Three-dimensional Phononic Crystals Research Articles

Refraction characteristics of phononic crystals

Some of the most interesting refraction properties of phononic crystals are revealed by examining the anti-plane shear waves in doubly periodic elastic composites with unit cells containing rectangular and/or elliptical multi-inclusions. The corresponding band structure, group velocity, and energy–flux vector are calculated using a powerful mixed variational method that accurately and efficiently yields all the field quantities over multiple frequency pass-bands. The background matrix and the inclusions can be anisotropic, each having distinct elastic moduli and mass densities. Equifrequency contours and energy–flux vectors are readily calculated as functions of the wave-vector components. By superimposing the energy–flux vectors on equifrequency contours in the plane of the wave-vector components, and supplementing this with a three-dimensional graph of the corresponding frequency surface, a wealth of information is extracted essentially at a glance. This way it is shown that a composite with even a simple square unit cell containing a central circular inclusion can display negative or positive energy and phase velocity refractions, or simply performs a harmonic vibration (standing wave), depending on the frequency and the wave-vector. Moreover, that the same composite when interfaced with a suitable homogeneous solid can display: (1) negative refraction with negative phase velocity refraction; (2) negative refraction with positive phase velocity refraction; (3) positive refraction with negative phase velocity refraction; (4) positive refraction with positive phase velocity refraction; or even (5) complete reflection with no energy transmission, depending on the frequency, and direction and the wavelength of the plane-wave that is incident from the homogeneous solid to the interface. For elliptical and rectangular inclusion geometries, analytical expressions are given for the key calculation quantities. Expressions for displacement, velocity, linear momentum, strain, and stress components, as well as the energy–flux and group velocity components are given in series form. The general results are illustrated for rectangular unit cells, one with two and the other with four inclusions, although any number of inclusions can be considered. The energy–flux and the accompanying phase velocity refractions at an interface with a homogeneous solid are demonstrated. Finally, by comparing the results of the present solution method with those obtained using the Rayleigh quotient and, for the layered case, with the exact solutions, the remarkable accuracy and the convergence rate of the present solution method are demonstrated. Anomalous refractive characteristics of phononic crystals are revealed using anti-plane shear waves in doubly periodic elastic composites. It is shown that negative or positive refraction can be accompanied by negative or positive phase velocity refraction, and/or the crystal can have complete refraction with no energy transmission, all depending on the wavelength and frequency of the plane wave that is incident from a homogeneous solid to its interface with the composite. A powerful computational tool is proposed which applies to one-, two-, three-dimensional phononic crystals yielding results with great accuracy.

Three-dimensional adaptive soft phononic crystals

We report a new class of three-dimensional (3D) adaptive phononic crystals whose dynamic response is controlled by mechanical deformation. Using finite element analysis, we demonstrate that the bandgaps of the proposed 3D structure can be fully tuned by the externally applied deformation. In fact, our numerical results indicate that the system acts as a reversible phononic switch: a moderate level of applied strain (i.e., −0.16) is sufficient to completely suppress the bandgap, and upon the release of applied strain, the deformed structure recovers its original shape, which can operate with a sizable bandgap under dynamic loading. In addition, we investigate how material damping significantly affects the propagation of elastic waves in the proposed 3D soft phononic crystal. We believe that our results pave the way for the design of a new class of soft, adaptive, and re-configurable 3D phononic crystals, whose bandgaps can be easily tuned and switched on/off by controlling the applied deformation.

A three-dimensional Bloch wave expansion to determine external scattering from finite phononic crystals.

External scattering from a finite phononic crystal (PC) is studied using the Helmholtz-Kirchhoff integral theorem integrated with a Bloch wave expansion (BWE). The BWE technique is used to describe the internal pressure field of a semi-infinite or layered PC subject to an incident monochromatic plane wave. Following the BWE solution, the Helmholtz-Kirchhoff integral is used to determine the external scattered field. For cubic PCs, the scattered results are compared to numerical treatments in both the frequency and time domain. The presented approach is expected to be valid when the PC size is larger than the acoustic wavelength. However, very good agreement in the spatial beam pattern is also documented for both large and small (with respect to the wavelength) PCs. The result of this work is a fully-analytical, efficient, and verified approach for accurately predicting external scattering from finite, three-dimensional PCs.

Directional asymmetry of the nonlinear wave phenomena in a three-dimensional granular phononic crystal under gravity.

