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Negative differential thermal resistance in a double-chain system
Physica B 406 (2011) 4639–4643
Contents lists available at SciVerse ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Negative differential thermal resistance in a double-chain system Mao-ping Zhang, Wei-rong Zhong n, Jian-yuan Qiu, Yin-ze Ding Department of Physics and Siyuan Laboratory, College of Science and Engineering, Jinan University, 510632 Guangzhou, China
a r t i c l e i n f o
abstract
Article history: Received 22 March 2011 Received in revised form 15 July 2011 Accepted 16 September 2011 Available online 22 September 2011
Using nonequilibrium molecular dynamics simulations, we study the phenomenon of negative differential thermal resistance (NDTR) in a double-chain system. We investigate the dependence of NDTR on the external potential, inter- and intra-chain interaction and the system size. It is reported that the NDTR can occur in a small double-chain system with weak external potential and weak interand intra-chain interaction. We also present the influence of the external potential, inter- and intrachain interaction and the system size on the heat current of the system through the phonon spectral analysis. & 2011 Elsevier B.V. All rights reserved.
Keywords: Negative differential thermal resistance Double chains Heat current Phonon spectra
1. Introduction In recent decades, heat transport in low-dimensional systems has been given widely attention, especially in one-dimensional single chain systems [1–6]. In these researches, there are two major models, which are Fermi–Pasta–Ulam (FPU) model without external potential and Frenkel–Kontorova (FK) model with external potential. It is fortunate that we have made great progress in theoretical and experimental study especially in recent years. Terraneo and his collaborators proposed a method of realizing thermal rectifier micro-mechanism in 2002 [7]. Li et al. modeled a thermal diode with two different parameter FK lattices in 2004 [8]. In 2006, Li et al. made a three-FK-lattice model of thermal transistor with different parameters and observed that the resulting heat current decreases with increase of the applied temperature difference [9]. They first presented a new concept of NDTR referring to this phenomenon. In the same year, a carbon nanotube heat rectifier had been realized in laboratory [10]. After that scientists raise models boldly, such as thermal logic gates and thermal computers [11,12]. In 2008, Li and co-worker presented that the phenomenon of NDTR also was observed in a model with two segments FK lattices of different parameters [12]. In 2009, He et al. declared that NDTR can occur in both the FK 4 model and the f model with the same parameters, respectively, and also renounced it can not occur in the FPU model [13]. Shao et al. announced that NDTR can be seen in a two-section system, which consists of a single FK lattice and a single FPU lattice [14]. The present research results made us increasingly recognize
n
Corresponding author. Fax: þ86 20 85220233. E-mail address: [email protected] (W.-r. Zhong).
0921-4526/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2011.09.050
NDTR. Asymmetric FK system at the beginning, symmetrical FK 4 system and symmetrical f system then, compounded system with FK lattice and FPU lattice recently, those results suggest that asymmetric structure is not necessary to NDTR. We could pay more attention to and study the phenomenon of NDTR for a more comprehensive understanding of it. Meanwhile we find these researches are only related to one-dimensional single chain systems. In fact, double-chain systems such as DNA or double-chain polymers are more common and closer to reality [15,16]. In the last century, with rapid development in nanotechnology, more and more nano-materials have been found and made. Lowdimensional thermal devices applications and developments will be a focus. For a double-chain system, the inter-chain interaction is closer to authentic material. Thus, it should be more meaningful to construct double chains system and probe NDTR. Moreover, it is investigated that the importance parameters of intrachain interaction, inter-chain interaction, external potential and finite size effect can influence the phenomenon of NDTR in a double-chain system.
