Molecular dynamics simulations of acoustic absorption by a carbon nanotube Physics of Fluids, 2018; 30(6):066101-1-066101-15

Acoustic absorption by a carbon nanotube (CNT) was studied using molecular dynamics (MD) simulations in a molecular domain containing a monatomic gas driven by a time-varying periodic force to simulate acoustic wave propagation. Attenuation of the sound wave and the characteristics of the sound ﬁeld due to interactions with the CNT were studied by evaluating the behavior of various acoustic parameters and comparing the behavior with that of the domain without the CNT present. A standing-wave model was developed for the CNT-containing system to predict sound attenuation by the CNT and the results were veriﬁed against estimates of attenuation using the thermodynamic concept of exergy. This study demonstrates acoustic absorption eﬀects of a CNT in a thermostatted MD simulation, quantiﬁes the acoustic losses induced by the CNT and illustrates their eﬀects on the CNT. Overall, a platform was developed for MD simulations that can model acoustic damping induced by nanostructured materials such as CNTs, which can be used to further understanding of nanoscale acoustic loss mechanisms associated with molecular interactions between acoustic waves and nanomaterials. simulation containing gas These results illustrate the considerable utility of exergy for analyzing nanoscale acoustic absorption. In the preceding estimates of acoustic parameters, only a subtle diﬀerence between the acoustic power curves observed between the simulations with and without the CNT, which were calculated based on the changes in the acoustic pressure and particle velocity. However, unlike the velocity components or acoustic power, substantial diﬀerences in the exergy and total power curves Figure 10 in the two simulations, particularly in the region containing the CNT, whereas no substantial changes occur in the gas region z < 90 nm. From the exergy, the attenuation coeﬃcient ( m cnte ) for the whole system with the CNT present was calculated using Equation (21). The value of m cnte = 2 . 17 × 10 7 m − 1 calculated form the exergy is approximately equal to the value of ( m c + m g ) = 2 . 26 × 10 7 m − 1 obtained from curve ﬁtting using the two-region approach. Note that the value of m c represents only the CNT region, hence the value of ( m c + m g ) represents the attenuation for the whole system both the gas and CNT regions Figure The discrepancy can be attributed to the ﬁtting methods of one-region and two-region approaches of standing wave equation used obtaining attenuation coeﬃcient m g for gas atoms only. In the estimates, g . 7 m − two-region approach


I. INTRODUCTION
Interest in carbon nanotubes (CNTs) for various applications has grown rapidly because of their extraordinary properties and versatility in forming composite nanostructures. In particular, structures can be fabricated with modified mechanical and thermal properties that show promise as sound-absorption materials for noise control. [1][2][3] In addition, the emergence of advanced manufacturing technologies offers exciting possibilities for creating tailored acoustic absorbers using CNTs. [2][3][4] The potential of CNTs and composite absorbers for use in noise-control applications has been investigated in various studies. [5][6][7][8][9] Moreover, the acoustic absorption properties of CNTs have been measured in several studies. [10][11][12] These investigations have yielded promising results for the absorption characteristics of CNTs and highlighted the need for improved understanding of the absorption mechanisms of nanoscopic fibers at the the nanoscale. [13][14][15] The mechanisms of sound absorption for conventional porous acoustic materials with fiber diameters or pores on the micro-scale (down to 1 µm) are currently well understood. The relative influence of the various mechanisms are, however, expected to change for materials with pores or fibers at the smaller nanoscale (down to 1 nm), while other mechanisms and nonlinear effects may also become significant. Theoretical and computational approaches such as molecular simulations can play an important role in shedding light on the detailed mechanisms by which nanomaterials absorb sound. However, conventional molecular dynamics (MD) simulations assume some conserved quantity and thus it is not straightforward to measure the dissipation of acoustic energy. Thus, a method to quantify the attenuation of acoustic energy during sound wave propagation in a medium (gas) interacting with a nanomaterial is needed in order to model sound absorption in a thermostatted molecular dynamics simulation.
The classical sources of attenuation in a sound wave are internal viscous friction and heat conduction. However, if the wave-propagation domain contains a solid material along with a gas, additional losses may occur due to the interactions between the two media. The interactions trigger an energy exchange between the fluid and the solid in the vicinity of the solid surface and creates a region, known as the viscous boundary layer, where the mean flow velocity varies from zero at the surface to the free stream velocity far from the surface. The presence of this region causes a viscous loss of the sound energy as viscous stresses oppose the fluid motion and dissipate the fluid kinetic energy as heat. 16 In addition, thermal losses can occur due to heat conduction to the solid surface within the thermal boundary layer, which results from the temperature gradient created by the transition of the wave-propagating media from the isothermal condition at the solid surface to the isentropic (adiabatic) condition far from the surface. 17 In conventional computer simulations of acoustic wave propagation (such as computational fluid dynamics), acoustic losses (thermal and viscous) are defined using continuum theory and the losses are quantified as a function of sound attenuation based on continuum fluid assumptions. 18,19 However, the theoretical approximations of attenuation for an acoustic system based on a fluid continuum are only applicable at low frequencies, where the relaxation time for absorption (viscous and thermal) is similar to the mean time between collisions. 13,20 Hence, estimates of acoustic losses based on the theory of continuum mechanics may not be applicable to acoustic wave propagation in a gas at the nanoscale.
