Heat conduction in coaxial nanocables of Au nanowire core and carbon nanotube shell: A molecular dynamics simulation

International Journal of Thermal Sciences 99 (2016) 64e70

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Heat conduction in coaxial nanocables of Au nanowire core and carbon nanotube shell: A molecular dynamics simulation Liu Cui a, Yanhui Feng a, b, *, Jingjing Tang a, Peng Tan a, Xinxin Zhang a, b a

School of Mechanical Engineering, University of Science and Technology Beijing, Beijing 100083, China Beijing Key Laboratory of Energy Saving and Emission Reduction for Metallurgical Industry, University of Science and Technology Beijing, Beijing 100083, China b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 12 November 2014 Received in revised form 8 June 2015 Accepted 12 August 2015 Available online xxx

Non-equilibrium molecular dynamics simulations have been employed to calculate the thermal conductivity of coaxial nanocables of Au nanowire core and carbon nanotube shell, i.e. nanotubes filled with nanowires. Our efforts are focused on how the thermal conductivity can be altered in nanocables. By performing analysis on the phonon vibrational density of states, we have revealed that the thermal conductivity of nanocables is 20-42% higher than the corresponding bare nanotubes, due to the interactions of CeC and CeAu and the mass transfer induced by nanowire axial motion. The dependences of thermal conductivity on the temperature, length, diameter, chirality and filling ratio have been investigated. It turns out that the tendencies of nanocable thermal conductivity changing with temperature, length and diameter are similar to bare nanotubes. In addition, the thermal conductivity of nanocables always goes up with the increasing filling ratio. For different chirality types, the zigzag (18, 0) nanocable has the largest thermal conductivity increments, followed by armchair (10, 10) and chiral (14, 7) nanocables. © 2015 Elsevier Masson SAS. All rights reserved.

Keywords: Carbon nanotubes Coaxial nanocables Molecular dynamics Thermal conductivity Phonon vibrational density of states

1. Introduction Since discovered in 1991 [1] carbon nanotubes (CNTs) have attracted a great deal of attention. Many investigations have indicated that CNTs possess various excellent properties including, for example, high Young's modulus [2], good field emission properties [3] and superior thermal conductivity [4]. CNTs are being considered as one promising candidate material for nanoscale device applications. Due to the unique quasi one-dimensional hollow structures, CNTs could be filled with other materials. In 1993, Ajayan and Iijima [5] firstly encapsulated Pt into CNTs. This immediately aroused widespread attention. Various studies have demonstrated that CNTs can be encapsulated with metals [6,7] and even fullerenes [8]. And the filled materials would have a great impact on CNT properties. Borowiak-Palen et al. [6] used a wet chemistry method to fill Fe into single-walled carbon nanotubes (SWCNTs) and found that Fe filled SWCNTs shown ferromagnetic behaviour even at room temperature. Kumar et al. [7] embedded tin

* Corresponding author. School of Mechanical Engineering, University of Science and Technology Beijing, Beijing 100083, China. Tel./fax: þ86 010 62334971. E-mail address: [email protected] (Y. Feng). http://dx.doi.org/10.1016/j.ijthermalsci.2015.08.004 1290-0729/© 2015 Elsevier Masson SAS. All rights reserved.

into multi-walled carbon nanotubes (MWCNTs) through a catalytic decomposition method. They revealed that the reversible capacities of the tin filled CNTs were remarkably high, stabilizing in the 720e800 mAh/g region over the first 20 cycles. Jo et al. [9] investigated the electronic and magnetic properties of Fe, Co, and Ni nanowires (NWs) encapsulated in SWCNTs through the spin polarized ab initio calculation. Their results indicated that the reduction of magnetic moment for Fe NWs filled SWCNTs was relatively smaller than that for Co and Ni NWs. However, there were few studies investigated the thermal conductivity of filled CNTs. The thermal conductivity of a material is a measure of how fast heat will flow in that material [10]. Vavro et al. [8] measured the thermal conductivity of buckypapers of carbon nanopeapods (CNTs embedded with fullerene C60 molecules). According to their experiments, there was little or no contribution by the C60 chains to the thermal conductivity of filled tubes. They pointed out three mechanisms and suggested that more accurate data and individual tube experiments were necessary to assess the relative importance of the various mechanisms. Noya et al. [11] and Kawamura et al. [12] computed the thermal conductivity of an individual carbon nanopeapod using molecular dynamics simulations. Both them indicated that thermal

