Thermal conductivity of single-walled carbon nanotubes

We study numerically the thermal conductivity of single-walled carbon nanotubes for the cases of an isolated nanotube and a nanotube interacting with a substrate. We employ two different numerical methods: (i) direct modeling of the heat transfer by molecular-dynamics simulations and (ii) analysis of the equilibrium dynamics by means of the Green-Kubo formalism. For the numerical modeling of the effective interatomic interactions, we employ both the Brenner potentials and the intermolecular potentials used in the study of the dynamics of large macromolecules. We demonstrate that, quite independently of the methods employed and the potentials used, the character of the thermal conductivity depends crucially on the interaction between a nanotube and a substrate. While an isolated nanotube demonstrates anomalous thermal conductivity due to ballistic transport of long-wave acoustic phonons, the nanotube interacting with a flat substrate displays normal thermal conductivity due to both the appearance of a gap in the frequency spectrum of acoustic phonons and the absorption of long-wave acoustic phonons by the substrate. We study the dependence of the thermal conductivity on chirality, radius, and temperature of the single-walled carbon nanotubes in both the regimes and compare our findings with experimental data and earlier theoretical results for the thermal conductivity.

Figure 1

(Color online) Schematic of a carbon nanotube with the chirality indices: (a) $\left(m,0\right)$ (zigzag SWCNT) and (b) $\left(m,m\right)$ (armchair SWCNT) and the numbering of atoms. Thick red (gray) lines mark the valent bond couplings, red (gray) arcs mark the valent angle couplings, and thin dashed lines show the foundation of two pyramids which form a dihedral angle along the valent bonds in the elementary cell $\left(n,l\right)$.

Figure 2

Geometry of the carbon structure near the valent bond connecting the carbon atoms with the indices $\mathbf{i}$ and $\mathbf{j}$.

Figure 3

(Color online) (a)–(c) Sixty dispersion curves of the (10,0) nanotube and (d)–(f) 36 dispersion curves of the (6,6) nanotube for (a) and (d) standard interatomic molecular-dynamics potential [Eq. (8)] and (b), (c), (e), and (f) Brenner potential with (b) and (e) original set of parameters [Eqs. (5)] and (c) and (f) modified parameters [Eqs. (6)]. Curves corresponding to phonons with zero momentum $\delta =0\left(j=0\right)$ are marked by red (dark gray), phonons with $\delta =2\pi /m\left(j=1\right)$ are marked by blue (dashed lines), and phonons with the momentum $\delta =\pi \left(j=m/2\right)$ are marked by green (light gray).

Figure 4

(Color online) Density of states $p\left(\omega \right)$ for small-amplitude oscillations of a carbon nanotube with the chirality indices (a) (10,0), (b) (6,6), and (c) (600,600) obtained for the model with the molecular-dynamics potential [Eq. (8)]. For comparison, the case (c) also shows the results obtained with the use of the Brenner potential with the parameters [Eqs. (6)] (curve 1 corresponds to the standard potential and curve 2 corresponds to the Brenner potential). The density of states is normalized by the condition ${\int }_0^{\infty }p\left(\omega \right)d\omega =1$.

Figure 5

Example of the nanotube (10,10) with $N$ longitudinal steps. First left $N_{+}$ segments are attached to the $T=T_{+}$ thermostat and the last right $N_{-}$ segments are attached to the $T=T_{-}$ thermostat.

Figure 6

(Color online) Distributions of (a) local heat flux $J_n$ and (b) local temperature $T_n$ in the nanotube (6,6) for relaxation times $t_r=0.4$, 0.2, 0.1, 0.05, and 0.025 ps (curves 1–5). The parameters are length $N\Delta h=29.52\text{nm}$ $\left(N=240,\Delta h=0.123\text{nm}\right)$, temperatures $T_{+}=310\text{K}$ and $T_{-}=290\text{K}$, and the number of cells in the thermostats, $N_{\pm }=40$. The averaging time is $t=1\text{ns}$. The regions placed in the thermostats are shown in gray, $1<n\le N_{+}$, $N-N_{-}<n\le N$.

Figure 7

(Color online) Correlation function $C\left(t\right)$ (in the units $\left[C\right]={\left(\text{eV}\text{Å}/\text{ps}\right)}^2$) for $T=300\text{K}$ and (a) nanotube (6,6) and (b) and (c) nanotube (10,0) calculated with the use of the molecular-dynamics potentials [Eq. (8)]. For comparison, in (d) and (e) we show the correlation function for the nanotube (10,0) obtained for the Brenner potential [red (gray) curve shows a smoothen dependence]. In calculations, we used $N=3000$ periods for the nanotube (6,6) and $N=2000$ periods for the nanotube (10,0).

Figure 8

(Color online) Density of states $p\left(\omega \right)$ for decaying oscillations of the correlation function $C\left(t\right)$ for the nanotube (10,0) calculated for the molecular-dynamics potential [Eq. (8)] (blue) and the Brenner potential [red (gray) curve]. The first peak corresponds to the frequency of $258{\text{cm}}^{-1}$, second $-847{\text{cm}}^{-1}$, and the third $-1670{\text{cm}}^{-1}$. The density is normalized by the condition ${\int }_0^{\infty }p\left(\omega \right)d\omega =1$.

