Topological mechanochemistry of graphene
Elena F. Sheka, Vera A. Popova, Nadezhda A. Popova
Peoples’ Friendship University of Russia
In view of a formal topology, two common terms, namely, connectivity and adjacency, determine the ‘quality’ of C-C bonds of sp2 nanocarbons. The feature is the most sensitive point of the inherent topology of the species so that such external action as mechanical deformation should obviously change it and result in particular topological effects. The current paper describes the effects caused by uniaxial tension of a graphene molecule in due course of a mechanochemical reaction. Basing on the molecular theory of graphene, the effects are attributed to both mechanical loading and chemical modification of edge atoms of the molecule. The mechanical behavior is shown to be not only highly anisotropic with respect to the direction of the load application, but greatly dependent on the chemical modification of the molecule edge atoms thus revealing topological character of the graphene deformation.
Key words: graphene, molecular theory, mechanochemical reaction; chemical modification; strength characteristics, Young’s modules
1. Introduction differently for different members of the sp2 nanocarbon family. Therefore, one may conclude that the spatially 2 The modern topology in chemistry covers two large extended sp nanocarbons present not only peculiarly valleys, namely, formal, mathematical and empirical, structural chemicals, but the class of species for which the chemical. The former is concerned with the description of formal and empirical topology overlap. At the first time, molecular structures on the basis of finite topological results, presented in [11, 12] have revealed this tight spaces. The space shows itself as a mathematical image or interconnection in terms of molecular quantum theory. Not instrument of theoretical study. A large collection of only fullerenes, but carbon nanotubes and graphene (their comprehensive reviews, related to a topological description fragments) have been considered at the molecular level. of fullerenes from this viewpoint, has recently been The obtained results are related to the computational study published [1]. The second field covers vastly studied of the intermolecular interaction between one of the above 2 topochemical reactions. The space in this case is a physical sp nanocarbon molecules and one of the other addends, 2 reality defining the real place where the reactions occur. If among which there are both sp nanocarbons and the appearance of mathematical topology in chemistry can monoatomic species. The intermolecular interaction lays be counted off the publication of the Merrifield and the foundation of any reaction, so that its topological Simmons monograph in 1989 [2], topochemical reactions peculiarities may evidence a topochemical character of the have been studying from the nineteenth century (see [3] reaction under study. However, since the ‘quality’ of C-C and references therein). The first stage of the study was bonds is the most sensitive point of the inherent topology of 2 completed in the late nineteenth-twenties [4] and then sp nanocarbons, external actions, say, mechanical obtained a new pulse after appearing the Woodward and deformation, on the bonds should obviously result in Hoffman monograph, devoted to the conservation of orbital particular topological effects that accompany the relevant symmetry, in 1970 [5]. Since then, topochemical reactions intramolecular reactions. The current paper is devoted to have become an inherent part of not only organic, but the discussion of such reactions that are presented by a inorganic chemistry, as well. The readers, who are mechanochemical one related to uniaxial tension of a interested in this topic, are referred to a set of graphene molecule. comprehensive reviews [3, 6-9], but a few. The current situation in this field can be seen by the example of a direct structural understanding of a topochemical solid state 2. Uniaxial tension of graphene as a photopolymerization reaction [10]. mechanochemical reaction Nowadays, we are witnessing the next pulse, stimulating investigations in the field, which should be Below we will consider a particular topological effect attributed to the appearance of a new class of spatially caused by the influence of both the loading direction and 2 extended molecular materials, such as sp nanocarbons. the graphene molecule edge termination on the inherited Obviously, the main members of the class such as topology of the molecule. As turned out, the graphene fullerenes, nanotubes, and numerous graphene-based deformation under external mechanical loading is species are absolutely different from the formal topology extremely sensitive to the state of the edge atoms and viewpoint. Thus, fullerenes exist in the form of a makes it possible to disclose a topological nature of this hollow sphere, ellipsoid, or tube consisting of differently sensitivity. packed benzenoid units. Carbon nanotubes present Oppositely to real physical experiments, when predominantly cylindrical packing of the units. In changing the object shape under loading is usually graphene, the benzenoid units form one-atom-thick planar monitored, computational experiments deal with the total honeycomb structure. If we address the common terms of energy response to the object shape deformation that the formal topology, namely, connectivity and adjacency, simulates either tension and contraction or bending, we have to intuitively accept their different amount in the screwing, shift, and so forth. As for graphene, whose above three species. In its