Tunable Mechanical and Thermal Properties of One-Dimensional

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Tunable Mechanical and Thermal Properties of One-dimensional Carbyne Chain: Phase Transition and Microscopic Dynamics Xiangjun Liu, Gang Zhang, and Yong-Wei Zhang J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b08026 • Publication Date (Web): 30 Sep 2015 Downloaded from http://pubs.acs.org on October 3, 2015

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The Journal of Physical Chemistry

Tunable Mechanical and Thermal Properties of One-dimensional Carbyne Chain: Phase Transition and Microscopic Dynamics Xiangjun Liu†, Gang Zhang†*, Yong-Wei Zhang† †

Institute of High performance Computing, A*STAR, Singapore 138632 Abstract

Recently, carbyne chain, the one-dimensional sp-hybridized carbon allotrope in the form of either α-carbyne (polyyne) with alternating single and triple bonds or β-carbyne (cumulene) with repeating double bonds, has attracted more and more attention. However, the mechanical and thermal properties of individual phase, their phase transition dynamics and defect formation remain largely unknown. Our molecular dynamics simulations show that the critical temperature for the phase transition from cumulene to polyyne is 499 K, and the phase transition is ultrafast and completed within 150 fs. To achieve perfect polyyne, however, refined temperature control is needed so as to avoid defective bonds. The bending stiffness and Young’s modulus of cumulene are significantly higher than those of polyyne, while both of them are comparable to the hardest natural materials. The large difference in the stress-strain behavior between cumulene and polyyne provides a novel route for storing mechanical energy. Furthermore, the thermal conductivity of cumulene is found to be two times higher than that of polyyne, and the defective bonds can dramatically decrease the thermal conductivity of polyyne to only 13% of that of pristine polyyne. The significant changes in the mechanical and thermal properties between the two phases of carbyne chain present a great opportunity for its use as strain sensors, mechanical connectors and mechanical/thermal energy storage devices.

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Introduction Carbon allotropes have attracted continuous interest due to their stability and various fascinating properties.1 Carbon atom is capable of forming various allotropes: three-dimensional sp3 diamond, two-dimensional sp2 graphene,2 quasi-one-dimensional carbon nanotube,3 or zerodimensional sp2 fullerenes.4 Each of them has notably different electronic and mechanical properties. Diamond is a wide band gap insulator and one of the hardest natural materials. On the contrary, graphene has a semimetal electronic structure with linear band dispersion and extraordinarily high electron mobility.5,6 One of the newest potential carbon allotropes is the monoatomistic chain of sp-hybridized carbon, and its bulk modification is known as carbyne. Although the existence of this exotic phase of bulk carbon is still debated until today,7 the isolated chains have been observed in several experiments, for example, in gas-phase deposition, epitaxial growth, electrochemical synthesis, or pulling the atomic chains from graphene or carbon nanotubes.8−19 The longest isolable carbon chains was produced in solution with a length of about 60 Å.17 Theoretical21−23 and experimental studies have revealed interesting properties for carbon chains. A sp carbon chain can have two possible types of structures: cumulene with repeating double bonds over the whole chain, ( = C = C =) n ; or polyyne with alternating single and triple bonds, ( −C ≡ C −) n . In the symmetric cumulene structure, it is characterized by σ-bonds along the axis (s-px orbitals), and two decoupled π-bonds per atom from the perpendicular py and pz orbitals. The π-electrons are uniformly distributed over the chain, which makes cumulene a metallic conductor. On the other hand, in the symmetry-breaking polyyne, the electrons are mostly localized at the triple bonds, thus polyyne is expected to be semiconducting. As a 1D conductor, cumulene can be destabilized by Peierls distortion, so that polyyne with alternating 2 ACS Paragon Plus Environment

