A Phenomenological Model of the Rotation Dynamics of Carbon Nanotube Gears with Laser Electric Fields

Deepak Srivastava
Computational Nanotechnology at NAS Systems Division
NASA Ames Research Center
Mail Stop T27A-1, Moffett Field, CA 94035-1000, USA

Abstract:

A phenomenological model for the rotational dynamics of a single laser powered molecular motor is discussed and tested through MD simulations. The motor is used to power carbon nanotube based gears. For a given laser power density and arrangement of free charges in the body of the gear we have defined an intrinsic frequency of the gear oscillatory rotations. The nanotube rotations are not of oscillatory nature if the laser field frequency is of the same order of magnitude as the intrinsic frequency of the tube and there is an additional phase match between the two. For the laser powered gear motor dynamics, the rotational angular momentum of the driven gear tend to stablize the rotational dynamics of the system, and unidirectional rotations for the entire duration of the simulations are observed.

Introduction

An important issue in the growing field of computational nanotechnology is the means for powering molecular machines in a controlled way. The simplest form of these machines would be nanoscale molecular bearings, shaft and gear, and multiple gear systems. Such a machine would operate by powering the input component from an external source with the resulting rotational or translational motion utilized to do mechanical work by the output component. In the initial stages, few examples of molecular gears such as diamondoid molecular planetary gears by Drexler and Merkle (1992) [1], fullerene gears by Robertson (1994) [2] and carbon nanotube gears by Han et. al. (1997), [3] have been comceptualized and simulated. In the carbon nanotube gears, as shown in Figure (1b),

  

benzyne molecules bonded to carbon nanotube form teeth while the nanotube forms the body about which the gear rotates. Through structural and dynamical calculations using the ab-initio electronic structure and analytic Brenner's [4] reactive hydrocarbon potential methods, respectively, Jaffe et. al. [5] have shown that carbon nanotube gears are simple in structure and could be synthetically accessible as well. Moreover, using molecular dynamics simulations with Brenner's potential Han et. al. [3] have also shown that the stable rotations of the driven gear are possible with the forced rotations of the powered gear. The benzyne molecular teeth do not break and suitable operating conditions for uniform rotations of the powered and driven gears can be identified. The question of how to force the rotations of the powered gear with an external power source was not addressed.

The first simulations of the laser driven molecular motor, such as graphitic molecular bearings, have been reported by Tuzun et. al. (1995). [6] The molecular bearings were formed by rolling two graphite sheets into two concentric cylindrical tubes. This structure is not a multi-walled carbon nanotube because the spacing between the tubes is much larger than the experimentally observed value for the carbon nanotubes. The external tube was held fixed while the internal tube was allowed to move under the forces of interaction between the applied laser electric field and two unit charges, one positive and the other negative, fixed on the circumference of the inner tube. A wide range of operating parameters, such as laser frequency and power, molecular bearing sizes and the positions of the charges within the molecular bearing, were investigated for stable unidirectional rotations of the inner bearing tube. For a single applied laser field under all investigated operating conditions the bearing oscillated with positive and negative angular velocities, i.e., for some time it would rotate in one direction and then would rotate in the opposite direction. Under two applied laser fields the duration of unidirectional rotation was increased, and an attempt to find the optimal set of operating parameters with a neural net simulator was made.

Another approach to investigating the laser powered rotational dynamics of nanomachines is to develop simplified heuristic models which can be used to minimize the parameter space that need to be searched for optimal operating conditions for the machines. Development of simplified models for the dynamics of complex nanotechnological systems is necessary because for many cases fully atomistic simulations would be computationally expensive and prohibitive. In this work we investigate the rotational dynamics of carbon nanotubes and carbon nanotube gears of Han et. al [3] under a single applied laser field. A phenomenological relationship between the laser operating parameters and an intrinsic nanotube or nanotube-gear frequency is discussed. We show that for a resonance between the laser operating frequency and the intrinsic gear frequency it is possible to achieve unidirectional rotations with a single laser field. In section 2 we discus the phenomenological model and in section 3 we also present fully atomistic simulation results which support the phenomenological model description of section 2.

