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Thermal rectification in a polymer-functionalized single-wall carbon nanotube

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 Nanotechnology 25 345401 (http://iopscience.iop.org/0957-4484/25/34/345401) View the table of contents for this issue, or go to the journal homepage for more

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Nanotechnology Nanotechnology 25 (2014) 345401 (8pp)

doi:10.1088/0957-4484/25/34/345401

Thermal rectification in a polymerfunctionalized single-wall carbon nanotube Souvik Pal1 and Ishwar K Puri2,3,4 1

School of Engineering, University of California, Merced, Merced, CA 95343, USA Department of Mechanical Engineering, McMaster University, Hamilton, ON L8S 4L7, Canada 3 Department of Engineering Physics, McMaster University, Hamilton, ON L8S 4L7, Canada 4 Department of Materials Science and Engineering, McMaster University, Hamilton, ON L8S 4L7, Canada 2

E-mail: [email protected] Received 18 March 2014, revised 1 June 2014 Accepted for publication 10 June 2014 Published 31 July 2014 Abstract

Thermal rectification occurs when heat current through a material is favored in one direction but not in the opposite direction. These materials, often called thermal diodes, have the potential to perform logic calculations with phonons. Rectification obtained with existing material systems is either too minor or too difficult to implement practically. Hence, we present a scheme to enable higher rectification using a single-wall carbon nanotube (SWCNT) that is covalently functionalized near one end with polyacetylene (PA) chains. This composite structure allows rectification R up to 204%, which is higher than the values reported for SWCNTs. Here, R = ( J+ − J − )/J − 100%, where J+ and J − are the heat currents for forward and reverse bias, respectively. The interatomic interactions in the SWCNT-PA nanocomposite are nonlinear, i.e., they are anharmonic, which is a requirement for thermal rectification. Through atomistic simulations, we identify two additional conditions to accomplish thermal rectification at the nanoscale, namely, (1) structural asymmetry, and (2) that the influence of this asymmetry on thermal transport is temperature dependent. The optimum temperature difference to achieve the highest thermal rectification with the structure is 40–80 K.

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S Online supplementary data available from stacks.iop.org/NANO/25/345401/mmedia Keywords: thermal rectification, thermal diode, phononics, covalent functionalization, carbon nanotube (Some figures may appear in colour only in the online journal) 1. Introduction

observed in 1936 [1] when the thermal conductance across a copper oxide-copper interface was found to be directionally dependent. Motivated by the need for better heat removal and thermal management in small electronic devices, research on thermal rectification resumed significantly only about a decade ago [2]. This has included theoretical [3, 4] and experimental [5] investigations, and molecular simulations [6–12] with single materials [8, 13], and composite materials and interfaces [7, 9, 14]. Thermal diodes could enable thermal logic gates and eventually, a thermal computer [15, 16]. The process of performing logic calculations by manipulating phonon transport through thermal logic gates is called phononics for which the possibilities of thermal memory [17] and a thermal

Our ability to regulate the flow of electricity is far superior to how we can control heat transfer. Directional control of electric current leads to functional units such as diodes and transistors, which are foundations for modern electronics. An electronic diode allows electric current to flow in a particular direction, but behaves as an electrical insulator in the other direction. For heat transfer, however, we are typically restricted to using materials that behave only as insulators or as conductors in all directions. The thermal counterpart of an electric diode is a thermal diode or rectifier, which transfers heat preferentially in a particular direction. Rectification of heat current was first 0957-4484/14/345401+08$33.00

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transistor [16, 18] have been proposed. Thermal rectification could also conceivably improve the efficiency of a solar water heater by reducing heat leakage [19] when there is darkness, or during winter when the temperature of the collector falls below the water temperature. Although the rectification of heat current has been established in a one-dimensional nonlinear lattice, such as a Fermi-Pasta-Ulam (FPU) [20], or a Frenkel-Kontorova (FK) lattice [20, 21], these idealized systems do not represent existing nanomaterials. When their diameter is reduced so that they can be considered quasi-one-dimensional, SWCNTs exhibit one-dimensional thermal behavior, since they exhibit a length-dependent thermal conductivity [22]. The interatomic interactions in an SWCNT close to room temperature are also anharmonic [23]. Such a quasi-one-dimensional structure, coupled with anharmonic interatomic interactions, makes SWCNTs an ideal choice to investigate thermal rectification [24–26]. Thermal rectification requires local modification of the intrinsic structure of a material [27], e.g., by varying the masses of carbon atoms [26] or the diameter [24] along the length of an SWCNT. This is extremely challenging in practice and, besides, the rectification obtained is minor. We describe a more practical and easier-to-implement thermal rectification scheme where an SWCNT is functionalized with PA near one end while its other end remains pristine. As for an SWCNT, the various bonded interactions in PA are also anharmonic [28]. Thus, an SWCNT-PA nanocomposite satisfies the anharmonicity requirement for thermal rectification. The functionalization of SWCNTs with polymers has been extensively investigated [29, 30] using well-established experimental methods [31], which makes implementation more practical. Here, we identify the principles that lead to thermal rectification in such a composite structure by employing nonequilibrium molecular dynamics simulations. We also characterize rectification as a function of the temperature difference across the device to determine optimal operating conditions.