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bond lengths becomes energetically favoured24−25 with a band gap at the edge of Brillouin zone to lower the total energy per atom by 2 meV. As the band gap increases with increasing strain,25−27 the Peierls distortion also increases in strained carbon chains compared with the freestanding ones, which has been observed experimentally in recent studies.28 These novel electronic properties and strain-induced metal-semiconductor transition are promising in the applications of electromechanical switching, strain sensor, and atomistic scale circuit26,29-31. To drive the progress of these applications, an in-depth theoretical understanding of the structural transition between cumulene and polyyne is necessary. Although polyyne is believed to be more energetically favorable, a recent theoretical study by Artyukhov et al. suggested that in unstrained carbyne chains, the energetic gain by the Peierls instability is so small that it can be overcompensated by zero-point vibrations.27 This finding is critically important since it predicts that metallic cumulene can exist under specified conditions. Thus the cumulene-polyyne transition temperature in mechanically relaxed chains is an important question that deserves further studies. Although many theoretical studies of carbon chains have been performed in the past decade, most studies focused on the strain-induced transition, while the temperature effect and dynamical information about the bond-rearranging process during phase transition is still very limited. Furthermore, compared with the large amount of studies of electronic properties, the thermal properties of 1D carbon chain remain elusive. Molecular dynamics (MD) simulation is a powerful tool to handle many-body problems at atomic level. Besides predicting the intrinsic thermal properties without the thermodynamic-limit assumption,32 the MD simulation is also very efficient for studying mechanical properties33 and structural changes34 of nanomaterials. In MD simulation, the Newton's equation of motion for each atom is numerically solved with the interatomic force derived from the system’s potential 3 ACS Paragon Plus Environment


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energy. In the present work, we perform molecular dynamics simulations using the firstprinciples based ReaxFF35,36 force field to address the questions raised above, aiming to not only reveal new insights into the fundamental properties of carbyne, but also provide useful design guidelines towards the functional use of this novel material. Computational Methods In this work, MD simulations are employed to study the mechanical and thermal properties of carbyne using the LAMMPS package.37 In all MD simulations performed here, the ReaxFF force field,35,36 an empirical bond-order-dependent potential, is used to describe the reactive, (single-, double-, and triple-) covalent bonding interactions, in which dissociation and reaction curves were obtained from fitting to quantum mechanics calculations. ReaxFF38 is based on a bond order/bond distance relationship, a concept introduced by Tersoff39 and first employed to carbon chemistry by Brenner40. Bond orders ( BOij' ), including contributions from σ, π, and π-π bonds, are calculated from the interatomic distances, using equation, '

σ

π

ππ

ij

ij

ij

ij

BO = BO + BO + BO

p p p     r ij  bo 2   r ij  bo 4   r ij  bo 6  .          = exp p + exp p + exp p  bo1  σ    bo 3  π    bo 5  ππ    r 0    r 0    r 0     

(1)

The instantaneous bond orders are updated every MD step, thus the bond-order-dependent ReaxFF provides accurate descriptions of bond breaking and bond formation during simulation. This allows us for the possibility to catch the instantaneous bond state in the simulation. In addition, ReaxFF also accounts for polarization effects by using a geometry-dependent charge calculation scheme. Thus, the bond-order based ReaxFF method is built to bridge the gap between Quantum mechanics method and conventional molecular dynamics method. ReaxFF partitions the overall system energy into contributions from various partial energy terms, include the terms of bond, lone pair, over-coordination, under-coordination, angle strain, 4 ACS Paragon Plus Environment

