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Phonon transport in helically coiled carbon nanotubes
CARBON
x x x ( 2 0 1 4 ) x x x –x x x
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Phonon transport in helically coiled carbon nanotubes Zoran P. Popovic´, Milan Damnjanovic´, Ivanka Milosˇevic´
*
University of Belgrade, Faculty of Physics, Studentski trg 12, 11001 Belgrade, Serbia
A R T I C L E I N F O
A B S T R A C T
Article history:
We perform theoretical studies on the phonon thermal transport in helically coiled carbon
Received 31 March 2014 Accepted 13 May 2014
nanotubes (HCCNTs). The Gru¨neisen parameter, as a function of the phonon wave vector and phonon branch, is numerically evaluated for each vibrational mode, so that the
Available online xxxx
three-phonon Umklapp scattering rates can be calculated exactly by taking into account all allowed phonon relaxation channels. We considered wide temperature range and heat conductor lengths from nano- to macro-scale. We examine the crossover from ballistic to diffusive transport regime and impact of HCCNT geometrical parameters on their heat conduction. Thermal conductivity in HCCNTs is found to be slightly lower than that in single walled carbon nanotubes (SWCNTs). This is interpreted by the competition among three factors. Firstly, threefold reduction of the Gru¨neisen parameter for the acoustic branches. Secondly, lower phonon group velocities. Finally, availability of purely acoustic scattering channels. Nevertheless, HCCNTs are predicted to be more suitable (than SWCNTs) for thermal management applications due to their spring-like shape. HCCNTs are extremely elastic, natural NanoVelcro material. Ó 2014 Elsevier Ltd. All rights reserved.
1.
Introduction
Since the early work of Ruoff and Lorents [1] where high thermal conductivity of carbon nanotubes have been predicted on the basis of the high in-plane thermal conductivity of graphite and of bulk and thin film diamond, wealth of experimental and theoretical research of heat transport in carbon nanotubes and related carbon nanostructured materials have been done [2]. It has been experimentally evidenced that, at all temperatures, thermal conductivity of single walled carbon nanotubes (SWCNTs) is dominated by phonons rather than electrons [3] and that in nanotube thermal conductors Fourier empirical law of thermal conduction is violated even when the phonon mean free path (MFP) is much shorter than the nanotube length [4].
However, with the exception of the recent reports on the molecular dynamics calculations of thermal conductivity [5] and thermal expansion [6] of helically coiled carbon nanotubes (HCCNTs) [7], theoretical research of their thermal properties is lacking, although the first synthesis [8,9] of the coiled nanotubes followed soon after the discovery of the straight ones [10]. Presumably, the lack of large-scale synthesis method of coiled CNTs over the past decades has hindered extensive research of their precious properties and further realizations of their potentials for the nano-technological applications [11–13]. However, several recent reports on the catalytic chemical vapor deposition production of the HCCNTs with controlled morphology [14,15] substantially improve the prospects of fundamental and applied physics research of these peculiar nanostructures.
* Corresponding author. E-mail address: [email protected] (I. Milosˇevic´). http://dx.doi.org/10.1016/j.carbon.2014.05.031 0008-6223/Ó 2014 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Popovic´ ZP et al. Phonon transport in helically coiled carbon nanotubes. Carbon (2014), http://dx.doi.org/ 10.1016/j.carbon.2014.05.031
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Here, using the upgraded Klemens treatment of heat conduction in basal planes of graphite [16–18] we perform a detail theoretical study of the intrinsic thermal conductivity of HCCNTs. Used is the model of HCCNTs based on the topological coordinate method [19,20] which has significant advantages over the scheme proposed in the pioneering work of Ihara
and Itoh [21]. Firstly, it allows for almost continual variation of a single geometrical parameter while all the other parameters are kept fixed. Secondly, it matches geometrical parameters of the as-synthesized HCCNTs [22,23]. Finally, apart from the previously defined smooth helical structure [19] we here introduce also corrugated model of HCCNTs (Fig. 1), as both circular and polygon-like shapes of the cross-sections
Fig. 1 – Model of HCCNTs. (a) Symmetry generating elements (helical rotational angle 2p=Q which is followed by a fractional translation F and twofold horizontal rotational axis, U-axis, which interlinks pairs of atoms within a monomer) and geometrical parameters of HCCNTs (tubular diameter d, coil pitch p, inclination angle v and helical diameter D); as a guide to the eyes one half of the monomer is highlighted; (b) Cohesive energies (upper panel) and relative difference of tubular radii (lower panel) of the corrugated and un-corrugated models of HCCNTs as a function of the graph parameter nr ; (c) On the left side: Initial LR triple connected graph (n6=1, ((3,0), (0,3))) of pentagons (black), hexagons (white) and heptagons (dark grey) [20]. Super-cell ((3,0), (0,3)) is shown; rectangle denotes its unit cell. On the right side: Augmented graph (1,2,2,2, ((3,0), (0,2))). Super-cell ((3,0), (0,2)) is shown; rectangle denotes its unit cell. Inserted are nr ¼ 2 rows of hexagons (between initial unit cells), n5 ¼ 2 columns of hexagons (between the pairs of pentagons within a unit cell) and n7 ¼ 2 columns of hexagons (between the pairs of heptagons of adjacent unit cells) [19]. Please cite this article in press as: Popovic´ ZP et al. Phonon transport in helically coiled carbon nanotubes. Carbon (2014), http://dx.doi.org/ 10.1016/j.carbon.2014.05.031
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are evidenced in the as-synthesized samples of the coiled carbon nanotubes [7,24]. We perform calculations of thermal conductivity of HCCNTs of finite lengths, from 10 nm to 10 mm. Apart from the anharmonic three-phonon Umklapp processes which are treated exactly, for all allowed phonon relaxation channels, backscattering is accounted for within relaxation time approximation. We also examine temperature dependant crossover from ballistic to diffusive transport regime, as well as dependance of thermal conductivity on geometrical parameters of HCCNTs. The breakdown of the Fourier law of heat conduction which was evidenced in SWCNTs [4] we also predict for HCCNTs. Namely, we find that regularly distributed pentagonal and heptagonal ‘‘defects’’ do not to scatter the long-wave length phonons to the extent enough to remove the thermal conductivity divergence with nanotube length. We predict that thermal conductivity of HCCNTs amounts 75% of that of the corresponding SWCNTs as the coil length exceeds 500 lm. So good heat conducting properties in combination with the unique mechanical qualities [13,25] make HCCNTs promising candidates for thermal management applications.
2.
3
Fig. 2 – Low energy phonon dispersion relations for a (2,2,0,0, ((1,0), (0,5))) HCCNT, with geometrical parameters D ¼ 2:2 nm; d ¼ 0:5 nm; p ¼ 2:6 nm; v ¼ 210 . Acoustic branches (green), the lowest optical branch (blue) and scattering channels AAA (green solid line) and AAO (blue solid line) are highlighted. Three-phonon Umklapp scattering of the type I (II) is shown by solid (dashed) line. Red diamond marks the considered phonon. (A colour version of this figure can be viewed online.)
