# 19. Phononic crystals for acoustic tweezers

Feiyan Cai

Paul C. Lauterbur Research Center for Biomedical Imaging, Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen 518055, People's Republic of China

E-mail: fy.cai@siat.ac.cn

## Status

Acoustic tweezers have garnered increasing attention as a non-contact method capable of manipulating objects in a controllable manner. The underlying physical mechanism involves the absorbing, scattering, and reflecting of acoustic waves by objects, leading to an exchange of momentum and energy between the objects and the waves, then generating an acoustic radiation force on the objects. Thus, the performance of acoustic tweezers is determined by the properties of the field, the objects, and their interactions (scattering or absorption). Phononic crystals, along with metamaterials and metasurfaces, represent a burgeoning class of artificially engineered materials designed to control, direct, and shape acoustic waves [253]. The field of phononic crystal engineering advances the capabilities of acoustic tweezers, offering innovative solutions for enhancing acoustic manipulation techniques.

In the past decade, various acoustic fields shaped by phononic crystals have been created to enable versatile particle manipulation, thereby advancing acoustic tweezer technology. For example, the distinctive bandgap of phononic crystal has been used for reflection waves and enhancing the intensity of acoustic energy in a tunable manner, benefiting activities such as drop manipulation and jet enhancement [254]. The structure-induced excitation of Lamb wave in phononic crystal plate generating enhanced localized field has proven for trapping, levitation, and transporting microparticles, as well as realizing repairable sonoporation for massive cells [255, 256], as shown in figure 29. The metamaterial bricks forming with a designed passive phase array can construct a bottle-shaped beam to achieve single-sided air-borne acoustic levitation [257]. Recently, based on the particle scattering-induced mode conversion in the phononic crystal waveguide, the forward momentum of the acoustic beam can be amplified, resulting in the particle pulling toward the source [258].

Aside from shaping incident acoustic fields for particle manipulation, metamaterials can also be as anomalous acoustic scattering (particles) for shaping acoustic radiation force to realize complex manipulation phenomena. Exploiting anomalous metasurface scattering, a metasurface object can self-guide in response to the acoustic wave or be pulled toward the acoustic source [259]. These anomalous phenomena stem from the subwavelength patterns on the metasurface, which induce scattering asymmetry and ensure the necessary momentum balance.

It is worth noting that acoustic tweezers can also be utilized to manipulate individual particles for the fabrication

of photonic crystals. By applying a high-amplitude acoustic standing wave field, a 3D colloidal phononic crystal can be dynamically reconfigured in real-time, showcasing clear band-pass and band-stop frequencies in the transmission spectrum [260]. Additionally, by utilizing a phononic crystal composed of periodically patterned bubbles, the acoustic radiation force can be augmented through bubble resonance. This amplification enables the effective use of the bubble array-based phononic crystal to assemble macrostructures [261].

## Current and future challenges

Despite the powerful acoustic field-shaping ability of phononic crystals, they have not yet emerged as a competitive alternative to traditional acoustic components used in acoustic tweezers. This is partly due to several remaining challenges that need to be addressed, including, but not limited to:

* **Identification of suitable materials or structures:** While phononic crystals possess exceptional physical properties through the optimization of their structure and materials, these optimized structures or materials may not be ideal for acoustic tweezer devices. Research is needed to identify materials or structures with superior properties specifically tailored for acoustic tweezers.

* **Computational complexity:** Achieving device optimization requires a comprehensive, full-wave model that takes into account all components of the acoustic tweezer system, including the acoustic source, channel, finite-size phononic crystal, and more. However, the current theoretical frameworks face challenges in accurately simulating the complete system, resulting in a less-than-ideal match with experimental observations. This gap hampers the effectiveness of the design-build-test-repeat loop.

