Molecular junctions for thermal transport between graphene nanoribbons:
covalent bonding vs. interdigitated chains
Alessandro Di Pierro, Guido Saracco, Alberto Fina*
Dipartimento di Scienza Applicata e Tecnologia, Politecnico di Torino, Alessandria Campus,
Viale Teresa Michel 5, 15121 Alessandria, Italy
*Corresponding author: Alberto Fina
e-mail address: alberto.fina@polito.it
Abstract
Proper design and manufacturing thermal bridges based on molecular junctions at the contact between
graphene platelets or other thermally conductive nanoparticles would provide a fascinating way to produce
efficient heat transport networks for the exploitation in heat management applications. In this work, using
Non Equilibrium Molecular Dynamics, we calculated thermal conductance of alkyl chains used as molecular
junctions between two graphene nanoribbons, both as covalently bound and Van der Waals interdigitated
chains. Effect of chain length, grafting density, temperature and chain interdigitation were systematically
studied. A clear reduction of conductivity was found with increasing chain length and decreasing grafting
density, while lower conductivity was observed for Van der Waals interdigitated chains compared to
covalently bound ones. The importance of molecular junctions in enhancing thermal conductance at
graphene nanoribbons contacts was further evidenced by calculating the conductance equivalence between
a single chain and an overlapping of un-functionalized graphene sheets. As an example, one single pentyl
covalently bound chain was found to have a conductance equivalent to the overlapping of an area
corresponding to about 152 carbon atoms. These results contribute to the understanding of thermal
phenomena occurring within networks of thermally conductive nanoparticles, including graphene
nanopapers and graphene-based polymer nanocomposites, which are or high interest for the heat
management application in electronics and generally in low-temperature heat exchange and recovery.
Keywords
Thermal Conductance, Molecular Junction, Thermal transport, Non Equilibrium Molecular Dynamics,
Graphene.
Introduction
In the last decade, a great deal of research interest has been devoted to graphene1-2, owing to its amazing
characteristics: electrical properties3-4, thermal conductivity5-7, mechanical stiffness and strength8 which are
on the top of the known materials. Since its discover, significant effort has also been devoted to exploring
graphene and graphene-related materials9 for various applications, such as in energy storage2,
nanoelectronics2, sensors and functional materials1-2 In several of these fields, thermal conductivity of
graphene may play an important role whenever dissipation of heat is crucial. In particular, in highperformance electronics, heat removal has become a critical issue7. Owing to their ultrahigh thermal
conductivity, graphene and graphene-related materials have been suggested as candidates in nanomaterials
for efficient thermal management.
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Significant efforts have been devoted to the preparation of nanocomposite materials embedding graphene
and graphene-related materials. However, graphene properties are only partially reflected in
nanocomposites10, where thermal transport is not limited by the thermal conduction of graphene itself but
rather by the high thermal resistance at the contact between nanoplatelets11-13. In fact, the reach of a
percolation threshold, a value over which the particles inside a composite material get physically in contact,
is not enough to improve radically the overall thermal conductivity14. Indeed, for every single contact,
physical limitations in phonon transfer have to be taken into account, leading to a highly inefficient heat
transfer across the nanoparticles network that characterize nanocomposites materials15. Compatibilization
between nanoparticles or between nanoparticles and matrix have been suggested by various authors16-18 as
a possible solution to decrease interfacial resistance. A chemical compatibilization is typically obtained by
the functionalization of nanoplatelets with molecules able to build chemical bridges towards the matrix.
Different functionalization methods are possible for graphene, either through covalent and non-covalent
surface modifications19. Covalent functionalization leads to rehybridization of a fraction of the carbon atoms
in graphene, thus introducing defects in the sp2 graphene bidimensional lattice20, creating an alteration of
the pristine phonon scatter modes and therefore reducing the conductivity of the nanoplatelets19. On the
other hand, the non-covalent interactions between the platelets and suitable molecules19 does not damage
the atomic structure, so the thermal proprieties of the graphene layers are preserved. However, the weak
forces involved between the molecules can still be a remarkable bottleneck in thermal conductivity15.
