1.

INTRODUCTION

The variety of structural forms and a wide range of physical and chemical properties have caused development of new scientific branch: electronics on carbon nanostructures. Torus-like structure, called carbon nanotori attract special attention among the family of carbon nanostructured materials [1]. For the first time theoretical model of a new class of carbon nanostructures with a torus-like form was proposed in Ref [2].

Geometric characteristics and topology of these forms of carbon were determined unsaturated bonds of carbon nanotubes. The broad prospects for potential applications of nanotori in various scientific and technological fields are caused Interest in the study of nanotori. The use of carbon nanotori as new forms of hydrogen storage is one of these applications. This problem is relevant, because hydrogen is a source of clean energy will play an important role in reducing the global use of hydrocarbon. The results of a theoretical investigation of the adsorption of hydrogen on the carbon nanotorus C₁₂₀ presented in [3]. Hydrogen was placed inside a nanotorus in two different ways: parallel and perpendicular to the central axis of nanotorus. It was established in consequence of the optimization of nanotorus that all adsorbed hydrogen molecules are evenly distributed along a path that is equidistant path for all points near the inner surface of the nanotorus. The results of calculations of the total energy and the binding energy of the investigated nanotorus C₁₂₀ showed that hydrogen is advantageous for accumulate carbon nanostructures. The total energy of the system decreases with increasing number of adsorbed hydrogen molecules.

Carbon nanotori have a number of notable magnetic properties due to the peculiarities of the atomic structure. In particular, in 1997 Hadd theoretically predicted that nanotorus C₅₇₆ has a very large value of the diamagnetic susceptibility, which can be about 130 times higher than that of benzene molecules [4]. Subsequently, a significant paramagnetic moment was found in metal torus-like carbon nanotubes [5]. The existences of a torus-like carbon nanotubes ferromagnetic moment at low temperatures were predicted by the tight-binding method [6]. This effect was caused by the presence of pentagons and heptagons in the structure of the nano torus. Aharonov-Bohm effect is another important magnetic phenomena found in torus-like carbon nanotubes[7].

Evaluation of mechanical properties of nanotori was not carried out despite a number of studies on the stability of the carbon nanotori [8-10]. It is necessary that the material nanotori met all performance requirements for its successful application in the design of electronic nanodevices. The aim of this work is a theoretical study of the strength characteristics of carbon by the tight-binding method and classical molecular dynamics method.

2.

THE MATHEMATICAL APPARATUS FOR THE STUDY OF CARBON NANOTORI

2.1

QUANTUM-CHEMICAL TIGHT-BINDING METHOD

Modification of the tight-binding method, developed in [11] was used in this study. The total energy of the system within the used model is the sum of two terms:

where Erep is repulsive energy, Ebond is filled electronic energy levels. Geometrical and energy parameters of the carbon nano-torus are determined by minimizing its total energy from the characteristic linear parameters of the framework.

Repulsive energy, which takes into account electron-electron and internuclear interaction, is the sum of the repulsive pair potentials

where i, j are number of interacting atoms; r_i, r_j are Cartesian coordinates. Function V_rep defined by the expression

where = 10.92 eV. Energy filled levels determined by the formula

where ε_n is energy electron state with the number n (eigenvalues of the Hamiltonian). «2» takes into account the electron spin. Interatomic matrix elements of the Hamiltonian are calculated by the following formula

where r is the distance between the atoms; i,j are orbital momenta of the wave functions; a is an index indicating a type of bond (σ and π). The values of the atomic terms ε_s, ε_p and equilibrium overlap integrals , , , are shown in Table 1 [11].

Table 1.

Parameter values for tight-inding method

εs, eV	εp, eV	, eV	, eV	, eV	, eV	p1	p2,nm	p3,nm	p4	p5,eV	p6
-10.932	-5.991	-4.344	3.969	5.457	-1.938	2.769	0.232	0.154	22	10.92	4.455

2.2.

MOLECULAR DYNAMICS METHOD

Dynamic compression of carbon nano tori we investigated by molecular dynamics method. Newton’s equations of motion for a system of interacting particles are solved within of this method.

where r_i is radius vector i atom, m_i is atom mass, F_i is the total force acting on the i-th atom from the other particles:

where U (r) is the potential energy, which depends on the mutual arrangement of all the atoms; n - number of atoms.