* **Ensuring device performance stability:** In traditional designs, the acoustic source, chamber, and passive phononic crystal are designed and manufactured separately. While this allows for the phononic crystal to be considered as a disposable chip, the separated design can result in less stable device performance. To address this, it is crucial to consider stability, compactness, and low-cost manufacturing as critical factors in the design of acoustic tweezer devices.

* **Unique application scenario:** Acoustic tweezers require a compact design and simple components for shaping the field. Therefore, traditional acoustic components are preferred for this purpose. However, in certain cases where unique application scenarios are being explored, the distinctive properties of phononic crystals cannot be replicated by traditional acoustic components. In such cases, the use of phononic crystals is preferred in acoustic tweezers systems.

Addressing these challenges will pave the way for further advancements in the field of acoustic tweezers, enabling more precise manipulation of particles and opening up exciting possibilities for various applications. Further research and technological developments are essential to overcome these obstacles and unlock the full potential of acoustic tweezers.
<figure><img src="image_38.png" /><figcaption>**Figure 29.** Dynamical control of microparticles and cells in an acoustofluidic channel using phononic crystal technology. (a) (Left) Schematic of the stop-and-go manipulation of particles in a microfluidic channel built on a phononic crystal plate (PCP), (Right) configuration of the PCP-based microfluidic device. (b) Trapping and levitation of polystyrene (PS) particles by frequency switching between two resonant frequencies. Reprinted (figure) with permission from [255], Copyright (2020) by the American Physical Society. (c) Double staining assay with PI/calcein-AM to verify cell permeability and viability after repairable sonoporation on the surface of PCP. Reprinted from [256], with the permission of AIP Publishing.</figcaption></figure>

## Advances in science and technology to meet challenges

To overcome the challenges associated with the use of phononic crystals in acoustic tweezers, several important advances in science and technology are being developed. These advancements are expected to have a significant impact on the field in the short to medium term. Some key areas of progress include:

* New physical mechanisms: Recent theoretical works have demonstrated the potential of utilizing the anomalous dispersion curves of phononic crystals for anomalous scattering and novel acoustic radiation force generation [259, 262]. These findings open up exciting possibilities for innovative applications of phononic crystals in acoustic tweezers.

* Novel materials: The advent of new materials with lower loss and externally controllable acoustic properties, such as

those with extremely higher acoustic impedance or matching acoustic impedance in water or air, is anticipated to contribute to the development of advanced acoustic components for acoustic tweezers. Additionally, the integration of active materials, such as piezoelectric phononic crystals and soft piezoelectric films, allows for the integration of the device components and further enhances the performance and functionality of acoustic tweezers.

* Advanced computational methods: More powerful computational techniques, including those based on artificial intelligence (such as deep learning) and inverse optimization methods, are expected to offer improved optimization capabilities for phononic crystal design and the modeling of acoustic tweezer systems. These methods can provide enhanced optimization capabilities and enable more accurate simulations and predictions.

* Refined fabrication methods: The development of more precise and cost-effective fabrication methods is crucial for the practical implementation of phononic crystals in acoustic tweezers. Integrated manufacturing approaches, including the fabrication of phononic crystals combined with acoustofluidic chips, can streamline the manufacturing process and enable efficient integration of components.

## Concluding remarks

The field of phononic crystal has emerged as a highly active research areas in both fundamental and applied acoustics. Building on foundations in phononic crystal, metamaterials, and metasurfaces offer exciting prospects for anomalous scattering, as well as the development of compact, scalable, and versatile acoustic components and devices. Integrating these advancements into acoustic tweezer systems holds tremendous potential for enhancing functionality while reducing size, complexity, and cost, making them more accessible for a wide range of applications. In addition, the use of acoustic tweezers for structure assembly offers an opportunity to produce phononic crystals with tailored properties. Given the rapid progress in the field and the immense flexibility in designing phononic crystal structures, there is significant room for innovative applications and further refinement of concepts and devices for acoustic tweezers.