turn, the connectivity and mechanical properties are amenable to experimental study adjacency determine the ‘quality’ of the C-C bond structure with difficult, the computational experiments takes on great of the species, thus, differentiating them by this mark. significance. Since non-saturated C-C bonds are the main target for A lot of works are devoted to the calculation of chemical reactions of any type, one must assume that mechanical properties of graphene due to which two identical reactions, involving the bonds, will occur approaches, namely, continuum and atomistic ones have been formulated. The continuum approach is based on the Eyring more than sixty years ago [33], suggested the use well developed theory of elasticity of continuous solid of a well developed quantum-chemical approach of the media applied to shells, plates, beams, rods, and trusses. reaction coordinate [34] in the study of atomic structure The latter are structure elements used for the continuum transformation under deformation. Firstly applied to the description. When applying to graphene, its lattice structure deformation of poly(dimethylsiloxane) oligomers [35], the is presented in terms of the above continuum structure approach has revealed a high efficacy in disclosing the elements and the main task of the calculation is the mechanism of failure and rupture of the considered reformulation of the total energy of the studied atomic- polymers. It has been successfully applied recently for the molecular system subjected to changing in shape in terms description of the uniaxial tension of both graphene [29, of the continuum structure elements. This procedure 30] and graphane [31] molecules, thus exhibiting itself as a involves actually the adaptation of the theory of elasticity significant part of the current molecular theory of graphene of continuous media to nanosize objects which makes [28]. allowance for introducing macroscopic basic mechanical parameters such as Young’s modulus (E), the Poisson ratio zigzag mode zigzag mode (ν), the potential energy of the elastic deformation, etc into the description of mechanical properties of graphene. Since the energy of graphene is mainly calculated in the framework of quantum chemistry, which takes the object atom structure into account, the main problem of the continuum approach is a linkage between molecular configuration and continuum structure elements. Nanoscale continuum methods (see Refs. 13-17 and references therein), among which those based on the structural Fig. 1. Six mechanochemical internal coordinates of mechanics concept [18] are the most developed, have uniaxial tension of the molecule (5,5) NGr for two shown the best ability to simulate nanostructure materials. deformation modes. F , F , F , F , F и F are forces of In view of this concept, graphene is a geometrical frame- 1 2 3 4 5 6 response along these coordinates. Blue atoms fix the like structure where the primary bonds between two coordinates ends. nearest-neighboring atoms act like load-bearing beam members, whereas an individual atom acts as the joint of The main point of the approach concerns the reaction the related beams [19-22]. coordinate definition. When dealing with chemical The basic concept of the atomistic approach consists reactions, the coordinate is usually selected among the in obtaining mechanical parameters of the object from internal ones (valence bond, bond angle or torsion angle) or results of the direct solutions of either Newton motion laws is presented as a linear combination of the latter. Similarly, [22, 23] or Schrödinger equations [24, 25] under changing mechanochemical internal coordinates (MICs) are the object shape following a particular algorithm of introduced as modified internal coordinates defined in such simulation of the wished type of deformation. It should be a way as to be able to specify the considered deformational necessary to issue a general comment concerning modes [35, 37]. Thus, uniaxial tension and contraction are calculations based on the application of the DFT described by linear MICs similar to valence bonds. In the computational schemes. All the latter, except the recent one case of tensile deformation, the benzenoid pattern of [26], were performed in the framework of restricted graphene sheets and a regular packing of the units versions of the programs that do not take into account spins predetermined the choice of either parallel or normal MICs of the graphene odd electrons and thus ignore the orientation with respect to the chain of C-C bonds. In the correlation interaction between these electrons. The rectangular nanographene sheets and nanoribbons the peculiarities of the graphene odd electron behavior are former orientation corresponds to tensile deformation connected with a considerable enlarging of its C-C bonds, applied to the zigzag edges (zigzag mode) while the latter which, in its turn, causes a noticeable weakening of the odd should be attributed to the armchair edges (armchair electron interaction and thus requires taking into account mode). The MIC configurations of the two tensile modes of these electrons correlation [27, 28]. the (5,5) NGr molecule are presented in Fig.1. The In the case of atomistic approach, not energy itself, but molecule has been led into the foundation of previously forces applied to atoms become the main goal of performed computational experiments [29-31] and presents calculations. These forces are input later into the relations a rectangular fragment of a graphene sheet cut along zigzag of macroscopic linear theory of elasticity and lay the and armchair edges and containing 