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Phonon wave-packet scattering and energy dissipation dynamics in carbon nanotubeoscillatorsMatukumilli V. D. Prasad and Baidurya Bhattacharya Citation: Journal of Applied Physics 118, 244906 (2015); doi: 10.1063/1.4939277 View online: http://dx.doi.org/10.1063/1.4939277 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/118/24?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The performance volatility of carbon nanotube-based devices: Impact of ambient oxygen Appl. Phys. Lett. 95, 123118 (2009); 10.1063/1.3236779 Deformation potential carrier-phonon scattering in semiconducting carbon nanotube transistors Appl. Phys. Lett. 90, 062110 (2007); 10.1063/1.2437127 Characterizing energy dissipation in single-walled carbon nanotube polycarbonate composites Appl. Phys. Lett. 87, 063102 (2005); 10.1063/1.2007867 Double‐Resonant Raman Scattering in an Individual Carbon Nanotube AIP Conf. Proc. 685, 225 (2003); 10.1063/1.1628023 Neutron scattering studies of carbon nanotubes AIP Conf. Proc. 486, 299 (1999); 10.1063/1.59801
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Phonon wave-packet scattering and energy dissipation dynamics in carbonnanotube oscillators
Matukumilli V. D. Prasad1 and Baidurya Bhattacharya2,a)
1Advanced Technology Development Centre, Indian Institute of Technology Kharagpur, Kharagpur,West Bengal 721302, India2Civil Engineering Department, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal 721302,India
(Received 15 July 2015; accepted 19 December 2015; published online 31 December 2015)
Friction in carbon nanotube (CNT) oscillators can be explained in terms of the interplay between
low frequency mechanical motions and high frequency vibrational modes of the sliding surfaces.
We analyze single mode phonon wave packet dynamics of CNT based mechanical oscillators, with
cores either stationary or sliding with moderate velocities, and study how various individual
phonons travel through the outer CNT, interact with the inner nanostructure, and undergo
scattering. Two acoustic modes (longitudinal and transverse) and one optical mode (flexural
optical) are found to be responsible for the major portion of friction in these oscillators: the
transmission functions display a significant dip in the rather narrow frequency range of 5–15 meV.
We also find that the profile of the dip is characteristic of the inner core. In contrast, radial
breathing and twisting modes, which are dominant in thermal transport, display ideal transmission
at all frequencies. We also observe polarization dependent scattering and find that the scattering
dynamics comprises of an oscillating decay of localized energy inside the inner CNT. This work
provides a way towards engineering CNT linear oscillators with better tribological properties.VC 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4939277]
I. INTRODUCTION
Friction during the relative motion of two atomic surfa-
ces arises from an exchange of energy between the surfaces.
For strongly interacting surfaces, friction involves large
stress and displacement fields, rearrangement of interatomic
bonds, and loss or exchange of atoms at the interface. In case
of weakly interacting interfaces, which is the focus of this
work, friction amounts to the dissipation of energy through
phonon excitations1–5 and electronic excitations.6,7
Cumings and Zettl’s experimental work8 on multi-
walled carbon nanotube (MWCNT) pull-out set the stage
for CNT based oscillators, actuators, and bearings. While
telescoping out the inner walls, they quantified the dynamic
(static) friction force between the walls as less than
4.3� 10�15 N/A2 (less than 6.6� 10�15 N/A2), which are
almost three orders smaller than friction forces in current
MEMS.9 Around the same time, Kolmogorov and Crespi’s10
computational work on smooth intershell mobility between
the walls of commensurate MWCNTs introduced the possi-
bility of NEMS operating at Gigahertz range.11 In a recent
experimental work,12 this ultra-smooth sliding under ambient
conditions is observed even in a centimeter long DWCNT.
And very recently, Ma et al.13 proposed an analytical model
for the critical length above which such superlubricity
breaks down. Various nanoscale mechanical devices have
been proposed since then that take advantage of the relative
motion of carbon nanotube walls, such as: nanobear-
ing,10,14–17 nanoactuator,18 nanoswitch,19 nanomotor,20,21
nanogear,22 nanobolt-nut pair,23,24 nanoaccelerometer,25
nanoresonator,26 nanothermometer,27 oscillator,8,28,29 vari-
able resistor,30 high performance FETs,31 Brownian motor,32
inertial measurement systems,33 non-volatile memory devi-
ces,34–36 nanoscale cargo transport,37 etc.
