Topological imbalanced phononic crystal with semi-enclosed defect for high-performance acoustic energy confinement and harvesting

Highlights

• Topological imbalance is proposed to design unit cell configuration.

• Mode jump is used to explain the band gap opening of PnCs.

• Semi-enclosed defect is proposed to enhance the energy confinement effect.

• The maximum output power is increased by 75.1 times from point defect case.

Abstract

Phononic crystals (PnCs) attract attentions in acoustic energy conversion and harvesting due to their excellent properties in regulating elastic and acoustic waves. Topological imbalance and semi-enclosed defect has been proposed in this work to systematically broaden the band gap of PnCs and improve the localization of elastic wave in defect. Parametric equations are introduced to generate nine square lattice unit cells with four-fold rotationally symmetric shapes. The imbalance between the long and short branches is utilized to describe the topological evolution of the perfect PnCs. With the increase of topological imbalance, the jump of the Bloch mode appears and thus leads to the band gap opening and expansion. The defect supercell with the highest topological imbalance has the best energy confinement effect. The 2D and 3D transient numerical simulations for the PnC of the highest topological imbalance indicate that the energy confinement effect of the semi-enclosed line defect is more intensive than the semi-enclosed arrow or bottle defect, which is confirmed by piezoelectric energy harvesting experiments. Under the excitation of 50 kHz, the period-2 nonlinear phenomena of the output voltage by piezoelectric disk are experimentally noticed in the PnC with the semi-enclosed line defect. The peak-to-peak power is 3.08 mW at the optimal resistance. Compared with the traditional point defect case, the voltage of the semi-enclosed line defect case is increased by 12.0 times and its power is increased by 75.1 times due to combination of perfect mirror effect and nonlinear defect state mechanism. This study provides a new avenue in design of high-frequency nonlinear acoustic devices and self-powered acoustic sensors.

Introduction

Phononic crystals are periodic artificial structures with exotic properties that are not available in natural materials, such as negative mass density, negative modulus [1], and zero or even negative refractive index [2], [3]. In their band gap of Bragg scattering, one of the interesting features is the defect state, appearing with periodicity broken. New Bloch modes generated by the defect state which can confine the energy within the defect are called defect modes. Compared with locally resonant phononic crystals, Bragg scattering phononic crystals can control elastic waves at higher frequencies with the same unit cell size, so PnCs with defect modes have potential ability to localize high-frequency elastic waves, e.g., ultrasonic waves, for non-destructive health monitoring of engineering structures [4], [5] and RF signal processing of surface acoustic wave (SAW) devices [6]. These applications are inseparable from the acousto-electric conversion.

Applications of PnCs are mainly dependent on the width of the band gap [7]. Before using the defect state characteristics of PnCs for acoustic-electric conversion, the band gap should be optimized first. The width of the band gap is mainly determined by the configuration of the unit cell. Existing studies have pointed out that the unit cell configuration with internal geometric imbalance, such as the cross [8], triangle [9], ellipse [10], [11] or grading structures [12], will increase the band gap width. Some scholars found some internal mechanisms for the influence of the unit cell configuration on the band structures through the investigation of Bloch mode: the unit cell configuration will affect the local resonance characteristics of the periodic masses in the entire PnCs, which in turn changes the band structure and band-gap width. Yu et al. [13] studied lamb wave propagation in PnCs formed by periodically embedding rectangular cylinders in pre-stressed concrete slabs, and found that the width, height and rotation angle of rectangular cylinders can significantly adjust neck stiffness and shape of larger mass in PnCs, leading to variations in lamb wave eigenmodes and band gap. Wang et al. [14] investigated the influence of geometric parameters of 2D PnCs with cross holes on band structure. The first band gap is caused by the local resonance of a periodically arranged mass connected to narrow connectors, and the lower edge of band gap can be predicted by a spring-mass model. Wang et al. [15] found that replacing square holes with cross holes in 2D PnCs slabs can generate multiple broad band gaps at lower frequencies. Through the analysis of the Bloch mode, it is found that the band gap of the periodically arranged intersecting holes is caused by the resonance of the periodic mass or connectors. Wen et al. [16] proposed a 2D square lattice PnCs whose unit cell consisting of cross concave holes and square convex holes, and investigated the band structure and the eigenmodes of the band gap edges. The band gap was formed by the local resonance of periodically arranged mass and the connectors. Wang et al. [17] studied the variations of geometric parameters in the band structure for a square lattice PnCs composed of zigzag unit cells. The bending of the zigzag arm structure or the rotation of the unit cell modifies the local resonance frequency, leading to the separation of the degeneracy points in the band structure and the appearance of complete band gap. Jiang et al. [18] proposed a 2D hexagonal lattice PnCs inspired by Hoberman spheres and analyzed the formation mechanism of band gaps using the Bloch modes of band gap boundaries. The results turned out that the first complete band gap is formed by the local resonance at the junction of unit cells. Wang et al. [19] designed a unit cell with a fractal porous structure based on the Sierpinski triangle. Complete band gaps were more potential to form in the fractal structure under the same porosity, and structures with higher fractal levels were more likely for multiple and wider complete band gaps. The complete band gap becomes wider when displacement fields of the upper and lower boundary vibrate symmetrically along the symmetric axis. These research works have pointed out that the unit cell configuration will affect the local resonance characteristics of periodic mass in the whole PnCs, and thus alters the band gap width, which provides a mechanism guide for the widening band gap and subsequent application of PnCs. However, the main local masses were located at the junction of the periodic unit cells, which limits the variety of local mass vibrations and the diversity of band gap formation. Other ways of band gap generation in PnCs containing pores, such as constructing mass which mainly vibrations within the unit cell, need further exploration.