Solution of the equations of motion was carried by a predictor-corrector scheme of the third order [12]. Modeling the compression process was carried out in steps of 1 fs. The forces acting on the atoms of the studied system, were calculated by quantum-chemical tight-binding method.

3.

TOPOLOGICAL MODEL NANOTORI

Several models of nanotori corresponding equilibrium configuration are discussed in this paper. The geometry of topological models is described by the point group of symmetry D_5d.

Nanotorus C₁₂₀

C₁₂₀ is the smallest torus of the group. Basic segment of this torus is presented in Fig. 1a. Atomic coordinates of framework of nanotorus were generated by the symmetry operations of C5, which acted on the atomic coordinates of the torus segment as a result of the four transformations over atoms segment. Atomic coordinates of segment are given in the initial approximation. Atomic framework of molecule C₁₂₀ are shown in Fig. 1b. Figure 1 shows that ten pentagons (light gray shading in Fig. 1b) form the outer circle, and ten heptagons (dark gray shading) form the inner circle.

Fig. 1.

Nanotorus C120: a) reference framework segment; b) the atomic framework.

Atomic coordinates nanotorus corresponding to ground state are determined by minimizing the total energy of the coordinates of the atoms within the tight-binding method. Various lengths of bonds are given numbers on the Figure 1: 1 – 1,38 Å, 2 – 1,46 Å, 3 – 1,46 Å, 4 – 1,47 Å, 5 – 1,46 Å, 6 – 1,47 Å, 7 – 1,46 Å, 8 – 1,43 Å, 9 – 1,46 Å, 10 – 1,40 Å, 12 – 1,46 Å. It can be seen, the shortest bond length is in the heptagon on the inner circle, where the framework structure in greatest deformation. The radius of the inner circle is equal to 2,03 Å. The largest bond lengths are located in the Pentagon of outside circle whose radius is equal to 5,86 Å.

Nanotorus C₃₄₀

C₃₄₀ is the next class of symmetry D5d of nanotori. Its difference consists not only in increasing the radii of the inner and outer circles, but also in positioning heptagons and pentagons. Basic segment of nanotorus C₃₄₀ is shown in Fig. 2a, and the atomic cell generally shown in Fig. 2b. The difference framework of nanotori C₁₂₀ and C₃₄₀ is clearly manifested in the process of formation of the cells of the inner circle heptagon. If heptagon of nanotorus C₁₂₀ are positioned adjacent to each, the heptagon of nanotorus C₃₄₀ are separated by one hexagon. Naturally, the tension of the inner circle in this case framework is markedly reduced.

Fig. 2.

Nanotorus C340: a) reference framework segment; b) the atomic framework.

Technique generate coordinate framework remains the same (the coordinate transformation of the base segment) (Fig. 2a). Let us consider the atomic structure nanotorus C₃₄₀. Lengths of bond to the base segment are equal: 1 – 1,38 Å, 2 – 1,40 Å, 3 – 1,40 Å, 4 –1,40 Å, 5 – 1,40 Å, 6 – 1,40 Å, 7 – 1,41 Å, 8– 1,44 Å, 9– 1,41 Å, 10 – 1,40 Å, 11 – 1,44 Å, 12 – 1,46 Å, 13 – 1,43 Å, 14 – 1,43 Å, 15 – 1,42 Å, 16 – 1,47 Å, 17 – 1,45 Å. Numbers of the various of bond lengths are given numbers in Fig. 2a. The shortest bond length of the torus C₃₄₀ is a heptagon on the inner circle as in the nanotorus C₁₂₀. The radius of the inner circle is equal to 6,09 Å. The largest bond lengths are located in the pentagon outside circle whose radius is equal to 10,68 Å.

Nanotorus C460

Feature of atomic carbon framework nanotorus C₄₆₀ consists in that the radius of the inner circle 4.97 nm and inside part of a framework formed by heptagon. Heptagon is separated hexagons (as in nanotora C340). The radius of the outer circle of the nanotorus C₄₆₀ is much higher than on the radius at the nanotorus C340: 14,88 Å. This increase in size of the outer circle of torus is provided significant removal of from each other pentagons. Basic segment of nanotorus C₄₆₀ are shown in Fig. 3a. Structure of the entire framework is shown in Fig. 3b. The number of different of bond lengths also was increased by increasing the number of atoms of the base segment. We present just a few of them: lowest bond length 1,40 Å is observed in hexagons, forming the surface of the torus, the highest 1,55 Å observed in the Pentagon of outer circle.