Hadjiconstantinou and Garcia 21 conducted direct simulation Monte Carlo (DSMC) simulations of sound wave propagation in the gigahertz (GHz) range without any solid present in the simulation domain. They calculated the sound speed and attenuation coefficient by non-linear fitting the simulated velocity amplitude, assuming plane-wave theory. The results were found to be significantly affected by free molecular flow near the sound source during a high-frequency sound wave propagation, for which the wavelength was comparable to the mean free path. Thus, the calculated sound speed and attenuation coefficient were sensitive to the distance from the sound source used for curve fits. 21 Therefore, a method that can accurately measure acoustic losses in a molecular system without relying on factors such as free molecular flow or continuum approximations is highly desirable. Here, a simulation framework for molecular dynamics was developed to study the sound field characteristics of high-frequency wave propagation in a simple monatomic gas in a simulation domain containing a CNT with the aim of investigating the acoustic losses due to the CNT and to capture the atomistic mechanisms involved. Losses occur both due to the conversion of the coherent acoustic energy into random thermal energy as the wave propagates and due to the atomistic interactions between the acoustic wave and the CNT. A notable contribution of this work is to identify the acoustic damping in the MD simulations and to quantify the acoustic absorption of the CNT using the thermoacoustic concept of exergy 22 , in which the dissipation of acoustic energy (acoustic losses) is defined in terms of entropy generation. Exergy is a thermodynamic potential that measures the ability to do useful work in a system in the presence of a freely accessible thermal reservoir at a particular temperature. 22 Acoustic losses evaluated using this method are compared with the estimates from standing-wave theory 15,21 developed for a system of acoustic wave propagation containing both the gas and the CNT. Additional MD simulations were also performed for sound wave propagation in a domain without the CNT present to distinguish the losses in the CNT-containing system from the classical acoustic losses in the fluid medium. The framework for MD simulations and the analysis of the sound field demonstrated in this study can be extended to investigate the loss mechanisms in systems containing more CNTs or other nanomaterials. It should also be noted that the MD simulation framework (without the CNT present) used here was validated by the authors in a previous publication 15,23 comparing Hadjiconstantinou and Garcia's 21 results from DSMC simulations and with the theoretical estimate of the transmission matrix method 24 of a duct system, which showed comparable particle velocity amplitudes in the simulation domain.
This paper is organized as follows. Details of the simulated system and simulation protocol are described in Section II. Section III presents the relevant theories and calculation methods used to estimate the acoustic losses and attenuation coefficient. The simulation results for the acoustic behavior and acoustic losses in systems with and without a CNT are compared in Section IV. The effects of interactions with the acoustic wave on the CNT are explained in Section V.

II. SIMULATION DETAILS
For simplicity, this work considers acoustic absorption effects of a CNT arising only from interactions between the acoustic wave and the outer surface of the CNT. Figure 1 shows a schematic of a system of nanotubes vertically aligned with respect to the substrate to represent a CNT forest with millions of nanotubes per square centimeter grown on a silicon substrate, in which interaction of sound waves occurring within and between the tubes. To understand the interaction of multiple tubes with acoustic waves, it is first necessary to determine the sound wave propagation behavior around a single tube, as the sound absorption by each tube is also expected to contribute significantly to the total absorption. Moreover, due to the limited availability of computational resources, simulations were restricted to a relatively short and narrow CNT. In the work presented in this study, the analysis of a single CNT fibre in a small acoustic domain was considered, where typical solution times were on the order of 25 days (per 100 periods of the wave cycle) using 96 CPU cores of a supercomputer.
MD simulations were performed for a plane sound-wave propagation in a monatomic argon gas in a rectangular domain at a frequency of f ≈1.5 GHz. The simulating domain is illustrated in Figure 1. An oscillating wall comprising a closed-packed FCC (face-centered cubic) lattice of "solid" argon atoms was used as the sound source and excited at z = 0 with a velocity assigned to wall atoms collectively by imposing a sinusoidally varying velocity in the z-direction. The piston wall was constructed by holding a collection of 35,645 argon atoms fixed with respect to one another in an FCC lattice with one face of the atoms exposed to the gas. The wall of solid argon, combined with the thermostat applied to the gas, was designed to mimic a diffuse reflection boundary condition. During the simulation, the piston was oscillated back and forth with a sinusoidally varying velocity in the direction perpendicular to its surface, while the thermostat imposed a Maxwell-Boltzmann distribution at the desired temperature on the velocity components of the gas atoms perpendicular to the wave propagation direction. The far end of the simulation domain in the z-direction was terminated by a specular reflecting wall and the system was replicated in the transverse directions using periodic boundary conditions. The domain was filled with argon gas at a density of ρ = 1.8 kg m −3 (20,402 molecules) to maintain the average gas pressure at P = 1 atm.
