J. Phys. Chem. C 2007, 111, 10855-10860
10855
New Carbon Molecules in the Form of Elbow-Connected Nanotori Barry J. Cox* and James M. Hill Nanomechanics Group, School of Mathematics and Applied Statistics, UniVersity of Wollongong, Wollongong, NSW, 2522, Australia ReceiVed: March 17, 2007; In Final Form: May 7, 2007
Toroidal caged molecules of carbon have been investigated previously as constructed from elbows formed from armchair (5,5) and zigzag (9,0) nanotubes connected through a pentagonal and heptagonal defect at each bend site. In this paper, we consider these elbows, and we describe two new elbows constructed from (3,3)-(5,0) and (4,4)-(7,0) nanotubes. By assuming the constituent atoms remain in their ideal positions on the nanotube sections, we determine the bend angle and nanotube lengths that characterize each elbow in the unconstrained state and also when the elbow is constrained as when incorporated in a toroidal caged molecule. Using these results, we describe specific examples of new families of carbon caged molecules and the associated geometric parameters of these molecules. In order to properly prescribe the new molecules, we need to introduce a new nomenclature N(n1, m1)pM(n2, m2)q..., where (n1, m1) and (n2, m2) designate the component nanotubes, p, q, ... are the number of atoms in each nanotube section, and N, M, ... are the number of each section of nanotube making up the complete molecule. Of the numerous molecules introduced here, only two molecules in the 5(5,5)p5(9,0)q family have been discussed in previous work, while the remainder are new. The modeling of these toroidal molecules as perfect tori is facilitated by the particular method which we adopt to determine the representative geometric parameters for these molecules. Illustrative values of these parameters for various example molecules are also given.
Introduction Dunlap1,2 considers connecting carbon tubules by means of crease defects comprising one pentagon and one heptagon, which translate the chirality of a tubule from one type to another. By continuing such crease defects all in the same plane, Dunlap1,2 imagines that small-scale cyclic caged molecules can be formed, which effectively are toroidal surfaces. Dunlap1,2 also predicts that the tubule bend angle is 30°, meaning 12 such elbow defects are required for one complete molecule. Fonseca et al.3 analyze the same type of defect, and their results indicate that a bend angle of 36° characterizes this structure, and therefore, only 10 elbows are necessary to construct a completely cyclic molecule. Fonseca et al.3 go on to specify some of the dimensions of such structures and, in particular, nanotoricomprising elbows constructed from (9,0) and (5,5) nanotube sections. The first is a C520 molecule and the other is a C900, both of which are constructed from carbon nanotube sections of approximately equal length. Itoh et al.4 describe a cage toroidal form of carbon C360 and use a Stillinger-Weber-type potential5 to calculate its cohesive energy. This is extended by Ihara et al.,6 who apply the Goldberg transformations7 to the C360 torus. Itoh and Ihara8 describe toroidal structures with elongated, elliptical cross sections and, in a subsequent work,9 describe various isomers of toroidal forms of carbon between C240 and C250. Meunier et al.10 determine the electronic structure of a C1960 polygonal torus formed from 10 elbows comprising (10,10) and (6,6) nanotube sections bent at an angle of 36°. Ceulemans et al.11 investigate the electronic structure of bent carbon nanotubes using a tight-binding approach and derive a theoretical condition for the metallic nature of polyhedral carbon * To whom correspondence should be addressed. E-mail address:
[email protected].
