Probing Thermal Conductivity of Fullerene C60 Hosting a

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Probing Thermal Conductivity of Fullerene C Hosting a Single Water Molecule Yuan Gao, and Baoxing Xu J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b05663 • Publication Date (Web): 20 Aug 2015 Downloaded from http://pubs.acs.org on August 21, 2015

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The Journal of Physical Chemistry

Probing Thermal Conductivity of Fullerene C60 Hosting a Single Water Molecule Yuan Gao1, and Baoxing Xu1, 2∗ 1

Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22904, USA 2

Institute for Nanoscale and Quantum Scientific and Technological Advanced Research, University of Virginia, Charlottesville, VA 22904, USA

Abstract: Encapsulation of a single water molecule inside the fullerene C60 can yield a significant change on its fundamental properties. Through comprehensive reverse nonequilibrium molecular dynamics (RNEMD) simulations, we investigate the thermal conductivity of endohedral C60 fullerene containing a single water molecule. The encapsulated single water molecule enhances the thermal phonon propagation resistance of the fullerene C60 due to its interaction with carbon atoms, and brings down the thermal conductivity of the fullerene C60. Further simulations indicate that the thermal conductivity will depend on the external electrical field due to the polarity of the encapsulated water molecule. A larger electrical intensity will lead to a smaller thermal conductivity. A hierarchical model is proposed to quantitatively understand thermal behavior of the single endohedral fullerene and its enabled single-chain structures, and shows good agreement with simulations. Keywords: Endohedral fullerene H2O@C60; Thermal property; Hierarchical thermal model; Electric field; Phonon spectrum; Reverse nonequilibrium molecular dynamics (RNEMD)

∗

Corresponding author, Tel: +1-434-924-1038, Email: [email protected] 1 ACS Paragon Plus Environment


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1. Introduction Endohedral fullerene that contains a single molecule or ion has attracted great attention since the discovery of fullerene,1,2 and has exhibited a booming growth in recent years with emerging synthetic approaches such as mechanical insertion and chemical routes.3-8 The encapsulation of a single neutral guest water molecule has been innovated through a facial molecular surgery technique,9 and has been modified very recently to avoid the use of high temperature and pressures.8 Through these techniques, only one single water molecule is trapped in the C60 cage. More importantly, there is no physical bonding reaction between the encapsulated water molecule and carbon atoms and only nonbonding interaction between them are expected.8-13

With the encapsulation of a single water molecule inside, fundamental

properties of C60 dramatically change, and even show completely novel and unique properties.1417

For example, Insulated by the outer spherical carbon shell, the encapsulated single water

molecule shows an enhanced self-diffusion coefficient in comparison with that in bulk water.18,19 More importantly, the endohedral fullerene H2O@C60 possesses a dipole moment due to the polar nature of the trapped water molecule, altering the inherent physical properties of the empty fullerene C60.20-23 For example, the encapsulation of the single water molecule has promoted the solubility of C60 in polar solvents.24 The introduction of such a dipole moment is more apparent than that of the neutral water molecule when an ion is imprisoned inside, and has intrigued several new electronic and magnetic features in the endohedral fullerenes counterpart.25,26 The transformation of H2O@C60 has also been reported due to the chemical reaction between polar water molecule and nonpolar carbon shell under mechanical compression,27 and is of fundamental value to reveal the interaction between encapsulated molecule and outer carbon shell and response of the overall structure to external stimuli.