Phenomenological approach to rotations of carbon nanotube gears with externally applied laser field

  The rotation of a carbon nanotube or a nanotube gear due to interaction with a laser electric field is accomplished through the forces of interactions between free charges in the body of the tube and the applied laser electric field. A completely neutral nanotube or a gear would be transparent to the laser field. Free charges in the body of the tube or a gear could be introduced by any suitable functional group substitution. For these simulations, however, we assume that two of the atoms along the circumference of the tube of the powered gear carry unit positive or negative charges. As shown in figures 1a and 1b, the field-nanotube or field-nanotube gear interaction is then due to the familiar field-dipole (unit positive and negative charges separated by a distance) interaction. For simplicity we have assumed that the charges are placed diametrically opposite to each other on the circumference of the tube. The classical equations of motion in three dimensions are

$${{{\bf I}{{d{{\bf\omega}}}\over{dt}}+{\bf\omega}{{d{{\bf I}}}\over{dt}}={{\bf\sum_{i}}{\bf r}_i{\times}{\bf F}_i}}}$$ (1)

Where ${\bf I}$ is the moment of inertia tensor, ${\bf \omega}$ is angular velocity, ${\bf r}$ and ${\bf F}$ are the radial arm and the force of the applied torque. If the electric field vector of the linearly polarized laser field is in a plane perpendicular to the long axis of the tube, or the gear, the rotational motion of the tube is essentially one dimensional, i.e., rotation about the long axis of the tube or the gear. The one dimensional phenomenological equation for the rotational or oscillatory motion around the long axis of the tube can be written as

$${{I{{d{{\bf\omega}(t)}}\over{dt}}={A(\omega,t)Cos(\Omega t + \phi)-D(\omega,t).}}}$$ (2)

where the moment of inertia about the long axis of rotation, I, may be an implicit function of time because at the molecular level the structure of the nanotube or the gear is quite flexible and may distort during the dynamics. In the first order approximation, the explicit dependence of I on time may be ignored. The first term on the right hand side of Eq.(2) is the laser field force term with time dependent amplitude $A(\omega,t)$ and $Cos(\Omega t + \phi)$ for laser electric field oscillations with frequency $\Omega$ and phase $\phi$. The second term is the time dependent dissipation or frictional drag on the rotational motion. There could be various contributations to the dissipation, such as, friction between different layers of a multi-walled gear, friction between teeth of multiple gears, and friction between the gear and any cooling medium. For simplicity of the model we include all such different sources of the frictional drag into a single dissipation term. The Eq.(2) for the rotational motion of the gear is a nonlinear equation with some characteristic features. In the absence of any frictional drag or dissipation the solution to the Eq.(2) is oscillatory because the force term on the right hand side is oscillatory. In reality, this means that the angular velocity can be both positive and negative, i. e., the tube can rotate in one direction as well as in the other.

For a simple distribution of the charges along the circumference of a nanotube or a powered gear shown in Fig. 1, it is possible to write the laser field force term in detail. The dissipation on the other hand, depends upon the details of the frictional forces through bonding and Van der walls interactions. To simplify the analysis we assume the dissipation to be zero and expand the forcing term as

$${{{d{\bf\omega}}\over{dt}}={-{2qE_0(t)}\over{I}}r(t)Sin{[{\bf\alpha}(t)]Cos(\Omega t + \phi)}}$$ (3)

