Transversal thermal transport in single-walled carbon nanotube bundles: Influence of axial stretching and intertube bonding Mohammad Reza Gharib-Zahedi, Mohsen Tafazzoli, Michael C. Böhm, and Mohammad Alaghemandi Citation: The Journal of Chemical Physics 139, 184704 (2013); doi: 10.1063/1.4828942 View online: http://dx.doi.org/10.1063/1.4828942 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/139/18?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Phonon scattering and thermal conductivity of pillared graphene structures with carbon nanotube-graphene intramolecular junctions J. Appl. Phys. 116, 014303 (2014); 10.1063/1.4885055 Introducing thermally stable inter-tube defects to assist off-axial phonon transport in carbon nanotube films Appl. Phys. Lett. 104, 191902 (2014); 10.1063/1.4874624 Thermal transport in double-wall carbon nanotubes using heat pulse J. Appl. Phys. 110, 074305 (2011); 10.1063/1.3641970 The interfacial thermal conductance between a vertical single-wall carbon nanotube and a silicon substrate J. Appl. Phys. 106, 034307 (2009); 10.1063/1.3191673 Electrical and thermal transport in metallic single-wall carbon nanotubes on insulating substrates J. Appl. Phys. 101, 093710 (2007); 10.1063/1.2717855
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THE JOURNAL OF CHEMICAL PHYSICS 139, 184704 (2013)
Transversal thermal transport in single-walled carbon nanotube bundles: Influence of axial stretching and intertube bonding Mohammad Reza Gharib-Zahedi,1 Mohsen Tafazzoli,1,a) Michael C. Böhm,2 and Mohammad Alaghemandi3 1
Department of Chemistry, Sharif University of Technology, 11365-9516 Tehran, Iran Eduard-Zintl-Institut für Anorganische und Physikalische Chemie, Technische Universität Darmstadt, D-64287 Darmstadt, Germany 3 Faculty of Chemistry and Biochemistry, Ruhr University Bochum, D-44801 Bochum, Germany 2
(Received 26 August 2013; accepted 22 October 2013; published online 8 November 2013) Using reverse nonequilibrium molecular dynamics simulations the influence of intermolecular bridges on the thermal conductivity (λ) in carbon nanotube (CNT) bundles has been investigated. The chosen cross linkers (CH2 , O, CO) strengthen the transversal energy transport relative to the one in CNT bundles without bridges. The results showed that λ does not increase linearly with the linker density. The efficiency of the heat transport is determined by the number of linkers in the direction of the heat flux, the type of the linker, and their spatial ordering. The influence of a forced axial stress on the transversal λ has been also studied. The observed λ reduction with increasing axial stretching in a neat CNT bundle can be (over)compensated by cross linkers. The present computational data emphasize the contribution of phonons to the transversal heat transport in CNT bundles with intertube bonds. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4828942] I. INTRODUCTION
The ongoing size reduction of nanomaterials as well as the increased tailoring of their functionalities have rendered possible an efficient transfer and removal of heat factors which are important for improvements in information technology.1, 2 The control of the energy transport in nanostructured materials is a key element for the thermal management of electronic devices,3–5 energy conversion applications,6, 7 or the fine-tuning of functional materials.8, 9 Carbon nanotubes (CNTs) have been recognized as promising candidates for the optimization of thermal management processes and other technological applications. Their high axial thermal conductivity (λ) is one of their most exceptional quantities.10–14 The energy transport in single-walled and multi-walled nanotubes (SWNTs, MWNTs) has been investigated extensively both by theoretical models and experimental techniques.10–26 The highest λ values of roughly 3500 W/mK have been measured in the axial direction of isolated tubes.10 They are a result of an efficient phonon support via longitudinal acoustic modes of large mean free paths.15, 16 Thermal conductivity calculations of isolated CNTs were performed as a function of the temperature,13, 14, 16–18 chain length,18–20 atomic defects,12, 21, 22 impurities,18, 23 tube diameter,14, 17 chirality of the tubes,17, 18 bond length alternation,24 and a forced axial strain.25–27 For applications, however, the thermal conductivity of (ordered) CNT ensembles (e.g., bundles, films ) is more important than λ numbers measured in isolated CNTs.28–30 A theoretically challenging high-symmetry arrangement are bundles of CNT tubes with parallel longitudinal axes. The nanotubes in these systems interact via weak van der Waals a) Author to whom correspondence should be addressed. Electronic mail:
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forces. In certain cases they can form intertube bonds leading to energetically stable ropes.31, 32 In contrast to the high longitudinal λ in a single CNT, the transversal thermal conductivity between the tubes is rather small.33, 34 For this transport direction an efficient phonon support via delocalized modes is missing. The intertube collisions allow only a weak λ enhancement. As a matter of fact, the thermal conductivity in CNT ensembles with parallel longitudinal axes is highly anisotropic. Experimental and theoretical studies have shown that the interlayer bonds in MWNTs play a crucial role in the improvement of their electronic and mechanical properties.35–39 Under the influence of electron beam irradiation or femtosecond laser excitation chemical bonds between the walls of MWNTs, interlayer “sp3 ” bonding in multilayer graphenes and covalent cross linking between neighboring tubes in SWNTs with parallel longitudinal axes can be formed. In the present work we have employed the label “sp3 ” center to discriminate (CNT) atoms with four bonded neighbors from central (CNT) atoms with only three bonded neighbors which will be classified as “sp2 ” sites. Note, however, that these labels should not be used as descriptors for the hybridization of a given CNT atom or – more generally – for an atom in a confined cylindrical or spherical arrangement.40 To re-emphasize, additional transfer channels for the heat are opened by intertube bonds in CNT ensembles or by interlayer bonds in graphenes. Rajabpour and Vaez Allaei, e.g., have demonstrated that with a fraction of 5% randomly distributed interlayer bonds, the thermal conductivity of double-layered graphene is reduced by roughly 70%.41 In theoretical publications with the aim to improve the transversal heat transport between two parallel CNTs, the influence of bridging units on λ has been studied.42, 43 Varshney et al.42 investigated the transversal thermal conductance between two parallel CNTs