Computational tools, including first principle quantum calculations and Molecular Dynamics (MD), have been
crucial to study and evaluate the interfacial properties between graphene platelets as well as between
graphene and the matrix in a nanocomposite. In an experimental and computational work, Han et al10
exploited silane functionalization to reduce thermal resistance between substrate and supported graphene
nanoflakes. Both computational and experimental results agree that the bridging molecule density tuned the
thermal conductivity leading to an heat conduction enhancement. A Reverse Non-Equilibrium MD (RNEMD)
approach was proposed from Gao and Müller21 using bonded fragments of Polyamide 6,6 backbone molecule
between graphene sheets. The interfacial and planar thermal boundary conductance (TBC) were evaluated
as a function of grafting density and molecule length effect, finding an optimum density value. These results
were used in EMA (Effective Medium Approach) to develop a predictive model to extrapolate the
macroscopic conductivity of a massive nanocomposite. Liu et al22 used C2H4 linkers to joint two partially
overlapping graphene nanoribbons. Thermal conductance and thermal jump were related to the position and
amount of linkers used in the simulations. An inversely proportional relation was found between the thermal
jump across the junction and the number of cross-linkers. The related phonon spectra were described as
effective mainly on the out of plane vibrational modes due to a mismatch between the in-plane vibrational
modes of the linkers and the graphene ones. Chen et al23 simulated the use of external force and chemical
functionalization through CH2 linkers across stacked nanotubes to investigate thermal transport, the
conductance going through a maximum at the optimal concentration of linkers. A single molecular junction
analysis was studied by first-principles full quantum calculations by Li et al24 to evaluate the thermal
conductance in short alkanes between two graphene nanoplatelets. The thermal conductance as a function
of the molecule stretching state was evaluated, determining that an extended junction almost doubles the
thermal conductance compared to the compressed form. In a more recent paper, the same group addressed
the comparison between covalently bound molecular junction vs. non-covalently bonded π-stacked systems,
showing strong suppression of phonon transport, leading to about 95% reduction in thermal conductance
for π-stacked junctions, compared to the covalent one25. In this work, we investigate the effect of edge
grafted short (CH2)n alkyl chains as a thermal bridge between two adjacent graphene nanosheets, by the
calculation in interfacial conductance via classic Non Equilibrium Molecular Dynamics simulations (NEMD).
In particular, the effect of chain length and grafting densities were systematically studied for alkyl chains
covalently bound on both graphene nanoribbons. Furthermore, comparisons were done between covalently
bonded molecules and interdigitated ones (chains covalently grafted on one graphene sheet and interacting
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with opposite chains via van der Waals forces), as well as with non-functionalized partially overlapped
graphene nanoribbons.
Theory and Computational Methods / MD modeling
Classical Molecular Dynamics (MD) calculations was used on LAMMPS (Large-scale Atomistic Molecular
Massively parallel Simulator) package code witch uses Velocity Verlet integration algorithm to recalculate
velocities and positions of the atoms. The 2nd generation of the AIREBO (Adaptive Intermolecular Reactive
Empirical Bond Order)26 force field was employed due to its thermal transport capability as described from a
wide literature on carbon and graphene family structures27-29 and hydrocarbons30.
The model topology was composed by two ≈100 Å graphene nanoribbons in zigzag configuration joint from
alkyl chains chemically grafted between the two graphene sheets, as shown in Figure 1C. The width (Y axis)
was set to ≈50 Å, while the total length of simulation box (X axis) varied from 193 Å to 200 Å, depending on
the length of the chains, in a stretched configuration, among the sheets. The thickness of the sheet (Z axis)
was assumed 3.45 Å from the Van der Waals (VdW) diameter of the carbon atom and the C-C is 1.41 Å as
average value, very close to which reported from Diao et al 29.
NEMD calculations were performed by the application of Nosé-Hoover thermostats at the two ends of the
simulation box, which coincides with the graphene sheets ends. The Hot (310 K) and the cold bath (290 K) of
the thermostats regions (last 10 Å) are set as NVT canonical ensemble (constant Number of atoms, Volume
and Temperature) while the slabs among the two thermostats ran under NVE (constant Number of atoms,
Volume and Energy) condition. All the simulations were carried out for 5 ns using a 0.25 fs timestep. An initial
500 ps of thermal equilibrium at 300 K was set as previously reported by Mortazavi et al31-32. A second 500
ps stage of transient non-equilibrium heating, followed with the purpose to reach a constant heat flux. After
those initial stages, the constant energy flowing through the thermostats33 started recording. The thermal
flow inside NVE regions is derived from the slope of energy versus time plots31. Additional investigations
involved different equilibrium and run temperatures: 200 K; 400 K; and 500 K while the ∆T at the thermostats
was kept constant to ±10 K as previously described. In this range the temperature quantum correction
suggested at lower temperatures by classic Molecular Dynamic simulation is considered neglibile34.