Figure 1. The base case structure to investigate thermal rectification. The front view is presented in (a) and the top view in (b). The left panel shows the initially prepared structure in Xenoview [35], while the right shows the corresponding structure after equilibration. The PA chains are bonded to SWCNT (zigzag, 8, 0) atoms that are represented in purple in the inset of (a). The carbon atoms of the pristine SWCNT and the PA molecules are in an sp2 hybridized state, indicated in green. The heat source and the sink are of the same dimension and represented by red and blue, respectively.

directions while the system is considered nonperiodic in the z direction. The equilibrated structure is shown in the right panel of figure 1. There, the polymers remain attached to the SWCNT and form a layer around it in the vicinity of the functionalization. The size of the initial simulation supercell is 100 Å × 100 Å × 150 Å, while the maximum extents of the initial structure are 72 Å in both x and y directions, as evident from figure 1. We have kept the size of the supercell significantly larger than the structure extents in order to prevent any interactions between the structure atoms and its images in the periodic cells in the x and y directions. The anharmonic bonded and nonbonded interatomic interactions in SWCNTpolymer composites are described using the PCFF forcefield [36]. This forcefield has been reported to accurately describe phonon transport in a pristine [37] as well as a chemically functionalized CNT better than empirical potentials such as REBO or Tersoff [38]. A comparison of phonon spectra of CNT atoms computed with PCFF and REBO forcefield has been provided as a supplementary document. The system energy is first minimized using the conjugate gradient technique [32]. The minimized structure is then initialized at 300 K. Next, the system is equilibrated under isothermal-isobaric conditions (NPT ensemble with T = 300 K and P = 1 atm.) for 0.4 nanoseconds (ns), with the pressure being controlled by a Nosé-Hoover barostat. The simulations continue (1) under a microcanonical ensemble (NVE) for 0.4 ns with the temperature held constant at 300 K using a

2. Methodology 2.1. Nonequilibrium molecular dynamics simulations

Molecular dynamics simulations can characterize thermal transport in SWCNT-polymer nanocomposites [30, 31]. We employ nonequilibrium simulations to explore thermal rectification using LAMMPS [32]. Our base case initial structure is presented in the left panel of figure 1. PA chains are attached to a ring of carbon atoms, indicated by purple in the inset of figure 1(a), in a zigzag (8, 0) SWCNT. Each PA chain consists of 24 carbon atoms. The end atoms of the SWCNT are bonded to hydrogen atoms [33]. The SWCNT atoms that are covalently bonded to PA are in an sp3 hybridized state [34]. Throughout this study, we use the color red to represent hotter temperatures, while blue represents those that are colder. All structures are generated in Xenoview [35]. Periodic boundary conditions are applied along the x and y 2

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Langevin thermostat, (2) under a canonical (NVT) ensemble for 0.4 ns when the temperature is again 300 K using a NoséHoover thermostat, and finally (3) as the system is equilibrated under an NVE ensemble for 0.2 ns. All simulations use a 1-femtosecond (fs) timestep. 2.2. Thermal transport and rectification ratio

