Research Papers

Nonlinear Phonon Modes in Second-Order Anharmonic Coupled Monoatomic Chains

[+] Author and Article Information
B. Dubus

Institut d'Electronique, de Microélectronique
et de Nanotechnologie,
UMR CNRS 8520,
Cité Scientifique,
Villeneuve d'Ascq Cedex 59652, France
e-mail: bertrand.dubus@isen.fr

N. Swinteck

Department of Materials
Science and Engineering,
University of Arizona,
Tucson, AZ 85721
e-mail: swinteck@email.arizona.edu

K. Muralidharan

Department of Materials
Science and Engineering,
University of Arizona,
Tucson, AZ 85721
e-mail: krishna@email.arizona.edu

J. O. Vasseur

Institut d'Electronique de,
Microélectronique et de Nanotechnologie,
UMR CNRS 8520,
Cité Scientifique,
Villeneuve d'Ascq Cedex 59652, France
e-mail: jerome.vasseur@univ-lille1.fr

P. A. Deymier

Department of Materials
Science and Engineering,
University of Arizona,
Tucson, AZ 85721
e-mail: deymier@email.arizona.edu

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received August 20, 2015; final manuscript received April 7, 2016; published online May 25, 2016. Assoc. Editor: Mohammed Daqaq.

J. Vib. Acoust 138(4), 041016 (May 25, 2016) (10 pages) Paper No: VIB-15-1331; doi: 10.1115/1.4033457 History: Received August 20, 2015; Revised April 07, 2016

We have used multiple-time-scales perturbation theory as well as the numerical methods of molecular dynamics and spectral energy density (SED) to investigate the phonon band structure of a two-chain model with second-order anharmonic interactions. We show that when one chain is linear and the other is nonlinear, the two-chain model exhibits a nonlinear resonance near a critical wave number due to mode self-interaction. The nonlinear resonance enables wave number-dependent interband energy transfer. We have also shown that there exist nonlinear modes within the spectral gap separating the lower and upper branches of the phonon band structure. These modes result from three phonon interactions between a phonon belonging to the nonlinear branch and two phonons lying on the lower branch. This phenomenon offers a mechanism for phonon splitting.

Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.


