Molecular Dynamics Study of Collective Behavior of Carbon Nanotori in Columnar Phase

Abstract

Supramolecular interaction of carbon nanotori in a columnar phase is described using the methods of classical molecular dynamics. The collective behavior and dynamic properties of toroidal molecules arising under the action of the van der Waals forces are studied. The conditions under which columnar structures based on molecular tori become unstable and rearrange into another structure are investigated. The reasons for the appearance of two types of directed rotational motion from the chaotic motion of molecules are discussed.

1. Introduction

Carbon nanomaterials are of great interest to researchers in connection with the discovery of many unique properties in them that are significantly superior to existing substances and allow the development of further miniaturization of electronic, photonic, plasmonic, and electromechanical devices. One of these materials is a carbon nanotorus [1,2], which is a nanotube with both ends connected together. Methods of laser ablation, sonication, organic reactions, and chemical deposition are used to create nanotori [3,4,5]. Molecular tori have many unique physical properties, such as high reversible tension, physical and chemical stability [6], high magnetic response [7,8,9], and large magnetic moments [10]. Nanotori can be used to transfer atoms and ions in the inner region [11], as well as to encapsulate atoms [12], molecular chains [13,14], and nanoparticles [15] into the inner cavity. Nanotori are capable of rotating in the megahertz and gigahertz frequency ranges [16,17], just like carbon nanorings [18] and fullerenes [19,20,21,22,23]. Due to the presence of these properties, nanotori have great prospects for use in modern nanoelectromechanical systems as a structural element. To expand the range of applications, it is necessary to further study the properties of molecular tori and ways to control these properties.

In this article, we investigate the group behavior of carbon nanotori in a columnar phase [24]. For this, we use the method of molecular dynamics, which consists of the numerical integration of the Newton–Euler equations [25,26] describing the combined translational and rotational dynamics. The interaction of molecular assembly is described using potential as a function of the coordinates of all the atoms in the molecular system. The stability of supramolecular structures consisting of toroidal molecules is investigated. The influence of the size of the supramolecular structure on the generation of two types of directed rotational motion from chaotic motion, as well as the mutual influence of rotational motions are studied.

2. Mathematical Statement of the Problem

As the initial moment of time, we take the moment of formation of an ideal columnar fragment in which the axes of all tori coincide (Figure 1). We are interested in the evolution of the presented simplest mechanical system in the case when no forces act on it, except for the internal forces of intermolecular interactions. Interactions between individual toroidal molecules are described as the totality of all cross-atom-atom interactions. Unlike paper [27], which used the continual model, in this paper, we use the full pair-additive Lennard–Jones (LJ) potential. The influence of the specific shape of a torus is taken into account by the spatial position of carbon atoms on the surface of a torus, which is considered as undeformed. We assume that the interaction of any two atoms depends only on their mutual arrangement and does not depend on the position of any other particles as in many-body potentials [28,29,30].

According to Euler’s approach, the movements of individual tori are determined by the displacements of their centers of mass and rotations around them. Moreover, it is convenient to describe the rotation of an individual molecule in a coordinate system rigidly connected with the moving molecule. Then the axial moments of inertia of the tori will remain constant throughout the movement. In this regard, the number of local coordinate systems will be equal to the number of tori in the considered mechanical system. Thus, Euler’s approach describes a combination of two groups of movements: translational and rotational movements. This approach is more convenient for interpreting the calculation results than the methods used by the authors in earlier papers [31,32,33].

The translational displacements of the kth nanotorus are determined by the equations [25,26]:

$$M\frac{d{v}_{kc}}{dt}=-{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^N\nabla U\left({\rho }_{ij}\right)}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\frac{d{r}_{kc}}{dt}=v_{kc},\hspace{0.17em}\hspace{0.17em}k=1,\dots ,S.$$

(1)

Here M is the mass of the torus; $v_{kc}$ is the velocity of the center of mass of the kth torus; $\nabla$ is gradient operator; U is the LJ potential [25]; ρ_ij is the distance between the ith carbon atom of the kth torus and the jth atom of the surrounding nanotori, not including the kth nanotorus; $r_{kc}$ is the position vector of the center of mass of the kth torus; N = n(S − 1); n is the number of carbon atoms in the molecular torus; S is the number of nanotori.