During the first half of the previous decade, various stud-
ies based on classical molecular dynamics have attempted
to address the role of commensuration,10,38–40 boundary condi-
tions,41 functionalization,42 defects,43–46 interwall dis-
tance,45,47 length of sliding surfaces/overlap/telescoping,45,48
and temperature38,49–52 in energy dissipation in CNT based
linear oscillators. There have been conflicting findings in
regard to sliding forces in commensurate vs. incommensurate
systems. Incommensurate systems were found less dissipative
than commensurate systems;10,38,39 however, Tangney et al.40
claimed that dynamical friction was independent of commen-
suration of bitubes and of the overlap area between tubes.
They further claimed that the source of friction was the motion
of tube ends (surface edges), which depended on the relative
velocity of tubes. Zhao and Cummings53 investigated these
differences and found the outcome to depend on temperature,
commensuration, and type of molecular dynamics ensemble
adopted. In an attempt to resolve the conflicting findings on
the effect of commensurability, Zhu et al.54 identified interac-
tion strength and intertube energy corrugation as two crucial
parameters in assessing the effect of commensurability on
energy dissipation, yet the underlying physics need to be
investigated.
Stone-Wales and vacancies have been found to excite
low frequency vibrational modes45 and to markedly increase
the dissipation rate.44,46 However, a single Stone-Wales
a)Author to whom correspondence should be addressed. Electronic mail:
0021-8979/2015/118(24)/244906/9/$30.00 VC 2015 AIP Publishing LLC118, 244906-1
JOURNAL OF APPLIED PHYSICS 118, 244906 (2015)
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defect was found to suppress dissipation in a linear oscillator
by avoiding rotational motion.43 Although dissipation rate
was typically found to increase linearly with temperature in
most studies, Chen et al.52 reported an inverse relation at
ultra-low temperature ranges (�0.01 K–1 K): the critical
temperature at which the inversion occurred corresponded to
the saturation of thermal lubrication and the appearance of
phononic friction. The critical temperature in turn was
related inversely with the CNT length.
Phononic friction at the interfaces involving adsorbed
monolayers has been studied both experimentally and theo-
retically. The experimental works involved hydrogen and
deuterium terminated Si surfaces,55 H2O, N, superheated He
films adsorbed on Pb(111) substrates,56 and adsorbed films
of several inert gas atoms and hydrocarbons on various me-
tallic substrates as reported in Ref. 57. The theoretical works
involved CO molecule adsorbed on the (100) face of Cu.58
Although a few pioneering experimental works8,12,28,59 have
addressed friction in sliding nanotubes, these are all phenom-
enological in nature, and to our knowledge, phononic friction
in CNT oscillators has not yet been subjected to theoretical
or experimental investigations.
Recent theoretical attempts60,61 on thermal energy
exchange at the interfaces of CNT confirmed a strong de-
pendence of frequency and polarization of phonons and
also of interface characteristics. In the area of interfacial
thermal transport, a variety of CNT interface configura-
tions such as surface-surface interface,60,62,63 end-end
interface64 and CNT-substrate interface61 have been stud-
ied. In order to understand the effect of interface bonding
and spacing on thermal transport, most of these studies an-
alyzed individual phonon scattering mechanism at the
interfaces. In an MD based study, Greaney and
Grossman63 observed a resonance based energy transfer
between CNTs when positioned side by side. Kumar and
Murthy60 used heat pulse propagation and wavelet analysis
to identify dominant phonon modes in thermal transport
across CNTs when CNTs were positioned perpendicular to
each other.
In CNT oscillators, deformations such as bending and
wavy modes of outer CNT, which are manifested by the
collective excitation of disorderly phonon modes, have
been identified as the major channels through which the
translational kinetic energy of the core dissipates as ther-
mal energy.48,50,65 Exchange of vibrational energy from
useful mechanical motion to thermal fluctuations and
vice-versa have so far not involved any phonon level
treatment. Phonon wave packet formalism66 provides a
powerful computational framework for treating single pho-
non transport across interfaces.64,66,67 Using this scheme,
the dynamics of single mode phonon at the sliding CNT
interface can be studied.
