Effective phonon theory of heat conduction in 1d nonlinear lattice chains

133 243 0
Effective phonon theory of heat conduction in 1d nonlinear lattice chains

Đang tải... (xem toàn văn)

Tài liệu hạn chế xem trước, để xem đầy đủ mời bạn chọn Tải xuống

Thông tin tài liệu

EFFECTIVE PHONON THEORY OF HEAT CONDUCTION IN 1D NONLINEAR LATTICE CHAINS LI NIANBEI A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF PHYSICS NATIONAL UNIVERSITY OF SINGAPORE 2007 Acknowledgements First I would like to thank my dear parent for their consistent support. They always have confidence on me and encourage me to go further and further along this academic avenue. There would be no this beautiful research work without the guidance of Prof. Li Baowen, my dear supervisor. He is such a tutor with great passion and enthusiasm acting like a high temperature thermostat with tremendous heat capacity. I can always gain momentum by absorbing the energy from him. Determination, focus, diligence, insight, , I have learned a lot from him. I would also like to thank Prof. Wang Wenge, Prof. Tong Peiqing, Dr. Wang Lei and Dr. Lan Jinghua for their valuable suggestions and comments. Many thanks to the colleagues under the same roof, Mr. Lo Wei Chung, Mr. Yang Nuo, Mr. Dario Poletti, Dr. Zhang Gang. Finally, I would like to thank all my friends for experiencing the four years in Singapore along with me. I really enjoy these days. i Summary This thesis deals with the classical heat conduction of 1D nonlinear lattices. A new theory of heat conduction, Effective Phonon Theory, has been developed based on the effective phonons. The effective phonons are the renormalized phonons due to the nonlinear interaction of nonlinear lattices. Their broad existence is found for lattices without on-site potential and lattices with on-site potential. For lattices without on-site potential, the resulted effective phonons are acoustic-like. For lattices with on-site potential, the effective phonons are optical-like. These properties are considered by the Debye formula of heat conductivity in terms of effective phonons and the anomalous/normal heat conduction for lattices without/with on-site potential is well explained by this effective phonon theory. A correlation between nonlinearity strength and heat conductivity has been found through numerical simulations. By incarnating this nonlinearity strength into the expression for the mean-free-path of effective phonons, the temperature dependence of heat conductivity is explained consistently by the effective phonon theory for lattices without on-site potential and lattices with on-site potential, at ii SUMMARY iii low temperature regimes and high temperature regimes. The effective phonon theory is applied to the 1D φ4 lattice with strong harmonic on-site potential. The parameter dependence of heat conductivity beyond the size and temperature dependence has been derived and compared with the numerical simulations performed by stationary Non-Equilibrium Molecular Dynamics. Contents Acknowledgements i Summary ii Contents iv List of Figures viii List of Tables x Introduction 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Lattice models . . . . . . . . . . . . . . . . . . . . . . . . . . iv CONTENTS v 1.2.2 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.3 Heat flux 1.2.4 Heat baths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Literature Review of Heat Conduction in 1D systems . . . . . . . . . 14 1.3.1 Breakdown of Fourier’s law . . . . . . . . . . . . . . . . . . . 14 1.3.2 Anomalous heat conduction . . . . . . . . . . . . . . . . . . . 