Energetic stability of graphene nanoflakes and nanocones

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Energetic stability of graphene nanoflakes and nanocones Natalie Wohner, Pui Lam, Klaus Sattler

*

Department of Physics and Astronomy, University of Hawaii at Manoa, 2505 Correa Road, Honolulu, HI 96822, United States

A R T I C L E I N F O

A B S T R A C T

Article history:

We investigate the relative energetic stability of a variety of nanographene structures such

Received 21 June 2013

as graphene nanoflakes, nanoribbons, nanodisks, and nanocones. We calculate the cohe-

Accepted 24 October 2013

sive energies with respect to hydrogen passivation, edge nature (zigzag versus armchair)

Available online 1 November 2013

and shape (triangular, rectangular, hexagonal). The cohesive energy is confirmed to increase with size for all these structures. We pay particular attention to optimally-compact circular flakes and compare our theoretical results with round disks produced in a plasma torch atmosphere. We find in the calculations that round shape does not have preferred relative stability. This suggests that the observed disks are grown under conditions where carbon atoms are highly mobile. For graphene nanocones we obtain a similar result. Experimentally, the open base of a 19-degree-cone is observed perpendicular to the cone axis, but this does not correspond to the most stable configuration as obtained by the calculations. Instead, we find that both, disks and cones, prefer minimal length of the edge termination rather than a maximum in the cohesive energy. With respect to our results we discuss for polycyclic aromatic hydrocarbons (PAH) and atomic clusters, as models for graphene flakes, the significance of the cohesive energy for the observed abundances.  2013 Elsevier Ltd. All rights reserved.

1.

Introduction

Hexagonal networks of sp2 carbons have attracted tremendous interest in recent years, with focus on planar single layers of graphite in two- (2-D), one- (1-D), and zero- (0-D) dimensions, which are exciting new nanomaterials with intriguing properties. Due to the confinement of electrons and quasi-particles in two dimensions and small lateral size and their large variety of possible topological structures, there are many open questions to be answered by theorists and experimentalists for these new materials. Graphene nanoflakes (GNFs) are compact structures of fused hexagonal rings of aromatic benzene. If hydrogenated, the GNFs are giant aromatic molecules which are termed polycyclic aromatic hydrocarbons (PAHs) [1]. Due to the extensive research of PAHs over several decades, PAH data

can now be used to better understand hydrogenated nanographene species. Compared to bulk graphene, nanoflakes exhibit additional properties due to quantum confinement and edge effects. Stability, electronic structure and transport properties depend sensitively on the GNF morphology and the type of surface structure and termination with functional groups. This has attracted considerable attention from chemists, physicists, and material scientists in recent years. In theoretical studies, graphene nanoflakes are usually chosen with the two fundamental edge geometries, zigzag (zz) and armchair (ac), and dangling bonds are saturated with a single hydrogen per carbon atom. GNFs with triangular (TGNFs), rectangular (RGNFs), and hexagonal (HGNFs) shapes show that their electronic, vibrational, optical and magnetic properties may have new and often unexpected features [2–8]. Graphene nanoflakes have the properties of

* Corresponding author. E-mail address: [email protected] (K. Sattler). 0008-6223/$ - see front matter  2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.carbon.2013.10.064

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semiconductor quantum dots, where electrons are confined in all three directions. They often are called ‘artificial atoms’ since the electrons occupy spectrally sharp energy levels [9]. All flakes are less stable than icosahedral fullerenes but are as stable as carbon nanotubes of similar size [5]. Small flakes tend to be semiconductors or insulators while large flakes tend to a gap closure approaching the graphene electronic structure [5]. The HOMO–LUMO gap can be engineered by changing size and shape [10]. Graphene nanoflakes are expected to have many exciting applications, such as photovoltaic cells [11], organic nonvolatile flash memory devices [12], supercapacitors [13], chemical sensors [14], and anodes in lithium ion batteries [15]. Also graphene nanoflakes could have important applications in the biomedical fields such as DNA cleavage [16], detection of human immunoglobulin [17], or for enhancing the efficiency of polymerase chain reaction [18]. Epoxy–graphene nanoflake composites have outstanding thermal conductivity and mechanical properties [19]. Polymer-based composites were found to be good materials for energy harvesting [20]. Porous graphene has been proposed for applications as atmospheric nanofilters, as gas sensors and as membranes in fuel cells [21]. For the development of formation techniques it is often crucial to know the relative stability between different nanographene structures. Most technological applications require graphene nanoflakes with specific shapes and edge structures. Substantial efforts have been made to produce homogeneous carbon nanoflakes experimentally [22–25], but their nucleation and growth is still not well understood. In particular, it is often discussed how the abundance of a particular species is correlated with its cohesive energy. If growth kinetics is not a factor, the one would expect that the most stable species are the ones which occur predominantly in the experiment. However, the growth conditions are usually complex, and properties other than stability may lead to preferred abundance. The problem of nucleation and growth is also complicated by the great variety of possible structures. A large number of structural isomers is possible and increases fast with the size of the flakes [26]. Also, since fullerenes and nanotubes are known to be thermodynamically preferred, graphene nanoflakes will only form under special nonequilibrium conditions. The formation methods for GNFs focus on two main approaches, top-down and bottom-up. Top-down approaches consist of exfoliation of graphene layers [27], etching flakes into small islands [28], the controlled carving of a graphene sheet [29], electrochemical expansion of graphite in an electrolyte [30], or lithographic patterning techniques [31]. Bottom-up methods often involve solution chemistry with growth on surfaces [23], or chemical vapor deposition [32]. Multi-layered graphene flakes have been fabricated using a thermal plasma jet system [33] and by pyrolyzing biodegradable composites [34]. Often, lateral size selection is required which has been achieved by controlled centrifugation [27] and size-exclusion chromatography [35]. A review of the preparation of graphene nanoflakes was recently given by Wang et al. [36]. Nanosized circular graphene flakes (CGNFs), also termed graphene nanodisks (GNDs), are the most compact structures