5 benzenoid units along foundation for the evaluation of micro-macroscopic each direction. The deformation proceeds as a stepwise mechanical parameters such as Young’s modulus (E*), the elongation of the MICs with the increment δL=0.1Å at each Poisson ratio (ν*), and so on. Nothing to mention that step so that the current MIC length parameters E and E* as well as ν and ν* are not the same so that their coincidence is quite accidental. Obviously, constitutes L = L0 + nδL , where L0 is the initial length atomistic approach falls in opinion comparing with the of the MIC and n counts the number of the deformation continuum one due to time consuming calculations and, as steps. Right ends of all the MICs are fixed so that these a result, due to applicability to smaller objects. However, it blue colored atoms are immobilized while atoms on the left possesses doubtless advantages concerning the description ends of MICs move along the arrows providing the MIC of the mechanical behavior of the object under certain successive elongation, once excluded from the optimization loading (shape changing) as well as exhibiting the as well. The relevant force of response is calculated as the deformation and failure process at atomic level. energy gradient along the MIC, while the atomic Recently a new atomistic approach has been suggested configuration is optimized over all of the other coordinates for the description of the graphene deformation based on under the MIC constant-pitch elongation. The results considering the failure and rupture process of graphene as presented in paper were obtained in the framework of the the occurrence of a mechanochemical reaction [29-32]. A Hartree-Fock unrestricted (UHF) version of the similarity between mechanically induced reaction and the DYQUAMECH codes [37] exploiting advanced first-type chemical ones, first pointed out by Tobolski and semiempirical QCh methods (PM3 version [37]). The corresponding forces of response F applied i Before deformation zigzag armchair along the ith MICs are the first derivatives of the total energy E(R) over the Cartesian coordinates [35]: dE∂∂ Hϕϕ ∂ dP (1) =<ϕϕ > +22
Here, ϕ is the wave function of atom of the ground state at fixed nucleus positions, Н presents the adiabatic electron Hamiltonian, and P is the nucleus momentum. When the force calculation is completed, the gradients are re- determined in the system of internal coordinates in order to proceed further in seeking the total energy minimum by atomic structure optimization. Forces Fi are used afterwards for determining all required micro-macroscopic mechanical characteristics, which are relevant to uniaxial tension, such as the total force of response , stress F = ∑ Fi i ⎛ ⎞ 2. Equilibrium structures of the (5,5) NGr with different σ = F / S = ⎜ F ⎟ / S , where S is the loading area, chemical modification of edge atoms before and after ∑ i completing tensile deformation in two modes of ⎝ i ⎠ deformation. Bare edges (top); H1-terminated edges the Young’s modulus , where both stress σ E = σ /ε (middle); H2-terminated edges (bottom). and the strain ε are determined within the elastic region of deformation. former case, the deformation is one-stage and is terminated on the 17th step of the deformation. In contrast, the deformational mode zigzag is multi-stage and consists of 3. Computational results 250 consequent steps with elongation of 0.1Å at each step [29, 30]. The formation of one-atom chain under zizgzag- Thus arranged computations have revealed that a high mode tension of a graphene sheet has been supported stiffness of the graphene body is provided by the stiffness experimentally [39]. of benzenoid units. The anisotropy of the unit mechanical Quite unexpectedly, the character of the deformation behavior in combination with different packing of the units has occurred to be strongly dependent on chemical situation either normally or parallel to the body C-C bond chains at the molecule edges [31]. As seen in Fig. 2b, the addition lays the ground for the structure-sensitive mechanism of the of one hydrogen atom to each of the molecule edge atoms mechanical behavior of the object that drastically depends does not change the general character of the deformation: it on the deformation modes [29-31]. The elastic region of remains a tricotage-like one so that there is still a large tensile deformation of both graphene and graphane (5, 5) difference between the behavior of zigzag and armchair NGr molecules is extremely narrow and corresponds to a modes. At the same time, the number of the deformation few first steps of the deformation. The deformation as a steps of zigzag mode reduces to 125. whole is predominantly plastic and dependent on many Even more drastic changes for this mode are caused parameters. Among the latter, the most important is by the addition of the second hydrogen atoms to the edge chemical state of the molecule edge atoms [32]. ones (Fig. 2c). Still, the armchair mode is quite Equilibrium structures of the molecule before and conservative while zigzag one becomes practically identical after uniaxial tension, which was terminated by the rupture to the former. The tricotage-like character of the of the last C-C bond coupling two fragments of the deformation is completely lost and the rupture occurs at the molecule, are shown in Fig. 2. Looking at the picture, two 20th step. main peculiarities of the molecule deformation should be Figure 3 presents a set of ‘stress-strain’ relations that notified. First concerns the anisotropy of the deformation fairly well highlight the difference in the mechanical with respect to two deformational