2D Phononic Crystals Research Articles

Topological insulator based locally resonant phononic crystals: Wave propagation and acoustic band gaps

ABSTRACTIn the present work the acoustic band structure of a two-dimensional phononic crystal (PC) containing an organic ferroelectric (PVDF- polyvinylidene fluoride) and topological insulator (Bi2Te3) were investigated by the plane-wave-expansion (PWE) method. Two-dimensional PC with square lattices composed of Bi2Te3 cylindrical rods embedded in the PVDF matrix are studied to find the existence of stop bands for the waves of certain energy. Phononic band diagram ω = ω(k) for a 2D PC along the Г-X-M-Г path in the square Brillouin zone show four stop bands in the frequency range 0.01–8.0 kHz.

Phononic band gap and wave propagation on polyvinylidene fluoride-based acoustic metamaterials

In the present work, the acoustic band structure of a two-dimensional phononic crystal (PC) containing an organic ferroelectric (PVDF-polyvinylidene fluoride) and topological insulator (SnTe) was investigated by the plane-wave-expansion (PWE) method. Two-dimensional PC with square lattices composed of SnTe cylindrical rods embedded in the PVDF matrix is studied to find the allowed and stop bands for the waves of certain energy. Phononic band diagram ω = ω(k) for a 2D PC, in which non-dimensional frequencies ωa/2πc (c-velocity of wave) were plotted vs. the wavevector k along the Г–X–M–Г path in the square Brillouin zone shows five stop bands in the frequency range between 10 and 110 kHz. The ferroelectric properties of PVDF and the unusual properties of SnTe as a topological material give us the ability to control the wave propagation through the PC over a wide frequency range of 103–106 Hz. SnTe is a discrete component that allows conducting electricity on its surface but shows insulator properties through its bulk volume. Tin telluride is considered as an acoustic topological insulator as the extension of topological insulators into the field of “topological phononics”.

Acoustic band gaps with diffraction gratings in a two-dimensional phononic crystal with a square lattice in water

The present work reports a combined experimental and theoretical study on the acoustic band gaps in a two-dimensional (2D) phononic crystal (PC) consisting of periodic square arrays of stainless-steel cylinders with diameters of 1.0 mm and a lattice constant of 1.5 mm in water. The theoretical band structure of the 2D PC was calculated along the ΓX direction of the first Brillouin zone. The transmission and the reflection coefficients were obtained both experimentally and theoretically along the ΓX direction of the 2D PC. The 2D PC exhibited 5 band gaps at frequencies below 2.0 MHz, with the first Bragg gap being around a frequency of 0.5 MHz. To understand the band gaps in the 2D PC, we calculated the acoustic pressure fields at specific frequencies of interest for normal incidence, and we explained them from the perspective of acoustic diffraction gratings.

Research on the tunability of point defect modes in a two-dimensional magneto-elastic phononic crystal

Point defect modes in a 2D phononic crystal with giant magnetostrictive material tuned by a magnetic field and compressive stress are investigated theoretically in this study. The 3D magnetostrictive constitutive model proposed by Liu and Zheng (2005 Acta Mech. Sin. 21 278–85) is adopted to develop effective elastic, piezomagnetic, and magnetic permeability constants. The finite element method, in combination with a supercell technique, is then applied to obtain the band structures and transmission spectra of the point defect modes in a 2D phononic crystal composed of Terfenol-D rods of circular cross section embedded in a polymethyl methacrylate matrix with a square lattice. The magnetic field not only enlarges the first band gap (FBG) but also opens up a new band gap of XY modes. New point defect modes are simultaneously trapped in the band gaps. The width of the FBG and the frequencies of the point defects of the Z mode decrease as the magnetic field increases.

Realization of Complex 3D Phononic Crystals with Wide Complete Acoustic Band Gaps.