compared to other shapes. They are considered to be among the most stable topologies, while the least stable structures are the narrow graphene nanoribbons [5]. GNDs have been studied theoretically by classical molecular dynamics [37] and quantum mechanical methods [5,38], and experimentally they were investigated by using transmission electron microscopy (TEM) and electron diffraction [39]. Carbon nanocones (CNCs) are curved graphene sheets with the addition of pentagons at the cone tips. They have first been produced in small quantities in 1994, and it was shown that there are five types of cones according to five possible cone angles [40]. The number of configurations of closed-apex cones depends on how many pentagons are located in the conical tip. In the applied vapor deposition method, performed in ultra-high vacuum (UHV), nanocones were formed together with nanotubes [41]. A few years later graphitic cones were produced in large quantities by a plasma torch technique [42]. In subsequent years the localized electron states near the pentagons [43] and the intriguing nonperiodic bulk structure of carbon nanocones have been studied [44,45]. Carbon cones are expected to have large electrostatic dipole moments [46] which depend on the apex angles of the cones. Also, resonance states dominate the electronic structure close to the Fermi level and depend sensitively on the number and location of the pentagons in the conical tip [44]. Carbon cones may also have interesting applications as possible candidates for hydrogen storage [47,48] and as field emission tips [49,50]. In a different development, the hexagonal–pentagonal cone structure, originally developed for the carbon network, was also used for explanation of the observed conical core structure of the HIV virus [51] and various retroviruses [52]. A number of theoretical techniques have been applied for the study of nanographene species. Among these are Hartree– Fock-based semiempirical methods [53], classical molecular dynamics simulations [37], density functional and tight binding methods [54], or studies based on the Clar sextet concept [55,56] and Kekule´ structures [57]. Enthalpies of formation of 77 PAHs were calculated with the semiempirical Austin Model 1 (AM1) [58] approximation, and good agreement was obtained between experimental and calculated values [59]. Semiempirical parametric method 3 (PM3) [60], an extension to AM1, is often used for large PAHs, for which a full ab initio calculation is not possible in a reasonable amount of time due to their large number of carbon atoms [61]. Fast calculation techniques such as AM1 or PM3 are also required in studies where the properties of a large number of nanoflakes or PAHs should systematically be compared. Besides using PM3 for stability determinations, is has been applied to calculate electronic, topological and other properties, such as HOMO and LUMO energies, molecular hardness, polarizability and atomic charges [62]. AM1 has been applied to develop various quantitative structure–property relationships (QSPR) [62], for transition state conformations [63], C–H bond energies [64], photolysis [65], and PM3 for reaction path analysis [66]. By using AM1 and PM3 excellent geometries for PAH minimum energy structures were obtained [67]. For 81 PAHs, metabolic activation mechanism calculated by PM3 resulted in accuracy of higher than 80% for identifying their carcinogenic activity [68]. Also, published toxic

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environmental equivalency factors (TEFs) for a variety of PAHs could correctly be reproduced by PM3 calculations [69]. These examples show that the PM3 technique can give excellent results for the study of two-dimensional structures of fused benzene rings, such as PAHs and flat nanographene. In this work we study the energetic stability of various nanographene clusters using PM3 molecular orbit theory. The binding energy per carbon atom (cohesive energy) is taken as a measure for their relative stability. We show that PM3 results for graphene nanoflakes are very close to those obtained recently with density functional theory (DFT) [7] (Section 3.1). We systematically change the sizes of triangular, rectangular, hexagonal and round graphene nanoflakes and find increasing stability with increasing size for all the structures. Hydrogen-passivated flakes have higher cohesive energy than unpassivated flakes (Section 3.2). We compare hexagonal flakes with only zigzag and only armchair edges and find that zigzag edges give the higher cohesive energy (Section 3.3). The preference of compactness is seen from studies of rectangular flakes with different aspect ratios, where square-shaped flakes exhibit the highest stability for a given number of carbon atoms (Section 3.4). Comparison of triangles, squares and hexagons shows increasing relative stability in this order (Section 3.5). Extrapolation toward large sizes of these structures yields the cohesive energy of bulk graphene. One might expect that round flakes have the highest cohesive energy of all shapes since they are the most compact structures. However, we find that round flakes have binding energies less than those of hexagonal flakes, for the same number of carbon atoms (Section 3.6). This may be due to the competition between compactness and edge structure for the stability of a flake. We compare the theoretical results with experiments where free carbon flakes were produced (Section 3.7). These were formed by condensation of a carbon–hydrogen mixture in a plasma torch, and the method yields a large amount of circular flakes. It is an intriguing result that only circular flakes are produced in the experiment while hexagonal flakes are calculated to have superior stability. In equilibrium situations, one would expect a correlation between stability and abundance of molecular species. These observations may be due to the ultrahigh-temperature non-equilibrium nucleation and growth conditions in the plasma torch method which involve complex time evolution, temperature and carbon density profiles, and therefore may not lead to the most stable structures obtained by our calculations. Also, the plasmatorch-produced disks are multilayered graphene structures which may form differently compared to single-layered ones. In order to probe these result for other nanographene structures, we analyzed graphene cones produced with the same plasma torch method (Section 3.8). For cones with 19 degree opening angles, our calculations give equal cohesive energies for base of the cone normal to the axis and for the tilted one (Section 3.9). In the experiments however we only observe one type (base-line normal to axis) and not the other type. Again, as for the flat flakes, the most stable structure is not preferred in the experiment. The observed disks and cones have in common the smallest possible length of the edge configuration. For disks this is given by the circular shape and for the cones by the orientation of the base

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perpendicular to the cone axis. We conclude that it is rather the minimum edge surface than the cohesive energy which determines the preferred structure of these species. We also calculate the cohesive energies of nanoflakes and nanocones and relate the results to their relative abundance (Section 3.10). In Section 3.11 we discuss the formation of carbon nanodisks and nanocones by the plasma torch method. Our studies open interesting questions about the nucleation and growth of nanographene and the correlation between stability and abundance.