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bonded by smaller molecular bridges. These authors have verified that the connectivity of the molecular linkers has a considerable impact on the transversal conductance between two parallel CNTs. In a more complete study, an increase in the transversal conductance between two CNTs with an increasing number of molecular linkers has been emphasized.43 The correlation between λ enlargement and the number of linkers, however, was not linear. The latter research also illustrated that the transversal conductance decreases with increasing chain length. In this work we present a molecular dynamics (MD) analysis of the transversal thermal conductivity of a neat SWNT sample with a hexagonal structure with parallel longitudinal axes as well as of a number of systems with different types of cross linkers (bridges) between neighboring nanotubes in a more or less square arrangement. For our computational study we have chosen methylene (CH2 ), oxygen (O), and carbonyl (CO) bridges. The influence of the intertube bonding on the transversal thermal conductivity will be quantified by correlating these data with the transversal λ of a neat CNT sample. Additionally we have performed MD simulations with the aim to analyze changes in the transversal heat transfer under the influence of longitudinal stretching. To our best knowledge, most of the recent investigations of strained samples have been restricted to single-tube models.26–29 Motivated by published theoretical results as well as by certain deficits of energy transport studies in CNT bundles, we have performed reverse nonequilibrium molecular dynamics (RNEMD) simulations44 to investigate the transversal thermal conductivity in cross linked CNT bundles. In addition to the influence of the intermolecular bonds we have analyzed modifications of the transversal λ under the influence of longitudinal bond stretching. The RNEMD results will be employed to formulate some general principles for the control of the heat transfer in CNT ensembles. II. SIMULATION DETAILS
A schematic view of a CNT bundle of forty (5,5) SWNTs in a dense hexagonal packing corresponding to 9600 atoms in the simulation cell is depicted in Fig. 1. We should mention that the cross sectional area of thermally equilibrated (5,5) CNTs is somewhat oval. This behavior has been described in several articles45–49 prevailingly for CNT bundles under transversal pressure. Zhang et al., e.g., have shown that an exact hexagonal lattice is only formed in (6n,6n) CNTs while an oval cross section has been observed in CNT bundles with
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FIG. 2. Frontal (left) and lateral (right) views of a CNT bundle with intertube bridges in an almost square arrangement. The united atom CH2 bridges are colored in yellow. The linker bridges have been tethered to diametrically opposite atoms of a CNT pair (left). The linked areas are equally spaced over the whole CNT length (right).
other chiralities.45 To sum up, the packing of the (5,5) CNTs in Fig. 1 can be considered as hexagonal-like with slightly deformed CNTs. The shortest tube–tube distances of 0.334 nm are similar to the calculated intertube spacing in other CNT bundles.50, 51 For reasons of simplicity, the studied CNTs with intertube linkers have been defined in an almost square arrangement rather than a hexagonal one; see left-hand side of Fig. 2. Although this ordering differs from the hexagonal structure mentioned above, it is sufficient to understand the influence of intertube linkers on the transversal thermal transport. To form the intertube linkers between a CNT and its four neighbors, an atom of each tube in a region facing the target neighbor is chosen. Then this atom is connected to the closest atom of that neighboring tube by an intermolecular bridge. Unless specified otherwise the linker bridges have been distributed regularly. They are tethered to diametrically opposite atoms of a CNT pair (Fig. 2, left). The linked areas are distributed regularly over the whole CNT length (Fig. 2, right). All molecular dynamics simulations have been performed with the simulation package YASP.52 The employed force field for the carbon nanotube has been taken from the literature.53, 54 It has been used previously by some of the present authors to study the thermal conductivity and thermal rectification in carbon nanotubes.17, 24, 34, 55 The CH2 bridges have been described by a united atom model. The geometrical and force field parameters for the chosen linker bridges were taken from Refs. 56 and 57; they are listed in Tables I and II. The force field parameters of the CNT carbon atom connected to a bridging unit have been modified in a way that this atom has sp3 -like characteristics. These atoms are no longer forced to contribute to a perfect cylindrical CNT shape. Additionally the bond lengths and bond angles at the CNT atoms involved in the bridge bonding have been reparametrized. In Table I, e.g., we see that the carbonyl groups between neighboring nanotubes can form a cross link with a bond length to the CNTs of 0.153 nm and C=O bonds of 0.123 nm. The angle in TABLE I. Bond lengths and bond angles in the cross linker groups. Cross linker
FIG. 1. Schematic representation of a bundle of forty (5,5) CNTs in a hexagonal arrangement. The bold tubes symbolize a distorted hexagon with a small oval deformation of the CNTs (hardly detectable in the figure).
CH2 O CO
Bond
Bond length (nm)
Angle
Bond angle (deg)
C(CNT)–CH2 C(CNT)–O C(CNT)–C(CO) C=O
0.153 0.141 0.153 0.123
C(CNT)–CH2 –C(CNT) C(CNT)–O–C(CNT) C(CNT)–C(CO)–C(CNT) C(CNT)–C–O
114 112 117 121
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TABLE II. Non-bonded Lennard-Jones parameters for the chosen cross linker groups. Cross linker
σ (nm)
ε (kJ mol−1 )
CH2 O C(CO) O(CO)
0.3851 0.6565 0.3324 0.2800
0.4396 0.3050 0.3820 0.4570
q (e) 0.000 −0.500a 0.424 − 0.424
a
The CNT carbon atoms involved in the intermolecular oxygen bridges have a charge of +0.250 to conserve a vanishing overall charge in the atomic triple forming the bridge.