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angle conjugation, C2 correction, torsion, torsion conjugation, hydrogen bond to properly handle the nature of preferred configurations of atomic and molecular orbitals, and other terms of van der Waals and Coulomb interactions.38-40 The latter two non-bonded interactions are calculated between each atom pair, irrespective of connectivity, and are shielded to avoid excessive repulsion at short distances. This treatment of non-bonded interactions allows ReaxFF to describe ionic, covalent, and intermediate materials, thus, greatly improving its transferability. In this paper, the version of ReaxFF force field (ReaxFFCHO) and its parameters are adopted from Ref. 36, which is developed for carbon-rich hydrocarbon materials. The ReaxFFCHO expanded the ReaxFF reactive force field training set to include additional transition states and chemical reactivity of systems relevant to these reactions and optimized the force field parameters against a quantum mechanics-based training set. The ReaxFFCHO can accurately provide the chemical reaction process of dehydrogenation, e.g. from H3 C − C H2 − C H2 − C H3 to HC ≡ C − C ≡ CH .36 It has also been demonstrated that the ReaxFF potential provides a reasonable

account of the chemical and mechanical behavior of many carbon-based materials,41−44 including carbyne.44 The velocity Verlet algorithm is employed to integrate Newton’s equations of atom motion numerically with a time step of 0.1 fs. First, the system was equilibrated at a constant target temperature T for 100 ps using Nosé-Hoover temperature thermostat45 (NVT ensemble). After the constant temperature relaxation, we continued to relax the system with NVE (constant volume and no thermostat) ensemble for 100 ps. During this stage, the total energy and temperature of the system were monitored. We found that the total energy was conserved and the temperature of the entire system remained constant with fluctuations around T, which means that the system has reached equilibrium. It is worth pointing out that, in order to take the quantum effects of phonon occupation into account, the temperature calculated from the MD simulation 5 ACS Paragon Plus Environment


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2 2 3 (TMD) has been corrected using the quantum correction scheme46,47 T MD = T2 ∫0T T dx xx , where e −1 TD D

TMD, TD (= 517 K for carbyne48) and T are the MD temperature, Debye temperature and corrected temperature, respectively. Results and Discussions Temperature Induced Phase Transition First, we explored the temperature induced phase transition of carbyne from cumulene to polyyne. In experiments, neutron-diffraction technique gives the most accessible information on the microscopic structures of a material. Usually, the bond-length distribution function calculated from MD simulations can provide comparable results to the neutron-diffraction data on atomistic structure.49 In our MD simulation, a cumulene chain is initially constructed and relaxed at 5 K. The temperature is then increased with a rate of approximately 0.04 K/fs. Figure 1a shows the bond-length distribution function of carbyne chain at 498 K (Similar distributions at temperature lower than 498K are not shown here). A single peak is clearly shown, with an equilibrium bond length of 1.37 Å. However, at a higher temperature such as 500 K, the bond-length distribution is quite different, as shown in Figure 1b (Similar distributions at temperature higher than 500K are not shown here). At this temperature, the distribution peak at 1.37 Å disappears, while two separated peaks appear at 1.21 and 1.58 Å, respectively. In DFT studies, the bond length alternation (BLA, defined as the difference between the long and short bonds) is usually used to describe the phase transition. An increase of BLA from cumulene (0.088Å) to polyyne (0.248Å) is predicted by HSE06 calculations.26 Our MD simulation results of single/double/triple bond lengths are in a reasonable agreement with those (1.54Å, 1.35Å, and 1.20Å for single, double, and triple bond, respectively) from first-principles calculations.50 6 ACS Paragon Plus Environment