nanotori. Oh et al.12 investigate geometries of various species of nanotori, which they classify into four main types, and analyze the stability of these structures using tight-binding and semiempirical quantum chemical methodologies. La´szlo´ and Rassat13 investigate the stability of the C120 toroidal molecule using ab initio, tight-binding, Tersoff14 and Brenner15 methods. They analyze these methods by comparing the square-root average bond distance between the various models. In addition to this, La´szlo´ and Rassat13 describe toroidal structures solely comprising 5- and 7-member rings. Liu et al.16 report laser-grown single-walled carbon nanotubes, which they term “crop circles” because they were initially sceptical that these were seamless toroidal nanotubes. However, in that paper, many or most of the crop circles they observe are indeed perfect tori. Han17 analyzes the stability of toroidal structures and finds that spherical tori are more stable than pentagonal tori like those described by Dunlap.1,2 This work focuses on laser-grown nanotubes, and the observed tori have radii of greater than 150 nm. Later work by Martel et al.18,19 and Sano et al.20 also reports the formation of tori of similar dimensions. Hilder and Hill21 propose a new nano-oscillator comprising a nanotori symmetrically placed around the outside of a carbon nanotube and performing oscillations along the length of the tube. In the present paper, we are motivated to use tori as a component of a nanomechanical system like that described in Hilder and Hill,21 in which the generating radii of the tori are assumed to be the same order of magnitude as the radii of typical carbon nanotubes, that is, a few nanometers, and we term tori of this size as “nanotori”. We wish to make the distinction between tori formed from bent nanotubes without defects, which have generating radii greater than 150 nm, and those described here, which have generating radii in the range of 0.7 to 3.5 nm,
10.1021/jp0721402 CCC: $37.00 © 2007 American Chemical Society Published on Web 06/29/2007
10856 J. Phys. Chem. C, Vol. 111, No. 29, 2007
Figure 1. Elbows formed from two nanotube sections.
Han17
a difference of 2 orders of magnitude. shows that for radii of the smaller size, perfect tori are not realizable due to the problems of the elliptical tube cross section, tube buckling, and bond breaking, which occur when the radius of the torus is decreased. Likewise, Meunier et al.10 examine the stability of large carbon nanotori using elastic theory and estimate the critical radius to be approximately 90 nm. They conclude that tori smaller than this cannot be formed by bending a straight nanotube because the strain energy would exceed the bondbreaking energy, and therefore, the molecular structure would be unstable. However, polygonal tori like the dodecagonal structure Dunlap1,2 proposes and the decagonal structure analyzed by Fonseca et al.3 are realizable at this scale, and while not true circular tori, they can nevertheless be considered effectively toroidal when the length of the sides are similar and small. By considering the elbow as formed from two distinct nanotube species, we perform a least-squares minimization calculation on the bonds linking one tube to the next. This calculation is performed with three parameters, namely, the half-lengths of the two constituent nanotubes l 1 and l 2 and the bend angle of the elbow θ. Using these three parameters, we are able to determine the geometric parameters of the nanotori constructed from the various elbows that we analyze. Structure of Elbow-Connected Nanotori The elbow proposed by Dunlap1,2 comprises a section of (5,5) armchair nanotube and a section of (9,0) zigzag nanotube, which are connected via pentagonal and heptagonal defects on the outside and inside of the bend, respectively. In this paper, we consider this elbow but also elbows constructed from (3,3) armchair and (5,0) zigzag nanotubes and elbows constructed from (4,4) armchair and (7,0) zigzag nanotubes. All three of these elbows are shown in Figure 1. We model the elbow structure mathematically by positioning atoms on ideal cylinders representing the two species of nanotube and then identifying the terminal atoms on each structure which bond with the corresponding terminal atoms on the other nanotube. The models are originally defined as centered on the origin and aligned with their axes as the z-axis. We then perform a translation of the armchair tube in the negative z-direction by a length l 1 and a corresponding translation of the zigzag tube in the positive z-direction by a length l 2. With the tubes so situated, we then perform a rotation of the zigzag tube by an angle θ around the x axis, as shown in Figure 2. We start by defining the ith terminal atom in the armchair nanotube by the position vector ai ) (aix, aiy, aiz) and the corresponding terminal atom in the zigzag tube by bi ) (bix, biy, biz). After the translations and rotation described
Cox and Hill
Figure 2. Elbow formed from two nanotube sections.