the triple of carbon atoms has a value of 117.2◦ . The modified CNT carbon atoms are shifted away from the tube cylinder to form a pyramidal shape,58, 59 see right plot in Fig. 2. Consequently, the majority of CNT–CNT axial distances are the same as found in the non-bridged system. In the manuscript at hand, the density of the cross linkers is defined by the ratio between the number of sp3 carbon atoms of a tube and its whole number of atoms. We have chosen linker densities between 3.3% and 10%. The length of the studied CNTs was 5 nm. This value coincides with the dimension of the MD simulation box. As mentioned above, each simulation box contains forty (5,5) CNTs. The box dimension for the bare CNT bundle with hexagonal packing amounts to 4.0 × 10.1 nm2 . This value increases to 4.3 × 10.4 nm2 in the quadratic bundle with CH2 and CO linkers. To describe the non-bonded Lennard– Jones interaction, we have used the Lorentz–Berthelot mixing rule.60 The leapfrog formalism and orthorhombic periodic boundary conditions have been employed.60 All MD simulations have been performed at constant volume and temperature after a foregoing equilibration of an isothermal–isobaric ensemble with an anisotropic coupling time for the pressure. We have chosen a target temperature (T) of 300 K. For the Berendsen thermostat,61 a coupling time of 1 ps has been used. This value was sufficient to keep the average temperature within 1 K of the target temperature. In the majority of calculations, a time step of 1 fs has been adopted. The nonbonded interactions were calculated using a Verlet neighbor list60 which was updated every 15 time steps. An interaction cutoff of 1.0 nm has been combined with a neighbor list cutoff of 1.1 nm. In the calculations of the thermal conductivity, the simulation box was divided into 20 slabs in the direction of the heat flux. We allowed an exchange of the atom velocities every 0.3 ps. It has been checked that the thermal conductivity converged with this exchange period. The last nanosecond from roughly 3 ns simulation time of the reverse nonequilibrium molecular dynamics approach has been used for the data production. Detailed information on the RNEMD formalism can be found in our pervious works.17, 24, 55 The adopted method to derive thermal conductivities makes use of the Fourier law, i.e., it is based on a linear T response. For the longitudinal λ in CNTs one has to accept certain deviations from the Fourier law in some marginal slabs.17, 24, 34 Thus calculations of λ had to be restricted to the inner core part of the CNTs. For the transversal conductivity analyzed in the present work we have also observed some deviations from the Fourier law in marginal slabs. In analogy to our recent studies
of CNTs17, 24, 34 we have restricted the evaluation of λ to the slabs with a linear T gradient; see supplementary material.62 In connection with phonon density of states (DOS) calculations, which have been performed to discuss the contribution of vibrational modes to λ in bridged samples, the power spectra were obtained from equilibrium MD simulations. The average velocity autocorrelation function was computed from three trajectories covering 9 ps; they were separated by 500 ps. The average velocity autocorrelation function was transformed with a fast Fourier algorithm to derive the power spectra. The time interval between two adjacent data points of the autocorrelation function was 1 fs. For more detailed information we refer to Ref. 24.
III. RESULTS AND DISCUSSION A. Influence of the cross linkers on the transversal thermal transport in (5,5) SWNT bundles
Before studying the transversal thermal conductivity of CNT bundles with cross linkers, let us investigate first a neat bundle of (5,5) CNTs in a hexagonal packing without covalent bridges (see again Fig. 1). The calculated transversal λ amounts to 0.26 W/mK, a value that is also characteristic for isotropic polymers.63, 64 This number is 3 to 4 orders of magnitude smaller than the axial thermal conductivity in isolated CNT tubes.17 As mentioned above, the large splitting between the axial and transversal λ values is an outcome of different heat transfer mechanisms. While the longitudinal heat transport profits from an efficient phonon mechanism, the heat transfer in the transversal direction is prevailingly supported by intermolecular collisions.34, 65 The insertion of covalent cross linkers between the tubes opens new channels for the heat transport into the transversal direction. Here a phonon support can be expected, too, which is of course less efficient than the influence of delocalized acoustic modes in the longitudinal direction. The vibrations in the bridging units are more or less localized. The normalized transversal thermal conductivity values λ/λ0 of CNT bundles with methylene, carbonyl, and oxygen cross linkers are depicted in Fig. 3 as a function of the density of bridges. We have used the label λ to abbreviate the thermal conductivity of the bridged system while λ0 is the transversal conductivity of the unbridged bare CNT parent. The figure shows that the insertion of additional cross linkers enhances the transversal thermal conductivity. This effect is more pronounced for the CH2 and CO units than for the oxygen bridges. In the first two samples both the central linker and the bonded CNT atoms are carbon. The oxygen hetero linkers couple thermally less efficient to the bonded CNTs atoms23 and hence cannot increase the thermal conductivity as much as the other two linkers. In our simulations we have also verified that the enhancement of the transversal λ is not linearly correlated with the bridging density. In fact, at least two different regions for the λ gradient can be observed in Fig. 3. The smallest one is found for low linker densities. A similar behavior was observed in the case of two bonded parallel CNTs.42, 43 We should mention that both enhancement factors for the heat transport mentioned above (phonons and
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FIG. 3. Normalized transversal thermal conductivity (λ/λ0 ) of CNT bundles with different types of bridges as a function of the cross linker density. λ0 denotes the transversal thermal conductivity of the neat CNT system without any cross linker while λ stands for the thermal conductivity of the bridged system. In all figures containing calculated thermal conductivities, the vertical bars symbolize the error bars of the simulations.
