-point, in an isolated cell which is periodic only in the direction of the dislocation line. Carbon atoms far from the core region have been fixed in single and bilayer configurations. and dangling carbon bonds have been passivated using hydrogen atoms. For additional details of the supercell used, see Supplementary Information Section 6 & Fig. S6)

We find that the bond-breaking process remains endothermic as a function of the number of broken bonds, with an average energy of approximately 0.4 eV/bond, which is larger than the nanoribbon cells due to conditions of strain and deformation along the dislocation line. The large and monotonic energy increase as a function of broken bonds is highly suggestive that processes such as kink and jog formation are required for dislocation motion. Local conditions of applied stress, the presence of environmental contaminants, edge reconstructions, and other processes are also likely to play an important role in realistic scenarios. Adjacent dislocations can strongly interact with one another, and dislocations of opposite signs should show strong, elastic-field driven attraction. At sufficiently high temperatures, this inter-core interaction necessitates the pile-up and eventual annihilation of oppositely-signed prismatic cores, a process which will release significant strain from the graphite lattice and anneal damage. We have examined this process using pairs of zigzag dislocation cores. In order to fully anneal and restore a perfect graphite lattice, it is necessary that both of the adjacent cores have the same (α-ZZ) core configuration. As annihilation must proceed through the breaking and rebonding of the edge structure in both adjacent cores, this guarantees that the carbon atoms along each zigzag edge inhabit opposite graphene sublattices, which is necessary to ensure perfect graphene structure in the reconstructed sheet. In principle this process can also occur for a pair of β-ZZ cores, although this is not considered here due to the higher energy of that structure.

Fig. 15 shows the annihilation barrier and corresponding core structures of two oppositely-signed α-ZZ prismatic dislocations. The annihilation barrier of this dislocation pair (0.28 eV/Å) is only moderately higher than the barrier for normal zigzag glide, and the transition state for core annihilation is simply the FSE configuration of one edge, which is the same as the transition state for normal prismatic diffusion. Once the interlayer bonds of one core are broken, the corresponding bonds in the adjacent prismatic spontaneously break, leaving perfect sp 2 graphene sheets in the layers above and below, and joining the two half-sheets together and creating a new, perfect sp 2 -bonded layer (Fig. 15c).

Notably, other low energy core structures are possible. For example, two adjacent α-ZZ cores can lower their energy by 0.3 eV/Å by converting one core to a β-ZZ structure through reorientation of the sp 3 bond (Fig. 15c). This structure can then form an sp 3 -bridge between the adjacent cores, creating a fully sp 3 -zone within a prismatic dislocation complex (Fig. 15d). This substantially lowers the energy compared to a pair of isolated dislocations, only requires a moderate energetic barrier to be circumvented and has a sufficiently high barrier for the reverse process that it is likely metastable. It is tempting to see such metastable structures in HRTEM images (see for example, the left circled region in Figure 19b, and the lower-right region in Figure 19c). It is likely that other metastable structures exist, and they may provide a partial explanation for the unannealable damage observed in high temperature irradiated graphite [START_REF] Heggie | Buckle, ruck and tuck: A proposed new model for the response of graphite to neutron irradiation[END_REF]. Such metastable structures lock in substantial amounts of energy into the lattice and may also be potential candidates for higher temperature energy release peaks in irradiated graphites.