Vakakis, A. F. , Gendelman, O. V. , Bergman, L. A. , McFarland, D. M. , Kerschen, G. , and Lee, Y. S. , 2009, Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems (Solid Mechanics and Its Application), Springer, Dordrecht, The Netherlands.
Vakakis, A. F. , and Rand, R. H. , 2004, “ Nonlinear Dynamics of a System of Coupled Oscillators With Essential Stiffness Nonlinearity,” Int. J. Nonlinear Mech., 39(7), p. 1079. [CrossRef]
Gendelman, O. V. , Sapsis, T. , Vakakis, A. F. , and Bergman, L. A. , 2011, “ Enhanced Passive Targeted Energy Transfer in Strongly Nonlinear Mechanical Oscillators,” J. Sound Vib., 330(1), pp. 1–8. [CrossRef]
Gendelman, O. V. , Manevitch, L. I. , Vakakis, A. F. , and M'Closkey, R. , 2001, “ Energy Pumping in Nonlinear Mechanical Oscillators: Part I-Dynamics of the Underlying Hamiltonian System,” Trans. ASME, 68(1), pp. 34–41. [CrossRef]
Kerschen, G. , Vakakis, A. F. , Lee, Y. S. , McFarland, D. M. , Kowtko, J. J. , and Bergman, L. A. , 2005, “ Energy Transfer in a System of Two Coupled Oscillators With Essential Nonlinearity: 1:1 Resonance Manifold and Transient Bridging Orbits,” Nonlinear Dyn., 42(3), pp. 283–303. [CrossRef]
Laxalde, D. , Thouverez, F. , and Simou, J.-J. , 2006, “ Dynamics of a Linear Oscillator Connected to a Small Strongly Nonlinear Hysteretic Absorber,” Int. J. Nonlinear Mech., 41(8), pp. 969–978. [CrossRef]
Panagopoulos, P. N. , Vakakis, A. F. , and Tsakirtzis, S. , 2004, “ Transient Resonant Interactions of Finite Linear Chains With Essential Nonlinear End Attachments Leading to Passive Energy Pumping,” Int. J. Solids Struct., 41(22), pp. 6505–6528. [CrossRef]
Vakakis, A. F. , Manevitch, L. I. , Gendelman, O. , and Bergman, L. , 2003, “ Dynamics of Linear Discrete Systems Connected to Local Essential Nonlinear Attachments,” J. Sound Vib., 264(3), pp. 559–577. [CrossRef]
Starosvetsky, Y. , Hasan, M. A. , Vakakis, A. F. , and Manevitch, L. I. , 2012, “ Strongly Nonlinear Beat Phenomena and Energy Exchanges in Weakly Coupled Granular Chains on Elastic Foundations,” SIAM J. Appl. Math., 72(1), pp. 337–361. [CrossRef]
Seidel, A. , Lin, H. H. , and Lee, D. H. , 2005, “ Phonons in Hubbard Ladders Studied Within the Framework of the One-Loop Renormalization,” Phys. Rev. B, 71(22), p. 22050.
Peyrard, M. , and Bishop, A. R. , 1989, “ Statistical Mechanics of a Nonlinear Model for DNA Denaturation,” Phys. Rev. Lett., 62(23), pp. 2755–2758. [CrossRef] [PubMed]
Bender, C. M. , and Orszag, S. A. , 1999, Advanced Mathematical Methods for Scientists and Engineers I, Asymptotic Methods and Perturbation Theory, Springer-Verlag, New York.
Kevorkian, J. , and Cole, J. D. , 1996, Scale and Singular Perturbation Methods, Springer-Verlag, New York.
Belhaq, M. , Clerc, R. L. , and Hartmann, C. , 1988, “ Multiple Scales Methods for Finding Invariant Solutions of Two Dimensional Maps and Application,” Mech. Res. Commun., 15(6), p. 361. [CrossRef]
Maccari, A. , 1999, “ A Perturbation Method for Nonlinear Two Dimensional Maps,” Nonlinear Dyn., 19(4), pp. 295–312. [CrossRef]
van Horssen, W. T. , and ter Brake, M. C. , 2009, “ On the Multiple Scales Perturbation Method for Difference Equations,” Nonlinear Dyn., 55(4), pp. 401–418. [CrossRef]
Helleman, R. H. G. , and Montroll, E. W. , 1974, “ On a Nonlinear Perturbation Theory Without Secular Terms,” Physica, 74(1), pp. 22–74. [CrossRef]
Lee, P. S. , Lee, Y. C. , and Chang, C. T. , 1973, “ Multiple-Time-Scale Analysis of Spontaneous Radiation Processes. I. One- and Two-Particle Systems,” Phys. Rev. A, 8(4), p. 1722. [CrossRef]
Khoo, I. C. , and Wang, Y. K. , 1976, “ Multiple Time Scale Analysis of an Anharmonic Crystal,” J. Math. Phys., 17(2), p. 222. [CrossRef]
Swinteck, N. , Muralidharan, K. , and Deymier, P. A. , 2013, “ Phonon Scattering in One-Dimensional Anharmonic Crystals and Superlattices: Analytical and Numerical Study,” ASME J. Vib. Acoust., 135(4), p. 041016. [CrossRef]
Überall, H. , 1992, Acoustic Resonance Scattering, Gordon and Breach Science Publishers, Philadelphia, PA, Chap. 4.
Rapaport, D. C. , 1995, The Art of Molecular Dynamics Simulation, Cambridge University Press, Cambridge, UK.
Thomas, J. A. , Turney, J. E. , Iutzi, R. M. , Amon, C. H. , and McGaughey, A. J. H. , 2010, “ Predicting Phonon Dispersion Relations and Lifetimes From the Spectral Energy Density,” Phys. Rev. B, 81(8), p. 091411.
Dot, A. , Borne, A. , Boulanger, B. , Segonds, P. , Félix, C. , Bencheikh, K. , and Levenson, J. A. , 2012, “ Energetic and Spectral Properties of Triple Photon Down Conversion in a Phase-Matched KTiOPO4 Crystal,” Opt. Lett., 37(12), p. 2334. [CrossRef] [PubMed]
Boitier, F. , Orieux, A. , Autebert, C. , Lemaître, A. , Galopin, E. , Manquest, C. , Sirtori, C. , Favero, I. , Leo, G. , and Ducci, S. , 2014, “ Electrically Injected Photon-Pair Source at Room Temperature,” Phys. Rev. Lett., 112(18), p. 183901. [CrossRef] [PubMed]


Grahic Jump Location
Fig. 1

Schematic illustration of the two coupled infinite atomic chains made of identical atoms of mass m with a lattice parameter a. β would correspond to the stiffness of the spring linking atoms in the two horizontal chains if these springs were linear. A degree of nonlinearity can be introduced inside the model via parameters δ and ε (see text for definition). β′ is the stiffness of the linear spring connecting together atoms n in the two horizontal chains.