We report the experimental observation of the gravity-induced asymmetry for the nonlinear transformation of acoustic waves in a noncohesive granular phononic crystal. Because of the gravity, the contact precompression increases with depth inducing space variations of not only the linear and nonlinear elastic moduli but also of the acoustic wave dissipation. We show experimentally and explain theoretically that, in contrast to symmetric propagation of linear waves, the amplitude of the nonlinearly self-demodulated wave depends on whether the propagation of the waves is in the direction of the gravity or in the opposite direction. Among the observed nonlinear processes, we report frequency mixing of the two transverse-rotational modes belonging to the optical band of vibrations and propagating with negative phase velocities, which results in the excitation of a longitudinal wave belonging to the acoustic band of vibrations and propagating with positive phase velocity. We show that the measurements of the gravity-induced asymmetry in the nonlinear acoustic phenomena can be used to compare the in-depth distributions of the contact nonlinearity and of acoustic absorption.

Extreme Low Thermal Conductivity in Nanoscale 3D Si Phononic Crystal with Spherical Pores

In this work, we propose a nanoscale three-dimensional (3D) Si phononic crystal (PnC) with spherical pores, which can reduce the thermal conductivity of bulk Si by a factor up to 10,000 times at room temperature. Thermal conductivity of Si PnCs depends on the porosity, for example, the thermal conductivity of Si PnCs with porosity 50% is 300 times smaller than that of bulk Si. The phonon participation ratio spectra demonstrate that more phonons are localized as the porosity increases. The thermal conductivity is insensitive to the temperature changes from room temperature to 1100 K. The extreme-low thermal conductivity could lead to a larger value of ZT than unity as the periodic structure affects very little the electric conductivity.

Tuning of Bandgap Structures in Three-Dimensional Kagome-Sphere Lattice

In this manuscript, acoustic wave propagation in a novel three-dimensional porous phononic crystal-Kagome lattice, is studied by using finite element method. Firstly, a Kagome-sphere structure is established based on Kagome truss. For lattice with fixed rods (sphere radius varied) or fixed spheres (rod radius varied), the band structures are calculated in order to clarify the influence of geometrical parameters (sphere and rod sizes) on the bandgap characteristics in Kagome-sphere lattice. The vibration modes at the band edges of the lowest bandgaps are investigated with the aim to understand the mechanism of the bandgap generation. It is found that the emergence of the bandgap is due to the local resonant vibration of the unit cell at the adjacent bands. The width and position of this bandgap can be tuned by adjusting the geometrical parameters. An equivalent mass-spring model is proposed and the equivalent system resonance frequency can be evaluated which predicts well the upper and lower edges of the complete bandgaps. Moreover, the critical geometrical parameter is formulated which gives the critical geometrical condition for the opening of the complete bandgaps. The results in this paper are relevant to the bandgap structure design of three-dimensional porous phononic crystals (PPCs).

Direct observation of the phonon dispersion of a three-dimensional solid/solid hypersonic colloidal crystal

We report on Brillouin measurements of the phonon dispersion relation of a three-dimensional hypersonic phononic crystal. The phononic crystal, comprised of close-packed polystyrene spheres embedded in a background matrix of solid polydimethylsiloxane, was fabricated using a self-assembly technique. The experimentally measured dispersion relation reveals the band folding effect and the formation of a band gap for longitudinal waves at the boundary of the Brillouin zone. Theoretical calculations, based on the finite element method and the finite-difference time-domain method, are employed to interpret experimentally observed phonon modes and provide insight on important features of the solid/solid hypersonic crystals.

On the quasistatic effective elastic moduli for elastic waves in three-dimensional phononic crystals

Effective elastic moduli for 3D solid–solid phononic crystals of arbitrary anisotropy and oblique lattice structure are formulated analytically using the plane-wave expansion (PWE) method and the recently proposed monodromy-matrix (MM) method. The latter approach employs Fourier series in two dimensions with direct numerical integration along the third direction. As a result, the MM method converges much quicker to the exact moduli in comparison with the PWE as the number of Fourier coefficients increases. The MM method yields a more explicit formula than previous results, enabling a closed-form upper bound on the effective Christoffel tensor. The MM approach significantly improves the efficiency and accuracy of evaluating effective wave speeds for high-contrast composites and for configurations of closely spaced inclusions, as demonstrated by three-dimensional examples.

Complete Bandgap in Three-Dimensional Holey Phononic Crystals With Resonators

In this paper, the bandgap properties of three-dimensional holey phononic crystals with resonators are investigated by using the finite element method. The resonators are periodically arranged cubic lumps in the cubic holes connected to the matrix by narrow connectors. The influence of the geometry parameters of the resonators on the bandgap is discussed. In contrast to a system with cubic or spherical holes, which has no bandgaps, systems with resonators can exhibit complete bandgaps. The bandgaps are significantly dependent upon the geometry of the resonators. By the careful design of the shape and size of the resonator, a bandgap that is lower by an order of magnitude than the Bragg bandgap can be obtained. The vibration modes at the band edges of the lowest bandgaps are analyzed in order to understand the mechanism of the bandgap generation. It is found that the emergence of the bandgap is due to the local resonance of the resonators. Spring-mass models or spring-pendulum models are developed in order to evaluate the frequencies of the bandgap edges. The study in this paper is relevant to the optimal design of the bandgaps in light porous materials.