where q is the positive charge on an atom on the circumference. Another atom on the diametrically opposite end has a q negative charge. E₀(t) is the time-dependent (due to pulse shaping) amplitude of the laser field, ${\bf\alpha}(t)$ is the angular displacement from the initial position, and $\Omega$ and $\phi$ are the frequency and the initial phase of the oscillating laser electric field. The moment of inertia I may be an implicit function of time through the distortions in radius r(t) of the nanotube, because the shape of molecular nanotube distorts considerably during the dynamics. In the above derivation we have ignored the implicit time-dependence in the moment of inertia. Even with no dissipation forces present, Eq.(3) is a nonlinear equation of Mathieu type [7] with a time-dependent oscillating forcing field and time-dependent coefficients. Eq.(3) can be linearized under the following approximations and a phenomenological intrinsic frequency of the oscillatory dynamics of the tube or the gear can be defined. First, if the time-scale of laser pulse shaping in E₀(t) is much larger than the time scale of the rotational or oscillatory motions of the tube, the laser field strength E₀ can assumed to be a constant. Second, the time dependence in the radius of the tube can be expanded as r(t) = r₀ + r₁ t + ..., and in the rigid body nanotube or gear (with frozen C-C bonds) approximation r(t) = r₀. Lastly at t = 0 and shortly thereafter the initial angular displacement is assumed to be small such that $Sin({\bf \alpha}) = \bf{\alpha}$. Thus, the linearized form of the non-linear Eq.(3) is

$${{{d^2{\bf\alpha}(t)}\over{dt^2}}={-{{2q r_0 E_0}\over{I_0}} {\bf\alpha}(t)}}$$ (4)

Where we have used ${\bf\omega}(t)={{d{\bf\alpha}(t)}\over{dt}}$. Eq.(4) is the equation of a simple rotational pendulum with frequency

$${f_{int}} = {1\over{2\pi}}{\sqrt{{2q r_0 E_0}\over{I_0}}}$$ (5)

where f_int is the intrinsic frequency of the rotational pendulum of Fig.1a under a non-oscillating forcing field of constant magnitude E₀. The constant intrinsic frequency in Eq.(4) does not depend upon the frequency $\Omega$ of the external laser field because we have also assumed that for a non-oscillating external field both $\Omega$ and $\phi$ are zero. The effect of the oscillating nature of the external laser field will be to impose non-linearlity on the system where the time-dependent phase and amplitude change with the oscillations of the field. Additional non-linearity could arise also because of the explicit time-dependencies due to pulse shaping as well as shape distortions in I and r(t). The term "intrinsic" frequency of the tube or the powered gear is not solely the property of the tube or the gear. As defined in Eq.(5), the intrinsic frequency is due to carbon nanotube or gear interaction with a constant electric field E₀ perpendicular to the long axis of the tube. For a more complex arrangement of the charges in the body of the tube expressions similar to Eqs. (4) and (5), in principle, can be derived. Additionally, according to Eq.(5) the intrinsic frequency of the tube is inversely proportional to square root of the mass and radius of the tube and directly proportional to the square root of the laser field strength E₀. The linearized definitions in Eqs.(4) and (5), thus can help in analyzing the limiting behavior of the non-linear dynamics under three possible scenarios discussed below.

First, when $\Omega \gt\gt {f_{int}}$, the changes in the external laser field are much more rapid than the rotational rate for a given field strength. The carbon nanotube or the gear in this case is not able to respond quickly to the field. The nanotube or the gear is then transparent to the laser electric field. Second, when $\Omega << {f_{int}}$, the changes in the laser field strength are very slow over the period of the intrinsic oscillations of the tube such that the laser field strength can be treated to be as constant. The angular velocity in this case should show steady oscillations with frequency close to f_int and superposed over slowly varying laser field oscillations. In the positive half of the laser field oscillations if the rotational pendulum, of equation (4), oscillates in one phase then in the negative half of the laser field oscillations it will oscillate in the opposite phase. For a single applied laser field the oscillations of the nanotube angular velocity between positive and negative values are, therefore, a natural consequence of operating under the second scenario. The third, most interesting, case is when $\Omega$ is nearly same as or is a harmonic of f_int. In this case the phase of the oscillating laser field changes approximately at the same rate as the rate with which the nanotube or the gear rotates. If there is proper phase matching between the two, it is possible to have unidirectional rotations of the tube or the gear. The angular velocity of the unidirectional rotations, however, will not be constant and will change phase with respect to the phase of the laser electric field. Unless there are damping or inertial forces present to oppose any changes in the rotational state of the system the dynamics of a single nanotube will show periods of unidirectional rotations when there is a phase match and periods of oscillations where there is no phase match. If the rotations of a powered gear are coupled to that of a driven gear there is a possibility that the driven gear gains enough rotational momentum to resist any further changes in the rotational state of the system. Under a quasi-resonance between the laser field and the gear intrinsic frequencies it is possible to have persistent unidiretional rotations albeit with oscillating angular velocity. The oscillations will not be between positive and negative angular velocities but the gear will acclerate and slow down as it continues to rotate consistently in one direction.

