ISSN 10637834, Physics of the Solid State, 2012, Vol. 54, No. 7, pp. 1507–1513. © Pleiades Publishing, Ltd., 2012. Original Russian Text © A.I. Podlivaev, L.A. Openov, 2012, published in Fizika Tverdogo Tela, 2012, Vol. 54, No. 7, pp. 1417–1423.
FULLERENES
Specific Features of the Formation of Defects in Fullerene C46 A. I. Podlivaev and L. A. Openov* National Research Nuclear University “MEPhI,” Kashirskoe sh. 31, Moscow, 115409 Russia * email:
[email protected] Received November 22, 2011
Abstract—The initial stage of the formation of defects in the fullerene C46 has been investigated using the ato mistic computer simulation. It has been found that the relatively low symmetry of this fullerene leads to the emergence of channels of defect formation, which have not been revealed in the fullerenes C20, C36, and C60. These channels consist in breaking a single C–C bond (in contrast to the simultaneous breaking of two bonds in the course of the Stone–Wales transformation, which is characteristic of highsymmetry fullerenes). For some typical channels, the paths of transformation of the C46 fullerene into the corresponding defect isomers have been determined and the heights of the potential barriers encountered in these paths have been calcu lated. DOI: 10.1134/S106378341207030X
1. INTRODUCTION The mechanism of formation of fullerenes—sphe roidal carbon clusters with the “surface” where the C– C covalent bonds form pentagons and hexagons adja cent to each other [1, 2]1—is not still clearly under stood. According to one of the existing hypotheses, fullerenes are formed from the initial strongly disor dered clusters as a result of the annealing of defects [2]. The reverse process—the accumulation of defects during the heating of the fullerenes—leads to a loss of stability of the fullerenes and to their decomposition [4, 5]. In this respect, the investigation of the defect fullerenes and the mechanisms of defect formation is an important problem from both the fundamental and practical points of view. The transformation of the fullerene into a defect isomer, which is similar to this fullerene in structure and energy, usually stimulates the formation of new defects and, eventually, leads to a complete loss of the spheroidal shape of the fullerene and/or to its frag mentation during heating [4, 5]. For this reason, the analysis of the initial stage of the defect formation has acquired particular importance. In the most studied fullerene C60 [1] (which includes twenty hexagons and twelve pentagons isolated from each other), the first defect, as a rule, is formed as a result of the Stone– Wales transformation [6], which involves the rotation of a single bond B662 through an angle of 90° (see Fig. 1). The Stone–Wales transformation involves the simultaneous breaking of two C–C bonds connecting the atoms of the rotating B66 bond with other atoms of 1 In
the smallest possible fullerene C20, there are only pentagons [3]. 2 Here and below, the symbols B nm stand for bonds that are com mon to the adjacent n and mgons.
the cluster (Fig. 1b) and, then (after the rotation), the formation of two new bonds but with other atoms (Fig. 1c). In the process of rotation of the B66 bond, the cluster passes through the saddle configuration (Fig. 1b), in which the potential energy of the cluster has a local maximum with a height U ≈ 6.5 eV [7]. This maximum is achieved by the rotation of the B66 bond through an angle of ~45°. As a result of the Stone– Wales transformation, the number of pentagons and hexagons on the “surface” of the cluster remains unchanged, but there arise two pairs of pentagons with common B55 bonds (Fig. 1c). In this case, the potential energy of the cluster increases by ΔEpot ≈ 1.5 eV [7]. In the C20 fullerene (Fig. 2a), the defect isomer with the lowest energy is also formed upon the rotation of one of the bonds (now, it is the B55 bond) not through an angle of 90° but through an angle of ~45° (as in the C60 fullerene, the rotation occurs with the simultaneous breaking of two other C–C bonds). This results in the formation of a configuration with two adjacent octagonal “windows” (Fig. 2b), and the potential energy increases by ΔEpot ≈ 3 eV [8]. The fun damental difference between this configuration and the intermediate saddle configuration formed with two nanogonal “windows” in the course of the Stone– Wales transformation in the C60 fullerene (Fig. 1b) lies in the fact that it is a metastable configuration; i.e., it corresponds to the local minimum of the potential energy of the cluster (in contrast to the local maximum of the potential energy, as is the case with the configu ration shown in Fig. 1b). Therefore, the lifetime of this configuration is relatively long (on atomic scales), which we directly observed in the molecular dynamics simulation of the evolution of the heated fullerene C20 [4, 8]. On the contrary, the saddle configuration in the
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Fig. 1. Stone–Wales transformation in the C60 fullerene: (a) perfect fullerene, (b) atomic configuration corresponding to the sad dle point on the hypersurface of the potential energy, and (c) final configuration. The heavy line shows the B66 bond, which during the Stone–Wales transformation is rotated through an angle of 90°.