Fig. 3.

Nanotorus C460: a) reference framework segment; b) the atomic framework.

We calculated the reaction enthalpy of formation of torus-like structures for a quantitative estimate of stability formed torus-like structures. The calculation results are in Table 2. It can be seen from the Table 2 that the most stable structure is nanotorus C₃₄₀. Enthalpy and energy per atom in these nanotori are practically identical with the same parameters C₆₀.

Table 2.

Some energy characteristics of nano tori

4.

STRENGTH CHARACTERISTICS OF CARBON NANO TORI

Strength characteristics of carbon nanotori were studied based on the behavior of the objects under deformation in real time. We have studied the process of axial compression. Deformation investigated nanotori was carried out at a speed of 20 m / s along the axis Z (Fig. 4). The pressure in the work was seen as applied load. Pressure was created by the graphene plate approaching to torus with a velocity of 20 m / sec. Carbon nanotori during the study were exposed to longitudinal compression by 1-5%. The value of the compressive force applied to the objects is fixed at each stage of compression. The calculation results for the case of deformation of nanotorus C₁₂₀, C₃₄₀, C₄₆₀ were shown in Fig. 5-7.

Fig.4.

The compression scheme of the carbon nanotorus.

Fig.5.

Dependence deforming force on the magnitude of axial compression nanotorus C120

Fig. 6.

Dependence deforming force on the magnitude of axial compression nanotorus C340

Fig. 7.

Dependence deforming force on the magnitude of axial compression nanotorus C460

It is seen from Figure 8 that with compression C₁₂₀ deforming force acting on the torus increases monotonically, approaching the saturation near values 120 nN. A similar pattern is observed during deformation carbon nanotori C₃₄₀ and C₄₆₀. Deforming force with increasing compression nanotorus C₃₄₀ increases under the linear law. The small drop in value of deforming force is observed when compression of the structure 3%. Deforming force at a higher compression is almost unchanged. Deforming force of nanotorus C₄₆₀, is growing rapidly in compression of object by 1-2%, and during subsequent deformation (3.5%) changes insignificant in the range of (160-180 nN).

Young’s modulus of compression was calculated for quantitative assessment strength properties of the investigated nanotori. Algorithm for computing pseudo Young’s modulus is as follows.

where S is the cross section nanotorus; the force required to compress the torus defined by the formula

where ΔE is the energy of elastic tension, ΔL is reducing the value of the structure.

Young’s modulus of compression for all studied in this paper nanotori was calculated by the algorithm described above. Results are presented in Table 3.

Table 3.

Calculated values of the Young’s modulus of compression for nanotori

It can be concluded from table data that nano torus C₄₆₀ has the highest modulus of elasticity. The order of the calculated modulus tells us that nanotori possess high strength properties comparable with the strength properties of carbon nanotubes [13]. Consequently, this carbon material can be considered as basic material for the base element of modern electronics.

5.

CONCLUSION

We have installed the most stable carbon nanotori during the theoretical study their electron energy and strength characteristics. Nanotorus C₃₄₀ has the most stable structure among the investigated objects. This fact is confirmed by the results of calculations of the enthalpy of reaction of structure formation (10.22 kcal • mol / atom) and the energy per atom (-43.02 eV). The values obtained are practically identical with the same parameters of the C₆₀ fullerene, which is one of the most stable members of the family of carbon nanoclusters.

It is shown that the process of dynamic compression nanotori is accompanied by constant increasing deforming force. Numerical estimation of Young’s modulus in compression of carbon nanotori possible to establish that the carbon nanotorus C₄₆₀ characterized by the greatest modulus of elasticity (0.65 TPa). The growth of the elastic modulus is observed with an increase in the geometric dimensions of the nanotorus.

The results obtained in this study indicate that carbon torus-like structures are characterized by stability and high strength characteristics. These structures can be used as the material of element base of modern nanoelectronics. In particular, their use as a material for the creation of cold cathodes is promising use in nanoelectronics.