Tunable phononic metamaterials for controlling the sound propagation
We describe some details on different heterogeneous devices that display new and unexpected responses such as negative index of refraction, subwavelength imaging and negative bulk modulus which, in general are referred as metamaterials, particularly the mechanical and acoustic properties of pentamode metamaterials, also known as phononic metamaterials (PnM). We present a short review on the principal advances in tunable PnM including own results on this topic. Some own experimental and simulation evidences of the tunability of metamaterials-like phononic crystals are presented. We describe the tunability of the sound propagation in two-dimensional phononic crystals (PnCs) based on steel cylinders periodically embedded in deionized water, comparing the steel cylinder diameter and lattice parameters and the effect of have a lens-like structure. Analysis is doing by observing the sound transmission spectrum. We observe band gaps dependent on the geometrical features of the PnCs.
Metamaterials (MMs) are 1D, 2D or 3D novel artificial material devices, displaying interesting, unexpected and unusual properties not existing in nature, which will be completely determined by the disposition, structure or composition of its components [1, 2]. Thanks to advances in theoretical physics, computational modeling techniques, fabrication and characterization tools, in the past decade many different MMs devices has been possible to engineering i.e., photonic metamaterials for the light propagation control , plasmonic MMs controlling plasmon resonances , an class of planar MMs with abilities to mold light flow, called plasmonic metasurfaces , tunable MMs based on liquid crystals where the main metamaterial ingredient is a liquid crystal included in the metamaterial structure , MMs based on graphene layers composites , superconducting  and quantum MMs presenting nonlinear behaviors or described by quantum structure (dots, wires) , phononic MMs for cloaking, tuning or collimating mechanic waves [10-12], and others.
Non-natural properties of metamaterial-like devices includes for instance, negative refractive index , ultra high  or zero refraction , subwavelength imaging , negative bulk modulus or mass density , artificial magnetism , tunable electric permittivity , and others. Even when the negative refractive index was reported by Veselago  in 60´s, metamaterial term was used firstly by Walser until ending 90´s to describe the material performance of structures with properties beyond the limitations of conventional composites , particularly the electromagnetic metamaterials design. The above mentioned unprecedented properties of MMs have open some current and possible applications like for example, perfect absorber incident waves for antennas or radar devices [21, 22], optical MMS for Raman scattering applications , MMs for nanophotonics and nanocircuits fabrication , MMs used in sensor technologies , MMs has been also used in different biomedical and applications , the construction of superlens, hyper and metalenses for super-resolution imaging [27, 28], the use of metamaterials for cloaking  and invisibility  or electromagnetically induced transparency components for slow light [31, 32], engineered devices for collimate, tune, filter or cloak thermal or sonic waves [11, 12, 33, 34], and among.
In this work authors are interested in describe the principal advances and some applications on phononic metamaterials, putting special attention on the tunability properties MMs-like structures, including own results. The paper is organized as follows: in the next section authors present some details about the general properties of phononic metamaterials. Section 3 is focus to describe the main advances on tunable phononic crystals, describing some works about the tunability process. Sections 4 and 5 contain the experimental and simulations details and the concerning result and discussion, which evidence the tunability effect. We end our paper with some conclusions.
Phononic metamaterials (PnM) are carefully man-engineered structures capable to control and tune both, sound (acoustic vibrations) [11, 35] and heat (thermal vibrations) [33, 36] propagation in ways not observed in nature. In that sense, one can referred to thermal or acoustic PnM depending on its internal structure by virtue of acoustic vibration are in the Hz-KHz frequency range while thermal vibration correspond to THz range [11, 33], such that, the PnM internal structure needs be sensible to these frequencies. Phononic Crystals (PnC) are referred as a PnM kind, an artificial periodic composite metamaterials consisting of sound or heat scatters, periodically disposed in a matrix with high impedance contrast of mass densities and/or elastic moduli with respect to that for scatters. PnC are the analogue photonic crystals but engineering for the phonons controls, photons in the second case. The contrast between the elastic features of the matrix and scatter can give rise to new and unusual phononic features and the existence of phononic band gaps (PnBG), coming from the periodic Bragg scattering as well as localized Mie scatterings from the individual scatters [36, 37]. Recent progress in nanofabrication technologies, however, has made it possible to engineering micro/nano-scale periodic structures allowing to the modulation of wavelength in the GHz frequency range, which are in the domain of thermal excitations . Thus, the manipulation of thermal properties, such as thermal conduction and heat capacity, are possible [39, 40].