20 1.4 Purpose and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Effective Phonon Theory of Heat Conduction 27 2.1 Concept of Effective Phonons . . . . . . . . . . . . . . . . . . . . . . 28 2.1.1 Renormalized phonon spectrum in general 1D nonlinear lattices 29 2.1.2 Quasi-Periodical oscillation of effective phonons . . . . . . . . 39 2.1.3 Sound velocity of effective phonons . . . . . . . . . . . . . . . 44 2.2 Formula of Heat Conductivity . . . . . . . . . . . . . . . . . . . . . . 48 2.2.1 Lattices with on-site potential . . . . . . . . . . . . . . . . . . 51 2.2.2 Lattices without on-site potential . . . . . . . . . . . . . . . . 53 2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 CONTENTS vi Temperature-dependent Thermal Conductivities 58 3.1 Failure of Phonon Collision Theory . . . . . . . . . . . . . . . . . . . 59 3.2 Nonlinearity and Heat Conductivity . . . . . . . . . . . . . . . . . . . 62 3.3 Temperature Behavior of Heat Conductivities . . . . . . . . . . . . . 65 3.3.1 Lattices without on-site potential . . . . . . . . . . . . . . . . 66 3.3.2 Lattices with on-site potential . . . . . . . . . . . . . . . . . . 72 3.3.3 Lattices with single scaling potential . . . . . . . . . . . . . . 78 3.3.4 Bulk materials, nanotubes and nanowires . . . . . . . . . . . . 80 3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Parameter-dependent Thermal Conductivities of 1D φ4 Lattice 85 4.1 Effective Phonon Theory . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.2.1 T dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.2.2 λ dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.2.3 µ dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.2.4 ω dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 CONTENTS vii Conclusions and Future Works 99 A Dimensionless Units in MD Simulations 103 B Specific Heat of 1D Nonlinear Lattices 109 C Publication List 113 List of Figures 1.1 Pictorial representation of a lattice chain . . . . . . . . . . . . . . . . 1.2 Schematic temperature profile for harmonic lattice . . . . . . . . . . . 17 2.1 Renormalized phonon spectrum of FPU-β lattice . . . . . . . . . . . 34 2.2 Renormalized phonon spectrum of H4 lattice . . . . . . . . . . . . . . 36 2.3 Renormalized phonon spectrum of φ4 lattice . . . . . . . . . . . . . . 37 2.4 Renormalized phonon spectrum of Quartic φ4 lattice . . . . . . . . . 38 2.5 Quasi-periodic oscillation of H4 lattice . . . . . . . . . . . . . . . . . 42 2.6 Quasi-periodic oscillation of quartic φ4 lattice . . . . . . . . . . . . . 43 2.7 Sound velocity of FPU-β lattice . . . . . . . . . . . . . . . . . . . . . 47 3.1 Temperature dependence of heat conductivity of FK lattice . . . . . . 61 viii LIST OF FIGURES ix 3.2 Temperature dependence of heat conductivity for FPU-β, symmetric FPU-α and FPU-αβ lattice . . . . . . . . . . . . . . . . . . . . . . . 70 3.3 Temperature dependence of heat conductivity for Quartic φ4 lattice . 76 4.1 Temperature dependence of heat conductivity of φ4 lattice with different parameter µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.2 Temperature dependence of the integral P . . . . . . . . . . . . . . . 92 4.3 Parameter λ dependence of heat conductivity of φ4 lattice with different µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.4 Parameter λ dependence of the integral P . . . . . . . . . . . . . . . 94 4.5 Parameter µ dependence of heat conductivity of φ4 lattice . . . . . . 95 4.6 Parameter ω dependence of heat conductivity of φ4 lattice . . . . . . 