Vector Equations (1) are integrated under the following initial conditions:

$$t=0:v_{kc}=v_{kc}^0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{r}_{kc}=r_{kc}^0, k=1,\dots ,S,$$

(2)

where $v_{kc}^0$, is the initial thermal velocity of an kth molecule, $r_{kc}^0$ is its initial position.

The rotational motion of the kth molecule can be determined by the dynamic Euler equations [25,26]:

$$\begin{gathered} A\frac{d{p}_k}{dt}+(C-B)q_k{r}_k=M_{{\xi }_k}, B\frac{d{q}_k}{dt}+(A-C)r_k{p}_k=M_{{\eta }_k}, \\ C\frac{d{r}_k}{dt}+(B-A)p_k{q}_k=M_{{\zeta }_k}, k=1,\dots ,S. \end{gathered}$$

(3)

Here p_k, q_k, r_k are the projections of the angular velocity vector ω onto the ξ_k-, η_k-, and ζ_k-axes associated with the moving kth torus; A, B, C are the principal moments of inertia for the center of mass of the toroidal molecule; $M_{{\xi }_k}$, $M_{{\eta }_k}$, $M_{{\zeta }_k}$ are the external axial moments with respect to the kth torus.

Equation (3) are supplemented by the Euler kinematic relations for each individual molecule:

$$\begin{gathered} p_k={\dot{\psi }}_k\sin {\theta }_k\sin {\varphi }_k+{\dot{\theta }}_k\cos {\varphi }_k, q_k={\dot{\psi }}_k\sin {\theta }_k\cos {\varphi }_k-{\dot{\theta }}_k\sin {\varphi }_k, \\ {r}_k={\dot{\psi }}_k\cos {\theta }_k+{\dot{\varphi }}_k, k=1,\dots ,S. \end{gathered}$$

(4)

In relations (4), the dot above the dependent variable means a derivative of a function with respect to time. These relations are differential equations for the Euler angles denoted as φ_k, ψ_k, and θ_k. Therefore, they require setting the initial conditions, which we take in the following form.

$$t=0:p_k=p_k^0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{q}_k=q_k^0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}r=r_k^0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\varphi }_k={\psi }_k=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\theta }_k=\pi /2,\hspace{0.17em}\hspace{0.17em}k=1,\dots ,S.$$

(5)

Here $p_k^0,\hspace{0.17em}{q}_k^0,\hspace{0.17em}{r}_k^0$ are the projections of the initial angular rotational velocity on the ξ_k-, η_k- and ζ_k-axes. Equations (4) have a coordinate singularity at θ_k = jπ (j = 0, ±1, +2). Therefore, the initial condition for the angle θ_k is set equal to π/2. In addition, in the course of calculations, changes in the angles θ_k of each nanotorus were monitored.

The mathematical model is based on evolutionary equations for the coordinates of the centers of mass of the considered tori and the angles that determine their orientation in space. Therefore, we have the Cauchy problem for 12S ordinary differential equations of the first order (1), (3), (4) with initial conditions (2), (5).

There are many ways to solve the problem numerically. However, it should be taken into account that approximation errors accumulate when solving evolutionary problems by step-by-step methods. Therefore, methods of a high order of accuracy are required. In carrying out these calculations, we used the fourth-order Runge–Kutta method [34]. The integration step was constant and amounted to Δt = 10^–6 ns.

3. Results and Discussion

We consider several identical molecular nanotorus located near each other (see Figure 1). Each nanotorus with an outer diameter of a = 1 nm consists of 158 carbon atoms. At the initial moment of time, the centers of mass of neighboring nanotori are at a distance of 0.69 nm from each other. These molecules have no initial translational velocity. The Cartesian (ξ_kη_kζ_k) coordinate system associated with the movable molecule is chosen so that the origin of coordinates coincides with the center of mass of the kth nanotorus, and the ζ_k-axis is directed perpendicular to the plane of the nanotorus (the ξ_kη_k-plane). At the initial moment of time, the origin of the global (xyz) coordinate system is set at the center of mass of the first molecular torus (k = 1). Thus, the y- and ζ_k-axes are parallel.

