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Phononic origins of friction in carbon nanotube oscillators Matukumilli V.D Prasad, and Baidurya Bhattacharya Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.6b04310 • Publication Date (Web): 24 Feb 2017 Downloaded from http://pubs.acs.org on February 24, 2017
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Phononic origins of friction in carbon nanotube oscillators Matukumilli V.D. Prasad
†Advanced
∗,†
and Baidurya Bhattacharya
∗,‡
Technology Development Centre, Indian Institute of Technology Kharagpur, Kharagpur, WB 721302, India
‡Civil
Engineering Department, Indian Institute of Technology Kharagpur, Kharagpur, WB 721302, India
E-mail:
[email protected];
[email protected] Phone: +91 (3222) 28-3422. Fax: +91 (3222) 28-2254
Abstract Phononic coupling can have a signi cant role in friction between nanoscale surfaces. We nd frictional dissipation per atom in carbon nanotube (CNT) oscillators to depend signi cantly on interface features such as contact area, commensurability and by endcapping of the inner core. We perform large-scale phonon wavepacket MD simulations to study phonon coupling between a 250 nm long (10,10) outer tube and inner cores of four dierent geometries. Five dierent phonon polarizations known to have dominant roles in thermal transport are selected, and transmission coe cient plots for a range of phonon energies along with phonon scattering dynamics at speci c energies are obtained. We nd that the length of interface aects friction only through LA phonon scattering and has a signi cant non-linear eect on total frictional force. Incommensurate contact does not always give rise to superlubricity: the net eect of two competing interaction mechanisms shown by longitudinal and transverse phonons decides the role 1
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of commensurability. Capping of the core has no eect on acoustic phonons but destroys the coherence of transverse optical phonons and creates diusive scattering. In contrast, the twisting and radial breathing phonon modes have perfect transmission at all energies and can be deemed as the enablers of ultra-low friction in CNT oscillators. Our work suggests that tuning of interface geometries can give rise to desirable friction properties in nanoscale devices.
Keywords nanoscale friction, CNT oscillators, phonon propagation, wave packets, phononic coupling Introduction.
Vinci,
1,2
Although friction has been explored since the times of Leonardo Da
understanding the fundamental mechanisms of friction is still a challenge.
3
Typically
friction involves the dissipation of mechanical energy into various vibrational excitations during sliding of the contact surfaces.
4,5
Development of low-friction surfaces by minimizing
the energy lost at the interface has tremendous economic and technological implications. With increasing miniaturization of mechanical devices, assumed an important role.
6
friction at the atomic scale has
7
Under particular conditions sliding interfaces can exhibit ultra-low friction (superlubricity). Such low friction values can be between per atom.
8
2.3 × 10−14
N per atom
7
and
1.4 × 10−15
N
Incommensurate contact is one such interface condition where unaligned peri-
odicity exists between surfaces and thus provides a relatively smaller potential barrier to slide.
9
By forming incommensurate interface, various systems (such as tungsten tip and Si
surface,
10
graphite surfaces,
11
7
multi-walled carbon nanotubes (CNTs) ) display superlubric-
ity. Superlubric sliding can also be attained by other means such as thermolubricity, external mechanical excitations.
13
Nigues et al.
14
12
and
reported a surprisingly dierent frictional
behaviour of multiwalled CNTs and multiwalled boron nitride nanotubes (BNNTs), despite their identical crystallography.