An open-end (uncapped) (5,5) single-wall carbon nanotube (SWCNT) of length L CNT = 50 nm and diameter d CNT = 0.69 nm was cantilevered at the specular wall and immersed in the simulation domain. One end of the CNT, consisting of atoms within 0.1 nm of the clamped boundary, was fixed and the remainder of the CNT atoms were allowed to move freely according to the interaction forces on them. The small size (diameter and length) of the CNT was chosen in order to reduce the computational expense and to simplify the atomistic mechanisms by ensuring that gas atoms not could enter the interior of the CNT.
The simulation domain had dimensions of L z = 150 nm in the wave-propagation direction and L x = L y = 70 nm transverse to the wave-propagation direction. The domain size was large enough to exclude inter-nanotube coupling interactions between periodic images. With the periodic boundary conditions applied in the directions perpendicular to the propagation direction, the simulated system represents an array of nanotubes with an area density of 0.0002 nm −2 and an intertube spacing of 70 nm, which is similar to an experimental CNT array investigated in the authors' previous work 10,12 . The evenly spaced CNT alignment would thus represent a simple cubic array (or a planar repeating pattern) of nanotubes. Although this arrangement of nanotubes is an approximation, it is a reasonably realistic representation of the types of nanotube arrays that can be achieved experimentally by growing patterned CNTs on a surface 2,3 .
It should also be noted that the potential of resonant interactions between nanotubes is small given that all simulations were performed with larger dimensions of 70 nm to ensure that the domain size was at least of similar size to one mean free path (≈72 nm) of the gaseous argon atoms. Having a large enough system size to accommodate a full collision length for A second generation REBO (reactive empirical bond order) 25 potential was used to model the inter-atomic interactions between carbon atoms in the CNT. The REBO potential has been used widely by researchers 26 to perform MD simulations of carbon nanotubes. It has been successfully used to calculate the thermal transport properties of carbon nanomaterials. 26 In addition, it is known for its accuracy to reproduce the phonon dispersion relations of CNTs, which is very important for heat transfer mechanisms. [26][27][28][29] The interactions between argon gas molecules was described by a Lennard-Jones (LJ) 12-6 potential with LJ parameters ε AA = 10.33 meV and σ AA = 3.40Å and a cut-off distance of 3σ AA . 26 The inter-atomic interactions between argon and carbon atoms (argon-carbon) were also represented by a LJ potential with parameters ε AC = 4.98 meV and σ AC = 3.38Å. 26 A short-range purely repulsive WCA (Weeks-Chandler-Andersen) potential 30 with the same LJ parameters was used for the interaction between the atoms of the solid wall and those of the propagating medium (gas) by truncating the LJ potential at 2  acoustic modeling of carbon nanotubes such as fluid/structure interactions, bi-directional heat transfer, and acoustic wave propagation. 15,[26][27][28][29]31 The total number of atoms in the simulation domain, N system = 60, 127 (with N wall = 35, 645, N argon gas = 20, 402, N free CNT = 3, 990, and N fixed end CNT = 90) was constant during the simulations.
Simulations were initiated with a small timestep of 0.01 fs using a velocity-Verlet algorithm in order to relax the system; the timestep was gradually increased to 0.4 fs. Thereafter, simulations were continued with a time step of 0.5 fs. The initial velocities of the gas molecules were chosen randomly from a Gaussian distribution consistent with a temperature of 273 K, while the CNT atoms were initially stationary (i.e., at a temperature of 0 K). The gas molecules and CNT were coupled to two separate Langevin thermostats, both at 273 K during an equilibration period of 15 ns. After the equilibration, the system was excited by an acoustic wave generated by a oscillating the solid wall with a velocity amplitude of v 0 = 0.15c = 49.69 m s −1 (where c is the classical sound speed) and frequency of 1.5 GHz (R = 1, where R is the acoustic Reynolds number 21 ). During the simulation of high-frequency sound-wave propagation, an auxiliary mechanism was required to remove the added heat resulting from the work done on the system by the rapid oscillation of the sound source. Hence, a Nosé-Hoover thermostat at T = 273 K was coupled loosely (with a thermostatting damping time (τ ) of 1 wave period) to the degrees of freedom of the gas molecules perpendicular (x and y) to the propagating wave (z-direction) to control the temperature. This thermostat mimics the interaction the gas with a thermal reservoir surrounding the simulated system in the directions perpendicular to the wave-propagation direction, as would be the case in a real physical system. The CNT was not directly thermostatted during the acoustic wave propagation. As the flow reached the steady state, sampling was carried out for a total duration of 40 wave-cycle time periods.