Grahic Jump Location
Fig. 2

Band structure associated with the zeroth-order solutions, i.e., ω0−(k) (black line) and ω0+(k) (gray line). Parameters a, m, β, and β′ (see text for definitions) were chosen to be a=1 m, m=1 kg, β=1 N · m−1, β′=6.4 N · m−1.

Grahic Jump Location
Fig. 3

ω−(k) (black lines) and ω+(k) (gray lines) as functions of wave number for three values of the spring constant β′, namely, β′=1 N · m−1 (a), 2 N · m−1 (b), and 7 N · m−1 (c). The dotted lines correspond to the linear–linear two-chain model. The system parameters are a=1 m, m=1 kg, β=1 N · m−1,ε=1.6 N · m−2, α0+=α0−=0.17 m. The black dashed line indicates the value of the critical wave number: kc=1.27 m−1 (a), kc=1.57 m−1 (b), and kc=2.63 m−1 (c).

Grahic Jump Location
Fig. 4

Complex pulsation ω− as function of the wave number around the critical wave number kc when the stiffness β′ of the spring coupling together the two horizontal chains takes complex values with real part equals to 7 N · m-1 and imaginary part equals to 3.5×10−2 N  · m-1 (black solid lines), 7.0×10−2 N  · m-1 (gray solid lines), and 1.4×10−1 N  · m−1 (black dotted lines). The top (resp. bottom) panel represents the behavior of the real (resp. imaginary) part of the pulsation ω−. The other parameters are the same as those of Fig. 3(c). The black dashed line indicates the value of the critical wave number: kc=2.63 m-1.

Grahic Jump Location
Fig. 5

Contributions of the lower and upper dispersion branches to the energy of the linear two-chain model (dotted lines), E0−(k) (black) and E0+(k) (gray), and the linear–nonlinear model (solid lines), E−(k) (black) and E+(k) (gray) normalized to their respective total energy, E0(k) and E(k). The system parameters are a=1 m, m=1 kg, β=1 N · m−1, β′=4 N · m−1,ε=1.6 N · m−2, α0+=α0−=0.17 m. The black dashed line indicates the value of the critical wave number: kc=2.0 m−1.

Grahic Jump Location
Fig. 6

ω0−(k) (black solid line), ω0+(k) (gray solid line), and ω0NL(k+kc)=ω0−(kc)+ω0−(k) (black dotted line) as functions of wave number for three values of the spring constant β′, namely, 5.4 N · m−1 (a), 6.4 N · m-1 (b), and 7.4 N · m−1 (c). The system parameters are a=1 m, m=1 kg, β=1 N · m−1. kℓ=kc and ku=(2π/a)−2kc are the wave numbers at which the nonlinear branch (black dotted line) intersects the lower and the upper linear bands. The black dashed lines indicate the values of the wave numbers: kℓ=2.27 m-1 and ku=1.74 m-1 (a), kℓ=2.48 m-1 and ku=1.32 m−1 (b), and kℓ=2.75 m−1 and ku=0.78 m−1 (c).

Grahic Jump Location
Fig. 7

SEDs in J.s calculated from the velocities of the atoms in the linear–nonlinear two chains model for three values of the coupling elastic constant β′=5.4 N · m−1 (a), β′=6.4 N · m−1 (b), and β′=7.4 N · m−1 (c). The system parameters are a=1 m, m=1 kg, β=1 N · m−1, ε=1.6 N · m−2, α0+=α0−=0.17m. The background scale corresponds to  log10(SED). The white arrow indicates the value of kuSED: kuSED ≈ 1.7 m−1 (a), kuSED ≈ 1.3 m−1 (b), and kuSED ≈ 0.7 m−1 (c). Meaning of the black and gray arrows is given in the text.



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In