Simulation Technique

  The laser powered rotational dynamics of a carbon nanotube and a carbon nanotube gear has been simulated with molecular dynamics (MD). The general MD technique is well known and described in detail elsewhere [8]. Of the many known interaction potentials [9] for hydrocarbon interactions we have used Brenner's reactive hydrocarbon potential [4] for bonded interactions as it has been used in many simulation studies involving hydrocarbon atoms, molecules, bulk, and surfaces [10] and diamond, fullerenes and carbon nanotube systems [11]. Recently the Brenner potential has been used in simulations of reactive dynamics of benzyne teeth formation in carbon nanotube gears [5] as well as forced rotational dynamics of different carbon nanotube gears [3]. The long range non-bonded interactions between different gear atoms are described by a Lennard-Jones (6-12) potential with parameters derived from fitting to graphite and C₆₀ experimental data [12]. Comparison with other non-bonded interactions did not yield any significant differences in the dynamics. [3] The laser field-nanotube or laser field-gear interaction was due to locating unit positive and negative charges on atoms along the circumference of the tube or gear. The unit positive and negative charges also interact with each other via Coulomb interactions. The strength and the time-dependence of the laser electric field used in the simulations are described in the next section.

MD simulations were performed on energy optimized nanotube or gear structures. For these simulations we use the (14,0) nanotube and the gear formed by symmetrically placing benzyne teeth on a (14,0) nanotube. These systems are same as those used by Han et. al. [3] for artificially forced rotations of the gears. The diameter of the (14,0) tube, which defines the separation of the unit positive and negative charges on the circurference of the tube, 1.096 nm, and we have used 2.123 nm long tube and gears in the simulations. The equations of motions were integrated using a 4th order predictor-corrector algorithm with a fixed time step of 0.5 femtosecond. The rotational velocity components were subtracted from all atomic velocities to define the thermodynamic temperature. Angular velocities were measured for each atoms and averaged for each rotating component. All MD simulations were done on NAS' (Numerical Aerodynamic Simulation program) IBM SP2 computer by a parallel MD algorithm using message passing interface (MPI). As the gears must be kept fixed relative to each other during the simulation, the end atoms of both the tube and the gears were allowed to move only in fixed planes perpendicular to the long axis of the tube or the gear. Additionally, the center of masses of both the gears were adjusted at each time step to maintain a fixed relative distance (1.895 nm) between the two interacting gears. As explained above, the interaction with the laser electric field causes the dipole in the body of the tube to rotate or oscillate. The rotational momentum transfer to the rest of the tube or gear atoms causes the tube or gears to rotate or oscillate as well.

Results

  The power density P of the laser used is related to the amplitude E₀ of the laser electric field vector by

$${P} = { 1 \over {2}} {\epsilon_0 c {E_0}^2}$$ (6)

where $\epsilon_0$ is the permitivity constant for the vacuum and c is the velocity of light. The choice of the amplitude of the laser electric field vector and hence of the power density of the laser is based on the criteria that if laser is very weak, it is unable to rotate or oscillate the nanotube. For similar system of comparative size Tuzun et. al. [6] have used lasers of power density 0.249 GW cm^-2 to 6.24 TW cm^-2 in their simulations. Continuous wave lasers with such high power are not currently realistic. The power delivered to a target, however, can be boosted by using a pulsed laser beam. We have used cw laser of 0.5 GW cm^-2 power density (E₀ = 61.4 mega volt/meter) to first understand the rotational dynamics of the simple nanotube and nanotube gear systems. The time-dependence in the laser electric field is of simple $Cos(\Omega t + \phi)$ form, where we have chosen a constant phase $\phi = 0$ and constant laser frequency $\Omega$ described below. The effect of pulsing the laser field under similar condition was also studied in detail[13]. The rotational dynamics of a carbon nanotube powered by an external laser field is discussed, first, under two limiting cases: a) when the laser electric field frequency $\Omega$ is much less than the intrinsic frequency f_int of the nanotube, and b) when the laser field frequency $\Omega$ is of the same order as the intrinsic frequency of the nanotube. The rotational dynamics of the gear under the second operating condition is discussed later.