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Clearly, the critical temperature of phase transition is between 498K and 500K. With further refined calculations, the critical temperature is found to be 499K, below which only carboncarbon double bonds are observed, while above which all double bonds change to alternating single and triple bonds. Using DFT calculations together with analytic modeling, Artyukhov et al.27 studied the strain-temperature “phase diagram” of the cumulene-polyyne transition and found in unstrained carbyne, the transition temperature is about 800K. The differences between our MD predicted transition temperature and the value from DFT could be due to the following reasons. First, in principle, each phonon mode is equally excited in classical MD simulations, which is different from the quantum statistical distribution. Furthermore, the periodic boundary condition is used along the axis of carbyne chain in our MD simulations. Note that adoption of different boundary conditions has been found to have certain influence on the atomic structure of carbyne chains in DFT calculations.24,51,52 Second, the ReaxFF potential could not exactly reproduce the force field predicted by first-principles calculations, while the strength of Peierls distortion is extremely sensitive to the force field.26 Third, in DFT calculations, the well-known underestimated band gap53,54 may result in a much weakened Peierls instability, which may lead to an overestimated transition temperature. Overall, the most important characteristic in the transition process is well revealed by our MD calculations. A significant advantage of MD study is its ability to provide a detailed description of dynamics process, including bond breaking and creating. To explore the dynamics picture of the phase transition process, we record the time-dependent bond-length distribution function at the critical temperature of 499 K. As shown in Figure 2a, initially, only double-bonds exist in the system. Local perturbations created by the vibrations of carbon atoms in the cumulene chain 7 ACS Paragon Plus Environment


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induce the nucleation of a pair of single- and triple-bonds at one carbon atom. This leads to its adjacent double bonds to continue to convert to triple-/single-bonds, causing a spontaneous phase transition from cumulene to polyyne. This phase transition process is reflected in the decreased distribution peak of double-bonds and the increased distribution peaks of single- and triple-bonds. It is interesting to note that the phase transition process is ultrafast. For a carbyne chain with 50 atoms, it only needs 145 fs to complete the phase transition process, as shown in Figure 2a−2e. The ideal polyyne atomic structure with alternating single- and triple-bonds is predicted when the temperature is increased from 5K to 499K with a sufficiently slow heating rate. However, if the heating rate is high (e.g. 2.5K/fs) or if the cumulene chain is initialized under a temperature over 499K, several non-ideal bonds are observed as shown in Figure 2j. The phase transition process is shown in Figure 2f−2j. Unlike the representative bond-length distribution function in perfect polyyne, there exists one additional peak at 1.45 Å, which is shorter than that of the single-bond, but longer than the double-bond and triple-bond. In this work, for simplicity, this bond is termed as “defective bond”. Similar “soliton” defects have also been reported in polyethylene chains.55 This phenomenon indicates that, during the fabrication/formation process, temperature control is a critical factor to produce perfect polyyne. In previous DFT studies 26, 27, the bond length alternation is used to evaluate the strain induced phase change, which is defined as the difference between the long and short bonds. However, as the length of the defective bond is within the lengths of single- and triple-bond, this defective bond was ignored by previous DFT studies. Mechanical Properties of Carbyne


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Next, we investigated the bending rigidity of both cumulene and polyyne chains. As the focus here is the bending rigidity of carbyne chains, a curvature is imposed on the carbyne chain by forming a ring structure. The potential energy of the ring (Er) is obtained by energy minimization with an energy tolerance of 4.3×10-8 eV. To directly assess the bending energy of the ring, we calculated the potential energy of a linear chain with the same atom number (Ec), as shown in Figure 3a. The difference between these two energies is the bending energy (Eb = Er – Ec). A series of ring structures are constructed with carbon atoms N from 8 to 50, with associated ideal radii (R) of approximately 1.70 to 10.66 Å for cumulene rings and 1.82 to 11.07 Å for polyyne rings, respectively. The results of bending energy of cumulene and polyyne are shown in Figure 3b. These carbyne rings represented a set of structures with constant curvatures, ρ (= R-1). Combing the bending energy with the minimized perimeter L of each ring resulted in the normalized energy per length, EL (= Eb/L eV/Å), as shown in Figure 3c. It is found that the bending energy of cumulene chain is higher than that of polyyne chain with the same atom number N, but as ρ decreases (N increases), the difference gradually decreases and it disappears when ρ is below 0.09 Å-1. In continuum elastic theory, the elastic bending energy Eb of a curved element can be calculated as26,44

Eb =

1 L 2 ∫0 Dρ dsˆ, 2

(2)

where D is the bending stiffness and ρ is the curvature. Assuming a constant bending stiffness D, the normalized bending energy per length is:



