G. I. Leach, R. E. Tuzun, D. W. Noid, B. G. Sumpter
Molecular dynamics simulations indicate positional stability to be an important issue in a wide variety of molecular nanotechnology applications. It can determine the difference between the success and failure of mechanical nanodevice designs. Diamondoid materials are proposed for many such designs in part because of the stiffness and strength of diamond compared to other materials. These properties should allow the problems of positional stability to be minimized while simultaneously minimizing atom count (bulk, molecular weight, etc.). Because of their synthetic availability and desirable mechanical properties, graphitic materials, in particular carbon nanotubes and buckyballs, are also components in many proposed nanomachine designs.
We present results of a molecular dynamics study of the positional stability of several diamondoid and carbon nanotube structures, including comparing similar functional designs made from both these materials. In particular, we investigate hollow tubular and solid rectangular ``struts'' -- which, for example, could form part of the linear actuators required in a Stewart platform molecular positioning device. The effects of constraints on positional stability and nanomachine performance are examined. Quantum mechanical results for some of these structures are briefly compared with molecular dynamics results.
Finally, we discuss the use of the web-based virtual reality modeling language -- VRML 2 -- to publish the results of molecular dynamics simulations on the world wide web as three-dimensional movies.
For the purpose of feasibility studies of proposed nanomachine designs, which have not yet been built, it is necessary to perform some sort of computational analysis [1,2]. One common, if not the most common, technique is molecular dynamics (MD) because it is quick and versatile and because it can be used to study processes at the atomic scale which cannot be directly observed experimentally. Within the last several years, several groups have performed MD simulations on bearings , motors , gears [5,6], and other systems in order to study their mechanical properties and to explore nanomachine design considerations.
One of the most important issues in nanomachine design is positional stability. Nanomachine components must be fairly stiff in order for bearings to rotate smoothly, for gears to mesh well, for sub-angstrom resolution on positional control devices, and so on. This issue can be investigated with MD. Results so far indicate that the geometry of a nanostructure plays as crucial role in its positional stability as its potential energy surface. For example, using MD Sumpter and Noid  recently found the dynamical stability of carbon nanotubes to depend in large part on the aspect ratio.
More recently, the correspondence of classical and quantum mechanics has begun to emerge as an area of interest to nanotechnology. Of particular interest to us is the well-known zero point energy problem. Classically, the flow of energy is unrestricted. Quantum mechanically, however, some energy must always remain locked within each vibrational mode. This could mean that classical simulations may sometimes contain extraneous vibrational motion from the flow of what should be zero point energy. Results of a new method for calculating vibrational ground states of systems with many atoms [8,9] indicate that this is indeed a serious issue, especially for systems with sparsely connected bond networks and with no constraints or external forces. This is actually good news for nanotechnology because it implies that many nanomachine designs may perform better than MD results indicate and/or require fewer atoms for the same positional stability.
Adding constraints or collapsing degrees of freedom can help reduce extraneous vibrational motion. This can be done in several ways using methods which are already used for computational savings or other reasons. Internal cordinate methods  have been used by the Goddard group  for several biological and nanotechnology applications. In recent nano-fluid dynamics simulations [12,13] the ends of a nanotube were frozen in order to not allow an axial fluid flow to drag the tube along. In the gear simulations performed at NASA Ames  the ends of the carbon nanotube shaft were constrained to not elongate but were still allowed to move within a plane transverse to the tube symmetry axis. Finally, in two recent papers rigid-body (quaternion) dynamics methods were compared to fully atomistic MD for the operation or assembly of molecular bearings [14,15].
In this paper we investigate the positional stability of diamondoid blocks and of carbon nanotubes by MD for several geometries, constraints, and initial conditions and using several potential energy surfaces. We include both small aspect ratios, which are so far the most commonly modeled, and large aspect ratiso, which have not been modeled as extensively and which would appear in some nano-hydraulic components or support struts for Stewart platforms or other positioning devices. As an illustration of the considerations for positional stability, we begin to address the question of how thick a support strut should be for a Stewart platform, given a target compliance on the order of a bond length (atomic precision).
The details of the molecular dynamics methds used in this study have been well-described elsewhere and are not repeated here. Essentially, MD consists of integrating the classical equations of motion over small time steps (in our case, 1 fs). Initial momenta are set to correspond to desired temperatures, rotational velocities, or other desired initial conditions. Overall translational and rotational motion are removed before the beginning of simulation so that only random thermal motion remains. External forces maybe applied at any time. The simulations presented here were performed at constant energy, using symplectic integration  and recent improvements to the bonded interaction portion of the calculation [18,19,20].