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Fig. 2. Complete and incomplete Stone–Wales transformations in the C20 fullerene: (a) perfect fullerene, (b) atomic configura tion corresponding to a local minimum of the potential energy (metastable state), and (c) final configuration. The heavy line shows the B55 bond, which during the incomplete and complete Stone–Wales transformations is rotated through an angle of ~45° and an angle of 90°, respectively.
C60 fullerene (Fig. 1b) is only a transient configura tion; hence, it exists only for a very short time (~1 fs) [5]. The metastable configuration of the C20 cluster (Fig. 2b) can be considered as a result of the “incom plete” Stone–Wales transformation [8]. After the completion of the Stone–Wales transformation, there arises a nonclassical fullerene C20 with two tetragons (Fig. 2c). The available data regarding the occurrence of the incomplete Stone–Wales transformation in the C36 fullerene are controversial and sensitive to the compu tational method [8]. This suggests that the potential energy of this cluster in the vicinity of the configura tion with two adjacent “windows” has an almost flat hypersurface, so that, if the hypersurface has a local minimum, it is very shallow, and, therefore, a defect isomer is actually formed by means of the conven tional Stone–Wales transformation [8]. Thus, using the fullerenes C20, C36, and C60 as an example, we can see that the first defect is formed as a result of either the complete Stone–Wales transforma
tion or the incomplete Stone–Wales transformation. However, it should be noted that all the aforemen tioned fullerenes have a high degree of symmetry. In the fullerenes with a lower symmetry, we can expect the emergence of new channels of defect formation. This problem is of interest in terms of the stability of not only fullerenes but also other sp2hybridized car bon allotropes (including graphene [9] and other quasitwodimensional nanocarbon systems [10]), in which the local symmetry breaking either can occur in the vicinity of the interfaces (for example, graphene/graphane interfaces [11, 12]) or can take place under deformation. The purpose of this work is to perform a detailed analysis of the initial stage of the formation of defects in the C46 fullerene (Fig. 3). The choice of this fullerene as the object of our investigation is motivated by two factors. First, the C46 fullerene has a large num ber of nonequivalent interatomic bonds, which con tributes to the diversity of channels of defect forma tion. Second, the potential energy of the C46 fullerene
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is higher than that of the nonclassical fullerene with one tetragon [13], which is a rare exception to the gen eral rule that the nonclassical fullerenes are energeti cally unfavorable. Apparently, this can be explained by a strong local deformation of the C46 fullerene due to the presence of a large number of adjacent pentagons in it (Fig. 3), which facilitates the formation of defects and favors the emergence of new channels of isomer ization. This paper is organized as follows. Section 2 con tains the description of the computational techniques. Section 3 presents the obtained results and their dis cussion. Section 4 contains the brief summary and conclusions. 2. COMPUTATIONAL TECHNIQUES Usually, the search for ways of transforming the cluster into other atomic configurations and the calcu lation of heights U of the potential barriers hindering these transformations have been performed using the static simulation method [14, 15]. This method con sists in analyzing the potential energy of the cluster Epot as a function of the coordinates {Ri} of the constit uent atoms. This makes it possible to determine the stationary points Epot({Ri}) and to elucidate the char acter of the change in the potential energy Epot along the reaction coordinate connecting the different iso mers of the cluster. The stationary points correspond to the atomic configurations in which the forces acting on the atoms are equal to zero. In this case, the meta stable configurations correspond to local minima of the potential energy Epot, whereas the saddle configu rations correspond to the local maximum of the potential energy Epot along one coordinate and to local minima along all the other coordinates. Since the hypersurface of the potential energy Epot({Ri}) in the cluster consisting of ~100 atoms has a quite complex shape and the detailed analysis of this hypersurface requires large computer resources, the search for stationary points has usually been based on a priori assumptions regarding the structure of meta stable configurations, into which the cluster can trans form, and the corresponding saddle configurations. This has often led to the fact that some of the ways of transforming the cluster into other atomic configura tions are “lost.” In the present work, we use a different approach [7, 16]. First, the time evolution of the C46 fullerene heated to a temperature T = 2500–3000 K is investigated using the molecular dynamics method. This makes it possible (after the visualization of the data in the form of a “computer movie”) to obtain visual information on the defect isomers, as well as on the intermediate atomic configurations formed upon the transitions into these isomers. In other words, we allow the cluster as a whole to “travel” over the hyper surface of the potential energy Epot({Ri}), keeping track of the sequence of the configurations thus PHYSICS OF THE SOLID STATE
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Fig. 3. Fullerene C46.