After equilibration, the atoms forming the heat source are heated to a temperature Th while the sink atoms are simultaneously cooled to Tl under an NVE ensemble. For the base case, Th = 350 K and Tl = 250 K. Both temperatures are held constant by rescaling the atomic velocities in the source and sink at each timestep [39]. This sets up a heat current J , which is calculated from the energy addition and subtraction at each timestep at the source and the sink, respectively. This mode of thermostatting establishes the desired heat bath temperature, eliminating temperature fluctuation observed in other popular thermostats such as Langevin [40]. Hence, it has been widely used in thermal transport studies with molecular dynamics simulation [41–44], especially in thermal rectification studies [40]. The energy added to the heat source equals that subtracted from the sink, keeping the system energy constant. To characterize thermal rectification, simulations with the equilibrated structure are conducted for two configurations, i.e., forward and reverse bias as shown in figures 2(a) and (b), respectively. For forward bias, the heat source is placed at the functionalized end, whereas for reverse bias, that end is instead now a sink. The direction of heat current and phonon propagation is opposite for the two cases, i.e., J+ for forward bias and J − for reverse bias. The rectification is characterized by the rectification parameter [6] R=

((J

+

)

− J − ) J − 100%

Figure 2. (a) The forward bias configuration. In forward bias, the

heat source is placed at the functionalized end, and the heat sink at the pristine end. The temperatures of the heat source and sink are maintained at Th and Tl , respectively. The heat current flowing from the source to sink due to the temperature difference ΔT = Th − Tl is denoted by J+. The ring of functionalized sp3 atoms shown in red is placed at a distance z f from the longitudinal center of the nanotube and at a distance z e from the nearest end. Another set of atoms, represented by blue, is selected near the cold end at the same displacement z f from the center. These two sets of atoms are used to calculate the phonon spectrum presented in figure 5(a). (b) The reverse bias configuration. In reverse bias, the heat source and sink are at opposite ends as compared to the schematic in (a). The source and sink temperatures are the same as for forward bias, leading to an identical ΔT . The heat current is denoted by J −. The red atoms for forward bias are now represented by blue, and the forward bias atoms that were colored blue are now red. They are used to calculate the phonon spectrum in figure 5(b).

(1)

The temperature difference ΔT = ( Th − Tl ) is held constant for both forward and reverse bias. When the heat transfer simulation is run for 9 ns, a steady state is achieved after 7 ns. The steady-state value of J is obtained from a temporal average over 10 samples recorded at regular intervals during the last 2 ns.

3. Results and discussion The steady-state temperature distributions along the SWCNT length for the base case under both forward and reverse bias are presented in figure 3. The two temperature profiles are dissimilar, implying that heat transfer for the two cases is different. Hence, the system exhibits thermal rectification (here, R = 173%). For both cases, the temperature changes rapidly at the location of functionalization. This stark alteration of the local temperature gradient is explained through the significant local change in thermal conductivity. Covalent functionalization of a SWCNT leads to a local thermal conductivity decrease because it produces greater phonon scattering and leads to a smaller phonon mean free path [45], as we will show next.

Figure 3. The steady-state temperature profiles along the length of

the PA functionalized SWCNT obtained after 9 ns of heat transfer simulations for forward and reverse bias. The temperatures are obtained by averaging over 10 such profiles at intervals of 0.2 ns over the last 2 ns of the simulation. We omit error bars for clarity, since the temperature fluctuations are within 0.1% of the average value, indicating a steady state.

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Figure 4. The top panel shows four cases with randomly selected

atoms for functionalization and the base case. In each case, exactly 8 carbon atoms are functionalized for the sake of comparison. The spread of the functionalized atoms along the z direction, denoted by Δz , varies for different cases. The bottom panel shows the value of R for each of these cases. Figure 5. The steady-state phonon power spectra for the base case

for (a) forward bias and (b) reverse bias. The red and blue spectra in (a) correspond to the red and blue atoms in figure 2(a), and likewise for (b) corresponding to figure 2(b). Since the velocity autocorrelation is normalized, the power spectrum obtained is a dimensionless quantity. The ordinate has a logarithmic scale.