collisions) play a role in the enhancement of the transversal thermal conductivity of bridged samples. On one hand, the insertion of an increasing number of bridges increases the heat transport via cross linker phonons. On the other hand, the deformation of the CNT walls and the formation of pyramidal structures around the CNT sp3 carbon atoms reduce some of the CNT–CNT surface distances. Thus also the energy transfer between two tubes via collisions can be enhanced although most of the axial CNT distances remain almost unchanged. Quite generally the sp3 atoms can be considered as defect sites in the CNT structure. These defects scatter the delocalized longitudinal phonons and hence reduce the axial thermal conductivity of the CNTs.12, 21, 22 To sum up, intertube linkers improve the transversal λ; however they reduce the axial λ and even more the electrical conductivity due to perturbations in the CNT structure. This behavior has been reported recently by Park et al.66 The distribution of the intertube bridges in the CNT systems of Fig. 3 was regular which means that all CNTs of a given linker density have the same number of bridges. Now let us consider random distributions of the linkers. Fig. 4 provides the transversal thermal conductivity of three CNT bundles with a random distribution of 5% CH2 cross linkers. The theoretical random data are related to the λ value of a sample with a regular distribution of the methylene units. Each random system has a different average number of bridges in the direction of the heat flux (nlinker ) but the same total linker density. The figure indicates that for the same total density of intertube bridges, different values of the transversal λ are observed. From the diagram we deduce that the key parameter for the thermal conductivity is the number of the linkers in the direction of the heat flux. The transversal thermal conductivity is enlarged with an increasing value of the parameter nlinker . Furthermore, we have found that for approximately the same nlinker value in the direction of the heat flux, the thermal conductivity of the ordered sample exceeds the one of a random network. Already in some of our recent articles we commented on the importance of highly ordered atomic arrange-
FIG. 4. Transversal thermal conductivity of CNT bundles with 5% randomly distributed CH2 linkers as a function of the average linker concentration per CNT in the direction of the heat flux nlinker . (i), (ii), and (iii) denote three random distributions of cross linkers in the bundle which differ in their nlinker index. In the diagram the random data are correlated with λ derived for a regular distribution of CH2 bridges.
ments for high λ values.23, 64 Structural perturbations imply a localization of vibrational modes and thus a weakening of the phonon support to the energy transfer.
B. Influence of axial stretching on the transversal thermal transport in (5,5) SWNT bundles
Axially stretched CNTs have been studied intensively in the literature.25–27 It has been found that beyond a critical tension (roughly 6%), the network releases its excess energy by a so-called Stone-Wales transformation67 which leads to 5-7-75 defects. In zig-zag tubes a fracture strain between 10% and 15% has been computed. An armchair arrangement leads to an even larger critical strain.68 In the present work we have restricted our simulations under longitudinal stress to values ≤8%. We have stretched CNT bundles with and without cross linkers to compute the transversal thermal conductivity as a function of the forced axial strain. Defect formation and its influence on the transversal λ remain unconsidered in the present simulations. Fig. 5 displays the normalized transversal thermal conductivity λ / λ0 as a function of the longitudinal stretching (in %) and as a function of the linker density. λ0 (λ) abbreviates the transversal thermal conductivity in the absence (presence) of strain. A strong reduction of λ/λ0 with increasing strain is predicted for the sample without intertube linkers (black curves in Fig. 5). The strain-induced λ/λ0 reduction can be prevented with an increasing number of cross linkers. For a linker concentration of 10%, the ratio λ/λ0 is almost strain independent and close to the transversal λ of the undistorted sample (i.e., ≈1.0). Exceptional in the figure are the methylene and oxygen bridged samples with a linker
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FIG. 5. Normalized transversal thermal conductivity λ / λ0 of CNT bundles connected by CH2 (left), CO (middle), and O (right) cross linkers as a function of an axial stretching (in %). λ (λ0 ) labels the transversal thermal conductivity of the deformed (non-deformed) system. The concentration of cross linkers (from 0% to 10%) has been given at the upper right.
density of 6.66%. Here the transversal λ of the stretched systems is larger than the one in the undistorted material. Axial stretching reduces the out-of-plane fluctuations of CNT atoms (not shown here) and hence the transversal heat transport between CNTs via collisions. CNT bundles containing intertube bonds, which open new channels for the heat transport via phonons, however, are less sensitive to an axial stretching. Prerequisite for an efficient phonon support to the transversal thermal conductivity between CNT molecules and cross linkers is a strong overlap of the phonon density of states profiles of the bridge and the sp3 sites of the tubes bonded to the linker units. A convenient measure for this effect is the DOS difference DOS = DOS (linkers) – DOS (sp3 tube atoms). The base line DOS = 0 symbolizes a complete (100%) overlap between the two DOS profiles and thus an optimum for the phonon support to λ. In Fig. 6 we have plotted DOS profiles for methylene bridged CNT bundles with linker densities of 3.33% and 6.66% at stretching ratios of 0% and 6%. We have restricted the DOS plot to smaller wave numbers. This region is relevant for the phonon support to the thermal conductivity in CNTs.69 Although this energy window has been verified only for the longitudinal heat transfer it can be assumed that such a “low-energy” limitation also holds for other vibrational transport channels. Modes with higher energies differ too much from thermal wavelengths; furthermore they seem to be too localized.23 For the bun-
FIG. 6. Difference in the phonon DOS between the CH2 linkers and the sp3 carbon atoms of the tubes for bundles with 3.33% (bottom) and 6.66% (top) linker densities under 0% and 6% axial strain.
dle with 3.33% linker density, the phonon overlap between the components of the bridges in the stretched arrangement is somewhat smaller than in the non-deformed structure (0% stretching). This grading is not only valid for the 300 cm−1 interval plotted in Fig. 6 but also for larger energy windows. The described ordering is changed in the two samples with a linker density of 6.66% where larger deviation from the base line (DOS = 0) is predicted for the non-stretched sample. The DOS values in Fig. 6 correlate with the λ numbers in Fig. 5. Hence, we can conclude that the transversal λ in stretched CNT bundles is strongly determined by the overlap between the phonon DOS profiles of the central bridging atoms and the sp3 tube carbons bonded to this linker unit. Thus we feel that the present simulations have given a clear evidence for a phonon support of the transversal λ in CNT bundles with intermolecular bridges.
IV. CONCLUSION
In the present article we have adopted reverse nonequilibrium molecular dynamics simulations to investigate the transversal heat transport in CNT bundles with and without intertube bridges. The rather low transversal λ of a neat CNT bundle without cross linkers (≈0.26 W/mK) can be enlarged by cross linkers. Their influence increases with an increasing linker density in the direction of the heat flux. These properly directed intertube bridges are of higher efficiency if they are ordered. Additionally we have discussed the transversal λ as a function of an axial stretching. In the absence of cross linkers the transversal λ is reduced with increasing strain. This λ reduction can be avoided in samples with intertube bonds, if the phonon densities of the central bridging unit and the bonded sp3 carbons of the tubes have a large overlap. By studying the difference between the two densities of states in the range of small wave numbers, we were able to identify the participation of vibrations as enhancement factor for the transversal λ in bridged CNT bundles. The MD results of the present contribution are helpful to optimize the heat transport in nanotube bundles with and without cross linkers. Key parameters for this purpose are overlapping vibrational DOS profiles between the atoms forming the bridging unit, cross linkers in the direction of the heat flux, an ordering of the linkers as well as
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the chemical nature of the bridging unit. Homopolar bridges allow a stronger phonon support than heteropolar fragments. ACKNOWLEDGMENTS
We are grateful to S. Philipp for a careful reading of the manuscript and useful suggestions. 1 L.