Phonon dispersion relations
We consider the model of HCCNTs which is obtained by rolling up a triple connected graph of pentagons, hexagons and heptagons, conventionally defined by a set of numbers ðn6 ; nr ; n7 ; n5 ; ðb1 ; b2 ÞÞ, where ðb1 ; b2 Þ are the super-cell vectors, while the tiling pattern is given by the first four parameters, Fig. 1. In addition to the relaxation procedure given in Ref. [19], tubular radial coordinates of carbon atoms are also varied and obtained is a model of corrugated HCCNTs. Comparison of the cohesive energies of the corrugated and uncorrugated HCCNTs together with the rate of corrugation are given in Fig. 1. Geometrical structure of HCCNTs is characterized by tubular diameter d, outer diameter of a helix D, helical step p and inclination angle v. Unlike the case of SWCNTs where phonon dispersion curves are evaluated using the graphite force constants [26] adopted to the nanotube geometry [27], dynamical sub-matrices of HCCNTs are numerically derived out of Brenner interatomic potential [28] by varying space coordinates of carbon atoms and calculating the corresponding energy changes. Since HCCNTs are complex systems which, in general, do not have translational periodicity and contain thousands of atoms within a single coil pitch, application of the line group symmetry [29] and related symmetry based techniques [30] as well as helical quantum number representation of phonon modes were essential for carrying out numerical computation1. Symmetry of HCCNTs [19], described by a 5th family LG L ¼ TQ ðFÞD1 , which is generated by helical transformation ðCQ jFÞ (i.e. rotation for 2p=Q followed by fractional translation
F) and by p-rotation around twofold horizontal axis (U-axis, Fig. 1). Generally, parameters Q and F (which is determined by the inclination angle v and monomer length a of the HCCNT: F ¼ a sin v) are real and only in the special case of the rational Q value (i.e. Q ¼ q=r, where q and r are co-primes satisfying condition q > r), there is a translational symmetry with period A ¼ qF. However, even in these special cases, helical quantum number representation of the phonon dispersions is more convenient since unit cell is always very large. In addition, helical angular momentum m is conserved in Umklapp processes, regardless of whether reduced or extended Brillouin zone scheme is applied [29]. Helical wave vector k reflects helical periodicity and runs over the irreducible domain ½0; p=F. Full symmetry assigned dispersion curves, when shown over the irreducible domain, correspond to doubly degenerated phonon states. On the other hand, first Brillouin zone is given by the interval ðp=F; p=F and C-point and edge-point phonon states are characterised by a parity with respect to the U-axis. Lack of the rotational symmetry implies vanishing helical angular momentum (m ¼ 0). Although helical quantum number representation2 is used, total number of phonon branches is still very large: three times the number of atoms within a monomer, nF . Therefore, for clarity, in Fig. 2, displayed is only the low energy part of phonon dispersions (from 0 to 501012 rad/s out of roughly 300 1012 rad=sÞ for the HCCNT with relatively small number of atoms in a monomer: nF ¼ 60. Being quasi-one dimensional systems, HCCNTs have four acoustic branches: In helical quantum numbers representa-
1 Quite recently, similar method which apply line group symmetry to empirical potential calculations has been used to study lattice vibrations in MoS2 NTs [31]. 2 ~ and m, ~ in order to distinguish from the linear quantum In the literature, helical quantum numbers are, by convention, denoted as k numbers. However, as herein we are dealing only with the helical quantum numbers, tilde symbol is omitted for convenience.
Please cite this article in press as: Popovic´ ZP et al. Phonon transport in helically coiled carbon nanotubes. Carbon (2014), http://dx.doi.org/ 10.1016/j.carbon.2014.05.031
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tion, transverse acoustic (TA) modes are characterized by finite wave number k ¼ 2p=Q (V point) and quadratic dispersion, while longitudinal acoustic (LA) and shear wave acoustic modes (SWA) are represented by k ¼ 0 (C point) and show linear dispersions (Fig. 2) with the sound velocities of the same intensity, vLA ¼ vSWA ¼ 3000 m/s, which is much lower than the sound velocities in SWCNTs [27]. However, in contrast to SWCNTs, the maximal acoustic band velocities in HCCNTs are not the sound velocities. As illustrated in Fig. 2, over a relatively large interval of the Brillouin zone, dispersions of the acoustic phonons are linear with the band velocity of, typically 9000 m/s, corresponding to the intensity of the TA sound velocity in SWCNTs [27] (whilst the TA acoustic 2 modes of HCCNTs have quadratic dispersion xTA ¼ ak ; a ¼ 106 m2 =s). Finally, the optical gap is an order of magnitude lower than in the case of straight SWCNTs.
3.
Phonon scattering
Different momentum non conserving scattering processes contribute to the thermal resistance: anharmonic phonon interaction, diffuse phonon scattering from rough boundaries, phonon-defect scattering, etc. In ideal cases (crystal structures without defects and without boundaries, e.g.) the intrinsic thermal conductivity is limited only by the phonon–phonon scattering due to the lattice anharmonicity which can be theoretically interpreted by considering threephonon Umklapp processes. Following the method of Klemens and Pedraza for evaluating theoretical values of phonon thermal conductivity (which was applied originally to single crystals of graphite [16] and later, by Balandin and co-workers [18], advanced and adapted for calculation of thermal transport in graphene) we derive expressions of intrinsic thermal conductivity of HCCNTs taking into account the three-phonon Umklapp scattering processes in which: (I) a phonon xðkÞ leaves the state k as being absorbed by another phonon from the heat flux, and (II) a phonon comes to the state k, due to the decay of a pho00 non from the state k . For these two types of Umklapp pro0 00 cesses energy conservation laws are: xðkÞ xðk Þ xðk Þ ¼ 0, where the upper sign corresponds to the first and lower sign to the second process. In Umklapp processes, in general, quasi-momenta and quasi-angular momenta are not conserved. However, unlike the case of the straight SWCNTs, where the scattering phase space is severely restricted [32] due to the chirality dependant quasi-angular momenta selection rules [29], there is no such a reduction of the scattering phase space of HCCNTs. Hence, three phonons which pertain to any combination of the branches can take part in an Umklapp process provided that the energy is conserved and the condition imposed on the 0 00 quasi-momenta is fulfilled: k k ¼ k þ K (where K is a reciprocal lattice vector). This is in contrast to the achiral SWCNTs where Umklapp scattering which involves three acoustic phonons is not allowed [32]. Consequently, while contribution of the normal scattering processes have consid-