3.1. Collective Behavior of the Nanotori Cluster

Figure 2 shows the time dependences of the y-coordinates of the centers of mass of all molecules in a cluster of 5 nanotori. It can be seen that the nanotori oscillate around their initial position up to t ≈ 8 × 10^–2 ns. Then the 1st nanotorus changes places with the 5th nanotorus, the 2nd changes places with the 4th, and the 3rd nanotorus remains near its original position.

We consider clusters consisting of S = 2,…, 5 nanotori. Figure 3a,b show the time dependences of the y-coordinates of the centers of mass of the first and last tori, which in Figure 1 are marked as k = 1 and k = S. As can be seen from Figure 3, the molecules perform chaotic movements at the initial time interval due to intermolecular interaction.

The first and last nanotori begin to change places when the time reaches t ≈ 5 × 10^–2 ns. Clusters consisting of 2 and 3 nanotori were able to change their position twice during the considered period of time. Such a consistent change of positions has similar features to the flip effect of one body, which is described by the intermediate axis theorem [35,36,37,38]. Note that the classical intermediate axis theorem explains the case of unstable rotational motion of a body around an axis with an intermediate moment of inertia. Even small perturbations along other axes are sufficient for the appearance of the flip effect through an angle π.

Although the superficially observed flip effect has many similarities, in contrast to the classical turning effect, this article discusses several isolated molecules. They interact with each other and are in continuous chaotic motion. The toroidal shape of the molecules in the cluster determines the unequal directions of motion in space. The rotational motion of the nanotorus is limited in all directions by the surrounding molecules, except for rotation around the ζ_k-axis with the maximum moment of inertia. The translational movements of the nanotorus have similar limitations. This causes its translational displacements mainly in directions parallel to the xz-plane. If the deviations from the initial position caused by translational motions in the xz-plane become large enough, this leads to a large asymmetry of the cluster structure. At the same time, a combination of moments of forces caused by intermolecular interaction and rotational motion of nanotori initiate the indicated rotational motion of the cluster, which we will call π-rotation. The observed effect is caused by the precessional motion arising from the coordinated action of the moments of forces at all rotating nanotori, which behave like gyroscopes. In other words, we observe how nanotori are controlled by a consistent gyroscopic effect [39]. Gyroscopic effects are widely used and have great prospects in micro- and nanodevices [26,40,41,42,43,44,45,46].

After performing the π-rotation, the nanotori are in a new position for some time, and then they again perform a similar π-rotation. Note that the more molecules there are in the cluster, the longer it will take for the combination of factors to initiate π-rotation (Figure 3). In this regard, the angular speed of π-rotation for a cluster of five nanotori decreases approximately 1.7 times compared to two nanotori (Figure 4). Figure 5 shows the π-rotation effect for a cluster of three molecules at different points in time.

Figure 6a,b illustrates the time dependence of the y-coordinates of centers of mass of the first and last tori, respectively, for clusters consisting of six or more molecules. In contrast to the results presented in Figure 3, two groups of clusters can be distinguished here, those with an even number of molecules and those with an odd number.

The first toroidal molecule in odd groups (the center of mass coincides with the central molecule) almost immediately leaves its initial position but does not reach the position of the last nanotorus. Even groups behave in a similar way as at S ≤ 5, but their rotation begins later at t ≈ 0.1 and 0.12 ns for groups with 8 and 6 molecules, respectively. Especially here a cluster with 6 molecules can be distinguished, which shows a significantly greater stability to the effect of π-rotation compared to all other variants. A detailed analysis shows that a columnar structure of odd clusters consisting of 7 and 9 nanotori is unstable under these conditions. At the initial moment, molecules are separated from the odd columnar structures and form groups with a smaller number of molecules. In the case of S = 7, the nanotori self-organize into two columns, consisting of 3 and 4 molecules (Figure 7). Molecules with k = 2,…, 5 remain in the primary column. Molecules with k = 1, 6, 7 move to the new column. Despite the division of the cluster into parts, each of the tori of the considered molecular system nevertheless performs π-rotation (see Figure 7).