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THE JOURNAL OF CHEMICAL PHYSICS 139, 184704 (2013)
Transversal thermal transport in single-walled carbon nanotube bundles: Influence of axial stretching and intertube bonding Mohammad Reza Gharib-Zahedi,1 Mohsen Tafazzoli,1,a) Michael C. Böhm,2 and Mohammad Alaghemandi3 1
Department of Chemistry, Sharif University of Technology, 11365-9516 Tehran, Iran Eduard-Zintl-Institut für Anorganische und Physikalische Chemie, Technische Universität Darmstadt, D-64287 Darmstadt, Germany 3 Faculty of Chemistry and Biochemistry, Ruhr University Bochum, D-44801 Bochum, Germany 2
(Received 26 August 2013; accepted 22 October 2013; published online 8 November 2013) Using reverse nonequilibrium molecular dynamics simulations the influence of intermolecular bridges on the thermal conductivity (λ) in carbon nanotube (CNT) bundles has been investigated. The chosen cross linkers (CH2 , O, CO) strengthen the transversal energy transport relative to the one in CNT bundles without bridges. The results showed that λ does not increase linearly with the linker density. The efficiency of the heat transport is determined by the number of linkers in the direction of the heat flux, the type of the linker, and their spatial ordering. The influence of a forced axial stress on the transversal λ has been also studied. The observed λ reduction with increasing axial stretching in a neat CNT bundle can be (over)compensated by cross linkers. The present computational data emphasize the contribution of phonons to the transversal heat transport in CNT bundles with intertube bonds. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4828942] I. INTRODUCTION
The ongoing size reduction of nanomaterials as well as the increased tailoring of their functionalities have rendered possible an efficient transfer and removal of heat factors which are important for improvements in information technology.1, 2 The control of the energy transport in nanostructured materials is a key element for the thermal management of electronic devices,3–5 energy conversion applications,6, 7 or the fine-tuning of functional materials.8, 9 Carbon nanotubes (CNTs) have been recognized as promising candidates for the optimization of thermal management processes and other technological applications. Their high axial thermal conductivity (λ) is one of their most exceptional quantities.10–14 The energy transport in single-walled and multi-walled nanotubes (SWNTs, MWNTs) has been investigated extensively both by theoretical models and experimental techniques.10–26 The highest λ values of roughly 3500 W/mK have been measured in the axial direction of isolated tubes.10 They are a result of an efficient phonon support via longitudinal acoustic modes of large mean free paths.15, 16 Thermal conductivity calculations of isolated CNTs were performed as a function of the temperature,13, 14, 16–18 chain length,18–20 atomic defects,12, 21, 22 impurities,18, 23 tube diameter,14, 17 chirality of the tubes,17, 18 bond length alternation,24 and a forced axial strain.25–27 For applications, however, the thermal conductivity of (ordered) CNT ensembles (e.g., bundles, films ) is more important than λ numbers measured in isolated CNTs.28–30 A theoretically challenging high-symmetry arrangement are bundles of CNT tubes with parallel longitudinal axes. The nanotubes in these systems interact via weak van der Waals a) Author to whom correspondence should be addressed. Electronic mail:
[email protected] 0021-9606/2013/139(18)/184704/6/$30.00
forces. In certain cases they can form intertube bonds leading to energetically stable ropes.31, 32 In contrast to the high longitudinal λ in a single CNT, the transversal thermal conductivity between the tubes is rather small.33, 34 For this transport direction an efficient phonon support via delocalized modes is missing. The intertube collisions allow only a weak λ enhancement. As a matter of fact, the thermal conductivity in CNT ensembles with parallel longitudinal axes is highly anisotropic. Experimental and theoretical studies have shown that the interlayer bonds in MWNTs play a crucial role in the improvement of their electronic and mechanical properties.35–39 Under the influence of electron beam irradiation or femtosecond laser excitation chemical bonds between the walls of MWNTs, interlayer “sp3 ” bonding in multilayer graphenes and covalent cross linking between neighboring tubes in SWNTs with parallel longitudinal axes can be formed. In the present work we have employed the label “sp3 ” center to discriminate (CNT) atoms with four bonded neighbors from central (CNT) atoms with only three bonded neighbors which will be classified as “sp2 ” sites. Note, however, that these labels should not be used as descriptors for the hybridization of a given CNT atom or – more generally – for an atom in a confined cylindrical or spherical arrangement.40 To re-emphasize, additional transfer channels for the heat are opened by intertube bonds in CNT ensembles or by interlayer bonds in graphenes. Rajabpour and Vaez Allaei, e.g., have demonstrated that with a fraction of 5% randomly distributed interlayer bonds, the thermal conductivity of double-layered graphene is reduced by roughly 70%.41 In theoretical publications with the aim to improve the transversal heat transport between two parallel CNTs, the influence of bridging units on λ has been studied.42, 43 Varshney et al.42 investigated the transversal thermal conductance between two parallel CNTs
139, 184704-1
© 2013 AIP Publishing LLC