Phononic crystals offer unique band structures for acoustic wave propagation. Fabricating intricate threedimensional phononic crystals allows a new class of devices with complex phononic band structures beyond capabilities of two-dimensional designs. We have successfully fabricated novel 3D phononic crystals with 1 mm lattice constant and minimum feature sizes as low as 100 micron using high-resolution stereolithography printing. Here we report the first theoretical calculations and experimental results demonstrating wide complete 3D phononic band gaps not attainable by corresponding 2D structures with the same lattice geometry. Longitudinal and shear wave propagation is suppressed by more than -60 dB in frequency bands as wide as 400 kHz to 1 MHz.

Coherent heat transport in 2D phononic crystals with acoustic impedance mismatch

In this work we have calculated the cumulative thermal conductivities of micro-phononic crystals formed by different combinations of inclusions and matrices at a sub-Kelvin temperature regime. The low-frequency phonon spectra (up to tens of GHz) were obtained by solving the generalized wave equation for inhomogeneous media with the plane wave expansion method. The thermal conductivity was calculated from Boltzmann transport theory highlighting the role of the low-frequency thermal phonons and neglecting phonon–phonon scattering. A purely coherent thermal transport regime was assumed throughout the structures. Our findings show that the cumulative thermal conductivity drops dramatically when compared with their bulk counterpart. Depending on the structural composition this reduction may be attributed to the phonon group velocity due to a flattening of the phonon dispersion relation, the extinction of phonon modes in the density of states or due to the presence of complete band gaps. According to the contrast between the inclusions and the matrices, three types of two dimensional phononic crystals were considered: carbon/epoxy, carbon/polyethylene and tungsten/silicon, which correspond respectively to a moderate, strong and very strong mismatch in the mechanical properties of these materials.

Band structure computation of in-plane elastic waves in 2D phononic crystals by a meshfree local RBF collocation method

In this paper, the band structures of in-plane elastic waves in two-dimensional (2D) phononic crystals are calculated by using a meshfree local radial basis functions (RBF) collocation method. In order to improve the stability of the local RBF collocation method, special techniques suggested in our previous work for anti-plane waves are further improved and extended for calculating the primary field quantities and their normal derivatives required by the treatment of the boundary conditions in the local RBF collocation method for computing the band structures of the in-plane elastic waves in 2D phononic crystals. The developed meshfree local RBF collocation method for the band structure calculations of in-plane elastic waves propagating in 2D phononic crystals is validated by using the corresponding numerical results obtained with the finite element method (FEM). The band structures of different material combinations or acoustic impedance ratios, different filling fractions, various lattice forms and scatterer shapes are computed numerically to show the accuracy and the efficiency of the meshfree local RBF collocation method for computing the band structures of in-plane elastic waves in 2D phononic crystals.

Acoustic band structures and time reversal of elastic waves in two- and three-dimensional phononic crystals

The present study aims to investigate experimentally the acoustic band structures of elastic waves in two-dimensional (2D) and three-dimensional (3D) phononic crystals (PCs) consisting of periodic square arrays of cylindrical air holes in a solid matrix made of cubic polyoxymethylene. In the 2D PC, the first band gap appeared at frequencies between 0.50 and 0.77 MHz, and the second pass band at frequencies between 0.77 and 1.17 MHz. On the other hand, the 3D PC exhibited a wider band gap extending from a frequency of 0.44 MHz with noise level transmission at a frequency around 0.93 MHz. The time compressions of the time-reversed waves through the 2D and the 3D PCs were also obtained by using a time reversal process. The 2D PC exhibited better time compression than the 3D PC in the time-reversed waves, consistent with the fact that the time compression achieved by time reversal is sensitive to the number of frequency information grains.

Plane Wave-Perturbative Method for Evaluating the Effective Speed of Sound in 1D Phononic Crystals

A method for calculating the effective sound velocities for a 1D phononic crystal is presented; it is valid when the lattice constant is much smaller than the acoustic wave length; therefore, the periodic medium could be regarded as a homogeneous one. The method is based on the expansion of the displacements field into plane waves, satisfying the Bloch theorem. The expansion allows us to obtain a wave equation for the amplitude of the macroscopic displacements field. From the form of this equation we identify the effective parameters, namely, the effective sound velocities for the transverse and longitudinal macroscopic displacements in the homogenized 1D phononic crystal. As a result, the explicit expressions for the effective sound velocities in terms of the parameters of isotropic inclusions in the unit cell are obtained: mass density and elastic moduli. These expressions are used for studying the dependence of the effective, transverse and longitudinal, sound velocities for a binary 1D phononic crystal upon the inclusion filling fraction. A particular case is presented for 1D phononic crystals composed of W-Al and Polyethylene-Si, extending for a case solid-fluid.