97 107 Deviding the above equation by ka2 and we obtain N H = i=1 K pi2 + (xi − xi−1 − 1)2 − cos 2πxi 2 (2π)2 (A.7) where a new dimensionless parameter K has been defined as K ≡ V /ka2 . The above equation is the dimensionless Hamiltonian of FK lattice which is actually used for calculation in MD simulation. Every parameter and variable is dimensionless inside this dimensionless Hamiltonian. The final physical quantity is recovered by the multiplication of calculated dimensionless number and its appropriate dimensionful unit. In the study of heat conduction which is in the area of statistical physics, we should introduce another fundamental unit of temperature by the Boltzmann constant kB T = p2i m (A.8) From the dimension analysis, the Boltzmann constant kB has the dimension of −1 has the [kB ] =Mass·Length2 ·Time−2 ·Temperature−1 . So the combination of ka2 kB −1 dimension of temperature. For convenience, we can chose this combination ka2 kB as the dimensionful unit of temperature. Therefore the dimensionful temperature T and the dimensionless temperature T has the following relationship −1 T = T × ka2 kB (A.9) Substituting the expression of dimensionful momentum and temperature into the definition of temperature of Eq. (A.8), we obtain kB T −1 ka2 kB = pi2 a2 mk m (A.10) 108 By dropping the factor ka2 from both side, we obtain the expression of dimensionless temperature T in terms of the dimensionless momentum p T = pi2 (A.11) Appendix B Specific Heat of 1D Nonlinear Lattices In classical statistical physics, the specific heat is defined as c= H NT (B.1) where the Boltzmann constant has been set to unit and H is the ensemble average of total energy at temperature T . It is convenient to split the total energy into kinetic energy K and potential energy V . The specific heat consists of contributions from kinetic energy as well as potential energy V K + NT NT c = cK + cV = (B.2) The ensemble average is usually calculated with the help of equipartition theorem which states T = qi ∂H ∂qi 109 (B.3) 110 where qi could be the displacement xi or momentum pi . The specific heat from kinetic energy is same for all systems as long as the kinetic energy has the quadratic form p2i N i=1 pi cK = NT = N i=1 pi ∂H ∂pi NT = (B.4) The determination of specific heat c is now simplified to the determination of specific heat from potential energy cV . For Harmonic lattice with Hamiltonian N H =K +V = i=1 p2i + N i=1 (xi − xi+1 )2 (B.5) The expression of equipartition theorem for displacement xi has the form N xi NT = i=1 N = ∂H ∂xi xi [(xi − xi+1 ) − (xi−1 − xi )] i=1 N = N xi (xi − xi+1 ) − i=1 N = xi (xi−1 − xi ) i=1 N xi (xi − xi+1 ) − i=1 N xi+1 (xi − xi+1 ) i=1 (xi − xi+1 )2 = i=1 = V (B.6) Thus the specific heat of potential energy is obtained cV = V = NT (B.7) 111 The total specific heat of Harmonic lattice c = cK + cV = which is the well known results of energy equipartition theorem. For any lattice system which can be described by the Harmonic lattice with small perturbation, the specific heat can be approximated as c ≈ for this perturbation regime. This is the case for low temperature regime of FPU-β lattice, symmetric FPU-α lattice and φ4 lattice. For FK lattice, the specific heat c ≈ for both low and high temperature regime. At high temperature, the nonlinear interaction usually can no longer be neglected such as FPU-β lattice and symmetric FPU-α lattice. However, the specific heat of potential energy can still be derived analytically for these lattices at high temperature limit T → ∞. Considering the Hs lattices with the following Hamiltonian N H =K +V = i=1 p2i + N i=1 (xi − xi+1 )s s (B.8) The equipartition theorem states N NT = xi i=1 N ∂H ∂xi xi [(xi − xi+1 )s−1 − (xi−1 − xi )s−1 ] = i=1 N = N xi (xi − xi+1 ) s−1 xi (xi−1 − xi )s−1 − i=1 N i=1 N xi (xi − xi+1 )s−1 − = i=1 N xi+1 (xi − xi+1 )s−1 i=1 (xi − xi+1 )s = i=1 = s V (B.9) 112 The total specific heat is thus obtained c = cK + cV = 1 + s (B.10) The specific heat for FPU-β lattice or symmetric FPU-α lattice can be approximated as a constant for both low and high temperature regime. However, the constant is at low temperature regime and or FPU-β and symmetric FPU-α lattices respectively. at high temperature regime for Appendix C Publication List 1. Nianbei Li, Peiqing Tong and Baowen Li, ”Effective phonons in anharmonic lattices: Anomalous vs. normal heat conduction”, Europhys. Lett., 75 (1), 49 (2006). 