2. Model and simulation method We consider heat conductance in a double-chain system, a non-integrable model called the FK model. The Hamiltonian of the whole system is H ¼ HA þ HB þHintðA,BÞ :
ð1Þ
As shown in Fig. 1, we connect the ith particle of A-chain with the same order particle of B-chain via a harmonic spring kab, where we call it inter-chain interaction. The coupling Hamiltonian of the
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Fig. 1. Diagram of a symmetric double-chain structure. TL and TR are the temperature of the left and right heat baths, respectively.
two chains, A-chain and B-chain, is HintðA,BÞ ¼
N X
½kab ðxA,i xB,i Þ2 :
ð2Þ
Fig. 2. Heat flux J and the corresponding boundary jump dT as a function of the temperature difference D for weak inter-chain interaction kab ¼ 0:01. Here, k ¼0.5, V¼ 5.0, N ¼ 32, T R ¼ 0:001.
i¼1
The Hamiltonian of A-chain or B-chain is ( ) N X p2M,i k V þ ðxM,i þ 1 xM,i Þ2 þ ½1cosð2 p x Þ , HM ¼ M,i 2m 2 ð2pÞ2 i¼1
ð3Þ
where m is the mass of the particle, N the number of the particle of A-chain or B-chain. M stands for A or B. xM,i and pM,i are, respectively, the displacement from equilibrium position and conjugate momentum of the ith particle in chain M. k and V are the strength of the intra-chain interaction and the external potential in M-chain, respectively. We set m¼1 and adjust the main parameters N, k, kab and V. In our simulations we use fixed boundary conditions, i.e., xM,0 ¼ xM,N þ 1 ¼ 0. The first particle and the last particle of A-chain and B-chain are connected to heat baths. The temperature of the left and the right heat baths is, respectively , T R ð ¼ 0:001Þ and TL. In our simulations we use Langevin thermostat and integrate the equations of motion by using the 4th-order Runge–Kutta algorithm. We choose a step size of the simulation Dt ¼ 0:01 and averaging over 2 109 time steps. We have checked that our results do not depend on the particular thermostat realization (for example, Nose–Hoover thermostat [17]). The local temperature is defined as T i ¼ /p2i S=mkB , /S means time average, where the Boltzmann constant kB is set as unit. The local heat flux in double-chain is defined as [18] ji ¼ ka /pA,i ðxA,i xA,i1 ÞSþ kb /pB,i ðxB,i xB,i1 ÞS þ kab /pA,i ðxA,i xB,i ÞS þ kab /pB,i ðxB,i xA,i ÞS:
ð4Þ
The simulations are performed long enough to allow the system to reach a steady state in which the local heat flux is constant along the double-chain. The thermal conductivity is defined as K ¼ Nj=D, where D ¼ T L T R the temperature difference, J¼Nj the total heat flux. The boundary temperature jump is defined as dT ¼ TðN1ÞT R , where TR and TðN1Þ are the temperature of the right heat baths and the n-th particle connected to it, respectively.
3. Results and discussions The present research results show that the phenomenon of NDTR is easily observed in a single chain with high external potential and small size [13,14,19,20]. So we choose parameters as follow: N ¼32, k¼ 0.5, V ¼ 5:0, kab ¼ 0:01. 3.1. Phenomena and explanation Fig. 2 shows the heat current increases first and then decreases with increase of temperature difference. The region where the
Fig. 3. Temperature profiles for different temperature difference: (a) D ¼ 0:129 corresponding to point p2 in Fig. 2, (b) D ¼ 0:019 corresponding to point p1 in Fig. 2.