III. CNT-INDUCED ATTENUATION: CALCULATION METHODS
Sound attenuation due to the interaction between the sound wave and the CNT was evaluated using both curve fitting based on standing wave theory and the thermodynamic concept of exergy. The development of the equations and the calculation methods are described in the following sections.

A. Standing-Wave Theory
Curve fitting of the waveform components over an entire domain (as discussed in previous studies 15, 21 ) is not appropriate in the current study because there are two coupled domains: one between the source and the tip of the CNT, which contains only argon atoms; and another for the remainder of the domain, which contains both argon and CNT atoms.
Hence, standing-wave equations based on classical acoustic wave theory were formulated for an acoustic domain incorporating a CNT by considering two separate domains between the source and receiver as illustrated in Figure 2.
The rectangular simulation domain of cross-sectional area S=(L x ×L y ) and length L z was subdivided into two separate regions of length L <z <L (Region 1 ) and 0 <z <L (Region 2 ), respectively, with the boundary between the two regions at z = L . Wave propagation was driven by a piston at z = 0. The domain was terminated by a reflecting wall at z = L.
If the piston oscillates harmonically at a frequency ω, then superposition of the incident and reflected waves on each side leads to a standing wave which can be expressed in a generalized form of wave equation as v g/c (z, t) = A g/c (z) sin ωt + B g/c (z) cos ωt. (1) Here, A g/c (z) and B g/c (z) are the components of the velocity amplitude of the standing wave on each side, where for the CNT region (Region 1 ), and for the gas region (Region 2 ), where k=ω/c is the acoustic wave number, ω is the angular frequency, c is the classical sound speed, and m is the attenuation constant. Derivation of these equations are described in Appendix A. These Equations (2)-(5) also can be rearranged to form the standing wave equation for a single region between the source and receiver which contains only gas atoms similar to the case discussed in the DSMC study by Hadjiconstantinou and Garcia 21 . By replacing m g = m c = m for the case without CNT, one may obtain the equations of velocity components as which represent velocity components of a standing wave for a single region of gas atoms between the source and receiver. Verification for two-region approach against that of a single region between the source and receiver are also discussed in Appendix A.
Furthermore, it should be noted that the attenuation coefficient m c represents sound attenuation in the CNT region, which has both gas and a CNT, hence the resultant attenuation by the CNT would only be (m c − m g ). If the CNT induces any acoustic absorption in the gas then m c has to be larger than m g . The acoustic absorption coefficient within the CNT region can be estimated from the calculated attenuation constant (m c ) using the relationship between the reflection coefficient (R α ) and attenuation constant as 32 Similarly, the reflection coefficient for CNT only can be expressed as The resultant absorption coefficient is then calculated as 19

B. Exergy Analysis
The acoustic energy losses in the system can also be evaluated using exergy, a thermoacoustic potential that measures the ability to do useful work in a system in the presence of a freely accessible thermal reservoir at a particular temperature T 0 . A detailed description of exergy can be found in the book by Swift 22 , in which the losses of acoustic energy in a system are considered as lost work (Ẇ lost ) and estimated as entropy generation, which is responsible for the irreversibility of processes in the system. A similar approach of energy balance has been applied here. A schematic of a generalized microscopic portion of an acoustic system in the presence of a CNT is shown in Figure 3. Following the method described by Swift 22 , the entropy generation in this domain can be expressed as whereẊ(z) is the time-averaged exergy flux in the z-direction (the subscripts "In" and "Out" denote fluxes into and out of the domain, respectively), which can be written aṡ whereĖ is the acoustic power flowing at a mean temperature T m along the wave path, which is equal to active acoustic intensity I ac when presented as an acoustic power flowing per unit area in a plane wave propagation. The active acoustic intensity I ac can be associated with the particle velocity, v(z, f ), and sound pressure, p(z, f ), of the standing wave as 19,33,34 Here, v * is the complex conjugate of the particle velocity v(z, f ).
The total power flux,Ḣ(z), in the z-direction is 35 where h (= U + P V = E k + E p + P/ρ) is the enthalpy per unit mass, κ is the thermal conductivity, A is the cross-sectional area, T m is the mean temperature of the gas and the subscript 'solid' denotes properties of whatever solid is present in the system. The first term on the right side of Equation (14) is the time-averaged enthalpy flux and the second term is the conduction of heat both in the gas and in the solid present in the system. Here, U is the sum of the total kinetic (E k ) and potential (E p ) energy, P is the pressure and V is the volume of the gas.