2. Nianbei Li and Baowen Li, ”Temperature dependence of thermal conductivity in 1D nonlinear lattices”, EPL, 78, 34001 (2007). 3. Nianbei Li and Baowen Li, ”Parameter-dependent heat conductivity of 1D φ4 lattice”, Phys. Rev. E 76, 011108 (2007). 4. Nuo Yang, Nianbei Li, Lei Wang and Baowen Li, ”Thermal rectification and negative differential thermal resistance in mass graded systems”, Phys. Rev. B 76, 020301(R) (2007). 113 Bibliography [1] S. Lepri, R. Livi and A. Politi, Phys. Rep. 377, (2003) and the references therein. [2] R. E. Peierls, Quantum Theory of Solids, Oxford University Press, London (1955). [3] B. Hu, B. Li and H. Zhao, Phys. Rev. E 61, 3828 (2000). [4] F. Zhang, D. Isbister and D. Evans, Phys. Rev. E 61, 3541 (2000). [5] H. Zhao, Z. Wen, Y. Zhang and E. Zheng, Phys. Rev. lett. 94, 025507 (2005). [6] K. Aoki and D. Kusnezov, Phys. Rev. Lett. 86, 4029 (2001). [7] N. Zabusky and M. Kruskal, Phys. Rev. Lett. 15, 240 (1965). [8] J. Ford, Phys. Rep. 213, 271 (1992) and the references therein. [9] S. Flach, C. R. Willis, Phys. Rep. 295, 181 (1998). [10] G. Tsironis, A. Bishop, A. Savin and A. Zolotaryuk, Phys. Rev. E 60, 6610 (1999). 114 BIBLIOGRAPHY 115 [11] A. Savin and O. Gendelman, Phys. Rev. E 67, 041205 (2003). [12] X. Zhang, S. Fujiwara, and M. Fujii, Int. J. Thermophys. 21, 965 (2000). [13] M. Fujii, et al., Phys. Rev. Lett. 95, 065502 (2005). [14] D. Morelli, J. Heremans, M. Sakamoto and C. Uher, Phys. Rev. Lett. 57, 869 (1986). [15] A. Smontara, J. Lasjaunas and R. Maynard, Phys. Rev. Lett. 77, 5397 (1996). [16] A. Sologubenko, et al., Phys. Rev. B 64, 054412 (2001). [17] H. Forsman and P. Anderson, J. Chem. Phys. 80, 2804 (1984). [18] M. Terraneo, M. Peyrard and G. Casati, Phys. Rev. Lett. 88, 094302 (2002). [19] B. Li, L. Wang and G. Casati, Phys. Rev. Lett. 93, 184301 (2004). [20] B. Li, J. Lan and L. Wang, Phys. Rev. Lett. 95, 104302 (2005). [21] D. Segal and A. Nitzan, Phys. Rev. Lett. 94, 034301 (2005). [22] J. Lan and B. Li, Phys. Rev. B 74, 214305 (2006). [23] B. Hu, L. Yang and Y. Zhang, Phys. Rev. Lett. 97, 124302 (2006). [24] J. Lan and B. Li, Phys. Rev. B 75, xxx (2007). (in press) [25] B. Li, L. Wang and G. Casati, Appl. Phys. Lett. 88, 143501 (2006). [26] C. W. Chang, D. Okawa, A. Majumdar and A. Zettl, Science 314, 1121 (2006). [27] E. Fermi, J. Pasta and S. Ulam, Los Alamos, Report No. LA-1940, (1955). BIBLIOGRAPHY 116 [28] P. Bak, Rep. Prog. Phys. 45, 587 (1982) and references therein. [29] W. Selke, Phase Transitions and Critical Phenomena, edited by C. domb and J. L. Lebowitz (Academic Press, London, 1992), Vol. 15. [30] S. Aubry, in Solitons and Condensed Matter Physics, edited by A. R. Bishop and T. Schneider (Springer-Verlag, Berlin, 1978). [31] S. Aubry, J. Phys. (France) 44, 147 (1983). [32] M. Peyrard and S. Aubry, J. Phys. C 16, 1593 (1983). [33] S. Aubry, Physica D 7, 240 (1983). [34] S. Aubry and P. Y. LeDaeron, Physica D 8, 381 (1983). [35] D. Chen, S. Aubry and G. P. Tsironis, Phys. Rev. Lett. 77, 4776 (1996). [36] K. Huang, Quantum Field Theory, Wiley-VCH (2004). [37] F. M. Izrailev and B. V. Chirikov, Sov. Phys. Dokl. 11, 30 (1966). [38] D. I. Shepelyansky, Nonlinearity 10, 1331 (1997). [39] P. Bocchieri, A. Scotti, B. Bearzi and A. Loinger, Phys. Rev. A 2, 2013 (1970). [40] E. Fucito, F. Marchesoni, E. Marinari, G. Parisi, L. Peliti, S. Ruffo and A. Vulpiani, J. Phys. (France) 43, 707 (1982). [41] G. Parisi, Europhys. Lett. 40, 357 (1997). [42] R. Livi, M. Pettini, S. Ruffo, M. Sparpaglione and A. Vulpiani, Phys. Rev. A 31, 1039 (1985). BIBLIOGRAPHY 117 [43] R. Livi, M. Pettini, S. Ruffo and A. Vulpiani, Phys. Rev. A 31, 2740 (1985). [44] L. Berchialla, L. Galgani and A. Giorgilli, Discrete Contin. Dyn. Syst., Ser. A 11, 855 (2004). [45] L. Berchialla, A. Giorgilli and S. Paleari, Phys. Lett. A 321, 167 (2004). [46] A. Carati, L. Galgani and A. Giorgilli, Chaos 15, 015105 (2005). [47] H. Kantz, R. Livi and S. Ruffo, J. Stat. Phys. 76, 627 (1994). [48] J. De Luca, A. Lichtenberg and S. Ruffo, Phys. Rev. E 51, 2877 (1995). [49] L. Casetti, C. Clementi and M. Pettini, Phys. Rev. E 54, 5969 (1996). [50] M. Pettini, L. Casetti, M. Cerruti-Sola, R. Franzosi and E. G. D. Cohen, Chaos 15, 015106 (2005). [51] Ph. Choquard, Helv. Phys. Acta 36, 415 (1963). [52] R. J. Hardy, Phys. Rev. 132, 168 (1963). [53] M. Allen and D. Tildesley, Computer Simulation of Liquids, Oxford University Press (1987). [54] S. Nose, J. Chem Phys. 81, 511 (1984). [55] W. Hoover, Phys. Rev. A 31, 1695 (1985). [56] P. Poggi and S. Ruffo, Physica D 103, 251(1997). [57] Z. Rieder, J. Lebowitz and E. Lieb, J. Math. Phys. 8, 1073 (1967). BIBLIOGRAPHY 118 [58] H. Matsuda and K. Ishii, Suppl. Prog. Theor. Phys. 45, 56 (1970). [59] J. Keller, G. Papanicolaou and J. Weilenmann, Comm. Pure. App. Math. Vol. XXXII, 583 (1978). [60] W. Visscher, Prog. Theor. Phys. 46, 729 (1971). [61] D. Payton III and W. Visscher, Phys. Rev. 156, 1032 (1967). [62] D. Payton III, M. Rich and W. M. Visscher, Phys. Rev. 160, 706 (1967). [63] A. Casher and J. Lebowitz, J. Math. Phys. 8, 1701 (1971). [64] A. OConnor and J .Lebowitz ,J. Math. Phys. 15, 692 (1974). [65] H. Zhao, L. Yi, F. Liu and B. Xu, Eur. Phys. J. B 54, 185 (2006). [66] A. Dhar, Phys. Rev. Lett. 86, 5882 (2001). [67] G. Casati, J. Ford, F. Vivaldi and W. Visscher, Phys. Rev. Lett. 52, 1861 (1984). [68] T. Prosen and M. Robnik, J. Phys. A 25, 2449 (1992). [69] H. Posch and W. Hoover, Phys. Rev. E. 58, 4344 (1998). [70] S. Lepri, R. Livi and A. Politi, Phys. Rev. Lett. 78, 1896 (1997). [71] B. Li, L. Wang and B. Hu, Phys. Rev. Lett. 88, 223901 (2002) [72] B. Hu, B. Li and H. Zhao, Phys. Rev. E 57, 2992 (1998). [73] K. Aoki and D. Kusnezov, Phys. Lett. A 265, 250 (2000). BIBLIOGRAPHY 119 [74] T. Prosen and D. Campbell, Phys. Rev. Lett. 84, 2857 (2000). [75] O. Narayan and S. Ramaswamy, Phys. Rev. Lett. 89, 200601 (2002). [76] C. Giardina, R. Livi, A. Politi and M. Vassalli, Phys. Rev. Lett. 84, 2144 (2000). [77] O. Gendelman and A. Savin, Phys. Rev. Lett. 84, 2381 (2000). [78] A. Savin and O. Gendelman, Physics of the Solid State 43, 355 (2001). [79] S. Lepri, R. Livi and A. Politi, Physica D 119, 140 (1998). [80] S. Lepri, R. Livi and A. Politi, Europhys. Lett. 43, 271 (1998). [81] S. Lepri, Eur. Phys. J. B 18, 441 (2000). [82] S. Lepri, R. Livi and A. Politi, Phys. Rev. E 68, 067102 (2003). [83] T. Hatano, Phys. Rev. E 59, R1 (1999). [84] M. Vassalli, Diploma Thesis, University of Florence, (1999). [85] S. Lepri, Phys. Rev. E 58, 7165 (1998). [86] R. Kubo, M. Toda and N. Hashitsume, Springer Ser. solid-State Sci. 31, 185 (1991). [87] A. Pereverzev, Phys. Rev. E 68, 056124 (2003). [88] J. Lukkarinen and H. Spohn, arXiv:0704.1607v1 (2007). [89] P. Grassberger, W. Nadler and L. Yang, Phys. Rev. Lett. 89,180601 (2002). [90] J. M. Deutsch and O. Narayan, Phys. Rev. E 68, 010201(R) (2003). BIBLIOGRAPHY 120 [91] J. M. Deutsch and