erable impact on the phonon transport in the straight SWCNTs, they do not have the same effect in HCCNTs. Namely, as normal processes indirectly affect thermal conductivity through redistribution of the phonon modes, if they would not be taken into account, many triples of phonons of SWCNTs would be left trapped in the states which fulfill momenta and energy conserving conditions but do not satisfy the angular momenta selection rules. This however is not the case with HCCNTs, where all the phonon scattering channels are open. Out of the general forms for matrix elements of the threephonon interaction [17], we evaluate the expressions for the three-phonon Umklapp scattering rates in HCCNTs taking into account all the phonon branches and their dispersions as well as energy and momenta selection rules. At the temperature T, Umklapp scattering rate s1 U (where sU is the phonon relaxation time) of a phonon from the state km of the branch m and with energy h xm is calculated according to the following expression (upper/lower sign correspond to first/ second type of the Umklapp process): s1 U ðm; km ; TÞ ¼
2c2m hxm ðkm ÞF X X xa ðka Þxb ðkb Þ 3nF Mv2m ðkm Þ a;b ka ;kb 2X jDx0 ðka Þj
1 1 ffT ½xa ðka Þ fT ½xb ðkb Þ þ g: 2 2 The summations are taken over the all 3nF phonon dispersion branches and over the set X ¼ fk 2 ðp; p j Dx ðkÞ ¼ 0 ^ Dx0 ðkÞ – 0g; Dx ðka Þ ¼ xm ðkm Þ xa ðka Þ xb ðkb Þ; cm is Gru¨neisen parameter of the branch m (calculated is modedependant Gru¨neisen parameter which is then averaged over the each phonon branch3); M is a carbon atom mass; vm ðkÞ is the phonon group velocity; fT is Bose–Einstein equilibrium distribution function at temperature T. Gru¨neisen parameter cmk of an individual phonon mode xm ðkÞ is defined as the negative logarithmic derivative of the frequency of the mode with respect to the volume. Overall Gru¨neisen parameter for the acoustic branches is calculated as the weighted average of the individual Gru¨nisen parameters in which the contribution of each phonon is weighted by its contribution to the specific heat [34,16] and the typical value obtained for HCCNTs is 0.5 (which is for a factor 3 lower than in the case of SWCNTs). The coiled nanotubes do not have surface which can provide boundary scattering except for the nanotube ends which may lead to the backscattering. The backscattering rate s1 B , which is of relevance in the ballistic regime only, is calculated within relaxation time approximation: s1 B ¼ vm ð1 sÞ=L, where vm is the group velocity of the phonon branch m; L is NT length and s is adjustable parameter giving fraction of phonons which scatter specularly from the boundaries (the values zero and one correspond respectively to the absolutely rough and ideally smooth boundaries). Value of the s-parameter depends on the quality of the end-contacts [35] and in
3 In the previous theoretical studies of thermal conductivity of graphene [18] and BN NTs [33] mode-specific Gru¨neisen parameter gave better agreement with experiment.
Please cite this article in press as: Popovic´ ZP et al. Phonon transport in helically coiled carbon nanotubes. Carbon (2014), http://dx.doi.org/ 10.1016/j.carbon.2014.05.031
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our calculations nearly ideal contacts are assumed by taking s ¼ 0:98. The scattering rates we consider to be mutually independent and apply the Matthiessen’s rule, taking thus the overall scattering rate as a sum of the particular scattering rates.
4.
Thermal conductivity
Thermal conductivity j is the sum of the individual phonon conductivities jm ðkÞ: j¼
XX 1 XX jm ðkÞ ¼ cm ðkÞvm ðkÞKm ðkÞ V m k m k
ð1Þ
where cm ðkÞ ¼ hxm ðkÞ @@T f T ðxm ðkÞÞ is the phonon specific heat, Km ðkÞ ¼ jvm ðkÞjsm ðkÞ is the phonon MFP and the summation is taken over the entire phonon dispersions m and over the wave-vectors k of the Brillouin zone (p=F; p=F]. Calculations of thermal conductivity we perform on a sample of HCCNTs with helical diameters D 5 nm, helical — tubular radii ratio D : d from 4 to 5.5 and helical steps p between 2 and 6.2 nm. Taking for the volume of the monomer of HCCNT ˚ and r are the nanotube wall thickV ¼ 2prda, where d ¼ 3:4 A ness and tubular radius (averaged over the radii of the all orbits of the corrugated model of HCCNT, Fig. 1), replacing the summation over the 1D Brillouin zone with the integration over the irreducible domain of the reciprocal space, Eq. (1) takes the form: Z 1 X p=F j¼ 2 cm ðkÞvm ðkÞKm ðkÞ dk ð2Þ 4p rda m 0 In order to avoid the problem of vanishing scattering of the long-wave length phonons [36] (which makes the thermal conductivity to diverge as the heat conductor length increases) we introduce the cut-off frequencies for the dispersions around C and V points, (xC and xV ) exempting from the heat flow the phonons with infinite MFP. The cut-off frequencies are determined by analysing the wave-vector dependance of the MFP, Fig. 3. In all the cases considered it holds xC ðLAÞ > xC ðSWAÞ > xV ðTAÞ. Frequency dependance of the phonon MFPs for the acoustic branches of a (2,2,0,0, ((1,0), (0,5))) HCCNT, and of the LA, TA and TWA modes of a (7,1) SWCNT are presented in Fig. 4. In the calculations, only the Umklapp scattering rates are accounted for. Generally, phonon MFP in the coiled tubes is of the same order of magnitude as in the straight tubes. If long wave length phonons which are characterized by infinite MFP are excluded, MFP of the acoustic modes of vibration at room temperature spans to 10 lm. As the system is cooled down to 100 K, acoustic phonon MFP is predicted to reach 1 mm. According to the empirical Fourier law of heat conduction, which refers to the diffusive regime of heat transport (i.e. under condition that length of the conductor is considerably larger than the phonon MFP), the thermal conductivity j directly relates the heat energy flux U to the temperature gradient: U ¼ j rT. The Fourier law implicitly states that the thermal conductivity is independent of the length of the conductor. This was experimentally proved in all the cases of three-dimensional (3D) materials considered thus far. How-
Fig. 3 – Phonon MFP K vs. phonon wave-vector k at room temperature. Dotted horizontal line indicates MFP threshold K0 . Arrows point to the particular values of the cut-off frequencies, xC ðLAÞ; xC ðSWAÞ and xj ðTAÞ, which correspond to the given cut-off in the k-space.
Fig. 4 – Intrinsic MFP for the first two phonon branches of a (1,3,2,0, ((1,0), (0,5))) HCCNT at temperatures of 100 K and 300 K. Dashed lines show the room temperature intrinsic MFP of the acoustic modes of a (7,1) SWCNT. (A colour version of this figure can be viewed online.)
ever, there are many indications that the Fourier law in 1D systems does not hold. Theoretical investigation of the heat conduction in 1D chains of anharmonic oscillators suggests that the chaotic behavior in 1D is not enough to attenuate the free traveling long wave-length phonons [37]. From the experimental side, the breakdown of the Fourier law in carbon and boron-nitride nanotubes has been evidenced [4]: measured thermal conductivity data were fitted to a function jðLÞ Lb ; and the b parameter, which measures the deviation from the Fourier low behavior, was determined to be b 2 ½0:6; 0:8. According to our calculations, HCCNTs are yet another variety of 1D systems that violates the law of heat conduction. Fit to the numerically results at zero cut-off frequencies, (which is essentially equivalent to the long wavelength approximation) predicts b 2 ½0:4; 0:6 for HCCNTs and b 2 ½0:3; 0:5 for straight SWCNTs.
