Frontiers of Physics

, 12:126301 | Cite as

Role of disorder in determining the vibrational properties of mass-spring networks

  • Yunhuan Nie
  • Hua Tong
  • Jun Liu
  • Mengjie Zu
  • Ning Xu
Research article
Part of the following topical collections:
  1. Soft-Matter Physics and Complex Systems

Abstract

By introducing four fundamental types of disorders into a two-dimensional triangular lattice separately, we determine the role of each type of disorder in the vibration of the resulting mass-spring networks. We are concerned mainly with the origin of the boson peak and the connection between the boson peak and the transverse Ioffe–Regel limit. For all types of disorders, we observe the emergence of the boson peak and Ioffe–Regel limits. With increasing disorder, the boson peak frequency ω BP , transverse Ioffe–Regel frequency ω IR T , and longitudinal Ioffe–Regel frequency ω IR L all decrease. We find that there are two ways for the boson peak to form: developing from and coexisting with (but remaining independent of) the transverse van Hove singularity without and with local coordination number fluctuation. In the presence of a single type of disorder, ω IR T ω BP , and ω IR T ω BP only when the disorder is sufficiently strong and causes spatial fluctuation of the local coordination number. Moreover, if there is no positional disorder, ω IR T ω IR L . Therefore, the argument that the boson peak is equivalent to the transverse Ioffe–Regel limit is not general. Our results suggest that both local coordination number and positional disorder are necessary for the argument to hold, which is actually the case for most disordered solids such as marginally jammed solids and structural glasses. We further combine two types of disorders to cause disorder in both the local coordination number and lattice site position. The density of vibrational states of the resulting networks resembles that of marginally jammed solids well. However, the relation between the boson peak and the transverse Ioffe–Regel limit is still indefinite and condition-dependent. Therefore, the interplay between different types of disorders is complicated, and more in-depth studies are required to sort it out.

Keywords

disorder boson peak Ioffe–Regel limit amorphous solid 

Notes

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 21325418 and 11574278), and the Fundamental Research Funds for the Central Universities (Grant No. 2030020028). We also thank the Supercomputing Center of University of Science and Technology of China for computer times.