Once the motion of core initiates, a typical oscillation
cycle involves: motion along the outer CNT where a part
of orderly translational energy dissipated in to disorderly
high frequency phonon modes; reversal of motion of core
at the ends of outer CNT. The focus of the present work
is on the energy exchange dynamics between orderly
translation motion of core and phonon modes of outer
CNT. The fundamental understanding of the interaction
mechanism between individual phonon modes and orderly
moving core is of crucial importance in nanoscale
tribology.
In this work, we present a phonon wave-packet simula-
tion based study of the scattering mechanism of individual
phonons in pristine CNTs, in order to elucidate their role in
phononic dissipation occurring in co-axially sliding oscilla-
tors. Although, due to sliding instabilities, all phonon modes
are involved to some degree in the dynamics of dissipation,
here we study only those modes individually that are known
to be dominant in thermal transport. High transmittance indi-
cates less dissipation at the encounter of a single phonon and
core.
II. COMPUTATIONAL DETAILS
Phonons exist over a wide range of frequencies. Low
frequency phonons show a plane wave type behavior because
of their long wavelength (as in the propagation of sound);
whereas phonons of high frequency show a particle type
behavior (as in the case of light-matter interaction). Phonon
modes and other physical properties such as specific heat,
dielectric,68 piezoelectric,69 elastic constants,70,71 etc., can
be explained in terms of harmonic travelling waves in the
atomic lattice (anharmonic forces would be required for
obtaining properties such as thermal conductivity, phase
transitions, etc.).
Phonon wave packet method is used in this work along
with molecular dynamics simulations to study the influence
of phonon scattering on the dynamics of the CNT linear os-
cillator. Phonon wave packet method has been used by
Schelling et al.66 to study scattering mechanism at the coher-
ent semiconductor interface, by Refs. 64, 66, and 67 to study
the influence of phonon scattering due to defects67 and func-
tionalization64 on the thermal transport along CNTs, and by
Helgee and Isacsson72 to investigate the phonon transmission
across the grain boundaries in graphene. However, to our
knowledge, phonon wave packet method has not been
applied to study energy exchange mechanism in CNT
oscillators.
The force constant matrix is obtained using the supercellmethod64 in which the atoms in the unit cell are displaced
one at a time (small enough displacement to maintain har-
monicity) and resultant forces on all atoms of the supercell
are computed. Fourier transform of this force-constant ma-
trix gives the dynamical matrix, diagonalizing which results
in phonon frequencies and phonon modes at a particular
wave vector. The relation between frequencies and wave
vector gives the dispersion relation.
Phonon wave packets are formed from the linear combi-
nation of vibrational eigenstates. To generate wavepacket
with wavevector q0 and centered at z0, the initial displace-
ment (u) and initial velocity (t) of the kth base atom in lthunit cell along the direction a, assuming a Gaussian spread,
are set as
uslka ¼
Affiffiffiffiffiffimkp
Xq
exp �q� q0ð Þ2
2r2
� �es
ka qð Þexp iq zl � z0ð Þ� �
;
244906-2 M. V. D. Prasad and B. Bhattacharya J. Appl. Phys. 118, 244906 (2015)
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tslka ¼
Affiffiffiffiffiffimkp
Xq
�i2pxs qð Þexp �q� q0ð Þ2
2r2
� �
� eska qð Þexp iq zl � z0ð Þ
� �;
where A is the wave amplitude, r is the broadening parame-
ter to control width of Gaussian phonon wave packet (we
choose r to be 2p/100a where a is size of unit cell, such that
the wave packet covers a width of 200 unitcells), zl is posi-
tion of the lth unitcell, and mk is the mass of the kth atom.
xsðqÞ, eskaðqÞ are the frequency and eigenvector correspond-
ing to polarization s and the wave vector q.
The propagation of the wave packet thus generated on
the outer CNT with its corresponding group velocity is simu-
lated. Molecular dynamics simulations are conducted using
LAMMPS.73 As in previous works on phonon wave packet
simulations of CNTs,64,74,75 the polymer consistent force
field (PCFF)76,77 is adopted to model the atomic interactions.