above, we derive the following expression for the Euclidean distance between the atoms
|bi - ai| ) {(bix - aix)2 + [biycos θ - (biz + l2)sin θ - aiy]2 + [(biz + l2)cos θ + biysin θ - (aiz - l1)]2}1/2 With this distance between matching atoms so defined, we wish to minimize the variance of this distance from the bond length σ. Therefore, in a least-squares sense, we are looking to minimize the following objective function
f(l1, l2, θ) )
1 2
∑i (|bi - ai| - σ)2
(1)
Once the parameters l 1, l 2, and θ are determined, we can then attribute a “natural” bend angle θ to the elbow configuration under consideration. This angle will be of interest if the elbow is not constrained, for example, if it is situated as a nanotube with a bend or a spiral. However, in the case of nanotori, and assuming that the torus remains symmetric in the plane with no buckling, the bend angle θ would necessarily be constrained to a value of θ ) 180°/n, where n ∈ {2, 3, 4, ...}. A separate analysis of the equation with this constraint on θ leads to slightly different values for l 1 and l 2, which apply when the elbow is fixed into a toroidal configuration. When modeling a nanotorus, it is useful to be able to make the continuum approximate that the atoms at discrete points, from which the nanotorus is comprised, are replaceable by an average atomic density uniformly distributed over the surface and also assume that the polygonal torus is a perfect torus. In doing so, it is necessary to be able to assign representative values for the generating torus radius c and the tube radius a. In the following section, we derive formulas which give reasonable representative values for the two geometric parameters c and a. The first step in this process is to determine the distance from the center of the torus to the center of each type of nanotube, which we will denote by r1 and r2. We note that an elbow can then be considered as two right triangles with a common hypotenuse R. The angle subtended at the center of the elbow is θ. We will denote the angle adjacent to the r1 side as θ1 and, likewise, the angle adjacent to r2 as θ2, and we note that θ ) θ1 + θ2. From the compound angle formula and elementary trigonometry, we can show that
sin θ ) (l1r2 + l2r1)/R2 and therefore
r2 ) (R2sin θ - l2r1)/l1
(2)
C Molecules in the Form of Elbow-Connected Nanotori
J. Phys. Chem. C, Vol. 111, No. 29, 2007 10857
Similarly, from the compound angle formula for cosines, we have
cos θ ) (r1r2 - l1l2)/R
which we can rearrange and substitute for r2 from eq 2, giving
l1R2cos θ ) r1R2sin θ - l2(r21 + l 21) where we note that R2 ) r21 + l 21 is strictly positive. By dividing by R2 and rearranging, we obtain
r1 ) l1cot θ + l2csc θ
(3)
and likewise
r2 ) l2cot θ + l1csc θ
(4)
We note that the equations 3 and 4 are exact and strikingly simple, though not immediately obvious from the geometry of the problem. With the value for the perpendicular distances r1 and r2 determined, we wish to assign a representative value for the toroid-generating radius c. One way to do this is to consider the following formula for a mean radius jr for a circle
r(φ)dφ ) jrφ0
0
In the case of a right triangle with sides r1 and l 1, we have φ0 ) tan-1(l 1/r1) and r(φ) ) r1sec φ, and therefore
r1θ1 ) r1
∫0φ
sec φdφ ) r1 sinh-1(l1/r1)
1
We can repeat the same process for the second right triangle and combine the two equations, and now, applying the same result to the combined section, we obtain
c ) [r1sinh-1(l1/r1) + r2sinh-1(l2/r2)]/θ
(5)
Now, we can extend this process to determine a representative expression for the tube radius a. Here, we consider the following expression for the mean tube radius bh
∫0φ ∫02π b(φ, ψ)[r(φ) + b(φ, ψ)cos ψ] dψdφ ) 2πbhφ0c 0
where b(φ, ψ) is the radius of the tube and as before, and r(φ) is the torus-generating radius. For the first section of tube, φ0 ) tan-1(l 1/r1), r(φ) ) r1sec φ, and b(φ, ψ) )