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As another crucial property, however, the thermal behavior of this class of endohedral fullerene molecules is still blank. In principle, the presence of the encapsulated molecule will interact with carbon atoms and may change the ability of thermal transport. Besides, the encapsulated molecule may provide an additional path for thermal transport through nonbonded molecule-carbon interactions, and contributes to the thermal behavior. Using extensive reserve nonequilibrium molecular dynamics (RNEMD) simulations, we present that the thermal conductivity of a single endohedral C60 fullerene is depressed by the encapsulated single water molecule. The depression also changes with an external electrical field due to the polar nature of the encapsulated water molecule, and higher magnitudes of electrical field will lead to smaller thermal conductivities. A hierarchical model of thermal resistance is proposed to quantitatively reveal thermal properties of the single endohedral fullerene and its single-chain structure, and agrees well with all simulations. The phonon density of states and vibrational spectrum of both C60 fullerene and encapsulated water molecule are also examined to qualitatively understand the underlying thermal flow mechanism. 2. Computational modeling and method Figure 1 gives the schematics of the computational modeling. Both hot and cold reservoirs were constructed by a 3.5nm × 3.0nm graphene sheet. After the verification of temperature maintenances, bilayer A-A stacked graphene structures were chosen for simplifying settings of force field and saving computational time, and will not affect the resolution of thermal properties of endohedral C60 fullerene.28,29 An endohedral C60 fullerene containing a single water molecule was positioned in the middle and connected with adjacent bilayer graphene through a double carbon-carbon covalent bonds with bond length of 0.142nm. The introduction of double bonds is to stabilize the C60 fullerene in simulations and the similar double-bonded sandwich 3 ACS Paragon Plus Environment


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structure of the C60 fullerene has also been employed in the study of electron-conduction properties.30 The outermost carbon atoms in each graphene layer were fixed to prevent atoms from sublimating. Bonded interactions among carbon atoms were modeled by using a Morse bond, harmonic cosine angle and 2-fold torsion full atomistic force field. The encapsulated water molecule was modeled by the flexible SPC/E model. Since there is only nonbond interatomic interaction between water molecule and carbon atoms,23, 27, 31-33 the 12-6 Lennard-Jones (LJ) empirical force field is employed to describe their interaction with the most popular parameters

σ c−o = 0.319nm , and ε c−o = 0.3126KJ / mol .34-36 Nonperiodic boundary condition was applied to all three directions. The recent developed reverse nonequilibrium molecular dynamics (RNEMD) method is applied to form a constant heat flow (q) from the hot reservoir to cold reservoir along the z-direction by rescaling the velocity of the atoms in both reservoirs.37 All simulations were performed by using LAMMPS.38,39 The simulation procedure consists of three steps: the simulation system was first equilibrated in NVT ensemble and the Nose-Hoover thermostat was used to maintain temperature to a desired environmental temperature (300K unless otherwise specified) for 0.25ns. Next, NVE ensemble was applied to relax the simulation system for 1.0 ns. Afterwards, a constant heat flow was applied to the system to generate a temperature gradient between hot and cold reservoirs. The data in the last 1.25ns after both hot and cold reservoirs reach equilibrium were taken for all calculations. 3. Results and discussion 3.1. Fundamental thermal properties of a single H2O@C60 After an initial equilibrium is performed in NVT ensemble, a heat flow of 0.007kcal/mol/fs is applied to the system. Figure 2a shows that the temperature in both hot and


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cold reservoirs is stabilized after a quick adjustment in NVE ensemble and is maintained well. The approximately stable evolution of temperature with simulation time indicates a constant temperature gradient between hot and cold reservoirs, and makes it possible to extract the thermal properties of H2O@C60 positioned in their middle by using Fourier’s law. Given the distribution of carbon atoms on the spherical shell with an additional water molecule inside, the temperature gradient in H2O@C60 does not show a good linearity along the heat flow direction (Figure 2b). This nonlinear temperature profile is mainly caused by the nonbonded interaction between encapsulated water molecule and carbon shell, and is similar with that observation in study of thermal properties for inhomogeneous or interfacial materials.29,40 Under this circumstance, the traditional definition of thermal conductivity based on Fourier’s law can be simplified to k = qL / (Th − Tc ) A ,41 where L is the sample length, and Th and Tc are the resultant temperature in the hot and cold reservoirs after thermal equilibrium, respectively. A is the crosssection area of heat flow through carbon shell thickness and is taken 2π drc , where rc is the van der Waals radius of carbon atoms (=0.17 nm),42 d is the averaged diameter of the carbon shell thickness along the heat flow z-direction, and can be calculated via d =

2 d /2 (d / 2)2 − z2 dz . ∫ − d /2 d

d is the diameter of H2O@C60. When the water molecule is encapsulated inside, the diameter of