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In Fig. 5 presented is length dependance of the room temperature thermal conductivity of a coiled (1,3,2,0, ((1,0), (0,5))) CNT and of a straight (7,1) CNT having similar tubular diameters. Dashed lines show the results obtained in the long wavelength approximation and the parameters giving deviation from the Fourier low are b ¼ 0:3 (SWCNT) and b ¼ 0:5 (HCCNT). If the frequency cut-off is introduced, thermal conductivity saturates to finite value: j ¼ 3300 W/Km (matching the conductivity of the purified diamond [38]) at L > 500 lm (for the HCCNT) and j ¼ 4300 W/Km at L > 100 lm (for the SWCNT). The latter is in excellent agreement with the previously reported results [39] (j 4000 W/Km, L > 100 lm, T ¼ 316 K, calculated for (10, 0) NT). The relative thermal conductivity of HCCNTs with respect to the corresponding straight SWCNTs is length dependant. Heat transfer through the coiled NTs over the distance of 10 nm is ; 50% lower than through the straight counterparts. However, relative thermal conductivity of HCCNTs rapidly increases with the conductor length, being 70% at 100 nm and reaching the plateau of 75% at the lengths of the order of 1 mm. For the conductors shorter than phonon MFPs, scatteringfree phonon transport leads to linear dependance of the thermal conductivity with length. The lower the temperature, the larger is the phonon MFP (Fig. 4) and thus the limit of the ballistic conductor length is larger (Fig. 6). Modifications of the helical-tubular diameter ratio, D : d, induce the modifications of the phonon dispersions of HCCNTs which are further reflected in the thermal and mechanical properties. Namely, the lower D : d is, the higher are the both thermal conductivity and elastic modulus. As apart from the high thermal conductance it is important that theraml interface materials are also highly elastic, in order to quantify the optimal helical-tubular diameter ratio for heat-sinking applications of HCCNTs, thorough calculations of the both proper-
Fig. 5 – Room temperature thermal conductivity j of (1,3,2,0, ((1,0), (0,5))) HCCNT (black) and (7,1) SWCNT (gray) as a function of the conductor length L. Solid lines show the results obtained introducing the cut-off frequencies (Fig. 3), while long wave-length approximation is indicated by the dashed lines. Dotted line gives thermal conductivity of the HCCNT along the helical axis. Inset highlights the conductor length range: 0–10 lm.
Fig. 6 – Length dependance of thermal conductivity of (1,3,2,0, ((1,0), (0,5))) HCCNT (solid line) and (7,1) SWCNT (dashed line) at T ¼ 200 K (navy), T ¼ 300 K (olive) and T ¼ 400 K (wine). Dotted lines indicate thermal conductivity of HCCNT along the helical axis. (A colour version of this figure can be viewed online.)
ties are to be performed on a large and representative sample of the modeled HCCNTs. However, this task is beyond the scope of this paper and will be presented elsewhere. As illustrated in Fig. 7, HCCNTs with lower helical-tubular diameter ratio are characterized by higher acoustic phonon velocities which are further reflected in the higher thermal conductivity (under condition that other structural parameters are not changed notably).
4.1.
Thermal conductance
The heat flow rate through a single ballistic phonon channel at temperature T is given by the quantum of thermal conduc2 tance [40]: rth ðTÞ ¼ p2 kB T=3h. In the ballistic limit, thermal
Fig. 7 – Thermal conductivity of HCCNTs: (3,3,0,0, ((1,0), (0,5))) (red); (1,3,2,4, ((1,0), (0,5))) (green); (1,4,2,2, ((1,0), (0,5))) (blue); with different helical-tubular diameter ratio D : d, as a function of the length of the helix (solid line) and of the length of the helical axis (dotted line). Inset shows the corresponding low energy phonon dispersions. (A colour version of this figure can be viewed online.)
Please cite this article in press as: Popovic´ ZP et al. Phonon transport in helically coiled carbon nanotubes. Carbon (2014), http://dx.doi.org/ 10.1016/j.carbon.2014.05.031
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Fig. 8 – Temperature dependance of thermal conductance r for 1 lm long (2,2,0,0, ((1,0), (0,5))) HCCNT. Ballistic r T (red) and diffusive r T1 (blue) regimes are highlighted. Arrow indicates that two quanta of conductance do not extrapolate to zero at T ¼ 0 K. (A colour version of this figure can be viewed online.)
conductance r is an integer multiple of rth and thus more appropriate measure of the heat transport than thermal conductivity j. The conductance is related to the conductivity as: r ¼ ðS=LÞ j, where S is the area of the cross-section of the heat conductor. Temperature dependance of the thermal conductance of one micrometer long (2,2,0,0, ((1,0), (0,5))) HCCNT is shown in Fig. 8. The quantized ballistic conductance is characterized by a linear temperature dependance rðTÞ T and holds until T 30 K. At temperatures higher than several Kelvins, HCCNTs carry two quanta of thermal conductance, rHCCNT ¼ 2rth ðTÞ, i.e. a half of the energy flux through the straight SWCNTs [41] in a scattering-free transport regime. Only in the zero temperature limit HCCNTs have four transport channels, Fig. 8. The peak of the conductance is reached at T ¼ 55 K when a balance between specific heat increase and increased scattering rate is established. At higher temperatures scattering processes dominate and at T > 200 K diffusive transport regime rule rðTÞ T1 holds.
5.
Conclusion
Presented is the theoretical study of phonon heat conduction in helically coiled carbon nanotubes. Flexible atomistic model in which the coiling arises as an intrinsic property of the structure [42] and which matches the geometrical parameters of the as-synthesized HCCNTs is proposed. Phonon dispersions of HCCNTs are found to substantially differ from that of SWCNTs: The coiled tubes are characterized by narrow acoustic bands and by relatively low phonon group velocities. In addition, since lacking pure rotational symmetry, they allow for three-phonon Umklapp scattering of solely acoustic modes of vibration (by contrast to the straight SWCNTs, where such kind of processes are forbidden by selection rules). Due to the balance of the low anharmonicity of acoustic modes, on the one side, and lack of the restrictions on the type of the scattering channels, on the other side, MFP of the acoustic phonons in HCCNTs is of the same order
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as in the case of straight SWCNTs: from one to ten micrometers at room temperature. Room temperature thermal conductivity of HCCNTs is predicted to match that of the purified diamond as the nanotube length exceeds 0.5 mm. For many applications, vertically aligned CNT arrays are promising as thermal interface materials and thermal conductance is a crucial measure of their performance. However the optimal interfaces need to have a variety of other characteristics like e.g. high elasticity. In this respect, the coiled nanotubes are superior to the straight ones. Besides, HCCNTs naturally form VelcroTM-like contact [43] where the tubes mechanically entangle, so that the load transfer efficiency in composites [11] made of the coiled CNTs (rather than of the straight ones) is expected to be substantially larger (as the coiled shape spontaneously induces mechanical interlocking when the composite is subjected to loading). In this paper we gave insight into the mechanisms which govern intrinsic heat conduction in HCCNTs. However, in order to come to real applications, many questions are still to be resolved. One of these, a heat dissipation along the nanotubes and from the nanotubes to the substrate, is critical to developing and improving CNT-based electronic devices [44,45].
Acknowledgments The authors acknowledge funding of Serbian Ministry of Science (ON171035) and Swiss National Science Foundation (SCOPES IZ73Z0–128037/1).
Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/ j.carbon.2014.05.031.
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x x x ( 2 0 1 4 ) x x x –x x x
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Phonon transport in helically coiled carbon nanotubes Zoran P. Popovic´, Milan Damnjanovic´, Ivanka Milosˇevic´
*
University of Belgrade, Faculty of Physics, Studentski trg 12, 11001 Belgrade, Serbia
A R T I C L E I N F O
A B S T R A C T
Article history:
We perform theoretical studies on the phonon thermal transport in helically coiled carbon
Received 31 March 2014 Accepted 13 May 2014
nanotubes (HCCNTs). The Gru¨neisen parameter, as a function of the phonon wave vector and phonon branch, is numerically evaluated for each vibrational mode, so that the
Available online xxxx
three-phonon Umklapp scattering rates can be calculated exactly by taking into account all allowed phonon relaxation channels. We considered wide temperature range and heat conductor lengths from nano- to macro-scale. We examine the crossover from ballistic to diffusive transport regime and impact of HCCNT geometrical parameters on their heat conduction. Thermal conductivity in HCCNTs is found to be slightly lower than that in single walled carbon nanotubes (SWCNTs). This is interpreted by the competition among three factors. Firstly, threefold reduction of the Gru¨neisen parameter for the acoustic branches. Secondly, lower phonon group velocities. Finally, availability of purely acoustic scattering channels. Nevertheless, HCCNTs are predicted to be more suitable (than SWCNTs) for thermal management applications due to their spring-like shape. HCCNTs are extremely elastic, natural NanoVelcro material. Ó 2014 Elsevier Ltd. All rights reserved.
1.
Introduction
Since the early work of Ruoff and Lorents [1] where high thermal conductivity of carbon nanotubes have been predicted on the basis of the high in-plane thermal conductivity of graphite and of bulk and thin film diamond, wealth of experimental and theoretical research of heat transport in carbon nanotubes and related carbon nanostructured materials have been done [2]. It has been experimentally evidenced that, at all temperatures, thermal conductivity of single walled carbon nanotubes (SWCNTs) is dominated by phonons rather than electrons [3] and that in nanotube thermal conductors Fourier empirical law of thermal conduction is violated even when the phonon mean free path (MFP) is much shorter than the nanotube length [4].
However, with the exception of the recent reports on the molecular dynamics calculations of thermal conductivity [5] and thermal expansion [6] of helically coiled carbon nanotubes (HCCNTs) [7], theoretical research of their thermal properties is lacking, although the first synthesis [8,9] of the coiled nanotubes followed soon after the discovery of the straight ones [10]. Presumably, the lack of large-scale synthesis method of coiled CNTs over the past decades has hindered extensive research of their precious properties and further realizations of their potentials for the nano-technological applications [11–13]. However, several recent reports on the catalytic chemical vapor deposition production of the HCCNTs with controlled morphology [14,15] substantially improve the prospects of fundamental and applied physics research of these peculiar nanostructures.
* Corresponding author. E-mail address: [email protected] (I. Milosˇevic´). http://dx.doi.org/10.1016/j.carbon.2014.05.031 0008-6223/Ó 2014 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Popovic´ ZP et al. Phonon transport in helically coiled carbon nanotubes. Carbon (2014), http://dx.doi.org/ 10.1016/j.carbon.2014.05.031
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Here, using the upgraded Klemens treatment of heat conduction in basal planes of graphite [16–18] we perform a detail theoretical study of the intrinsic thermal conductivity of HCCNTs. Used is the model of HCCNTs based on the topological coordinate method [19,20] which has significant advantages over the scheme proposed in the pioneering work of Ihara
and Itoh [21]. Firstly, it allows for almost continual variation of a single geometrical parameter while all the other parameters are kept fixed. Secondly, it matches geometrical parameters of the as-synthesized HCCNTs [22,23]. Finally, apart from the previously defined smooth helical structure [19] we here introduce also corrugated model of HCCNTs (Fig. 1), as both circular and polygon-like shapes of the cross-sections
Fig. 1 – Model of HCCNTs. (a) Symmetry generating elements (helical rotational angle 2p=Q which is followed by a fractional translation F and twofold horizontal rotational axis, U-axis, which interlinks pairs of atoms within a monomer) and geometrical parameters of HCCNTs (tubular diameter d, coil pitch p, inclination angle v and helical diameter D); as a guide to the eyes one half of the monomer is highlighted; (b) Cohesive energies (upper panel) and relative difference of tubular radii (lower panel) of the corrugated and un-corrugated models of HCCNTs as a function of the graph parameter nr ; (c) On the left side: Initial LR triple connected graph (n6=1, ((3,0), (0,3))) of pentagons (black), hexagons (white) and heptagons (dark grey) [20]. Super-cell ((3,0), (0,3)) is shown; rectangle denotes its unit cell. On the right side: Augmented graph (1,2,2,2, ((3,0), (0,2))). Super-cell ((3,0), (0,2)) is shown; rectangle denotes its unit cell. Inserted are nr ¼ 2 rows of hexagons (between initial unit cells), n5 ¼ 2 columns of hexagons (between the pairs of pentagons within a unit cell) and n7 ¼ 2 columns of hexagons (between the pairs of heptagons of adjacent unit cells) [19]. Please cite this article in press as: Popovic´ ZP et al. Phonon transport in helically coiled carbon nanotubes. Carbon (2014), http://dx.doi.org/ 10.1016/j.carbon.2014.05.031
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are evidenced in the as-synthesized samples of the coiled carbon nanotubes [7,24]. We perform calculations of thermal conductivity of HCCNTs of finite lengths, from 10 nm to 10 mm. Apart from the anharmonic three-phonon Umklapp processes which are treated exactly, for all allowed phonon relaxation channels, backscattering is accounted for within relaxation time approximation. We also examine temperature dependant crossover from ballistic to diffusive transport regime, as well as dependance of thermal conductivity on geometrical parameters of HCCNTs. The breakdown of the Fourier law of heat conduction which was evidenced in SWCNTs [4] we also predict for HCCNTs. Namely, we find that regularly distributed pentagonal and heptagonal ‘‘defects’’ do not to scatter the long-wave length phonons to the extent enough to remove the thermal conductivity divergence with nanotube length. We predict that thermal conductivity of HCCNTs amounts 75% of that of the corresponding SWCNTs as the coil length exceeds 500 lm. So good heat conducting properties in combination with the unique mechanical qualities [13,25] make HCCNTs promising candidates for thermal management applications.
2.
3
Fig. 2 – Low energy phonon dispersion relations for a (2,2,0,0, ((1,0), (0,5))) HCCNT, with geometrical parameters D ¼ 2:2 nm; d ¼ 0:5 nm; p ¼ 2:6 nm; v ¼ 210 . Acoustic branches (green), the lowest optical branch (blue) and scattering channels AAA (green solid line) and AAO (blue solid line) are highlighted. Three-phonon Umklapp scattering of the type I (II) is shown by solid (dashed) line. Red diamond marks the considered phonon. (A colour version of this figure can be viewed online.)