To evaluate the temperature variation inside the sample along the thermal flow, the simulation box length
was virtually split transversally into 22 thermal layers. The temperature of each thermal layer was then
computed using the relation in Equation 1,
𝑇𝑖 (𝑠𝑙𝑎𝑏) =
2
3𝑁𝑖 𝑘𝐵
∑𝑗
𝑝𝑗2
(1)
2𝑚𝑗
In Equation 1, Ti(slab) is the temperature of ith slab, Ni is the number of atoms in Ith slab, kB is the
Boltzmann’s constant, mj and pj are atomic mass and momentum of atom j, respectively. All the slabs
temperatures were then time-averaged to the simulation runtime excluding the non-linear regions at the
nanoribbon boundary, i.e. the ones close to the thermostats and the ones coupled to the junction. The
thermal boundary conductance (TBC) of the model, G, expressed in pW/K, was evaluated using the
Equation 2.
𝑞
𝐺 = ∆𝑇𝑥
(2)
Here qx is the thermal flow derived from the energy versus time plot slope and ∆𝑇 is the thermal difference
across the jump, as projection of the two linear fit of the temperature-length plot in the junction middle
point.
The single chain thermal conductance, Gchain, expressed in pW/K was calculated through the equation 3.
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𝑞
𝑥
𝐺𝑐ℎ𝑎𝑖𝑛 = 𝑛∙∆𝑇
(3)
where n is the number of molecules bridging the two adjacent graphene nanoribbons.
Molecular Dynamics modeling and results
To study the effect on thermal properties of organic chains bridging between two adjacent nanoribbons, MD
calculation were carried out as function of alkyl chain length, grafting density and type of bonding (covalent
or Van der Waals interaction) between the organic chains and the graphene nanoribbons.
As a first parameter evaluated, the length of the alkyl chains connecting the two graphene sheets via covalent
bonding on both nanoribbons was addressed as illustrated in Figure 1A-A’. Five different backbones chain
length were tested (in brackets the short name and the molecule length in elongated conformation): methyl
(C1; 2.46 Å), propyl (C3; 4.97 Å), pentyl (C5; 7.49 Å), eptyl (C7; 10.00 Å) and tridecyl (C13, 18.53 Å), with a
constant grafting density, defined as the number of chains per unit length of graphene armchair edges. The
second parameter addressed is the grafting density, while keeping constant the length of the covalently
bound chains. The maximum grafting density in this work was defined as one chain per aromatic ring along
the armchair graphene edge (4.26 Å spacing distance), referred to as 111. Lower grafting densities were
defined as one chain every two aromatic rings, referred to as 101, and one chain every 3 aromatic ring,
referred to as 100, as depicted in Figure 1A-A’’. In the third part of this work, alkyl chains covalently bound
on a single side were addressed, producing an interdigitated structure (Figure 1B) of alkyl chains grafted
alternatively on the two nanoribbons, aiming at the study of thermal conductance via Van der Waals forces
between the chains, in comparison with covalent bonding of chains on both nanoribbons.
Figure 1. Molecular junctions graphical description. A) The 101 C3 model top view shows for the C3 chain
length an alternate configuration between chains and empty sites while A’) represents the same
configuration for the longer C5 chain length and A’’) shows denser grafting scheme 111 is for the C3 chain
length. B) The interdigitated model junction detail with pending pentyl chains. C) The top view Molecular
Dynamics model for the evaluation of thermal conductance with highlighted reservoirs. VMD software is
used for all graphical representations35.
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The Energy added to the hot reservoir and removed from the cold reservoir versus time were found linear
and symmetric, with slopes dependent on chain length and grafting density. These observations evidence
that the total energy inside the ensemble is kept constant and a constant heat flux passes through the model.
Plots obtained for covalently bound chains C1 and C13 with the maximum grafting density are reported in
Figure 2A, as an example. Quantitatively, the slopes varied from 0.19 eV/ps for the longer C13 chain to 0.40
eV/ps for the shortest C1 chain (see Figure S1 for other chain lengths and grafting densities). However,
dependency of the slope with the chain length is not linear, as the slope is almost constant for C5 chains or
longer. Heat flux, qx, defined as the energy transferred from the atoms inside the hot reservoir to those inside
the cold reservoir at each timestep, was calculated from these curves.