3.2. Directed Rotational Motion of Individual Nanotori

Chaotic translational and rotational vibrations initiate directional movement not only through the collective effect of cluster rotation. The cluster of nanotori also generates a directed rotational motion of each molecule around its ζ_k-axis, which for brevity, we will call 2π-rotation. Consider a group consisting of S = 5 toroidal molecules.

Figure 8 shows the time dependence of the magnitude of the angular velocity component (r) along the ζ_k-axis. It can be seen that in the molecules with the number k = 1, 4 have a predominantly positive value of r, and the molecules with the number k = 2, 5 have a negative value of r. Moleculewith number k = 3 changes the direction of 2π-rotation from positive to negative when performing collective π-rotation. The maximum frequency of the 2π-rotation is observed for the first torus and is 19.9 GHz. Note that researchers using molecular dynamics methods also observed frequencies in the gigahertz range for nanorings [18] and fullerenes [19,20,23], which are in qualitative agreement with experimental data [21,22]. Oscillators based on nanotori in a nanotube showed frequencies in the megahertz and gigahertz ranges [16,17].

Table 1. The maximum number of revolutions in one direction (2π-rotation).

S	Molecule Number k
	1	2	3	4	5	6	7	8	9
2	0.81	0.61	–	–	–	–	–	–	–
3	0.83	0.95	0.48	–	–	–	–	–	–
4	1.76	0.55	1.37	0.69	–	–	–	–	–
5	1.68	0.43	0.50	0.68	0.67	–	–	–	–
6	0.83	0.58	1.21	0.43	2.40	1.72	–	–	–
7	0.72	1.03	2.66	1.36	0.68	2.32	0.77	–	–
8	0.46	0.48	1.72	0.75	0.97	1.16	0.57	0.54	–
9	1.07	0.82	2.57	1.26	0.52	1.27	1.10	0.56	0.48

shows the maximum number of revolutions (φ/2π) that the nanotori made in one direction around the ζ_k-axis during the considered time interval. It is seen that an increase in the number of nanotori in the cluster as a whole leads to an increase in the number of revolutions. This is due to the fact that a large number of system elements have a greater total impact. At the same time, π-rotation in clusters with fewer nanotori contributes to a change in the direction of 2π-rotation (see Figure 5 and Figure 8). As seen from Table 1, the largest number of revolutions of the nanotorus reaches 2.66 at S = 7 and k = 3.

3.3. Effect of 2π-Rotation on π-Rotation

We consider three problems with S = 2 to evaluate the effect of 2π-rotation on π-rotation. In all these problems, the last torus (k = 2) at the initial moment of time has an angular velocity ω⁰ along ζ₂-axis. Let us set the following initial conditions for the first torus (k = 1):

• The first and second tori rotate the same angular velocity: ${\omega }_1^0={\omega }_2^0={\omega }^0$;
• The first torus does not rotate: ${\omega }_1^0=0,\hspace{0.17em}{\omega }_2^0={\omega }^0$;
• The first torus rotates in the opposite direction: ${\omega }_1^0=-{\omega }^0,\hspace{0.17em}{\omega }_2^0={\omega }^0$.

Figure 9 shows the dependence of the maximum precession angle ψ_m (ω⁰), which characterizes the rotation of the first nanotorus.

$${\psi }_m=\underset{t\in [0,\hspace{0.17em}\hspace{0.17em}{t}_z]}{\max }\psi (t),$$

(6)

where t_z is the final time.

Due to the complexity of the three-dimensional motion, this parameter does not allow one to fully track the number of revolutions of the system of molecules performed within the π-rotation. However, this parameter makes it possible to estimate the presence or absence of the π-rotation by comparing it with the value ψ = π (dashed curve). The condition ψ_m(ω⁰) < π indicates that there is no π-rotation for a given initial angular velocity.