The reasons for the enhanced scattering are as follows. First, the carbon atoms to which the PA chains are covalently bonded are in an sp3 hybridized state, while the remaining Catoms are in an sp2 hybridized state. Therefore, the sp3 Catoms behave just as defects would in a pristine nanotube, causing phonons to scatter away from them. Second, the longitudinal phonon transport in an SWCNT is coupled with transport in the transverse direction [46]. The transverse phonon scattering at the PA-SWCNT interface interacts with phonons traveling in the longitudinal direction, also scattering these and causing a further reduction in the thermal conductivity. Interfacial phonon scattering is temperature dependent since the Kapitza resistance due to phonon scattering at the interface of two materials is a function of temperature [47, 48]. In forward bias, the functionalized C-atoms and the PA molecules are both at higher temperatures than their corresponding conditions during reverse bias. Thus, the transverse scattering of phonons between the PA molecules and the functionalized SWCNT atoms, as well as how this scattering is coupled with longitudinal phonon transport, is different for the two cases. These effects lead to thermal rectification. Since J+ > J −, R is positive, i.e., the SWCNT favors heat transfer from its functionalized to its pristine end. For the base case, PA chains are attached to a ring of 8 carbon atoms, a precision that is challenging to achieve in reality. Hence, we consider cases where random atoms near a selected end of the nanotube are functionalized, a situation that most likely corresponds to a real experiment. We consider four random cases as shown in figure 4 (top panel). In each case, we randomly select 8 carbon atoms near one end of the SWCNT and attach PA chains to them. The spread of the selected atoms along the length, i.e., the z axis denoted by Δz, varies for the 4 cases, being narrowest for case 1, with Δz = 4.4 Å, and widest for case 4, with Δz = 21.3 Å. The base case corresponds to Δz = 0. The bottom panel in figure 4 shows the corresponding values of R for these four random cases along with the base case. The values of R vary from

163% to 192%, with that for the base case being 173%. For case 4, the atoms are functionalized over Δz = 21.3 Å, a spatial resolution that is possible to achieve experimentally [49], which still enables a high degree of rectification with R = 186%. This demonstrates the potential of our proposed scheme in terms of practical implementation and high rectification. The only constraint is that the functionalization must be performed asymmetrically, i.e., near one end. The role of asymmetry in our scheme in achieving rectification is discussed in a later section. 3.1. The origin of thermal rectification: the phonon power spectrum

To further understand the directional difference in heat currents, we calculate the phonon power spectrum at the two ends of the SWCNT. A spectral overlap over different portions of a material diminishes scattering and facilitates more efficient phonon energy exchange between these segments. A mismatch in the spectra enhances phonon scattering on the other hand, which lowers the thermal conductivity [50]. The steady-state spectra at the two ends of the SWCNT in forward and reverse bias are shown in figures 5(a) and (b), respectively. A spectrum is calculated through the Fourier transform of the velocity autocorrelation function obtained from the corresponding MD trajectories [7, 51]. In forward bias, two spectra are obtained, one for the ring of functionalized sp3 atoms near the hot end of the nanotube and another for the ring of regular sp2 carbon atoms near its pristine cold end. Both atom sets lie equidistant from the center of the SWCNT, as shown in figure 2(a). In reverse bias, the spectra for the two ends are based on the same sets of atoms, although for this 4

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Figure 7. The rectification parameter R (equation (1)) with respect to

Figure 6. Schematic of the longitudinal and transverse phonon

z f . Error bars are determined from 10 samples recorded at intervals of 0.2 ns during the last 2 ns of the simulations after steady state is achieved.

modes. For the sake of clarity, only a single PA chain is highlighted along with functionalized sp3 C-atoms of the SWCNT.

case, the sp3 and sp2 atoms instead lie at the cold and hot ends, respectively. The spectra for the cold ends in both forward and reverse bias are similar, while those at the hot ends differ significantly for the two bias cases. For instance, as shown in figure 5(b), during reverse bias, the hot end spectrum contains a distinct low frequency peak, which is missing from the corresponding hot end spectrum for forward bias that is shown in figure 5(a). Hence, the spectral overlap between the two ends of the SWCNT is different in forward and reverse bias. There is greater spectral mismatch during reverse bias, which reduces the heat exchange between the hot and cold ends due to increased phonon-phonon scattering. Consequently, J − < J+ . Since the spectral mismatch between the ends of a pristine SWCNT is uninfluenced by switching the locations of the heat source and the sink, there is no thermal rectification. Functionalization imposes additional constraints on the motion of the C-atoms by modifying the phonon spectrum by filtering out certain phonon frequencies. In effect, functionalization manifests itself as a band-pass filter. Earlier, we explained how the transverse phonons from PA chains scatter the longitudinal phonons in CNT. Now we explain how this mechanism produces band-pass filtering of phonon frequencies. To illustrate this latter mechanism, the longitudinal and transverse phonon transport trajectories are shown in figure 6, where only one PA chain and the functionalized C atoms of the CNT are highlighted. Near the functionalization, phonon vibrational energies associated with phonons of CNT with only certain frequencies are transferred to transverse phonon modes in the PA chains. In essence, the transverse phonon modes absorb energies only from longitudinal phonon modes with those frequency ranges. Consequently, peaks corresponding to those frequencies are removed from the phonon spectrum of the functionalized CNT atoms and are essentially filtered. The range of allowable frequencies for this filter is temperature dependent. The low-frequency peak in the hot end spectrum is filtered in forward bias when the functionalization, or filter, is placed near the heat source. When placed near the heat sink in