As the parameters in this potential are developed using abinitio computations corresponding to 0 K, PCFF is a better
choice for MD simulations involving ground state systems.
All simulations are conducted in NVE ensemble with a time
step of 1 fs. To ensure that anharmonic effects are com-
pletely avoided, the ground state (�0 K) configuration is
adopted in all simulations. Scattering due to the inner nano-
structure is visualized. Energy transmission coefficient is
computed as the ratio of transmitted energy to the incident
phonon wave packet energy. We simulated the scattering
process until a stable kinetic energy is attained on both sides
of the interface region to ensure that the process of interac-
tion is completed within the simulation time. Thus, the simu-
lation time span of a molecular dynamics run varied from
30 ps for high group velocity phonons to 300 ps for low
group velocity phonons. Wave packet amplitude of 0.5 A is
used throughout all the simulations. (For a detailed sensitiv-
ity study to arrive at this 0.5 A amplitude, see Fig. S1.78)
Fig. 1 shows the phonon dispersion plot for a (10,10)
SWCNT unit cell (2.45 A long and consisting of 40 atoms),
calculated using the supercell method. Of the 120 possible
phonon modes, only four are acoustic in nature, and being re-
sponsible for almost half of the thermal energy transport,79
are studied here in detail: the longitudinal acoustic mode
(LA), one of doubly degenerate transverse acoustic mode
(TA) and the twisting mode (TW), which is unique to nano-
tubes due to their symmetry. The remaining 116 are the opti-
cal modes; of these, two are picked up for further study on
account of having the highest group velocities: the radial
breathing mode (BR) which extends to most part of the fre-
quency spectrum than any other mode, and the first order
flexural optical mode (FO). A micron-long version of the
same unit cell forms the outer CNT of the oscillator used in
this study, as discussed next.
III. RESULTS AND DISCUSSION
The oscillator consists of an open ended 50 nm long
(5,5) SWCNT core within a micron long (10,10) SWCNT
outer tube. Fig. 2 shows the initial locations of the wave
packet and the inner core. Along with moving cores with
sliding velocities of 100 m/s and 300 m/s, we also studied
stationary cores. We restrict the core velocity to 300 m/s in
order to avoid resonance effects. Corresponding to each of
the five phonon modes selected for study (LA, TA, TW, BR,
and FO), a phonon wave packet is generated in the outer
tube far away from the inner core (500 unit cells to the left
of the centre) and we study how the individual phonon trav-
els through the outer CNT, interacts with the inner nano-
structure, and undergoes scattering. A typical phonon
wave packet interaction process with inner core is depicted
in Fig. 2 using wave packet displacement components corre-
sponding to before, during, and after the interaction with
inner core.
A. The enablers of ideal transmission
Fig. 3 shows the transmission coefficients of the five
phonon modes for the oscillator system as a function of the
FIG. 1. Phonon dispersion and symme-
try of phonon modes in (10,10) CNT
calculated from the PCFF force con-
stants using supercell method. The
highlighted five dispersion curves (LA
in orange, TA in cyan, TW in red, BR
in green, and FO in pink) are studied in
this work. v is frequency, “a” is uni-
tcell size, and h is Planck’s constant.
Symmetry of phonon modes (LA, TA,
TW, BR, and FO modes in right
panel).
244906-3 M. V. D. Prasad and B. Bhattacharya J. Appl. Phys. 118, 244906 (2015)
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phonon energy. The group velocity (the slope of the disper-
sion plot in Fig. 1) is plotted along the axis on the right.
Clearly, the TW and BR modes show near ideal transmission
throughout the entire frequency range. In fact, the radial
breathing mode is the most characteristic vibrational mode
for CNTs and can help identify the chirality and diameter of
the CNT through Raman scattering experiments. In this
mode, all atoms vibrate radially in-phase and resemble the
breathing effect.
Like the radial breathing mode, the twisting mode too
shows near ideal transmission (0.95 and above) in almost the
entire range of frequencies (Fig. 3(c)). With minimal radial
displacements, atoms in this mode vibrate tangential to the
surface of the CNT. Importantly, group velocity80 and pho-
non transmission (even in the presence of a vacancy
defect67) in the low energy region of TW mode have been
found to be unaffected by the diameter of the outer CNT,
which is unlike in other acoustic phonon modes. We conjec-
ture that such diameter-independence extends to phonon
transmission in the presence of inner core as well, provided
the inner core maintains a stable interwall distance. BR and
TW modes show ideal transmission irrespective of the slid-
ing velocity of the core.