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EL =

E b = 1 Dρ 2 . L

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(3)

2

Figure 3d shows the normalized bending energy per length EL versus ρ2. When ρ2 is less than 0.1 Å-2, the calculated normalized bending energy depends linearly on ρ2, which is in good agreement with the relationship described in Eq. 3. In this curvature range, the fitted bending stiffness D are 8.5 eV·Å for cumulene chain, and 6.7 eV·Å for polyyne chain. Hence, cumulene has a 27% higher bending stiffness than polyyne, which results from the fact that the sp2 conjugated double-bond in cumulene forms a more rigid linear structure with a less amplitude of bending/torsional motion than the alternating single and triple bonds in polyyne. First-principles method has been used to calculate the bending stiffness of polyyne chain with DDFT =

1 ∂ 2 EL , where a = 2.565Å is the unit cell length, with DDFT=3.56 eV·Å.26 Using a ∂ρ 2

the same definition, the calculated bending stiffness in our work is 2.6 eV·Å , which is close to that presented in Ref. 26. Using MD simulation with ReaxFF potential, Nair et al. studied the length-dependent vibrational frequencies of carbyne chains.41 In their simulation, a carbyne chain with two ends fixed was excited by perturbing the central atoms and allowed to freely vibrate. The fundamental frequencies of vibration were obtained by analyzing the oscillations via a discrete Fourier transform. The highest frequency was found to be close to 6THz in the shortest carbyne chain of 5 Å. Then the chain was assumed to be an elastic beam, and the frequency f was related to the stiffness D by the relation 2πf = α 2 D m L3 , where m is the mass, α is a constant of 4.43 for the chain with a load at the center. As the vibration frequency f is dependent on the length of carbyne chain, the estimated bending stiffness is also length-dependent. For a chain with length of 8 Å, 10 ACS Paragon Plus Environment

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the bending stiffness was found to be about 8.5 eV·Å , which is close to our predicted value of 6.7 eV·Å for polyyne chain. To use carbyne as a structural connection in mechanical components, persistence length is an important parameter that dictates its usability. The bending stiffness can be evaluated using the concept of persistence length Lp in polymer physics: Lp = D/kBT, where kB is the Boltzmann constant and T is the temperature.56 The persistence length Lp for carbyne chain is found to be about 33 nm at 300 K, which is in the same order of magnitude as those of double strand DNA (45−50 nm)57 and graphene nanoribbons (10−100 nm).58 This result indicates that, for a carbyne chain shorter than Lp, it behaves like a flexible elastic rod; while for a carbyne chain longer than Lp, its properties can only be described statistically. Since the longest carbyne chain observed to date is only 6 nm, its mechanical properties can be well described by a flexible elastic rod model. Then we apply uniaxial tensile strain along the carbyne axis to determine the maximum strength/strain. Uniaxial tension strain is applied by displacing both ends of the chain in opposite directions, at a constant rate of 2.5×10-5 Å/fs. We also select two different strain rates: 2.5×10-4 Å/fs, and 2.5×10-6 Å/fs, and found the stress-strain behavior of the carbyne chain is independent of the strain rate in this range. To obtain a complete stress vs. strain curve, we increase the applied strain until failure occurs, resulting in the maximum strain and stress. In the calculations, stress is calculated using the virial stress formulation. The usual definition of stress faces difficulty for the mono-atomistic chain due to the ambiguity in defining its cross section area. Here, we adopt a diameter of 3.35 Å59 to calculate the cross sectional area of carbyne chain, similar to the approach in defining the cross section of graphene and carbon nanotubes.60,61 To alleviate this ambiguity, we also report the equivalent 1D stress (units of force), which does not need the cross-sectional area. 11 ACS Paragon Plus Environment