a1xsec2φcos2ψ+sin2ψ. Using these definitions with equivalent expressions for the second type of tube and then combining, we derive the following formula for the representative tube radius
a)
1 2cπθ
∫02π
(
a1r1
∫0θ
1
x1 - sin2 φ sin2 ψ
dφ +
cos2 φ a2r2
∫0θ
2
x1 - sin2 φ sin2 ψ
)
dφ dψ
cos2 φ
where we note that the ψ integration is an elliptic integral, which upon the substitution of k ) sin φ, gives
a)
[
2 ar cπθ 1 1
∫0l /R 1
E(k) dk + a2r2 k′3
base unit
incremental unit
nanotube radius (Å) number atoms number atoms half-length (Å)
2
∫0φ
TABLE 1: Fundamental Parameters of Nanotube Elbows
∫0l /R 2
]
E(k) dk k′3
(6)
(3,3) (5,0) (4,4) (7,0) (5,5) (9,0)
2.0963 2.0550 2.7586 2.8096 3.4273 3.5769
+12 +20 +16 +28 +20 +36
18 30 32 42 50 54
+1.2103 +2.0550 +1.2190 +2.0930 +1.2229 +2.1079
TABLE 2: Bend Angles and Base Unit Section Half-Lengths θ unconstrained
θ constrained
elbow type
θ (°)
l 1 (Å)
l 2 (Å)
θ (°)
l 1 (Å)
l 2 (Å)
(3,3)-(5,0)
32.80
1.7755
3.1497
(4,4)-(7,0) (5,5)-(9,0)
35.39 35.89
2.4434 2.9737
3.1700 3.2599
30 36 36 36
1.8242 1.7100 2.4390 2.9701
3.0946 3.2196 3.1758 3.2636
where E(k) is the complete elliptic integral of the second kind with modulus k, and k′ ) x1-k2 is the complementary modulus. The integral in eq 6 can be expanded as an infinite series in which each term can be evaluated exactly. However, doing so increases the algebraic workload without elucidating the physical situation, and therefore, in the present paper, we simply evaluate this integral numerically using the software program MAPLE, but the analytical expression of this integral as a series is given in Appendix A. Results In this section, we begin by considering elbows made from the smallest possible nanotube sections. In the case of an armchair section, this means that if two elbows are as close as possible on the armchair section of the tube, the heptagon defects on the inner side of the bend must share a common side. In the case of a zigzag section, the carbon atom forming the top of a heptagon ring must bond with the carbon atom forming the top of the next heptagon. In the armchair tube, two rings consisting of a total of 4n atoms can be added to extend the tube by the longitudinal dimension of two rings, where n ∈ {2, 3, 4, ...}. With zigzag tubes, a complete ring of hexagons consisting of 4n atoms can be added to the tube to change its length without changing the orientation of the atoms at the end of the elbow. Table 1 contains the basic parameters for elbows, including the number of atoms in the basic section as well as the increase in the number of atoms and the half-length, which applies when each incremental unit is added. We now examine some numerical results for our procedure when applied to various different species of nanotube elbows. In this section, we consider three species of nanotube elbow, one constructed from (3,3)-(5,0) tubes, another constructed from (4,4)-(7,0) nanotubes, and last one constructed from (5,5)-(9,0) nanotubes. Cox and Hill22 present a new facetted model of carbon nanotubes, which properly incorporates curvature and from which all of the conventional carbon nanotube formulas emerge as the leading order term in an asymptotic expansion of the exact formulas. In the present paper, the tube radii used are calculated using the geometrically precise method described by Cox and Hill22 and using a value for the bond length of σ ) 1.42 Å. Once the positions of atoms are determined, the physical parameters l 1, l 2, and θ are determined by a minimization process both for an unstrained θ and again with the constraint θ ) 180°/n, where n ∈ {2, 3, 4, ...}. The results are presented in Table 2. In the case of the (5,5)-(9,0) elbow, our results are in excellent agreement with those of Fonseca et al.3 in that the
10858 J. Phys. Chem. C, Vol. 111, No. 29, 2007
Cox and Hill
Figure 3. C240 molecule with structure 5(3,3)185(5,0)30.