Geometric tuning of thermal conductivity in three-dimensional anisotropic phononic crystals.

Molecular dynamics simulations are performed to investigate the thermal transport properties of a three-dimensional (3D) anisotropic phononic crystal consisting of silicon nanowires and films. The calculation shows that the in-plane thermal conductivity is negatively correlated with the out-of-plane thermal conductivity upon making geometric changes, whether varying the nanowire diameter or the film thickness. This enables the anisotropy ratio of thermal conductivity to be tailored over a wide range, in some cases by more than a factor of 20. Similar trends in thermal conductivity are also observed from an independent phonon ray tracing simulation considering only diffuse boundary scattering effects, though the range of anisotropy ratios is smaller than that obtained in MD simulation. By analyzing the phonon dispersion relation with varied geometric parameters, it is found that increasing the nanowire diameter increases the out-of-plane acoustic phonon group velocities, but reduces the in-plane longitudinal and fast transverse acoustic phonon group velocities. The calculated phonon irradiation further verified the negative correlation between the in-plane and the out-of-plane thermal conductivity. The proposed 3D phononic crystal may find potential application in thermoelectrics, energy storage, catalysis and sensing applications owing to its widely tailorable thermal conductivity.

Elastic wave band gaps tuned by configuring radii of rods in two-dimensional phononic crystals with a hybrid square-like lattice

The plane wave expansion (PWE) method is used to calculate the band gaps of two-dimensional (2D) phononic crystals (PCs) with a hybrid square-like (HSL) lattice. Band structures of both XY-mode and Z-mode are calculated. Numerical results show that the band gaps between any two bands could be maximized by altering the radius ratio of the inclusions at different positions. By comparing with square lattice and bathroom lattice, the HSL lattice is more efficient in creating larger gaps.

A meshfree local RBF collocation method for anti-plane transverse elastic wave propagation analysis in 2D phononic crystals

In this paper, a meshfree or meshless local radial basis function (RBF) collocation method is proposed to calculate the band structures of two-dimensional (2D) anti-plane transverse elastic waves in phononic crystals. Three new techniques are developed for calculating the normal derivative of the field quantity required by the treatment of the boundary conditions, which improve the stability of the local RBF collocation method significantly. The general form of the local RBF collocation method for a unit-cell with periodic boundary conditions is proposed, where the continuity conditions on the interface between the matrix and the scatterer are taken into account. The band structures or dispersion relations can be obtained by solving the eigenvalue problem and sweeping the boundary of the irreducible first Brillouin zone. The proposed local RBF collocation method is verified by using the corresponding results obtained with the finite element method. For different acoustic impedance ratios, various scatterer shapes, scatterer arrangements (lattice forms) and material properties, numerical examples are presented and discussed to show the performance and the efficiency of the developed local RBF collocation method compared to the FEM for computing the band structures of 2D phononic crystals.

The Calculation of the Band Structure in 3D Phononic Crystal with Hexagonal Lattice

Abstract In this article, the propagation of acoustic waves in the phononic crystals (PCs) of three dimensions with the hexagonal (HEX) lattice is studied theoretically. The PCs are constituted of nickel (Ni) spheres embedded in epoxy. The calculations of the band structure and the density of states are performed using the plane wave expansion (PWE) method in the irreducible part of the Brillouin zone (BZ). In this study, we analyse the dependence of the band structures inside (the complete band gap width) on c/a and filling fraction in the irreducible part of the first BZ. Also, we have analysed the band structure of the ALHA and MLHKM planes. The results show that the maximum width of absolute elastic band gap (AEBG) (0.045) in the irreducible part of the BZ of HEX lattice is formed for c/a=6 and filling fraction equal to 0.01. In addition, the maximum of the first and second AEBG widths are 0.0884 and 0.0474, respectively, in the MLHKM plane, and the maximum of the first and second AEBG widths are 0.0851 and 0.0431, respectively, in the ALHA plane.