They reported an ultralow friction force in CNT pullout
that was independent of interface contact area (which agrees well with earlier reports
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7,15
);
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whereas for BNNTs, a higher friction (by a few orders) that was proportional to the contact area was found. The ionic character of BNNTs with partial charges in the positive boron and negitive nitrogen atoms (unlike in CNTs) explains this ultrahigh friction. CNTs, by possessing a uniquely large disparity among its intertube and intratube interaction strengths, have been established as ultra-low friction nanostructures tuning frictional response. Sokolo
17
7,8
9
and are serving as testbeds for
15,16
in his pioneering work, brought out a connection between inverse phonon
lifetime and spacing between modes and argued that, for energy dissipation to occur in a nite harmonic solid, inverse phonon lifetime should be larger than the spacing between modes. In phonon dispersion, spacing between modes widens as the system size goes down. It is not surprising that ad hoc surfaces with ne-tuned phononic coupling have been advocated by several researchers
13,18,19
as a means to attaining e cient miniaturized devices. However,
the ne-tuning of phononic coupling still suers from substantial drawbacks such as: lack of knowledge on how individual phonons behave at the interface since anharmonic environment prevails, and the extremely challenging nature of in situ measurement of energy dissipation during sliding and the way phonons carry away the dissipated energy. In spite of several experimental eorts
4,1921
including a very recent observation of phonon
generation and propagation in sliding frction at MoS 2 (0001) surface,
22
current theoretical
approaches are still unable to provide a generalized mode-wise interaction of phonons in nanoscale systems. Our recent work on coaxial CNT oscillators has revealed how energy is transferred between the tubes,
23
how individual phonons in the outer tube near the interface
respond to the sliding inner CNT, sliding motion of the core.
25
24
and how longitudinal acoustic phonon mode can initiate
We found the radial breathing and twisting acoustic modes with
high group velocity, which exhibit no coupling at the interface, as the enablers of ultra-low friction in CNT oscillators. Longitudinal acoustic (LA), transverse acoustic and rst-oder optical modes, on the other hand, are chie y responsible for dissipation in CNT oscillators by causing substantial scattering in the 515 meV range.
24
LA phonons have recently been
3
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hν
(a) (5,5)CNT-L50
(b) (5,5)CNT-L100
(c) Capped (5,5)CNT-L50
(d) (9,0)CNT-L50
Figure 1:
Interaction of a typical phonon wave packet with four dierent interfaces that are
comprised of four dierent inner CNT con gurations (not drawn here with actual lengths) present inside a (10,10) CNT. (a) 50 nm long (5,5) CNT (b) 100 nm long (5,5) CNT (c) end-capped 50 nm long (5,5) CNT and (d) 50 nm long (9,0) CNT. Outer tube is shown in cut-out section view. White and black colors represent maximum and minimum atomic displacement, respectively, corresponding to the imposed wave packet
hν .
found to enhance the ow of con ned water in multi-walled CNTs by causing an oscillatory shear stress at the interface.
26
In this work, we study the eect of interface geometry on phonon coupling between two 1
coaxial CNTs and identify the speci c phonon modes and surface conditions responsible for friction.
We looked at four inner nanostructures as shown in Figure 1 such that the
eects of dierent interface features (contact length, chirality and end con guration) can be investigated. The four core structures are: a 50 nm long (5,5) CNT (5,5)CNT-L50, a 100 nm long (5,5) CNT (5,5)CNT-L100, an end-capped 50 nm long (5,5) CNT capped(5,5)CNTL50 and a 50 nm long (9,0) CNT (9,0)CNT-L50. The outer tube in each case is a 250 nm long (10,10) CNT.(Computational details are given in Materials and Methods section). Frictional analysis. We start by estimating the frictional force between the oscillator
tubes. We extrude half the length of the inner CNT out of the left end of the outer tube and release it with zero initial velocity. The unbalanced vdW forces between the tubes generate the initial driving force. As the core enters the outer tube and continues its journey toward the right end, a part of the kinetic energy of the core is transferred as heat into the outer
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Figure 2:
Dissipated energy as a function of core center of mass displacement in CNT
oscillators for the four dierent cores (Figure 1) inside the (10,10) CNT of 250 nm long. The outer tube is thermostatted at 0.01K in order to avoid the eect of thermally induced motions. The trends are similar at higher temperaturesdissipated energy doubling every 100-200 K(not shown here).
tube. The outer tube in turn is connected to a Langevin thermostat and is thus maintained at a constant average temperature,
T0 .