L. Cui et al. / International Journal of Thermal Sciences 99 (2016) 64e70

conductivity of the carbon nanopeapod was higher than CNT due to the motion of C60 molecules. Noya's results showed that the thermal conductivity of the carbon nanopeapod increased first and then decreased with rising temperature. However, Kawamura's results turned out just to the contrary. Toprak et al. [13] predicted that the thermal conductivity of Cu NWs filled SWCNTs was 24% higher than the corresponding SWCNTs and estimated to be 40% lower than pure copper NWs. The length dependence of thermal conductivity for the Cu NWs filled CNTs was similar to that of analogous bare SWCNTs and Cu NWs whereas the length increased, the thermal conductivity also increased. The composite of CNT filled with NW can be named as coaxial nanocable [14], with the NW representing the conducting core and the CNT representing the insulating sheath. The coaxial nanocable of Au NW core and CNT shell was systematically investigated in this work, with the aim to reveal the dependence of thermal conductivity on temperatures, lengths, diameters, chiralities and filling ratios. Using the phonon vibrational density of states (VDOS), the heat transfer mechanisms for the change in the thermal conductivity of nanocables were also explored. This study is an attempt to explore possible structures and thermal properties to satisfy requirements of different nanoscale devices, such as thermoelectric devices requiring strongly suppressed thermal conductivity, whereas electronic or optoelectronic devices demanding efficient heat removal. 2. Method 2.1. Model structures The simulation models of coaxial nanocables are CNTs embedded by gold NWs. It is know that bulk gold crystallizes in face-centered cubic (fcc) lattices. However, the gold NWs filled in CNTs are found to have “weird” structures that differ from the crystalline bulk. They spontaneously exhibit multishell packs consisting of coaxial cylindrical shells [15]. Each shell consists of helical atom rows coiled round the wire axis and can be viewed as a triangular (1 1 1) lattice sheet that is folded on to itself to form a cylinder. KT-indices nen0 en00 [16] were employed to describe a nanowire consisting of coaxial shells with n, n0 and n00 helical atom rows (n > n0 > n00 ), which can be counted at the cross section. For example, an SWCNT with the chirality of (10, 10) is filled with a

65

same length gold NW, as shown in Fig. 1. The corresponding KTindex of the filled gold NW is 9-3. That is, the outer and inner gold shells are composed of 9 and 3 atom rows, respectively. Similar behaviour was also observed by Xiao et al. [17]. 2.2. Thermal conductivity Heat conduction actually results from the random motion of the carriers in the solid system transporting thermal energy from one location to another. The heat carriers include electrons, atoms and molecules in gases, liquids and solids. In dielectric solid materials like CNTs, heat is conducted through the vibration of atoms. The discrete units of vibrational energy that arises from oscillating atoms within a crystal can be defined as phonons [18]. That is, contributions to CNT thermal conductivity mostly come from phonons (i.e. phonon thermal conductivity, PTC). In metals such as Au NWs, the thermal conductivity is usually dominated by electrons (i.e. electronic thermal conductivity, ETC). In their composites, i.e. nanocables, both PTC and ETC should be taken into consideration. 2.2.1. Phonon thermal conductivity Non-equilibrium molecular dynamics (NEMD) simulations have been proven to be a very useful technique to predict thermal transport properties of nanostructured materials [19e22]. In this work, NEMD simulations were used to calculate the PTC of CNTs and nanocables. The periodic boundary condition was used in the longitudinal direction of CNTs. The CeC bonding interactions among CNT atoms were determined by the Tersoff potential [23]. The embedded atom method (EAM) potential [24] was used to describe the AueAu interactions. The CeAu interactions were assumed as the LeJ potential, and LeJ size and energy parameters are 0.29943 nm and 0.01273 eV [25], respectively. In this work, the Muller-Plathe method [26] was used to establish a temperature gradient parallel to the tube axis. The CNTs and nanocables were axially partitioned into 40 slabs of equal length for temperature recording and control. As illustrated in Fig. 1, the first slab and the 21st slab were chosen as heat sink and heat source, respectively. A heat flux transferred between these two slabs through exchanging momentum between the ‘hottest’ atom in heat sink and the ‘coldest’ atom in the heat source. The

Fig. 1. Model of the coaxial nanocable of Au nanowire core and carbon nanotube shell.