Dislocation climb

Finally, we consider the complementary diffusion mechanism of dislocation climb. Prismatic climb is an integral part of the motion of the prismatic core, and it is not possible to fully consider the unconstrained motion of a prismatic core without it. Due to the Bernal stacking of the surrounding graphite lattice, a prismatic dislocation in the lowest-energy C-stacked configuration cannot glide indefinitely between layers, and reorientation of the prismatic edge so that the edge carbon atoms align with those on adjacent layer is necessary for completely free motion. This is shown schematically for the zigzag configuration in Fig. 16. While the prismatic core can easily bond into the adjacent B layer, due to the overlap of carbon atoms on the C and B layers, the A layer is distinct in that the edge atoms of the prismatic layer lie above a hollow site rather than a line of carbon atoms. Hence, in order to continue to perform glide in this direction, either the adjacent A-stacked sheet or the C-stacked prismatic sheet must climb -execute motion perpendicular to the c-axis direction -to facilitate further glide. The entire process is depicted for the armchair (zigzag) core in Fig. 17. A prismatic core which is initially bonded at the lowest-energy AC (α-ZZ) site, moves to an adjacent AC' (β-ZZ) site via a three-step barrier. The initial and final steps simply consist of the bond-breaking mechanism. The intermediate steps are sp 2 -hybridised "dangling" edges, which climb simply by the passage of the flat bottom layer against the edge so as to alter the stacking. This process has a small barrier in our cell of 0.050 eV/Å. In reality, interlayer sliding is an extensive quantity, with an energetic penalty which is proportional to the area within a prismatic dislocation loop. In the bulk of graphite, this process is likely to proceed via the creation of a stacking fault and the emission of a basal dislocation, which as previously discussed are large and beyond the capability of conventional plane-wave DFT. However, our calculation does demonstrate the basic mechanism of prismatic climb, and is indicative that the climb -and therefore glide -should proceed relatively freely once interlayer bonds can be broken. Other processes, such interactions with vacancies and interstitials, are also likely to be relevant to prismatic climb. In order to determine the degree that different core structures will be visible experimentally, Fig. 18 shows the results of simulated TEM images for ZZ-ZZ-ZZ ILB and FSE cores, from both smaller monoclinic DFT cells, and a larger MD simulation using LAMMPS [START_REF] Plimpton | Fast Parallel Algorithms for Short-Range Molecular Dynamics[END_REF] with the multilayer graphene hybrid neural network potential [START_REF] Wen | Hybrid neural network potential for multilayer graphene[END_REF]. While the DFT structures can be clearly distinguished, the ILB is less evident in the MD simulated structure (top yellow square, Fig. 18(c)). The contrast for bonded reconstructed edges is also dependent on the degree of defocus (see Supplementary Materials), and this suggests that in non-C s corrected microscopes it may be difficult to distinguish FSE and ILB cores. Reprinted from Ref. [START_REF] Johns | Formation of carbon nanostructures in nuclear graphite under high-temperature in situ electron-irradiation[END_REF], Copyright (2019), with permission from Elsevier.

Simulated TEM Images

Discussion and Conclusions

In this work, we have performed extensive ab-initio calculations of prismatic dislocated cores, in isolated and bulk supercells. There are a number of implications of the calculations which are potentially relevant to models of graphite deformation subject to irradiation and other damage sources. For both isolated and bulk cells, the lowest energy structure lies along the armchair direction, which agrees well with previous theoretical and experimental results for prismatic loops, graphite step edges and graphene nanoribbons edges [START_REF] Suarez-Martinez | Dislocations of Burgers vector c/2 in graphite[END_REF][START_REF] Koskinen | Self-Passivating Edge Reconstructions of Graphene[END_REF][START_REF] Zobelli | A comparative study of density functional and density functional tight binding calculations of defects in graphene[END_REF]. The lowest energy armchair core structure involves the formation of interlayer bonds in order to eliminate dangling bonds, while for the zigzag core we find low energy free-standing Dienes reconstructed cores, and zig-zag interlayer bonded cores. Along both crystallographic orientations, differences in line energy are relatively small, and it is likely that multiple edge configurations can occur.

Local stacking conditions influence the prismatic core structure. Our calculations of isolated cores find higher energies for all AB-stacked prismatic cores due to strain and disruption of perfect aromaticity within the core. The small line energy of a basal dislocation in graphite, of the order of 0.07 eV/Å, is substantially less than the energy differences between AA-and AB-stacked prismatic cores [START_REF] Dai | Structure and energetics of interlayer dislocations in bilayer graphene[END_REF], suggesting that AA-stacked cores with associated basal dislocations to restore AB stacking in the lattice are most likely to be the dominant reconstruction. In the zigzag case the free-standing edge is stabilised by bond rotations along the core, which cannot be transferred during dislocation motion and will effectively freeze dislocation glide until the zigzag edge is restored.

Bulk calculations of prismatic cores were performed in quadrupole and grain boundary configurations. Importantly, we find that while high-angle grain boundaries result in stable, low-energy prismatic dislocation arrays, at low angles the quadrupolar configuration is preferred, suggesting that regular grain boundary arrays are less likely at smaller inter-grain angles. Notably, at intermediate angles we find bonding configurations with local double bonds which are extremely similar to the reconstructed Pandey diamond surfaces, occurring within the bulk of rotationally disordered graphite.