The heat of friction between the inner core and the
outer tube is thus dissipated to the Langevin thermostat. By measuring the incremental heat gain,
∆H , by the Langevin thermostat with distance, ∆L, travelled by the core, we estimate
the eective frictional force,
fe ,
as:
fe = ∆H/∆L.
MD simulation details are presented in
Materials and Methods section. Similar approaches have been successfully implemented in a few previous studies.
27,28
Although there is known temperature dependence of friction in CNT oscillators (e.g., a linear relation between friction and pre-heating temperature in microcanonical ensemble;
29,30
a linear relationship at higher temperatures with an inverse relation at the ultra-low 0.01-1 K temperatures in canonical ensembles;
28
and a linear temperature dependence for all cores
observed in the course of our work (not detailed here)), we conduct our study of friction, phonon propagation and coupling only at the ground state con guration of CNTs ( ∼ 0.01 K). We do so in order to eliminate anharmonic eects and to facilitate the observation
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Figure 3:
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Eective friction force between outer and inner CNTs for the four dierent os-
cillator systems computed from the dissipation trends in Figure 2: (a) before and (b) after the core completely enters into the outer CNT. The outer tube is thermostatted at 0.01K in these plots in order to eliminate the eect of thermally induced motions. At higher temperatures (not shown here) the trends are similar with friction force doubling every 40-60 K in the rst regime and every 100 K in the second.
of precisely how the weak vdW interactions between the outer surface and the end atoms in uence the propagating phonons. Figure 2 shows the energy dissipated,
Σ∆H ,
due to sliding motion of inner core to the
outer CNT. The two distinct regimes in the dissipation trends correspond to before and after the core has completely entered the outer CNT. The sudden jump in the dissipation
corresponds to the the point of complete entry and is the result of the strong interactions between the dangling end atoms of either tube. Within the simulated time span, the energy dissipated and distance traveled by the core varies according to core geometry. We captured the friction forces separately for the two regimes. Figure 3(a) corresponds to the regime before the core completely enters the outer CNT while Figure 3(b) corresponds to the regime after the core is completely encompassed by the outer tube. These eective friction values are closer to experimental results ( 2.3 Zettl
7
and
CNTs (2
1.4 × 10−15
× 10−17
× 10−14
N/atom by Cummings and
8
N/atom by Kis et al. ) than to previous MD results on friction in
N/atom by Ma et al.
30
and
4.43 × 10−16
N/atom by Chen et al.
28
).
The 50 nm long (5,5) CNT has the smoothest sliding inside the (10,10) CNT. It exhibits the least friction among the four cores both before and after entry into the outer tube. When
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the length of the (5,5) core is doubled, the friction force per atom is found to increase substantially (by as much as 60% before and 40% after). A linear relation between friction and contact area would have left the force/atom value unchanged. The source of such signi cant non-linearity in contact length dependence of friction needs to be explained. Chirality ((9,0) vs. (5,5) inner cores) has an even more pronounced eect on frictionthe force per atom increases 3 times after entry.
Both (5,5) and (9,0) CNTs have almost the same diameter
(6.79 Åvs. 7.05 Å), but the former (armchair) makes a commensurate pair with the (10,10) outer tube while the latter (zigzag) is perfectly incommensurate with it. This higher friction exhibited by incommensurate contact is interesting and also needs investigation. The most striking feature is revealed by the capped (5,5) inner core. In comparison to its uncapped counterpart, it experiences about 1.3 times greater frictional force before (Figure 3(a)) and about 5 times greater friction after complete entry (Figure 3(b)), and travels a much shorter distance as a result (Figure 2).
It is well known that the surfaces of C60 fullerene and
fullerene-like tips of CNTs are more reactive than the cylindrical walls of CNTs.