heat current increases as temperature difference increases is the positive differential thermal resistance (PDTR) and the region where the heat current decreases as temperature difference increases is the NDTR. In other words, the phenomenon of NDTR can be observed in a double-chain system. Like the single-chain system, we also hold the same viewpoint that there exist two competing effects: one is the enhancement effect by the thermal gradient and the other is the suppression effect by the phonon scattering. The heat current of the system depends on the winner of the effects. Therefore, the thermal gradient is the winner at low temperature but the phonon scattering at high temperature. The relevant temperature profiles are plotted in Fig. 3. When temperature difference is small, as shown in Fig. 3(b), the transport mode is superdiffusive [24]. However, at large temperature difference, Fig. 3(a) illustrates that the transport mode is
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more diffusive. This indicates the change of transport mode from superdiffusive to diffusive induces NDTR. We would also like to discuss the influence of the boundary thermal resistance on the NDTR. As shown in Fig. 1, the relationship of the heat flux J and the boundary temperature jump dT is similar to that of the heat flux vs temperature difference. It is believed that NDTR may be caused by some boundary effects, for example, the phonon-boundary scattering or thermal boundary resistance. 3.2. kab effect We change the inter-chain interaction kab and keep the other parameters unchanged. As shown in Fig. 4, we find that the heat current increases little in low temperature but much in high temperature as inter-chain interaction increases. Obviously, interchain interaction has a positive effect on the heat current. This positive heat current leads to the disappearance of NDTR. We choose three different inter-chain interactions and get the phonon spectra of the middle particle of each inter-chain interaction at the temperature difference D ¼ 0:129. As shown in Fig. 5, the phonon spectra show that its area increases and its frequency is invariant with the increase of inter-chain interaction. Usually phonon spectra is related to phonon frequencies and quantity. So we choose five different inter-chain interactions and get the areas of phonon spectra of the middle particle of each inter-chain interaction at the temperature difference D ¼ 0:02 and D ¼ 0:129. As shown in Fig. 6, the area of the phonon spectra increases with increase of inter-chain interaction at the same temperature difference. The two kinds of growth, at low and high temperature difference, are different as following: the former rises slowly and the latter grows fast. The results provide an explanation on the different increase of the heat current at low temperature difference and high temperature difference.
Fig. 5. Power spectra of the middle particle of A-chain for kab ¼ 0:1, 0.2, 0.3. Here, k¼0.5, V ¼5.0, N ¼ 32, T R ¼ 0:001, D ¼ 0:129.
3.3. k effect In order to study the effects of intra-chain interaction k, we adjust intra-chain interaction without changing the other parameters. Fig. 7 shows that the heat current of the system increases with the intra-chain interaction. NDTR can occur under weak intra-chain interaction. Considering the change of intra-chain interaction, we also study phonon spectra of the middle particle of each A-chain for various intra-chain interactions at the temperature difference D ¼ 0:129. The results are shown in Fig. 8.
Fig. 6. Power spectra area as a function of the inter-chain interaction kab for various values of the temperature difference D ¼ 0:02,0:13. Here, k¼ 0.5, V ¼5.0, N ¼32, T R ¼ 0:001.
Also, as expected, the area of phonon spectra increases with the intra-chain interaction. We find that they are different in the phonon spectra . As shown in Fig. 8, when intra-chain interaction increasing, the frequency is wide focusing on the previous frequency. A wider frequency range and a larger area of the phonon spectra mean more phonons. 3.4. V effect Fig. 4. Heat flux J as a function of the temperature difference D for various values of the inter-chain interactions kab ¼ 0:01,0:1,0:3,1:0. Here, k¼0.5, V ¼ 5.0, N ¼32, T R ¼ 0:001.
Similarly, altering the external potential V and maintaining other parameters, we get the relationship between the heat current and
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M.-p. Zhang et al. / Physica B 406 (2011) 4639–4643
Fig. 7. Heat flux J as a function of the temperature difference D for various values of the intra-chain interactions k¼ 0.3, 0.4, 0.5, 0.7, 1.0. Here, kab ¼ 0:01, V ¼5.0, N ¼32, T R ¼ 0:001.