Our NEB calculations provide significant insight into the microscopic processes underlying different material deformation modes. Calculated energy barriers indicate that interlayer bonds are highly stable, and significant bond breaking can only occur at high temperatures. This barrier also sets constraints on the motion of the prismatic dislocation. Prismatic dislocations and grain boundaries of c-axis character are therefore likely to be highly immobile until high temperatures (T > 2000K), a result which agrees well with experimental results for prismatic dislocations in carbon nanotubes. [START_REF] Huang | Self-Templated Growth of Carbon-Nanotube Walls at High Temperatures[END_REF] The observation that a prismatic core can climb via interlayer sliding, rather than (or in addition to) the classical vacancy and interstitial trap-and-release mechanism seen in isotropic 3D materials may have important implications for irradiation-induced changes. In principle, prismatic climb, by the sliding of small loops against adjacent layers, could lead to the generation of basal dislocations in the vicinity of prismatic loops in bulk. This suggests a possible mechanism which could lead to the generation of basal dislocations, explaining the star-shaped patterns seen on the surface of irradiated HOPG at low temperatures [START_REF] Hinks | Dynamic microstructural evolution of graphite under displacing irradiation[END_REF], and which can have a profound effect on irradiation-induced volumetric change. [START_REF] Wen | Hybrid neural network potential for multilayer graphene[END_REF][START_REF] Mchugh | Ripplocations in layered materials: Sublinear scaling and basal climb[END_REF] Finally, our calculations of the barrier to anneal a pair of prismatic cores is essentially the same as the barrier to perform glide, and we conclude that these defects are unlikely to anneal out until the same high temperatures where prismatic glide can occur. While we find very little additional energy is required in order for pairs of oppositely-signed cores to annihilate, we do find metastable structures which can impose substantial energy barriers toward the realignment and annealing in the core region. These more disordered sp 3 -bonded regions are a likely contributor to the unannealable damage evident in graphite irradiated at high temperatures. [START_REF] Heggie | Buckle, ruck and tuck: A proposed new model for the response of graphite to neutron irradiation[END_REF] The energy release involved when these anneal out could be significant, and may also be a contributing factor to Wigner energy.

There are a number of interesting future research directions on the basis of this work. The biggest limitation of our ab-initio calculations is the cell size. Currently, the most commonly used graphene potentials do not correctly capture the sp 3 bonding, and consequently cannot model the prismatic dislocation core [START_REF] De Tomas | Graphitization of amorphous carbons: A comparative study of interatomic potentials[END_REF]. This severely inhibits our ability to examine certain properties, which in principle would require multiple simulations with, minimally, thousands of carbon atoms to accurately map out the entire energy range. Future work which could both accurately and more efficiently model the chemical interactions of graphene edges, for example through the development and use of more accurate molecular dynamics potentials, would do much to illuminate issues related to the diffusion and kinking of dislocation cores.

This would also open the door to examine a wider variety of dislocation structures, with intermediate combinations of dislocation line and Burgers vector, and would allow the direct simulation of prismatic loops. There are a number of other interesting questions suggested by our work, including questions regarding the full range of grain boundary structures at intermediate angles, the interaction of environmental impurities with dislocation cores, kinks, jogs and diffusion, and the magnetic and electronic properties of prismatic cores, which have the potential to be similar to the novel states observed at graphene nanoribbon edges. The study of the prismatic dislocation is overall of immense foundational interest, is likely immediately relevant to models of graphite damage, and is deserving of significant further work.

Fig. 1 :

 1 Fig. 1: Graphite lattice with different Bravais lattices (hexagonal in blue and orthorhombic in red), and high-symmetry directions indicated.

Fig. 2 :

 2 Fig. 2: Prismatic dislocations of a) vacancy character, which incorporates a basal component. This allows both "BC"-stacking across the dipole stacking fault and"AB"-stacking in adjacent sheets far from the core and b) interstitial character,

Fig. 3 :

 3 Fig. 3: a) Reconstructed free-standing edge (FSE) AA-stacked zigzag core. In the pictured cell, Dienes bond rotations have occurred along the top layer edge to reduce dangling carbon bonds. b) Interlayer-bonded (ILB) AA-stacked zigzag dislocation core. Dangling bonds in the top layer are eliminated by bonding into the layer below. The dislocation line direction, , which lies 𝑙 →

. 4 :

 4 Fig. 4: Non-functionalised terminations of graphene free-standing edges (FSE). Edge atoms are marked in red. Lowest energy reconstruction along both crystallographic orientations have been highlighted.. a) Zigzag FSE-ZZ, with glide and shuffle planes are indicated by dotted yellow and green lines, respectively b) Dienes reconstructed FSE-57 c) Reconstructed Klein FSE-RK d) Armchair edge FSE-AC e) Shifted armchair edge FSE-AC' f) FSE-56 edge, resulting from gain or loss of carbon atoms g) FSE-677 edge from a bond rotation along the armchair edge.