The TW and BR modes are the in-phase circumferential
and in-phase radial/breathing displacements respectively as
shown in Fig. 1. We believe that this symmetry in displace-
ments, coupled with the rotational symmetry in the defect-
free core, is responsible for the complete absence of scatter-
ing in the TW and BR modes. In contrast, a significant
amount of scattering has been observed in the presence of
mismatched symmetries where these TW and BR modes
interact with defects81 or functionalizations75 instead of pris-
tine inner CNT.
The three other modes—two acoustic (LA and TA) and
one optical (FO)—are responsible for major part of the fric-
tion in CNT oscillators. These modes are studied in detail.
FIG. 2. Top: Snapshots of a propagating LA wave packet along the outer CNT
at times 1 ps, 11 ps, and 20 ps. Transverse and axial displacements are drawn in
horizontal (blue) and vertical (red) planes respectively. The position of the
inner CNT is represented by the two dotted lines at the center. Bottom: The
atomic configuration of coaxially sliding CNTs with imposed wave packet on
left side of the moving core. Outer tube is shown in cut-out section view.
FIG. 3. Phonon transmission functions
for (10,10) CNT with (5,5) CNT-L50
as stationary and moving core.
Transmission coefficients (left vertical
scale) and phonon group velocity (blue
lines) (right vertical scale) as a func-
tion of incident phonon energy, hv. (a)
Longitudinal acoustic phonons. (b)
Transverse acoustic phonons. (c)
Twisting acoustic phonons. (d) Radial
breathing optical phonons. (e) First
order flexural optical phonons.
244906-4 M. V. D. Prasad and B. Bhattacharya J. Appl. Phys. 118, 244906 (2015)
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B. The dissipative modes
In case of CNTs with surface defects,64,67,75,81 the
acoustic phonon modes are known to transmit ideally near
the C-point (i.e., at low wave numbers) irrespective of their
group velocity, which is exactly what we see in Figs. 3(a)
and 3(b) of this work. A similar behavior is also seen in pho-
non transmission at Si/Ge interface.82 As the frequency
increases, however, there appears to be no consensus regard-
ing the extent of correlation between transmission coefficient
and group velocity in acoustic modes (Refs. 67, 75, and 81
as opposed to Refs. 64 and 74). Our study (Fig. 3(a)) shows
negligible correlation between either nLA or nTA and vg in
low energy regions, but a significant positive correlation
beyond the point of lowest group velocity.
The acoustic phonons with wavelengths very close to
C-point are long enough to not interact with the inner nano-
structure. Moving just right of the C-point, in the low energy
range (5–15 meV), they are able to interact with the core and
thereby lead to significant scattering (down to �0.7 for LA).
nTA is even smaller than nLA in this region, and can come
down to 0.2. As one moves up the energy scale, nLA and nTA
feature a sharper dip that is centered around the energy cor-
responding to the least group velocity. Finally, an ideal trans-
mission region is realized in the high energy region
(23–35 meV). The core velocity is not influencing the trans-
mission in low energy region. In the high energy region, the
transmission functions are deviating from the stationary
case. In Fig. 3(a), the small dip at 25 meV in nLA is amplified
significantly for both sliding velocities. Moreover, for
300 m/s case, the amplified dip is shifted towards the low fre-
quency. In contrast, the dip in nTA at 18 meV (Fig. 3(b))
found evaded for moving core cases. The changes in the
high frequency range transmission functions of moving cores
indicate the strong interaction of high frequency phonon
modes with the moving core.
Bending and wavy modes has been identified as the
major source of energy dissipation in CNT oscillators.48,50,65
In general, TA mode vibrations would manifest as bending
of CNT by transverse deformation of the tube structure.
Although the role of TA phonon modes has been studied for
CNT resonators,83 thermalization of CNTs,84 and thermal
energy transport in CNTs with defects67,81 and functionaliza-
tions,75 their role in energy exchange in CNT oscillators has
not be studied until now.