Phonon dispersion relations
We consider the model of HCCNTs which is obtained by rolling up a triple connected graph of pentagons, hexagons and heptagons, conventionally defined by a set of numbers ðn6 ; nr ; n7 ; n5 ; ðb1 ; b2 ÞÞ, where ðb1 ; b2 Þ are the super-cell vectors, while the tiling pattern is given by the first four parameters, Fig. 1. In addition to the relaxation procedure given in Ref. [19], tubular radial coordinates of carbon atoms are also varied and obtained is a model of corrugated HCCNTs. Comparison of the cohesive energies of the corrugated and uncorrugated HCCNTs together with the rate of corrugation are given in Fig. 1. Geometrical structure of HCCNTs is characterized by tubular diameter d, outer diameter of a helix D, helical step p and inclination angle v. Unlike the case of SWCNTs where phonon dispersion curves are evaluated using the graphite force constants [26] adopted to the nanotube geometry [27], dynamical sub-matrices of HCCNTs are numerically derived out of Brenner interatomic potential [28] by varying space coordinates of carbon atoms and calculating the corresponding energy changes. Since HCCNTs are complex systems which, in general, do not have translational periodicity and contain thousands of atoms within a single coil pitch, application of the line group symmetry [29] and related symmetry based techniques [30] as well as helical quantum number representation of phonon modes were essential for carrying out numerical computation1. Symmetry of HCCNTs [19], described by a 5th family LG L ¼ TQ ðFÞD1 , which is generated by helical transformation ðCQ jFÞ (i.e. rotation for 2p=Q followed by fractional translation
F) and by p-rotation around twofold horizontal axis (U-axis, Fig. 1). Generally, parameters Q and F (which is determined by the inclination angle v and monomer length a of the HCCNT: F ¼ a sin v) are real and only in the special case of the rational Q value (i.e. Q ¼ q=r, where q and r are co-primes satisfying condition q > r), there is a translational symmetry with period A ¼ qF. However, even in these special cases, helical quantum number representation of the phonon dispersions is more convenient since unit cell is always very large. In addition, helical angular momentum m is conserved in Umklapp processes, regardless of whether reduced or extended Brillouin zone scheme is applied [29]. Helical wave vector k reflects helical periodicity and runs over the irreducible domain ½0; p=F. Full symmetry assigned dispersion curves, when shown over the irreducible domain, correspond to doubly degenerated phonon states. On the other hand, first Brillouin zone is given by the interval ðp=F; p=F and C-point and edge-point phonon states are characterised by a parity with respect to the U-axis. Lack of the rotational symmetry implies vanishing helical angular momentum (m ¼ 0). Although helical quantum number representation2 is used, total number of phonon branches is still very large: three times the number of atoms within a monomer, nF . Therefore, for clarity, in Fig. 2, displayed is only the low energy part of phonon dispersions (from 0 to 501012 rad/s out of roughly 300 1012 rad=sÞ for the HCCNT with relatively small number of atoms in a monomer: nF ¼ 60. Being quasi-one dimensional systems, HCCNTs have four acoustic branches: In helical quantum numbers representa-
1 Quite recently, similar method which apply line group symmetry to empirical potential calculations has been used to study lattice vibrations in MoS2 NTs [31]. 2 ~ and m, ~ in order to distinguish from the linear quantum In the literature, helical quantum numbers are, by convention, denoted as k numbers. However, as herein we are dealing only with the helical quantum numbers, tilde symbol is omitted for convenience.
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tion, transverse acoustic (TA) modes are characterized by finite wave number k ¼ 2p=Q (V point) and quadratic dispersion, while longitudinal acoustic (LA) and shear wave acoustic modes (SWA) are represented by k ¼ 0 (C point) and show linear dispersions (Fig. 2) with the sound velocities of the same intensity, vLA ¼ vSWA ¼ 3000 m/s, which is much lower than the sound velocities in SWCNTs [27]. However, in contrast to SWCNTs, the maximal acoustic band velocities in HCCNTs are not the sound velocities. As illustrated in Fig. 2, over a relatively large interval of the Brillouin zone, dispersions of the acoustic phonons are linear with the band velocity of, typically 9000 m/s, corresponding to the intensity of the TA sound velocity in SWCNTs [27] (whilst the TA acoustic 2 modes of HCCNTs have quadratic dispersion xTA ¼ ak ; a ¼ 106 m2 =s). Finally, the optical gap is an order of magnitude lower than in the case of straight SWCNTs.
3.
Phonon scattering
Different momentum non conserving scattering processes contribute to the thermal resistance: anharmonic phonon interaction, diffuse phonon scattering from rough boundaries, phonon-defect scattering, etc. In ideal cases (crystal structures without defects and without boundaries, e.g.) the intrinsic thermal conductivity is limited only by the phonon–phonon scattering due to the lattice anharmonicity which can be theoretically interpreted by considering threephonon Umklapp processes. Following the method of Klemens and Pedraza for evaluating theoretical values of phonon thermal conductivity (which was applied originally to single crystals of graphite [16] and later, by Balandin and co-workers [18], advanced and adapted for calculation of thermal transport in graphene) we derive expressions of intrinsic thermal conductivity of HCCNTs taking into account the three-phonon Umklapp scattering processes in which: (I) a phonon xðkÞ leaves the state k as being absorbed by another phonon from the heat flux, and (II) a phonon comes to the state k, due to the decay of a pho00 non from the state k . For these two types of Umklapp pro0 00 cesses energy conservation laws are: xðkÞ xðk Þ xðk Þ ¼ 0, where the upper sign corresponds to the first and lower sign to the second process. In Umklapp processes, in general, quasi-momenta and quasi-angular momenta are not conserved. However, unlike the case of the straight SWCNTs, where the scattering phase space is severely restricted [32] due to the chirality dependant quasi-angular momenta selection rules [29], there is no such a reduction of the scattering phase space of HCCNTs. Hence, three phonons which pertain to any combination of the branches can take part in an Umklapp process provided that the energy is conserved and the condition imposed on the 0 00 quasi-momenta is fulfilled: k k ¼ k þ K (where K is a reciprocal lattice vector). This is in contrast to the achiral SWCNTs where Umklapp scattering which involves three acoustic phonons is not allowed [32]. Consequently, while contribution of the normal scattering processes have consid-
erable impact on the phonon transport in the straight SWCNTs, they do not have the same effect in HCCNTs. Namely, as normal processes indirectly affect thermal conductivity through redistribution of the phonon modes, if they would not be taken into account, many triples of phonons of SWCNTs would be left trapped in the states which fulfill momenta and energy conserving conditions but do not satisfy the angular momenta selection rules. This however is not the case with HCCNTs, where all the phonon scattering channels are open. Out of the general forms for matrix elements of the threephonon interaction [17], we evaluate the expressions for the three-phonon Umklapp scattering rates in HCCNTs taking into account all the phonon branches and their dispersions as well as energy and momenta selection rules. At the temperature T, Umklapp scattering rate s1 U (where sU is the phonon relaxation time) of a phonon from the state km of the branch m and with energy h xm is calculated according to the following expression (upper/lower sign correspond to first/ second type of the Umklapp process): s1 U ðm; km ; TÞ ¼
2c2m hxm ðkm ÞF X X xa ðka Þxb ðkb Þ 3nF Mv2m ðkm Þ a;b ka ;kb 2X jDx0 ðka Þj
1 1 ffT ½xa ðka Þ fT ½xb ðkb Þ þ g: 2 2 The summations are taken over the all 3nF phonon dispersion branches and over the set X ¼ fk 2 ðp; p j Dx ðkÞ ¼ 0 ^ Dx0 ðkÞ – 0g; Dx ðka Þ ¼ xm ðkm Þ xa ðka Þ xb ðkb Þ; cm is Gru¨neisen parameter of the branch m (calculated is modedependant Gru¨neisen parameter which is then averaged over the each phonon branch3); M is a carbon atom mass; vm ðkÞ is the phonon group velocity; fT is Bose–Einstein equilibrium distribution function at temperature T. Gru¨neisen parameter cmk of an individual phonon mode xm ðkÞ is defined as the negative logarithmic derivative of the frequency of the mode with respect to the volume. Overall Gru¨neisen parameter for the acoustic branches is calculated as the weighted average of the individual Gru¨nisen parameters in which the contribution of each phonon is weighted by its contribution to the specific heat [34,16] and the typical value obtained for HCCNTs is 0.5 (which is for a factor 3 lower than in the case of SWCNTs). The coiled nanotubes do not have surface which can provide boundary scattering except for the nanotube ends which may lead to the backscattering. The backscattering rate s1 B , which is of relevance in the ballistic regime only, is calculated within relaxation time approximation: s1 B ¼ vm ð1 sÞ=L, where vm is the group velocity of the phonon branch m; L is NT length and s is adjustable parameter giving fraction of phonons which scatter specularly from the boundaries (the values zero and one correspond respectively to the absolutely rough and ideally smooth boundaries). Value of the s-parameter depends on the quality of the end-contacts [35] and in
3 In the previous theoretical studies of thermal conductivity of graphene [18] and BN NTs [33] mode-specific Gru¨neisen parameter gave better agreement with experiment.
