PHYSICAL REVIEW E 91, 012136 (2015)

Tamable heat conduction through coupled Fermi-Pasta-Ulam chains Ruixia Su,1 Zongqiang Yuan,2 Jun Wang,3’* and Zhigang Zheng1,1 1Department o f Physics and the Beijing-Hong Kong-Singapore Joint Centre fo r Nonlinear and Complex Systems (Beijing), Beijing Normal University, Beijing 100875, China 2Institute o f Applied Physics and Computational Mathematics, Beijing 100088, China 3Key Laboratory o f Enhanced Heat Transfer and Energy Conservation, Ministry o f Education, College o f Environmental and Energy Engineering, Beijing University o f Technology, Beijing 100124, China (Received 6 August 2014; revised manuscript received 24 November 2014; published 21 January 2015) We conduct a study on heat conduction through coupled Fermi-Pasta-Ulam (FPU) chains by using classical molecular dynamics simulations. Our attention is dedicated to showing how the phonon transport is affected by the interchain coupling. It has been well accepted that the heat conduction could be impeded by the interchain interaction due to the interface phonon scattering. However, recent theoretical and experimental studies suggest that the thermal conductivity of nanoscale materials can be counterintuitively enhanced by the interaction with the substrate. In the present paper, by consecutively varying the interchain coupling intensity, we observed both enhancement and suppression of thermal transport through the coupled FPU chains. For weak interchain couplings, it is found that the heat flux increases with the coupling intensity, whereas in the case of strong interchain couplings, the energy transport is found to be suppressed by the interchain interaction. Based on the phonon spectral energy density method, we attribute the enhancement of the energy transport to the excited phonon modes (in addition to the intrinsic phonon modes), while the upward shift of the high-frequency phonon branch and the interface phonon-phonon scattering account for the suppressed heat conduction. DOI: 10.1103/PhysRevE.91.012136

PACS number(s): 44.10.+i, 63.20.Ry, 63.22.-m

I. INTRODUCTION

Heat conduction is a rather well known problem related to the microscopic foundation of Fourier’s law, but still a long-standing open problem of nonequilibrium statistical physics [1-3]. The modern theory of heat conduction started from the famous work by Fermi, Pasta, and Ulam (FPU) [4], where an abnormal process of heat conduction was revealed in one-dimensional FPU lattices associated with diverging thermal conductivity with the system size [5]. In past decades, much effort has been focused on the validity of Fourier’s law in the low-dimensional systems [6-26]. Apart from these fundamental explorations, the thermal transport properties of nanomaterials, for example nanowires, carbon nanotubes, graphene, etc., have attracted much attention recently, partly because the nanomaterials are naturally related to the low­ dimensional system, and also because understanding energy transport through nanoscale structures is of great impor­ tance in microelectronics, photovoltaics, and thermoelectrics [27-29]. Considering that the nanodevices are always placed and connected to a certain substrate in real applications, it is reasonable and necessary to investigate the influence of the interface interactions on the energy transport through a nanomaterial. In a common picture of phonon transport, the thermal trans­ port of a nonmetallic or semiconductor material is supposed to be suppressed due to the phonon scattering at the interfaces upon connecting with the substrate [30-32]. On the other hand, it has recently been reported that the coupling to substrates counterintuitively enhances the thermal conductivity through double wall carbon nanotubes [33], supported graphene [34],