For the sake of comparison, several potential energy surfaces were used in this study. For terminated diamond blocks, we used a modified MM2 potential  with and without non-bonded interactions. The unmodified potential, which is most suitable for molecular mechanics calculations, has a cubic bond stretch term, which is physically unreasonable at moderate to large bond distances. For our MD simulations we used a Morse functional form, which is well-behaved at all bond distances, with the same equilibrium position and curvature as the cubic potential. For unterminated diamond blocks, we used a special potential with stretch and bend interactions. For carbon nanotubes, we used the graphite potential energy surface of Guo et. al.  with and without torsion interactions.
Studying positional stability using MD requires starting at the equilibrium geometry or at least near a local minimum in the potential energy surface. Two methods may be used to obtain initial geometry: molecular mechanics , or, as in this work, annealing (also called dynamical steepest descent). If a structure is away from equilibrium and at rest, it will move toward equilibrium. Annealing consists of performing an MD simulation in which kinetic energy is gradually removed until the structure is nearly motionless and almost completely relaxed. We demonstrate later the dramatic effects of incomplete annealing on subsequent MD simulation.
Unterminated diamondoid rectangular blocks with approximately square cross section were generated for three aspect (length to width) ratios: 1, 10, and 100 (hereafter referred to as short, medium length, and long) using the CrystalSketchpad  package. In all of these blocks, the surfaces were either (110) or (100) planes. To test the effects of bond network connectivity two thicknesses were chosen: one unit cell (about 0.4 nm) and 1 nm.
|(a) AR=1, CS=uc.||(b) AR=10, CS=uc.|
|(c) AR=100, CS=uc.|
|(d) AR=1, CS=nm.||(e) AR=10, CS=nm.|
|(f) AR=100, CS=nm.|
These blocks were then used to generate terminated blocks (no dangling bonds). The (110) surface was terminated with hydrogens. Rather than terminating with hydrogens on the (100) surface using the postulated surface rearrangement , the dangling bonds were bridged with oxygen atoms. Similar strategies have been used in several proposed nanomachine designs. When inserting the bridging oxygen atoms, due to the geometrical constraints from the diamond surface it was possible to insert the oxygen at a point where either the C-O bond lengths or the O-C-O bond angle was correct, but not both. We chose to begin with correct bond lengths. The final adjustment of the geometry was performed by annealing, as described above.
We use the following abbreviations to describe simulation conditions:
|AR||aspect ratio||1, 10, or 100|
|CS||cross section||uc (unit cell) or nm|
|S||surface termination||C (unterminated) or O,H (oxygen,hydrogen)|
|T||temperature||150K or 300K|
|NB||non-bondeds||on or off|
|NC||constraints||0, 1, or 2 constrained ends|
All of the above blocks were run for 20 ps. In some cases, simulations were run with one or both ends of the block frozen. The two key indicators of position stability of interest, end-to-end distance and maximum transverse displacement, were obtained from the saved trajectory files.
Similar runs were performed for carbon nanotubes with 20 atoms per ring (about 8 Åin diameter). This diameter was chosen because it has been used in many of our previous simulations. Tubes with aspect ratios of 1 (5 rings), 10 (39 rings), and 100 (375 rings) were generated and annealed with and without torsion interactions.
Figure 4 shows profiles of maximum transverse displacement for several diamondoid block simulations. Figure 4(a) shows the effects of temperature and aspect ratio for blocks with an unterminated nm cross section. Initial conditions for the 150K and 300K simulations are exactly the same except for a proportionality factor for the momenta. Gross features of the dynamics, and some fine features, are therefore similar for both temperatures. The maximum transverse displacement is about 6 Åin this set of simulations, for the longest block and highest temperatures. Constraining the ends changes this very little; when both ends are constrained, the maximum transverse displacement drops to about 4 Å.
Similar trends are observed for end-to-end distance (Figure 5). Results are shown for medium and long unterminated diamondoid blocks at temperatures of 150K and 300K, for unit cell and 1 nm cross sections. Here, refers to the change in end-to-end distance from the equilibrium value.
As expected, the unit cell thick diamondoid block simulations show far more flexibility (Figure 6).
Unlike the nm cross section, the unit cell diamond blocks build up appreciable torsional motion. The blocks with aspect ratios of 1 and 10 remain more or less straight. However, the longest block partially coils, and so the maximum transverse displacement continually increases throughout the simulation. Movies of this simulation show that several flexion modes become excited. Unsurprisingly, constraining both ends (not just a single end) reduces the maximum transverse displacement considerably, down to about 5 Å (Figure 4(b)).
For the 1 nm cross section blocks, surface termination appears to have very little effect on positional stability (Figure 4(c)).
Turning non-bonded interactions on or off also appears to have little effect. This is important because the non-bonded calculations are the most expensive portion of the simulation and turning off non-bonded interactions can save considerable time in initial feasibility studies.