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our calculations nearly ideal contacts are assumed by taking s ¼ 0:98. The scattering rates we consider to be mutually independent and apply the Matthiessen’s rule, taking thus the overall scattering rate as a sum of the particular scattering rates.
4.
Thermal conductivity
Thermal conductivity j is the sum of the individual phonon conductivities jm ðkÞ: j¼
XX 1 XX jm ðkÞ ¼ cm ðkÞvm ðkÞKm ðkÞ V m k m k
ð1Þ
where cm ðkÞ ¼ hxm ðkÞ @@T f T ðxm ðkÞÞ is the phonon specific heat, Km ðkÞ ¼ jvm ðkÞjsm ðkÞ is the phonon MFP and the summation is taken over the entire phonon dispersions m and over the wave-vectors k of the Brillouin zone (p=F; p=F]. Calculations of thermal conductivity we perform on a sample of HCCNTs with helical diameters D 5 nm, helical — tubular radii ratio D : d from 4 to 5.5 and helical steps p between 2 and 6.2 nm. Taking for the volume of the monomer of HCCNT ˚ and r are the nanotube wall thickV ¼ 2prda, where d ¼ 3:4 A ness and tubular radius (averaged over the radii of the all orbits of the corrugated model of HCCNT, Fig. 1), replacing the summation over the 1D Brillouin zone with the integration over the irreducible domain of the reciprocal space, Eq. (1) takes the form: Z 1 X p=F j¼ 2 cm ðkÞvm ðkÞKm ðkÞ dk ð2Þ 4p rda m 0 In order to avoid the problem of vanishing scattering of the long-wave length phonons [36] (which makes the thermal conductivity to diverge as the heat conductor length increases) we introduce the cut-off frequencies for the dispersions around C and V points, (xC and xV ) exempting from the heat flow the phonons with infinite MFP. The cut-off frequencies are determined by analysing the wave-vector dependance of the MFP, Fig. 3. In all the cases considered it holds xC ðLAÞ > xC ðSWAÞ > xV ðTAÞ. Frequency dependance of the phonon MFPs for the acoustic branches of a (2,2,0,0, ((1,0), (0,5))) HCCNT, and of the LA, TA and TWA modes of a (7,1) SWCNT are presented in Fig. 4. In the calculations, only the Umklapp scattering rates are accounted for. Generally, phonon MFP in the coiled tubes is of the same order of magnitude as in the straight tubes. If long wave length phonons which are characterized by infinite MFP are excluded, MFP of the acoustic modes of vibration at room temperature spans to 10 lm. As the system is cooled down to 100 K, acoustic phonon MFP is predicted to reach 1 mm. According to the empirical Fourier law of heat conduction, which refers to the diffusive regime of heat transport (i.e. under condition that length of the conductor is considerably larger than the phonon MFP), the thermal conductivity j directly relates the heat energy flux U to the temperature gradient: U ¼ j rT. The Fourier law implicitly states that the thermal conductivity is independent of the length of the conductor. This was experimentally proved in all the cases of three-dimensional (3D) materials considered thus far. How-
Fig. 3 – Phonon MFP K vs. phonon wave-vector k at room temperature. Dotted horizontal line indicates MFP threshold K0 . Arrows point to the particular values of the cut-off frequencies, xC ðLAÞ; xC ðSWAÞ and xj ðTAÞ, which correspond to the given cut-off in the k-space.
Fig. 4 – Intrinsic MFP for the first two phonon branches of a (1,3,2,0, ((1,0), (0,5))) HCCNT at temperatures of 100 K and 300 K. Dashed lines show the room temperature intrinsic MFP of the acoustic modes of a (7,1) SWCNT. (A colour version of this figure can be viewed online.)
ever, there are many indications that the Fourier law in 1D systems does not hold. Theoretical investigation of the heat conduction in 1D chains of anharmonic oscillators suggests that the chaotic behavior in 1D is not enough to attenuate the free traveling long wave-length phonons [37]. From the experimental side, the breakdown of the Fourier law in carbon and boron-nitride nanotubes has been evidenced [4]: measured thermal conductivity data were fitted to a function jðLÞ Lb ; and the b parameter, which measures the deviation from the Fourier low behavior, was determined to be b 2 ½0:6; 0:8. According to our calculations, HCCNTs are yet another variety of 1D systems that violates the law of heat conduction. Fit to the numerically results at zero cut-off frequencies, (which is essentially equivalent to the long wavelength approximation) predicts b 2 ½0:4; 0:6 for HCCNTs and b 2 ½0:3; 0:5 for straight SWCNTs.