The temperature profiles along the graphene nanoribbons and across the interface between those are
reported in Figure 2B for C1 and C13, 111 grafted chains, taken as an example. Despite a non-perfect
convergence to the thermostats value due to the exclusion of the boundary regions which classically shows
non-linearity close to the thermostats32, linear fitting of the temperatures along the graphene sheets allow
to extrapolate the temperature jump across the interface, which is a function of the length of the bridging
chain (see Figure S2 for all chain lengths and grafting densities). As expected, shorter chain bonding exhibits
a lower thermal jump between the graphene sheets compared to the longer ones, reflecting a better thermal
exchange through the junction, i.e. a higher TBC, accordingly with Equation 2. Similar temperature plots are
obtained for all of the chain length explored and the three different grafting schemes. Temperature jump
dependency on grafting density is in qualitative agreement with previous report form Liu et al22 in partially
overlapped nanoribbons.
Thermal boundary conductance was calculated from equation 2 and the related plot of the values is reported
in Figure 2C. The TBC rapidly decreased with increasing length up to C5 chain, whereas TBC for longer chains
appear to level off around 150 pW/K. In the diffusive regime, a decrease in the thermal conductance is
obviously expected with increasing chain length. In the case of an ideal material observing Fourier’s law, the
conductance should decrease linearly with increasing the chain length. However, it was previously reported
that the decrease in conductance of alkyl chains is not linear with the chain length36. Examples of asymptotic
decrease of conductance with increasing alkyl chain length were reported by several authors, both from
experimental measurements36 and computational work37-38. Different explanations have been proposed,
including quasi-ballistic transport, effect of direct mechanical coupling or the variation of phonon modes as
a function of chain length36. Figure 2C also reports the effect of the different grafting schemes: as expected,
the reduction in the number of bridging chains per unit length leads to a rapid decay of the interfacial thermal
conductance. Table 1 reports Gchain values calculated from Equation 3 for all the cross-linkers schemes and
chain lengths. Gchain values are found independent on the grafting scheme and therefore reflect the
contribution of single chains to the interface between two nanoribbons, confirming the additive effect of the
single linker reported in previous works10, 22. Quantitatively, conductance values are aligned with results
previously reported in the literature. Indeed, Li et al.24 reported a Gchain of about 160 pW/K at 300K for edgegrafted C11 alkane chain bridging two graphene nanoribbons, while Gchain of about 330 pW/K was calculated
for -CH2-CH2- linkers intercalated between graphene planes22. Conductance values of the same order of
magnitude (about 250 pW/K) were also reported for methylene linkers between carbon nanotubes23. Figure
2D reports the TBC as a function of the equilibrium temperature in the range between 200 and 500 K. Limited
variations (± 10% on the average values) were obtained and no clear trend was observed for thermal
conductance values with increasing temperature. We therefore assume thermal conductance approx.
constant in the explored temperature range.
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(B) 310
2000
C1
C13
1000
0
-1000
-2000
1000
2000
3000
Time (ps)
4000
5000
304
302
296
294
292
20
6000
111
101
100
6000
4000
3000
2000
40
60
80
100
120
140
160
(D)
111 C1
111 C3
111 C5
101 C1
101 C3
101 C5
100 C1
100 C3
100 C5
6000
5000
4000
3000
2000
1000
1000
0
0
C1
C3
C5
Chain backbone
C7
180
length (Å)
G (pW/K)
5000
G (pW/K)
306
290
0
(C)
C1
C13
308
Temperature (K)
NVT Energy (eV)
(A)
C13
200
300
400
500
Temperature (K)
Figure 2. (A) Energy added to the hot reservoir (positive slope) and removed from the cold reservoir (negative
slope) for C1 and C13 chains, as an example. (B) temperature trend along the nanoribbons for the shortest
C1 and the longest C13 chain lengths. (C) Thermal boundary conductance as a function of the backbone chain
length, each series represent a different grafting scheme. The lines guide the eye on the chain length effect.
Alternatively, a vertical reading direction highlights the thermal conductance as a function of grafting density.
The TBC uncertainty is estimated from variations obtained between simulations made using three different
seeds per run in velocities assignments, on selected chain length grafting configurations (See Table S1 in
supporting information for details). (D). Dependency of thermal conductance for C1, C3 and C5 chain lengths
in all grafting schemes at 200 K,300 K, 400 K and 500 K.
Table 1. Average single chain conductance (Gchain) for covalently bound chains.