Fig. 8. Power spectra of the middle particle of A-chain for various values of the intra-chain interactions k ¼0.3, 0.5, 0.7. Here, kab ¼ 0:01, V ¼5.0, N¼ 32, T R ¼ 0:001, D ¼ 0:129.
the external potential. As shown in Fig. 9, it is found that the heat current decreases as the external potential increases. In brief, the NDTR disappears in the system with low external potential. For the sake of comparison, we would like to consider phonon spectra under different external potentials. As shown in Fig. 10, the peak of phonon spectra moves to higher frequency and the area shrinks with the increase of the external potential. Using the self-consistent phonon (SCP) theory [21] and Debye formula, one can obtain the relationship of K vs k and V as follows: 3=2
Kp
k
V2
ð5Þ
Fig. 9. Heat flux J as a function of the temperature difference D for various values of the external potentials V ¼3.0, 5.0, 8.0, 10.0. Here, kab ¼ 0:01, k¼ 0.5, N ¼ 32, T R ¼ 0:001.
Fig. 10. Power spectra of the middle particle of A-chain for various values of the external potentials V¼ 3.0, 5.0, 10.0. Here, kab ¼ 0:01, k¼0.5, N ¼32, T R ¼ 0:001, D ¼ 0:129.
More details are reported in Refs. [18,22,23]. From the above results, we can also get a conclusion: the intra-chain interaction plays a positive role and the external potential performs a negative influence on the heat current of the system. Likewise, we can interpret the changing heat current in Figs. 7 and 9. 3.5. Size effect At last, we also discuss the influence of the finite size on the NDTR. Fig. 11 shows the relationship between the heat current of system and the number of particle in each chain. It is clearly
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the external potential plays a negative role on the heat current of the system.
Acknowledgments We would like to thank Siyuan clusters for running part of our programs. This work was supported in part by the National Natural Science Foundation of China (Grant No. 11004082), the National Natural Science Foundation of Guangdong Province (Grant No. 01005249) and the Fundamental Research Funds for the Central Universities, JNU (Grant No. 21609305). References
Fig. 11. Heat flux J as a function of the temperature difference D for various values of the scales N ¼32, 64, 128, 256. Here, kab ¼ 0:01, V ¼ 5.0, k¼ 0.5, T R ¼ 0:001.
presented that the heat current decreases as the system size increases. Just like the single chain system, NDTR disappears for the large system.
4. Conclusion In this paper, it is reported that the phenomenon of NDTR can be seen in a double-chain system. We investigate the influence of the parameters of N, V, k and kab on the NDTR. It is observed that the phenomenon of NDTR depends on the system with finite size and weak intra-chain interaction. For the system with weak external potential or strong inter-chain interaction the NDTR cannot occur. The heat current of the system, which related to the NDTR, is also studied. The inter- and intra-chain interactions have positive effects on the heat current of the system. However,
[1] S. Lepri, R. Livi, A. Politi, Phys. Rep. 377 (2003) 1; A. Dhar, Adv. Phys. 57 (2008) 457. [2] A.F. Neto, H.C.F. Lemos, E. Pereira, J. Phys. A 39 (2006) 9399. [3] F. Bonetto, J.L. Lebowitz, J. Lukkarinen, J. Stat. Phys. 116 (2004) 783. [4] B. Hu, B. Li, H. Zhao, Phys. Rev. E 57 (1998) 2992. [5] B. Li, H. Zhao, B. Hu, Phys. Rev. Lett. 86 (2001) 63. [6] B. Li, J. Wang, Phys. Rev. Lett. 91 (2003) 044301; B.W. Li, J.H. Lan, L. Wang, Phys. Rev. Lett. 95 (2005) 104302. [7] M. Terraneo, M. Peyrard, G. Casati, Phys. Rev. Lett. 88 (2002) 094302. [8] B. Li, L. Wang, G. Casati, Phys. Rev. Lett. 93 (2004) 184301. [9] B. Li, L. Wang, G. Casati, Appl. Phys. Lett. 88 (2006) 143501. [10] C.W. Chang, D. Okawa, A. Majumdar, A. Zell, Science 314 (2006) 1121. [11] L. Wang, B. Li, Phys. Rev. Lett. 99 (2007) 177208. [12] L. Wang, B. Li, Phys. Rev. Lett. 101 (2008) 267203. [13] D.H. He, B.Q. Ai, H.