C60 fullerene might have a slight change due to the charge transfer between water molecule and carbon atoms.43,44 However, such a small change is usually not considered in classic MD simulation,22,31 and here d is taken as the diameter of C60 fullerene (0.70 nm) for simplification.45,46 The effect of the heat flux, q , on the thermal conductivity of H2O@C60, k , is first examined. Figure 2c shows that k is independent of the magnitude of the applied heat flux,



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q . The heat flux of 0.007kcal/mol/fs will be employed in all following simulations, unless

otherwise specified. Figure 3 shows that the variation of thermal conductivity of H2O@C60, k , with the environmental temperature (T). As T increases, k increases, behaving an obvious temperature dependence. For comparison, by following the similar procedure with the same computational settings, the RNEMD simulations on an empty C60 is also performed and its thermal conductivity is plotted in Figure 3. Similar with that in H2O@C60, an increase of k with temperature is observed in the empty C60. These strong temperature dependences are also similar with the findings in narrow graphene nanoribbons.47,48 In addition, at the same temperature T, the thermal conductivity of H2O@C60 is lower than that of empty C60, suggesting that the encapsulation of water molecule in the fullerene suppresses the thermal transport through the non-bond interaction with carbon shell. The similar reduction in thermal conductivity has also been found in the carbon nanotube filled with water molecules due to the interaction between water molecules and carbon atoms in the confined environments. 49 3.2. Thermal properties of series H2O@C60 With the extraction of the thermal conductivity in a single H2O@C60, a number of H2O@C60 are connected with each other through a double carbon bond and used to investigate the thermal property of H2O@C60 single-chain structure. The length of the chain structure is Ln . Figure 4a shows the effect of the number of H2O@C60 elements, n , on thermal conductivity of H2O@C60 single-chain structure, kn . kn increases with the increase of n , indicating a strong size effect. The finite-size effect of the single-chain structure can be understood through the Matthiessen rule

49

that predicts a linear relationship between 1/ kn and Ln . Hence, the thermal 6 ACS Paragon Plus Environment

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conductivity for a single-chain structure with an infinity length, k ∞ , can be obtained by using a linear extrapolation to the curve of 1/ kn - 1/ Ln . Figure 4a shows that the extracted k ∞ of the H2O@C60 single-chain structure is 5.06 W/mK, on the same order with that of findgins in molecular chain structures,50 thus verifying the present scaling effect. In order to make further comparisons with the thermal properties of H2O@C60 single-chain structure, RNEMD simulations on the C60 single-chain structure were also carried out by replacing H2O@C60 with empty C60. Other computational conditions are kept the same. For a single-chain structure with empty C60 elements, Figure 4b shows the thermal conductivity increases with the increase of the number of C60 elements, n , indicating a similar trend with that of H2O@C60 single-chain structure. Following the same procedure with that performing on H2O@C60 chain structure, the thermal conductivity of infinity long C60 chain structure, k∞ , is obtained and is 11.78 W/mK, which is higher than that of H2O@C60 chain. The reduced k∞ in H2O@C60 chain implies that the encapsulated water molecule suppresses the thermal transport of carbon shells and is consistent with previous observations in the single element in Figure 3. 3.3. Thermal resistance model and theoretical prediction To quantitatively understand the thermal conductivity of H2O@C60, we estimate the thermal resistance via Rn = Ln / kn A . Figure 5a plots the relationship between the thermal resistance of the H2O@C60 chain structure, Rn , and the number of H2O@C60 elements, n , and a nonlinear variation is observed. This non-linear relationship of thermal resistance is also found in the empty C60 single-chain versus the number of C60 elements in Figure 5b. Since thermal resistance directly represents the resistance to the heat flow through an object or material under a temperature difference, an equivalent heat flux circuit can be used to characterize the thermal 7 ACS Paragon Plus Environment