Fig. 5 :

 5 Fig. 5: Link between prismatic dislocation core edge configuration and Burgers vector. Core structures with different Burgers vector have been constructed through different combinations of graphene edges. This allows distinct stacking across the dislocation line and bonding within the core. a) A core constructed from the intersection of three ZZ edges leads to AA stacking in the bilayer region. b) The RK edge is shifted by a bond length with respect to a ZZ edge. This gives both a different core structure and AB stacking in the bilayer region.

  armchair cases, respectively. Negative values imply stability of the ILB core relative to the free-standing edge.Selected structures are shown in Fig.6& Fig.7, and additional structures are shown in Fig.S1.It is interesting that the RK-ZZ-ZZ configuration, where a monolayer reconstructed Klein edge bonds into bilayer zigzag edges in the core region, (Fig.7a) has the lowest energy of all FSE and ILB zigzag structures considered in this work (i.e. a RK edge binding into the basal plane below it). The resulting structure gives a line of shared 5-8 defects at the boundary. Every other zigzag-oriented core has a Burgers vector which lies only in the x-z plane and therefore contains only edge components. However the RK-ZZ-ZZ is the only zigzag core we have considered which also includes a screw component (i.e. movement along the dislocation line), in the lowest-energy configuration. In the absence of a screw component there is a triangular arrangement in the core (see inset of Fig.7a) which is sheared away upon full relaxation.

Fig. 6 :

 6 Fig. 6: a) ZZ-ZZ-ZZ-α core, where ML and both BL regions have zigzag edges, and b) ZZ-ZZ-ZZ-β core, which again has the same edge structure but bonds into the bottom layer along the opposite sublattice through a relative displacement of one bond length. C-C bonds are coloured depending on the degree of bond strain as shown in the key on the left, with white matching the pristine graphene bond length (1.42 Å).

Fig. 7 :

 7 Fig. 7: a) RK-ZZ-ZZ core, featuring one pentagonal bond reconstruction in the ML region, inset shows the initial core before shearing in the y-direction and b) RK-ZZ-RK core, which allows AB stacking. Reconstructed pentagons are indicated for clarity.

Fig. 8 :

 8 Fig. 8: a) AC-AC-AC dislocation core structure b) AC-AC-AC' dislocation core structure (square structure marked in yellow as a guide to the eye). C-C bonds are coloured depending on the degree of bond strain as shown in the key on the left, with white matching the pristine graphene bond length (1.42Å).

Fig. 9 :

 9 Fig. 9: Schematics of different simulation cells. a) Prismatic loops in bulk graphite, each nanoribbons inserted into the bulk giving a pair of prismatic dislocations. b) A prismatic dislocation array at a graphite c-axis tilt grain boundary. c) Relaxed supercells containing a quadrupole configuration of zigzag dislocation cores in an orthorhombic supercell and

Fig. 10 :

 10 Fig. 10: Side and end-views of the a,b) zigzag (α-ZZ), c,d) β-ZZ and e,f) armchair (AC-AC-AC) ILB dislocation cores in bulk graphite.

Fig. 11 :

 11 Fig. 11: Different bulk zigzag prismatic configurations, calculated in a monoclinic cell a) Unreconstructed ZZ-RK-ZZ in an ILB and b) FSE configuration c) ILB ZZ-RK-ZZ core with pentagonal reconstruction. d) The lowest-energy ZZ-57-ZZ core. Bonds that are stretched (compressed) are coloured green (red).

Fig. 12 :

 12 Fig. 12: Armchair prismatic core configurations a) Initial hexagonally-bonded configuration (AC-AC-AC ILB) b) AC-AC-AC FSE c) AC-AC-AC', which features the 4-8 square structure.