-K. Chan, B. Hu, Phys. Rev. E 81 (2010) 041131. [14] Z.-G. Shao, L. Yang, H.-K. Chan, B. Hu, Phys. Rev. E 79 (2009) 061119. [15] K. Drukker, G. Wu, G.C. Schatz, J. Chem. Phys. 114 (2001) 579. [16] X. Yu, D.M. Leitnera, J. Chem. Phys. 122 (2005) 054902. [17] S. Nose, J. Chem. Phys. 81 (1984) 511; W.G. Hoover, Phys. Rev. A 31 (1985) 1695. [18] W.R. Zhong, Phys. Rev. E 81 (2010) 061131; B.Q. Ai, W.R. Zhong, B. Hu, Phys. Rev. E 83 (2011) 052102. [19] D.H. He, S. Buyukdgli, B. Hu, Phys. Rev. B 80 (2009) 104302. [20] W.R. Zhong, P. Yang, B.Q. Ai, Z.G. Shao, B. Hu, Phys. Rev. E 79 (2009) 050103R. [21] Z.G. Shao, L. Yang, W.R. Zhong, D.H. He, B. Hu, Phys. Rev. E 78 (2008) 061130. [22] T. Dauxois, M. Peyrard, A.R. Bishop, Phys. Rev. E 47 (1993) 684. [23] B. Hu, D. He, L. Yang, Y. Zhang, Phys. Rev. E 74 (2006) 060101R. [24] B. Li, J. Wang, L. Wang, G. Zhang, Chaos 15 (2005) 015121.
Contents lists available at SciVerse ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Negative differential thermal resistance in a double-chain system Mao-ping Zhang, Wei-rong Zhong n, Jian-yuan Qiu, Yin-ze Ding Department of Physics and Siyuan Laboratory, College of Science and Engineering, Jinan University, 510632 Guangzhou, China
a r t i c l e i n f o
abstract
Article history: Received 22 March 2011 Received in revised form 15 July 2011 Accepted 16 September 2011 Available online 22 September 2011
Using nonequilibrium molecular dynamics simulations, we study the phenomenon of negative differential thermal resistance (NDTR) in a double-chain system. We investigate the dependence of NDTR on the external potential, inter- and intra-chain interaction and the system size. It is reported that the NDTR can occur in a small double-chain system with weak external potential and weak interand intra-chain interaction. We also present the influence of the external potential, inter- and intrachain interaction and the system size on the heat current of the system through the phonon spectral analysis. & 2011 Elsevier B.V. All rights reserved.
Keywords: Negative differential thermal resistance Double chains Heat current Phonon spectra
1. Introduction In recent decades, heat transport in low-dimensional systems has been given widely attention, especially in one-dimensional single chain systems [1–6]. In these researches, there are two major models, which are Fermi–Pasta–Ulam (FPU) model without external potential and Frenkel–Kontorova (FK) model with external potential. It is fortunate that we have made great progress in theoretical and experimental study especially in recent years. Terraneo and his collaborators proposed a method of realizing thermal rectifier micro-mechanism in 2002 [7]. Li et al. modeled a thermal diode with two different parameter FK lattices in 2004 [8]. In 2006, Li et al. made a three-FK-lattice model of thermal transistor with different parameters and observed that the resulting heat current decreases with increase of the applied temperature difference [9]. They first presented a new concept of NDTR referring to this phenomenon. In the same year, a carbon nanotube heat rectifier had been realized in laboratory [10]. After that scientists raise models boldly, such as thermal logic gates and thermal computers [11,12]. In 2008, Li and co-worker presented that the phenomenon of NDTR also was observed in a model with two segments FK lattices of different parameters [12]. In 2009, He et al. declared that NDTR can occur in both the FK 4 model and the f model with the same parameters, respectively, and also renounced it can not occur in the FPU model [13]. Shao et al. announced that NDTR can be seen in a two-section system, which consists of a single FK lattice and a single FPU lattice [14]. The present research results made us increasingly recognize
n
Corresponding author. Fax: þ86 20 85220233. E-mail address: [email protected] (W.-r. Zhong).
