Introduction

Sound is, by nature, a concept that is defined in continuous media. Typical treatments begin with the perturbation of a continuous fluid with some density and pressure [ 1,2]. These lead to a wave equation for pressure variations in which the speed of propagation (speed of sound) can be related to fluid density and other parameters. From these results and thermodynamic arguments other important quantities, such as the energy density of a sound wave, can be determined.

The smallest size scale so far at which sound has been experimentally observed is on the order of tenths of microns. For example, in a recent review Maris [3] discussed a "nanoxylophone", made of gold bars 150 atoms thick, which emitted sound waves with frequencies of about 8 THz for about 2 ns [4]. Within the last decade and a half, picosecond ultrasonic laser sonar (PULSE) [5] has begun to emerge as a technique for the investigation of thin multilayer films, most specifically thicknesses and interfacial properties [6]. Such studies are of crucial importance in the semiconductor industry.

Recent concerns about overly chaotic motion in classical system [7] and the zero point energy problem [8] have led us to begin to explore the limitations of classical MD simulation. In the context ofnano-fluidic systems, this would translate into whether or not it would be possible to study shock and pressure waves at the nanometer size scale using MD simulation. Of course, MD simulation has enabled reasonably accurate calculations of fluid viscosities [9]; however, the phenomenon of fluid Viscosity, at the atomic level, does not require coherent energy transfer.

In this paper we report MD investigations of coherent fluid motion in a system we have previously simulated, namely helium inside a carbon nanotube. It is argued that MD simulation is unsuitable for studying such phenomena.

Simulation Methods

The MD simulations reported here are very similar to helium-carbon nanotube simulations ina previous paper inthis journal [10]; only the differences will be described. Figure 1 shows the.

essentials of the system: a carbon nanotube which is static throughout the simulation and a region in which fluid atoms may move. The boundaries ofthis region are denoted z1 and z2, where z is the direction of the nanotube symmetry axis. The fluid atoms are initially placed in a hexagonal close packed lattice which does not approach within 3 A of the nanotube. This entire system (fluid plus nanotube) is near mechanical equilibrium. In simulations in which the fluid atoms are initially motionless and no external forces are exerted on the system, the fluid temperature never rises above a few degrees K.

Figure 1 Sideways view of a fluid-carbon nanotube system. The solid lines represent the nanotube. The dashedlines represent either minimum image boundaries (shock wave simulations) or initial locations of driving and driven membranes (pressure wave simulations). Fluid atoms are enclosed within.

Two types of simulations, namely shock and pressure wave, are reported. In the shock wave simulations, zt and z3 are fixed at the tenth and tenth to last rings of the nanotube and represent locations of minimum image boundaries described previously. A 10 A plug of fluid is given an initial z velocity. Subsequent fluid motion is tracked very carefully through visualization and through extensive analysis.

In the pressure wave simulations, the boundaries represent idealized membranes located, respectively, at z,(t) and z2(t). These membranes have Lennard-Jones interactions with the fluid atoms with identical constants as the fluid-fluid interactions, but in the z direction only. The location of one membrane is varied sinusoidally z1(t) = Zio + Asin (2nt/rl (1)

where zla, A, and t are constants. The other membrane, in addition, has an interaction term of the form k

and the force constant k and idealized membrane mass are chosen so that the vibrational period is

Results and Discussion

The nanotube was kept static in each simulation reported here so that the motion of the nanotube would not perturb the fluid flow. In other words, conditions were made as favorable as possible for coherent energy transfer. Also, the system was equilibrated for 2 ps before the introduction of any shock waves or external forces. The flow region was about 100 A, long;

Figure 2 Typical shock wave simulation results (fraction of kinetic energy in the direction of flow) for helium within a 19.9 A diameter nanotube Solid line: central region atoms. Dashed line: adjacent region atoms.

nanotube diameters ranged from 16 A (259 helium atoms inside the nanotube) to 32 A, (2157 helium atoms or 1 g/ml).

Shock wave simulations were performed in which the 10 A slice of fluid at the center of the nanotube was given an initial velocity of 10 A/ps (near the speed of sound in helium at 100 K). The number of fluid atoms in this plug ranged from 25 for the smallest radius nanotubes to 211 for the largest. The initial fluid temperature (from random initial momenta) was about 100 K. Two groups of atoms were singled out for analysis: those in the central and adjacent fluid slices at the beginning of the simulation. The fraction of kinetic energy in the z direction was followed in each group of atoms. Typical results are shown in Fig. 2. The fiaction of z kinetic energy steadily decreases in the central region atoms, averaging about the thermal equilibrium value, 1/3, within less than 5 ps. The adjacent region atoms suddenly gain z kinetic energy within about the first 0.1 ps and then a decreasing trend similar to that for the central region atoms. No coherent transfer of z kinetic energy is observed. In short, the motion very rapidly thermalizes, which is consistent with the results of visualization. In several simulations, positions of every fluid atom were saved every 0.05 ps. The movies showed that within 2 ps, the atoms in the initial plug essentially interspersed themselves within the adjacent fluid volume and then randomly dispersed. This behavior was observed at each nanotube radius.