The optical phonon modes with low group velocity has
been predicted earlier60 as strongly interacting modes
between two CNTs. The optical phonon modes with wave-
lengths in the intermediate regime and having large enough
group velocity are found responsible for non-Fourier heat
conduction85 in SWCNTs. In Fig. 3, the optical modes, how-
ever, behave differently near the C-point compared to the
acoustic modes. The general characteristic of optical pho-
nons—low transmission near C-point—is displayed by both
FO and BR modes (Figs. 3(d) and 3(e)). The strong scatter-
ing of optical modes near the C-point is due to their low
group velocities causing them to be confined in the defective
or edge region of the inner CNT. Therefore, the optical pho-
nons of long wavelength cause substantial heat generation
through electron-phonon interactions. Similar to LA and TA,
nFO also show a sharp dip in low energy region at 11 meV.
The next dip at 15 meV is found shifted towards the high fre-
quency side due to the core motion (the 100 m/s case). For
higher core velocity, this dip is less pronounced. The small dip
in nBR at 110 meV corresponds to the band anticrossing region
(where these BR modes mix strongly with other bands).
For dissipative modes (LA, TA, and FO), the transmis-
sion functions in the low energy region are almost unaffected
by the core velocity. In the moderate energy region, trans-
mission functions are changed significantly due to strong
interaction of phonons with the moving core (nLA in
22–28 meV, nTA and nFO in 13–22 meV). Similar trends have
been observed in earlier studies86,87 at finite temperatures.
We conjecture this coupling at moderate frequencies occurs
due to resonance. Further, in the high energy region, the
transmission functions are unaffected and all three dissipa-
tive modes transmit ideally irrespective of core velocity.
The increased scattering of slower LA phonons is well
agreeing with earlier studies.75,81 The interface scenario in
the CNT oscillator system provides a new perspective to
comprehend this behavior. The severity of scattering in low
group velocity region for LA phonons in our oscillator can
be explained by considering both phase velocity vp and
group velocity vg as presented in Fig. 4. However, in an ear-
lier study by Wang and Wang,81 the scattering of slower LA
phonons in CNTs with Stone-Wales defects (no inner core)
has been investigated. Using an infinite atomic chain as simpli-
fied model, they demonstrated the correlation between low
transmission and low group velocity as a general characteristic
of phonon transport across CNT with broken translational
invariance. Their classical acoustics based reasoning is that a
low group velocity in the LA mode causes the phonon wave
impedance to be low, which in turn causes slower phonons to
almost completely be reflected by the defect.
From the Fig. 4, we conjecture that a high value of the
ratio vp/vg is the reason for the huge scattering at low group
velocities. In the LA mode, atoms vibrate mainly along ra-
dial and axial directions. For the duration of time when the
packet passes over the inner tube, packets with the slower
group velocity have a higher number of radial oscillations.
FIG. 4. Ratio of phase velocity to group velocity. vp/vg (red) on right y axis;
and on left y axis, phase velocity (black) and group velocity (blue) of LA
mode as a function of incident phonon energy, hv.
244906-5 M. V. D. Prasad and B. Bhattacharya J. Appl. Phys. 118, 244906 (2015)
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This number of oscillations over the inner tube is directly
correlated with (vp/vg). Fig. 4 clearly shows a sharp increase
of vp/vg corresponding to the least group velocity.
From the results of transmission functions of five impor-
tant phonon modes, we can observe ideally transmitting BR
and TW modes and a narrow frequency band (5–15 meV) in
which the major scattering occurs for three dissipative
modes. The TA and FO modes show a severe scattering in
this frequency band. The sliding velocity significantly modi-
fying the transmission functions of all three dissipative
modes, however not in the peak scattering region.
C. Dynamics of the scattering events
Further, to elucidate the frictional dissipation mecha-
nism, we now look at the scattering events created as a
phonon wave packet travels along the outer tube of the oscil-
lator and encounters the inner core. In each case, the excita-
tion starts at 500 unit cells left of the center of the stationary
core and travels right. Fig. 5 shows the time evolution of ki-
netic energy along the length of the oscillator (taken to be
the kinetic energy of the first basis atom of each unit cell).