Figure 8. The dependence of R on the length of SWCNT. The

dimensionless number η is also provided.

reverse bias, the filter does not significantly alter the cold end spectrum. This band-pass filtering is a fundamental cause of the thermal rectification.

3.2. The role of structural asymmetry

Structural asymmetry with respect to the heat source and sink is also a necessary condition for nanoscale thermal rectification [24]. The asymmetry shown in figure 2 depends upon zf. Structural symmetry increases as zf is reduced, with the material being fully symmetric when zf = 0. Then, the functionalization is equidistant from the source and the sink. The variation of R with respect to zf is presented in figure 7. As expected, the reduction in R with decreasing zf reinforces the inference that thermal rectification is a strong function of symmetry. Again, the modification of the phonon spectrum due to functionalization differs for forward and reverse bias, except, of course, when zf = 0.

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3.3. The dependence of rectification on SWCNT structure: length and diameter

The thermal conductivity of an SWCNT varies based on whether its length is smaller than or larger than 100 nm [52]. For the range of lengths (7–12 nm) considered herein, phonon transport is primarily ballistic at room temperature so that there is a rapid increase in thermal conductivity with increasing SWCNT length. This length dependence is presented in figure 8. In order to explain the results, we use a dimensionless number η = z e /(z f + z e ), where z e and z f are the distances of the functionalized atoms from the nearest end and the middle, respectively, as defined in figure 2(a). For an SWCNT of specified length, z f + z e too is fixed and equals half the length of SWCNT. η = 1 corresponds to the perfectly symmetric case discussed in section 2, i.e., z f = 0. Decreasing η below 1 implies increasing asymmetry. We simulated four lengths for which z e was held constant at 11.7 Å. The corresponding η values for these cases are presented in figure 8. Initially, as η decreases, asymmetry increases, and in turn, there is larger rectification in a manner consistent with the results of section 2. However, for the 9.5 nm-long system, a higher asymmetry decreases R. At this longer length, the reduction in thermal conductivity and of J in reverse bias becomes less significant relative to a very rapid increase in the intrinsic thermal conductivity of the SWCNT. With further increases in the length, the thermal conductivity increase greatly diminishes the rectification effect. The thermal conductivity of an SWCNT at room temperature increases with diameter, irrespective of its chirality [52]. This is seen in figure 9(a) where the heat currents J+ and J − are plotted with respect to SWCNT diameter. The corresponding R values are plotted in figure 9(b), which demonstrate a decrease in R with increasing diameter. Similar to the reasons for the length dependence of R, with increasing diameter, the heat currents and thermal conductivity also increase gradually, suppressing the influence of functionalization, which in turn reduces R. In this context, previous use of an SWCNT nanocomposite to demonstrate thermal rectification [5] demonstrated rectification up to only 10%. There are a few key differences between that lower rectification system and ours, primarily regarding the rectification mechanism and the system length and diameter. Our SWCNTs are significantly smaller than the ∼10 μm length and 30–40 nm diameters [5]. It is apparent from figures 8 and 9 that for larger lengths and diameters, rectification is less significant, irrespective of the mechanism. Next, deposition of C9H16Pt on one side of an SWCNT creates a mass graded system [5] where the deposited material becomes part of the heat bath at that end. In contrast, our PA chains are not part of the heat bath so that rectification originates from structural asymmetry instead of from mass asymmetry.

Figure 9. (a) The heat currents J+ and J − with respect to the SWCNT diameter. (b) The corresponding R values are presented with respect to the SWCNT diameter.