O. Narayan, Phys. Rev. E 68, 041203 (2003). [92] L. Delfini, S. Lepri, R. Livi and A. Politi, Phys. Rev. E 73, 060201(R) (2006). [93] T. Mai and O. Narayan, Phys. Rev. E 73, 061202 (2006). [94] J. S. Wang and B. Li, Phys. Rev. Lett. 92, 074302 (2004). [95] J. S. Wang and B. Li, Phys. Rev. E 70, 021204 (2004). [96] B. Li and J. Wang, Phys. Rev. Lett. 91, 044301 (2003). [97] D. Alonso, R. Artuso, G. Casati and I. Guarneri, Phys. Rev. Lett. 82, 1859 (1999). [98] B. Li, H. Zhao and B. Hu, Phys. Rev. Lett. 86, 63 (2001). [99] B. Li, H. Zhao and B. Hu, Phys. Rev. Lett. 87, 069402 (2001). [100] B. Li, G. Casati and J. Wang, Phys. Rev. E 67, 021204 (2003). [101] B. Li, G. Casati, J. Wang and T. Prosen, Phys. Rev. Lett. 92, 254301 (2004). [102] D. Alonso, A. Ruiz and I. de Vega, Phys. Rev. E 66, 066131 (2002). [103] B. Li, J. Wang, L. Wang and G. Zhang, CHAOS 15, 015121 (2005). [104] J. Mao, Y. Li and Y. Ji, Phys. Rev. E 71, 061202 (2005). [105] X. Zhang and J. Bao, Phys. Rev. E 73, 061103 (2006). [106] S. Denisov, J. Klafter and M. Urbakh, Phys. Rev. Lett. 91, 194301 (2003). [107] P. Cipriani, S. Denisov and A. Politi,Phys. Rev. Lett. 94, 244301 (2005). BIBLIOGRAPHY 121 [108] H. Zhao, Phys. Rev. Lett. 96, 140602 (2006). [109] C. Alabiso, M. Casartelli and P. Marenzoni, J. Stat. Phys. 79, 451 (1995). [110] C. Alabiso and M. Casartelli, J. Phys. A 33, 831 (2000). [111] C. Alabiso and M. Casartelli, J. Phys. A 34, 1223 (2001). [112] B. Gershgorin, Y. Lvov and D. Cai, Phys. Rev. Lett. 95, 264302 (2005). [113] C. Kittel, Introduction to Silid State Physics, seventh edition (Wiley) 1996. [114] J. Hone, M. Whitney, C. Piskoti and A. Zettl, Phys. Rev. B 59, R2514 (1999). [115] Savas Berber, Young-Kyun Kwon and David Tomnek, Phys. Rev. Lett. 84, 4613 (2000). [116] P. Kim, L. Shi, A. Majumdar and P. L. McEuen, Phys. Rev. Lett. 87, 215502 (2001). [117] Z. Yao, J. S. Wang, B. Li and G-R. Liu, Phys. Rev. B 71, 085417 (2005). [118] G. Zhang and B. Li, J. Chem. Phys. 123, 114714 (2005). [119] J. Wang and J. S. Wang, Appl. Phys. Lett. 88, 111909 (2006). [120] A. Balandin and K. L. Wang, Phys. Rev. B 58, 1544 (1998). [121] S. G. Volz and G. Chen, Appl. Phys. Lett. 75, 2056 (1999). [122] D. Li et.al., Appl. Phys. Lett. 83, 2934 (2003). [123] D. G. Cahill et.al., J. Appl. Phys. 93, 793 (2003). BIBLIOGRAPHY 122 [124] D. Lacroix, K. Joulain, D. Terris and D. Lemonnier, Appl. Phys. Lett. 89, 103104 (2006). [125] L. Liang and B. Li, Phys. Rev. B 73, 153303 (2006). [126] R. Lefevere and A. Schenkel, J. Stat. Mech., L02001 (2006). [...]... dependence of heat conductivity of 1D nonlinear lattices In this theory, phonon gas scatterings in lattices are attributed to the transport processes of heat conduction The transport coefficient (heat conductivity) will be the sum of contribution from phonons covering the whole frequencies of the phonon spectrum The size dependence of heat conductivity in 1D nonlinear lattice will be deduced by analyzing the... Fourier’s heat conduction law These striking results bring the imminent challenge: what is the reason for 1D lattices to exhibit anomalous heat conduction in stead of normal heat conduction? Efforts have been done in this direction [1] but full understanding of the mechanisms responsible for anomalous heat conduction or normal heat conduction in 1D lattices is still absent The stunning results of anomalous heat. .. choices of heat baths, one can get the “Normal heat conduction α = 0 where κ is size independent Chaos Chaos, in the sense of positive Lyapunov exponent, was thought as another candidate for the cause of normal heat conduction in 1D lattice models In 1984, Casati et al 1.3 Literature Review of Heat Conduction in 