References

  1. 1.
    C. Kittel, Introduction to Solid State Physics, John Wiley & Sons, Inc., 2005MATHGoogle Scholar
  2. 2.
    N. W. Ashcroft and N. D. Mermin, Solid State Physics, Thomson Brooks/Cole, 1976MATHGoogle Scholar
  3. 3.
    A. F. Ioffe and A. R. Regel, Non-crystalline, amorphous and liquid electronic semiconductors, Prog. Semicond. 4, 237 (1960)Google Scholar
  4. 4.
    T. Nakayama, K. Yakubo, and R. L. Orbach, Dynamical properties of fractal networks: Scaling, numerical simulations, and physical realizations, Rev. Mod. Phys. 66(2), 381 (1994)ADSCrossRefGoogle Scholar
  5. 5.
    E. Duval, A. Boukenter, and T. Achibat, Vibrational dynamics and the structure of glasses, J. Phys.: Condens. Matter 2(51), 10227 (1990)ADSGoogle Scholar
  6. 6.
    T. Keyes, Instantaneous normal mode approach to liquid state dynamics, J. Phys. Chem. A 101(16), 2921 (1997)CrossRefGoogle Scholar
  7. 7.
    W. Schirmacher, G. Diezemann, and C. Ganter, Harmonic vibrational excitations in disordered solids and the “boson peak”, Phys. Rev. Lett. 81(1), 136 (1998)ADSCrossRefGoogle Scholar
  8. 8.
    J. W. Kantelhardt, S. Russ, and A. Bunde, Excess modes in the vibrational spectrum of disordered systems and the boson peak, Phys. Rev. B 63(6), 064302 (2001)ADSCrossRefGoogle Scholar
  9. 9.
    T. S. Grigera, V. Martin-Mayor, G. Parisi, and P. Verrocchio, Vibrations in glasses and Euclidean random matrix theory, J. Phys.: Condens. Matter 14(9), 2167 (2002)ADSGoogle Scholar
  10. 10.
    T. S. Grigera, V. Martin-Mayor, G. Parisi, and P. Verrocchio, Phonon interpretation of the “boson peak” in supercooled liquids, Nature 422(6929), 289 (2003)ADSCrossRefGoogle Scholar
  11. 11.
    V. L. Gurevich, D. A. Parshin, and H. R. Schober, Anharmonicity, vibrational instability, and the boson peak in glasses, Phys. Rev. B 67(9), 094203 (2003)ADSCrossRefGoogle Scholar
  12. 12.
    A. P. Sokolov, U. Buchenau, W. Steffen, B. Frick, and A. Wischnewski, Comparison of Raman- and neutronscattering data for glass-forming systems, Phys. Rev. B 52(14), R9815 (1995)ADSCrossRefGoogle Scholar
  13. 13.
    J. Wuttke, W. Petry, G. Coddens, and F. Fujara, Fast dynamics of glass-forming glycerol, Phys. Rev. E 52(4), 4026 (1995)ADSCrossRefGoogle Scholar
  14. 14.
    P. Lunkenheimer, U. Schneider, R. Brand, and A. Loid, Glassy dynamics, Contemp. Phys. 41(1), 15 (2000)ADSCrossRefGoogle Scholar
  15. 15.
    T. Nakayama, Boson peak and terahertz frequency dynamics of vitreous silica, Rep. Prog. Phys. 65(8), 1195 (2002)ADSCrossRefGoogle Scholar
  16. 16.
    W. A. Phillips (Ed.), Amorphous Solids: Low Temperature Properties, Berlin: Springer-Verlag, 1981CrossRefGoogle Scholar
  17. 17.
    N. Xu, M. Wyart, A. J. Liu, and S. R. Nagel, Excess vibrational modes and the boson peak in model glasses, Phys. Rev. Lett. 98(17), 175502 (2007)ADSCrossRefGoogle Scholar
  18. 18.
    M. Wyart, On the rigidity of amorphous solids, Ann. Phys. 30(3), 1 (2005)CrossRefGoogle Scholar
  19. 19.
    H. Shintani and Y. Tanaka, Universal link between the boson peak and transverse phonons in glass, Nat. Mater. 7(11), 870 (2008)ADSCrossRefGoogle Scholar
  20. 20.
    Y. M. Beltukov, C. Fusco, D. A. Parshin, and A. Tanguy, Boson peak and Ioffe-Regel criterion in amorphous siliconlike materials: The effect of bond directionality, Phys. Rev. E 93(2), 023006 (2016)ADSCrossRefGoogle Scholar
  21. 21.