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momentum exchanging was performed every 10 fs. The heat flux J (W/m2) can be accurately computed through

P J¼

transfer

m 2



v2h  v2c

 (1)

2At

where t (s) is the simulation time, m (kg) is the mass of the atoms, nh and nc (m/s) are the velocities of selected atoms in heat source and heat sink, respectively. A (m2) is the circular cross-sectional area and calculated by using the external radius of the CNT, i.e. the average CNT radius plus half the CNT wall thickness. The wall thickness of CNTs is equal to the interlayer spacing of graphite (0.34 nm) [27]. After the system reached equilibrium, that is, the heat transferred along the nanocable between the two regions equaled the heat transfer created by the velocity switching, a temperature gradient dT/dz (K/m) was obtained, as illustrated in Fig. 2. Then, the PTC can be calculated by linear-fitting of the temperature distribution according to Fourier's law

J ¼ k

transport equation (see the reference [33] for mathematical details) and WiedemanneFranz law [31].

dT dz

s 12 z ¼ 1 p kbulk sbulk ke

Temperature [K]

450 400

0

0

where ke and kbulk (W/m$K) are the respective ETC of NWs and bulk materials. s and sbulk (S/m) are electrical conductivity of NWs and bulk materials, respectively. g ¼ d/Le, where d (m) is the equivalent diameter of NWs. Le (m) is the mean free path of electrons in the bulk materials. Le values of bulk Au at different temperatures are taken from the work of Qiu and Tien [34]. 2.3. Phonon vibrational density of states The VDOS is proportional to the Fourier-transform of the velocity autocorrelation function (VACF) averaged over all atoms [35,36]. Thus, the VDOS can be given by

Zþ∞ VDOSðvÞ ¼

VACFðtÞe2pivt dt

(4)

∞

where n (Hz) is the frequency of phonons, t (s) is the simulation time. And for a system contains N atoms, the VACF can be calculated as

* VACFðtÞ ¼

2.2.2. Electronic thermal conductivity Generally, the ETC is deduced from the electrical conductivity according to the WiedemanneFranz law [31]. The WiedemanneFranz law is valid for materials in bulk form. However, Kumar and Vradis [32] pointed out that, in the first-order approximation, it is also suitable for metallic materials in micro- and nano-scale. In the same way as Kumar and Vradis used [32], this work was basing upon Dingle's work [33] on the electrical conductivity of metallic NWs. The gold NWs are assumed to be free of any grain boundaries or impurities. The elastic scattering and diffuse reflections on the nanowire surface are taken into account. The ETC of gold NWs with a circular cross section was given by the solution of Boltzmann

p=2   pffiffiffiffiffiffiffiffiffiffiffiffi Z gx dx 1x2 dqsin q cos2 q exp sin q

(3)

(2)

All NEMD calculations were done using the LAMMPS [28]. The simulated systems were first equilibrated for 100 ps in the NPT -Hoover (constant pressure and temperature) ensemble with Nose thermostat and barostat [29,30], followed by NVT (constant volume -Hoover thermostat [29,30] and temperature) ensemble with Nose at 300 K for 200 ps, and NVE (constant volume and energy) ensemble for additional 600 ps.

Z1

N 1 X . . v ðt Þ$ v i ðt0 þ tÞ N i¼1 i 0

+ (5)

where ni ðt0 Þ (m/s) is the velocity of the i-th atom at time t0 (s), angular brackets indicate the time-average. In the one-dimensional limit, the phonon thermal conductivity P can be expressed as kph ¼ Cn2t [37]. C (J/m3$K), n (m/s) and t (s) are the heat capacity per unit volume, phonon group velocity along the tube axis and the relaxation time of a given phonon mode, respectively; and the sum in this equation is over all phonon modes. In an anisotropic material like CNTs, the weighting of each phonon mode by the factor n2t becomes especially important. The thermal conductivity is most sensitive to the phonon modes with the highest group velocities and the longest scattering times [38]. The lager the number of these phonon modes, the higher the thermal conductivity. The phonon mode number per unit frequency can be described by the VDOS. Therefore, the thermal conductivity is strongly influenced by the VDOS. The VDOS has been used in many literature to reveal the underlying mechanisms of heat transfer [39e42]. 3. Results and discussion

350

3.1. Temperature dependence

300 250 200

0

10

20 30 Number of Segments

Fig. 2. Temperature distribution along the length direction.