Figure 4. C360 molecule with structure 6(3,3)306(5,0)30.
unconstrained bend angle θ is very close to 36°. The (4,4)(7,0) elbow is also shown to have an unconstrained bend angle close to 36°. The unconstrained bend angle of the (3,3)-(5,0) elbow is shown to differ the most from a valid constrained angle. In this case, when the bend angle is constrained, the least-squares minimization process produces a preferred bend angle of 30°. However, this is not an order of magnitude improvement over using a bend angle of 36°, and therefore, for this elbow type, we consider nanotori constructed from both 10 and 12 elbows. The purpose of the remainder of this section is to introduce a nomenclature for the new carbon molecules. First, we use the notation (n,m)p to refer to a section of an (n,m) nanotube that is constructed from p atoms, which we assume consists of one base unit plus an integer multiple of incremental units arising from additional rings, as specified, for example, in Table 1. From Table 1, we see that (3,3)30 signifies a base unit (comprising 18 atoms) plus one increment unit (comprising 12 atoms) of a (3,3) armchair nanotube, whereas (5,0)30 signifies one base unit only (comprising 30 atoms) of the zigzag (5,0) nanotube. When constructing nanotori or other shapes from the elbow sections, we prefix the nanotube chiral numbers with the number of units making up the shape. Therefore, 5(3,3)185(5,0)30would be a molecule consisting of 5 base units of (3,3) armchair and 5 base units of (5,0) zigzag nanotube. This molecule would therefore comprise 240 atoms (that is, C240), and this is the smallest nanotorus possible with the elbows considered here. Therefore, the molecules proposed by Fonseca et al.3 are C520, which in our notation is 5(5,5)505(9,0)54, and C900, which we denote by 5(5,5)905(9,0)90. Two of these nanotori are illustrated graphically
TABLE 3: Examples of Other Ring Structures for the (3,3)-(5,0) Elbow a
b
(3,3)18 (3,3)18 (3,3)18 (3,3)18 (3,3)18 (3,3)18 (3,3)30
(5,0)30 (3,3)90 (5,0)30 (5,0)30 (5,0)90 (5,0)90 (5,0)30
c
d
(5,0)90 (5,0)30 (5,0)90 (5,0)90 (5,0)90 (5,0)150 (5,0)90
structure
shape
formula
abacabacabac acbcadacbcad abacacabacac abacacacabad ababababab abababababab ababacababac
triangle rectangle rhombus trapezium pentagon hexagon oval
C468 C552 C528 C588 C540 C648 C480
in Figures 3 and 4. Figure 3 shows a C240 molecule with structure 5(3,3)185(5,0)30, and Figure 4 shows a C360 molecule with structure 6(3,3)306(5,0)30. Using the parameters for the constrained elbows, we calculate the toroidal parameters r1 and r2 from equations 3 and 4, and finally, values for the mean torus-generating radius c and mean tube radius a are derived from the expressions 5 and 6 given in the previous section. We summarize these results in Table 4 for nanotori constructed from basic units and where l 1 ∼ l 2 and l 1, l 2 < 10 Å. We comment that the construction scheme presented here is limitless, and many more such caged molecules can be obtained using this scheme, both when l 1 is not similar to l 2 and when l 1 or l 2 > 10 Å. For reasons of space and divergence from the basic toroidal shape, which occurs when l 1 and l 2 differ significantly or when they become large, we do not attempt to fully delineate all of these molecules here. However, we comment that Table 4 describes four families of molecules, (i) 5(3,3)p5(5,0)q, which contains molecules with 240 + 60j + 100k atoms, (ii) 6(3,3)p6(5,0)q, which contains
C Molecules in the Form of Elbow-Connected Nanotori