Chain Backbone
C1
C3
C5
C7
C13
111 Gchain (pW/K)
455
252
169
169
163
101 Gchain (pW/K)
504
230
149
147
147
100 Gchain (pW/K)
442
215
137
151
156
Average Gchain (pW/K)
467 ± 31
232 ± 19
152 ± 16
156 ± 11
156 ± 8
Beside the effect of chains covalently bound on two adjacent nanoribbons, in this work the effect of chains
covalently bound on one graphene sheet and interacting via VdW bonding with chains attached on another
graphene sheet (Figure 1B) was also addressed. In fact, from the application point of view, this scenario is
much more relevant to the interaction between functionalized graphene nanosheets, provided
functionalization is able to induce a self-assembly of nanoplatelets. In the study of VdW interactions, the
chain length was fixed as C5, while the equilibrium distances between the nanoflakes was stepwise increased
at 9.8, 10.6, 11.5, 12.4 and 13.0 angstrom, directly reflecting a decrease in the interdigitation depth between
chains, as depicted in Figure 3.
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(A)
(B)
(C)
(D)
(E)
Figure 3. Interface detail of interdigitated models with various distance between nanoribbons: (A) 9.8 Å, (B)
10.6 Å, (C) 11.5 Å, (D) 12.4 Å and (E) 13.0 Å.
Figure 4 depicts the single chain conductance of the interdigitated models as a function of the platelets
distance. The single chain thermal conductance exhibits a linear decrease, from 131 to 24 pW/K when the
nanoribbons distance increases from 9.8 to 13.0 Å, which clearly depends on the extent of interaction surface
between interdigitated molecules. It is worth noting that conductivity values are lower than the primary
bonded C5 chain (152 pW/K) evaluated previously, because of the weaker interaction via VdW forces
compared to covalent bonding.
140
Gchain (pW/K)
120
100
80
60
40
20
0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
Platelets distance (Å)
13.0
13.5
Figure 4. Single chain thermal conductance of interdigitate models as a function of platelets distance.
A set of partially overlapped graphene nanoribbons with the same dimensions as the set used before (about
100 Å in length x 50 Å in width) was simulated to compare the value of their surface specific interfacial
conductance as a function of the overlapped surface (Figure 5) The platelets distance (3.40 Å) and offset
between layers were set corresponding to the actual graphite crystal lattice stacking. Overlapped area was
varied from one-half, to one-third and one-quarter of the nanoribbons surface, corresponding to 4395 Å2,
2973 Å2 and 2198 Å2, respectively. Thermal conductance for these overlapped areas therefore calculated as
1681, 1024 and 907 pW/K, respectively, yielding to an average of 0.38±0.04 pW/Å2K. Based on this value, an
equivalence in conductance was obtained between a single C5 chain (152 pW/K) and an area of 400±42Å2,
corresponding to 152±16 atoms per layer, assuming the previously reported39 atomic density value of 0.38
atoms/Å2. Such a comparison provides an estimation of correspondence between TBC through simple
graphene overlapping and trough covalent crosslinking.
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Figure 5. A one third partial overlapped nanoribbons, corresponding to a contact area of 2973 Å2 in VMD35
orthographic view.
Conclusions
The thermal boundary conductance between graphene nanosheets bonded by covalent molecular junctions
or via VdW interdigitated chains in head-head configuration was evaluated using classical non-equilibrium
Molecular Dynamics simulations. The role of differently bonded aliphatic chains of different length, grafting
density and bond type was evaluated. The simulation results highlighted a strict dependence of the TBC on
the bridging molecule length. Additionally, the individual thermal conductance for the alkyl chains was found
also in the range of 102 pW/K. The highest conductance was obtained for shorter chains (up to about 470
pW/K for C1 junction) with a dramatic decay with the chain length increase, until a conductance plateau is
reached (at about 150 pW/K). Moreover, the grafting density for this model was found linearly correlated to
the interfacial thermal conductance. The analysis of Van der Waals bonded interdigitated molecules revealed
a lower thermal conductance compared to covalent bonded ones, linearly decreasing as a function of the
platelets distance, i.e. the reduction in molecules interdigitation. Equivalence in terms of conductance was
also calculated for partially overlapping non-functionalized graphene, showing equivalence between a single
C5 chain with an overlapped area of 400±42 Å2, which corresponds to 152±16 carbon atoms per layer.