40

The PTC was calculated in the temperature range of 100e500 K in increments of 100 K. For comparison, two cases were run for each temperature, one was a nanocable with the length of 9.7 nm and chirality (10, 10), the other was a corresponding bare CNT. The results are summarized in Fig. 3(a), together with the thermal conductivity of the bare CNT reported by Lukes et al. [35] and Pop et al. [43]. The temperature dependence of PTC are similar for bare CNTs and nanocables, increasing first and then decreasing with the temperature rising. The peak temperature, i.e. the temperature at which the peak of the thermal conductivity

L. Cui et al. / International Journal of Thermal Sciences 99 (2016) 64e70

67

Fig. 3. Unnormalized (a) and normalized (b) phonon thermal conductivity (kph), electronic thermal conductivity(ke) and total thermal conductivity under different temperatures. Insets: normalized electronic thermal conductivity of Au nanowire.

3.2. Length dependence Fig. 4 illustrates the length dependence of the PTC of (10, 10) nanocables in the range from 9.7 to 19.5 nm. It is shown that the PTC of nanocables increases as the tube length expands, just like CNTs. This tendency is similar to Toprak's report on CNTs filled with Cu NWs [13]. It is attributed to the following factors. 1) It is similar to bare CNTs, when the tube length of nanocables is much smaller than the phonon mean free path, the highly ballistic thermal transport is observed. That is, phononephonon scattering is negligible, and the thermal transport is dominated by the phononboundary scattering. With increasing tube length, the effect of phonon-boundary scattering is less severe [35]. This behaviour arouses the thermal conductivity to enhance with the tube length

increasing. 2) The atomic oscillations of filled Au NWs are conducive to carbon atoms of CNTs to vibrate along the longitudinal direction, which reduces the phonon scattering and enlarges the phonon mean free path. This mechanism gives a further increase in the PTC of nanocables, relative to bare CNTs. In addition, the length of filled Au NWs increases with the enlarging tube length. Corresponding, larger contact areas between gold NWs and CNTs are provided. The enhancement effect in the PTC of nanocables is strengthened. As a result, the PTC of nanocables varies more rapidly than bare CNTs with the increasing length.

3.3. Diameter dependence To study the effects of tube diameters on the PTC of nanocables intuitively, five systems were considered. All the lengths of them are 9.7 nm, and the respective tube chiralities are (8, 8), (10, 10), (12, 12), (14, 14) and (16, 16). Correspondingly, KT-indices of embedded Au NWs are 6-1, 9-3, 11-6-1, 15-9-3 and 17-11-6-1. It is interesting to find that the distance between the outermost atom layer of NWs and the wall of CNTs remains constant at about 0.3 nm. In Fig. 5, we show the PTC of nanocables with different diameters and compare them with unfilled CNTs. Both the PTC values of nanocables and bare CNTs reduce with the increasing diameter. It is attributed to the fewer phonon branches and lower occurrence of phonon umklapp processes for nanocables and bare CNTs with a smaller diameter [46]. Additionally, the PTC increments between the

450

Thermal Conductivity [W/m·K]

