Diameter and Length Effect on Diffusive-ballistic Phonon Transport in a Carbon Nanotube

We report a non-equilibrium molecular dynamics (MD) study on heat conduction of finite-length single-walled carbon nanotubes (SWNTs). The length and diameter dependences of the thermal conductivity are quantified for a range of nanotubelengths up to a micrometer at room temperature using two different temperature control techniques. A thorough investigation was carried out on the influence of intrinsic thermal boundary resistance between the temperaturecontrolled layers and the rest of the SWNT. The trend of length effect indicates a gradual transition from nearly pure ballistic phonon transport to diffusive-ballistic phonon transport. The nearly pure ballistic phonon transport was also confirmed by the minor diameter-dependence of thermal conductivity for short SWNTs. For longer SWNTs with stronger diffusive effect, the thermal conductivity is larger for SWNTs with smaller diameters. INTRODUCTION The ever-expanding expectations for single-walled carbon nanotubes (SWNTs) include applications for various electrical and thermal devices due to their unique electrical and thermal properties [1]. SWNTs are expected to possess high thermal conductivity due to their strong carbon bonds and the quasione-dimensional structure [2]. On considering the actual applications, one of the essential tasks is to characterize the thermal properties not only for thermal devices but also for electrical devices since they determine the affordable amount of electrical current through the system. With advances in SWNT synthesis and MEMS techniques, thermal conductivity (or thermal conductance) measurements of individual SWNTs have been recently reported [3, 4]. Measurements were made for 2.76-μm-long SWNT suspended across a gap between two thermal reservoirs and thermal conductance of 4 nW/K was obtained at room temperature [3]. Later, thermal conductivity of a 2.6-μm-long individual suspended SWNT with diameter of about 1.7 nm was extracted from I-V electrical characteristics to be about 3500 W/m K at room temperature. The value is similar to those of individual multi-walled carbon nanotubes [5, 6] and about an order of magnitude larger than that of bulk carbon nanotubes in forms of mats and bundles [7]. Experiments with temperature variation suggest that the thermal conductivity (or conductance) increases with temperature in temperature range between 110 K and 300 K [3] and it decreases in temperature range between 300 K and 800 K [4]. These observations suggest a critical temperature (~300 K) above which Umklapp scattering becomes important. The thermal property measurements of SWNTs mentioned above are extremely challenging as there are potential uncertainties residing in the technicality for instance related to the contact resistances between thermal reservoirs and an SWNT. Therefore, the demands for reliable theories and numerical simulations are greater than ever for validations of the experimental results and for investigation of detail heatconduction characteristics that are not accessible in experiments. One of such heat conduction characteristics with practical importance is the size dependence of thermal conductivity. In general, the size-dependence of the thermal conductivity appears when the system characteristic length is smaller or comparable to the phonon mean free path [8]. For SWNTs, due to the expected long phonon mean free path, the regime of the length effect stretches beyond the realistic length in many applications. The length effect has been demonstrated using MD simulations [9, 10] and the power-law divergence was discussed with analogy to the convectional one1 Copyright © 2007 by ASME dimensional models [11]. More recently, the length effect was investigated up to fully diffusive phonon transport regime using a kinetic approach [12], where the divergence was shown to disappear with presence of the second order (or higher) 3phonon scattering processes. The issue of the transition from the pure ballistic to diffusive-ballistic phonon transport was discussed by Wang and Wang [13] by modeling the energy transmission based on the ratio of the overall average phonon mean free path to L. In this study, we aim to investigate phonon transport in SWNTs for a range of SWNT lengths by performing nonequilibrium MD simulations at room temperature. MD simulations are capable of handling phonon transport for all the phonon branches, unlike the kinetic approach with relaxation approximations [12]. As shown later, this aspect is important for SWNTs with significant ballistic phonon transport, especially at room temperature where a wide range of phonon branches is populated. NOMENCLATURE A= nanotube cross-sectional area Ac-c= interatomic distance a= unit cell thickness B= many body term Eb= binding energy E = phonon energy spectrum K= thermal conductance k= wave number L= nanotube length Lc= thermostat length N= number of unit cells n= number of atoms per unit cell q= heat flux rij = distance between atom i and j VA = attractive term VR = repulsive term Greek Symbols α= cylindrical coordinates ΔT= temperature drop λ= thermal conductivity ω= frequency τ= relaxation time Subscripts P= phatom thermostat NH= Nose-Hoover thermostat CLASSICAL MOLECULAR DYNAMICS The Molecular Dynamics Potential Function In current MD simulations, the carbon-carbon interactions were modeled using Brenner potential [14] in a simplified form [15] where the total potential energy of the system is expressed as, [ ] ∑ ∑