The top row corresponds to the LA phonons, the second row
to the TA phonons, and the last row to the FO phonons. In
each of these nine panels, the square plot corresponds to
energy time history of the outer CNT while the companion
strip corresponds to the core. It should be noted that the color
scales corresponding to the nine panels are all different.
Phonons with three different energies are studied for
each mode. For the LA mode, we choose three phonons
(5.76, 9.18, and 12.3 meV) that outline the profile of the first
dip in the transmission plot occurring to the right of the
FIG. 5. Phonon scattering dynamics. Temporal evolution of the phonon wave packets while interacting with (5,5) CNT-L50 as inner core. LA phonons with
energy (a) 5.76 meV, (b) 9.18 meV, and (c) 12.3 meV. Decay of localized phonon state (Quasi bound state) in inner CNT can be clearly seen. TA phonons with
energy (d) 2.4 meV, (e) 8.15 meV, and (f) 17.76 meV. Both extremes of scatterings can be seen from (d) and (e). FO phonons with energy (g) 8.78 meV, (h)
11.3 meV, and (i) 13.54 meV. The kinetic energy per atom (meV) is represented according to the color bar. Contour maps for both outer and stationary inner
CNTs are shown separately. Vertical dashed lines indicate the position of inner CNT, contour map of which presented separately on right side.
244906-6 M. V. D. Prasad and B. Bhattacharya J. Appl. Phys. 118, 244906 (2015)
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gamma point in Fig. 3(a). For the TA mode, we choose one
close to the gamma point (2.4 meV), one at the first dip
(8.15 meV), and one near the lowest vg (17.76 meV) in the
transmission plot (Fig. 3(b)). For the FO mode, we again
choose three phonons (8.78 meV, 11.3 meV, and 13.54 meV)
that outline the profile of the first dip in the transmission plot
(Fig. 3(c)). Although these three modes are responsible for
energy dissipation in CNT oscillators, each mode has its own
characteristic behaviour distinctly different from the other
two.
In the LA mode, the wave packet creates the first scatter-
ing event in the outer tube only when it travels over the right
end of the core. A substantial part of the packet is reflected
back first through the inner CNT, part of which then is sent
back along the outer CNT as it reaches the left end of the
core. This creates a cycle of slowly decaying scattering
events and represents anomalous reflection of LA phonon by
the inner core.
Although the two dips in the transmission function are
very similar to those in the LA mode, the scattering events in
the TA mode are significantly different from those in LA.
The packet starts at 500 unit cells to the left of the centre
as before. While the long wavelength TA phonon shows no
scattering (Fig. 5(d)) as it passes over the core, the one near
low vg shows diffused scattering (Fig. 5(f)). For the TA pho-
non with energy near the first dip (middle panel), scattering
occurs as soon as the packet reaches the left end of the core,
whereas for the LA phonon with the same energy profile,
scattering occurs only when the packet has travelled the
length of the core.
The first order optical phonons too behave very differ-
ently from the LA mode. Not only is there a substantial
reflection as soon as the phonon encounters the inner core,
but also when the phonon passes over the right end of the
core. The diffused scattering however in the right panel is
similar to that observed for TA phonons with the lowest
group velocity.
Similar to the TW and BR modes, the LA mode also has
rotational symmetry in match with the symmetry of inner
CNT. Surprisingly, the LA mode is not showing ideal trans-
mission. From the scattering events (Figs. 5(a)–5(c)), first we
can observe the unscattered passage of the LA mode phonon
wave packet over the left end atoms of inner CNT which is
similar to the TW and BR modes. Afterwards, the dominated
longitudinal component of the LA mode wave packet causes
an intense shearing that leads to transmission of wave packet
energy into inner CNT. The transmitted part of wave packet
continues to propagate in the inner CNT and causes an end
scattering due to right end atoms. The reflected part of wave
packet propagates towards left end while part of its energy
transmitting to the outer CNT. This oscillating decay of the
LA mode wave packet in the core, which can be clearly seen
in the contour strips corresponding to the inner core.