Covalent chain bridging is therefore a rather efficient way, yet experimentally challenging, to produce
molecular junction aimed at the efficient heat exchange into a percolating network of graphene nanosheets,
to be exploited in graphene nanopapers as well as in graphene-based polymer nanocomposites.
Authors’ contributions
A. Fina conceived the experiments, interpreted results and coordinated the project, A. Di Pierro carried out
MD simulations and post processing, G. Saracco contributed to the discussion of the results. Manuscript was
written by A. Di Pierro and A. Fina. All the authors have approved the final article.
Acknowledgements
This work has received funding from the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation programme grant agreement 639495 — INTHERM — ERC-2014-STG.
The authors gratefully acknowledge Dr. Bohayra Mortazavi at Bauhaus University Weimar (D) for continuous
advices and support during the preparation of this work. Dr. Diego Martinez Gutierrez at Politecnico di Torino
is also acknowledged for the useful discussions.
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References
1.
Geim, A. K., Graphene: status and prospects. Science 2009, 324 (5934), 1530-4.
2.
Novoselov, K. S.; Fal'ko, V. I.; Colombo, L.; Gellert, P. R.; Schwab, M. G.; Kim, K., A roadmap for
graphene. Nature 2012, 490 (7419), 192-200.
3.
Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Zhang, Y.; Dubonos, S. V.; Grigorieva, I. V.;
Firsov, A. A., Electric field effect in atomically thin carbon films. Science 2004, 306 (5696), 666-9.
4.
Mayorov, A. S.; Gorbachev, R. V.; Morozov, S. V.; Britnell, L.; Jalil, R.; Ponomarenko, L. A.; Blake, P.;
Novoselov, K. S.; Watanabe, K.; Taniguchi, T.; Geim, A. K., Micrometer-Scale Ballistic Transport in
Encapsulated Graphene at Room Temperature. Nano letters 2011, 11 (6), 2396-2399.
5.
Munoz, E.; Lu, J.; Yakobson, B. I., Ballistic thermal conductance of graphene ribbons. Nano letters
2010, 10 (5), 1652-6.
6.
Balandin, A. A., Thermal properties of graphene and nanostructured carbon materials. Nat Mater
2011, 10 (8), 569-81.
7.
Ghosh, S.; Calizo, I.; Teweldebrhan, D.; Pokatilov, E. P.; Nika, D. L.; Balandin, A. A.; Bao, W.; Miao, F.;
Lau, C. N., Extremely high thermal conductivity of graphene: Prospects for thermal management applications
in nanoelectronic circuits. Applied Physics Letters 2008, 92 (15).
8.
Lee, C.; Wei, X.; Kysar, J. W.; Hone, J., Measurement of the elastic properties and intrinsic strength
of monolayer graphene. Science 2008, 321 (5887), 385-8.
9.
Bianco, A.; Cheng, H. M.; Enoki, T.; Gogotsi, Y.; Hurt, R. H.; Koratkar, N.; Kyotani, T.; Monthioux, M.;
Park, C. R.; Tascon, J. M. D.; Zhang, J., All in the graphene family - A recommended nomenclature for twodimensional carbon materials. Carbon 2013, 65, 1-6.
10.
Han, H.; Zhang, Y.; Wang, N.; Samani, M. K.; Ni, Y.; Mijbil, Z. Y.; Edwards, M.; Xiong, S.; Saaskilahti, K.;
Murugesan, M.; Fu, Y.; Ye, L.; Sadeghi, H.; Bailey, S.; Kosevich, Y. A.; Lambert, C. J.; Liu, J.; Volz, S.,
Functionalization mediates heat transport in graphene nanoflakes. Nat Commun 2016, 7, 11281.
11.
Pollack, G. L., Kapitza Resistance. Reviews Of Modern Physics 1969, 41 (1), 34.
12.
Butler, S. Z.; Hollen, S. M.; Cao, L.; Cui, Y.; Gupta, J. A.; Gutierrez, H. R.; Heinz, T. F.; Hong, S. S.; Huang,
J.; Ismach, A. F.; Johnston-Halperin, E.; Kuno, M.; Plashnitsa, V. V.; Robinson, R. D.; Ruoff, R. S.; Salahuddin,
S.; Shan, J.; Shi, L.; Spencer, M. G.; Terrones, M.; Windl, W.; Goldberger, J. E., Progress, challenges, and
opportunities in two-dimensional materials beyond graphene. ACS nano 2013, 7 (4), 2898-926.
