NANO LETTERS
Thermal Conductivity of Diamond Nanorods: Molecular Simulation and Scaling Relations
2006 Vol. 6, No. 8 1827-1831
Clifford W. Padgett,*,† Olga Shenderova,†,‡ and Donald W. Brenner† Department of Materials Science and Engineering, North Carolina State UniVersity, Raleigh, North Carolona, 27695, and International Technology Center, Raleigh, North Carolina 27617-7300 Received March 15, 2006; Revised Manuscript Received June 5, 2006
ABSTRACT Thermal conductivities of diamond nanorods are estimated from molecular simulations as a function of radius, length, and degree of surface functionalization. While thermal conductivity is predicted to be lower than carbon nanotubes, their thermal properties are less influenced by surface functionalization, making them prime candidates for thermal management where heat transfer is facilited by cross-links. A scaling relation based on phonon surface scattering is developed that reproduces the simulation results and experimental measurements on silicon nanowires.
Diamond nanorods (DNRs) are an emerging class of carbon nanostructure with potential applications for mechanical reinforcement and thermal management in nanocomposites. Molecular modeling and first principles calculations have established the energetic stability of these structures1,2 and have suggested that DNRs would have a brittle fracture force and stiffness that exceeds carbon nanotubes (CNTs) for radii greater than about 1-3 nm depending on the orientation of the DNR.2 Motivated by the modeling studies, Dubrovinskaia et al. synthesized and measured the mechanical properties of needles composed of aggregated DNRs. They showed that this new structure has the lowest measured compressibility of any known material and is able to scratch natural diamond single crystals.3 DNRs and related structures have been produced experimentally by several methods. Aligned diamond whiskers 60 nm in diameter were formed by air plasma etching of asgrown polycrystalline diamond films and films with molybdenum deposited as an etch-resistant mask.4 Ordered arrays of polycrystalline diamond nanocylinders 300 nm in diameter and 5 µm in length have been grown in anodic aluminum oxide templates using microwave plasma assisted chemical vapor deposition.5 Monocrystalline DNRs that were 50-200 nm in diameter and several micrometers in height were fabricated using reactive ion etching of synthetic singlecrystal diamond substrates containing either oxide impurities or Al dots that act as micromasks.6 The aggregated DNRs * Corresponding author. E-mail:
[email protected]. † North Carolina State University. ‡ International Technology Center. 10.1021/nl060588t CCC: $33.50 Published on Web 07/04/2006
© 2006 American Chemical Society
discussed above were created from C60 fullerenes using high temperature and pressure in a multianvil apparatus. The individual DNRs that make up the aggregates were 5-20 nm in diameter and longer than 1 µm.3 On the basis of the high thermal conductivity of bulk diamond, DNRs may have thermal conductivities that rival CNTs. Moreover, the thermal conductivity of DNRs may be less sensitive to surface functionalization than modeling suggests for CNTs,7 thereby providing a potentially important mode for enhancing heat transfer within a nanocomposite via cross-linking. On the other hand, the nanometer-scale dimensions of DNRs may severely reduce their thermal conductivity compared to bulk diamond. Experiments and theoretical analysis by Novikov et al.,8 for example, show that thermal conductivity in polycrystalline diamond thin films is severely reduced as grain sizes approach the nanometer scale due to phonon scattering. In previous work, Moreland, Freund, and Chen used simulations to characterize the thermal conductivity of a (10,10) CNT and a diamond “nanowire”.9 They showed that the conductivity of the diamond nanowire is significantly less than that of the CNT but that the calculated values for the thermal conductivity for both structures depends on the choice of thermostat. Unfortunately, the simulations were performed on an unstable diamond structure. A number of groups have experimentally and theoretically characterized the vibrational properties of silicon nanowires, including the role of phonon confinement and heating on vibrational spectra and the influence of surface scattering and confinement on thermal conductivity.10-21 Li and co-workers, for example, measured the thermal conductivities of single-
Figure 1. Illustrations of the simulated DNRs: top, images of the cross sectional areas of the three DNRs simulated; bottom, illustration of a DNR where 1.0% of surface atoms have been functionalized with phenyl groups.