Figure 2,

  

In figure 2a, when the laser field frequency (5 GHz) is much less than the intrinsic frequency of the nanotube (75 GHz), we observe rotational pendulum like oscillatory behavior. The angular velocity of the nanotube oscillates between positive and negative values. For positive angular velocities the nanotube rotates in counter clockwise direction and for negative angular velocity the nanotube rotates in clockwise direction. The rotational displacements from the starting position are also shown. For a uniformly rotating tube the rotational displacements should vary uniformly and continously between 0 and 1. For clarity, we have shifted and plotted the rotational displacement between 0.5 and 1.5. The rotational displacements also show oscillatory behavior, and the oscillations change phase in accordance with the phase of the external laser field. Beacuse of the non-linear nature of field-nanotube interaction dynamics the magnitude of the oscillations as well their speed vary considerably. For example, between 350 and 450 ps the oscillations decay in magnitude and slow down considerably. However, at about 450 ps as the phase of the laser field oscillations changes the oscillations grow back in magnitude. This behavior of the nanotube oscillations is similar to that observed in laser powered molecular bearing simulations of Tuzun et. al. [6]. For a laser of given power we also note that the frequency of the nanotube oscillations does not changes much with the changes in the laser electric field. This means that throughout 500 ps of simulation the frequency of the nanotube oscillations roughly remains same. This is in agreement with the linearized definition of an intrinsic frequency in Eq.(5).

In Fig.2b, we show the dynamics of the nanotube for a case when the frequency of the laser field at 140 GHz is close to the first harmonic of the intrinsic frequency of the tube. There are two distinct regions of nanotube rotational behavior. For up to about 200 ps the nanotube oscillations are not in phase with the oscillations of the laser field. The angular velocity of the tube oscillates between positive and negative values with non-uniform periodicity. Between 200 and 450 ps the nanotube and the laser field oscillations are in phase with each other. The angular velocity of the tube then oscillates with consistently negative values. This means that between 200 and 450 ps the tube rotates in one direction. During each cycle, oscillating angular velocity goes through minimun and maximum negative values which means that the unidirectionally rotating tube acclerates and deacclerates but continues to rotate in one direction. At about 450 ps, the nanotube rotation and the laser field oscillation get out of phase with each other. The tube then starts to rotate with oscillating positive and negative angular velocities. We repeated the trajectory with different starting configuration and varied the laser field frequency between 50 to 150 GHz. We found that due to the changing phase of the nanotube rotations and thermal fluctuations, the laser field oscillations and the nanotube rotations randomly get in and out of phase with each other. When the two are in proper phase with each other unidirectional rotation is observed and when the phases do not match the nanotube rotates with alternating positive and negative angular velocities.

The nanotube gear(Fig.1b) rotations with laser field frequency varying between 100 to 150 GHz were also simulated. The simulation results for the laser field frequency at 140 GHz are shown in Figure 3,

  