    U. Tanaka, Physical origin of the boson peak deduced from a two-order-parameter model of liquid, J. Phys. Soc. Jpn. 70(5), 1178 (2001)ADSCrossRefGoogle Scholar
  22. 22.
    E. Duval, A. Boukenter, and T. Achibat, Vibrational dynamics and the structure of glasses, J. Phys.: Condens. Matter 2(51), 10227 (1990)ADSGoogle Scholar
  23. 23.
    C. A. Angell, Formation of glasses from liquids and biopolymers, Science 267(5206), 1924 (1995)ADSCrossRefGoogle Scholar
  24. 24.
    L. E. Silbert, A. J. Liu, and S. R. Nagel, Vibrations and diverging length scales near the unjamming transition, Phys. Rev. Lett. 95(9), 098301 (2005)ADSCrossRefGoogle Scholar
  25. 25.
    E. DeGiuli, A. Laversanne-Finot, G. Düring, E. Lerner, and M. Wyart, Effects of coordination and pressure on sound attenuation, boson peak and elasticity in amorphous solids, Soft Matter 10(30), 5628 (2014)ADSCrossRefGoogle Scholar
  26. 26.
    W. Schirmacher, G. Ruocco, and T. Scopigno, Acoustic attenuation in glasses and its relation with the boson peak, Phys. Rev. Lett. 98(2), 025501 (2007)ADSCrossRefGoogle Scholar
  27. 27.
    W. Schirmacher, Thermal conductivity of glassy materials and the “boson peak”, Europhys. Lett. 73(6), 892 (2006)ADSCrossRefGoogle Scholar
  28. 28.
    A. Ferrante, E. Pontecorvo, G. Cerullo, A. Chiasera, G. Ruocco, W. Schirmacher, and T. Scopigno, Acoustic dynamics of network-forming glasses at mesoscopic wavelengths, Nat. Commun. 4, 1793 (2013)ADSCrossRefGoogle Scholar
  29. 29.
    F. Léonforte, A. Tanguy, J. P. Wittmer, and J. L. Barrat, Inhomogeneous elastic response of silica glass, Phys. Rev. Lett. 97(5), 055501 (2006)ADSCrossRefGoogle Scholar
  30. 30.
    G. Monaco and S. Mossa, Anomalous properties of the acoustic excitations in glasses on the mesoscopic length scale, Proc. Natl. Acad. Sci. USA 106(40), 16907 (2009)ADSCrossRefGoogle Scholar
  31. 31.
    C. A. Angell, Y. Z. Yue, L. M. Wang, J. R. D. Copley, S. Borick, and S. Mossa, Potential energy, relaxation, vibrational dynamics and the boson peak, of hyperquenched glasses, J. Phys.: Condens. Matter 15(11), S1051 (2003)Google Scholar
  32. 32.
    D. A. Parshin, H. R. Schober, and V. L. Gurevich, Vibrational instability, two-level systems, and the boson peak in glasses, Phys. Rev. B 76(6), 064206 (2007)ADSCrossRefGoogle Scholar
  33. 33.
    L. Wang and N. Xu, Probing the glass transition from structural and vibrational properties of zerotemperature glasses, Phys. Rev. Lett. 112(5), 055701 (2014)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    S. Singh, M. D. Ediger, and J. J. de Pablo, Ultrastable glasses from in silico vapour deposition, Nat. Mater. 12(2), 139 (2013)ADSCrossRefGoogle Scholar
  35. 35.
    S. N. Taraskin, Y. L. Loh, G. Natarajan, and S. R. Elliott, Origin of the boson peak in systems with lattice disorder, Phys. Rev. Lett. 86(7), 1255 (2001)ADSCrossRefGoogle Scholar
  36. 36.
    A. I. Chumakov, G. Monaco, A. Monaco, W. A. Crichton, A. Bosak, R. Rüffer, A. Meyer, F. Kargl, L. Comez, D. Fioretto, H. Giefers, S. Roitsch, G. Wortmann, M. H. Manghnani, A. Hushur, Q. Williams, J. Balogh, K. Parliński, P. Jochym, and P. Piekarz, Equivalence of the boson peak in glasses to the transverse acoustic van hove singularity in crystals, Phys. Rev. Lett. 106(22), 225501 (2011)ADSCrossRefGoogle Scholar
  37. 37.
    H. Tong, P. Tan, and N. Xu, From crystals to disordered crystals: A hidden order-disorder transition, Sci. Rep. 5, 15378 (2015)ADSCrossRefGoogle Scholar