crystal silicon nanowires with diameters ranging from 22 to 115 nm.22 They report that the thermal conductivity of the nanowires is reduced from bulk silicon by over 2 orders of magnitude, with a strong dependence on the radius of the nanostructure that they attribute to phonon surface scattering and possibly frequency shifts due to phonon confinement. Reported in this Letter are thermal conductivities calculated using classical trajectories of hydrogen-terminated and functionalized DNRs with a [110] long axis, and cross sectional radii and lengths ranging from 0.578 to 1.606 nm and from 0.016 to 0.128 µm, respectively (Figure 1). The simulations predict that thermal conductivities for DNRs with hydrogen surface termination are about a factor of 4 less than previously calculated values for pristine (10, 10) CNTs.7 To study the effect of surface functionalization on thermal conductivity, structures on which attached phenyl groups replace surface hydrogen have been modeled. The simulations indicate that the thermal conductivities of DNRs are much less influenced by surface functionalization than are thermal conductivities of CNTs, suggesting that DNRs are a viable alternative to CNTs for thermal management in nanocomposites. The simulation results show a strong dependence of thermal conductivity on length and radius of the DNRs for both the hydrogen-terminated and surfacefunctionalized structures. A scaling relation derived from a straightforward analytic model based solely on the geometry of nanorods that considers phonon surface scattering is proposed that explains the dependencies of thermal conductivity on radius and length. The same scaling relation is also shown to reproduce reasonably well experimental data for silicon nanowires. Simulations were performed on three DNRs with hydrogen termination (Figure 1, top) and on functionalized variants of these three DNRs on which 1%, 5%, or 10% of the hydrogen atoms are replaced with phenyl groups (Figure 1, bottom). The interatomic forces used in the simulations were modeled by a many-body bond order function.23 This potential function, which has been employed for a wide range of hydrocarbon systems,24 analytically describes energies and forces for both sp2 and sp3 bonding depending on local 1828
coordination and the degree of conjugation. The classical equations of motion for the atoms were numerically integrated using a time step size of 0.25 fs. A periodic boundary condition was applied along the rod axis, and the length of the supercell in this direction was varied to explore the dependence of the thermal conductivity on DNR length. Mu¨ller-Plathe’s method25 was used to calculate the thermal conductivities of the DNRs. In this method cold and hot regions are created by switching the velocity of the hottest atom in the cold region with the velocity of the coldest atom in the hot region. The thermal conductivity is calculated from the known heat flux due to the velocity switching, the temperature gradient at equilibrium between the two regions, and the cross sectional area of the structure. Velocity exchange rates, which are dependent on the radius of the DNR, were chosen to maintain a temperature difference between hot and cold regions of ∼100 K. All of the calculated thermal conductivities decay to steady-state values within less than 115 ps, after which they fluctuate around a constant value. This is consistent with prior simulations by Schelling et al. who observed a steady-state value after 110 ps for silicon nanowires. 26 While Mu¨ller-Plathe’s method is only formally correct in the limit that the temperature gradient approaches zero and in the limit that time approaches infinity, for practical applications it can produce reasonable values provided that the system is adequately equilibrated and the simulation is sampled over a sufficiently long time. These classical trajectories do not include electronic contributions or quantum corrections to the phonon thermal conductivity. Che et al. have argued, however, that both effects are largely unimportant for diamond simulations.27 On the other hand, quantum effects set an upper limit on the thermal conductivity of CNTs, and as a result classical simulations can predict overly large values for the thermal conductivities of CNTs.28 The simulations described here are intended to explore the relative thermal conductivities of CNTs and DNRs with similar radii, the size dependencies for DNRs, and to illustrate the qualitative effects of functionalization on the thermal conductivities of DNRs as compared to CNTs. For this reason the same potential function and calculational methodology is used as was used previously for calculating thermal properties of CNTs.7 We note that an alternate computational approach based on Green-Kubo relations reportedly eliminates the length dependence of the thermal conductivity.29 It is shown below, however, that this length dependence is a physically meaningful scaling parameter necessary for understanding thermal conductivity in nanorods. Plotted as the solid lines in the top panel of Figure 2 are the calculated thermal conductivities for the hydrogenterminated DNRs as a function of the length of the periodic supercell used in the simulations (twice the distance between the hot and cold regions). For comparison, given by the dotted line is the thermal conductivity of a (10, 10) CNT calculated as above,7 while the dashed line corresponds to a “bulk diamond” (BD) supercell created via periodic boundaries as a function of the periodic cell length in the direction of heat flow, [110]. For the CNT, the cross sectional area is Nano Lett., Vol. 6, No. 8, 2006
Figure 3. Thermal conductivity versus DNR radius for various degrees of functionalization. The DNRs all have a supercell length of ∼64 nm. Surface functionalization of 0% (circles), 1% (squares), 5% (diamonds) and 10% (triangles). For clarity the points are connected with a gray line. Error bars plotted for the 0% line are typical of all of the data points.