Equation (12) for the exergy fluxẊ denotes the associated power to do work in the presence of a thermal reservoir at T 0 . 22 Dividing the full simulation domain into M "bins" in the wave-propagation direction and expressingẊ is the bin number index, Equation (11) can be expressed as The entropy generation calculated using Equation (15) gives the rate of energy lost per unit length of the microscopic portion of the thermoacoustic system. For a simulation domain of a thermoacoustic system of plane wave propagation of volume V , cross-sectional area A and length L z containing of a CNT of length L CNT = L z and cross-sectional area A CNT , the rate at which the energy per unit volume is lost from the wave iṡ In the current simulation, the CNT was placed at the termination wall and the length of the tube L CNT was smaller than the total length of the simulation domain L z . Hence heat conduction between the gas and the CNT can be considered to be limited to a region of length L CNT . Therefore the simulation domain can be subdivided, similarly to the schematic in Figure 2, into two adjacent regions: one containing only gas atoms in which heat conduction occurs only in the gas, and another containing both gas atoms and the CNT, in which the conduction of heat occurs both in the gas and the CNT. This can be realized by separating the heat conduction terms in the calculation of the total power flux using Equation (14).
The total power flux can be expressed aṡ whereḢ g is the total power flux without the heat conduction term for the CNT andḣ CNT is the heat conduction term for the CNT. These terms can be written aṡ Combining Equations (17)- (19), the thermoacoustic approximation for the exergy fluxẊ in Equation (12) can be rearranged for the gas in the absence of any solid in the system aṡ Substitution into Equation (16) gives a generalized equation for the entropy generation (lost work) in a simulation domain containing a CNT: where (L z − L CNT ) is the length of the domain containing gas atoms only, which represents a domain similar to Region 2 in Figure 2. Equation (21) gives the power lost per unit volume, which can then be used to obtain the attenuation coefficient m cnt e as, where I(= 1 2 ρcv 2 0 ) is the classical acoustic intensity for the simulated wave, v 0 = 49.69 m s −1 is the particle velocity amplitude chosen for the study and the subscript 'e' is used to indicate an estimate based on the exergy.

IV. SIMULATION RESULTS AND DISCUSSIONS
Following equilibration, the simulations were run for 67.35 ns, which is equivalent to 100 periods of the propagating wave cycle, which was at a frequency of f ≈1.5 GHz. The energy balance of the system, the variation of the temperature of both the gas and the CNT, and of the gas pressure during the wave propagation were monitored to observe the state of the flow and to check the effect on the sound speed due to dissipation. Details of these parameters can be found in Appendix B.

A. Sound Field
To observe the effect of the interaction between the CNT and the gas on acoustic parameters such as the velocity profile, acoustic active and reactive intensity, and absorption coefficient, the acoustic sound field behavior was compared between the cases with and without the CNT for the same simulation conditions. Figure 4 shows comparisons of the velocity components, A(z) and B(z), of the particle velocity amplitudes v(z) and and the real and imaginary components of acoustic pressure p(z) as a function of distance for both cases. It can be observed that the velocity and pressure components do not reveal any substantial differences between the two cases. Similarly no considerable differences are observed in the active and reactive intensities 33 of the sound field shown in Figure 5. The reason for these insubstantial differences in the acoustic components between the simulations with and without the CNT can be attributed to the small differences between the attenuation coefficients for the regions of the gas only (Region 2 ) and the gas with CNT (Region 1 ). So to observe any considerable changes in the acoustic variable profiles, the attenuation coefficient in the CNT region must be significant compared with that in the gas region. However, as demonstrated below, an analysis of the power balance in the system in terms of exergy reveals considerable differences between the simulations with and without the CNT. A similar comparison of the transfer function, H p 0 pz , between the acoustic pressure at the sound source (p 0 ) and that away from the source (p z ), for the simulations with and without the CNT is displayed in Figure 6. The local acoustic absorption coefficient as a function of distance along the wave path is also compared for the two simulations and presented in Figure 7. Again, no considerable changes in the absorption coefficient are observed even near to the tip of the CNT (z ≈ 99.9 nm) or within the distance from the tip of the CNT to the termination wall.
However, it can also be seen that the components of the acoustic intensities and transfer function between acoustic pressures at the source and away from the source, as shown in Figures 5 and 6, in the presence of CNT show a small but discernible changes in the corresponding profiles compared with those in the simulation without the CNT, which can be attributed to the interaction between the acoustic wave and the CNT and the subsequent changes to the acoustic energy in the system. These indicate the introduction of attenuation due to the presence of the CNT and its interaction with the wave, much like what would be expected as a result of increasing noise (due to attenuation) in the signal as a function of frequency for high-frequency wave propagation. These responses also show that some of the energy in the system must have gone into the structure, which therefore alters the resulting pressure response and ensuring effective re-distribution of total energy into two systems of gas and CNT, hence the discernible changes in pressure dependent quantities once the CNT   These findings can be verified by estimating the attenuation coefficients using a more rigorous approach for the energy balance provided by the exergy, described in Section III B.