J. Phys. Chem. C, Vol. 111, No. 29, 2007 10859
TABLE 4: Geometric Parameters for New Nanotori Carbon Molecules Constructed from Elbows (* Denotes the Two Molecules Studied by Fonseca et al.3) formula
l 1 (Å)
l 2 (Å)
r1 (Å)
r2 (Å)
R (Å)
a (Å)
c (Å)
5(3,3)185(5,0)30 5(3,3)305(5,0)30 5(3,3)545(5,0)50 5(3,3)785(5,0)70 5(3,3)905(5,0)90
C240 C300 C520 C740 C900
1.7100 2.9203 5.3409 7.7615 8.9718
3.2196 3.2196 5.2746 7.3296 9.3846
7.8311 9.4970 16.3248 23.1526 28.3147
7.3406 9.3997 16.3463 23.2930 28.1805
8.0156 9.9358 17.1763 24.4190 29.7021
2.0929 2.0926 2.0936 2.0941 2.0931
7.6730 9.6064 16.6111 23.6134 28.7232
6(3,3)186(5,0)30 6(3,3)306(5,0)30 6(3,3)546(5,0)50 6(3,3)666(5,0)70 6(3,3)906(5,0)90
C288 C360 C624 C816 C1080
1.8242 3.0345 5.4551 6.6654 9.0860
3.0946 3.0946 5.1496 7.2046 9.2596
9.3488 11.4451 19.7477 25.9540 34.2566
9.0084 11.4290 19.8296 25.8095 34.2101
9.5251 11.8406 20.4873 26.7962 35.4411
2.0852 2.0877 2.0885 2.0872 2.0877
9.2620 11.5699 20.0180 26.1811 34.6311
5(4,4)325(7,0)42 5(4,4)485(7,0)42 5(4,4)645(7,0)70 5(4,4)965(7,0)98 5(4,4)1285(7,0)126
C370 C450 C670 C970 C1270
2.4390 3.6580 4.8770 7.3150 9.7530
3.1758 3.1758 5.2688 7.3618 9.4548
8.7600 10.4378 15.6764 22.5929 29.5093
8.5206 10.5945 15.5491 22.5777 29.6062
9.0932 11.0602 16.4175 23.7476 31.0793
2.8127 2.8066 2.8091 2.8081 2.8076
8.7775 10.6905 15.8748 22.9662 30.0559
5(5,5)505(9,0)54 5(5,5)705(9,0)54 5(5,5)905(9,0)90 5(5,5)1305(9,0)126 5(5,5)1505(9,0)162
C520* C620 C900* C1280 C1560
2.9701 4.1930 5.4159 7.8617 9.0846
3.2636 3.2636 5.3715 7.4794 9.5873
9.6404 11.3235 16.5929 23.5454 28.8148
9.5450 11.6255 16.6073 23.6696 28.6514
10.0875 12.0749 17.4544 24.8232 30.2129
3.5358 3.5245 3.5318 3.5303 3.5342
9.7533 11.6578 16.8801 24.0049 29.2166
molecular structure
molecules with 288 + 72j + 120k atoms, (iii) 5(4,4)p5(7,0)q, which contains molecules with 370 + 80j + 140k atoms, and (iv) 5(5,5)p5(9,0)q, which contains molecules with 520 + 100j + 180k atoms, for any j, k ∈ {0, 1, 2, ...}. For the toroidal results in Table 4, we note that the mean generating radius c for all nanotori lies between the values of min (r1, r2) and R, as one would expect. Furthermore, the value of the mean tube radii of the nanotori are also generally between a1 and a2 but somewhat closer to the larger of these two values. This is again as expected since the constituent tubes are sliced further from the perpendicular point; the tube cross section becomes more elliptical, which has a tendency to increase the mean value of a. In addition to circular nanotori, the 5-fold symmetry of the 5(i,i)p5(j,0)q shapes allows a pentagonal-shaped cage molecules to be generated by including only base units of one type of tube and many incremental units of the other. Other shapes such as rhombi and trapezia can also be achieved by carefully matching the lengths of the armchair and zigzag nanotubes. The 6-fold symmetry of the 6(3,3)p6(5,0)q shape has even greater versatility for generating regular shapes. Using three longer sections of one tube species alternately in the structure would lead to an equilateral triangular shape, for example, 6(3,3)183(5,0)303(5,0)90. Other shapes such as rectangles, rhombi, trapezia, hexagons, and ovals are all achieved by simply varying the tube lengths appropriately. Some illustrative structures are listed in Table 3. We comment that, for example, the first row of Table 3 can be readily generalized as follows