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transport and is illustrated in Figure 6. This heat model is analogous to an electrical circuit, where the electrical resistance is heat resistance. For a single H2O@C60 (i.e. n = 1 ), parallel with the path via C60 shell, the encapsulated water molecule provides the second heat flow path through the carbon shell/water molecule/carbon shell composite interface. The corresponding thermal resistance for this path, Rcwc , is similar with that for the interface of graphene/water 2 molecules/graphene, Rgwg , and can be estimated approximately by Rcwc = Rgwg / π rw , where rw is

the van der Waals radius of a single water molecule. Taking rw = 0.14nm ,51 and

Rgwg = 6.8×10−8 m2 KW −1 ,52 one can have Rcwc of 1102GK / W . Meanwhile, the heat flux through the C60 shell is effected by the encapsulated water molecule, and the resultant thermal resistance, RC60− H O , can be estimated approximately through an effective thickness of C60 shell 2

via RC60− H O = 2

rc RC , where RC60 is the thermal resistance of a single empty fullerene C60 (Figure re 60

3, or Figure 5b at n = 1 ), and re is a half of the effective thickness of C60 shell in the presence of the water molecule. Taking re = 0.125nm , half of equilibrium distance between water molecules and graphene,53 one can have RC60−H O = 1.52GK / W , several orders of magnitude lower than 2

Rcwc . Therefore, the heat flow through the interface of carbon shell/water molecule/carbon shell is nearly “blocked”, and mainly goes through the C60 shell with an effective thickness re . The total thermal resistance of a single H2O@C60 is R1 = RH2O @C60 = RC60−H O Rcwc / ( RC60−H O + Rcwc ) , and is 2

2

1.52GK / W , consistent with the simulation result 1.37GK / W ( Figure 3, or Figure 5a at n = 1 ). We should note the effective thickness re depends on the species of encapsulated molecule and


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their polarity. For instance, a stronger polarity will lead to a smaller effective thickness of C60 shell re due to stronger interactions,54,55 and thus a smaller thermal resistance is expected. When a number of H2O@C60 are stringed to form a single-chain structure ( n > 1 ), the interaction among H2O@C60 needs to be included. For instance, when there are two H2O@C60 elements in the chain structure, the total thermal resistance Rn = n( R1 − r1 ) + r1 , at n = 2 , where

r1 represents the thermal resistance due to their interaction between each other. Given the cutoff radius (1.0nm) used in simulations, when there are more than two H2O@C60 elements, we consider the effect of interaction among only three H2O@C60 elements, and the thermal resistance caused by interactions from every other element is defined as r2 . The total thermal resistance is Rn = n( R1 − r1 − r2 ) + r1 + 2r2 , at n ≥ 3 . This resistance model predicts a good linear increase at the number of H2O@C60 number, n ≥ 3 , and agrees with simulations (Figure 5a). The agreement between the theoretical predication and simulation has also been confirmed in the empty C60 single- chain structure (Figure 5b), where R1 corresponds to the thermal resistance of empty C60. Generally, if the size of a single element in the single-chain structure is smaller, it will interact with more neighbors, and vice versa.56 The linear dependence of total thermal resistance, Rn , with the number of elements, n , will start from m , where m is the maximum number of neighbors in one direction that the single element can interact with. 3.4. Effect of external electrical fields on thermal conductivity Given the polarity of encapsulated water molecule, a dipole moment for endohedral fullerene H2O@C60 exists and thus endohedral fullerene is sensitive to an external electrical field.20, 22, 57 We further investigate the variation of thermal conductivity of a single endohedral 9 ACS Paragon Plus Environment


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fullerene H2O@C60 with electrical intensity. An external uniform electrical field, E , was applied to the simulation system in parallel with the heat flux z-direction, and all other computational conditions were kept the same. When an external uniform electrical field, E , was applied, its reduction due to water polarization and C60 shell was not considered.58,59 Simulations (Figure 7) show there is a slight decrease in k with the increase of electrical intensity, and this decrement is independent of the direction of electrical field. To reveal mechanism of thermal flow through H2O@C60 in the presence of electrical filed, the polar orientation of the encapsulated H2O molecule is studied after the hot and cold reservoirs reach equilibrium. Its inclination angle with regard to the electrical field direction (i.e. z) , α , is −1 defined via α = cos ( pµˆ / | p |) .60 Here, µˆ is a unit vector along the z -direction, and p is the