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The stress-strain curve for polyyne chain at 500 K is shown in Figure 4a. It is found that the chain was broken when the strain reached about 8.0% with an ultimate stress of 67.1 GPa or a pulling force of 7.53 nN, which is in perfect match with the experimental value of ~7.916 nN reported in previous work 62. According to the linear elastic regime of the curves, the Young’s modulus γ can be calculated by: γ = (dσ/dԑ)|ԑ=0

(4)

where σ is the tensile stress, and ԑ is the extensional strain. Here the Young’s modulus is approximated by using a linear fit of the stress versus strain data up to 2.0% strain. The calculated Young’s modulus of polyyne is about 1345 GPa, which is higher than that of diamond (1220 GPa)63 and that of graphene (1000 GPa)26. The estimated breaking force for polyyne chain is 7.53 nN, or a specific strength of 5.7×107 N·m/kg, which again significantly outperforms graphene (4.7~5.5×107 N·m/kg), carbon nanotubes (4.3~5.0×107 N·m/kg), and diamond (2.5~6.5×107 N·m/kg).64−68 It should be noted that, to calculate the cross sectional area of carbyne chain, the diameter of the chain is adopted as 0.335 nm here, similar to the approach in defining the cross section of graphene and carbon nanotubes. Based on this cross sectional area, a breaking stress (67.1 GPa) is obtained. However, in a previous theoretical and experimental study of the mechanical properties of carbyne chain, the diameter of 0.2 nm was also adopted, which is the cutoff distance in Tersoff-Brenner bond order potential62. If we also adopt 0.2nm to calculate the cross sectional area, the obtained breaking stress is about 200 GPa, which is very close to the experimental value of 245 GPa69.


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Clearly, the perfect polyyne possesses an ultra-high Young’s modulus. However, as expected, defects in the polyyne chain are able to reduce the mechanical strength. For instance, the polyyne chain with 4% defective bonds (here, two defective bonds existed in the polyyne chain) exhibits the ultimate strain of about 7.3%, and the maximum stress of 66.3 GPa, which are lower than those of pristine polyyne at the same temperature. It is interesting to find that unlike the polyyne chain, the cumulene chain has a totally different mechanical response under strain at 66 K. As shown in Figure 4b, there are obviously three steps observed during the elongation process of cumulene. The first one is a linear elastic process, providing a Young’s modulus of about 1753 GPa, which is 30% higher than that of perfect polyyne. This elastic stage corresponds to the elastic deformations arising from chain uncoiling and stretching. Subsequently, the carbon-carbon double-bonds are being stretched, resulting in the phase transition from cumulene to polyyne, but the carbyne chain is not broken. It is known that the double bonds are made up of one σ bond and one π bond. Before transforming to singleand triple-bonds, π bond must be broken. Because carbon-carbon double-bond is energetically favorable compared to carbon-carbon single-bond, it is energetically costly to break the double bond, thus a large external work is required to overcome this energy barrier. Therefore, in the stress-strain curve, the stress increases sharply until the π bonds break and simultaneously rearrange themselves to form single- and triple-bonds. As the length of two carbon-carbon double-bonds is 2.74Å, while the length of a pair of single- and triple-bond is 2.79Å, there is a rapid decrease in stress during the transition from double-bonds to single- and triple-bonds. This is the origin of the sharp peak in the stress-strain curve. In the third step, since the chain has been transformed to polyyne with alternating single- and triple-bonds, upon further straining, the chain keeps its structure until it is broken at about 11%. The sharp increase in stress to resist the chain 13 ACS Paragon Plus Environment


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deformation provides new physical insight into the bond breaking/rearrangement in monoatomic structures. The above results also indicate that cumulene and polyyne are super strong 1D materials along the axial direction. Thermal Conductivity of Carbyne Chains Finally, we explored the thermal transport in the carbyne chains using equilibrium molecular dynamics (EMD) simulations, which is based on the Green-Kubo formula (GKF) derived from the fluctuation-dissipation theorem and linear response theory. In GKF, the thermal conductivity is related to heat current autocorrelation function (HCACF),70