occurs, observed in this paper is 300 K. Its value reported for bare CNTs varies with the tube diameters, chiralities and lengths [43e45]. However, there is no coincident quantitative conclusion on the peak temperature. In addition, our results of bare CNT are qualitatively similar to experimental data of Pop et al. [43] and theoretical results of Lukes et al. [35]. Due to CNTs have a unique small dimension effect, our simulation data cannot be quantitatively compared with the experimental results whose CNT length reaches several micrometers. On the other hand, our results are greater than those of Lukes et al. [35], because they simplified the calculation of heat current, making the whole CNT having no translational or rotational movement. Moreover, they used different methods and potentials, that is, equilibrium molecular dynamics (EMD) method and REBO potential for the CeC interactions. The normalized data by using the values of bare CNTs are further calculated and shown in Fig. 3(b). The PTC of nanocable is consistently 20e42% larger than bare CNT. Noya et al. [11] computed the thermal conductivity of a carbon nanopeapod, and found an overall increase of 100% in the thermal conductivity as compared with the bare CNT. The increase is larger than our results. Noya et al. attributed their results to two factors. The interactions between the fillers and CNT are favorable for heat conduction. On the other hand, the motion of filled C60 molecules induces mass transfer. However, in our case, the filled Au NW barely moves, so the PTC of the nanocable has a relatively small enhancement. It is also shown in Fig. 3(b) that the ETC of Au NW is so small that it only amounts to 1e8% of the total thermal conductivity of nanocables. The diameter of the filled Au NW is rather small compared to the electron mean free path of bulk Au, which leads to the increase in the electron scattering and reduction in the ETC. The PTC and the total thermal conductivity of the nanocable are comparable. Accordingly, only the PTC is taken into account in the following simulations.

Au@CNT CNT

400 350 300 250 200 150 100

8

10

12

14

16

18

Tube Length [nm] Fig. 4. Thermal conductivity under different lengths.

20

22

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L. Cui et al. / International Journal of Thermal Sciences 99 (2016) 64e70

KT: 6-1

300

KT: 9-3

250

450

Au@CNT CNT

Thermal Conductivity [W/m·K]

Thermal Conductivity [W/m·K]

350

KT: 11-6-1 KT: 15-9-3 KT: 17-11-6-1

200

(8,8)

150

(10,10) 100

(12,12)

50 0.8

1.2

(14,14)

(16,16)

1.6 2.0 Diameter [nm]

2.4

Fig. 5. Thermal conductivity under different diameters.

392.1

350

Au@CNT CNT

300 250

202.9

200 150

146.2

157.0

123.4

142.1

100 50 0

(10,10)

(18,0) Chirality

(14,7)

Fig. 6. Thermal conductivity under different chiralities.

nanocables and CNTs nearly maintain a constant for the five systems, as depicted in Fig. 5. It is due to the similar interactions between the Au NWs and CNTs, induced by the mentioned constant distance, make almost the same contributions to heat transfer. 3.4. Chirality dependence Three types of nanocables were studied to indicate the chirality effect on the PTC in the present work: a (10, 10) armchair type, a (18, 0) zigzag type and a (14, 7) chiral type. These three nanocables are selected because they have the nearly equal diameters, which is convenient for comparison. Structural characteristics of three nanocables are given in Table 1. The PTC results of bare CNTs and nanocables are depicted in Fig. 6, respectively. They show that chiralities have an important effect on the PTC. Specifically, compared with the corresponding bare CNTs, the increment percentage of PTC is the highest about 150% in the (18, 0) zigzag nanocable and the lowest about 15% in the (14, 7) chiral nanocable. However, for zigzag nanocable, the increment is so dramatic that it needs to be confirmed in additional studies.

axially translational motion. A similar effect was predicted by Schoen [48] who studied the motion of Au nanoparticles encapsulated in CNTs. The filling ratio dependence of PTC of nanocables with tube chirality (10, 10) and length 14.62 nm is displayed in Fig. 7. The results indicate that the increase in the PTC of nanocables along with the increase in the filling ratio. As the filling ratio rising, the contact area between Au NWs and CNTs enlarges. Accordingly, the AueC interactions are enhanced and benefit for heat transfer. On the other hand, when the CNTs are not fully filled, Au NWs would be in axial motion, which induces the mass transfer and increases the energy transportation. However, once the tubes are fully filled with Au NWs (s ¼ 100%), NW motion is so severely restricted that their contributions to heat transfer almost vanish. Interestingly, the PTC of nanocable with the filling ratio 100% is still the largest one as shown in Fig. 7. We concluded that the contributions of AueC interactions to heat transfer are greater than that of the mass transfer induced by the NW motion.