instantaneous dipole of the H2O molecule pointing from the hydrogen shell to the oxygen shell. In the absence of electrical field, α varies randomly between 0o to 180o (Figure 8a). At an electrical field (3 V/nm), α is approximately stabilized to an average of 24.2o with a perturbation of 13.33 o. The decreased α indicates that the motion of water molecule is restricted by the alignment effect on dipole from the applied electronic field, and the perturbation is caused by thermal gradient due to the interaction between water molecule and carbon shell. When E increases to 6 V/nm, α shows a decrease in both average (~19.34 o) and perturbation (~10.28 o) due to a stronger alignment effect on dipole of water molecule closer to the direction of the electrical field (Figure 8b). Besides, the encapsulated water molecule will vibrate cooperatively with the outer carbon shell owning to the interaction between them. Once its motion is influenced by the external electric field, the vibration motion of the carbon atoms and the resultant thermal transport are expected to be restricted. This restriction further suppresses the


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effective thickness of thermal flow, re (corresponding to a reduced equilibrium distance between water molecule and carbon wall in an external electrical field),55 thus leading to a higher RC60− H O , 2

and a lower k . In addition, the symmetric fluctuation of α with regard to 90o (i.e. perpendicular to the heat flow direction) shows a good consistence with the approximately symmetric variation of k with regard to the external electrical direction. 3.5. Thermal transport mechanism and phonon spectrum In order to understand the molecular/atomistic mechanism of thermal conductivity and heat resistance of the endohedral C60 fullerene, the commonly used physical parameters- the phonon spectrum is examined. Since the heat flow is generated in the z-direction, the phonon ∞

spectrum in the z-direction is first investigated via Gz (ω) = ∫ e−iωt 0

< vz (t ) ⋅ vz (0) > dt , where Gz is < vz (0) ⋅ vz (0) >

the phonon spectrum in the z-direction, ω is the frequency, v z (t ) is the z-component of atomic velocity vector and ⋅ denotes the average over associated atoms. To study the influence of the encapsulated water molecule, its vibrational spectrum is also plotted by replacing vz with the velocity vector of the water molecule v . Figure 9a shows the phonon spectra of carbon atoms in the z-direction for both empty fullerene C60 and endohedral fullerene H2O@C60 accompanied by vibrational spectrum of the water molecule. In low frequencies (< 25 THz), the phonon modes occur at almost the same frequency. However, the average magnitude of phonon spectrum is slightly higher in the empty C60 than that in the H2O@C60, indicating the heat capacity of phonon modes is reduced in the presence of water molecule. In higher frequencies (>40THz), the magnitude of each phonon mode for H2O@C60 and empty C60 is close, but there are obvious shifts of modes to low frequencies when the water molecule is encapsulated in the empty C60.



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For instance, the modes at the frequency 59.3THz and 57.5 THz in the empty C60 have shifted to 56.0 THz and 54.2 THz in the H2O@C60, respectively. The shifting phenomenon of the phonon modes to low frequencies usually is referred to as the red shift and corresponds to the reduction of group velocity of phonon groups. According to the lattice thermal transport theory,61 either a smaller group velocity at high frequencies or reduced heat capacity at low frequencies will lead to a lower thermal conductivity. Therefore a lower thermal conductivity of H2O@C60 than that of the empty fullerene C60 is observed in Figure 3. When an external electrical field is applied to the H2O@C60 system along with the temperature gradient, Figure 9b shows that the vibrational spectrum of the water molecule varies, indicating that the motion of water molecule is influenced by the applied electrical field. Besides, the magnitudes of phonon modes seem to decrease, particularly at 22.5THz and 25.3THz. A lower magnitude implies a lower phonon heat capacity, and thus a lower thermal conductivity. Further comparisons show that phonon modes at the electrical field of 6.0 V/nm and -6.0 V/nm appear almost at the same frequencies with similar averaged magnitudes, suggesting independence of the thermal conductivity of the direction on the electrical field. These findings are also in consistence with simulation results in Figure 7. Considering the spherical shell structure of the fullerene C60, phonon modes in other directions may also play an important role in thermal transport. To study their effects, the total ∞