0921-4526/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2011.09.050
NDTR. Asymmetric FK system at the beginning, symmetrical FK 4 system and symmetrical f system then, compounded system with FK lattice and FPU lattice recently, those results suggest that asymmetric structure is not necessary to NDTR. We could pay more attention to and study the phenomenon of NDTR for a more comprehensive understanding of it. Meanwhile we find these researches are only related to one-dimensional single chain systems. In fact, double-chain systems such as DNA or double-chain polymers are more common and closer to reality [15,16]. In the last century, with rapid development in nanotechnology, more and more nano-materials have been found and made. Lowdimensional thermal devices applications and developments will be a focus. For a double-chain system, the inter-chain interaction is closer to authentic material. Thus, it should be more meaningful to construct double chains system and probe NDTR. Moreover, it is investigated that the importance parameters of intrachain interaction, inter-chain interaction, external potential and finite size effect can influence the phenomenon of NDTR in a double-chain system.
2. Model and simulation method We consider heat conductance in a double-chain system, a non-integrable model called the FK model. The Hamiltonian of the whole system is H ¼ HA þ HB þHintðA,BÞ :
ð1Þ
As shown in Fig. 1, we connect the ith particle of A-chain with the same order particle of B-chain via a harmonic spring kab, where we call it inter-chain interaction. The coupling Hamiltonian of the
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M.-p. Zhang et al. / Physica B 406 (2011) 4639–4643
Fig. 1. Diagram of a symmetric double-chain structure. TL and TR are the temperature of the left and right heat baths, respectively.
two chains, A-chain and B-chain, is HintðA,BÞ ¼
N X
½kab ðxA,i xB,i Þ2 :
ð2Þ
Fig. 2. Heat flux J and the corresponding boundary jump dT as a function of the temperature difference D for weak inter-chain interaction kab ¼ 0:01. Here, k ¼0.5, V¼ 5.0, N ¼ 32, T R ¼ 0:001.
i¼1
The Hamiltonian of A-chain or B-chain is ( ) N X p2M,i k V þ ðxM,i þ 1 xM,i Þ2 þ ½1cosð2 p x Þ , HM ¼ M,i 2m 2 ð2pÞ2 i¼1
ð3Þ
where m is the mass of the particle, N the number of the particle of A-chain or B-chain. M stands for A or B. xM,i and pM,i are, respectively, the displacement from equilibrium position and conjugate momentum of the ith particle in chain M. k and V are the strength of the intra-chain interaction and the external potential in M-chain, respectively. We set m¼1 and adjust the main parameters N, k, kab and V. In our simulations we use fixed boundary conditions, i.e., xM,0 ¼ xM,N þ 1 ¼ 0. The first particle and the last particle of A-chain and B-chain are connected to heat baths. The temperature of the left and the right heat baths is, respectively , T R ð ¼ 0:001Þ and TL. In our simulations we use Langevin thermostat and integrate the equations of motion by using the 4th-order Runge–Kutta algorithm. We choose a step size of the simulation Dt ¼ 0:01 and averaging over 2 109 time steps. We have checked that our results do not depend on the particular thermostat realization (for example, Nose–Hoover thermostat [17]). The local temperature is defined as T i ¼ /p2i S=mkB , /S means time average, where the Boltzmann constant kB is set as unit. The local heat flux in double-chain is defined as [18] ji ¼ ka /pA,i ðxA,i xA,i1 ÞSþ kb /pB,i ðxB,i xB,i1 ÞS þ kab /pA,i ðxA,i xB,i ÞS þ kab /pB,i ðxB,i xA,i ÞS:
ð4Þ
The simulations are performed long enough to allow the system to reach a steady state in which the local heat flux is constant along the double-chain. The thermal conductivity is defined as K ¼ Nj=D, where D ¼ T L T R the temperature difference, J¼Nj the total heat flux. The boundary temperature jump is defined as dT ¼ TðN1ÞT R , where TR and TðN1Þ are the temperature of the right heat baths and the n-th particle connected to it, respectively.