31,32
Gener-
ally this does not make much impact on overall chemical or mechanical properties of CNTs as the tip region is signi cantly smaller than the whole CNT structure, but appears to have a major role in friction which needs to be revealed. The phononic mechanism behind the eects of contact length, commensurable contact and end-capping will be explained in the next section. Phonon wave packet scattering.
All phonon modes of CNT participate to some
degree in the dynamics of dissipation. Of these, we focus on vethree acoustic and two opticalthat are dominant in thermal transport (as explained in our earlier work
24
): the
longitudinal acoustic mode (LA), twisting acoustic mode (TW), the doubly degenerate transverse acoustic mode (TA), the radial breathing mode (BR) and the rst order exural optical mode (FO).
Typical wave packet propagation.
Figure 4 illustrates the typical wave packet propa-
gation and interaction of single mode phonons with the inner nanostructure. A phonon wave
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Figure 4: Wave packet propagation with uncapped 50 nm long (5,5) CNT as core. Snapshots of the displacements for wave packets before, during and after interface scattering. (a) 5.76 meV LA phonon (b) 12.3 meV LA phonon and (c) 11.3 meV FO phonon. Vertical dashed lines indicate the position of inner CNT. The axial and two transverse components of wave packet displacement are represented in blue, green and red respectively.
packet with a well-de ned polarization centered at a narrow frequency range is generated on the outer CNT far away from the stationary core tube and allowed to propagate toward the interface. After the interaction is complete, we compute the transmission coe cient,
ξ,
de ned as the ratio of the transmitted energy to the incident energy. The simulation details are provided in Materials and Methods section. As stated above, we conduct these phonon simulations at the ground state con guration of CNTs ( ∼ 0.01 K). Atomic displacements of the rst basis atom in each unitcell along the length of CNTs are presented at dierent times (before, during and after collision process) in Figure 4. The dotted vertical lines indicate the range of interaction zone with the inner nanostructure. Figure 4(a) shows that the low energy LA phonon wave packet transmits through the interface region almost without any re ection (ξ = 0.96). The same LA phonon with 12.3 meV energy (Figure 4(b)) undergoes
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signi cant scattering by the same core ( ξ = 0.75). The FO phonon with 11.3 meV energy in comparison undergoes almost complete re ection (Figure 4(c) with complement of the transmission coe cient,
1 − ξ,
ξ
= 0.06). Clearly, the
is a relative measure of the strength of
phonon coupling between the two structures and thus oers a measure of phononic dissipation. At the same time, the energy,
hν ,
at which such strong coupling occurs gives an index
of the intensity of phononic dissipation. In the rest of this paper, therefore, we take a detailed look at the transmission coe cient as phonon packets, with a range of energies, encounter the stationary core in the middle of the outer tube (Figure 5). As stated above, ve phonon modes are considered and for each, four cores geometries (Figure 1) are studied. The range of phonon energies for each mode covers the entire spectrum of the phonon dispersion plot.
24
Among the ve modes shown in Figure 5, BR and TW modes display near ideal transmission irrespective of the core geometry or phonon energy. These two phonon modes have group velocities comparable to those of the LA mode, however, their coupling with any of the four inner cores is insigni cant. The matched radial symmetry of these two vibrational modes with the inner CNT (unlike the dissipative modes) helps eliminate scattering. Hence, such phonon modes can be recognized as the enablers of ultra-low dissipation in coaxial sliding of CNTs. The three other phonon modesLA, TA and FO show signi cant phonon scattering activity through most part of their frequency ranges, primarily in the low 5-15 meV range, in the form a striking dip in the transmission plot. The parameters
ξ
and
hν
corresponding to this dip, which indicates the strength of coupling and thus the contribution to phononic dissipation, depend on the geometry of the inner core.
Length of interface aects friction only through LA phonon scattering.
Figure
5(a) shows the LA phonon transmission function corresponding to the four dierent cores. The acoustic nature of LA phonons is a rmed by the display of ideal transmission near the
Γ-point
for all core geometries.