수중에서 정방형 격자를 갖는 2차원 포노닉 크리스탈의 음향 밴드 구조

포노닉 크리스탈이란 기저물질 내에 주기적으로 배열된 산란체로 구성된 복합물질로서 포노닉 크리스탈에 입사된 음파가 특정 주파수 대역에서 차단되는 현상인 밴드 갭이라는 중요한 특성을 갖는다. 본 연구에서는 수중에서 산란체로서 1 mm의 직경을 갖는 원기둥 형태의 스테인리스 스틸 막대가 1.5 mm의 격자상수를 가지며 정방형으로 배열된 2차원 포노닉 크리스탈의 음향 밴드 구조를 이론 및 실험적으로 고찰하였다. 2차원 포노닉 크리스탈의 밴드 구조를 예측하기 위해 유한요소법을 이용하여 첫째 브릴루앙 영역의 <TEX>${\Gamma}X$</TEX> 방향에 대해 주파수와 파동벡터에 대한 분산관계를 계산하였다. 초음파가 입사되는 방향과 수직한 스테인리스 스틸 막대 층의 개수를 1, 3, 5, 7, 9개로 변화시켜가며 투과계수 및 반사계수를 측정하였다. 계산된 분산관계로부터 2 MHz 이하의 주파수 대역에서 5개의 밴드 갭이 존재하는 것으로 예측되었으며, 첫째 밴드 갭은 0.5 MHz를 중심으로 나타났다. 투과계수 및 반사계수로부터 실험적으로 확인된 밴드 갭은 분산관계로부터 예측된 밴드 갭과 잘 일치하는 것으로 나타났다. Phononic crystals are composite materials consisting of a periodic arrangement of scattering inclusions in a host material. One of the most important properties of phononic crystals is the existence of band gaps, i.e., ranges of frequencies at which acoustic waves cannot propagate through the structure. The present study aims to investigate theoretically and experimentally the acoustic band structures in two-dimensional (2D) phononic crystals consisting of periodic square arrays of stainless steel solid cylinders with a diameter of 1 mm and a lattice constant of 1.5 mm in water. The theoretical dispersion relation that depicts the relationship between the frequency and the wave vector was calculated along the <TEX>${\Gamma}X$</TEX> direction of the first Brillouin zone using the finite element method to predict the band structures in the 2D phononic crystals. The transmission and the reflection coefficients were measured in the 2D phononic crystals with 1, 3, 5, 7, and 9 layers of stainless steel cylinders stacked in the perpendicular direction to propagation at normal incidence. The theoretical dispersion relation exhibited five band gaps at frequencies below 2 MHz, the first gap appearing around a frequency of 0.5 MHz. The location and the width of the band gaps experimentally observed in the transmission and the reflection coefficients appeared to coincide well with those determined from the theoretical dispersion relation.

Surface/Interface Effect on the Band Gap of an Antiplane Wave in a 2D Phononic Crystal with Parallel Nanoholes or Nanofibers

The dispersion relations and band gap properties of 2D phononic crystals with periodically arranged nanoholes or nanofibers are studied based on the multiple scattering method. Elastic waves polarized along the axes of the nanoholes or nanofibers are considered. Gurtin’s surface/interface models are used to deal with surfaces of the nanoholes and interfaces between the nanofibers and the host material. Some numerical examples are given, and the results obtained are illustrated graphically. It is found that the surface/interface effect influences the dispersion relations and band gaps considerably. The band gaps may become wide or narrow depending on the surface parameter.