Figure 2. Thermal conductivity as a function of supercell length: top panel, bulk diamond (solid circles), (10, 10) CNT (solid diamonds), and the three DNRs (open triangles, diamonds, circles, in order of decreasing radius); bottom panel, data for a DNR with radius ∼6 Å for functionalization of 0% (circles), 1% (squares), 5% (diamonds), and 10% (triangles). For clarity the points are connected with a gray line. Error bars plotted on the 0% line are typical of all of the data.
approximated by an annular ring with width 3.4 Å centered at the radius of the CNT. For the BD supercell the two periodic directions normal to the heat flow form a square with a cross sectional area of 2.43 nm2. The calculated thermal conductivities of the CNT and BD are roughly comparable to one another. The DNRs, on the other hand, all have significantly smaller thermal conductivities than the CNTs and BD. For the largest DNR studied, which contains ∼100000 atoms, the thermal conductivity is about a factor of 4 less than the BD with a comparable cell length along the direction of heat flow. Plotted in the bottom panel of Figure 2 are the thermal conductivities versus supercell length for the DNR with the smallest radius (0.578 nm) for different degrees of functionalization. At the longest DNR simulated, functionalizing 1% of the surface atoms reduces the thermal conductivity by ∼25%. Additional functionalization to 5% further reduces the thermoconductivity to about one-half of the initial value. Doubling the functionalization to 10% has little additional Nano Lett., Vol. 6, No. 8, 2006
influence on the thermal conductivity. This is in contrast to CNTs, where similar modeling predicts that functionalizing 1% and 5% of carbon atoms reduces the CNT thermal conductivity by almost factors of 4 and 7, respectively.7 Plotted in Figure 3 is the calculated thermal conductivity as a function of radius for the DNRs with a constant supercell length of 64 nm. The thermal conductivity for all of the structures increases roughly linearly with radius, with surface functionalization reducing the conductivity value as discussed above. To better understand the relationship between radius, length, and functionalization suggested by the simulations, we consider a straightforward model in which it is assumed that phonons propagate down a cylinderical nanorod (NR) at different angles with respect to the long axis of the structure. These propagation modes contribute to the overall thermal conductivity along the NR axis as long as either their associated propagation vector does not terminate due to scattering with a surface, it is shorter than the NR length, or it is shorter than the intrinsic scattering length if the NR length or radius exceeds the intrinsic scattering length. The contribution of each propagation mode to the overall thermal conductivity wNR of a nanorod is equal to the sum of the components of the phonon propagation vector qm with respect to the NR axis wNR )
wmqm cos θ ∑ m
(1)
where θ is the angle of the propagation vector with respect to the axis of the NR and wm is a proportionality factor that includes the heat capacity and phonon group velocity. We further assume that the system is a straight continuum cylinder in which noninteracting phonons propagate in all directions with equal velocities starting from the center of the cylinder at the hot end of the NR. Implicit in these assumptions is that the heat capacity and group velocity are 1829
independent of the phonon wave vector (i.e., a Debye model in the long wavelength limit). With these assumptions the sum in eq 1 can be replaced with an integration over all propagation directions for which the component along the NR axis is in the direction of heat flow. For the case where both the radius R and the length L of the cylinder are less than the intrinsic phonon scattering length qsc, eq 1 becomes wNR ) w
∫0π q cos(θ) dθ ) w[∫0θ q cos(θ) dθ + ∫θπ q cos(θ) dθ] c
(2)
c
where w is a proportionality factor that includes a 2π from integration around the circumference of the NR. The angle θc, which is given by tan-1(R/L), corresponds to the angle of the vector for which a phonon propagating from the center of a hot region first reaches the distance between the hot and cold regions. For the first integral on the right side of eq 2 the component along the NR axis of the phonon propagation vector is constant and equal to the supercell length L. For the second integral on the right side of eq 2, the phonon propagation vectors are constrained so that their component in the direction perpendicular to the NR axis is equal to the NR radius q sin(θ) ) R
Figure 4. Thermal conductivity from the simulations as a function of the scaling quantity in eq 4. The symbols are related to the degree of functionalization and are the same as in the bottom panel of Figure 2. The lines are a least-squares fit to the data. Inset: Illustration from the simulation of the DNR with the highest aspect ratio.