2.

Methods

2.1.

Calculations

2.1.1.

Computer modeling

Computer modeling was used to obtain initial atomistic structures for carbon nanoflakes and nanocones. Two-dimensional single-layer graphene flakes were constructed with various sizes and shapes. These flakes are cut-out flat sheets of the honeycomb graphene network, containing only hexagons as structural units. We constructed four types of carbon flakes; with triangular, rectangular, hexagonal, and spherical shapes. We chose nanocones with 19-degree opening angle, with five pentagons at the apex, in order to compare the results with the observations. The tip of the cone was constructed with five- and six-membered rings according to the C60 fullerene structure and the isolated pentagon rule. The graphene network was conically bent to form the bulk of the cone with the requirement of not leaving any defects at the closure line [40].

2.1.2.

Calculation methods

The atomic structures of flakes and cones were first geometry-minimized employing force field molecular mechanics (MM) theory. This is a fast method which can be applied to structures containing several thousand atoms. For disks and cones up to about 300 atoms we subsequently used quantum mechanical PM3 [60] restricted Hartree–Fock (RHF) calculations for geometry optimization and calculation of cohesive energies. For all flakes, studied with these computations, all bonding orbitals were completely filled and all antibonding orbitals were empty. PM3 parameters are fitted to the experimental enthalpy of formation of organic compounds. The parameters incorporate electron correlation and zero-point corrections. We show that PM3 gives results very close to those obtained with density functional theory (DFT) for nanographene flakes [7]. All species were edge-passivated with hydrogen in order to saturate the dangling surface bonds. The binding energies, obtained for the H-passivated species are corrected by 112 kcal/mol per C–H bond [70–72]. Variation of values between 110 and 114 kcal/mol (approximately the range of uncertainties in the published data) resulted in negligible change in our results. After correction for the C–H edge bond energies, the obtained energy per carbon atom represents the cohesive energy of the carbon skeleton [7]. Neglecting the contribution from the zero-point energy, the cohesive energy

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is defined as the energy difference of the compound compared to the energy of the free carbon atoms, and is taken as a measure for its relative energetic stability.

2.2.

Experiments

2.2.1.

Samples used in this work

Powder samples produced by the modified Kvaerner Carbon Black and Hydrogen (CBH) production method [73] have been used for our experimental studies. The samples were fabricated in a carbon-arc plasma generator torch configuration under a continuous flow of hydrocarbon (heavy oil). The CBH method is a two step process, where the hydrocarbons do not reach pyrolysis temperature in the first step and are only partially decomposed to form polycyclic aromatic hydrocarbons (PAHs). In the second step, the PAHs are mixed with the plasma gas and exposed to the intense heat in the plasma arc zone which causes the PAHs to be converted into graphitic particles. The effective plasma temperature was estimated to be at least 2000 C [42]. The method yields about 10% carbon black, 20% carbon cones, and 70% carbon disks [74]. Carbon black essentially consists of carbon particles with sizes less than about 100 nm [75]. Small crystallites within a single carbon black particle are randomly oriented. Carbon blacks have turbostratic structure where layers of graphene flakes are parallel but rotated around the c-axis [75]. It is remarkable that the CBH plasma process developed for carbon black production, leads to the formation of well-ordered atomic layers in graphene disks and cones [74].

2.2.2.

Sample analysis with SEM

For microscopic analysis, the powders were dispersed in ethyl alcohol and after sonication, the dispersions were placed on warm metal substrates and dried. The metal substrate had been polished and cleaned prior to the deposition of the suspension. Then the samples were transferred to the vacuum chamber of a Hitachi S-2400 scanning electron microscope (SEM) where the morphologies were determined.

3.

Results and discussion

3.1. Comparison nanoflakes

of

PM3

with

DFT

for

graphene

In Fig. 1 we show the structures of graphene nanoflakes (GNF) with their respective cohesive energies, after the species were fully optimized with molecular mechanics (MM) and PM3. We choose nanoflakes for which we can compare our calculated energies with those obtained from density functional theory (DFT) by Ricca et al. [7]. In the first row we display structures and cohesive energies of hexagonal graphene nanoflakes (HGNFs); C54H18, C94H24, C150H30, C216H36, and C294H42. In the second row we add the data of nanoflakes with various shapes; C78H22, C82H24, C84H24, C90H24, and C94H26. It can be seen that the cohesive energies obtained by PM3 are very close to those obtained by DFT [7]. They differ only by 0.05– 0.13 eV, depending on size and shape of the structure. For the HGNFs the differences become less with increasing size of the flakes, with C294H42 showing a deviation of only

0.05 eV. This is a remarkable equivalence of values considering the substantial differences in geometry optimization methods and algorithms and the applied energy calculation techniques. Comparison of various theoretical techniques (AM1, PM3, DFT) for carbon nanocones also revealed that the optimized geometries were similar at each level of theory, which may be due to the high rigidity of nanocarbons [76]. We conclude that PM3 is a useful technique for energy calculations of graphene nanoflakes and nanocones.

3.2.