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bonded by smaller molecular bridges. These authors have verified that the connectivity of the molecular linkers has a considerable impact on the transversal conductance between two parallel CNTs. In a more complete study, an increase in the transversal conductance between two CNTs with an increasing number of molecular linkers has been emphasized.43 The correlation between λ enlargement and the number of linkers, however, was not linear. The latter research also illustrated that the transversal conductance decreases with increasing chain length. In this work we present a molecular dynamics (MD) analysis of the transversal thermal conductivity of a neat SWNT sample with a hexagonal structure with parallel longitudinal axes as well as of a number of systems with different types of cross linkers (bridges) between neighboring nanotubes in a more or less square arrangement. For our computational study we have chosen methylene (CH2 ), oxygen (O), and carbonyl (CO) bridges. The influence of the intertube bonding on the transversal thermal conductivity will be quantified by correlating these data with the transversal λ of a neat CNT sample. Additionally we have performed MD simulations with the aim to analyze changes in the transversal heat transfer under the influence of longitudinal stretching. To our best knowledge, most of the recent investigations of strained samples have been restricted to single-tube models.26–29 Motivated by published theoretical results as well as by certain deficits of energy transport studies in CNT bundles, we have performed reverse nonequilibrium molecular dynamics (RNEMD) simulations44 to investigate the transversal thermal conductivity in cross linked CNT bundles. In addition to the influence of the intermolecular bonds we have analyzed modifications of the transversal λ under the influence of longitudinal bond stretching. The RNEMD results will be employed to formulate some general principles for the control of the heat transfer in CNT ensembles. II. SIMULATION DETAILS
A schematic view of a CNT bundle of forty (5,5) SWNTs in a dense hexagonal packing corresponding to 9600 atoms in the simulation cell is depicted in Fig. 1. We should mention that the cross sectional area of thermally equilibrated (5,5) CNTs is somewhat oval. This behavior has been described in several articles45–49 prevailingly for CNT bundles under transversal pressure. Zhang et al., e.g., have shown that an exact hexagonal lattice is only formed in (6n,6n) CNTs while an oval cross section has been observed in CNT bundles with
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FIG. 2. Frontal (left) and lateral (right) views of a CNT bundle with intertube bridges in an almost square arrangement. The united atom CH2 bridges are colored in yellow. The linker bridges have been tethered to diametrically opposite atoms of a CNT pair (left). The linked areas are equally spaced over the whole CNT length (right).
other chiralities.45 To sum up, the packing of the (5,5) CNTs in Fig. 1 can be considered as hexagonal-like with slightly deformed CNTs. The shortest tube–tube distances of 0.334 nm are similar to the calculated intertube spacing in other CNT bundles.50, 51 For reasons of simplicity, the studied CNTs with intertube linkers have been defined in an almost square arrangement rather than a hexagonal one; see left-hand side of Fig. 2. Although this ordering differs from the hexagonal structure mentioned above, it is sufficient to understand the influence of intertube linkers on the transversal thermal transport. To form the intertube linkers between a CNT and its four neighbors, an atom of each tube in a region facing the target neighbor is chosen. Then this atom is connected to the closest atom of that neighboring tube by an intermolecular bridge. Unless specified otherwise the linker bridges have been distributed regularly. They are tethered to diametrically opposite atoms of a CNT pair (Fig. 2, left). The linked areas are distributed regularly over the whole CNT length (Fig. 2, right). All molecular dynamics simulations have been performed with the simulation package YASP.52 The employed force field for the carbon nanotube has been taken from the literature.53, 54 It has been used previously by some of the present authors to study the thermal conductivity and thermal rectification in carbon nanotubes.17, 24, 34, 55 The CH2 bridges have been described by a united atom model. The geometrical and force field parameters for the chosen linker bridges were taken from Refs. 56 and 57; they are listed in Tables I and II. The force field parameters of the CNT carbon atom connected to a bridging unit have been modified in a way that this atom has sp3 -like characteristics. These atoms are no longer forced to contribute to a perfect cylindrical CNT shape. Additionally the bond lengths and bond angles at the CNT atoms involved in the bridge bonding have been reparametrized. In Table I, e.g., we see that the carbonyl groups between neighboring nanotubes can form a cross link with a bond length to the CNTs of 0.153 nm and C=O bonds of 0.123 nm. The angle in TABLE I. Bond lengths and bond angles in the cross linker groups. Cross linker
FIG. 1. Schematic representation of a bundle of forty (5,5) CNTs in a hexagonal arrangement. The bold tubes symbolize a distorted hexagon with a small oval deformation of the CNTs (hardly detectable in the figure).
CH2 O CO
Bond
Bond length (nm)
Angle
Bond angle (deg)
C(CNT)–CH2 C(CNT)–O C(CNT)–C(CO) C=O
0.153 0.141 0.153 0.123
C(CNT)–CH2 –C(CNT) C(CNT)–O–C(CNT) C(CNT)–C(CO)–C(CNT) C(CNT)–C–O
114 112 117 121
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TABLE II. Non-bonded Lennard-Jones parameters for the chosen cross linker groups. Cross linker
σ (nm)
ε (kJ mol−1 )
CH2 O C(CO) O(CO)
0.3851 0.6565 0.3324 0.2800
0.4396 0.3050 0.3820 0.4570
q (e) 0.000 −0.500a 0.424 − 0.424
a
The CNT carbon atoms involved in the intermolecular oxygen bridges have a charge of +0.250 to conserve a vanishing overall charge in the atomic triple forming the bridge.