C288+60k ) 6(3,3)183(5,0)303(5,0)30+20k where k is the number of incremental units used to form the triangular sides. The first row of Table 3 arises from the value k ) 3, and there are similar generalizations for the other rows of the table. Concluding Remarks In this paper, we take the elbow described by Dunlap,1,2 involving armchair (5,5) and zigzag (9,0) carbon nanotubes, and define similar elbows for (3,3)-(5,0) and (4,4)-(7,0) configurations. For elbows constructed from two nanotubes, we assume that the atoms remain in their preferred position on the nanotube sections and join them in such a way that the bond lengths
between adjacent atoms on the two sections are as close as possible to the carbon-carbon bond length in a least-squares sense. On the basis of these assumptions, we are then able to determine the bend angle θ and half-lengths l 1 and l 2, which characterize the elbow when the joint is unconstrained. The analysis is repeated with the constraint that the bend angle is θ ) 180°/n, where n ∈ {2, 3, 4, ...}. The outcome of this analysis is that for the previously identified (5,5)-(9,0) elbow, the unconstrained bend angle is very close to 36°. This supports the finding of Fonseca et al.,3 who find the same angle, but not that of Dunlap,1 who predicts a bend angle of 30°. For the new (4,4)-(7,0) elbow, an unconstrained angle very close to 36° is also found, but for the new (3,3)-(5,0) elbow, we find an angle close to 33°. Following this analysis of the three elbow types, we then consider toroidal cage molecules of carbon formed from these elbows, where each section of nanotubes is less than 10 Å long. In this investigation, it is possible to construct a large number of such molecules, and in Table 4, we detail the geometric parameters of twenty such possible nanotori. Two of these molecules, namely, those with structures 5(5,5)505(9,0)54 and 5(5,5)905(9,0)90, are the C520 and C900 molecules previously described by Fonseca et al.3 However, the remainder are new, and using the geometric parameters given here, it is possible to describe other molecules involving various combinations of elbow base and incremental units. In particular, we comment that the C240 and C360 molecules contained in this work are structurally distinct from those of Itoh et al.4 and Itoh and Ihara.8,9 We also describe a procedure for determining the mean torusgenerating radius c and the mean tube radius a for any of these nanotori. Once the bend angle and tube lengths are known, the perpendicular lengths from the torus center can be determined, and these can then be used to calculate the geometric parameters mentioned above. These parameters are useful in incorporating such nanotori in nanomechanical systems, such as the oscillating system described by Hilder and Hill,21 where it is assumed that the approximately toroidal molecules can be modeled as perfect tori with the dimensions as given in Table 4. Acknowledgment. The support of the Australian Research Council, both through the Discovery Project Scheme and for providing an Australian Professorial Fellowship for J.M.H., is gratefully acknowledged.
10860 J. Phys. Chem. C, Vol. 111, No. 29, 2007
Cox and Hill
Appendix Evaluation of Equation 6. In eq 6, we are required to integrate an elliptic function of the second kind in the form
I)
I)
E(k)
∫0l/R (1 - k2)3/2dk
where l is either l 1 or l 2, and we note that the upper limit is 0 < l /R < 1. Using the series expansion of the elliptic function given as eq (900.07) in Byrd and Friedman23 gives
I)
∞
( )
-1/2 2 m)0 1 - 2m m
π
1
∑
2
∫0l /R