  38. 38.
    A. J. Liu and S. R. Nagel, Nonlinear dynamics: Jamming is not just cool any more, Nature 396(6706), 21 (1998)ADSCrossRefGoogle Scholar
  39. 39.
    A. J. Liu and S. R. Nagel, The jamming transition and the marginally jammed solid, Annu. Rev. Condens. Matter Phys. 1(1), 347 (2010)ADSCrossRefGoogle Scholar
  40. 40.
    M. van Hecke, Jamming of soft particles: Geometry, mechanics, scaling and isostaticity, J. Phys.: Condens. Matter 22(3), 033101 (2010)ADSGoogle Scholar
  41. 41.
    N. Xu, Mechanical, vibrational, and dynamical properties of amorphous systems near jamming, Front. Phys. 6(1), 109 (2011)CrossRefGoogle Scholar
  42. 42.
    C. S. O’Hern, L. E. Silbert, A. J. Liu, and S. R. Nagel, Jamming at zero temperature and zero applied stress: The epitome of disorder, Phys. Rev. E 68(1), 011306 (2003)ADSCrossRefGoogle Scholar
  43. 43.
    S. Torquato and F. H. Stillinger, Jammed hard-particle packings: From Kepler to Bernal and beyond, Rev. Mod. Phys. 82(3), 2633 (2010)ADSCrossRefGoogle Scholar
  44. 44.
    G. Parisi and F. Zamponi, Mean-field theory of hard sphere glasses and jamming, Rev. Mod. Phys. 82(1), 789 (2010)ADSCrossRefGoogle Scholar
  45. 45.
    M. Müller and M. Wyart, Marginal stability in structural, spin, and electron glasses, Annu. Rev. Condens. Matter Phys. 6(1), 177 (2015)ADSCrossRefGoogle Scholar
  46. 46.
    M. Wyart, L. E. Silbert, S. R. Nagel, and T. A. Witten, Effects of compression on the vibrational modes of marginally jammed solids, Phys. Rev. E 72(5), 051306 (2005)ADSCrossRefGoogle Scholar
  47. 47.
    M. Wyart, S. R. Nagel, and T. A. Witten, Geometric origin of excess low-frequency vibrational modes in weakly connected amorphous solids, Europhys. Lett. 72(3), 486 (2005)ADSCrossRefGoogle Scholar
  48. 48.
    H. Tong and N. Xu, Order parameter for structural heterogeneity in disordered solids, Phys. Rev. E 90, 010401(R) (2014)ADSCrossRefGoogle Scholar
  49. 49.
    http://www.caam.rice.edu/software/ARPACKGoogle Scholar
  50. 50.
    X. Wang, W. Zheng, L. Wang, and N. Xu, Disordered solids without well-defined transverse phonons: the nature of hard-sphere glasses, Phys. Rev. Lett. 114(3), 035502 (2015)ADSCrossRefGoogle Scholar
  51. 51.
    J. P. Hansen and I. R. McDonald, Theory of Simple Liquids, Amsterdam: Elsevier, 1986MATHGoogle Scholar
  52. 52.
    J. Liu, Y. Nie, and N. Xu (in preparation)Google Scholar
  53. 53.
    E. Bitzek, P. Koskinen, F. Gahler, M. Moseler, and P. Gumbsch, Structural relaxation made simple, Phys. Rev. Lett. 97(17), 170201 (2006)ADSCrossRefGoogle Scholar
  54. 54.
    E. D. Cubuk, S. S. Schoenholz, J. M. Rieser, B. D. Malone, J. Rottler, D. J. Durian, E. Kaxiras, and A. J. Liu, Identifying structural flow defects in disordered solids using machine-learning methods, Phys. Rev. Lett. 114(10), 108001 (2015)ADSCrossRefGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Yunhuan Nie
    • 1
  • Hua Tong
    • 1
    • 2
  • Jun Liu
    • 1
  • Mengjie Zu
    • 1
  • Ning Xu
    • 1
  1. 1.CAS Key Laboratory of Soft Matter Chemistry, Hefei National Laboratory for Physical Sciences at the Microscale and Department of PhysicsUniversity of Science and Technology of ChinaHefeiChina
  2. 2.Institute of Industrial ScienceUniversity of Tokyo, Meguro-kuTokyo 153-8505Japan

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