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Front. Phys. 11(2), 114401 (2016) DOI 10.1007/s11467-015-0548-z RESEARCH ARTICLE Interface-facilitated energy transport in coupled Frenkel–Kontorova chains Rui-Xia Su1 , Zong-Qiang Yuan2 , Jun Wang3,∗ , Zhi-Gang Zheng4,† 1 Department of Physics and the Beijing–Hong Kong–Singapore Joint Centre for Nonlinear and Complex Systems (Beijing), Beijing Normal University, Beijing 100875, China 2 Science and Technology on Plasma Physics Laboratory, Research Center of Laser Fusion, China Academy of Engineering Physics, Mianyang 621900, China 3 Key Laboratory of Enhanced Heat Transfer and Energy Conservation (Ministry of Education), College of Environmental and Energy Engineering, Beijing University of Technology, Beijing 100124, China 4 College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China Corresponding authors. E-mail: ∗ jwang@bjut.edu.cn, † zgzheng@hqu.edu.cn Received November 5, 2015; accepted December 17, 2015 The role of interface couplings on the energy transport of two coupled Frenkel–Kontorova (FK) chains is explored through numerical simulations. In general, it is expected that the interface couplings result in the suppression of heat conduction through the coupled system due to the additional interface phonon–phonon scattering. In the present paper, it is found that the thermal conductivity increases with increasing intensity of interface interactions for weak inter-chain couplings, whereas the heat conduction is suppressed by the interface interaction in the case of strong inter-chain couplings. Based on the phonon spectral energy density method, we demonstrate that the enhancement of energy transport results from the excited phonon modes (in addition to the intrinsic phonon modes), while the strong interface phonon–phonon scattering results in the suppressed energy transport. Keywords interface couplings, energy transport, heat conduction, phonon-phonon scattering, Frenkel–Kontorova (FK) chains, excited phonon modes, phonon spectral energy density PACS numbers 44.10.+i, 63.20.Ry, 63.22.-m, 05.60.-k 1 Introduction Recent years have witnessed an increasing number of studies on the thermal transport properties of nanostructured materials because understanding heat transfer at the nanoscale is of growing importance in the development of energy-conversion applications and the thermal management of microelectronic and optoelectronic devices [1–12]. Previous studies have all focused on a single free-standing nanostructure, such as isolated silicene [13–15], carbon nanotubes [16–21], graphene [22–26], and nanoporous Si [27]. However, it is inevitable that most nanostructures have to be supported or surrounded by environmental materials in practical applications, and the heat-transfer process may be significantly affected by the interface. Usually, the scattering of phonons at interfaces leads to a decreased phonon mean free path and suppressed thermal conductivity [28–30]. However, recent experimental and numerical results demonstrate that contact with a substrate counterintuitively enhances the thermal conductivity through double-wall carbon nanotubes [31], supported graphene [32], supported silicene structures [33], and a bundle of nanoribbons [34]. Therefore, the interface effect can improve the thermal conductivity under certain conditions. However, a comprehensive theoretical explanation of this counterintuitive physical phenomenon is still lacking. In 1938, Frenkel and Kontorova first proposed the FK model to study surface phenomena [35]. Despite the deceptively simple form of the model, it has been applied in a wide variety of physical systems, such as adsorbed monolayers, Josephson junctions, and DNA denaturation [36]. In condensed-matter physics and nonlinear physics, FK chains are also widely used to depict the heat conduction and phonon transport process of real materials [37–46]. Thus, we adopt a coupled FK chain model as a simplified working bench of the coupled nanostructures c Higher Education Press and Springer-Verlag Berlin Heidelberg 2016  RESEARCH ARTICLE and investigate the effect of interface couplings on the phonon transport process. In the present paper, we employ numerical simulations to elucidate the thermal transport mechanisms of two coupled FK chains. It is found that the heat flow through the interacting FK chains vary nonlinearly