*[email protected] [email protected] 1539-3755/2015/91(l)/012136(8)

and coupled Frenkel-Kontorova (FK) lattices [35], where the enhancement of energy transport has been attributed to the shift of the phonon band [33], the coupling of graphene ZA modes to the substrate Rayleigh waves [34], and the additional highfrequency phonon branch [35], respectively. The enhancement of the thermal conductivity of interacting nanoribbons has also been observed in Yang et al. ’s recent experiment, whereby the authors attribute the enhancement of the energy transport to the interlayer van der Waals interaction [36], However, the phonon transport is found to be enhanced even with strong interface coupling [33-35], Physically, for strong couplings, the interface phonon scattering should play an important role in the phonon transport through interacting materials, and the heat conduction is expected to be suppressed. Therefore, conflicting conclusions exist in the previous research, and a more in-depth understanding of how the phonon transport is affected by the interface coupling is needed before one can apply this effect to manipulate and design thermal devices. In this paper, we perform molecular dynamics (MD) simulations to explore the influence of interchain coupling on the heat conduction property of two coupled FPU chains. It is found that the heat flow through the interacting FPU chains depends nonlinearly on the interchain coupling strength. For weak interchain interactions, the energy transport can be enhanced by increasing the interchain coupling. However, the enhancement of the heat conduction cannot be observed for stronger interchain couplings, where the heat flux is suppressed by the interchain interaction, as expected by the common understanding of interface phonon scattering. Based on the phonon spectral energy density (SED) method [37-40], we combine phonon band theory and MD simulations to explore how interchain couplings influence the phonon transport. It is found that the enhancement of the heat conduction is mainly related to the additionally excited phonon modes (in addition to the intrinsic phonon modes), while the upward

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©2015 American Physical Society

RUIXIA SU, ZONGQIANG YUAN, JUN WANG, AND ZHIGANG ZHENG

PHYSICAL REVIEW E 91, 012136 (2015)

In our sim ulations, w e em ploy the R unge-K utta algorithm with a tim e step o f 0.01 to evolve the system [43]. Long enough sim ulations are necessary to reach a nonequilibrium steady state before the data collection over 1 x 109 tim e steps.

III. DEPENDENCE OF HEAT FLUX ON THE INTERCHAIN COUPLING

shift o f the high-frequency phonon branch and the phononphonon scattering due to interchain interactions account for the suppressed energy transport. T he rest o f the present paper is organized as follows. In Sec. II, we describe the coupled FPU chain model. Section III is devoted to the dependence o f the heat flux on the interchain coupling strength. In Sec. IV, the phonon dispersion relationship is calculated based on the SED m ethod, by w hich the enhancem ent and suppression o f heat conduction through the coupled FPU chains are investigated. In Sec. V, w e present the results o f coupled harm onic and FK systems. Then, w e conclude our paper in Sec. VI.

II. COUPLED FPU MODEL In the present paper, the system consists o f two coupled FPU chains connected w ith the interchain coupling strength kc (as depicted in Fig. 1). T he H am iltonian o f the w hole system can be w ritten as

H = H l + H 1 + Hc,

R

£ II

T

1

H

OCXI^ +

04

(2)

= H \ ^ ( x2 ~ x i ) 2 7=1 L

1

n = 1,2,

7 *"'»R

l + 2!L(yj k fpj a— _2mn 2 "

r 7=1

for

(1)

(3)

J

Here, H | and H2 denote the H am iltonians o f chain 1 and chain 2, respectively, w hile Hc represents the coupling contribution to the total H am iltonian; x„ and pi are position and m om en­ tum , respectively, o f the / th particle in chain n. For simplicity, w e set the values o f all the m asses m\ — m2 = 1.0 and the quartic param eters /3\ = /32 = 2.0. N onequilibrium m olecular dynam ics (NEM D ) sim ulations have been em ployed to study the heat conduction properties. Fixed boundary conditions are adopted, and the two ends O' = 1 and j = N) o f each chain are connected to N oseH oover therm ostats [41,42] at tem peratures TH = 1.05 and Tl — 0.95, respectively. The Langevin heat bath has also been adopted in our sim ulations and the sam e results can be observed. T he length o f the FPU chain is set to be N — 50. The local tem perature at site j is defined as the average o f the kinetic energy T j = {(pi)2) / m n, w here (•) stands for tem poral average. T he local heat flux is defined by Jn = (pJ„ dV„/dxi).

