Journal Club Theme of May 2012: Hierarchical and Quasiperiodic Phononic Crystals and Metamaterials

Following last month’s blog by Mahmoud on the role of structure in the design of materials capable of controlling the propagation of mechanical waves, I would like to delve into the characteristics of the structure itself. The propagation of mechanical waves can be controlled via scattering induced by a material’s structure.The propagation of waves for example is significantly disturbed when their wavelength and direction become correlated with the medium’s structure. This is known as Bragg scattering and can be achieved simply by embedding voids or inclusions (of different physical properties) in a material [1]. As Mahmoud explained, phononic crystals (PCs) exploit this type of scattering to obtain band gaps [2], but they also feature negative refraction [3], and have been used to focus sound [4]. If the structure contains elements that may resonate within the frequency spectrum of mechanical waves, the resulting scattering is said to be resonant (Mie scattering is an example) [5]. Acoustic metamaterials (AMs) exploit resonant scattering to obtain effective negative stiffness [6-7], negative mass [5,8], or both [9] for example.
Among many aspects of a material’s structure, periodicity, order or the lack of either are strong factors in the dynamic response that can be expected. Structure for example can also determine whether a solid will behave more like a liquid [10]. The useful operational regime of PCs and AMs however is often limited to a single frequency or a narrow frequency range. PC and AM prototypes often include a single length scale [5], say a characteristic distance between features, or a single/few internal resonant frequencies [10, 11]. Low-frequency operation also remains very challenging as it requires either low stiffness/high mass or a large characteristic length. These are significant drawbacks.
Breaking a PC or AM’s structural periodicity however may provide broad-band effectiveness and the capability to control mechanical waves even at very low frequencies [12-13]. The lack of periodicity in the material’s structure does not necessarily mean disorder, which generally induces a broad-band diffusive-like behavior [12] or localized waves [14], but can lead to phononic quasicrystals (PQCs) with high symmetry [12]. Note that symmetry and periodicity are not synonyms here. These quasi-periodic structures often retain strong amplitude in the Fourier spectrum of periodic average structures (PAS), approximations of PQCs, but can have broad-band effectiveness unlike PAS [12]. PQCs moreover, can exhibit Bragg scattering for frequencies near zero [12].
Structure at various scales, finally, can also provide broad band effectiveness: fractal [12, 13], hierarchical [15] and optimized phononic crystals [16] may lead the way for structured materials with much improved capabilities.
While fractal features have made their way into electromagnetics for example (see NOVA: the hunt for the hidden dimension http://video.google.com/videoplay?docid=-6917200224135375895), what would it take to manufacture a PC or AM with fractal dimensions?

[1] J. Zhu, et al, “Holey-structured metamaterial for acoustic deep-subwavelength imaging.” Naturel Physics, 7, 52-55, 2011.

[2] M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B. Djafari-Rouhani, Phys.Rev. Lett. 71, 2022, 1993.

[3] X. Zhang and Z. Liu, Appl. Phys. Lett. 85, 341, 2004.

[4] S. Yang, J. H. Page, Z. Y. Liu, M. L. Cowan, C. T. Chan, and P. Sheng, Phys. Rev. Lett. 93, 024301, 2004.

[5] M.H. Lu, L. Feng and Y. F. Chen, “Phononic crystals and acoustic metamaterials.” Materials Today, 2009.

[6] Liu, Z., et al., Science (2000) 289, 1734.

[7] N. Fang et al, “Ultrasonic metamaterials with negative modulus.” Nature Materials, 5(6), 452-456.

[8] H. Huang, C. Sun and G. Huang, “On the negative effective mass density in acoustic metamaterials.” International Journal of Engineering Science, 47(4), 610-617, 2009.

[9] J. Li and C. T. Chan, “Double-negative acoustic metamaterial.” Phys. Rev. E, 70, 055602, 2004.

[10] Y. Lai, Y. Wu, P. Sheng and Z. Q. Zhang, “Hybrid elastic solids.” Nature Materials, 10(8), 620-624, 2011.

[11] : Z. Liu et al, “Locally Resonant Sonic Materials.” Science, 289(5485), 1734-1736, 2000.

[12] W. Steurer and D. Sutter-Widmer, “Photonic and Phononic quasicrystals.” J. Phys. D: Appl. Phys., 40, 229-247, 2007.

[13] F. Axel and D. Gratias, Beyond quasicrystals, Springer, 1995.

[14] H. Hu et al, “Localization of ultrasound in a three-dimensional elastic network.” Nature Physics, 4, 945 – 948, 2008.

[15] Y.L. Xua, X.G. Tiana, C.Q. Chenb, “Band structures of two dimensional solid/air hierarchical phononic crystals.” Physica B: Condensed Matter, 407(12), 1995–2001, 2012.

[16] O R. Bilal and M. I. Hussein, “Ultrawide phononic band gap for combined in-plane and out-of-plane waves.” Phys. Rev. E, 84, 065701, 2011.