The effective moment of inertia of the single (powered) gear is not much different from that of the single nanotube. Moreover, the diameter along which unit positive and negative charges are located is the same. We expected the intrinsic frequency of the single nanotube gear to be close to that of the single nanotube. The gear configuration (Fig.1b) has the powered gear rotated by the laser electric field and the driven gear due to rotational energy transfer from the powered gear. The damping term in Eq.(2) thus has frictional contribution due to meshing of the teeth of the powered and driven gears during the rotation. For up to about 50 ps, the phase of the powered gear rotation does not match well with the phase of the oscillating electric field, the gear rotates with positive and negative angular velocities as the rotational angular momentum builds up. After 50 ps both the powered and the driven gears show steady unidirectional rotations for the remainder of the simulation trajectory. Other trajectories with different starting configurations showed similar rotational behavior, i.e., in all cases once the rotational angular momentum builds up sufficiently both the powered and the driven gears showed steady unidirectional rotation for the duration of the simulations. The continuous wave laser field was replaced with the pulsed laser field by switching on and off the laser field at random locations. When the laser field was off the rotational angular momentum decayed, i.e.,gear rotations slowed down. When the field was turned back on during the decay the unidirectional rotations resumed. The details of the pulsed laser simulations and the role of damping forces in stablizing the dynamics is currently under investigation and will be published elsewhere [13]. Under quasi-resonance between the laser field frequency and nanotube gear intrinsic frequency the steady unidirectional rotations thus are possible. The angular velocity of the powered and driven gears shows oscillations in the magnitude but does not change direction.

Discussion

  A phenomenological model for the rotational dynamics of a single laser powered molecular motor, in the form of carbon nanotube and carbon nanotube gears, is discussed and tested through MD simulations. For given laser field strength and location of the charges in the body of the nanotube we have defined an intrinsic frequency of the nanotube or gear oscillations. If the laser field frequency is much less than the intrinsic frequency of the nanotube or the gear, the motor shows rotational oscillations with positive and negative angular velocities. When the laser field frequency is of the same order of magnitude as the intrinsic frequency a quasi-resonance between the nanotube oscillations and the laser field oscillations sets up. If the phases of the two oscillations are also properly matched the unidirectional rotation of the nanotube, though with acclerating and deacclerating angular velocities, are possible. Additionally, simulations also showed that due to thermal fluctuation the two rotations or oscillations can randomly get in and out of phase with each other.

Simulations of the nanotube gear rotations under similar laser operating condition showed much stable unidirectional rotation for the entire duration of the simulations. The angular velocity fluctuation (oscillations) of both the powered and driven gear were much less than the magnitude of the angular velocity fluctuations (oscillations) observed in the single nanotube case. This is primarily because the rotational angular momentum of the driven gear, once built up to a significant value, not only forces the gear system to continue to rotate in one direction but also reduces the fluctuations in the magnitude of the angular velocities of the driven and powered gear system.

There are two significant implications of the postulated phenomenological model and the simulation observations. First, for more complicated motor system with complex arrangement of free charge distribution it is not necessary to find an expression or the value of the intrinsic frequency as defined in Eq.(5). One could instead expose the system to an approximate low frequency laser field. The fourier power spectrum analysis of the rotational oscillations would give the magnitude of the intrinsic frequency of the system. A tunable laser around the intrinsic frequency can then be used to successfully excite and control unidirectional rotations of the tube or the gear motor system as discussed above. Second, the inertial or damping effect of the driven gear rotations on the oscillating angular velocity of the powered gear rotations can be used to reduce the fluctuations in the magnitude of the rotational angular velocity of the entire system. For example, in the gear rotation simulation of this work, the fluctuations in the angular velocity of the system would be further reduced in a three coupled gear system if the output work is drawn from a second driven gear instead of the first. The power required to rotate a complex arrangement of three or more coupled gears or gear axel system, however, would be considerably more than the laser power density used in this work.

Finally, for a heuristic model the definition of the fundamental frequency gives an idea about how to tune the single laser frequency for unidirectional rotations. The drag and dissipation terms are not explicitly defined. A further investigation into those may yield predictions about the detailed features of the dynamics such as the beat patterns in the angular velocities and how to control those. Further work along these lines and investigations to control the direction of motion - clockwise vs. conterclockwise rotations - is currently in progress.

Acknowledgement

  The author thanks Jie Han and Al Globus for many useful discussions and providing the coordinates for the nanotube gear system used in testing the model. D. Srivastava is an employee of MRJ Inc. Financial support to the Computational Nanotechnology program by NAS systems division at NASA ARC is gratefully acknowledged.

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