Fig. 13 :

 13 Fig. 13: Fully-relaxed prismatic dislocation quadrupole cells a) Formation line energy vs angle for the armchair and zigzag cores, 0° corresponds to the quadrupole/isolated value. b) 19 o armchair grain boundary c) sp 2 capped 90° armchair grain boundary.

Fig. 14 :

 14 Fig. 14: a) Energy vs transition coordinate for the spontaneous breaking process of the zigzag (α-ZZ) and armchair (AC) prismatic cores b) Total energy fraction of broken bonds for an isolated zigzag (α-ZZ) prismatic core of length 1.9nm (12 interlayer bonds).

3. 3 . 2 Fig. 15 :

 3215 Fig. 15: Energy barriers to fully anneal (dark blue) and to form a metastable bound state (light blue) vs transition coordinate, from the same initial state. a) In order to fully anneal and restore perfect graphite structure, the two adjacent cores must adopt the same initial (α-ZZ) configuration, so that they have a net Burgers vector of zero, and the two bonded-in carbon atoms (highlighted, red) in each core lie on different sublattices. b) The adjacent pair of dislocations can then annihilate, resulting in three perfect graphene

Fig. 16 :

 16 Fig. 16: Schematic depiction of prismatic glide and climb a) Top down view showing the initial structure of a prismatic core where the prismatic (C stacked) layer edge is on the same sublattice (α atom) as the lower, (B stacked layer) b) Side view of the same structure in a. The prismatic core can glide freely in this configuration between layers C and B. c) Further glide is inhibited because the prismatic edge atoms are above the hollow site of the A layer, this necessitates climb or interlayer sliding of the A layer.

Fig. 17 :

 17 Fig. 17: Prismatic climb barriers for the glide and climb processes, and (insets) armchair core configurations Note that the migration climb energy is dependent on the stacking fault width, which is small in these current cells, but is in general extensive and proportional to the area of stacking fault. a) Initial AC-AC-AC bonded core configuration. b) After bond-breaking, the core can climb by sliding adjacent graphene sheets. c) The re-bonding process which takes the core to the higher energy AC-AC-AC' square structure. (Only around half the calculated points are marked for visual clarity. Lines connect directly calculated points and are a guide for the eye.)

Fig. 18 :

 18 Fig. 18: Simulated TEM images of prismatic cores in graphite: (a) DFT optimized bulk monoclinic cell with ZZ-ZZ-ZZ-α ILB cores, (b) DFT-optimized bulk monoclinic cell with ZZ-ZZ-ZZ FSE cores, (c) Molecular dynamics bulk structure, optimized using LAMMPS [54] and the multilayer graphene hybrid neural network potential [55], having both (upper) ZZ-ZZ-ZZ-α ILB and (lower) ZZ-ZZ-ZZ FSE cores, as highlighted by yellow squares. Full defocus sequences are given in Supplementary Materials.

Fig. 19 :

 19 Fig. 19: Experimental literature HRTEM images showing: (a) two snapshots of prismatic dislocation glide in a carbon nanotube. Reprinted from Ref. [16], Copyright (2007), with permission from John Wiley & Sons. This structure is similar to Fig.17(a-c). (b)

  Fig.15(a). (d) The presence of sp 3 zone with dislocation climb within prismatic dislocation complexes (image reproduced from [57]). These structures match the modelled counterparts illustrated in Fig.15(d) and Fig.17(b), respectively. (e) A region of large

  (b), respectively. (e) A region of large c-axis tilt in irradiated graphite, accompanied by material which is visibly similar to the bonded prismatic dislocation core.

Line energy (eV/Å) 1010

			AA	AB	Δ AA-AB
	FSE ZZ	1.357	1.317	1.319	-0.002
	FSE RK	1.379	1.353	1.363	-0.009
	FSE 57	0.991	0.993	1.002	-0.009
	<	> Armchair Isolated	AA	AB	Δ AA-AB
	AC FSE	1.064	1.112	1.038	0.073
	AC 56	1.506	1.559	1.491	0.068
	AC 677	1.155	1.240	1.181	0.026
	Table 1 Free-standing edge line energies for different reconstructions in finite-width graphene nanoribbon
	geometries. We include results for isolated nanoribbon edges, and nanoribbons deposited on a graphene substrate in
	AA and AB stackings (broken bond prismatic dislocation configurations)

Table 2 :