modes. Second exhibits a behavior of all the three molecules. Table 1 presents strong dependence of the deformation on the chemical Young’s modules that were defined in the region of the composition of the molecule edge atoms. As mentioned elastic deformation. As seen from the table, the Young’s above, the deformation anisotropy of graphene has been modules depend on the character of edge atom chemical attributed to mechanical anisotropy of the constituent modification. As shown in [31], elastic properties of big benzenoid units [29, 30]. The dependence of the molecules such as polymers [35, 40] and nanographenes deformation on the chemical modification of framing edge [31] are determined by dynamic characteristics of the atoms has been revealed for the first time. objects, namely, by force constants of the related vibrations. As seen in Fig. 2, when the edge atoms are bare and Since benzenoid units have a determining resistance to any not terminated by other chemicals, the deformation deformation of graphene molecules, the dynamic behavior is the most complex. The mechanical behavior of parameters of stretching C-C vibrations of the units are the (5, 5) NGr molecule is similar to that of a tricotage mainly responsible in the case of uniaxial tension. sheet when either the sheet rupture has both commenced Changing in Young’s modules means changing in the force and completed by the rupture of a single stitch row constants (and, consequently, frequencies) of these (armchair mode) or the rupture of one stitch is ‘tugging at vibrations. The latter are attributed to G-band of graphene thread’ the other stitches that are replaced by still elongated that lays the foundation of a mandatory testing of any one-atom chain of carbon atoms (zigzag mode). In the Fig. graphenium system by Raman spectrum. In many cases, the relevant band is quite wide which might indicate the Changing in ND reveals changing in the molecule chemical chemical modification of the edge zone of the graphene activity induced by deformation. objects under investigation.
zigzag mode armchair mode
45 45 zigzag mode armchair mode 40 40 35 35 160 e 160 e 30 30 , , D D 9 25 25 9 N N
120 20 20
*10 120 *10 2 2 15 15
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N/m 10 10
, 80 , 0 2 4 6 8 101214161820222426 00,511,522,53 ∀ ∀ Elongation 2 L , A Elongation 2 L , A 40 40 Stress Stress
Stress Stress 45 45
0 0 40 40 0,0 0,1 0,2 0,3 0,4 0,0 0,4 0,8 1,2 1,6 2,0 2,4 2,8 35 35
St rain ∀ Strain ∀ e 30 e 30 , , D 25 D 25 N N
160 160 20 20 15 15 9 9 120 120 10 10 *10 *10 0123 45678910111213 00,511,522,53 2 2 Elongation 2 L , A Elongation 2 L , A N/m 80 N/ m 80 , , ∀ ∀ 45 45
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Stress Stress Stress 35 35 e 0 0 e 30 30 , , D 0,00,40,81,21,62,02,42,8 0,00,10,20,30,4 D 25 25 N N
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10 10 160 160 00,40,81,21,622,42,8 00,511,522,53 Elongation 2 L , A 9 9 Elongation 2 L , A 120 120 *10 *10
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N/m 80 N/ m 80 , , ∀ ∀ Fig. 4. Evolution of the odd electrons correlation interms of 40 40 Stress Stress
St ress the total numbers of effectively unpaired electrons under
0 0 0,00,40,81,21,62,02,42,8 00,10,20,30,4 tensile deformation of the (5, 5) NGr at two deformation Strain ∀ Strain ∀ modes. Bare edges (top); H1-terminated edges (middle); H2-terminated edges (bottom). Fig. 3. Stress-versus-strain dependences of tensile deformation of the (5, 5) NGr molecule with different Figure 4 presents the evolution of ND for the three chemical modification in two deformation modes. Bare studied molecules. Since breaking of each C-C bonds edges (top); H1-terminated edges (middle); H2-terminated causes an abrupt changing in ND, a toothed character of the edges (bottom). relevant dependences related to zigzag mode of the molecule with bare and H-terminated edges is quite evident. Since the molecule deformation is mainly provided by One should draw attention to the ND absolute values as well basal atoms, so drastic changes in the deformation behavior as to their dependence on both chemical modification of the points to a significant influence of chemical state of edge edge atoms and the deformational modes. Evidently, atoms on the electronic properties in the basal plane. The chemical activity of the molecules is drastically changed in observed phenomenon can be understood if suggest that 1) due course of a mechanically induced transformation, This the deformation and rupture of the molecule is a collective changing is provided by the redistribution of C-C bond event that involves the electron system of the molecule as a lengths caused by the mechanical action. This action whole; 2) the electron system of the graphene molecule is combines positions of both basal plane and edge atoms into highly delocalized due to extreme correlation of odd united whole and is topologically sensitive. Therefore, the electrons; and 3) the electrons correlation is topologically redistribution of the C-C bonds over their lengths causes sensitive due to which chemical termination of edge atoms changing in a topological ‘quality’ of individual bonds. To so strongly influences the behavior of the entire molecule. illustrate the latter, let us look at not the total number of The latter has turned out to be a reality, indeed. effectively unpaired electrons ND, but at the atomic chemical susceptibility distribution over the molecule Table 1. Young’s modules for (5,5) NGr with different configuration of edge atoms that is determine by a fractional number of the atoms, TPa effectively unpaired electrons NDA at atom A [28].