κ=

V

kBT

∫ J i (0 ) J i (t ) dt , ∞

2 0

(5)

where kB is the Boltzmann constant, V is the volume of the system, and J i (0) J i (t ) is the average heat flux autocorrelation function along the carbyne chain direction, respectively. The heat flux J is calculated by70 N

J = ∑ ε i vi + i

1 N 1 N ∑ (Fij ⋅ vi )rij + ∑ (Fijk ⋅ vi )(rij + rik ), ij ; i ≠ j 2 6 ijk ;i≠ j ≠k

(6)

where, ԑi and vi are the energy and velocity associated with atom i, respectively. Vector rij denotes the interatomic distance between two atoms, and Fij and Fijk denote the two-body and three-body force, respectively. The carbyne (both cumulene and polyyne) chains are constructed with 50 carbon atoms. Periodic boundary condition (PBC) is applied along the axis of the chain. Free boundary condition (FBC) is applied in directions perpendicular to the carbyne chain. Once the chain is relaxed and the temperature reaches the targeted value, the heat flux is collected at each step for 10 ns. After that,


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the thermal conductivity is calculated according to Eq. 5. It should be noted that, since PBC is adopted along the axis of the chain during the thermal transport simulation, the edge/end effect and length effect on the thermal transport are not considered in our results. The calculated thermal conductivity of cumulene is about 83 W/mK at 480 K, which is comparable to that of the widely used thermal and electric conductor, copper nanowire (75 W/mK for copper nanowire with 20 µm long and 180 nm wide71), but significantly higher than that of semiconductor silicon nanowires72,73. The thermal conductivity of the polyyne chain is about 42 W/mK at 500 K, which is about half of that of cumulene. For the polyyne with 4% defects, its thermal conductivity dramatically decreases to 5.5 W/mK, which is only about 13% of that of pristine polyyne. It should be noted that the thermal conductivity calculated from MD simulations only includes the phonon contribution and does not include the electronic contribution. Since cumulene is metallic, we estimate the electronic contribution to the thermal conductivity of cumulene using the Wiedemann-Franz law: κe = σLwT, where, Lw (= 2(kB/e)2 = 1.45×10-8 V2/K2) is the Lorenz number, T is 480 K, and σ is the electrical conductivity. According to experimental measurements28, σ = ~1.60×102 S/m. Thus, the electronic contribution to the thermal conductivity κe is about 1.11×10-3 W/mK, which is negligible compared to the phonon thermal conductivity. In order to understand the underlying mechanism of this remarkable thermal transport difference among cumulene, polyyne, and polyyne with 4% defects, we calculated the heat current autocorrelation function (HCACF), C JJ (t ) = J i (0) J i (t ) , of these three chains. It is known that the decay of HCACF in bulk material is exponential.74 Also, the initial fast decay is due to



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the high frequency phonon modes while the slow decay arises from the low frequency phonon modes. From the HCACF of cumulene as shown in Figure 5a, it is seen that there is a rapid initial decay followed by a long time decay (15 ps) of the correlation function in cumulene. The HCACF of polyyne is similar to that of cumulene, but its decay time is shorter, which is about 10 ps. In general, high frequency (short wavelength) phonons have a limited contribution to thermal conductivity due to their low group velocity. Thus the large difference in relaxation time for the long wavelength phonons along the chain is responsible for the remarkable difference in thermal conductivity between cumulene and polyyne. It is also seen that the HCACF of polyyne with defective bonds is obviously lower than that of the pristine polyyne. Interestingly it shows a regular high frequency oscillation. We note that a similar oscillation was also observed in heterogeneous systems with different atomic masses.75,76 From Figure 5b, it is also seen that the decay time in the polyyne with defects is remarkably shorter than that of pristine cumulene and polyyne, which is clearly due to the presence of defective bonds. These defected atoms lead to increased phonon scattering, which introduces localized vibrational modes and thus reduces the thermal conductivity. Tunability of thermal conductivity, which can control or rectify heat fluxes, may have important applications in thermal devices.77,78 Recently, a graphene-based thermal modulator is predicted theoretically, in which the heat flux is controlled by applying a tunable external pressure.79 On the other hand, temperature-induced change in thermal conductivity is important for application in thermal memory. Perhaps a better choice for tuning thermal conductivity is through using phase change materials.80 In fact, phase change materials have been widely applied to store and release heat energy due to their high latent heat. Although the latent heat in existing phase change materials can be huge, the change in their thermal conductivity at phase 16 ACS Paragon Plus Environment