 2 Formation

line energy for different interlayer-bonded prismatic dislocation cores, for both the zigzag and armchair cores. Cores which preserve aromaticity are highlighted in green. The last column gives the energy difference compared to the lowest energy free-standing configuration, FSE 57 and AC FSE, for the zigzag and

are the ZZ-RK-ZZ cores without a pentagonal reconstruction.(see Fig. 11b)Table 5 :

 5

	RK	ZZ	ZZ	ILB	1.897	0.574
	ZZ	RK	ZZ	ILB*	2.172	0.849
	ZZ	RK	ZZ	FSE*	2.212	0.889
			Armchair Line energy (eV/Å)
	ML	Top	Bottom	Type	E f	Δ
	AC	AC	AC	ILB	1.263	0.000
	AC	AC	AC'	ILB	1.429	0.165
	AC	AC	AC	FSE	1.439	0.176
	AC	AC	AC'	FSE	1.382	0.119
		Top	Bottom	Type	E f	Δ
	ZZ	57	ZZ	FSE	1.323	0.000
	ZZ	ZZ	ZZ	ILB	1.409	0.086
	ZZ	ZZ	ZZ	ILB-β	1.593	0.270
	ZZ	ZZ	ZZ	FSE	1.632	0.309
	ZZ	RK	ZZ	ILB	1.672	0.349
	ZZ	ZZ	ZZ	FSE-β	1.675	0.352
	ZZ	RK	ZZ	FSE	1.702	0.379

* These .Formation line energies, E F , (eV/Å), for both interlayer bonded (ILB) and free-standing edge (FSE) armchair and zigzag cores in a bulk monoclinic cell. Δ gives the difference in line energy between each configuration and the lowest-energy. Low-energy ILB ZZ cores are highlighted in green and are shown in Fig.

10

, AC are highlighted in yellow and shown in Fig.

12

)

Perfect basal dislocation glide occurs within this plane with Burgers vector b = 1/3< >. These perfect

Acknowledgements

This work was supported by the United Kingdom EPSRC grant EP/R005745/1, Mechanisms of Retention and Transport of Fission Products in Virgin and Irradiated Nuclear Graphite. Kenny Jolley and Pavlos Mouratidis also gratefully acknowledge funds from EDF energy generation 2016-2021. The authors gratefully acknowledge the use of Athena at HPC Midlands+, which was funded by the EPSRC grant EP/P020232/1 as part of the HPC Midlands+ consortium. CE and AI acknowledge ANR-16-CE24-0008-01 "EdgeFiller" and ANR-20-CE08-0026 "OPIFCat" for funding. DE acknowledges support from the TUBITAK-2219 post-doctoral research abroad fund.

SUPPLEMENTARY MATERIALS Prismatic Edge Dislocations in Graphite

James G. McHugh 1.2.3* , Pavlos Mouratidis 1 , Anthony Impellizzeri 4 , Kenny Jolley 1 , Dogan Erbahar

Quadrupole & Monoclinic Boundary Conditions

Having considered the relative energies of different prismatic cores for the same unit cell, we briefly examine the dependence of the calculated energies on cell dimensions for the two low energy bonded configurations (ZZ-ZZ-ZZ and AC-AC-AC) in the quadrupole configuration, using different orthorhombic and monoclinic boundary conditions. These energies are summarised in Supplementary Table 1 (Fig. 9g defines the structural parameters L 1 and L 2 ). There is moderate variation in the predicted core line energy between different cell configurations due to varying elastic energy, but we generally observe consistent formation line energies within a range of 0.1 eV/Å of one another, with a markedly smaller variation for the armchair cores.

Angular Variation of Grain Boundary Structures

We include here some additional images of simulated grain boundary structures. Supplementary Figure 4: 60° zigzag grain boundary

Bond-breaking probability

Supplementary Figure 5: Bond-breaking probability vs temperature, calculated using Equation ( 2) and CI-NEB barriers for the armchair and zigzag cores.

Isolated, extended dislocation supercells

To examine the dynamics of the bond-breaking process, we have created supercells which are long in the direction of the dislocation line. These cells are created using the initial isolated core structures, which have been repeated along the dislocation line (y) direction. Even for isolated cells this results in very large supercells when a full dislocation dipole is considered. To circumvent this, we have created a smaller cell containing only one dislocation core by cutting out a segment of the full isolated cell, freezing carbon atoms