For the TA and FO modes, such a matching symmetry
does not exist. Once the propagating phonon encounters the
mismatched symmetry or augmented degrees of freedom due
to the presence of inner CNT (at the ends itself), it generates
quasibound vibrational states in the inner CNT. The gener-
ated low quasibound state acts as localized scatterer and
causes mixing of modes and severe scattering. From the
trends of TA and FO modes, we can predict the dissipation
due to other unstudied optical modes is in proportion with
their transverse atomic displacements. The BR and TW
modes are found to display near ideal transmission irrespec-
tive of the sliding velocity which indicates that the phonon
vibrational modes with matched symmetry with core (unlike
dissipative modes) causes no scattering. Hence, such phonon
modes can be recognized as the enablers of ultra-low dissipa-
tion in coaxial sliding of CNTs.
IV. CONCLUSIONS
We have studied single mode phonon scattering dynam-
ics in co-axially sliding CNTs. The radial breathing and
twisting modes (which are relatively high group velocity
phonons) are potentially the major contributors to ultra-low
friction in CNT oscillators. Over a wide range of frequencies
and for core velocities up to moderate values (�300 m/s),
they show near ideal transmission. Extensive testing with
wavepacket amplitudes in the feasible range (0.1–1 A) con-
firmed that the BR mode’s ideal transmission is not affected
by wavepacket amplitude.
In contrast, the major participants in phonon dissipation
in CNT oscillators are two acoustic (longitudinal and trans-
verse) and one optical (first order) modes. We found the
energy ranges of phonons that can prevent ideally smooth
sliding of CNTs. The transmission plot in the LA mode fea-
tures two dips, and the first one (at low phonon energies, in
the 5–15 meV range) is characteristic of the core. The maxi-
mum scattering observed in the TA mode (�0.9) is higher
than that in the LA mode (�0.5), which suggests the
dominance of out of plane deformations in the damping of
oscillations. An anomalous reflection of LA phonon is
observed. Up to moderate velocities of the moving core (up
to 300 m/s), the essential scattering behavior in the LA, TA,
and FO modes is not changed; however, an amplification or
suppression of few high frequency modes is observed due to
the moving core.
We restricted the core velocity to a moderate value of
300 m/s in this work since at very high core velocities
(between 800 and 1100 m/s), we found the BR mode of the
outer CNT to become excited (not shown in this paper) simi-
lar to other works.87 We conjecture that such high amplitude
resonance in the BR mode engenders LA phonons, which in
turn causes a high frictional dissipation.
Although the present simulations are conducted on a
ground state configuration, the deformations imposed by the
thermal effects, wavy and sagging deformations of long
CNTs (which essentially create the out of plane deformation
on the walls of outer CNT) causes further scattering. Thus,
the dissipation behavior could be affected by the deforma-
tions due to structural instabilities. Since the anharmonic
effects indulge in observing single mode wave packet, study-
ing finite temperature scattering behavior of single mode
phonons is not possible. However, the present trends, in
essence, corroborating the previous MD simulations based
results.40
244906-7 M. V. D. Prasad and B. Bhattacharya J. Appl. Phys. 118, 244906 (2015)
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In this study, we restricted the core geometry to an
uncapped (5,5) CNT of 50 nm long. Our future work will
address the effect of core geometry in detail by considering
different core structures. Preliminary results suggest that a
certain amount of core geometry dependence may exist on
phonon scattering dynamics. For example, when the length
of open ended (5,5) CNT is doubled the first dip in the LA
transmission mode appears to shift towards the C-point and
the corresponding phonon scattering increases in magnitude.
Also, when the armchair CNT is replaced by a zigzag CNT
of almost the same diameter, the first dip in LA transmission
appears to become more pronounced indicating an increased
scattering by the zigzag core. Preliminary results also appear
to suggest the exciting possibility that the transmission plot
can be used to identify the inner core structure using its
unique signature.
Our findings may lead to better design of sliding surfa-
ces and to make highly efficient functional devices. Being
able to drive or cool specific phonon modes88,89 in order to
attain high phonon mean free path or to have controlled
anharmonic phonon dynamics is a promising way to engi-
neer the performance of CNT coaxial oscillators. However,
controlling individual phonons in situ remains extremely
challenging.
ACKNOWLEDGMENTS
We thank Dr. Vikas Varshney and Dr. Jonghoon Lee for
insightful discussions. Some of the MD runs were performed
on PARAM YUVA HPC for which we thank Centre for
Development of Advanced Computing (CDAC), Pune.
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