the triple of carbon atoms has a value of 117.2◦ . The modified CNT carbon atoms are shifted away from the tube cylinder to form a pyramidal shape,58, 59 see right plot in Fig. 2. Consequently, the majority of CNT–CNT axial distances are the same as found in the non-bridged system. In the manuscript at hand, the density of the cross linkers is defined by the ratio between the number of sp3 carbon atoms of a tube and its whole number of atoms. We have chosen linker densities between 3.3% and 10%. The length of the studied CNTs was 5 nm. This value coincides with the dimension of the MD simulation box. As mentioned above, each simulation box contains forty (5,5) CNTs. The box dimension for the bare CNT bundle with hexagonal packing amounts to 4.0 × 10.1 nm2 . This value increases to 4.3 × 10.4 nm2 in the quadratic bundle with CH2 and CO linkers. To describe the non-bonded Lennard– Jones interaction, we have used the Lorentz–Berthelot mixing rule.60 The leapfrog formalism and orthorhombic periodic boundary conditions have been employed.60 All MD simulations have been performed at constant volume and temperature after a foregoing equilibration of an isothermal–isobaric ensemble with an anisotropic coupling time for the pressure. We have chosen a target temperature (T) of 300 K. For the Berendsen thermostat,61 a coupling time of 1 ps has been used. This value was sufficient to keep the average temperature within 1 K of the target temperature. In the majority of calculations, a time step of 1 fs has been adopted. The nonbonded interactions were calculated using a Verlet neighbor list60 which was updated every 15 time steps. An interaction cutoff of 1.0 nm has been combined with a neighbor list cutoff of 1.1 nm. In the calculations of the thermal conductivity, the simulation box was divided into 20 slabs in the direction of the heat flux. We allowed an exchange of the atom velocities every 0.3 ps. It has been checked that the thermal conductivity converged with this exchange period. The last nanosecond from roughly 3 ns simulation time of the reverse nonequilibrium molecular dynamics approach has been used for the data production. Detailed information on the RNEMD formalism can be found in our pervious works.17, 24, 55 The adopted method to derive thermal conductivities makes use of the Fourier law, i.e., it is based on a linear T response. For the longitudinal λ in CNTs one has to accept certain deviations from the Fourier law in some marginal slabs.17, 24, 34 Thus calculations of λ had to be restricted to the inner core part of the CNTs. For the transversal conductivity analyzed in the present work we have also observed some deviations from the Fourier law in marginal slabs. In analogy to our recent studies
of CNTs17, 24, 34 we have restricted the evaluation of λ to the slabs with a linear T gradient; see supplementary material.62 In connection with phonon density of states (DOS) calculations, which have been performed to discuss the contribution of vibrational modes to λ in bridged samples, the power spectra were obtained from equilibrium MD simulations. The average velocity autocorrelation function was computed from three trajectories covering 9 ps; they were separated by 500 ps. The average velocity autocorrelation function was transformed with a fast Fourier algorithm to derive the power spectra. The time interval between two adjacent data points of the autocorrelation function was 1 fs. For more detailed information we refer to Ref. 24.
III. RESULTS AND DISCUSSION A. Influence of the cross linkers on the transversal thermal transport in (5,5) SWNT bundles
Before studying the transversal thermal conductivity of CNT bundles with cross linkers, let us investigate first a neat bundle of (5,5) CNTs in a hexagonal packing without covalent bridges (see again Fig. 1). The calculated transversal λ amounts to 0.26 W/mK, a value that is also characteristic for isotropic polymers.63, 64 This number is 3 to 4 orders of magnitude smaller than the axial thermal conductivity in isolated CNT tubes.17 As mentioned above, the large splitting between the axial and transversal λ values is an outcome of different heat transfer mechanisms. While the longitudinal heat transport profits from an efficient phonon mechanism, the heat transfer in the transversal direction is prevailingly supported by intermolecular collisions.34, 65 The insertion of covalent cross linkers between the tubes opens new channels for the heat transport into the transversal direction. Here a phonon support can be expected, too, which is of course less efficient than the influence of delocalized acoustic modes in the longitudinal direction. The vibrations in the bridging units are more or less localized. The normalized transversal thermal conductivity values λ/λ0 of CNT bundles with methylene, carbonyl, and oxygen cross linkers are depicted in Fig. 3 as a function of the density of bridges. We have used the label λ to abbreviate the thermal conductivity of the bridged system while λ0 is the transversal conductivity of the unbridged bare CNT parent. The figure shows that the insertion of additional cross linkers enhances the transversal thermal conductivity. This effect is more pronounced for the CH2 and CO units than for the oxygen bridges. In the first two samples both the central linker and the bonded CNT atoms are carbon. The oxygen hetero linkers couple thermally less efficient to the bonded CNTs atoms23 and hence cannot increase the thermal conductivity as much as the other two linkers. In our simulations we have also verified that the enhancement of the transversal λ is not linearly correlated with the bridging density. In fact, at least two different regions for the λ gradient can be observed in Fig. 3. The smallest one is found for low linker densities. A similar behavior was observed in the case of two bonded parallel CNTs.42, 43 We should mention that both enhancement factors for the heat transport mentioned above (phonons and
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FIG. 3. Normalized transversal thermal conductivity (λ/λ0 ) of CNT bundles with different types of bridges as a function of the cross linker density. λ0 denotes the transversal thermal conductivity of the neat CNT system without any cross linker while λ stands for the thermal conductivity of the bridged system. In all figures containing calculated thermal conductivities, the vertical bars symbolize the error bars of the simulations.
collisions) play a role in the enhancement of the transversal thermal conductivity of bridged samples. On one hand, the insertion of an increasing number of bridges increases the heat transport via cross linker phonons. On the other hand, the deformation of the CNT walls and the formation of pyramidal structures around the CNT sp3 carbon atoms reduce some of the CNT–CNT surface distances. Thus also the energy transfer between two tubes via collisions can be enhanced although most of the axial CNT distances remain almost unchanged. Quite generally the sp3 atoms can be considered as defect sites in the CNT structure. These defects scatter the delocalized longitudinal phonons and hence reduce the axial thermal conductivity of the CNTs.12, 21, 22 To sum up, intertube linkers improve the transversal λ; however they reduce the axial λ and even more the electrical conductivity due to perturbations in the CNT structure. This behavior has been reported recently by Park et al.66 The distribution of the intertube bridges in the CNT systems of Fig. 3 was regular which means that all CNTs of a given linker density have the same number of bridges. Now let us consider random distributions of the linkers. Fig. 4 provides the transversal thermal conductivity of three CNT bundles with a random distribution of 5% CH2 cross linkers. The theoretical random data are related to the λ value of a sample with a regular distribution of the methylene units. Each random system has a different average number of bridges in the direction of the heat flux (nlinker ) but the same total linker density. The figure indicates that for the same total density of intertube bridges, different values of the transversal λ are observed. From the diagram we deduce that the key parameter for the thermal conductivity is the number of the linkers in the direction of the heat flux. The transversal thermal conductivity is enlarged with an increasing value of the parameter nlinker . Furthermore, we have found that for approximately the same nlinker value in the direction of the heat flux, the thermal conductivity of the ordered sample exceeds the one of a random network. Already in some of our recent articles we commented on the importance of highly ordered atomic arrange-
FIG. 4. Transversal thermal conductivity of CNT bundles with 5% randomly distributed CH2 linkers as a function of the average linker concentration per CNT in the direction of the heat flux nlinker . (i), (ii), and (iii) denote three random distributions of cross linkers in the bundle which differ in their nlinker index. In the diagram the random data are correlated with λ derived for a regular distribution of CH2 bridges.
ments for high λ values.23, 64 Structural perturbations imply a localization of vibrational modes and thus a weakening of the phonon support to the energy transfer.