Estimates of the acoustic powerĖ, total powerḢ and exergyẊ for the simulation containing the CNT are shown in Figure 9. The largest differences between the exergy and acoustic power can be observed in the CNT region of the domain along the wave path of z >90 nm, indicating additional consumption of acoustic energy by the CNT compared with the CNTfree region. These differences are more pronounced when these curves are compared with that of the simulation containing gas only, as shown in Figure 10. These results illustrate the considerable utility of exergy for analyzing nanoscale acoustic absorption. In the preceding estimates of acoustic parameters, only a subtle difference between the acoustic power curves ( Figure 5) was observed between the simulations with and without the CNT, which were calculated based on the changes in the acoustic pressure and particle velocity. However, unlike the velocity components or acoustic power, substantial differences in the exergy and total power curves can be seen in Figure 10 in the two simulations, particularly in the region containing the CNT, whereas no substantial changes occur in the gas region z <90 nm. From the exergy, the attenuation coefficient (m cnt e ) for the whole system with the CNT present was calculated using Equation (21). The value of m cnt e = 2.17 × 10 7 m −1 calculated form the exergy is approximately equal to the value of (m c + m g ) = 2.26 × 10 7 m −1 obtained from curve fitting using the two-region approach. Note that the value of m c represents only the CNT region, hence the value of (m c + m g ) represents the attenuation for the whole system combining both the gas and CNT regions in Figure 2. The discrepancy can be attributed to the curve fitting methods of one-region and two-region approaches of standing wave equation used for obtaining attenuation coefficient m g for gas atoms only. In the preceding estimates,  (13)) with (a) total power (Equation (17)) and (b) exergy (combination of Equations (19) and (20) estimate seems reasonable considering that the experimental absorption coefficient obtained for a 3 mm CNT forest (see authors' previous article 10 ), which includes millions of CNT per unit area, is 5% ∼ 10% for an audible frequency range of 125 Hz to 4 kHz. The value estimated here is for a CNT of 50 nm and an acoustic wave of a very high frequency at 1.5 GHz.

V. EFFECTS ON CNT
The transfer of acoustic energy from the gas atoms into the CNT during the wave propagation can also be confirmed by analyzing the atomistic behavior of the CNT. Several aspects of molecular-scale changes to the CNT during wave propagation were analyzed to determine how the attenuation (if any) of the acoustic wave energy is related to the interactions between the CNT and gas atoms. Two primary loss mechanisms 20,36 were anticipate to dominate for sound absorption by the CNT: • Damping of the wave due to the induced structural vibrations of the material caused by sound pressure and velocity fluctuations within the material. This effect was studied by analyzing the vibrational behavior of the CNT.
• Viscous and thermal losses caused by collisions of the oscillating gas atoms with the CNT atoms and heat conduction due to differences between the thermal conductivity of the wave medium and the CNT. This effect was studied by analyzing the phonon spectrum behavior of the CNT as inelastic collisions between the gas atoms and the CNT would excite phonons along the nanotube and the absorbed energy would be converted to heat by structural damping mechanisms.

A. Vibrational Behavior of Carbon Nanotube
The vibrational behavior of the CNT was analyzed to investigate any significant changes in its structural modes as a results of acoustic wave propagation. To gain further insight into the excitation of structural vibrations in the CNT, a principal component analysis for CNT vibration modes are shown in Figure 11, which reveals no significant changes in the most dominant vibrational modes with or without acoustic excitation.
However, the deflection frequency and energy of the CNT were were found to be amplified with acoustic excitation compared with the case without excitation, which can be verified by examining the deflection of an atom at the tip and an atom in the middle of the CNT (as illustrated in Figure 12). The displacement of the tip and midpoint of the CNT is The single-sided auto-spectral density of the deflection amplitudes of the midpoint atom in the x-, y-, and z-directions in Figure 15 reveals the presence of additional peaks at high frequency due to acoustic excitation. Without acoustic excitation, the first three peaks are at frequencies of approximately 1 GHz, 7 GHz and 40 GHz, which can be attributed to random fluctuations of CNT atoms. With acoustic excitation, an additional peak can be seen at approximately 20 GHz along with the harmonics of this peak at even higher frequencies.
Furthermore, the amplitudes of each peak (including those due to thermal fluctuations of CNT atoms) in the auto-spectral densities increased with acoustic excitation. This indicates that a portion of the acoustic energy contributes to amplifying the deflection of the CNT and is also transferred into the normal modes of the CNT that are excited by random fluctuations at equilibrium.