−iωt spectrum of carbon atoms is further investigated via G(ω) = ∫ e 0

< v(t ) ⋅ v(0) > dt . Figure 10a < v(0) ⋅ v(0) >

shows the total phonon spectrum of carbon atoms in both empty fullerene C60 and endohedral fullerene H2O@C60. Similar with observations in Figure 9a, in low frequencies, the frequency of phonon modes is close in the empty C60 and H2O@C60, but the magnitude is lower in H2O@C60; 12 ACS Paragon Plus Environment

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and in high frequencies, the presence of the water molecule have shifted the phonon modes to low frequencies. However, for the phonon modes of the empty C60 at the frequencies from 25 THz to 40 THz, the magnitude is relatively higher in total spectrum than that in the z-direction, which is expected to be contributed by phonon activities in the x and y -directions. When the water molecule exists, the magnitude enhancement of these activities shows an obvious decrease due to the interaction between water molecule and carbon shell, lowering the phonon group heat capacity of H2O@C60. The reduced phonon activities in the x and y -directions also imply the lower thermal conductivity of H2O@C60 than that of the empty C60, and are consistent with the MD simulations (Figure 3) Under an external electrical field, Figure 10b shows that the magnitude of majority of modes in the total spectrum of H2O@C60 decreases while the number and frequency of modes remain nearly the same, indicating a lower heat capacity of phonons and a lower thermal transport performance. These results are also similar with observations of the phonon models in the z-direction (Figure 9b). Further comparison between Figures 9b and 10b demonstrates that the magnitude of phonon models at the frequencies less than 40THz shows more severe suppressions in the total spectrum than that only in the z-direction. Such aggravated suppressions are also expected to be led by the reduced phonon activities in the x and y -direction due to the presence of the water molecule, and agree well with a decreased thermal conductivity in the presence of electrical field (Figure 7). 4. Concluding remarks We present extensive evidences on the thermal properties of a single endohedral C60 fullerene containing a single water molecule and its single-chain structure through RNEMD



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simulations. Simulations show that the thermal conductivity of H2O@C60 is lower than that of the empty C60 due to the interaction of encapsulated water molecule and carbon atoms. The presence of the encapsulated water molecule suppresses the effective radius of thermal transport through the carbon C60 shell and decreases the thermal conductivity of C60 fullerene. The influence of the thermal transport through the interface of carbon shell/the encapsulated water molecule/carbon shell can be neglected due to the weak nonbonded interaction. Besides, the thermal conductivity of H2O@C60 shows dependence on an external electrical field due to the polar nature of the encapsulated water molecule. As the magnitude of the electrical field increases, the thermal conductivity of H2O@C60 decreases. For the chain structures with a series of unit H2O@C60 elements, the thermal conductivity increases with the increase of the number of units. This size effect is captured through the proposed hierarchical thermal resistance model by considering the interactions among units, and the predicted linearity at the number n ≥ 3 confirms the simulation results. The underlying mechanism underpinned by the encapsulated molecule is further revealed through a systemic study of phonon density of states of carbon atoms and vibrational spectrum of the encapsulated water molecule. Analysis on the spectrum of carbon atoms in both heat flow z-direction and all three directions indicates that the presence of the encapsulated water molecule leads to the red shift of phonon models of C60 fullerene and decreases the activities of phonon groups. The present study is expected to facilitate the applications of endohedral fullerenes with decreased thermal properties in thermal management system. In particular, the electrically dependent thermal conductivity of endohedral C60 fullerenes shed lights on materials design with adjustable thermal properties through inner or external fields. Acknowledgement This work is supported by the start-up funds at the University of Virginia. 14 ACS Paragon Plus Environment

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Figure 1: Molecular simulation model to probe the thermal conductivity of endohedral C60 fullerene. The endohedral C60 fullerene hosting a single water molecule is connected to two adjacent graphene layers by a double carbon bond. Each two-layer 3.5nm×3.0nm graphene with a separation of 0.34nm was to model the hot (red) and cold (blue) reservoir, and a heat flux (q) was introduced by scaling velocity of carbon atoms on the graphene layers. The outermost carbon atoms (grey) of all graphene layers were fixed. A nonperiodic boundary condition was used in all three directions.