B. Influence of axial stretching on the transversal thermal transport in (5,5) SWNT bundles
Axially stretched CNTs have been studied intensively in the literature.25–27 It has been found that beyond a critical tension (roughly 6%), the network releases its excess energy by a so-called Stone-Wales transformation67 which leads to 5-7-75 defects. In zig-zag tubes a fracture strain between 10% and 15% has been computed. An armchair arrangement leads to an even larger critical strain.68 In the present work we have restricted our simulations under longitudinal stress to values ≤8%. We have stretched CNT bundles with and without cross linkers to compute the transversal thermal conductivity as a function of the forced axial strain. Defect formation and its influence on the transversal λ remain unconsidered in the present simulations. Fig. 5 displays the normalized transversal thermal conductivity λ / λ0 as a function of the longitudinal stretching (in %) and as a function of the linker density. λ0 (λ) abbreviates the transversal thermal conductivity in the absence (presence) of strain. A strong reduction of λ/λ0 with increasing strain is predicted for the sample without intertube linkers (black curves in Fig. 5). The strain-induced λ/λ0 reduction can be prevented with an increasing number of cross linkers. For a linker concentration of 10%, the ratio λ/λ0 is almost strain independent and close to the transversal λ of the undistorted sample (i.e., ≈1.0). Exceptional in the figure are the methylene and oxygen bridged samples with a linker
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FIG. 5. Normalized transversal thermal conductivity λ / λ0 of CNT bundles connected by CH2 (left), CO (middle), and O (right) cross linkers as a function of an axial stretching (in %). λ (λ0 ) labels the transversal thermal conductivity of the deformed (non-deformed) system. The concentration of cross linkers (from 0% to 10%) has been given at the upper right.
density of 6.66%. Here the transversal λ of the stretched systems is larger than the one in the undistorted material. Axial stretching reduces the out-of-plane fluctuations of CNT atoms (not shown here) and hence the transversal heat transport between CNTs via collisions. CNT bundles containing intertube bonds, which open new channels for the heat transport via phonons, however, are less sensitive to an axial stretching. Prerequisite for an efficient phonon support to the transversal thermal conductivity between CNT molecules and cross linkers is a strong overlap of the phonon density of states profiles of the bridge and the sp3 sites of the tubes bonded to the linker units. A convenient measure for this effect is the DOS difference DOS = DOS (linkers) – DOS (sp3 tube atoms). The base line DOS = 0 symbolizes a complete (100%) overlap between the two DOS profiles and thus an optimum for the phonon support to λ. In Fig. 6 we have plotted DOS profiles for methylene bridged CNT bundles with linker densities of 3.33% and 6.66% at stretching ratios of 0% and 6%. We have restricted the DOS plot to smaller wave numbers. This region is relevant for the phonon support to the thermal conductivity in CNTs.69 Although this energy window has been verified only for the longitudinal heat transfer it can be assumed that such a “low-energy” limitation also holds for other vibrational transport channels. Modes with higher energies differ too much from thermal wavelengths; furthermore they seem to be too localized.23 For the bun-
FIG. 6. Difference in the phonon DOS between the CH2 linkers and the sp3 carbon atoms of the tubes for bundles with 3.33% (bottom) and 6.66% (top) linker densities under 0% and 6% axial strain.
dle with 3.33% linker density, the phonon overlap between the components of the bridges in the stretched arrangement is somewhat smaller than in the non-deformed structure (0% stretching). This grading is not only valid for the 300 cm−1 interval plotted in Fig. 6 but also for larger energy windows. The described ordering is changed in the two samples with a linker density of 6.66% where larger deviation from the base line (DOS = 0) is predicted for the non-stretched sample. The DOS values in Fig. 6 correlate with the λ numbers in Fig. 5. Hence, we can conclude that the transversal λ in stretched CNT bundles is strongly determined by the overlap between the phonon DOS profiles of the central bridging atoms and the sp3 tube carbons bonded to this linker unit. Thus we feel that the present simulations have given a clear evidence for a phonon support of the transversal λ in CNT bundles with intermolecular bridges.
IV. CONCLUSION
In the present article we have adopted reverse nonequilibrium molecular dynamics simulations to investigate the transversal heat transport in CNT bundles with and without intertube bridges. The rather low transversal λ of a neat CNT bundle without cross linkers (≈0.26 W/mK) can be enlarged by cross linkers. Their influence increases with an increasing linker density in the direction of the heat flux. These properly directed intertube bridges are of higher efficiency if they are ordered. Additionally we have discussed the transversal λ as a function of an axial stretching. In the absence of cross linkers the transversal λ is reduced with increasing strain. This λ reduction can be avoided in samples with intertube bonds, if the phonon densities of the central bridging unit and the bonded sp3 carbons of the tubes have a large overlap. By studying the difference between the two densities of states in the range of small wave numbers, we were able to identify the participation of vibrations as enhancement factor for the transversal λ in bridged CNT bundles. The MD results of the present contribution are helpful to optimize the heat transport in nanotube bundles with and without cross linkers. Key parameters for this purpose are overlapping vibrational DOS profiles between the atoms forming the bridging unit, cross linkers in the direction of the heat flux, an ordering of the linkers as well as
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the chemical nature of the bridging unit. Homopolar bridges allow a stronger phonon support than heteropolar fragments. ACKNOWLEDGMENTS
We are grateful to S. Philipp for a careful reading of the manuscript and useful suggestions. 1 L.
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