As we mentioned before, higher transmission indicates a
weaker coupling. A sharply increased coupling can be observed as we move to the right of
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Figure 5:
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Single mode phonon transmission as a function of incident phonon energy h ν .
for (10,10) CNT with dierent cores.
Transmission coe cients along left vertical scale
and phonon group velocity (blue lines) is on right vertical scale.
(a) Longitudinal acous-
tic phonons, (b) Transverse acoustic phonons, (c) Twisting acoustic phonons, (d) Radial breathing optical phonons, and (e) First order exural optical phonons.
the
Γ-point.
LA phonons display signi cant scattering (down to
∼
0.5) in this range. The
pro le of the dip in the transmission plot at low energy range (515 meV) is the characteristic of the inner core geometry and as suggested in our earlier work,
24
can serve as an identifying
signature of the core. Of all the core geometry features studied here (Figure 5a), the length of the interface has the most eect on LA phonon scattering. Conversely, a careful look at Figures 5(b)-(e) suggests that length of the interface aects only the LA mode and no other mode.
We
therefore conclude that length (or more generally area) of interface aects friction at the atomic scale only through LA phonon scattering.
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Visualization of the detailed wave packet scattering dynamics using spatio-temporal evolution of kinetic energy per atom (separately for outer and inner CNTs) in Figure 6 reinforces this nding. The scattering event in Figure 6(a) clearly shows that the longer core allows wave packets to interact for a longer duration and absorb more energy from the travelling phonon. Our earlier study
25
on thermodiusion also described this leftward and downward
shift in the rst dip. Such length dependence of friction has been reported in earlier MD studies
28,3337
as well, however, no phonon-based explanation has been oered until now.
The relation between commesurability and frictionit's complicated.
It is gen-
erally thought that commensurate contact leads to signi cantly higher friction than incommensurate contact, and has been reported for various material systems such as mica, graphite sheets,
11
W(011)-Si(001),
10
and CNTs sliding over graphite substrate.
39
oscillators, some studies have shown commensurate surfaces to exhibit higher friction while some authors
30,42,43
38
In CNT
34,36,40,41
have questioned the claim.
As stated above, although both (5,5) and (9,0) CNTs have almost the same diameter, the former (armchair) makes a commensurate pair with the (10,10) outer tube while the latter (zigzag) is perfectly incommensurate with it. By studying the phonon scattering behaviour of these two inner coresone commensurate and the other incommensurate, we nd two distinct coupling mechanisms for longitudinal and transverse phonon modes.
For a given
interface condition, the net eect of these two competing mechanisms would decide the role of commensurability. The LA mode couples more strongly with the zigzag core than with the armchair core in the 5-15 meV range (note that the orange curve in Figure 5(a) is consistently below the black curve around the dip), that is, incommensurate contact causes more friction when the driver is low-energy LA polarization. This is corroborated by the greater re ection at right end shown by Figure 6(b) than by Figure 6(a). TA and FO modes, having dominant transverse displacements, are scattered by the ends of the inner tube due to the broken translational symmetry. In 5-15 meV range, they display
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Figure 6:
Phonon scattering dynamics.
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Temporal evolution of the phonon wave packets
while interacting with dierent cores. (a) LA phonon with energy 9.18 meV interacting with (5,5)CNT-L100 (b) LA phonon with energy 12.3 meV interacting with (9,0)CNT-L50 (c) TA phonon with energy 10.5 meV interacting with (9,0)CNT-L50. FO phonons of energy 11.3 meV while interacting with (d) open ended and (e) capped (5,5)CNT-L50 and (f ) (9,0)CNTL50. 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.
a severe scattering (almost a complete re ection) by all the cores.