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Figure 2: Temperature gradient on a single H2O@C60, and its thermal conductivity, k , as a functional of heat flux, q . (a) Evolution of temperature in hot and cold reservoirs with an initial environmental temperature of 300K. A steady temperature in both hot and cold reservoirs is obtained after a heat flux of 0.007kcal/mol/fs is applied at t=1.25ns. (b) Temperature gradient on a single H2O@C60 along the heat flow z-direction. Two groups of local temperatures are extracted, one is 10 carbon atoms per group and the other is 5 carbon atoms per group. The red and blue squares indicate the position of hot and cold ends respectively, and the black circle show the position of encapsulated single water molecule which is approximately in the center of C60. (c) Variation of thermal conductivity, k , of a single H2O@C60 with the applied heat flow, q .


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Figure 3: Comparison of thermal conductivity of a single empty C60, k, and endohedral fullerene H2O@C60 at elevated environmental temperatures, T .

Figure 4: Thermal conductivity of series H2O@C60 and empty C60 single chain structure. (a) Thermal conductivity of a single-chain H2O@C60 structure, k n , where n is the number of H2O@C60 elements, and Ln is the total length of the chain structure. The inset shows the H2O@C60 single chain structure with the number of elements. (b) Variation of thermal conductivity of the single-chain empty C60 structure, k n , with the number of C60 elements n and total length of chain, Ln . The inset shows the single-chain empty C60 structure with the number of elements.



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Figure 5: Thermal resistance of series H2O@C60 and empty C60 single-chain structure. (a) Variation of thermal resistance of H2O@C60 single-chain structure, Rn , with the number of H2O@C60 elements, n .(b) Thermal resistance of C60 single-chain structure, Rn , as a function of number of C60 elements, n .

Figure 6: Hierarchical thermal resistance model of H2O@C60, where R1 , R2 , Rn represent the thermal resistance of chain structure with the number of H2O@C60 elements, 1, 2 and n , respectively, RC60 −H2O represents the thermal resistance of C60 fullerene with an effective C60 shell thickness in the presence of the encapsulated water molecule, and Rcwc represents the interfacial resistance of C60 shell/water/C60 shell structure. r1 and r2 represent the thermal resistance due to force interactions of H2O@C60 from its first and second nearest neighbors.


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Figure 7: Variation of thermal conductivity of a single H2O@C60, k , with an external electrical intensity, E . The inset shows the direction of applied electrical field in parallel with the direction of the applied heat flux.

Figure 8: Variation of polar orientation of the encapsulated H2O molecule, α , with regard to the applied electrical field in the z-direction under both thermal gradient and electrical intensity of (a) ± 3.0 V/nm and (b) ± 6.0 V/nm.



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Figure 9: Phonon spectrum of carbon atoms in the thermal flow z-direction and the vibrational spectrum of the encapsulated water molecule. (a) Phonon spectrum of carbon atoms in the empty C60 and endohedral fullerene H2O@C60 (top), and the spectrum of the encapsulated single water molecule (bottom). (b) Phonon spectrum of carbon atoms in H2O@C60 (top), and the spectrum of the encapsulated single water molecule (bottom) with/without an external electrical field.

Figure 10: Total phonon spectrum of carbon atoms. (a) Phonon spectrum of carbon atoms in the empty C60 and endohedral fullerene H2O@C60. (b) Phonon spectrum of carbon atoms in H2O@C60 with/without an external electrical field.


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Table of Contents (TOC)

Probing Thermal Conductivity of Fullerene C60 Hosting a Single Water Molecule Yuan Gao1, and Baoxing Xu1, 2∗ 1

Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22904, USA 2

Institute for Nanoscale and Quantum Scientific and Technological Advanced Research, University of Virginia, Charlottesville, VA 22904, USA

∗

Corresponding author, Tel: +1-434-924-1038, Email: [email protected] 25 ACS Paragon Plus Environment

Probing Thermal Conductivity of Fullerene C60 Hosting a

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