13.
Bigdeli, M. B.; Fasano, M., Thermal transmittance in graphene based networks for polymer matrix
composites. International Journal of Thermal Sciences 2017, 117, 98-105.
14.
Colonna, S.; Monticelli, O.; Gomez, J.; Novara, C.; Saracco, G.; Fina, A., Effect of morphology and
defectiveness of graphene-related materials on the electrical and thermal conductivity of their polymer
nanocomposites. Polymer 2016, 102, 292-300.
15.
Han, Z. D.; Fina, A., Thermal conductivity of carbon nanotubes and their polymer nanocomposites: A
review. Progress in Polymer Science 2011, 36 (7), 914-944.
16.
Alexeev, D.; Chen, J.; Walther, J. H.; Giapis, K. P.; Angelikopoulos, P.; Koumoutsakos, P., Kapitza
Resistance between Few-Layer Graphene and Water: Liquid Layering Effects. Nano letters 2015, 15 (9), 57449.
17.
Wang, Y.; Zhan, H. F.; Xiang, Y.; Yang, C.; Wang, C. M.; Zhang, Y. Y., Effect of Covalent Functionalization
on Thermal Transport across Graphene-Polymer Interfaces. J Phys Chem C 2015, 119 (22), 12731-12738.
18.
Shen, X.; Wang, Z. Y.; Wu, Y.; Liu, X.; Kim, J. K., Effect of functionalization on thermal conductivities
of graphene/epoxy composites. Carbon 2016, 108, 412-422.
19.
Georgakilas, V.; Otyepka, M.; Bourlinos, A. B.; Chandra, V.; Kim, N.; Kemp, K. C.; Hobza, P.; Zboril, R.;
Kim, K. S., Functionalization of graphene: covalent and non-covalent approaches, derivatives and
applications. Chem Rev 2012, 112 (11), 6156-214.
©2017. This manuscript version is made available
http://creativecommons.org/licenses/by-nc-nd/4.0/
under
the
CC-BY-NC-ND
4.0
license
20.
Kim, J. Y.; Lee, J. H.; Grossman, J. C., Thermal transport in functionalized graphene. ACS nano 2012, 6
(10), 9050-7.
21.
Gao, Y.; Muller-Plathe, F., Increasing the Thermal Conductivity of Graphene-Polyamide-6,6
Nanocomposites by Surface-Grafted Polymer Chains: Calculation with Molecular Dynamics and EffectiveMedium Approximation. The journal of physical chemistry. B 2016, 120 (7), 1336-46.
22.
Liu, X. J.; Zhang, G.; Zhang, Y. W., Thermal Conduction Across Graphene Cross-Linkers. J Phys Chem C
2014, 118 (23), 12541-12547.
23.
Chen, W.; Zhang, J. C.; Yue, Y. A., Molecular dynamics study on thermal transport at carbon nanotube
interface junctions: Effects of mechanical force and chemical functionalization. International Journal of Heat
and Mass Transfer 2016, 103, 1058-1064.
24.
Li, Q.; Duchemin, I.; Xiong, S. Y.; Solomon, G. C.; Donadio, D., Mechanical Tuning of Thermal Transport
in a Molecular Junction. J Phys Chem C 2015, 119 (43), 24636-24642.
25.
Li, Q.; Strange, M.; Duchemin, I.; Donadio, D.; Solomon, G. C., A Strategy to Suppress Phonon
Transport in Molecular Junctions Using pi-Stacked Systems. J Phys Chem C 2017, 121 (13), 7175-7182.
26.
Brenner, D. W.; Shenderova, O. A.; Harrison, J. A.; Stuart, S. J.; Ni, B.; Sinnott, S. B., A secondgeneration reactive empirical bond order (REBO) potential energy expression for hydrocarbons. J PhysCondens Mat 2002, 14 (4), 783-802.
27.
Barbarino, G.; Melis, C.; Colombo, L., Effect of hydrogenation on graphene thermal transport. Carbon
2014, 80, 167-173.
28.
Barbarino, G.; Melis, C.; Colombo, L., Intrinsic thermal conductivity in monolayer graphene is
ultimately upper limited: A direct estimation by atomistic simulations. Physical Review B 2015, 91 (3).
29.
Diao, C.; Dong, Y.; Lin, J., Reactive force field simulation on thermal conductivities of carbon
nanotubes and graphene. International Journal of Heat and Mass Transfer 2017, 112, 903-912.