Tuning of Acoustic Bandgaps in Phononic Crystals With Helmholtz Resonators

In this paper, acoustic wave propagation in a two- or three-dimensional phononic crystal consisting of Helmholtz resonators embedded in a fluid matrix is studied. The band structures are calculated to discuss the influence of the geometry topology of Helmholtz resonators on the bandgap characteristics. It is shown that a narrow bandgap will appear in the lower frequency range due to the resonance of the Helmholtz resonators. The width and position of this resonance bandgap can be tuned by adjusting the geometrical parameters of the Helmholtz resonator. The position of the resonance bandgap can be evaluated by the resonance frequency of the Helmholtz resonator. A decrease in the size of the opening generally results in a lower position and a smaller width of the bandgap. The system with one opening exhibits a wider bandgap in a lower position than the system with two openings.

Effects of material parameters on elastic band gaps of three-dimensional solid phononic crystals

In this paper, we study the influences of material parameters on the phononic band gaps of three-dimensional (3D) solid phononic crystals (PCs) based on the finite difference time domain (FDTD) method. We begin with the basic wave equations and the FDTD formulation to derive the material parameters directly determining the band gaps of the 3D solid PCs. The parameters include the transverse velocity ratio, the acoustic impedance ratio and the Poisson ratios (or equivalently, the mass density ratio, the shear modulus ratio and the Poisson ratios) of the scatterers and host materials. The negative Poisson ratio is also considered in our investigation. The effects of these material parameters on band gap width are discussed based on detailed numerical calculations for systems with three typical lattices. The generation mechanism of band gaps (Bragg scattering or local resonance) which is determined by the material parameters is also discussed. The analysis is expected to be applied to the artificial design of 3D phononic band gap materials.

Reduction of thermal conductivity by nanoscale 3D phononic crystal.

We studied how the period length and the mass ratio affect the thermal conductivity of isotopic nanoscale three-dimensional (3D) phononic crystal of Si. Simulation results by equilibrium molecular dynamics show isotopic nanoscale 3D phononic crystals can significantly reduce the thermal conductivity of bulk Si at high temperature (1000 K), which leads to a larger ZT than unity. The thermal conductivity decreases as the period length and mass ratio increases. The phonon dispersion curves show an obvious decrease of group velocities in 3D phononic crystals. The phonon's localization and band gap is also clearly observed in spectra of normalized inverse participation ratio in nanoscale 3D phononic crystal.

Fabrication and Modelling of Three-Dimensional Sub-kelvin Phononic Crystals

We have investigated the fabrication and computational modelling of three-dimensional phononic crystals for the observation of full band gaps for thermal phonons at sub-kelvin temperatures. Self-assembled arrays of monodisperse polystyrene nano-spheres have been fabricated using a vertical deposition technique. Optimal conditions for increasing crystal domain size and crystalline quality have been studied. In addition, the phononic band structure has been computed using the finite element method for the simple cubic lattice. The dependence of band structure on contact area between spheres has also been studied. For small enough contact area a large band gap is observed, predicting a strong influence on sub-Kelvin thermal transport.

Perpendicular transmission of acoustic waves between two substrates connected by sub-wavelength pillars

We discuss theoretically the acoustic resonant transmission and zeros of transmission between two substrates connected by sub-wavelength pillars. The features of the transmission coefficient are explained in terms of the coupling of the incident waves with the Fabry–Perot oscillations inside the pillars and with the surface waves of both substrates. We discuss the dependence of the selective and zero transmission frequencies, in particular Fano resonances resulting from the proximity of a resonance to a zero of transmission, on the geometrical and physical parameters of the materials constituting the pillars and the substrate. These phenomena are studied in both one- and two-dimensional periodicities where the substrates are connected by a series of parallel plates or by a square lattice of cylindrical pillars, respectively. Finally, the calculation is extended to a periodic stacking of slabs and pillars that constitute a type of three-dimensional phononic crystal.

An effective medium approach and elastic metamaterials

Metamaterials are artificial materials that are designed to manipulate waves as desired. The unusual properties of a metamaterial are derived from its complex building blocks which make the development of effective medium theory challenging. In this work, effective medium parameters, such as anisotropic mass density and moment of inertia, are obtained for metamaterials based on mechanical models consisting of masses and springs. The effective moment of inertia is capable of providing a clear understanding of the mechanism of rotational modes which are widely observed in both two- and three-dimensional phononic crystals and metamaterials. The effective medium descriptions are served as guidance in the engineering of the building block to achieve rich resonances that lead to intriguing properties of the metamaterial. (Support of this work comes from KAUST Start-up Package)

Thermal characterization of nanoscale phononic crystals using supercell lattice dynamics