Mode Bare edges Н1-terminated edges Н2- terminated edges 4.1. Graphene molecule with bare edges
‘zigzag’ 1.05 1.09 0.92 Figure 5 presents a set of the NDA image maps related to the first 17 steps of the armchair mode of the molecule uniaxial ‘armchair’ 1.06 1.15 0.95 tension. The set corresponds to gradually increased NDA th values up to the 17 set shown at the right-hand top panel of Fig.4. The maps are accompanied by the molecule 4. Topological character of the odd equilibrium structures. For the maps to be presented in one scale, the N data were normalized by equalizing electron correlation in graphene DA maximum NDA values at each map to that one at the zeroth step that corresponds to the non-deformed molecule. White The performed computations have revealed that the figures present the equalizing coefficients. correlation of the odd electrons of the studied molecules As seen in the figure, the segregation of atoms into changes quite remarkably in due course of the deformation. two groups related to edge and basal plane areas, This result can be illustrated by the evolution of the total respectively, which is characteristic for the non-deformed number of effectively unpaired electrons ND during the molecules, takes place up to the 16th step inclusively. deformation. The ND value is a direct characteristic of the Obviously, since simultaneously, the total number of the extent of the electron correlation, on one hand, [2] and effectively unpaired electrons N grows, a redistribution of molecular chemical susceptibility, on the other, [41]. D the NDA values should occurs. However, the latter mainly
Fig. 5. Image NDA maps and equilibrium structures of the bare-edge (5, 5) NGr molecule in due course of the first stage armchair- mode tensile deformation. Black and white figures number steps and equalizing coefficients, respectively. Scale is related to all the maps. the bonds are ruptured homolytically, one should expect a concerns the basal plane atoms leaving the edge atoms only radicalization of the molecule at these points. Providing the slightly changed due to practically constant maximum NDA NDA indicative ability of just the very events, the values of the latter, which follows from the presented appearance of white spots on the maps should be expected. equalizing coefficient. Actually, Fig. 6 shows the However, until the 17th step no such spots are observed. distribution of the absolute NDA values for the zeroth and This shows that the bonds breaking occurs at much longer 16th steps alongside with their ratio. As seen in the figure, interatomic distance in comparison with standard tabulated the redistribution concerns basal plane atoms mainly, data, which supports a conclusion made in [42]. indeed. Figure 7 presents a similar picture describing the behavior of the odd electron correlation during zigzag- mode tension of the graphene molecule. The first 17 steps cover the first-stage zigzag-mode deformation and correspond to the first tooth of the strain-stress and ND- elongation dependences shown in the left-hand top panels in Figs. 3 and 4, respectively. In contrast to the previous case, the redistribution of the values occurs quite differently. Although the general image of the maps is kept up to the 16th step, the appearance of strongly stretched C-C bonds can be already noticed at the 15th step while a complete bond breaking becomes absolutely evident at the 17th step only. The effect of stretched C-C bonds in the basal plane on the NDA value redistribution is well seen in Fig. 8. About 5.5-fold changing in the NDA value is Fig. 6. Histograms of the NDA distribution over atoms of the characteristic for one of them while edge atoms keep only bare-edge (5, 5) NGr molecule related to non-deformed slightly changed N values. th DA (light) and 16 step-deformed (dark) armchair-mode tensile As was mentioned previously [29, 30], the perstep deformation. Curve plots the presented values ratio when elongation (stretching) of C-C bonds under armchair- and n=16. zigzag-mode tension is quite different from a geometrical viewpoint. Evidently, this might explain the difference in Important to draw attention to both image NDA maps the N values related to the same step but cannot evidence th th DA and the corresponding structures related to the 15 and 16 why the character of the redistribution for the two modes is steps. The structures are drawn by a visualization standard completely different thus attributing the difference to a program that is based on tabulated values of chemical bond topological nature of the considered mechanochemical lengths. According to the data, there is one broken C-C reaction. bond at the 15th step and five of them at the 16th step. Since Fig. 7. Image NDA maps and equilibrium structures of the bare-edge (5, 5) NGr molecule in due course of the first stage zigzag- mode tensile deformation. Black and white figures number steps and equalizing coefficients, respectively. Scale is related to all the maps. pristine atoms. Further elongation does not change the situation. 