with respect to the interface coupling intensity. For weak interface interactions, the energy transport is enhanced by the interface couplings, while for strong interface couplings, the heat flux is suppressed by the interface interaction. Based on the phonon spectral energy density (SED) method [19, 47–49], we obtained exact knowledge of the phonon spectra. By combining phonon band theory with numerical simulations to analyze the phonon spectra, we found that the enhancement of heat conduction mainly results from the additionally excited phonon modes, while strong interface phonon scattering leads to the suppression of thermal transport through strongly coupled FK chains. simplicity, we set m1 = m2 = 1.0, k1 = k2 = 1.0, and V1 = V2 = 1.0. Fixed boundary conditions are adopted in nonequilibrium molecular dynamics (NEMD) simulations and two Nose–Hoover thermostats [50, 51] at TH = 0.06 and TL = 0.04 are applied on the first and last particle of each chain, respectively. The size of each lattice is set to be N = 50. In our simulations, the equations of motion are integrated using a fourth-order Runge–Kutta algorithm with a small time step of 0.01 [52]. After performing simulations for a sufficiently long time (the total integration is for 109 time steps), the system reaches a stationary state in which the heat flux is constant along the longitudinal direction of lattices. The local heat flux is defined by Jn = pjn ∂Vn /∂xjn  based on the continuity equation, where Vn denotes intra-chain interactions and 2 Vn = k2n (xjn − xj−1 n ) in this paper. 3 Dependence of heat flux on intensity of interface couplings 2 Model of coupled FK chains In this study, we investigate heat conduction in two coupled FK chains with linear inter-chain coupling strength kc (see Fig. 1). The Hamiltonian of the entire system is H = H1 + H2 + Hc , (1)  2  N  pj k V n n n 2 + (xin − xj−1 cos(2πxjn ) , Hn = n ) − 2 2m 2 (2π) n j=1 for n = 1, 2,    kc j (x2 − xj1 )2 . Hc = 2 j (2) (3) Here, H1 and H2 are the Hamiltonians of chain-1 and chain-2, respectively, and Hc denotes the Hamiltonian of the interfacial coupling term. Further, xjn and pjn are the relative displacement and momentum of the j-th atom in the n-th chain, respectively, and mn , kn , and Vn represent the mass, harmonic coupling constant, and strength of the on-site potential of chain n, respectively. For Figure 2(a) shows the dependence of heat flux on the inter-chain couplings kc , where J1 , J2 , and Jt (Jt = J1 + J2 ) denote the heat fluxes of chain-1, chain-2, and the total heat flux, respectively. Because chain-1 and chain-2 are identical, J1 should be equal to J2 (thus, we only analyze the phonon-transport properties of chain-1 below). Figure 2(a) clearly shows that the heat flux increases with increasing kc for weak interface couplings. The heat flux reaches its maximum at an intermediate strength of the inter-chain couplings. When kc is further increased, the heat current decreases monotonously, which indicates that heat conduction is suppressed by the strong interfacial coupling. Considering the scale dependence of the thermal conductivity of FK chains [36, 45, 53, 54] (nanomaterials such as carbon nanotubes and graphene also show a certain scale dependence [56, 56]), (a) Fig. 1 114401-2 Schematics of the coupled FK chains. Fig. 2 (a) Dependence of the heat flux of coupled FK chains on the inter-chain coupling strength kc ; (b) Heat flux of different system sizes N = 50, 200, 500. J1 and J2 denote the heat flux of chain-1 and chain-2, respectively. Jt is the total heat flux of the coupled system. Rui-Xia Su, et al., Front. Phys. 11(2), 114401 (2016) RESEARCH ARTICLE we plot the the effect of size on this phenomenon in Fig. 2(b). The results show that the nonlinear dependence of heat conductivity on the intensity of interface coupling is size-insensitive. Thus, it is concluded that the heat conduction through two interacting FK chains can be manipulated by varying the strength of inter-chain couplings. 