From the scattering
dynamics in Figure 6(c-d), we can see an immediate re ection of TA and FO phonons from the left end of the core whereas the LA phonons showed an unscattered passage over the left end (Figure 6(a-b)). An important feature in Figures 5(b) and (e) is that, contrary to LA phonons, the orange curve intersects with the black curve for TA and FO phonons in the 5-15 meV range. Further, the energy of maximum coupling is higher for the incommensurate tube (orange) in both phonon modes.
Thus, it is the phonon wavelength in TA and FO
modes that determines whether incommensurate contact causes more friction or not than commensurate contact, and the dierence may be as much as an order of magnitude.
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Capping aects only FO phonons and destroys their coherence.
The transmis-
sion plots in Figure 5 (a)-(d) show that capping the core (olive curve) has practically no eect on phonon dissipation at almost all wavelengths for LA, TA, TW and BR modes. The FO phonon in contrast is severely aected by the presence of the cap (Figure 5(e)).
The
scattering dynamics of 11.3 meV FO phonon for the armchair, zigzag and capped cores in Figures 6(d-f ) explain why. In case of the open ended (5,5) CNT-L50, the FO phonon re ects immediately after it interacts with the left end structure and with a severity similar to that of TA phonons. The capped version of the same core, however, evades this intense re ection by displaying a diusive type scattering of the FO mode (unique among the studied cases) due to its optical nature. This is the only situation where capped end con guration of the inner core causes a signi cant dierence in scattering. End caps are the most reactive sites in pristine CNTs.
The distorted carbon atoms'
orbitals, depending on the local curvature, decides the chemistry of the end caps. local curvature on the caps cause conjugation of atomic spatial overlap, and a change in hybridization from sp
2
π -orbitals
31,44
The
due to the loss of their
to an intermediate between sp
2
and
3 4547 sp . These orbital eects lead to an increase in local energy and induce a partial radial character to
π -bonding
electrons.
While sliding through the outer CNT, these end caps are prone to reacting with local deformations on the outer CNT induced by defects and/or thermal eects. Such strongly interacting conditions do not provide a smooth sliding as evident in Figure 3(b) and in ground state conditions, destroys the coherence of the individual phonons as evident in Figure 5(e)(olive curve) and Figure 6(e). We believe this interesting feature of capped inner nanostructures in FO mode could be common to all other optical modes with transverse displacements as well, and cumulatively develop the intense diusive phonon activity which re ects in the enhanced friction. Further, this in uence of reactive capped-end could inspire more e cient manipulation of the frictional dissipation by attaching more reactive sites to the ends of the moving core.
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Conclusions. This paper demonstrated how interface geometry governs frictional dissi-
pation in CNT oscillators and for the rst time oers a phonon based explanation. The ability to observe single mode phonon propagation in CNT oscillators, using phonon wavepacket MD simulations, allows us to precisely investigate frequency-wise scattering behaviour of phonons and their interaction mechanisms at the interfaces. We found that the length of interface has a non-linear eect on frictional force per atom: this is because contact length aects friction only through LA phonon scattering and the longer core allows wave packets to interact for a longer duration and to absorb more energy from the travelling phonon. It is known in the literature that incommensurate contact does not always give rise to superlubricity and there is evidence to both sides of the claim.
We were able to provide
the following explanation: there are two distinct coupling mechanisms for longitudinal and transverse phonon modes. For a given interface condition, the net eect of these two competing mechanisms decide the role of commensurability.
It should be noted here that for
interfaces involving one or few layers of atoms, especially in the scenario of co-axially sliding CNTs, frictional dynamics strongly depends on the continually evolving contact area and vibrational dynamics. For graphene systems, a recent study
48
highlights the importance of
this evolving quality of contact area in estimating friction while sliding. Although small compared to the size of the rest of the CNT, end caps are the most reactive sites owing to the distorted orbitals caused by the local curvature. All else being equal, capped inner cores suer the greatest frictional resistance. We found that even though capping has no eect on acoustic phonons, it destroys the coherence of transverse optical phonons and creates a diusive scattering which manifests as gross frictional drag. We hope that these results on phononic origins of dissipation will help design graphene/CNT based systems with tunable frictional and thermal properties. Acknowledgments. We thank Dr. Vikas Varshney and Dr. Jonghoon Lee for insightful
discussions. All computations were performed on PARAM YUVA II HPC for which we thank Centre for Development of Advanced Computing (CDAC), Pune.