Bare versus H-passivated GNFs

The unreconstructed bare graphene nanoflakes have edge atoms with singly-occupied electron orbitals. Patterned adsorption of hydrogen has been studied for triangular and hexagonal shape, controlling the H coverage from bare to fully hydrogenated GNFs [77]. For nanoflakes with zigzag and armchair edges, mono-hydride and di-hydride terminated flakes have been proposed [78]. Also, the nature of carbon atoms at GNF edges has been studied, both for flat and curved graphenes. It has been postulated that the active sites for chemisorptions can be carbene- and carbyne-type carbon atoms [79]. Free zigzag sites may stabilize by forming carbene-like edges and free armchair sites by forming carbynelike edges [80]. Also, the formation of pentagon–heptagon pairs has been considered to occur at the bare GNF edge [81]. Such a process transforms the bare edge sites from zigzag to armchair [80] and keeps the planarity of the GNF. In this work, the dangling bonds are passivated with single hydrogen atoms in order to avoid the effect of the open edge on the cohesive energy of the GNFs. Since all investigated flakes have the same type of functionalization, we can compare cohesive energy results for a variety of flakes with different shapes and edge structures, and compare flakes with carbon nanocones. In an initial study we compare the cohesive energies of triangular graphene nanoflakes (TGNFs), with unpassivated and passivated edges. The relative stability comparison is straightforward. In both cases, we find the same trend, namely the cohesive energy increases with size. In addition, it can be seen in Fig. 2, that H-passivated species have higher cohesive energies than the bare flakes, for the same number of carbon atoms. This result is obtained with the PM3 semiempirical method, using the sigma–pi separability approximation. Sigma–pi mixing has been discussed for hydrogenated sp2/sp3 amorphous carbon films [82], organic molecules such as cyclodimes of benzenes [83], heteroaromatic complexes of benzene [84], and long polyenes [85]. While the sigma–pi separability seems to be a good approximation for flat graphene it has been found that sigma–pi hybridization is induced by departure from planar morphology [86], and it becomes important for strongly-curved carbon networks such as small-diameter carbon nanotubes [87] and fullerene C60 [88]. It may also play a role for carbon nanocones, which may be checked using first principle calculations. The comparison of bare and passivated GNFs is of great importance since a large fraction of carbon atoms are located at the free edge, and therefore edge atoms of the bare GNFs are under compression [89]. The resulting stress is

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Fig. 1 – Structures and cohesive energies of selected graphene nanoflakes (GNFs). PM3 results (this work) are compared with DFT results by Ricca et al. [7].

Fig. 2 – Cohesive energies as a function of size for bare and passivated graphene nanoflakes (GNFs).

determined from the difference of edge energies between strained and unstrained systems. The broken bonds of an unreconstructed graphene edge generate compressive edge stresses leading to edge warping which becomes tensile [90]. Compression for the bare species applies to any type of edge, and there is only a small difference in stress between different edge types. The stress for armchair edge is slightly larger than that of zigzag edge. When a single hydrogen terminates each of the dangling bonds at the edge, the unreconstructed edge energies drop by 1-2 orders of magnitude [91], and hydrogenation of the edges virtually eliminates both the edge energy and edge stresses [90]. Edge covalent functionalization stabilizes the GNFs and also affects their electronic, optical and magnetic properties [92]. The degree of edge passivation depends on the amount of hydrogen available in the growth process. By controlling the H-coverage from bare GNFs to half hydrogenated and then

to fully hydrogenated GNFs, the transformation of small-gap semiconductor to half-metal to wide-gap semiconductor occurs [77]. In our studies, the edges are completely passivated by hydrogen in order to avoid dangling bonds and edge reconstructions. The choice of full passivation makes sense since we later will compare the theoretical results with experiments where the species were produced in a high-density atomic hydrogen atmosphere which likely leads to full H-passivation of the surface atoms. The triangular GNFs of Fig. 2 have zigzag edges on all sides. When dangling r bonds at the edges are saturated with hydrogen, all the carbon atoms are sp2 hybridized. This leads to the stabilization of the GNFs with respect to the unpassivated edges. In order to avoid the stress-induced contributions to the cohesive energies, all other GNFs studied in this work were passivated with a single hydrogen per carbon atom. We note that although hydrogen adsorption has a substantial influence on the thermodynamic properties, it has little effect on the fundamental bandgap, ionization potential and electron affinity, as calculated for both armchair and zigzag converged structures [8]. For both, unpassivated and passivated flakes our calculations predict that the cohesive energies of small GNFs are lower than those of larger GNFs. This is consistent with computer model calculations of PAH destruction from exposure to cosmic rays which show that small PAHs are destroyed faster than large ones [93].

3.3.

Armchair- versus zigzag-terminated GNFs

The atomic structure at the edge of a GNF may have different configurations, since the graphene network can be cut in various ways. For rectangular shapes, two sides are armchairand the other two are zigzag-terminated. For triangular and hexagonal flakes, all sides can be either zigzag or armchair. In Fig. 3a, we show the cohesive energies of hexagonal GNFs with armchair- (ac-HGNFs) and zigzag-terminated edges

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comparison makes sense if the nanoflakes are edge-passivated with single hydrogen atoms because in this case GNFs and PAHs are identical species. In hydrocarbon flames, PAHs have been analyzed to possess zigzag edges [26], consistent with our stability predictions. Based on calculations, zigzagterminated GNFs are often considered the most stable and abundant GNFs, surviving extreme destructive conditions in interstellar space [7]. On the other hand, entirely armchair-terminated PAHs were produced by chemical synthesis. Hexagonal armchair PAHs with 90 (hexabenzocoronene, HBC) [97] and up to 222 [98] carbon atoms have also been produced. It is remarkable that armchair rather than zigzag edge configuration was synthesized in these experiments. It suggests that growth kinetics overrides the influence of cohesive energy in this case.

3.4.

Fig. 3 – (a) Size dependence of cohesive energies for zigzagand armchair-terminated hexagonal graphene nanoflakes. (b) Atomistic structures of the armchair-flakes in Fig. 3a.