4 Analysis and discussion on the mechanism of heat-conductivity variation To understand the underlying mechanism of the enhancement or suppression of thermal conductivity for two coupled FK chains, we calculate the phonon dispersion relation by using the SED method [19, 47–49]. The SED method can predict fully anharmonic phonon properties by using particle velocities obtained from the numerical simulations as follows: mn Φ n (ω, q) = 4πτ0 N N 2   τ0  j   pjn exp(i2π q − iωt)dt , (4) × N 0 j=1 where ω and Q = 2πq/(aN ) = 2πq/N (q = − N2 , − N2 + 1,..., N2 − 1, N2 , and the lattice constant a is set to be a = 1) are the phonon frequency and wavevector, respectively, a is the equilibrium distance of adjacent atoms in chains, and i and τ0 are the imaginary unit and total time of calculation, respectively. It is noted that the SED method is based on the equilibrium MD simulations; therefore, we must calculate SED under the periodic boundary condition and the equilibrium state at T = (TH + TL )/2 = 0.05 in the numerical simulations. Figure 3 shows the SED pattern with different interchain couplings: (a) without inter-chain interactions (kc = 0.0), (b) weak inter-chain interactions (kc = 0.13), (c) moderate inter-chain interactions (kc = 0.3), and (d) strong inter-chain interactions (kc = 2.0). As the parameters of chains 1 and 2 are identical, their SED patterns should be identical; therefore, we only show the SED patterns of chain-1 here. As shown in Fig. 3, with increasing interface couplings, more additional phonon modes are excited, and meanwhile, the high-frequency phonon band shows an upward shift. When the strength of interchain couplings is sufficiently high, more side peaks (the messy bright lines in SED patterns) emerge, which implies strong phonon scattering. At the low-temperature regime, the particles are confined at the bottom of the on-site potential owing to the small kinetic energy. Thus, the corresponding equation of motion of the coupled FK chians can be expressed as Rui-Xia Su, et al., Front. Phys. 11(2), 114401 (2016) Fig. 3 Contour plot of phonon SED of chain-1 with different inter-chain couplings: (a) kc = 0.0, (b) kc = 0.13, (c) kc = 0.3, and (d) kc = 2.0. The contour plots are the results by SED method and the red and black dotted lines are the analytical result. j j j j mẍj1 = k1 (xj+1 + xj−1 1 1 −2x1 )−kc (x1 − x2 ) − V1 x1 , (5) j j j j mẍj2 = k2 (xj+1 + xj−1 2 2 −2x2 )−kc (x2 − x1 ) − V2 x2 . (6) By setting the trial solutions in the forms xj1 = A1 ei(ωt−jaQ) = A1 ei(ωt−jQ) and xj2 = A2 ei(ωt−jaQ) = A2 ei(ωt−jQ) and by substituting them into Eqs. (5) and (6), we obtain the following: (−mω 2 −2k1 cos Q + 2k1 + kc + V1 )A1 −kc A2 = 0, (7) (−mω 2 −2k2 cos Q + 2k2 + kc + V2 )A2 −kc A1 = 0. (8) The above equations can be regarded as linear homogeneous equations with unknowns A1 and A2 , and the existence condition of the traveling-wave solution can be given as     C −k c   1  = 0,   −kc C2  where C1 = −mω 2 − 2k1 cos Q + 2k1 + kc + V1 , C2 = −mω 2 − 2k2 cos aQ + 2k2 + kc + V2 , and they satisfy the following relation: C1 × C2 − kc2 = 0. (9) The solution of Eq. (9) is 2(k1 + k2 )sin2 Q 2 + kc + V m  2 4(k1 − k2 )2 sin4 Q 2 + kc . ± m 2 ω± = As k1 = k2 ,  Q 2 ω± = 4k1 sin2 + V + kc ± kc /m. 2 (10) (11) 114401-3 RESEARCH ARTICLE Here, ω+ and ω− denote the high-frequency and lowfrequency phonon modes, respectively. The analytical phonon dispersion relations are plotted in Fig. 3(c) by red and black dotted lines, and they are in qualitative agreement with the result obtained using the SED method. It is worth noting that the phonon frequency in the analytical result is shifted upward relative to the SED patterns, which can be ascribed to the contribution from the nonlinearity of the on-site potential. Now, we analyze the mechanism of thermalconductivity enhancement. The relaxation-time approximation of the Boltzmann transport equation approach yields the formula of thermal conductivity [47]:  cph vg2 (Q, γ)τ (Q, γ), (12) κ= Q γ where cph is the specific heat of the phonon, Q is the wave vector, γ denotes the phonon branch, vg (Q, γ) = ∂ω/∂Q is the group velocity of the phonon, and τ (Q, γ) is the phonon lifetime. In the classical limit, the phonon specific heat, cph , could be considered a constant. The two summations in Eq. (12) include all phonon modes. The spectral energy density is a function of wave vector and frequency. Moreover, SED is a highly peaked function that should satisfy a Lorenzian distribution, and the peaks are centered at values of wave vector Q (Q = 2πq/N ) and frequency ω at which phonons exist. Thus, we are able to confirm the phonon modes by drawing the SED peaks in the entire frequency range with a given Q. The full phonon spectra obtained in this manner are shown