3. Results and discussions The present research results show that the phenomenon of NDTR is easily observed in a single chain with high external potential and small size [13,14,19,20]. So we choose parameters as follow: N ¼32, k¼ 0.5, V ¼ 5:0, kab ¼ 0:01. 3.1. Phenomena and explanation Fig. 2 shows the heat current increases first and then decreases with increase of temperature difference. The region where the
Fig. 3. Temperature profiles for different temperature difference: (a) D ¼ 0:129 corresponding to point p2 in Fig. 2, (b) D ¼ 0:019 corresponding to point p1 in Fig. 2.
heat current increases as temperature difference increases is the positive differential thermal resistance (PDTR) and the region where the heat current decreases as temperature difference increases is the NDTR. In other words, the phenomenon of NDTR can be observed in a double-chain system. Like the single-chain system, we also hold the same viewpoint that there exist two competing effects: one is the enhancement effect by the thermal gradient and the other is the suppression effect by the phonon scattering. The heat current of the system depends on the winner of the effects. Therefore, the thermal gradient is the winner at low temperature but the phonon scattering at high temperature. The relevant temperature profiles are plotted in Fig. 3. When temperature difference is small, as shown in Fig. 3(b), the transport mode is superdiffusive [24]. However, at large temperature difference, Fig. 3(a) illustrates that the transport mode is
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more diffusive. This indicates the change of transport mode from superdiffusive to diffusive induces NDTR. We would also like to discuss the influence of the boundary thermal resistance on the NDTR. As shown in Fig. 1, the relationship of the heat flux J and the boundary temperature jump dT is similar to that of the heat flux vs temperature difference. It is believed that NDTR may be caused by some boundary effects, for example, the phonon-boundary scattering or thermal boundary resistance. 3.2. kab effect We change the inter-chain interaction kab and keep the other parameters unchanged. As shown in Fig. 4, we find that the heat current increases little in low temperature but much in high temperature as inter-chain interaction increases. Obviously, interchain interaction has a positive effect on the heat current. This positive heat current leads to the disappearance of NDTR. We choose three different inter-chain interactions and get the phonon spectra of the middle particle of each inter-chain interaction at the temperature difference D ¼ 0:129. As shown in Fig. 5, the phonon spectra show that its area increases and its frequency is invariant with the increase of inter-chain interaction. Usually phonon spectra is related to phonon frequencies and quantity. So we choose five different inter-chain interactions and get the areas of phonon spectra of the middle particle of each inter-chain interaction at the temperature difference D ¼ 0:02 and D ¼ 0:129. As shown in Fig. 6, the area of the phonon spectra increases with increase of inter-chain interaction at the same temperature difference. The two kinds of growth, at low and high temperature difference, are different as following: the former rises slowly and the latter grows fast. The results provide an explanation on the different increase of the heat current at low temperature difference and high temperature difference.
Fig. 5. Power spectra of the middle particle of A-chain for kab ¼ 0:1, 0.2, 0.3. Here, k¼0.5, V ¼5.0, N ¼ 32, T R ¼ 0:001, D ¼ 0:129.
3.3. k effect In order to study the effects of intra-chain interaction k, we adjust intra-chain interaction without changing the other parameters. Fig. 7 shows that the heat current of the system increases with the intra-chain interaction. NDTR can occur under weak intra-chain interaction. Considering the change of intra-chain interaction, we also study phonon spectra of the middle particle of each A-chain for various intra-chain interactions at the temperature difference D ¼ 0:129. The results are shown in Fig. 8.
Fig. 6. Power spectra area as a function of the inter-chain interaction kab for various values of the temperature difference D ¼ 0:02,0:13. Here, k¼ 0.5, V ¼5.0, N ¼32, T R ¼ 0:001.