By consecutively varying the interchain coupling kc, w e are able to exam ine the the influence o f the interchain coupling on the heat conduction o f the coupled FPU chains. Figures 2(a), 2(b), and 2(c) show the dependence o f the heat flux on the coupling strength kc, w here J u J2, and J, (J, = T, + J2) denote the heat flux o f chain 1, chain 2, and the total heat flux, respectively. In Fig. 2(a), J\ is exactly equal to J2 because the param eters o f chains 1 and 2 are definitely identical to each other ( k\ = k2). If k2 > k\, the heat flux through chain 2, J2, is m uch higher than that through chain 1 ow ing to stronger interm olecular interactions in chain 2 [see Fig. 2(b)], O ur M D sim ulation results show that the heat flux assum es a m axim um value w ith increasing kc. For low kc, the heat flux rises up w ith increasing kc. Thus, the phonon transport through both chains 1 and 2 can be enhanced by the w eak interchain coupling. If kc is further increased, it is clearly noticeable that the heat flux goes down after a m axim um point, and an inverse relationship betw een the heat flux and kc is observed; i.e., the heat conduction is suppressed by the strong interchain coupling. T he size dependence o f this nonlinear behavior is plotted in Fig. 2(c), and the results show that the tendency o f the nonlinear dependence o f heat flux on the interchain coupling is insensitive to the system size. In Fig. 2(d), w e show the dependence N J (the product o f the ch ain ’s length and heat flux) on N (the ch ain ’s length) w hen k\ = 1, k2 = 5, and kc — 0. It is know n that the conductivity o f the one-dim ensional chain diverges as k (N) oc N J oc N a, w here k is the therm al conductivity. Here our data in doubly logarithm ic scales o f Fig. 2(d) are fitted by the curves o f the exponent a — 0.35, w hich is consistent w ith the open literature [1].

IV. ANALYSIS AND DISCUSSION ON THE MECHANISM OF THERMAL CONDUCTIVITY VARIATION A. Phonon dispersion relationship To understand the enhancem ent o r suppression o f the heat conduction through the coupled FPU chains, w e turn to the phonon spectral energy density calculation [37,38]. By projecting the m om enta o f the atom s onto norm al m odes o f vibration, we can obtain the SED as follows:

„(a),q) =

mn

Arc tqN

/•T o

l



M

Y.Pn j= 1

x exp i2n~^q — icot dt

(4)

w here co and Q = 2 n q / N (q is - j , - f + l , . . . , f - 1, j ) denote the phonon frequency and wave vector, respectively; i is the im aginary unit; and t0 is the total integration tim e for

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TUNABLE HEAT CONDUCTION THROUGH COUPLED . ..

0.21 (c) k = 1, k f 5

— »/|(AA=50 ) ’ HD-

J2(N=50)

-»-c/,(Afc100) 0.14 H > V 2( / A 100 ) A J^N = 500 ) -A -

0.07

0.00 -1

0

1

2

3

4

5

6

J2(A\+). Figure 6 shows the summation over all the phonon wave vectors for both the intrinsic branch and the excited branch with kc < 0.3. It can be seen that the contribu­ tion from the intrinsic branch is almost independent of kc. This is because neither the phonon group velocity nor the lifetime of the phonon in the intrinsic branch can be affected by the weak interchain coupling. On the other hand, from Figs. 3 and 5(b), it can been seen that with increasing kc, the excited phonon branch emerges from low-wave-vector to high-wave-vector