(zz-HGNFs). In both cases the cohesive energy increases with size. In addition, it can be seen that zigzag gives a higher cohesive energy than armchair, for the same number of carbons. This is consistent with previous theoretical studies of hexagonal graphene flakes [5]. Fig. 3b shows the atomistic structures of the four armchair flakes, for which we obtained the cohesive energies, plotted in Fig. 3a. The flakes C42H18 and C114H30 are completely planar whereas C222H42 is slightly bent. It is remarkable that the edge of C270H54 is twisted and results in a non-planar flake. As can be seen in Fig. 3b, C270H54 shows a different edge structure compared to the other three flakes. While C42H18, C114H30 and C222H42 have entirely regular armchair bays, C270H54 additionally has two deep bays on each side. These deep cuts into the graphene sheet might be the reason for twisting of the C270H54 flake. This effect lowers the cohesive energy of the C270H54 flake with respect to the planar flakes, as shown in Fig. 3a. Mixed results have been reported concerning the stabilities of ac- and zz-GNFs. For the pristine, unpassivated GNF edges, armchair was predicted more stable than zigzag in some computational studies [53,94], and zigzag more stable than armchair in others [4,37]. Also, for hydrogen-terminated graphene structures, theoretical studies yielded mixed results. Armchair GNFs were found more stable than zigzag, by the works of Wassmann et al. [95] and Gan et al. [94], but zigzag GNFs turned out to be more stable in the works of Girit et al. [96] and Ricca et al. [7]. From our results we might expect that zigzag GNFs are found more frequently in experiments due to their higher cohesive energy. For a comparison with experiments, we may consider the occurrence of polycyclic aromatic hydrocarbons (PAHs) as candidates for graphene nanoflakes. This

Square and rectangular GNFs

Now we address the influence of compactness for the stability of graphene nanoflakes and choose rectangular shapes with different aspect ratios. Fig. 4 shows the cohesive energies for square graphene nanoflakes (SGNFs) and rectangular graphene nanoflakes (RGNFs). The size of the SGNF is increased by adding successively further layers of benzene rings on two sides, as indicated in the figure. The size of the RGNF is increased by adding rows of benzene rings to one side, successively increasing its aspect ratio. We note that the curve for the square GNFs shows small periodic fluctuations due to the periodic change of the edge structure with size. It can be seen in Fig. 4, that for both, the rectangle and the square, the cohesive energy increases with size. However, the ribbons are clearly less favorable for the stability than the compact squares. The cohesive energy of the RGNFs is strongly affected by the increasing aspect ratio with increasing size, as seen in Fig. 4. This is consistent with previous studies of graphene nanoribbons (GNRs) with large aspect ratio [99]. GNRs are quasi-one-dimensional species showing peculiar electronic and optical properties due to the lateral confinement of

Fig. 4 – Cohesive energies for square and rectangular graphene nanoflakes.

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electrons and quasiparticles [100] and have promising applications [9]. Armchair and zigzag terminated nanoribbons have different electron band structures resulting in different magnetic properties [9]. The preference of compactness compared to long or branched shapes is also seen in studies of PAHs in interstellar clouds [7] and in hydrocarbon flames [26]. Also, the observation of interstellar C60 is an example for preference of compact structure [101]. The higher stability of compact PAHs leads to their predominant abundance. This correlation between stability and occurrence, as observed in these cases, does not apply however in general, as we will see in following paragraphs.

3.5.

Triangular, square, and hexagonal GNFs

If compactness of the GNFs is favorable for their stability, we might expect that flakes with hexagonal shapes have the highest cohesive energies relative to triangles and squares. This is indeed the case. In Fig. 5 we compare triangular, square and hexagonal GNFs as a function of size. All three structures show higher cohesive energy with increasing size. Among these three configurations, the hexagons have the highest cohesive energies, for equal carbon atoms per flake. This is consistent with previous studies where highly symmetric, compact GNFs showed the highest stability [7,37]. In Fig. 6 we plot the cohesive energies for triangles, squares and hexagons as a function of the hydrogen/carbon ratio. This represents the ratio between the number of carbon atoms at the edge and the bulk atoms, since every edge carbon is passivated by one hydrogen atom. Straight lines are obtained for the three types of GNFs. Extrapolation towards zero hydrogen/carbon ratio leads to the cohesive energy of bulk graphene, which is about 7.3 eV per carbon atom. This suggests that our calculated data are quantitatively valuable. Even though the cohesive energy is not a perfect measure of nanographene or PAH stability, it often is used to predict possible experimental outcomes in bottom-up formation. For example, highly stable, hexagonal PAHs are expected to

Fig. 5 – Hexagonal, square, and triangular GNFs; comparison of cohesive energies.

Fig. 6 – GNF cohesive energies as a function of their hydrogen/carbon ratio, for hexagonal, square, and triangular GNFs.

be resistant to destruction in the harsh environment of the interstellar medium (ISM) [102].

3.6.

Hexagonal versus circular GNFs

In Fig. 5 we have seen that there is a rise of cohesive energy from triangles to squares to hexagons. The structures become increasingly compact and the length of the edge becomes subsequently shorter. Following this trend, we may expect that round shape is the ultimately best structure giving the highest binding energy among all the other shapes since it is the most compact structure. In order to probe this expectation we calculated the cohesive energies of hexagonal graphene nanoflakes (HGNFs) and circular graphene nanoflakes (CGNFs), and the result is given in Fig 7a. We note that the flakes cannot be constructed perfectly round due to the hexagonal structure of the benzene units. However we can approximate the round shape by adding an increasing number of armchair sites at the corners of the zigzag hexagonal flakes. In Fig. 7b we show the structures of the calculated round flakes, separated in three groups; (1) C84H34, C138H30, C258H42; (2) C234H42, and (3) C132H36, C252H48. In the first group the edge is composed of alternating armchair and zigzag sites. These flakes are perfectly flat. The cohesive energy data points are located very close to the curve for the hexagonal flakes. In group two, we show one flake which contains zigzag, armchair, and protruding trio carbon sites. The cohesive energy of this flake is clearly below the HGNF curve. It appears that the reduced cohesive energy is due to the protruding trio carbon atoms. We display two structures from the third group. These flakes have no zigzag sites but only armchair and protruding trio carbon sites. Their cohesive energies are significantly below those for the hexagonal flakes, for equal numbers of carbons. We note that these flakes are not perfectly flat but are twisted at six edge sites, as shown in the figure. The results displayed in Fig. 7a show that circular flakes have cohesive energies less than those of hexagonal flakes,

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circular flakes have a variety of different armchair and corner configurations at the edge which should lead to favorable growth pathways. Various types of radicals present in the growth zone, with C1 , C2 , C3 or C5 carbon skeletons, such as CH3 [108], C2H and C2H3 [109] or C3H3 and C5H5 [110] could react with the diverse edge structure of the circular GND with high efficiency because of the large number of possible ways to match the radicals with the edge. These many possibilities may lead to effective and fast growth of circular disks within a diverse molecular reaction zone.