Also, as expected, the area of phonon spectra increases with the intra-chain interaction. We find that they are different in the phonon spectra . As shown in Fig. 8, when intra-chain interaction increasing, the frequency is wide focusing on the previous frequency. A wider frequency range and a larger area of the phonon spectra mean more phonons. 3.4. V effect Fig. 4. Heat flux J as a function of the temperature difference D for various values of the inter-chain interactions kab ¼ 0:01,0:1,0:3,1:0. Here, k¼0.5, V ¼ 5.0, N ¼32, T R ¼ 0:001.
Similarly, altering the external potential V and maintaining other parameters, we get the relationship between the heat current and
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M.-p. Zhang et al. / Physica B 406 (2011) 4639–4643
Fig. 7. Heat flux J as a function of the temperature difference D for various values of the intra-chain interactions k¼ 0.3, 0.4, 0.5, 0.7, 1.0. Here, kab ¼ 0:01, V ¼5.0, N ¼32, T R ¼ 0:001.
Fig. 8. Power spectra of the middle particle of A-chain for various values of the intra-chain interactions k ¼0.3, 0.5, 0.7. Here, kab ¼ 0:01, V ¼5.0, N¼ 32, T R ¼ 0:001, D ¼ 0:129.
the external potential. As shown in Fig. 9, it is found that the heat current decreases as the external potential increases. In brief, the NDTR disappears in the system with low external potential. For the sake of comparison, we would like to consider phonon spectra under different external potentials. As shown in Fig. 10, the peak of phonon spectra moves to higher frequency and the area shrinks with the increase of the external potential. Using the self-consistent phonon (SCP) theory [21] and Debye formula, one can obtain the relationship of K vs k and V as follows: 3=2
Kp
k
V2
ð5Þ
Fig. 9. Heat flux J as a function of the temperature difference D for various values of the external potentials V ¼3.0, 5.0, 8.0, 10.0. Here, kab ¼ 0:01, k¼ 0.5, N ¼ 32, T R ¼ 0:001.
Fig. 10. Power spectra of the middle particle of A-chain for various values of the external potentials V¼ 3.0, 5.0, 10.0. Here, kab ¼ 0:01, k¼0.5, N ¼32, T R ¼ 0:001, D ¼ 0:129.
More details are reported in Refs. [18,22,23]. From the above results, we can also get a conclusion: the intra-chain interaction plays a positive role and the external potential performs a negative influence on the heat current of the system. Likewise, we can interpret the changing heat current in Figs. 7 and 9. 3.5. Size effect At last, we also discuss the influence of the finite size on the NDTR. Fig. 11 shows the relationship between the heat current of system and the number of particle in each chain. It is clearly
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the external potential plays a negative role on the heat current of the system.
Acknowledgments We would like to thank Siyuan clusters for running part of our programs. This work was supported in part by the National Natural Science Foundation of China (Grant No. 11004082), the National Natural Science Foundation of Guangdong Province (Grant No. 01005249) and the Fundamental Research Funds for the Central Universities, JNU (Grant No. 21609305). References
Fig. 11. Heat flux J as a function of the temperature difference D for various values of the scales N ¼32, 64, 128, 256. Here, kab ¼ 0:01, V ¼ 5.0, k¼ 0.5, T R ¼ 0:001.
presented that the heat current decreases as the system size increases. Just like the single chain system, NDTR disappears for the large system.
4. Conclusion In this paper, it is reported that the phenomenon of NDTR can be seen in a double-chain system. We investigate the influence of the parameters of N, V, k and kab on the NDTR. It is observed that the phenomenon of NDTR depends on the system with finite size and weak intra-chain interaction. For the system with weak external potential or strong inter-chain interaction the NDTR cannot occur. The heat current of the system, which related to the NDTR, is also studied. The inter- and intra-chain interactions have positive effects on the heat current of the system. However,
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