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In Fig. 5 presented is length dependance of the room temperature thermal conductivity of a coiled (1,3,2,0, ((1,0), (0,5))) CNT and of a straight (7,1) CNT having similar tubular diameters. Dashed lines show the results obtained in the long wavelength approximation and the parameters giving deviation from the Fourier low are b ¼ 0:3 (SWCNT) and b ¼ 0:5 (HCCNT). If the frequency cut-off is introduced, thermal conductivity saturates to finite value: j ¼ 3300 W/Km (matching the conductivity of the purified diamond [38]) at L > 500 lm (for the HCCNT) and j ¼ 4300 W/Km at L > 100 lm (for the SWCNT). The latter is in excellent agreement with the previously reported results [39] (j 4000 W/Km, L > 100 lm, T ¼ 316 K, calculated for (10, 0) NT). The relative thermal conductivity of HCCNTs with respect to the corresponding straight SWCNTs is length dependant. Heat transfer through the coiled NTs over the distance of 10 nm is ; 50% lower than through the straight counterparts. However, relative thermal conductivity of HCCNTs rapidly increases with the conductor length, being 70% at 100 nm and reaching the plateau of 75% at the lengths of the order of 1 mm. For the conductors shorter than phonon MFPs, scatteringfree phonon transport leads to linear dependance of the thermal conductivity with length. The lower the temperature, the larger is the phonon MFP (Fig. 4) and thus the limit of the ballistic conductor length is larger (Fig. 6). Modifications of the helical-tubular diameter ratio, D : d, induce the modifications of the phonon dispersions of HCCNTs which are further reflected in the thermal and mechanical properties. Namely, the lower D : d is, the higher are the both thermal conductivity and elastic modulus. As apart from the high thermal conductance it is important that theraml interface materials are also highly elastic, in order to quantify the optimal helical-tubular diameter ratio for heat-sinking applications of HCCNTs, thorough calculations of the both proper-
Fig. 5 – Room temperature thermal conductivity j of (1,3,2,0, ((1,0), (0,5))) HCCNT (black) and (7,1) SWCNT (gray) as a function of the conductor length L. Solid lines show the results obtained introducing the cut-off frequencies (Fig. 3), while long wave-length approximation is indicated by the dashed lines. Dotted line gives thermal conductivity of the HCCNT along the helical axis. Inset highlights the conductor length range: 0–10 lm.
Fig. 6 – Length dependance of thermal conductivity of (1,3,2,0, ((1,0), (0,5))) HCCNT (solid line) and (7,1) SWCNT (dashed line) at T ¼ 200 K (navy), T ¼ 300 K (olive) and T ¼ 400 K (wine). Dotted lines indicate thermal conductivity of HCCNT along the helical axis. (A colour version of this figure can be viewed online.)
ties are to be performed on a large and representative sample of the modeled HCCNTs. However, this task is beyond the scope of this paper and will be presented elsewhere. As illustrated in Fig. 7, HCCNTs with lower helical-tubular diameter ratio are characterized by higher acoustic phonon velocities which are further reflected in the higher thermal conductivity (under condition that other structural parameters are not changed notably).
4.1.
Thermal conductance
The heat flow rate through a single ballistic phonon channel at temperature T is given by the quantum of thermal conduc2 tance [40]: rth ðTÞ ¼ p2 kB T=3h. In the ballistic limit, thermal
Fig. 7 – Thermal conductivity of HCCNTs: (3,3,0,0, ((1,0), (0,5))) (red); (1,3,2,4, ((1,0), (0,5))) (green); (1,4,2,2, ((1,0), (0,5))) (blue); with different helical-tubular diameter ratio D : d, as a function of the length of the helix (solid line) and of the length of the helical axis (dotted line). Inset shows the corresponding low energy phonon dispersions. (A colour version of this figure can be viewed online.)
Please cite this article in press as: Popovic´ ZP et al. Phonon transport in helically coiled carbon nanotubes. Carbon (2014), http://dx.doi.org/ 10.1016/j.carbon.2014.05.031
CARBON
xxx (2014) xxx–xxx
Fig. 8 – Temperature dependance of thermal conductance r for 1 lm long (2,2,0,0, ((1,0), (0,5))) HCCNT. Ballistic r T (red) and diffusive r T1 (blue) regimes are highlighted. Arrow indicates that two quanta of conductance do not extrapolate to zero at T ¼ 0 K. (A colour version of this figure can be viewed online.)
conductance r is an integer multiple of rth and thus more appropriate measure of the heat transport than thermal conductivity j. The conductance is related to the conductivity as: r ¼ ðS=LÞ j, where S is the area of the cross-section of the heat conductor. Temperature dependance of the thermal conductance of one micrometer long (2,2,0,0, ((1,0), (0,5))) HCCNT is shown in Fig. 8. The quantized ballistic conductance is characterized by a linear temperature dependance rðTÞ T and holds until T 30 K. At temperatures higher than several Kelvins, HCCNTs carry two quanta of thermal conductance, rHCCNT ¼ 2rth ðTÞ, i.e. a half of the energy flux through the straight SWCNTs [41] in a scattering-free transport regime. Only in the zero temperature limit HCCNTs have four transport channels, Fig. 8. The peak of the conductance is reached at T ¼ 55 K when a balance between specific heat increase and increased scattering rate is established. At higher temperatures scattering processes dominate and at T > 200 K diffusive transport regime rule rðTÞ T1 holds.
5.
Conclusion
Presented is the theoretical study of phonon heat conduction in helically coiled carbon nanotubes. Flexible atomistic model in which the coiling arises as an intrinsic property of the structure [42] and which matches the geometrical parameters of the as-synthesized HCCNTs is proposed. Phonon dispersions of HCCNTs are found to substantially differ from that of SWCNTs: The coiled tubes are characterized by narrow acoustic bands and by relatively low phonon group velocities. In addition, since lacking pure rotational symmetry, they allow for three-phonon Umklapp scattering of solely acoustic modes of vibration (by contrast to the straight SWCNTs, where such kind of processes are forbidden by selection rules). Due to the balance of the low anharmonicity of acoustic modes, on the one side, and lack of the restrictions on the type of the scattering channels, on the other side, MFP of the acoustic phonons in HCCNTs is of the same order
7
as in the case of straight SWCNTs: from one to ten micrometers at room temperature. Room temperature thermal conductivity of HCCNTs is predicted to match that of the purified diamond as the nanotube length exceeds 0.5 mm. For many applications, vertically aligned CNT arrays are promising as thermal interface materials and thermal conductance is a crucial measure of their performance. However the optimal interfaces need to have a variety of other characteristics like e.g. high elasticity. In this respect, the coiled nanotubes are superior to the straight ones. Besides, HCCNTs naturally form VelcroTM-like contact [43] where the tubes mechanically entangle, so that the load transfer efficiency in composites [11] made of the coiled CNTs (rather than of the straight ones) is expected to be substantially larger (as the coiled shape spontaneously induces mechanical interlocking when the composite is subjected to loading). In this paper we gave insight into the mechanisms which govern intrinsic heat conduction in HCCNTs. However, in order to come to real applications, many questions are still to be resolved. One of these, a heat dissipation along the nanotubes and from the nanotubes to the substrate, is critical to developing and improving CNT-based electronic devices [44,45].
Acknowledgments The authors acknowledge funding of Serbian Ministry of Science (ON171035) and Swiss National Science Foundation (SCOPES IZ73Z0–128037/1).
Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/ j.carbon.2014.05.031.
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Please cite this article in press as: Popovic´ ZP et al. Phonon transport in helically coiled carbon nanotubes. Carbon (2014), http://dx.doi.org/ 10.1016/j.carbon.2014.05.031