3.7.

Fig. 7 – (a) Cohesive energies of hexagonal (HGNFs) and round graphene nanoflakes (RGNFs). (b) Computer generated models of round graphene nanoflakes (RGNFs), separated in three categories. Group 1, alternating sections of zigzag- and armchair edge sites; group 2, zigzag-, armchair-, and protruding trio carbon edge sites; group 3, armchair and protruding trio carbon edge sites. The flakes of groups 1 and 2 are perfectly flat, while the flakes of group 3 are sixfold distorted with atoms twisted relative to the plane of the flakes.

for equal numbers of atoms. Greater compactness of the round GNFs obviously does not lead to higher stability. While increased compactness stabilizes a flake, the inclusion of armchair sites has the opposite effect (see Fig. 3a). This leads to smaller cohesive energies for the circular flakes, compared to hexagonal flakes with only zigzag edges. Large pericondensed PAHs are usually modeled in calculations as hexagonal flakes, such as circumcoronene (C54H18) or circumcircumcoronene (C96H24), and have attracted great interest with respect to their electronic absorption spectra [103] and their interesting optical [104] and magnetic [2] properties. They have motivated suggestions for applications in spintronics [2], quantum computing [105], bioimaging [106], catalysis [107], and photovoltaics [104]. Circular flakes, on the other hand, have little been studied, but may have interesting functional properties and applications as well. In fact,

SEM Studies of Graphene Disks

We now compare the theoretical result of Fig. 7a with our experimental observations. We used samples produced by the Kvaerner Carbon Black and Hydrogen Process (CBH) [73] and analyzed the species by scanning electron microscopy (SEM). In Fig. 8 an electron micrograph containing predominantly carbon disks is given. The diameters of the discs range between about 0.3 and 1.3 lm. SEM cannot resolve the atomistic edge structure but is sufficiently accurate in showing the shape of a flake. For example, hexagonal or spherical shapes can well be distinguished. It can be seen that none of the disks have a hexagonal form. We analyzed several hundred disks in the SEM images and found all of them with circular shapes. This is consistent with the observations in other studies of samples produced by the CBH method [42,46,74,111]. Obviously, the preferred stability of hexagonal with respect to round flakes, as found in the calculations, is not reflected in the experiments. Round flakes are found despite their lower cohesive energies. It shows that cohesive energy and abundance are not correlated in this case. This may be due to the fact that our theoretical calculations yield results for zero temperature whereas experimentally the disks form at high temperatures. In addition, our computer calculations are performed for monolayer structures while multilayer disks are produced in the experiments. Yet, the overall shape of a few-layer graphene disk is likely to be controlled by the shape preference of the single graphene layer, since the coupling between the layers is very weak.

Fig. 8 – 7.5 lm · 6.0 lm scanning electron microscopy image of a sample containing predominantly carbon disks, and in addition some carbon black particles and graphene nanocones.

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The association of theoretical stability with experimental abundance is an interesting issue to consider in general. One would expect that high cohesive energy of a molecule or atomic cluster also leads to preferred occurrence in nature, providing entropy and kinetic factors are negligible. Indeed, this has been verified by numerous experiments. In particular, many such results have been obtained from mass spectrometry. For example, clusters with magic numbers of atoms have been identified for many different atoms and molecules. Higher stability of the cluster is directly seen by an enhanced peak in the mass spectrum, i.e. preferred abundance. This has been found for clusters from various atoms and molecules, such as inert gases [112], metals [113], intermetallic compounds [114], metal carbides [115], alkali halides [116], molecular hydrogen [117], water [118], and many others. Correlation between abundance and stability has been found for clusters in a wide size range, for example for sodium clusters up to 22,000 atoms [119]. In addition, abundances of multiply-charged clusters [120] as well as their critical sizes for Coulomb explosion [121] were directly related to the stability of the clusters. In the case of carbon, the exceptionally stable C60 molecule has been discovered because the carbon cluster mass spectrum showed a strongly enhanced peak at 720 amu [122]. Other pronounced peaks in the carbon cluster spectrum, such as C50 and C70, have also been explained by enhanced stability [123]. These observations were made with fullerenes generated by different methods, such as supersonic expansion [122], or hydrocarbon combustion [124]. Magic numbers have also been identified for surface-supported clusters of metals [125], C60 [126] and other materials. The enhanced stability was related to processes of growth [117], solidification [127] and melting [128]. We also note that twodimensional nanographene flakes, such as polycyclic aromatic hydrocarbons, prefer the energetically favored compact structures both in hydrocarbon flames [26], and in the interstellar medium [7]. We conclude that in many experiments stability and abundance of species were correlated. On the other hand, there are many cases where stability and occurrence are not related. Many physical and chemical pathways for PAH formation are possible providing numerous ways for reaction mechanisms, intermediates, and rate limiting steps. Aromatic ring formation may occur via HACA (hydrogen abstraction C2H2-addition) [129], PAC (phenyl addition/cyclization +C6H5 followed by hydrogen elimination (–H)) [129], HAVA (hydrogen abstraction and vinyl +C2H3 radical addition), and many other processes. Stability of nanographene species may be considered with respect to processes of isomerization, degradation, ionization and complete dissociation. Since many PAHs are known to be toxic and carcinogenic for humans, numerous studies have been performed which can give essential information about degradation of nanographene. PAH degradation can occur for example in chemical [130], photochemical [131], radiation–induced [132], ultrasonic [133], bacterial [134], and microbial [135] processes. The degradation probability often depends on the specifics of the applied environment and method. Coming back to the results of Fig. 8, we see that the experiment yields round carbon disks. Comparison with the cohesive energies of graphene flakes in Fig. 7a however shows that

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round disks do not have preferred stability with respect to hexagonal flakes. Various reasons for this observation may be considered. First, the species are produced under very high temperatures and the flakes initially may be in a liquid state, as previously been proposed to explain the formation of spherical soot particles [136]. In general, a free liquid droplet acquires spherical shape in order to minimize its surface area. Similarly, graphene nanoflakes may be liquid initially before cooling to become solid platelets. This may lead to the minimum length of the edge as observed in the experiment.