4.2. Single-hydrogen terminated edges of graphene In the case of zigzag mode, the map appearance has molecule been kept up to the 14th step after which highly stretched C- C bonds are observed in the middle of the molecule basal th The evolution of the NDA image maps in due course of the plane at the 15 step, whose complete breaking is followed armchair- and zigzag-mode uniaxial tension of the (5, 5) at the 16th step. The sample becomes highly radicalized. NGr molecule with single-hydrogen terminated edges is presented in Fig. 9. The data are related to the first stage of deformation for both modes. The picture in the figure drastically differs from that one shown in Figs. 5 and 7. The difference starts from the non-deformed molecule. As seen in the figure, the NDA distributions differ remarkably for the two modes. The two maps are obtained in due course of the following procedure. Firstly, the equilibrium structure of the free standing H1-terminated (5, 5) NGr molecule was obtained. Afterwards, a set of carbon atoms which fix six selected MICs (see blue atoms in Fig.1) was fixed, differently for two deformation modes. Then the optimization procedure was repeated for adopting calculation to the new conditions. If in the case of the bare edge molecule considered in the previous section, this Fig. 8. Histograms of the NDA distribution over atoms of the procedure caused only a homogeneous scaling of the N bare-edge (5, 5) NGr molecule related to non-deformed DA th values (see scales in Figs. 5 and 7), a complete (light) and 16 step-deformed (dark) zigzag-mode tensile reconstruction of the image NDA map occurs in the current deformation. Curve plots the presented values ratio when case while the maximum NDA values are practically n=16. identical. Both findings mean that fixation of the graphene sheet edges provides a considerable change in its electron 4.2. Double-hydrogen terminated graphene molecule distribution. The latter is additionally greatly influenced by the chemical composition of the sheet edge atoms. Addition of the second hydrogen atom to the edge ones Coming back to Fig.9, one can see that, as previously, drastically changes the image maps again as seen in Fig. 10. in the case of armchair mode the image maps keep In contrast to the previous case, the initial fixation of edge practically unchanged appearance until the 19th step in spite atoms does not cause changing in both the map appearance of highly stretched C-C bonds at the latter step. The next and absolute NDA value. However, the molecule becomes 0.1Å elongation provides simultaneous breaking of six C-C non-planar, which greatly influence further deformation. bonds followed with 3-fold increasing of the NDA values. Thus, in due course of the armchair-mode tension, the The sample becomes highly radicalized. Concentration of difference in the values of eight framing basal atoms and high N values in the area of broken bonds drastically remainders is gradually smoothed once equalizing at the DA th th changes the image map fully suppressing much less active 19 step. The situation remains the same for the 20 step in Fig. 9. Image NDA maps and equilibrium structures of the H1-terminated-edge (5, 5) NGr molecule in due course of the first stage of the armchair (left) and zigzag (right) -mode tensile deformation. Black and white figures number steps and equalizing coefficients, respectively. Scales are related to all the maps within the deformational mode.
Fig. 10. Image NDA maps and equilibrium structures of the H2-terminated-edge (5, 5) NGr molecule in due course of the first stage of the armchair (left) and zigzag (right) -mode tensile deformation. Black and white figures number steps and equalizing coefficients, respectively. Scales are related to all the maps within the deformational mode
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