in Fig. 4, where ω1− and ω1+ denote the low-frequency phonon branch and high-frequency phonon branch of chain-1, respectively. From Fig. 4(a), it is seen that the low-frequency phonon branch ω− remains invariant with varying kc . In the above expression of analytical dispersion relations, Eq. (11), it is also found that ω− is independent of kc . Actually, ω− produces the exact phonon spectra for the corresponding single freestanding FK chain; thus, ω− can be viewed as the intrinsic phonon branch, and the phonon group velocity does not vary with interface interactions. On the other hand, the phonon scattering originating from weak Fig. 4 114401-4 Phonon SED peaks for the chain-1. interface couplings is too weak to affect the phonon transport significantly at low kc ; i.e., the influence of interchain couplings on the mean free time or the mean free path of intrinsic phonons can be ignored. Therefore, for low kc , the contribution of the intrinsic phonon branch to the total thermal conductivity does not change apparently with varying interface interactions. From Fig. 3 and Fig. 4(b), it can be seen that an increasing number of phonon modes of the high-frequency phonon branch ω1+ are excited with increasing kc , and they emerge one by one with increasing wave vectors. For example, for kc = 0.05, only the modes from q = 0 to q = 2 are excited, while for kc = 0.3, the phonon modes of the entire high-frequency branch are fully excited. This implies that more phonons contribute to the energy-transport process of the coupled FK chains as kc increases. This effect can also be seen from Eq. (12), in which more terms are involved in the summation operation with increasing kc ; thus, the thermal conductivity of the coupled system increases with increasing inter-chain couplings. It is noted that the excited highfrequency phonon branch slowly shifts upward with increasing kc , as shown in Fig. 3, which leads to a smaller phonon group velocity and accordingly a suppression effect on the thermal transport. However, for low kc , the shift of the high-frequency phonon branch is very weak, as shown in Fig. 4(b), and does not significantly influence the energy transport. Therefore, the dominant effect of the phonon excitation results in the thermal-transport enhancement of coupled system. In the regime of strong interfacial couplings, the emergence of side peaks (messy bright lines) in the SED patterns in Fig. 3(d) indicates strong phonon scattering due to the large interfacial couplings. To demonstrate this in more detail, we present in Fig. 5 the SED distribution of the coupled FK chains with variation in frequency when q = 3. It is quite clear that many small thorns of side energy peaks appear for strong inter-chain couplings (kc = 4), further illustrating the strong interface phonon scattering, which results in the reduced phonon lifetime; consequently, the thermal transport through Fig. 5 Dependence of the energy density on the frequency while q = 3. Energy-density distributions of chain-1 for (a) kc = 0.3 and (b) kc = 4.0. Rui-Xia Su, et al., Front. Phys. 11(2), 114401 (2016) RESEARCH ARTICLE strongly coupled FK chains is suppressed by strong interface interactions. skutterudite CoSb3 from first principles: Substitution and nanoengineering effects, Sci. Rep. 5, 7806 (2015) 7. J. S. Wang, B. K. Agarwalla, H. Li, and J. Thingna, Nonequilibrium Green’s function method for quantum thermal transport, Front. Phys. 9(6), 673 (2014) 8. N. P. Dasgupta and P. 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In the regime of weak inter-chain couplings, the thermal transport can be enhanced by increasing the strength of interface interactions. However, for strong inter-chain couplings, heat flux is suppressed by the interface interaction. The analysis of the phonon SED patterns demonstrates that the emergence of additionally excited phonon modes is the primary mechanism for the enhancement of thermal transport for weak interface interactions, while strong interface phonon scattering leads to the the suppressed heat conduction through strongly coupled FK chains. For real applications, interface interactions always exist in nature. The results of the present study pave the way for manipulating the energy transport of nanoscaled materials, which is necessary for designing thermal devices. 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