After obtaining a wide band gap, the energy localization can be achieved through constructing defects in perfect PnCs to generate defect states. The current configurations for energy harvesting using PnCs and acoustic metamaterials can be classified into the following categories: configurations using defect states of PnCs [20], [21]; wave focusing configurations using mirror [22] and lens [23], [24] effect of PnCs, and configurations using the rainbow trapping effect of graded PnCs [25], [26]. Among them, although the idea of using defect states to harvest energy was first put forward, related research using defect states to harvest energy has been very active in recent years, and still has great research value. Yang et al. [27] proposed an energy harvester consisting of PnCs with point defect and an electromechanical Helmholtz resonator. The maximum output power of 429 μW is harvested under the excitation frequency of 5.545 kHz. Yang et al. [28] proposed a high-quality factor PnCs coupled with an electromechanical Helmholtz resonator, which improves the localization of sound waves and energy harvesting efficiency. An output power of 578 μW was obtained at a resonance frequency of 6.975 kHz. Ma et al. [29] designed a porous elastic metamaterial with multiple point defect modes to collect flexural wave energy, and achieved a maximum output power of about 6.2 μW at the excitation frequency of 14.59 kHz. Zhang et al. [30] underwent topology optimization for the geometric configuration of point defects to obtain multi-channel PnCs resonant cavities, which harvested an output power of 2.5 mW at the excitation frequency of 44.2 kHz. Park et al. [31] designed 2D PnCs of octagonal holes and obtained a maximum power of 1.59 mW under Lamb waves with 50 kHz. A trade-off relationship between the attenuation of the evanescent wave in the PnCs and the realization of resonance formation was reported by Jo et al [32]. The acousto-electric conversion of the PnCs can be improved to harvest a maximum power of 1.392 mW at the resonance frequency of 55.95 kHz through appropriately designing the supercell size and defect location. Jo et al. [33] discussed the defect band splitting phenomenon of two coupled point defects in a PnCs. The maximum power density of 7.894 mW/cm³ was achieved under the excitation frequency of 60.19 kHz.

In other words, the defect mode can be excited only when the external excitation frequency is near the defect one. Besides, the voltage cancellation phenomenon caused by point defect [33], [34] further limits the practical application of energy harvesting based on PnCs. To overcome these shortages, the PnCs waveguide based on the defect state mechanism provides the possibility for multi-modes and large-scale energy harvesting. Shao et al. [35] employed the band gap difference caused by the piezoelectric and line defects to create the wave localization, and numerically obtained an output power of 3.7 mW at 2257 Hz. The band-gap range induced by the extra stiffness and mass of the piezoelectric patch is still narrow. Once the wave frequency does not locate in the range of the piezoelectric defect’s band gap, the wave will be completely prohibited to enter into the line defect or propagate through it, which limits the wave localization in the line defect and thus reduces the energy harvesting efficiency.

In this paper, a topologically imbalanced configuration of the unit cell controlled by the shape parameter equations [36], [37], [38] has been proposed. The unit cell topology design is combined with vibration mechanics to generate a unit cell topology configuration with local resonant masses mainly distributed inside the unit cell to create more vibration modes. The concepts of topological imbalance and Bloch mode jump are introduced to systematically describe the ability for gradually widening band gap of the unit cell, which provides a new method for the design of porous PnCs with local resonance effect. Compared with PnCs containing pores without using topological imbalance, the concept of topological imbalance can be introduced to regulate the generation of band gap with in a simpler and more straightforward way.

In order to expand the frequency range for energy harvesting, the concept of semi-enclosed defects has been proposed by combining the design of point defects and waveguides. When the external frequency is close to the defect one, the defect state can be employed to localize the energy. When the wave frequency is not close to defect one but within the band gap, the perfect mirror effect [39] of PnCs can be used to collect the wave energy. Three super cells with different semi-enclosed defects are formed using the unit cell with the strongest topological imbalance (the largest band gap width). The concepts of topological imbalance, Bloch mode jump and semi-enclosed defect proposed in this paper can be utilized to design non-destructive health monitoring of engineering structures [40], [41], phonon transmission inside the chip [42], [43], elastic resonators with high-quality factors [44], non-reciprocal acoustic components [45], ultrasound imaging [46], acoustically driven microfluidic devices [47], [48], high field intensity topological insulators [49] and self-powered microstructures.