These analyses suggest that the additional energy stored in the CNT atoms during acoustic excitation, due to acoustic energy transfer from the gas atoms into the CNT structure, induces a dramatic increase of the frequency of atom deflections ((dx, dy, dz)) of the nantoube. This also indicates that the loss of acoustic energy may be attributed to viscous and heat conduction losses due to the collision of gas atoms with the nanotube, thus resulting in the traveling of a phonon down the nanotube. Analysis of the phonon spectrum behavior of the CNT can confirm this hypothesis, as explained in the following section. It should also be noted here that nonlinear interactions with the nanotubes with respect to the applied forcing may exist, which are beyond the scope this study and hence are not addressed here. However, investigations of vibrational modes using principal component analysis (PCA) shows no significant changes in the eigenvector and eigenvalues with or without acoustic excitation, as shown in Figure 11. Overall, the structural response resembled a regular linear system and did not show signs of non-linear distortion. Moreover, analysis of the auto-spectral density (ASD) of the deflections of CNT atoms at the tip and in the middle of the CNT, as displayed in Figure 15, show that the first two peaks (attributed to the random fluctuations of CNT atoms) are at frequencies of approximately 1 GHz and 7 GHz with or without acoustic excitations and no harmonics of the driving frequency of 1.5 GHz are evident. This analysis indicates that additional resonant interactions due to the additional forcing of the piston at the driving frequency of 1.5 GHz were not significant in the simulation.

B. Phonon Spectrum behavior of Carbon Nanotube
The excitation of phonons by the acoustic wave can be measured by comparing the phonon spectra of the CNT for the cases with and without acoustic excitation. The phonon energy spectrum of the nanotube was calculated from the power spectral density of the velocity fluctuations of carbon atoms in the nanotube as 26 where f is the frequency at which the velocities are sampled from the simulation, v i is the velocity of carbon atom i, and N is the total number of atoms in the tube. The velocities were sampled at 133 THz for 66,667 trajectory snapshots during equilibrium (i.e., without acoustic excitation) and for 89,866 trajectory snapshots during acoustic wave propagation (i.e., with excitation) and ensemble averaged in 2048 sample windows for both cases. The phonon spectrum with acoustic excitations was calculated for only one wave period (between 100 and 101 wave periods) as it was a memory-intensive process to record the velocities of each of the 3990 CNT atoms for a repeated number of wave cycles. Figure 16 • Length and positioning of the CNT: The absorption of an absorbent material for any acoustic frequency of interest is greatest when the material is placed at a distance a quarter of the wavelength from a wall, as this is where the particle velocity of the vibrating medium is its highest 38,39 . In the current simulation, the CNT was positioned near the termination wall (reflection wall) and the length of the CNT was L CNT = 50 nm (approximately the distance from the CNT tip to the termination wall), which was smaller than a quarter of a wavelength (λ ≈ 290 nm) of the simulated acoustic frequency. Hence, a CNT either positioned at λ 4 away from the wall or with a length greater than a quarter of a wavelength of the acoustic frequency would potentially achieve greater acoustic absorption in the CNT.
• Cross-sectional area of the CNT: The diameter of the CNT simulated in this study was d CNT = 0.69 nm and the mean free path of the wave propagation medium (argon gas) was ≈ 72 nm. Since the simulation was performed for plane wave propagation with a normal incidence sound source, the wave strikes the cross-sectional area at the tip of the CNT. This means that the CNT would provide a cross-sectional area considerably smaller than a fraction of a mean free path required for a single collision to occur with gas atoms. Hence, interactions between the gas and CNT atoms were infrequent as observed from the atomic trajectory.
Unfortunately, an MD simulation of a much longer or wider CNT than used in the current work would be computationally prohibitive. Additional MD simaultion results of a smaller CNT of 25 nm can be found in the supporting information in Appendix C, which demonstrate the variation in acoustic absorption with the variation in the size of the CNT.

VI. SUMMARY
A molecular system was modeled in this study to investigate high-frequency acoustic wave propagation and sound absorption in the presence of a CNT using molecular dynamics propagation was also studied. The frequency of deflections of CNT atoms were found to increase dramatically due to its interactions with the acoustic wave. However, only weak coupling was found between the CNT structure and the propagating acoustic wave as no significant change in the vibrational modes was observed in the CNT structure. in summary, a platform for MD simulations was developed to model and employed to quantify significant acoustic absorption by a nanomaterial. This platform could be extended to investigate loss mechanisms in the audible frequency range and would be beneficial to demonstrate the acoustic behavior of CNT absorbers similar to those studied experimentally. and Similarly at the termination wall at z = L, v c = 0, which can be expressed as At the boundary between the regions, the acoustic impedance (as a ratio of pressure and particle velocity) for both regions near to z = L would be equal, which can be expressed as where A g/c (z) and B g/c (z) are the components of the velocity amplitude of the standing wave on each side and can be expressed separately for the CNT (Region 1 ) and the gas regions (Region 2 ) as written in Equations (2)-(5).