As can be seen from Figure 9, an increase in the initial angular velocity of the 2π-rotation leads to the disappearance of the π-rotation in problem # 1 (solid curve). This is due to the fact that the rotational motion reduces the chaotic motion and asymmetry of the cluster, which are necessary for the initiation of π-rotation. At the same time, the rotation of two nanotori in the same direction with an increase in the initial angular velocity does not allow one to assume such a mutual position and orientation that the acting moments of forces would lead to a coordinated precessional motion. At the same time, the 2π-rotation of two nanotori in the same direction with an increase in the initial angular velocity does not allow one to assume such a mutual position and orientation that acting moments of forces would lead to the π-rotation. Since unidirectional rotation and multidirectional moments of forces of the nanotorus lead to inconsistent precessional movements of the nanotorus that block each other

In problem # 2 (dash–dot curve), an increase in the initial angular velocity also leads to the suppression of π-rotation, but at larger values than in problem # 1. This is due to the fact that one rotating molecule suppresses π-rotation to a lesser extent, since a large initial angular velocity is required to mismatch the precessional motions.

In problem # 3 (dashed curve), the vectors of the initial angular velocity are directed in opposite directions. Note that the forces of intermolecular interaction (attraction or repulsion) are also directed in opposite directions. Based on this, an increase in the initial angular speed of 2π-rotation is not able to prevent π-rotation.

These results also confirm the conclusions made in Section 3.1 that the cluster asymmetry, chaotic motions in the corresponding directions, and coordinated precessional motion of molecules cause the π-rotation.

At present, a large number of ring nanostructures have been experimentally discovered [47,48,49]. Theoretically, their ring shape suggests similar dynamic properties in supramolecular systems. Note that the approach used in this article is applicable to any other systems of nanotori, nanorings, and so on. However, the mathematical model must necessarily be corrected to study, for example, a system of nanotori with a large ratio of the diameter of the ring to the diameter of the tube. Since in this case it is necessary to take into account the deformation of the nanoobject.

Our calculations show that taking into account the classical forces of intermolecular interaction is insufficient to ensure the stability of large columns of tori. However, the van der Waals forces provide some semblance of system stability. In reality, intermolecular interactions in supramolecules involve a combination of different types of interactions such as electric forces, magnetic forces, and so on. In this regard, the stability of the supramolecule and its dynamic properties due to the van der Waals forces will ultimately depend on the set of additional conditions and the molecular environment around a supramolecular system.

4. Conclusions

In this paper, we investigated the group behavior of molecular tori in the columnar phase. The calculation results showed that in the process of intermolecular interaction, the torus system performs coordinated rotational motions of two types. These types of rotational motion are due to the toroidal shape of the molecule, which allows transforming the chaotic motion of nanotori into directional motion. These movements occur in systems with any number of nanotori.

The first type of directed rotational motion is performed by the nanotori around an axis perpendicular to their plane. Some of the molecules in a cluster can rotate in a positive direction, while others rotate in a negative direction.

The second type of directed rotational motion is performed by all nanotori around a common center of mass. Collective rotation begins sometime after reaching the required level of chaotic movement. It is a consequence of the coordinated precessional motion caused by the action of the coordinated moments of forces. The beginning of collective rotation for clusters with a large number of molecules begins later. The rotation lasts longer due to the lower angular speed. In systems with more than five nanotori, the behavior of nanotori becomes more complex. If the number of nanotori is odd (i.e., the center of mass of the cluster coincides with the center of mass of the central nanotorus), then from the first moments of time, the nanotori tend to rearrange themselves into a new configuration of several columns. If the number of nanotori is even, then no rearrangement of the system is observed, but a large delay appears at the beginning of the collective rotation.

Both types of rotational motion have a strong mutual influence. It is revealed that the collective rotational effect is facilitated by low angular velocities, different-speed, and multidirectional rotational motion of nanotori. It was found that an increase in the cluster size leads to a decrease in the angular speed of collective rotation.