3.4. The diode characteristics

Diode performance is depicted through a typical characteristic, which plots J versus ΔT , as shown in figure 10(a). Positive and negative values of ΔT correspond to the forward and reverse bias conditions, respectively. The range of ΔT spans two regions, i.e., operating and breakdown. In the operating region, whereas J+ increases with increasing ΔT , the magnitude of J − remains relatively small and does not change as significantly. Therefore, the difference (J+ − J −) increases with increasing ΔT . This also escalates R, as shown in figure 10(b). In the breakdown region, increasing ΔT increases J − but here, J+ also rises at the same rate. Consequently, increasing ΔT decreases R in this region. This characteristic resembles that of a typical electrical diode that has distinct operating and breakdown regions. The critical value of ΔT for transition from the operating to the breakdown region is ∼80 K, as is evident from figures 10(a) and (b). The optimum values of ΔT for the highest rectification lie in the range 40–80 K. The maximum value of R is 204%, which is higher than values reported in the literature [25, 26, 53, 54] for thermal rectification with SWCNTs.

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80 K when R is maximum. Beyond this range, increasing ΔT leads to a breakdown of thermal rectification.

Acknowledgement This research was supported by a grant from the National Science Foundation (CBET 1246611). We thank the Virginia Tech Department of Engineering Science and Mechanics for the use of its Linux Computing Cluster.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

Figure 10. (a) The diode characteristic, i.e., a plot of heat current J vs

ΔT . Positive and negative ΔT values represent forward and reverse bias, respectively. (b) The variation of R with respect to | ΔT |. Error bars are calculated in a similar manner as for figure 6.

[22] [23]

4. Conclusion

[24] [25]

An SWCNT functionalized by PA can function as a thermal diode. This structure shows larger thermal rectification than do existing concepts. The rectification occurs due to the difference in heat currents originating from the dissimilar spectral mismatches during forward and reverse bias. Functionalization behaves as a band-pass filter, allowing a range of phonon frequencies to pass through and thus alter the phonon spectrum in the vicinity of functionalization. Such an alteration of the phonon spectrum is temperature dependent. When placed near the sink, i.e., in reverse bias, the filter alters the cold end phonon spectrum so that there is a greater spectral mismatch between the hot and cold ends, thereby increasing phonon scattering in contrast to its placement near the source during forward bias. Structural asymmetry also influences the rectification. Placing functionalization at the center of the nanotube makes the structure symmetric, resulting in no rectification at all. The J- ΔT characteristic obtained for the thermal diode resembles that for an electrical diode. The operating range is ΔT = 0–80 K when R increases with respect to ΔT , with optimal rectification between 40 and

[26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]

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Starr C 1936 Physics 7 15 Roberts N A and Walker D G 2011 Int. J. Therm. Sci. 50 648 Zeng N and Wang J-S 2008 Phys. Rev. B 78 024305 Pereira E 2010 Phys. Lett. A 374 1933 Chang C W, Okawa D, Majumdar A and Zettl A 2006 Science 314 1121 Yang N, Zhang G and Li B 2009 Appl. Phys. Lett. 95 033107 Hu M, Keblinski P and Li B 2008 Appl. Phys. Lett. 92 211908 Yang N, Zhang G and Li B 2008 Appl. Phys. Lett. 93 243111 Hu M, Goicochea J V, Michel B and Poulikakos D 2009 Appl. Phys. Lett. 95 151903 Alaghemandi M, Leroy F, Müller-Plathe F and Böhm M C 2010 Phys. Rev. B 81 125410 Murad S and Puri I K 2012 J. Chem. Phys. 137 081101 Murad S and Puri I K 2012 Appl. Phys. Lett. 100 121901 Hu J, Ruan X and Chen Y P 2009 Nano Lett. 9 2730 Murad S and Puri I K 2014 Appl. Phys. Lett. 104 211601 Wang L and Li B 2008 Phys. World 21 27–9 Murad S and Puri I K 2013 Appl. Phys. Lett. 102 193109 Wang L and Li B 2008 Phys. Rev. Lett. 101 267203 Li B, Wang L and Casati G 2004 Phys. Rev. Lett. 93 184301 Norton B and Probert S D 1983 Appl. Energy 14 211 Lan J, Wang L and Li B 2007 Int. J. Mod. Phys. B 21 4013 Hu T, Hu K and Tang Y 2010 Physica B: Condensed Matter. 405 4407 Chantrenne P and Barrat J-L 2004 Superlattice Microst. 35 173 Yu C H, Shi L, Yao Z, Li D Y and Majumdar A 2005 Nano Lett. 5 1842 Wu G and Li B 2007 Phys. Rev. B 76 085424 Alaghemandi M, Leroy F, Müller-Plathe F and Böhm M C 2010 Phys. Rev. B 81 125410 Alaghemandi M, Leroy F, Algaer E, Böhm M C and Müller-Plathe F 2010 Nanotechnology 21 075704 Balasubramanian G, Puri I K, Böhm M C and Leroy F 2011 Nanoscale 3 3714 Hsieh W P, Losego M D, Braun P V, Shenogin S, Keblinski P and Cahill D G 2011 Phys. Rev. B 83 174205 Zheng Q B, Xia D, Xue Q Z, Yan K Y, Gao X L and Li Q 2009 Appl. Surf. Sci. 255 3534 Frankland S J V, Caglar A, Brenner D W and Griebel M 2002 J. Phys. Chem. B 106 3046 Sahoo N G, Rana S, Cho J W, Li L and Chan S H 2010 Prog. Polym. Sci. 35 837 Plimpton S J 1995 J. Comput. Phys. 117 1 Stepanian S G, Karachevtsev M V, Glamazda A Y, Karachevtsev V A and Adamowicz L 2008 Chem. Phys. Lett. 459 153 Campidelli S, Klumpp C, Bianco A, Guldi D M and Prato M 2006 J. Phys. Org. Chem. 19 531 Shenogin S and Ozisik R 2005 Polymer 46 4397 Pal S, Balasubramanian G and Puri I K 2012 J. Chem. Phys. 136 044901