k2m
∞
(1 - k2)3/2
( )∫
-1/2 2 m)0 1 - 2m m
π
1
∑
2
sin2m x dx cos2 x
R
0
To avoid the case where m ) 0, we extract this term from the series and, by doing so, obtain
I)
πl
∑ ( 2 m)1 2m - 1 ∞
π
-
2r
-1/2 m
1
)∫
2m x R sin dx 0 cos2 x
2
where r ) (R2 - l 2)1/2. We now adjust the series to by substituting m + 1 for m, giving ∞
(
-1/2 I) 2r 2 m)0 2m + 1 m + 1 πl
π
∑
1
)∫ 2
R
0
sin2m+2 x dx cos2 x
and employing the eq 2.518(1) given in Gradshteyn and Ryzhik,24 we can express this as
I)
πl 2r
-
π
∞
∑ 2 m)0
( )[ -1/2 m+1
l 2m+1
2
-
(2m + 1)rR2m
∫0R sin2m x dx
]
whereupon we again wish to avoid the m ) 0 term, and therefore, we extract the first term from the series, which gives
I)
()
( )
π l π ∞ -1/2 + sin-1 8r 8 R 2 m)1 m + 1
3πl
∑
[
2
×
l 2m+1
(2m + 1)rR
2m
()
∫0R sin2m x dx
]
( )
l π ∞ -1/2 π + + sin-1 8r 8 R 2 m)1 m + 1
3πl
{
∑
()
2
×
[
()
(2m - 1)!! -1 l r2 l 2m+1 sin 1+ × R l 2mm! (2m + 1)rR2m m-1
∑ k)0
dk
By making the substitution k ) sin x and R ) sin-1(l /R), we derive
I)
and, employing eq 2.511(2) from Gradshteyn and Ryzhik,24 finally yields
( ) ]}
(m - k - 1)!(2m + 1)!! R
2k
2k+1m!(2m - 2k - 1)!! l
where the double factorial (2n - 1)!! denotes (2n - 1)(2n 3) ‚‚‚ 5 ‚ 3, and this equation can be used to numerically evaluate the integrals in equation 6. We also comment that, for the nanotori considered here, the ratio l /R ∼ 1/3, and for these parameters, a result for I correct to 8 significant digits can be determined by summing only the first terms of the series up to m ) 5; the leading two terms alone, outside of the summation, are sufficient for an answer accurate to 3 significant digits. References and Notes (1) Dunlap, B. I. Phys. ReV. B 1992, 46, 1933. (2) Dunlap, B. I. Phys. ReV. B 1994, 49, 5643. (3) Fonseca, A.; Hernadi, K.; Nagy, J. B.; Lambin, P.; Lucas, A. A. Carbon 1995, 33, 1759. (4) Itoh, S.; Ihara, S.; Kitakami, J.-I. Phys. ReV. B 1993, 47, 1703. (5) Stillinger, F. H.; Weber, T. A. Phys. ReV. B 1985, 31, 5262; Erratum Phys. ReV. B 1986, 33, 1451. (6) Ihara, S.; Itoh, S.; Kitakami, J.-I. Phys. ReV. B 1993, 47, 12908. (7) Goldberg, M. Tohoku Math. J. (1) 1937, 43, 104. (8) Itoh, S.; Ihara, S. Phys. ReV. B 1993, 48, 8323. (9) Itoh, S.; Ihara, S. Phys. ReV. B 1994, 49, 13970. (10) Meunier, V.; Lambin, Ph.; Lucas, A. A. Phys. ReV. B 1998, 57, 14886. (11) Ceulemans, A.; Chibotaru, L. F.; Bovin, S. A.; Fowler, P. W. J. Chem. Phys. 2000, 112, 4271. (12) Oh, D.-H.; Park, J. M.; Kim, K. S. Phys. ReV. B 2000, 62, 1600. (13) La´szlo´, I.; Rassat, A. Int. J. Quantum Chem. 2001, 84, 136. (14) Tersoff, J. Phys. ReV. Lett. 1988, 61, 2879. (15) Brenner, D. W. Phys. ReV. B 1990, 42, 9458; Erratum Phys. ReV. B 1992, 46, 1948. (16) Liu, J.; Dai, H.; Hafner, J. H.; Colbert, D. T.; Smalley, R. E.; Tans, S. J.; Dekker, C. Nature 1997, 385, 780. (17) Han, J. Toroidal Single Walled Carbon Nanotubes in Fullerene Crop Circles; Technical Report, NASA Advanced Supercomputing, 1997. (18) Martel, R.; Shea, H. R.; Avouris, P. Nature 1999, 398, 299. (19) Martel, R.; Shea, H. R.; Avouris, P. J. Phys. Chem. B 1999, 103, 7551. (20) Sano, M.; Kamino, A.; Okamura, J.; Shinkai, S. Science 2001, 293, 1299. (21) Hilder, T. A.; Hill, J. M. Phys. ReV. B 2007, 75, 125415. (22) Cox, B. J.; Hill, J. M. Carbon 2007, 45, 1453. (23) Byrd, P. F.; Friedman, M. D. Handbook of Elliptic Integrals for Engineers and Scientists; Springer-Verlag: Berlin, Germany, 1971. (24) Gradshteyn, I. S.; Ryzhik, I. M. Table of Integrals, Series, and Products; Academic Press: San Diego, CA, 2000.