formed. Then, based on the obtained results, we use the static simulation method in order to determine the stationary configurations Epot({Ri}), which, as a rule, tend to be close to those found in the visualization of the “life” of the cluster. In order to describe the interatomic interaction in the molecular dynamics calculations, we use the tight binding model [17], which, even though inferior to the ab initio approaches in accuracy, but is considerably less demanding on computing resources, and there fore, makes it possible to investigate the evolution of a cluster consisting of approximately 100 atoms for a sufficiently long (on atomic scales) time t ~ 10 ns and to gain sufficient statistics (t ~ 10 ps for ab initio meth ods). For more information about the details of the calculations, see [16] and references therein. The local minima and saddle configurations Epot({Ri}) were determined by the static simulation using the struc tural relaxation method and the method of search in normal coordinates, respectively [14, 15]. 3. RESULTS AND DISCUSSION We have revealed that, in the C46 fullerene, the first defect is most frequently (in approximately 80% of all the cases) formed in the course of the complete (as in the C60 fullerene) or incomplete (as in the C20 fullerene) Stone–Wales transformation; as a result, one of the interatomic bonds is rotated through an angle of 90° or ~45°, respectively (see Section 1). In this case, we have observed only the rotations of the B66 and B56 bonds (the rotations of the B55 bonds, most likely, are hindered by high energy barriers). The num ber of different imperfect fullerenes formed at the first stage of the defect formation is rather large. This is associated with the fact that the initial fullerene con tains the B66 bonds, which are nonequivalent to each other (i.e., have different local environments), and the B56 bonds, which are nonequivalent to each other as
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Fig. 5. Change of the potential energy Epot of the C46 fullerene as a function of the reaction coordinate upon breaking of the B66 bond and subsequent transitions to dif ferent defect isomers. Numerals 1–4 correspond to the numbers of the metastable configurations shown in Fig. 4. S1, S2, and S3 are the saddle configurations that visually differ only slightly from configurations 2, 3, and 4, respec tively. The origin is taken as the energy of the perfect fullerene. The reaction coordinate is chosen as the sum of increments in the lengths of all broken bonds.
Fig. 4. Sequence of formation of defect isomers of the C46 fullerene upon breaking of the B66 bond: (1) perfect fullerene, (2) isomer with a decagonal “window,” (3) iso mer with two adjacent “windows” (decagonal “window” and nanogonal “window”), and (4) isomer with three adja cent “windows” (one decagonal “window” and two nan ogonal “windows”). The heavy line shows the B66 bond, which during the incomplete Stone–Wales transformation is rotated through an angle of ~45°.
well. For example, the complete Stone–Wales trans formation can result in the formation of classical fullerenes (which differ from the initial fullerene in the rearrangement of several pentagons and hexagons) and nonclassical fullerenes (containing tetragons and/or heptagons), whereas the incomplete Stone– Wales transformation can lead to the formation of iso mers with different numbers of atoms in two adjacent “windows” (8 and 10, 9 and 9, 9 and 10). An exhaus tive statistical analysis of the whole variety of defect configurations and the ways of their formation in the fullerenes is beyond the scope of our present work. We will restrict our consideration to the discussion of a new channel of defect formation, which has not been revealed in the fullerenes C20, C36, and C60. It has been found that, in the C46 fullerene, the first defect can be formed upon the breaking of a single C– C bond rather than upon the simultaneous breaking of a pair of bonds, from which the complete Stone– Wales transformation (Fig. 1) and the incomplete Stone–Wales transformation (Fig. 2) begin to occur. Furthermore, depending on what type of bond is bro ken (i.e., the B66, B56, or B55 bond), one decagonal, nanogonal, or octagonal “window” is formed on the
“surface” of the fullerene, respectively. The performed analysis of the hypersurface of the potential energy Epot({Ri}) in the vicinity of the configurations with one “window” has demonstrated that these configurations are metastable; i.e., they correspond to local minima of the potential energy Epot. Because of the nonequivalence of the interatomic bonds of one type, the defect isomers with “windows” of the same size can differ slightly from each other in the potential energy ΔEpot (measured from the energy of the perfect fullerene) as well as in the height U of the energy barriers that prevent the defect formation. The processes of further defect formation can also occur in different ways. Below, this will be illustrated using the examples of “windows” of different sizes. 3.1. The Decagonal “Window” (Breaking of the B66 Bond) Figure 4 shows an example of a typical sequence of defect formation, which begins with the formation of a decagonal “window” as a result of the breaking of the B66 bond. We should emphasize that the configuration with a decagonal “window” is metastable (Fig. 5). The subsequent evolution of the cluster leads to the break ing of another bond and to the formation of two adja
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Fig. 6. Sequence of formation of defect isomers of the C46 fullerene upon breaking of the B56 bond: (1) perfect fullerene, (2) iso mer with a nanogonal “window,” (3) isomer with two adjacent “windows” (decagonal “window” and nanogonal “window”), (4) isomer with a decagonal “window,” and (5) nonclassical fullerene with one heptagon. The heavy line shows the B66 bond, which during the incomplete and complete Stone–Wales transformations is rotated through an angle of ~45° and an angle of 90°, respectively.