Nanotechnology 25 (2014) 345401

S Pal and I K Puri

[45] Pan R Q, Xu Z J, Zhu Z Y and Wang Z X 2007 Nanotechnology 18 285704 [46] Verma A, Kauser M Z and Ruden P P 2005 J. Appl. Phys. 97 114319 [47] Balasubramanian G, Banerjee S and Puri I K 2008 Interfacial thermal resistance in nanoscale heat transfer, IMECE Proc. 2008 ASME Int. Mechanical Engineering Congress and Exposition (Boston, MA, 31 October–6 November 2008) IMECE2008-69152 pp 969–973 [48] Murad S and Puri I K 2008 Appl. Phys. Lett. 92 133105 [49] Tan Y-Z, Yang B, Parvez K, Narita A, Osella S, Beljonne D, Feng X L and Müllen K 2013 Nat. Commun. 4 1–7 [50] Li N, Ren J, Wang L, Zhang G, Hänggi P and Li B 2012 Rev. Mod. Phys. 84 1045 [51] Bhowmik R, Katti K S and Katti D 2007 Polymer 48 664 [52] Wang J and Wang J-S 2006 Appl. Phys. Lett. 88 111909 [53] Chien S K, Yang Y T and Chen C K 2010 Phys. Lett. A 374 4885 [54] Ni X X, Zhang G and Li B W 2011 J. Phys.: Condens. Mater. 23 215301

[37] Lee J, Varshney V, Brown J S, Roy A K and Farmer B L 2012 Appl. Phys. Lett. 100 183111 [38] Varshney V, Lee J, Roy A K and Farmer B L 2011 J. Appl. Phys. 109 084913 [39] Balasubramanian G, Banerjee S and Puri I K 2008 J. Appl. Phys. 104 064306 [40] Vallabhaneni A K, Hu J, Chen Y P and Ruan X 2011 Thermal rectification in graphene and carbon nanotube systems using molecular dynamics simulations Proc. of ASME/JSME 8th Thermal Engineering Joint Conf. (Honolulu, March 13–17 2011) AJTEC 2011–44521 [41] Balasubramanian G, Kappiyoor R and Puri I K 2010 Multiscale thermal transport across solid-solid interfaces Proc. ASME 2010 Int. Mechanical Engineering Congress and Exposition (Vancouver, November 12–18 2010) pp 197–201 [42] Pang A L J, Sorkin V, Zhang Y-W and Srolovitz D J 2012 Phys. Lett. A 376 973 [43] Termentzidis K, Barreteau T, Ni Y, Merabia S, Zianni X, Chalopin Y, Chantrenne P and Volz S 2013 Phys. Rev. B 87 125410 [44] Noorian H, Toghraie D and Azimian A R 2014 Heat Mass Transfer 50 95

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Thermal rectification in a polymer-functionalized single-wall carbon nanotube.

Thermal rectification occurs when heat current through a material is favored in one direction but not in the opposite direction. These materials, ofte...
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