Negative refraction of acoustic waves in phononic crystals using recursive algorithms for block toeplitz matrices

Recent improvements in design and manufacturing have significantly improved our ability to manage phononic crystals (PC). Researchers have been studied the negative refraction in PC experimentally and theoretically using multiple scattering (MS) theory. This problem requires solving a large complex valued linear system that has a special multilevel block Toeplitz (BT) structure. We study negative refraction of acoustic waves in 2D PC by means of MS theory, by taking advantage of the PC structure and using specific recursive algorithms for BT matrices. We present new efficient and accurate algorithms for solving acoustic MS problem by the cluster of closely spaced cylinders to design acoustic negative refraction imaging in a PC. The unit cell of the PC consists, in general, of a solid cylinder in an acoustic medium. The dispersion curves of PC have a negative refraction dispersion branch producing the focusing effect. Particular attention is given to the dynamic behavior in the vicinity of dispersion branches with negative slope, and to the band gaps. We explore the effect of focal point on structural parameters, and employ a parallelization technique that allows efficient application of the proposed recursive algorithms for solving BT systems on high performance computer clusters. Numerical comparisons of CPU time and total elapsed time taken to solve the linear system using the direct LAPACK and TOEPLITZ libraries on Intel FORTRAN show the advantage of high performance recursive algorithms over the Gaussian elimination.

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.

Thermal transport in phononic crystals and the observation of coherent phonon scattering at room temperature.

Large reductions in the thermal conductivity of thin silicon membranes have been demonstrated in various porous structures. However, the role of coherent boundary scattering in such structures has become a matter of some debate. Here we report on the first experimental observation of coherent phonon boundary scattering at room temperature in 2D phononic crystals formed by the introduction of air holes in a silicon matrix with minimum feature sizes >100 nm. To delaminate incoherent from coherent boundary scattering, phononic crystals with a fixed minimum feature size, differing only in unit cell geometry, were fabricated. A suspended island technique was used to measure the thermal conductivity. We introduce a hybrid thermal conductivity model that accounts for partially coherent and partially incoherent phonon boundary scattering. We observe excellent agreement between this model and experimental data, and the results suggest that significant room temperature coherent phonon boundary scattering occurs.

Lamb wave band structure in phononic crystal composed of multiple square stubs symmetric or anti-symmetric deposited on a homogeneous plate

This paper theoretically investigates the propagation of Lamb waves in a two-dimensional (2D) phononic crystal (PC) structure composed of multiple square stubs symmetric or anti-symmetric deposited on a thin homogeneous plate. The dispersion relations, the power transmission spectra and the displacement fields of the eigenmodes are studied by using the finite-element method (FEM). We investigate the effect of the distance between adjacent square stubs on the band gaps (BGs). The results show that the distance between adjacent square stubs can remarkably change the location and the width of the BGs. Meanwhile, we conducted a numerically study on the different results between the anti-symmetric and symmetric distribution of the scatterers. The band structure of the anti-symmetric structure PC is significantly different from the symmetric structure. In general, the distribution of the square stubs has an influence on the band structures. We further identify the physical mechanism of the distinction in the formation of the BG between the anti-symmetric and the symmetric model through the displacement fields of the eigenmodes. In addition, we find that the variation of the PC BGs in anti-symmetric and symmetric structure are sensitive to the distance between adjacent square stubs, which could be indispensable to practical applications such as BG tuning.

A novel modal calculation method of 1-D phononic crystal band gap

Purpose – The purpose of this paper is to propose a modal method to calculate the band gaps of one-dimensional (1D) phononic crystals. Design/methodology/approach – The phononic crystals have modes with exponential form envelope in the band gaps, however, outside the band gaps the modes are of amplitude modulation periodic form. Thus the start and end frequencies of band gaps can be determined from the existence conditions of periodic modes. So, the band gaps calculation of 1D phononic crystal is transformed into the existence discussion of periodic solution of mode shapes equation. The results are verified by finite element harmonic response analysis. Findings – At the start and end frequencies of the band gap, the mode equation have solution with period of lattice constant. Originality/value – Compared with the traditional theoretical methods, the proposed modal method has a clearer principle and easier calculation.