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change temperature is usually insignificant. For example, the thermal conductivity of PEG4000, a representative phase change material, is 0.308 W/mK in solid state and drops to 0.216 W/mK after solid-to-liquid phase transition, giving rise to the tunability (high/low ratio) of 143%.81,82 For the cumulene-polyyne phase transition, the thermal conductivity is reduced from 83 W/mK to 42 W/mK, which gives rise to the tunability of 200%. In addition, this type of phase transition can be induced by both temperature and tensile strain, thus carbyne chain is a promising candidate for the realization of mechanically and/or thermally tuned solid-state thermal memories. Conclusions In summary, to fundamentally understand the characteristics of carbyne chain and vigorously expedite its practical applications, here, we performed systematic molecular dynamics simulations to investigate the mechanical and thermal properties of carbyne with two different phase forms. Our results indicate that the critical temperature for carbyne to transform from cumulene to polyyne is 499 K. Both cumulene and polyyne have a high bending stiffness and Young’s modulus, which are comparable with those of the hardest natural materials. Furthermore, cumulene has a fairly high thermal conductivity, which is two times higher than that of polyyne. A small amount of defects in polyyne is able to significantly reduce its thermal conductivity. For example, a 4% of defective bonds can reduce the thermal conductivity dramatically to only 13% of that of pristine polyyne. These unusual mechanical and thermal properties suggest that carbyne is promising for applications in electromechanical switching, mechanical connections and storage of mechanical and thermal energy.



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AUTHOR INFORMATION Corresponding Author E-mail: [email protected]; Phone: 65-64191583 Notes The authors declare no competing financial interests. Acknowledgement: The authors gratefully acknowledge the financial support from the Agency for Science, Technology and Research (A*STAR), Singapore and the use of computing resources at the A*STAR Computational Resource Centre, Singapore.


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Figure 1. Bond-length distribution of carbyne chains with 50 atoms: (a) Cumulene and (b) Polyyne. Insets show the integral of bond number. It is clear that in polyyne, 50% bonds are single bonds and the rest 50% are triple bonds.



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Figure 2. Phase transition process of carbyne chains with 50 atoms at the critical temperature of 499 K. (a) – (e) Evolution of bond numbers from cumulene to pristine polyyne with a low heating rate of 0.04K/fs. (f) – (j) Evolution of bond numbers from cumulene to polyyne with defective bonds (22 single-bond, 24 triple-bond, and 2 defective bonds) with a heating rate of 2.5 K/fs.


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Figure 3. (a) A straight carbyne chain (lower panel) and a carbyne ring representing a system with a constant curvature (upper panel). (b) Bending energy of carbyne chain Eb as a function of number of atoms in the chain. (c) Bending energy EL as a function of the carbyne chain curvature ρ. (d) Bending energy EL as a function of ρ2.



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Figure 4. Stress-strain curves of carbyne chains. (a) Polyyne chain with or without defects at 500K. (b) Cumulene chain at 66K.


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Figure 5. The heat flux autocorrelation functions for carbyne chains. (a) Cumulene and perfect polyyne chains. (b) Polyyne chain with 4% defective bonds. The temperature of cumulene is 480K, and temperature of polyyne is at 500K.



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Tunable Mechanical and Thermal Properties of One-Dimensional

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