One of the main and first observed properties of phononic crystals is concerning with the fact that these composite media typically exhibit stop bands or phononic band gaps (PnBG) in its transmission spectra, i.e., frequency ranges where the propagation of sound and vibrations is strictly forbidden, like in the photonic band gaps in photonic crystals. The first reports on phonons propagating in 1D-periodic elastic composites were done by Dobrzynski and Djafari-Rouhani group in 80s, describing in a first paper an Al-W superlattice made from alternate layers of Al and W , founding the importance of the layer thickness on the vibrational properties of the composite. The same group reported the existence of phononic gaps and surface localized phonons in superlattice consisting of alternating slabs of two different crystals . They develop the theory to easily study all the bulk and surface vibrational properties of a superlattice like the described. This last report is probably the first study on the existence of vibrational gaps.
Inspired in the existence of photonic crystals-like structures, starting 90s, Kushwaha and coworkers developed the first theoretical studies on the existence of phononic band gaps in 2D-periodic structures [43, 44].
Different studies have been theoretically and experimentally performed to insight the tunability of phononic crystal. Vasseur and coworkers in 2001 [A] demonstrated the existence of absolute stop bands in triangular array of parallel circular steel cylinders in an epoxy matrix, band gaps are independent of the direction of propagation in the plane perpendicular to the cylinders. Khelif [B] showed that the pass band could be tune by controlling the inner radius of the tubular steel inclusions in two-dimensional composite media composed of a square array of hollow steel cylinders embedded in water. The effect of the tube inner radius on the transmission spectrum was founded semiquantitatively separable from the effect of the composite periodicity. Tsong [C] work showed that the elastic band gaps can be enlarged or reduced by adjusting the temperature of the phononic structure. The temperature effect can potentially be utilized for fine-tuning of the phononic band gap frequency and the precise design of filters. Dehesa group [D, E] showed that specially designed two-dimensional arrangements of full elastic cylinders embedded in a nonviscous fluid or gas, define a new class of acoustic metamaterials characterized by a dynamical effective mass density that is anisotropic. Dehesa and colleagues showed an explicit dependence on the lattice filling fraction, the elastic properties of cylinders relative to the background, their positions in the unit cell, and their multiple scattering interactions.
More recently, Robillard [F] founded the feasibility of tuning the band structure of phononic crystals composed of an epoxy matrix and Terfenol-D inclusions, by employing magnetostrictive materials and applying an external magnetic field, which show the contactless tunability of the absolute band gaps of a two-dimensional phononic crystal. The tunable phononic crystal behaves like a transmission switch for elastic waves when the magnitude of an applied magnetic field crosses a threshold. Chin [G] and coworkers presented in 2011 a theoretical study on the tunability of phononic band gaps in two-dimensional phononic crystals consisting of various anisotropic cylinders in an isotropic host. The anisotropic materials used in their work include cubic, hexagonal, trigonal, and tetragonal crystal systems showing that by reorienting the anisotropic cylinders, we show that phononic band gaps for bulk acoustic waves propagating in the phononic crystal can be opened, modulated, and closed.
In the past two years many works have been reported on the tunability of PnC properties using different mechanisms. Bertoldi [H], for instance demonstrated the ability to tune the phononic band gaps of 3D periodic elastomeric structures using deformation. The elastomeric nature of the material makes the transformation of the band gaps a reversible and repeatable process, providing avenues for the design of tunable 3D phononic crystals like sonic switches. The influences of geometrical perturbations from rows to rows in PnC were investigated by Leonoir [I] and colleagues, showing that when either the inner radius of the shells between two adjacent rows is varied, or the spacing between the rows is changed, proposing the change of the steel rows can also by steel-polyethylene bilayers. Feng [J], investigated the compression effect on the tunability of phononic crystals consisting of arrays of cylindrical elements when these systems are excited by a continuous dynamic signal under large static precompression, they support a characteristic band structure whose cutoff frequency can be controlled by changing the alignment angles or the static precompression. The past year, Wang [K] investigated the effects of geometric and material nonlinearities introduced by deformation on the linear dynamic response of two-dimensional phononic crystals. He displayed out that not only the deformation can be effectively used to tune the band gaps and the directionality of the propagating waves, but also revealed how geometric and material nonlinearities contribute to the tunable response of phononic crystals. Pashchenko [P] reports a novel type of tunable phononic crystal based on an electric field-induced piezoelectric effect in a thin ferroelectric film. Using finite element simulation their group revealed the possibility of applying the ferroelectric phononic crystal as an electrically tunable surface acoustic wave filter.
Authors [Q] reported theoretical and experimental results on a novel tunable phononic crystal based on thermosensitive hydrogel and steel cylinders. The reported PnC is an optically responsive sonic structure that can modulate the filtering or transmission of ultrasonic waves through it. The modulation of the ultrasonic waves occurs due the electromagnetically induced volume phase transition of the PNIPAm hydrogel infiltrated within the periodic phononic structure.