30.
Henry, A.; Chen, G., High thermal conductivity of single polyethylene chains using molecular
dynamics simulations. Phys Rev Lett 2008, 101 (23), 235502.
31.
Mortazavi, B.; Ahzi, S., Thermal conductivity and tensile response of defective graphene: A molecular
dynamics study. Carbon 2013, 63, 460-470.
32.
Mortazavi, B.; Rabczuk, T., Multiscale modeling of heat conduction in graphene laminates. Carbon
2015, 85, 1-7.
33.
Cao, A., Molecular dynamics simulation study on heat transport in monolayer graphene sheet with
various geometries. Journal of Applied Physics 2012, 111 (8), 083528.
34.
Khan, A.; Navid, I.; Noshin, M.; Uddin, H.; Hossain, F.; Subrina, S., Equilibrium Molecular Dynamics
(MD) Simulation Study of Thermal Conductivity of Graphene Nanoribbon: A Comparative Study on MD
Potentials. Electronics 2015, 4 (4), 1109-1124.
35.
Humphrey, W.; Dalke, A.; Schulten, K., VMD: visual molecular dynamics. J Mol Graph 1996, 14 (1),
33-8, 27-8.
36.
Meier, T.; Menges, F.; Nirmalraj, P.; Holscher, H.; Riel, H.; Gotsmann, B., Length-dependent thermal
transport along molecular chains. Phys Rev Lett 2014, 113 (6), 060801.
37.
Segal, D.; Nitzan, A.; Hanggi, P., Thermal conductance through molecular wires. Journal of Chemical
Physics 2003, 119 (13), 6840-6855.
38.
Duda, J. C.; Saltonstall, C. B.; Norris, P. M.; Hopkins, P. E., Assessment and prediction of thermal
transport at solid-self-assembled monolayer junctions. The Journal of chemical physics 2011, 134 (9), 094704.
39.
Kurata, H.; Isoda, S.; Kobayashi, T., Quantitative Elemental Distribution Image of a Carbon Nanotube.
Microscopy Microanalysis Microstructures 1995, 6 (4), 405-413.
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http://creativecommons.org/licenses/by-nc-nd/4.0/
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SUPPORTING INFORMATION:
Molecular junctions for thermal transport between graphene nanoribbons:
covalent bonding vs. interdigitated chains
Alessandro Di Pierro, Guido Saracco, Alberto Fina*
Dipartimento di Scienza Applicata e Tecnologia, Politecnico di Torino, Alessandria Campus, Viale Teresa Michel 5, 15121
Alessandria, Italy.
*Corresponding author: Alberto Fina; e-mail address: alberto.fina@polito.it
0.45
100
101
111
0.40
NVT Slope (eV/ps)
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
C1
C3
C5
C7
C13
Chain length (Carbon atoms)
Figure S1. Energy versus time plot slopes for 300K simulation set. All the three grafting schemes are
represented.
©2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license
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100
101
111
20
19
Thermal Jump (K)
18
17
16
15
14
13
12
11
10
C1
C3
C5
C7
C13
Chain length (Carbon atoms)
Figure S2. Thermal jumps as a function of chain length for 300K simulation set. All the three grafting schemes
are represented.
©2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license
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Table S1. Thermal Boundary Conductances and single chains thermal conductances, energy versus time slopes
and thermal jumps for three different velocity seeds at 300 K: average values and deviations.
Grafting density
C3 chain
111
101
100
Grafting density
C3 chain
111
101
100
Grafting density
C3 chain
111
101
100
Grafting density
C3 chain
111
101
100
Seed 0
3045
1379
820
Seed 0
254
230
205
Seed 0
0.280
0.150
0.090
Seed 0
14.50
17.03
17.87
Seed 1
3150
1366
757
TBC [pW/K]
Seed 2
3200
1431
885
Average
3132
1392
821
Deviation
77
33
64
Seed 1
263
228
215
Gchain [pW/K]
Seed 2
267
238
221
Average
261
232
214
Deviation
7
6
9
Seed 1
0.270
0.140
0.100
NVT slope [eV/ps]
Seed 2
0.270
0.150
0.100
Average
0.273
0.147
0.097
Deviation
0.005
0.005
0.005
Seed 1
13.59
16.98
18.05
Thermal Jump [K]
Seed 2
13.64
16.79
18.09
Average
13.91
16.93
18.00
Deviation
0.46
0.12
0.11
©2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license
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