3.8.

SEM studies of a 19-degree carbon cone

It is interesting to compare the properties of carbon disks with those of carbon cones. True periodicity is absent in graphitic cones since the curvature of the network and the bond lengths change consistently along the direction of the cone axis. Therefore, electronic properties of carbon nanocones are quite different from those of graphite and nanotubes [46]. In addition, the inclusion of pentagonal defects at the apex of the cone is affecting the properties of this structure. Fig. 9 shows the electron microscopy image of a 19-degree carbon cone, produced by the Kvaerner plasma-torch method. The cone is about 1.2 microns long and its width at the base is about 0.5 microns. We focus our attention on the orientation of the base at the open end of the cone. It can be seen for the cone in Fig. 9 that the termination at the open side of the cone is perpendicular to the cone axis. We have found the perpendicular orientation of the base for all five 19-degree cones which we imaged on the CBH sample. The same orientation has been reported in previous work where cone samples produced with the CBH method were analyzed [42,46,74,111]. In order to probe if this orientation of the base represents the lowest energy structure we performed theoretical PM3 studies on 19-degree carbon nanocones.

3.9.

Base orientation for 19-degree carbon cones

Fig. 10 shows computer-generated carbon nanocone models with a graphene network structure and two types of base terminations, only zigzag (Fig. 10a) or half zigzag half armchair

Fig. 9 – 3.8 lm · 3.0 lm scanning electron microscopy image of a carbon cone with a 19-degree opening angle.

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(Fig. 10b), with the base line tilted or perpendicular to the cone axis, respectively. We may expect that the cone of Fig. 10a is more stable since it has entirely zigzag coordinated carbons at the edge. On the other hand, the cone in Fig. 10b has a shorter base line, which may favor its stability. In order to compare the relative stabilities of the cones in Fig. 10, we calculated their cohesive energies starting with pure zigzag edges, then adding subsequently armchair sites until half- armchair, half-zigzag was obtained. This study gives the gradual transition between the two types of edges. For each edge configuration we did the calculations for increasing size of the flakes. The result is given in Fig. 11. We find that the two terminations yield virtually the same cohesive energies for the carbon skeleton, if cones with the same number of carbons are compared. From the calculated results we therefore may expect that both terminations, shown in Fig. 10a and b, are realized in the experiment. However, none of the cones which we analyzed had a tilted base but all were found with the base perpendicular to the cone axis. The same result had been obtained in previous studies, with nanocones produced by carbon vapor deposition [40]. These cones also had open-end bases with the baseline perpendicular to the orientation of the axis. Again there is a remarkable difference between the predictions from stability calculations and the actual species formed in the experiment. It may be explained by the fact that in both, the vapor deposition and the plasma torch method, the temperatures were very high. We suggest that the species in both experiments were initially in a liquid state and therefore the length of the edges is minimal. As mentioned in Section 3.7, liquid hydrocarbon nuclei have been invoked to explain the formation of uniformly spherical soot particles [136]. It is interesting to note that high temperatures during growth may allow the migration of individual carbon atoms along the edge, a process which has been observed with a transmission electron microscope in real time at the edge of a single atomic layer of graphene [96]. Such migrating atoms

In Fig. 12 we compare the stabilities of graphene nanocones and nanoflakes. Cohesive energy as a function of number of carbons is plotted. For both structures, the cohesive energies gradually increase with the size of the species. This is consistent with the observations for the graphene nanoflakes discussed before. In addition, we observe superior relative

Fig. 10 – Two computer generated atomic models of carbon nanocones with base terminations tilted and perpendicular to the cone axis, shown in Figs. 10 a and b, respectively.

Fig. 12 – Comparison of cohesive energies for zigzag 19degree carbon nanocones (zz-CNCs) and zigzag hexagonal graphene nanoflakes (zz-HGNFs).

Fig. 11 – Cohesive energies for carbon nanocones with pure zigzag edges (no armchairs; 0 AC), and one through four armchairs (1 AC-4 AC), plotted for structures with increasing size.

may fill vacant sites at the edge eliminating sharp corners and protruding edge regions. This eventually may smooth an irregularly shaped edge and lead to the observed round shapes of the graphene flakes. In the case of graphene nanoflakes the assumption of a liquid-like state could lead to the observed round shape and for nanocones to the baseline perpendicular to the axis.

3.10.

Cones versus flakes

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cohesive energies of cones when compared to disks. This may be due to a combination of the curvature of the graphene network, the presence of the fullerene-type tip, and the lower hydrogen/carbon ratio for the nanocones. It is interesting to consider in this respect the relative number of carbon disks and cones in the samples produced by plasma torch method. The results of Fig. 12 would suggest, if cohesive energy is relevant, that more cones are produced during formation under equal growth conditions. However, several studies have shown that samples fabricated with the Kvaerner CBH method, consist predominantly of disks (70%) and only of few cones (10%) [42,73,74]. Again, experimental abundance and calculated energies are not correlated.

3.11.