(3)
This leads to the analytic relation between thermal conductivity and the length L and radius R of a NR wNR ) w[
∫0θ L dθ + ∫θπ R cot(θ) dθ] ) c
[
c
w L tan-1(R/L) - R ln
(|x
|)]
R
(4)
R + L2 2
when both R and L are shorter than the intrinsic scattering length qsc. For a NR with L greater than qsc, the angle θc is given by sin-1(R/qsc) and eq 2 leads to
[ ( )]
wNR ) wR 1 - ln
R qsc
(5)
When both the radius and length are greater than the phonon scattering length, the model gives wNR ) wqsc
(6)
Plotted in Figure 4 is the thermal conductivity from the molecular dynamics simulations as a function of the quantity in the brackets on the right side of eq 4 for all 53 of the DNR systems simulated. Data from each degree of surface functionalization follows the scaling relation predicted by this simple model reasonably well. Notable exceptions are the two data points given by the circles that appear above the lines. These correspond to the two unfunctionalized DNRs with the longest length and smallest radii. Animations 1830
Figure 5. Thermal conductivity versus temperature for silicon nanowires. Symbols are taken from ref 22. The bottom three solid lines are the prediction from eq 4 with the slope w for each temperature fit to the experimental data indicated by the filled circles. The radii from bottom to top are 22, 37, 56, and 115 nm, respectively.
made from the simulations of these two structures, a snapshot from which is given as in inset in Figure 4, indicate significant bending and twisting, and therefore our assumption of a straight cylinder breaks down. The solid lines in the figure are least-squares fits to the data for each degree of functionalization. The higher degrees of functionalization lead to smaller slopes. In our scaling model, the role of the surface functionalization is apparently to change the scaling factor w in eqs 2 and 4-6. The scaling relation eq 4 in principle applies to any generic NR system. To further explore this, we have analyzed recent experimental measurements of the thermal conductivity of silicon nanowires using eq 4. Plotted in Figure 5 as the symbols are the measured conductivities reproduced from ref 22 as a function of temperature for four different radii. Given by the lines are the results from eq 4 using L ) 2.5 µm (estimated from Figure 1 in ref 22) and fitting w for each temperature to the data for the largest radius. The model is able to reproduce the radius scaling of the thermal conductivity. This supports the suggestion in ref 22 that the reduction in thermal conductivity for the silicon nanowires Nano Lett., Vol. 6, No. 8, 2006
is due mainly to surface scattering. Our scaling relation reproduces reasonably well thermal conductivities for both the molecular modeling results on DNRs and the measurements on silicon nanowires. We note that for both of these cases R/L is small, consistent with a nanorod geometry. The validity of this scaling relation for aspect ratios closer to 1 has yet to be explored. Acknowledgment. D.W.B. and C.W.P. are supported by the Office of Naval Research and the National Science Foundation. Helpful discussions with D. Irving, J. A. Harrison, J. D. Schall, and G. T. Gao are acknowledged. References (1) Barnard, A. S. ReV. AdV. Mater. Sci. 2004, 6, 94-119. (2) Shenderova, O.; Brenner, D. W.; Ruoff, R. S. Nano Lett. 2003, 3, 805-809. (3) Dubrovinskaia, N.; Durbrovinsky, L.; Crichton, W.; Langenhorst, F.; Richter, A. Appl. Phys. Lett. 2005, 87, 083106. (4) Baik, E. S.; Baik, Y. J.; Lee, S. W.; Jeon, D. Thin Solid Fims 2000, 377, 295-298. (5) Masuda, H.; Yanagishita, T.; Yasui, K.; Nishio, K.; Yagi, I.; Rao, T. N.; Fujishima, A. AdV. Mater. 2001, 13, 247-249. (6) Ando, Y.; Nishibayashi, Y.; Sawabe, A. Diamond Relat. Mater. 2004, 13, 633-637. (7) Padgett, C. W.; Brenner, D. W. Nano Lett. 2004, 4, 1051-1053. (8) Novikov, N. V.; Podoba, A. P.; Shmegera, S. V.; Witek, A.; Zaitsev, A. M.; Denisenko, A. B.; Fahrner, W. R.; Werner, M. Diamond Relat. Mater. 1999, 8, 1602-1606. (9) Moreland, J. F.; Freund, J. B.; Chen, G. Microscale Thermophys. Eng. 2004, 8, 61-69. (10) Volz, S. G.; Chen, G. Appl. Phys. Lett. 1999, 75, 2056-2058.
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