cent “windows” (decagonal and nanogonal “win dows”). In this case, one of the B66 bonds is rotated through an angle ~45° (see Fig. 4); i.e., there occurs an incomplete Stone–Wales transformation, as in the C20 fullerene (Fig. 2b). The reason why two bonds are broken simultaneously in the C20 fullerene and two bonds are broken sequentially in the C46 fullerene lies in the lower symmetry of the latter fullerene and, con sequently, in the different strengths of the broken bonds because of their different local environments. It should be noted that, in the example presented in Fig. 4, the Stone–Wales transformation is not com pleted (as in the C20 fullerene; Fig. 2); i.e., the rotation of one of the bonds through an angle of 90° does not occur. Instead, we have dealt here with the formation of the third (nanogonal) “window.” However, this sce nario is not the only possible one: in the simulation of the dynamics of the C46 fullerene, we have also observed the completion of the Stone–Wales transfor mation after the formation of two adjacent “win dows.” However, in all these cases, the defect forma tion begins to occur with the breaking of one bond and with the transition to the metastable isomer with one decagonal window. PHYSICS OF THE SOLID STATE
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As can be seen from Fig. 5, the potential energy of the isomer with a decagonal “window” is ΔEpot = 2.93 eV. The height of the energy barrier for the transi tion to this isomer is U12 = 3.00 eV (the numbers of the corresponding configurations are indicated in Fig. 4). The energy barrier for the reverse transition is signifi cantly lower (U21 = 0.07 eV), which, as a rule, leads to a rapid (for a time of ~10 fs) recovery of the original defectfree configuration. The heights of the energy barriers for the subsequent transitions to the isomers with two (ΔEpot = 3.35 eV) and three (ΔEpot = 3.65 eV) adjacent “windows” are U23 = 0.65 eV and U34 = 0.58 eV (in this case, U32 = 0.24 eV and U43 = 0.27 eV), respectively. The return of the isomer with two “win dows” into the initial configuration was observed sev eral times. After the formation of the third “window,” no return has already occurred: the cluster rapidly loses the spheroidal shape and transforms into differ ent quasionedimensional or quasitwodimensional configurations. For this reason, the height of the energy barrier for the irreversible decomposition of the C46 fullerene through this channel can be considered to be equal to the energy of the saddle point for the
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such transition is rather small because of the relatively large height of the corresponding energy barrier; see below). The character and scale of the changes in the potential energy in the course of the transitions between different isomers (Fig. 7) are the same as those observed upon the breaking of the B66 bond: the potential energies of isomers with one nanogonal “window,” two adjacent “windows,” and one decago nal “window” are ΔEpot = 2.97, 3.49, and 3.64 eV, respectively. The heights of the energy barriers for the transitions to these isomers are as follows (the num bers of the corresponding configurations are indicated in Fig. 6): U12 = 3.01 eV, U23 = 0.86 eV, and U34 = 0.42 eV, respectively. The heights of the energy barriers for the reverse transitions of these isomers are as fol lows: U21 = 0.04 eV, U32 = 0.35 eV, and U43 = 0.27 eV, respectively. Noteworthy is the small value of the energy barrier height U21, which is even smaller than that for the isomer with a decagonal “window.” The potential energy of the nonclassical fullerene is ΔEpot = 1.58 eV, the height of the energy barrier for the transition to this isomer from the configuration with a decagonal “window” is U45 = 0.25 eV, and the height of the energy barrier for the reverse transition is U54 = 2.31 eV. The height of the energy barrier for the transi tion of the classical fullerene C46 to the nonclassical fullerene (through the given channel) is equal to the energy of the saddle point for the transition to the con figuration with one decagonal “window” (S4 in Fig. 7), U = 3.91 eV.