As observed previously, the long unit cell cross section block partially coils. This shows up in the end-to-end distance as well as the transverse displacement. In addition, as long as the block doesn't coil, the end-to-end distance is more obviously periodic than the transverse displacement. It is interesting to note that the dominant vibrational frequency is inversely proportional to the length, as observed previously for carbon nanotubes .
Trends in the classical dynamics of carbon nanotubes are similar to those for the diamondoid struts. Figure 7 shows profiles of maximum transverse displacement for nanotubes with diameters of about 8 Å. Only aspect ratios of 10 and 100 are shown for the sake of clarity. In the first 25 ps, the maximum transverse displacement reaches a maximum of about 2.5 Åfor the 300 K simulation. After this, it reaches 4.3 Å(a 5% strain). The aperiodicity of the transverse displacement profile appears to indicate that multiple modes are excited.
The end to end distance profiles are consistent with this interpretation. For the longest tube, the deviation from equilibrium end to end distance has a period of about 8.2 ps. However, the minimum and maximum deviation change between periods, meaning that some energy is transferring between modes.
In these simulations, torsion interactions were included. Simulations in which torsion interactions were excluded showed similar dynamics as the above results.
In Figure 8 we illustrate the effects of incomplete annealing on subsequent MD results. When the terminated, 1 nm thick, medium length block was only partially annealed, a spurious flexion (S) mode developed. This occurred because in the partially annealed structure all of the O-C-O bond angles were too large. At the start of the MD simulation, these bond angles begin to decrease, forcing the tube into an elongation mode and then a flexion mode.
One reliable indicator of incomplete annealing is too high a temperature in the simulation. Initially, for an average target temperature of 150K, the initial momenta are set to give a temperature of 300K. This is done because it is normally expected for energy to be evenly distributed among potential and kinetic energy on the average. After an initial transient, the temperature decreases, but settles about an average of about 180K (Figure 9). When the structure is completely annealed, the average temperature settles around an average of 150K, and no flexion mode develops.
The quantum mechanical ground state wavefunction of a molecule contains probability distributions for bond lengths and angles and for overall features such as end-to-end distances. From these distributions the dynamical stability at the low temperature limit can be estimated.
We would expect the positional stability of diamondoid blocks to be greater than that for more sparsely connected bond networks such as polyethylene chains. The ground state of model polyethylene chains has been explored elsewhere . The quantum mechanical half-width in end-to-end distance is on the order of 0.1 Å, which is far less than the classical results for diamondoid blocks -- most especially for those with a unit cell cross section. Although the end-to-end distance oscillations from the diamondoid simulations are larger by more than an order of magnitude, they would probably have little effect on the overall performance of nanomachines using these blocks -- at least for those with a 1 nm cross section. Transverse displacement, however, may still be problematically large in some MD simulations.
The classical and quantum simulation results for carbon nanotubes are compared in detail in . Essentially, the disagreement between classical and quantum end-to-end distance distributions appears to not seriously affect overall simulation results as long as the tubes are well enough constrained.
So far the MPEG digital movie format has been the most common mechanism for publishing MD trajectories on the web. The use of VRML 2.0 to publish MD trajectories offers the same advantages over MPEG movies that providing 3D object data has over 2D image data in general: it gives back to the viewer control of viewing, lighting, culling and other parameters and allows them to dynamically position and orient the objects for improved comprehension and focus.
Our simulation results have been published as VRML 2 trajectories on the web . A comparison of the use of VRML 2 and MPEG for this purpose may be found in .
The positional stability of diamondoid blocks and carbon nanotubes in a wide variety of sizes and initial conditions has been investigated using MD. Two major criteria for positional stability are end-to-end distance and maximum transverse displacement. Both of these showed reasonable trends for temperature, cross-sectional thickness, and length. Details such as surface termination and functional form of the potential energy surface played a minor role compared to the arrangement of the bond network.
So far, MD has been used for initial feasibility studies for several types of nanomachine components. Later, when the designs are further refined, it will be necessary to study finer features of the dynamics. At this level of detail, the simulation data may reflect more about the limitations of MD than about the dynamics of the problem. In the nanotechnology simulation results reported so far, constraints and external forces have tended to limit the magnitude of structural changes that would harm simulation results for overall trends in nanomachine performance. However, in future simulations new correction strategies may be needed.
For the Stewart platform, MD simulations indicate that a support strut should be at least 1 nm square in cross section. However, previous QMC results appear to indicate that this thickness could serve to overengineer the design requirements, and a smaller thickness may suffice. To refine this estimate it would be necessary to perform additional simulations, under different compressive and transverse loads and other operating conditions.
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