1D systems 19 introduced the 1D ding-a-ling model [67] H= 1 2 N 2 2 p2 + ωi qi + hard point... ratio of around 10% To increase 6 1.2 Basic Definitions the efficiency of these thermal devices, the better understanding of heat conduction is necessary As what we have mentioned above, an investigation of heat conduction theory in 1D lattice chains is of theoretical importance as well as practical importance Before we go through the literature review of the normal and anomalous heat conduction in 1D lattice. .. of effective phonon spectrum of lattice models Based on this, the Effective Phonon Theory is able to predict the kind of conditions under which the lattice will show normal or anomalous heat conduction in classical 1D nonlinear lattice And the temperature dependence of heat conductivity for classical 1D anharmonic lattice will be linked implicitly to the dynamical properties of lattices by this new theory. .. distribution 1.3 Literature Review of Heat Conduction in 1D systems 1.3 14 Literature Review of Heat Conduction in 1D systems 1.3.1 Breakdown of Fourier’s law The starting point for the study of heat conduction in 1D lattice systems comes from the consideration of the simplest harmonic lattice model: N H= i=1 p2 1 i + (xi − xi−1 )2 2 2 (1.21) where the mass of particle and the coupling strength have been scaled... of Heat Conduction in 1D systems 18 was a common belief that the disorder will play the same role for ideal phonon gas just like the defects to the electron gas The phonons in harmonic lattice acquire the finite lifetime by the scattering of disorder and will eventually yield normal heat conduction through this kind of energy diffusion The introduction of disorder did bring some kinds of “disorder” In. .. and heat conductivity can be expressed explicitly The Effective Phonon Theory of heat conduction with the capability of predicting size and temperature dependence of 1D lattice models may shed some light on the choice of materials of low-dimensional systems such as nanotubes and nanowires in which the consideration of properties of heat conduction is the priority The extension of this theory from 1D. .. physicists that the heat conduction of 1D lattices with on-site potential is caused by the interaction between phonons and different type of breathers [10, 11] Right now the type of energy carriers is still under debate Besides theoretical importance of the study of heat transport in 1D lattices, there 1.1 Motivation 5 are also practical requests for the development of heat conduction theory In nowadays techniques,... finite-size heat conductivity of solid polymers has been observed experimentally [17] Although the microscopic mechanism of heat conduction is unclear, the potential applications of nonlinear lattice chains as thermal devices have already been put into investigation with the aid of computer simulations By using the nonlinear properties of 1D lattice models, the prototypes of solid state thermal diode . EFFECTIVE PHONON THEORY OF HEAT CONDUCTION IN 1D NONLINEAR LATTICE CHAINS LI NIANBEI A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF PHYSICS NATIONAL UNIVERSITY OF SINGAPORE 2007 Acknowledgements First. experiencing the four years in Singapore along with me. I really enjoy these days. i Summary This thesis deals with the classical heat conduction of 1D nonlinear lattices. A new theory of heat conduction, . Effective Phonon Theory of Heat Conduction 27 2.1 Concept of Effective Phonons . . . . . . . . . . . . . . . . . . . . . . 28 2.1.1 Renormalized phonon spectrum in general 1D nonlinear lattices

Ngày đăng: 13/09/2015, 19:52

Tài liệu cùng người dùng

  • Đang cập nhật ...

Tài liệu liên quan