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Fig. 7. Change of the potential energy Epot of the C46 fullerene as a function of the reaction coordinate upon breaking of the B56 bond and subsequent transitions to dif ferent defect isomers. Numerals 1–5 correspond to the numbers of the metastable configurations shown in Fig. 6. S1, S2, S3, and S4 are the saddle configurations that visu ally differ only slightly from configurations 2, 3, 4, and 5, respectively. The origin is taken as the energy of the perfect fullerene. The reaction coordinate is chosen as the angle of rotation of the bolded B66 bond during the Stone–Wales transformation.
transition to the configuration with three “windows” (S3 in Fig. 5), U = 3.92 eV. 3.2. The Nanogonal “Window” (Breaking of the B56 Bond) The breaking of the B56 bond leads to the formation of a metastable isomer with one nanogonal “window” (Figs. 6 and 7). As in the case with the breaking of the B66 bond, the next stage proceeds with the formation of two adjacent “windows” (nanogonal and decagonal “windows”); i.e., there occurs an incomplete Stone– Wales transformation (the B66 bond is rotated through an angle of ~45°; Fig. 6). In this situation, the further evolution of the cluster does not result in the forma tion of a third “window”: initially, the isomer with one decagonal “window” is formed; then, the Stone– Wales transformation is completed (the B66 bond is rotated through an angle of 90°), and the nonclassical fullerene with one heptagon is formed. The lifetime of the aforementioned isomer is sufficiently long (at least, the formation of this isomer does not lead to a rapid loss of the spheroidal shape, as in the above case of the isomer with three “windows”). Furthermore, the reverse transition of this isomer to the perfect fullerene was observed only once (the probability of
3.3. The Octagonal “Window” (Breaking of the B55 Bond) In contrast to the aboveconsidered cases with the breaking of the B66 and B56 bonds, the breaking of the B55 bond, which is common to two adjacent penta gons, is accompanied by changes in their local envi ronment, namely, by a “switching” of one of the outer (with respect to these pentagons) C–C bonds in which the adjacent pentagon and hexagon exchange their places (here, the “switching” is considered to mean the breaking of one bond with the subsequent forma tion of another bond). The resulting isomer with one octagonal “window” is shown in Fig. 8. The potential energy of this isomer ΔEpot = 3.13 eV is only slightly higher than the potential energy of the isomers with nanogonal and decagonal “windows,” whereas the energy barrier for the transition to this isomer is signif icantly higher (3.58 eV), which, most likely, is associ ated with the rearrangement of the bonds nearest to the “window.” The height of the energy barrier for the reverse transition is sufficiently large (0.45 eV); how ever, the metastable configuration with an octagonal “window” is not a longlived configuration, because it is separated by a very low energy barrier (~0.01 eV) from the nonclassical isomer with one tetragon
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of their formation should also have analogues in other fullerenes with low symmetry. ACKNOWLEDGMENTS We would like to thank M.M. Maslov for his assis tance in performing this work and for participation in discussions of the results. This study was supported by the Russian Founda tion for Basic Research (project no. 120200561). REFERENCES
Fig. 8. Defect isomer of the C46 fullerene formed upon breaking of the B55 bond with one octagonal “window.”
(ΔEpot = 0.92 eV). The transitions to this nonclassical isomer have been observed in our simulation of the dynamics of the C46 fullerene. 4. CONCLUSIONS Thus, it has been found that the presence of defect isomers with octagonal, nanogonal, and decagonal “windows” in the C46 fullerene significantly extends the range of possible channels of defect formation, particularly at the initial stage. Since the formation of these defect isomers requires the breaking of only one C–C bond (rather than two bonds, as in the case of the Stone–Wales transformation), the energy barriers for the transitions to the corresponding configurations appear to be relatively low and, therefore, the proba bility of the formation of such isomers is sufficiently high. Although these isomers are usually shortlived, they play an important role in the process of disorder ing of the cluster; more precisely, they fulfill the func tion of “catalysts” for the transitions to other defect configurations, thus stimulating the formation of new defects and, eventually, the decomposition of the clus ter itself. Most likely, it is the high intensity of the defect formation that is the main cause of the extremely low percentage of fullerenes C46 in the gas phase. In conclusion, we note that the defects of the C46 fullerene investigated in the present work and channels
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Translated by O. BorovikRomanova