The purpose ofthe pressure wave simulations (the previously described moving membrane simulations) was to determine if molecular dynamics would allow the coherent transfer of mechanical energy across a fluid. In these simulations, if energy transmission is to occur, the membrane vibrational periods must be commensurate with the length of the nanotube and speed of transmission.

Typical results are shown in Fig. 3, for which the bottom membrane vibrates with an amplitude of 0.5 A and a period of 1 ps. An amplitude of 0.5 A was chosen because at larger amplitudes, the fluid temperature built up, usually within 50 ps, to several thousand degrees. The top (driven) membrane, which has a mass of 40 amu and a fundamental vibrational period of 1 ps, shows no coherent motion even after 100 oscillations of the driving membrane. Other simulations, in which the period ofvibration was up to 5 ps and the mass ofthe driven membrane up to 100 amu, yielded virtually identical looking results.

Figure 3 Typical pressure wave simulation results for helium within a 19.9 A diameter nanotube: locations of driving (solid line) and driven (dashed line) membranes. The coordinates of the driving membrane are displaced 90 A for the sake of easy visual comparison with the driven membrane coordinates.

Figure 3 Typical pressure wave simulation results for helium within a 19.9 A diameter nanotube: locations of driving (solid line) and driven (dashed line) membranes. The coordinates of the driving membrane are displaced 90 A for the sake of easy visual comparison with the driven membrane coordinates.

The comparison of classical and quantum mechanical results, beginning 25 years ago with few atom systems, has so far yielded two important drawbacks: the zero point energy problem and chaos. The topic of classical-quantum correspondence has been recently studied in single polyethylene chains in vacuum Calculations from the recently developed internal coordinate quantum Monte Carlo (ICQMC) method [8, 11] indicate that the ground vibrational state of polyethylene is extremely stiff; the probability distribution of end to end distance in a 100 monomer unit chain lies within a range of several tenths of an angstrom and is centered on the equilibrium planar zigzag value. In classical calculations at the quantum ground state energy, the polymer chain coiled; in other words, the flow of energy which quantum mechanically should have been locked in place was unrestricted m classical simulations. Even at energies at a fraction ofthe ground state energy, classical simulations showed large excursions in end to end distance and other indicators of overall positional stability. Classical simulations at very low energies (in which the temperatures reached only 2 K) showed high dimensional chaos [7].

The polyethylene system, a loosely connected bond network, was chosen as a worst possible case in order to illustrate the limitations of classical MD simulation. Systems with external constraints (such as nearby chains in a crystal) or cyclical bond networks would be expected to exhibit significantly restricted motion. In fact, classical simulations of carbon nanotubes, which have a two-dimensional bond network, showed a sizeable but significantly smaller disagreement with quantum results.

Of course, many of the essential features of a liquid are preserved in classical simulations. What determines these liquid properties more than the total energy is energy differences with respect to thermal motion, external forces, and so on. In the context of classical-quantum correspondence, it is important to note that any liquid is an unbound system and so the zero point energy problem has far less significance in considerations of liquid structure than in the problems previously discussed. However, for cases of coherent motion such as superfluidity, chaos is still an important consideration, and MD simulation will likely fail to describe these phenomena.

Conclusions

In two types of helium-nanotube setups, no coherent transmission of mechanical energy occurred in classical MD simulations. The fluid motion thermalized too rapidly to allow this to happen. This occurred despite the fact that conditions were made as favorable as possible for energy transmission (low temperatures, static nanotubes, near liquid densities).

We emphasize that MD simulation has worked well to predict some fluid properties such as viscosity and diffusion coefficient. However, it appears to be unsuitable for the study of phenomena requiring coherent fluid motion. Because of the inherent mechanical constraints, classical MD treatments may suffice for similar studies in solid nanostructures, depending on the degree of external mechanical constraints.

To what extent shock and pressure wave propagation occurs in nano-fluidic systems cannot be determined by classical simulation, nor has it been studied experimentally. If, in the future, such behavior were observed, a quantum mechanical treatment or some other method would be required for explanation.

Acknowldgments

Research sponsored by the Division of Materials Sciences, Office of Basic Energy Sciences, U.S. Department of Energy under contract DE-AC05-960R22464 with LockheedMartin Energy Research Corp. Also supported in part by an appointment to the ORAU Postdoctoral Research Associates program administered jointly by the Oak Ridge Intitute for Science and Engineering and Oak Ridge National Laboratory.

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