3.6. Phonon vibrational density of states The PTC increments of nanocables are attributed to the enhancement in the VDOS. Fig. 8(a) shows the VDOS of the isolated bare CNT and nanocable with chirality (10, 10) at 300 K. Both the primary peaks of VDOS of CNT and nanocable are located in the

3.5. Filling ratio dependence We define the filling ratio as

s¼

400

VAu ðR  0:3 þ 0:073Þ2 $LAu ¼ CNT VCNT ðRCNT þ 0:17Þ2 $LCNT

(6)

where VAu and VCNT (nm3) are volumes of the filled Au NW and CNT, respectively. RCNT (nm) is the average radius of CNT. LAu and LCNT (nm) are the lengths of Au NW and CNT. 0.073 nm is one half of the radius of the Au atom, which is equal to 0.146 nm [47]. It should be noted that the thermal gradient drives the filled Au NWs moving along the direction of heat flux if the filling ratio is less than 100%. That is, when the filling ratio s < 100%, Au NWs have Table 1 Structural parameters of nanocables. Tube chirality

Radius (nm)

Length (nm)

KT-index of filled Au NW

Armchair (10,10) Zigzag (18,0) Chiral (14,7)

0.6780 0.7046 0.7249

9.7 9.7 9.7

9-3 9-3 9-3

Fig. 7. Thermal conductivity under different filling ratios.

L. Cui et al. / International Journal of Thermal Sciences 99 (2016) 64e70

69

Fig. 8. Phonon vibrational density of states (VDOS) for bare CNTs and nanocabels under chiralities of (a) (10, 10), (b) (18, 0) and (c) (14, 7). (d) VDOS for nanocables under different filling ratios in the dominant frequency range of 12e30 THz.

frequency of 52 THz, which is believed to be the phonon spectrum characteristic of two-dimensional graphene sheet [49]. However, the VDOS of the nanocable shows a clear enhancement with respect to that of the bare CNT in the range of 12-30 THz. Wu and Li indicated that the phonon modes at 12-30 THz are most important for CNT heat transfer, while higher-frequency modes have much smaller group velocities and lower-frequency modes behave ballistically [50]. Therefore, the increment of phonon modes in 1230 THz impacts the thermal conductivity favourably. Moreover, the high frequency part of VDOS (50e54 THz) also increases, which contributes to the further enhancement in heat transfer. Similar results were obtained for CNTs and nanocables with chiralities of (18, 0) and (14, 7), as shown in Fig. 8(b) and (c). In addition, Fig. 8 indicates that the zigzag (18, 0) nanocable has the largest VDOS increments, followed by armchair (10, 10) and chiral (14, 7) nanocables. This result is coincident with the conclusion in section 3.4. The VDOS in the dominant frequency range of 12e30 THz of nanocables with different filling ratios at 300 K is displayed in Fig. 8(d). It shows that the VDOS enhances with the increasing filling ratio. This supports the discussion in section 3.5.

4. Conclusions Non-equilibrium molecular dynamics simulations have been performed to investigate the thermal conductivity of coaxial nanocables of Au nanowire core and carbon nanotube shell. Nanowires filled in nanotubes spontaneously exhibit helical cylindrical multi-shelled structures. The distance between the outermost atom layer of nanowires and the wall of nanotubes is found to remain constant at about 0.3 nm. The main conclusions are as follows: (1) Due to the interactions of CeC and CeAu are favorable for heat transfer, the phonon vibrational density of states of the nanocables show an enhancement with respect to bare

nanotubes in the range of 12-30 THz, correspondingly, the thermal conductivity of nanocables is 20-42% higher than that of nanotubes. The ETC of Au nanowire accounts for 1-8% of the total thermal conductivity of nanocables and is negligible. (2) The temperature, length and diameter dependences of thermal conductivity of nanocables are similar to bare nanotubes. (3) The thermal conductivity of nanocables rises with the increasing filling ratio. When the nanotubes are not fully filled, the indwelling nanowires would be in axial motion, which induces mass transfer and increases the energy transportation. (4) For different chirality types, the zigzag (18, 0) nanocable has the largest PTC increment up to 150%, followed by armchair (10, 10) and chiral (14, 7) nanocables. Acknowledgement This work is supported by the National Natural Science Foundation of China (No. 51176011 and 51422601), National Basic Research Program of China (2012CB720404), National Key Technology R&D Program of China (2013BAJ01B03) and Fundamental Research Funds for the Central Universities (FRF-TP-15-003C1). References [1] [2] [3] [4]

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