The non-linear fitting of these equations to the components of the velocity amplitudes for a theoretical standing wave can be used to compare the acoustic variables obtained from the one-region (Equations (6)- (7)) and two-region (Equations (2) and frequency f ≈ 2.5 GHz. Figure 17 shows the fitting of the wave forms to the theoretical standing wave using both approaches. The predicted values of the acoustic variables such as the sound speed and attenuation were found to be the same as the values used to formulate the theoretical standing wave. This confirms that the two-region approach can predict the same value of the attenuation coefficients for two adjacent regions if both have the same medium.
The two-region approach was also used to predict the sound speed (c) and attenuation coefficient (m) from the simulation results of the velocity components for an acoustic domain without the CNT present for a wave of frequency f ≈ 1.5 GHz. A comparison of the two approaches (one and two regions) for the non-linear fits of the waveforms is shown in Figure   18. approaches. However, if a constant value of c were used for the prediction in the two-region approach, then both attenuation coefficients would have been the same. It can be seen from Figure 18 that both approaches give a good fit to the waveforms for constant values of the sound speed and attenuation coefficient within the data fitting region at distances further than one mean free path (z>λ mfp =7.28 × 10 −8 m).

Appendix B: Steady state condition during wave propagation
The variation of the temperature of both the gas and the CNT, of the gas pressure during the wave propagation are shown in Figures 19(a) and 19(b), respectively. It can be seen that both the gas and the CNT reached a steady mean temperature after 40 periods. The increase in the average temperature and pressure of the gas, which reached 284 K and 1.09 bar (≈ 1 atm), was within 5% of the equilibrium temperature and pressure in the absence of wave propagation (273 K and 1 atm, respectively), which confirms that the change of sound speed would be less than 2.5%, since the sound speed varies as √ T . 21 The energy balance of the system was investigated to ensure consistency between the en- ergy input, the energy stored, and the energy dissipated. The acoustic energy input into the system is equal to the work done by the piston, which can be calculated from the measured normal force on the gas and the oscillating displacement of the piston. Figure 20 shows a complete history of the energy due to the work done by the piston and the energy extracted by the thermostat as a function of the oscillation period of the wave cycle during the acoustic wave propagation. The curves for similar slopes, with a linear fit giving the work done by the piston and the energy extracted by the thermostat as approximately 509 kcal/mol and 507 kcal/mol per period, respectively. This indicates that an approximate energy balance per acoustic wave period is maintained between the energy input by oscillations of the piston and the extraction of energy from the system by the thermostat. The small discrepancy can be attributed to the linear fitting of the curve for all wave periods, which includes the transition periods between the steady state condition and equilibration.
Appendix C: Comparison with MD simulation results of a 25 nm-long CNT Due to the limited availability of computational resources, simulations were restricted to a relatively short and narrow CNT 50 nm in length and 0.69 nm in diameter in a small acoustic domain. An MD simulation of a significantly longer or wider CNT than used in the current work would be computationally prohibitive using the available computational resources. Research is currently being carried out in our group to speed up the simulations 40 .
To demonstrate the variation in acoustic absorption with CNT length, we have instead compared simulation results of the 50 nm-long CNT for an acoustic frequency of 1.5 GHz with that those for a shorter 25 nm-long CNT of the same diameter for wave propagation at a higher acoustic frequency of 2.57 GHz. Figure 21 compares the acoustic power and exergy for the two cases. The significantly different behaviour of the acoustic power and exergy for the different nanotubes indicates differences in the acoustic wave energy in the two system.
The differences between the exergy and acoustic power for wave propagation at 2.57 GHz for the simulation of the 25 nm CNT are not statistically significant in the region where the CNT is situated (z >125 nm) as the acoustic energy in that region is not strong enough to have a considerable interaction of the acoustic wave with the CNT. On the other hand, the differences between the exergy and acoustic power become significant and more pronounced with the longer 50 nm CNT at the lower frequency of 1.5 GHz in the region where the CNT is situation (z >100 nm), indicating an increase in wave attenuation due to higher absorption by the longer CNT and higher acoustic energy.
A similar difference between the acoustic power and total power can be seen in Figure 22, which compares the acoustic power and total power for the two cases. The differences are largest near the CNT region (z >125 nm for the 25 nm CNT and z >100 nm for the 50 nm CNT) of the domain and are more pronounced for the longer CNT. These variations in the computational results between two cases with two different length of CNTs demonstrate that a CNT longer than one quarter of a wavelength would achieve greater absorption.