Formation of disks and cones in the plasma torch

We now consider the formation of nanographene in very hot environments, beyond the temperatures present in most combustion processes. We consider high-temperature formation in a plasma, which by definition is a fully or partially ionized gas, consisting of a mixture of electrons, ions, and neutral species [137]. One of the plasma types, the so-called ‘thermal plasma’, is characterized by temperatures up to 15,000 K, high energy density, and large abundance of highly reactive charged and neutral species [138,139]. Such a ‘thermal plasma’ is produced from both, arc-discharge devices or plasma torches [140]. The arc-discharge device for nanocarbon production operates by electric discharge in a gas atmosphere, usually He or Ar, and species are continuously produced through the sublimation of the carbon electrodes. The plasma torch consists of a cathode, an anode, and a gas injection channel. Nanocarbon species such as fullerenes, nanodisks and nanocones are formed by decomposition of the carbon-containing gas in the electric arc. Hydrocarbons are known to dehydrogenate at such high temperatures, yielding the same types of species as a pure carbon plasma [141]. We note that in the plasma formation of nanotubes, with arc-discharge between carbon rods, the temperature in the arc is about 3700 C [141]. At similarly high temperature, 3400 C-hot carbon vapor was deposited on a cold substrate in ultrahigh-vacuum (UHV), also leading to the formation of carbon nanotubes [142]. The same hot-temperature vapor deposition method yielded the formation of carbon nanocones [40]. The hot growth environment is characterized by a large variety of molecular radicals. These can be due to PAHs decomposing in the hot environment. PAH stability with respect to isomerization and decay will depend on the degree of their vibrational excitation which can be very high due to the low heat capacity. Many types of PAH fragments may be present in the hot environment. In hydrocarbon flames, radicals such as C2H and C2H3 [109] as well as C3H3 and C5H5 [110] and others [143] were identified as playing a role in PAH growth. The competition between growth and degradation has been studied in hot benzene flames, where molecules and radicals such as C2H2, C2H, and C2 are constantly released by hot fragmenting PAHs. These molecules may again be consumed for further PAH growth [144]. Formation of PAHs in hot combustion areas is often ascribed to the H-abstraction-C2H2-addition (HACA) mechanism

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[145]. Further pathways have been proposed [146], in addition to standard HACA. Hydrogen abstraction (-H) followed by methyl radical addition (+CH3) was considered for PAH growth in the gas phase [108]. For dimethyl ether (DME) diffusion flames, the methyl addition/cyclization (MAC) mechanism has been proposed for PAH growth [147]. We suggest that growth processes of PAHs found in hot combustion zones may also be relevant for nanographene growth in the plasma torch environment. In the CBH growth method, with which disks and cones in this work were produced, PAH’s are the precursors for the all-carbon structures. PAHs are also the precursors for growth of soot in hydrocarbon flames [148] and for the formation of carbonaceous grains [101] in interstellar dust clouds. In these hot environments, the presence of molecular radicals is considered to be of fundamental importance [149]. This leads us to propose a process for the preferred formation of round graphene nanoflakes. Circular flakes have a variety of different armchair and corner configurations at the edge which should lead to favorable growth pathways. Various types of radicals present in the growth zone, with C1 , C2 , C3 or C5 carbon skeletons, such as CH3 [108], C2H and C2H3 [109] or C3H3 and C5H5 [110] could react with the diverse edge structure of the circular GND with high efficiency because of the large number of possible ways to match the radicals with the edge. These many possibilities may lead to effective and fast growth of circular disks within a diverse molecular reaction zone.

4.

Summary

We have calculated the cohesive energy, as a measure for stability, of graphene nanoflakes (GNFs) with different shapes, and carbon nanocones (CNCs). The use of the same calculation technique for these various nanographene structures gives valuable information about their relative energetic stabilities, as a function of size. Unpassivated structures are stabilized by hydrogen termination of the edge atoms. Zigzag edge-termination is confirmed to be favorable for the stability compared to armchair. Increasing symmetry and compactness yields favorable structures, and square shape gives larger cohesive energies compared to rectangular shape. Higher compactness generally leads to increased binding energy, but this does not apply to the transition from hexagonal to circular shape. While hexagonal flakes are more stable than triangular and square flakes, the circular flakes are found to have lower cohesive energy due to their different edge structure. We compare the results of our calculations with experiments. Carbon flakes, grown in a plasma torch method, show circular shape and not the hexagonal shapes which one might expect from their superior stability. Also, carbon cones are not observed according to high stability consideration. We conclude that the cohesive energy is not the decisive factor for the observed shapes of these structures. It is rather a minimum of the hydrogenated edge surface which determines the preferred species. This seems to occur for formation processes where the temperatures are very high, and we suggest that the carbon atoms during growth are highly mobile similar to being in a liquid state. With respect to the

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interpretation of our results, we consider the significance of the cohesive energy of the species for their abundance in experiments. We discuss various formation and degradation processes with respect to stability for polycyclic aromatic hydrocarbons (PAHs), which can be considered as hydrogenated graphene nanoflakes.

5.

Conclusions

We conclude that our computational studies give valuable results concerning the energetic stability of graphene nanoflakes and nanocones. While cohesive energy and abundance are correlated in many systems such as in atomic cluster beams, with magic numbers as markers for high stability, such correlation is often not found. The growth process and abundance may be influenced by many conditions and parameters such as temperature, time- and space-varying molecular density profiles, the participation of hydrogen, oxygen, PAHs or other molecules, the presence of molecular radicals, and others. Despite these many possible influences in the formation and degradation of nanographene structures, the comparison of stability and abundance can be valuable for finding general principles for nanographene growth. While PAHs are naturally present in nature and have been studied experimentally for many years, graphene nanoflakes do not occur naturally and are very difficult to synthesize. This is why most of the results on graphene nanoflakes have been produced by computations. In order to test such studies, which often give mixed results, efforts need to be undertaken to produce graphene nanoflakes experimentally. Also, further computational studies, possibly with different methods, are required in order to deeper analyse the properties of the flakes. In this work, we suggest that graphene nanoflakes tend to have spherical shapes when grown in high-temperature reaction zones. This may be of importance for studies and applications of graphene quantum dots, whose electronic structures and functional properties depend on their shape. It also may better explain the electromagnetic emission bands of PAHs in interstellar space, the PAH formation in hydrocarbon flames, and the interaction between hydrogenated and all-carbon graphene nanostructures with respect to stability and abundance.

Acknowledgements We would like to thank Danilo Pescia and Urs Ramsperger at ETH Zurich for providing assistance with the scanning electron microscope (SEM).

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