Figure 9

(Color online) Power-law decay of the heat flux autocorrelation function $C\left(t\right)$ for (a) nanotubes (20,0) and (10,0) (curves 1 and 2) and (b) nanotubes (6,6) and (3,3) (curves 3 and 4) (dimensions are $\left[t\right]=\text{ps}$ and $\left[C\right]={\left(\text{eV}\text{Å}/\text{ps}\right)}^2$). Straight lines define the dependence $t^{-\alpha }$ with exponents $\alpha =0.87$, 0.72, 0.69, and 0.5 (lines 1–4). Temperature $T=300\text{K}$.

Figure 10

(Color online) Dependence of the coefficient of thermal conductivity $\kappa$ on the length of the central part of the nanotube, $L$. Dimensions are $\left[\kappa \right]=\text{W}/\text{mK}$ and $\left[L\right]=\text{Å}$. Markers 1–5 correspond to nanotubes (5,0), (10,0), (20,0), (3,3), and (6,6), respectively. Three lines show the dependencies $\kappa \sim L^{\beta }$ with $\beta =0.38$, 0.21, and 0.14. Marker 6 shows the dependence $\kappa /1.3$ for nanotube (10,0) calculated with the help of the Brenner potential.

Figure 11

(Color online) Dependence of the renormalized coefficient of thermal conductivity ${\kappa }_c$ on the length of the central part of the nanotube $L$. Dimensions are $\left[{\kappa }_c\right]=W\times {10}^{-10}/\text{K}$ and $\left[L\right]=\text{Å}$. Markers 1–3 to the nanotubes (5,0), (10,0), (20,0), respectively. Three lines show the dependencies ${\kappa }_c\sim L^{\beta }$ with $\beta =0.38$, 0.21, and 0.14.

Figure 12

(Color online) Schematic of a (6,6) nanotube interacting with the atoms of the substrate.

Figure 13

(Color online) Power-law decay of the heat flux autocorrelation function $C\left(t\right)$ of the nanotube on a plane substrate for the chirality index (3,3). Straight lines show dependence $t^{-\alpha }$ for $\alpha =1.3$. Temperature $T=300\text{K}$, stiffness $K_0=300\text{N}/\text{m}$, and dimensions are $\left[t\right]=\text{ps}$ and $\left[C\right]={\left(\text{eV}\text{Å}/\text{ps}\right)}^2$.

Figure 14

(Color online) Dependence of the thermal conductivity coefficient $\kappa$ on the nanotube length $L$ for an isolated nanotube (3,3) (marker 1) and for the nanotube on a plane substrate (marker 2). Temperature $T=300\text{K}$, stiffness $K_0=300\text{N}/\text{m}$, and dimensions are $\left[L\right]=\text{Å}$ and $\left[\kappa \right]=\text{W}/\text{mK}$. Line shows dependence $\kappa \sim L^{\beta }$ with $\beta =0.33$.

Figure 15

(Color online) Distribution of (a) local heat flux $J_n$ and (b) local temperature $T_n$ in the nanotube (3,3) in the presence of the energy exchange with a flat substrate (temperature of the left edge $T_{+}=310\text{K}$, temperature of the right edge $T_{-}=290\text{K}$, and temperature of the substrate $T=T_{-}$). The parameters are length $N\Delta h=88.56\text{nm}$ $\left(N=720,\Delta h=0.123\text{nm}\right)$, the number of cells in the thermostats, $N_{\pm }=40$, and interaction potential coefficient $K_0=300\text{N}/\text{m}$. The regions placed in the thermostats are shown in gray. Green (light gray) curve in (a) shows the power-law dependence [Eq. (37)] for $\beta =0.025$ and $\alpha =1.30$.

Figure 16

(Color online) Dependence of the thermal conductivity coefficient $\kappa$ on the nanotube length $L$ for (a) nanotube (3,3) and (b) nanotube (6,6) interacting with plane substrate for stiffness of interaction $K_0=300\text{N}/\text{m}$ (marker 1), 30 N/m (marker 2), and 3 N/m (marker 3) and for $K_0=0$ (isolated nanotube, marker 4). Straight lines show the dependences $\kappa \sim L^{0.38}$ and $\kappa \sim L^{0.24}$ for isolate nanotubes. Temperature $T=300\text{K}$; dimensions are $\left[L\right]=\text{Å}$ and $\left[\kappa \right]=\text{W}/\text{mK}$.

Figure 17

(Color online) Decay of the correlation function $C\left(t\right)$ (in the units of $\left[C\right]={\left(\text{eV}\text{Å}/\text{ps}\right)}^2$ and $\left[t\right]=\text{ps}$) for the nanotube (6,6) and temperatures $T=100$, 200, 300, 400, and 500 K (curves 1–5). Straight lines show the power-law dependence $t^{-\alpha }$ with the exponents $\alpha =0.63$, 0.68, 0.72, 0.75, and 0.75.

Figure 18

(Color online) Dependence of the thermal conductivity coefficient $\kappa$ on temperature $T$ for the nanotube (10,0) and the lengths $L=34$, 68, and 136 nm (curves 1–3, respectively).