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Materials and Methods Phonon wave packet simulation details
Phonon wave packet formalism
49
provides a powerful computational framework for treating
single phonon transport across interfaces.
4951
Constructing a phonon wave packet using a
single band of speci c polarization and narrow frequency range from the dispersion plot, and then studying the interaction of the wave packet with the interface using MD simulations are the typical steps in phonon wave packet simulations. Here, we obtained the force constant matrix using the supercell method
50
in which the atoms in the unit cell are displaced one at a
time (small enough displacement to maintain harmonicity) and resultant forces on all atoms of the supercell are computed.
Fourier transform of this force-constant matrix gives the
dynamical matrix, diagonalizing which resulted in phonon frequencies and phonon modes at a particular wave vector. Relation between frequencies and wave vector gives the dispersion relation. Phonon wave packets can be formed from the linear combination of vibrational eigen states. In order to generate a wave packet with polarization, s, wave number, position,
z0
, move all the atoms from their equilibrium positions by the displacement:
uslkα
where
uslkα
cell along
q0 , at a physical
" # (q − q0 )2 s A X exp − ekα (q) exp[iq(zl − z0 )] =√ 2σ 2 M q
is the displacement from the ground state position of
α
direction,
A
k th
(1)
base atom in
is the wave amplitude or energy content of wave packet,
phonon eigenvector corresponding to polarization, s, wave number, parameter to control width of Gaussian phonon wave packet and
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zl
q. σ
lth
unit
eskα (q)
is
is the broadening
position of
lth
unitcell.
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And set the initial velocities of all the atoms with:
s υlkα
" # 2 A X (q − q ) 0 =√ −i2πω s (q) exp − 2σ 2 M q
(2)
eskα (q) exp[iq(zl − z0 )] where
ω s (q)
is the phonon eigen frequency corresponding to wave number,
q,
and polariza-
tion, s. The propagation of the wave packet thus generated on the outer CNT with its corresponding group velocity is simulated using molecular dynamics in LAMMPS package.
52
The
polymer consistent force eld (PCFF) is adopted to model the atomic interactions. PCFF was originally developed based on rst-principles calculations at Hartree-Fock level of theory with Gaussian 6-31* basis set.
53,54
Thus, it is a better choice for low temperature calculations
than more popular empirical force elds. All simulations are conducted in NVE ensemble with a time step of 1 fs.
To ensure that anharmonic eects are completely avoided, the
ground state( ∼0.01K) con guration is adopted in all simulations.
Scattering due to the
inner CNT is visualized. For each computation, involving more than
∼1,63,000
atoms, we
carry out separately an extensive MD simulation of phonon propagation. A similar set of simulations are performed again for each of four core geometries. Energy transmission coe cient is computed as the ratio of transmitted energy to the incident phonon wave packet energy. The simulation time span of a molecular dynamics run depends on propagation time or group velocity of the phonon wave packet.
Details of MD simulations for friction computation
MD simulations are conducted in LAMMPS
52
using PCFF
54
to model atomic interactions
with a time step of 1 fs. The oscillator system comprises of outer (10,10) CNT of 250 nm long and a shorter (50 or 100 nm) inner CNT, together having close to 50,000 atoms. Initially the system is equilibrated under NVE for 200 ps.
Next, by extruding the inner core, a
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long NVT equilibration at desired temperature is carried out for 600 ps in order to attain constant temperature system with out any uctuations. Then, shifted to NVE ensemble to simulate the sliding process for 300 ps. While inner core is sliding, a Langevin thermostat is connected to outer tube to record the dissipated energy as a function of sliding distance.
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