FIG. 6. (Color online) Plot of k ' versus kc for the coupled FPU chains with k\ = k2 = 1.0. phonon modes. This indicates that more phonons are involved in the energy transport process of the coupled FPU chains besides the phonon in the intrinsic phonon branch. Hence, the contribution of the additionally excited phonon modes can give rise to the increase of the heat flux at low kc (positive effect). It is worth noting that the emergence process of the excited phonon branch is accompanied by the upward shift of the excited phonon band, which is actually the formation process of the optical phonon branch. From Fig. 5(b), it is obvious that the group velocity of the excited phonon decreases. Therefore, the upward shift of the excited phonon band suppresses the thermal transport (negative effect). Two such effects compete with each other. In order to determine the relative importance of the two competitive effects in thermal transport, we also calculated the dependence of heat conductivity of the excited phonon branch o>i+ (red dotted line in Fig. 6) on the interchain couplings and found that the heat conductivity monotonically increases with interchain couplings increasing from kc = 0.0 to kc — 0.3. Hence, the phonon excitation effect (positive effect) is dominant and the thermal conductivity can be enhanced by weak interchain couplings. C. Enhancement of heat conduction through weakly coupled FPU chains with nonidentical parameters

For the case of k\ = 1.0, k2 = 5.0, according to the SED results in Fig. 4, the results for the SED peaks versus q are given in Fig. 7; &>i _ and o)2+ are the intrinsic branches of chains 1 and 2, respectively. Analogous to the case of FPU chains with identical parameters, the emergence of the excited branch cui+ (the high-frequency branch of chain 1) and oo2- (the low-frequency branch of chain 2) also behaves in a relatively gradual way (one by one from low-wave-vector to high-wavevector phonon modes) associated with the upward shift of the phonon band. Moreover, it is found that the shift of the excited modes, o)]+ and co2- , follows the shift of the intrinsic modes, co2 + and o>i_, respectively, owing to the interchain coupling. As a result, the phonon modes of chains 1 and 2 are consistent with each other at high kc, as they should be.

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PHYSICAL REVIEW E 91, 012136 (2015)

FIG. 7. (Color online) Phonon SED peaks for the coupled FPU chains with k\ = 1.0, k2 = 5.0. For chain 1, the intrinsic branch &q_ shifts slightly upw ard w ith increasing kc as show n in Fig. 7(a), so there is a very sm all increase in the phonon velocity o f the intrinsic phonon mode. This leads to a quite w eak positive effect on the energy transport, as show n in Fig. 8(a), w hich plots k[ versus kc for the coupled FPU chains w ith k\ = 1.0, k2 = 5.0 (0 < kc < 0.3). A distinct increase o f heat conduction can result from the fact that m ore phonon m odes o f the excited branch (&>1+) em erge w ith increasing kc (positive effect). Being sim ilar to the case of coupled identical FPU chains, the upw ard shift o f the excited phonon band suppresses the therm al transport (negative effect). However, according to Fig. 8(a), this negative effect is too w eak to dom inate over the role o f the phonon excitation. M eanw hile, the phonon scattering due to w eak interchain couplings is too w eak to affect the phonon transport at low kc. T herefore, the therm al transport enhancem ent o f chain 1 results from the dom inant effect o f the phonon excitation. For chain 2, there is an obvious shift o f the intrinsic branch a>2+ w ith increasing kc [Fig. 7(d)] and it is apparent that the phonon group velocity decreases w ith interchain couplings. H ence, the upw ard shift o f intrinsic branch a>2+ suppresses the therm al transport o f chain 2, as shown in Fig. 8(b). However, from Fig. 7(c), an additional low -frequency branch 0 )2 - can be excited by the interchain coupling and more lowfrequency phonon m odes em erge as kc increases. T he excited low -frequency phonon m odes have high phonon velocity and long phonon lifetim e ( r ~ 1/cu). From Fig. 8(b), it can be seen that the increm ent o f therm al conductivity (owing to the excited low -frequency phonon m odes cu2_) prevails over the decrem ent o f therm al conductivity (owing to the shift o f the intrinsic branch

Tunable heat conduction through coupled Fermi-Pasta-Ulam chains.

We conduct a study on heat conduction through coupled Fermi-Pasta-Ulam (FPU) chains by using classical molecular dynamics simulations. Our attention i...
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