The concept of a phononic crystal can in principle be realized at the nanoscale whenever the conditions for coherent phonon transport exist. Under such conditions, the dispersion characteristics of both the constitutive material lattice (defined by a primitive cell) and the phononic crystal lattice (defined by a supercell) contribute to the value of the thermal conductivity. It is therefore necessary in this emerging class of phononic materials to treat the lattice dynamics at both periodicity levels. Here we demonstrate the utility of using supercell lattice dynamics to investigate the thermal transport behavior of three-dimensional nanoscale phononic crystals formed from silicon and cubic voids of vacuum. The periodicity of the voids follows a simple cubic arrangement with a lattice constant that is around an order of magnitude larger than that of the bulk crystalline silicon primitive cell. We consider an atomic-scale supercell which incorporates all the details of the silicon atomic locations and the void geometry. For this supercell, we compute the phonon band structure and subsequently predict the thermal conductivity following the Callaway-Holland model. Our findings dictate that for an analysis based on supercell lattice dynamics to be representative of the properties of the underlying lattice model, a minimum supercell size is needed along with a minimum wave vector sampling resolution. Below these minimum values, a thermal conductivity prediction of a bulk material based on a supercell will not adequately recover the value obtained based on a primitive cell. Furthermore, our results show that for the relatively small voids and void spacings we consider (where boundary scattering is dominant), dispersion at the phononic crystal unit cell level plays a noticeable role in determining the thermal conductivity.

Finite difference time domain calculation of three-dimensional phononic band structures using a postprocessing method based on the filter diagonalization

If the band structure of a three-dimensional (3D) phononic crystal (PNC) is calculated by using the finite difference time domain (FDTD) method combined with the fast Fourier transform (FFT)-based postprocessing method, good results can only be ensured by a sufficiently large number of FDTD iterations. On a common computer platform, the total computation time will be very long. To overcome this difficulty, an excellent harmonic inversion algorithm called the filter diagonalization method (FDM) can be used in the postprocessing to reduce the number of FDTD iterations. However, the low efficiency of the FDM, which occurs when a relatively long time series is given, does not necessarily ensure an effective reduction of the total computation time. In this paper, a postprocessing method based on the FDM is proposed. The main procedure of the method is designed considering the aim to make the time spent on the method itself far less than the corresponding time spent on the FDTD iterations. To this end, the FDTD time series is preprocessed to be shortened significantly before the FDM frequency extraction. The preprocessing procedure is performed with the filter and decimation operations, which are widely used in narrow-band signal processing. Numerical results for a typical 3D solid PNC system show that the proposed postprocessing method can be used to effectively reduce the total computation time of the FDTD calculation of 3D phononic band structures.

Calculational and Experimental Investigations into the Effects of the Scatterer and Matrix on Phononic Crystals Properties

Three-dimensional phononic crystals based on the locally resonant structure were fabricated in this work. The effects of the physical properties of the scatterer and matrix on the acoustic absorption coefficient (AAC) were investigated via calculational and experimental methods simultaneously. The consistency of calculational and experimental results was examined. Two kinds of backing of standing wave tube, air and steel-air backing, were used to validate the changing discipline of AAC. The results suggested that the location, extension and intensity of resonant peak had the close relationship with the modulus and thickness of the scatterer's coating layer, and the modulus of matrix. The type of backing also influenced the AAC of phononic crystals.

Dispersion of elastic waves in three-dimensional noncohesive granular phononic crystals: Properties of rotational modes

The dispersion relations of bulk modes propagating within a hexagonal close-packed structure of noncohesive monodisperse spherical elastic beads are derived. The contacts are modeled by two springs with stiffnesses given by the Hertz-Mindlin theory, one for normal interactions and one for transverse interactions. The existence of the transverse interaction requires to take into account the rotational degrees of freedom of the beads in the analysis. This leads to the prediction of translational modes and, due to the rotational degrees of freedom, of rotational modes and coupled rotational and translational modes. The study of the dispersion relations in a direction of high symmetry allows to identify the different modes and the influence of the rotational degrees of freedom on the bulk mode propagation. The evaluated dispersion relations provide guidelines for the experimental observation of rotational modes. Opportunities for controlling the dispersion laws of the modes by an external loading on the granular structure are discussed.

Octave omnidirectional band gap in a three-dimensional phononic crystal

We report on the experimental observation of a one-octave-large omnidirectional elastic band gap for longitudinal waves in a three-dimensional phononic crystal consisting of face-centered-cubic arrays of close-packed steel beads in a solid epoxy matrix. The 60% relative width of the band gap is larger than expected based on the contrast in material properties, and is confirmed by band structure diagram and transmission spectra computations. The coupling between shear and longitudinal polarizations is pointed out as a mechanism for enlarging the band gap by comparing with the steel beads in water matrix case.