## Acknowledgments

This work was supported by the National Key R&D Program of China (No. 2021YFB3801802) and the Shenzhen Science and Technology Program (Grant No. RCJC20221008092808013).
## 20. Lattice structures for sound absorption

Xinwei Li¹ and Wei Zhai²

¹ Faculty of Science, Agriculture & Engineering, Newcastle University, 567739 Singapore, Singapore

² Department of Mechanical Engineering, National University of Singapore, 117575 Singapore, Singapore

E-mail: xinwei.li@newcastle.ac.uk and mpezwei@nus.edu.sg

### Status

In modern engineering, there is a need for advanced materials optimized for mass and volume, providing superior acoustic properties and customizable design options. The rise of additive manufacturing has introduced new possibilities for creating functional materials based on structures rather than chemistry. One notable class of advanced materials that caters to the prior-mentioned needs is lattice structures. These three-dimensional arrangements consist of interconnected struts, shells, plates, or hybrids forming a repeating pattern (figure 30(A)) [263-269]. Lattice structures offer extensive design freedom with their customizable feature-pore morphology and interconnectivity, making them highly adaptable for achieving specific physical properties. Research has extensively explored the physical properties of lattice structures, showcasing their potential applications as lightweight materials, energy absorbers, thermal and electromagnetic metamaterials [270].

Given the potential of lattice structures, researchers have also delved into investigating the sound absorption properties of 3D printed lattice structures. These investigations encompass lattice structures made up solely of trusses, plates, a combination of trusses and plates, tubes, tubes combined with plates, smooth skeletal structures, triply periodic minimal surface, and bioinspired features [263]. Beyond the individual cell level, diversities such as parallel placement of heterogeneously-structured unit cells [271] and the introduction of additional acoustic meta-features have been suggested [272]. At present, the unit cell sizes studied extend up to 10 mm, while the overall sample thickness, influenced by both the unit cell size and the number of layers, ranges from approximately 20-60 mm [263]. Lattice structures with homogeneous unit cells exhibit either distinct resonance peaks or gradual absorption curves, depending on their structural morphology (figure 30(B)). In contrast, lattices composed of heterogeneously structured cells display broadband absorption curves (figure 30(B)), with effective bandwidths having absorption coefficients >0.5 typically spanning the frequency range of 1000-6000 Hz [271, 272]. Several analytical models have been suggested to anticipate absorption properties, encompassing mechanisms ranging from multi-layered Helmholtz resonance (MLHR), cavity resonance, effective acoustical properties, and slow-sound effects [263]. However, for most lattices, the most effective representation involves

modeling them as a series of Helmholtz resonators using the MLHR model. These models have demonstrated high fidelity in accurately predicting sound absorption coefficients.

Lattice structures demonstrated advantages over conventional absorbers. First, they display comparable absorption characteristics, owing to them having similar acoustical geometries and thus adhering to the same sound absorption principles. Second, they are highly customizable, for instance, they can be designed with heterogeneous features and optimized for broadband absorption (figure 30(B)). Additionally, lattice structures display potential for multifunctionalities, for instance, various works demonstrated that highly sound-absorbing lattice structures also serve as lightweight structures [272-274] or tough materials [271, 275]. The broad design flexibility inherent in lattice structures not only provides avenues for customization and optimization but also opens possibilities for multifunctionality.

### Current and future challenges

The primary challenge lies with analytical modeling. Thus far, most lattices are modeled using the MLHR method (figure 31(A)). One important aspect of the MLHR model lies with the resistance and reactance end corrections. Traditionally, end correction values are based on values proposed by Maa and other scientists [276]. These correction values work perfectly when modeling microperforated panel absorbers. However, when used in lattice structures, recent works reveal that the end correction factors are highly dependent on the acoustical geometries, including both the pore and cavity morphologies. For instance, works have revealed that if the pores are placed closer to the cavity wall, dissipation is enhanced and thus both the resistance and reactance end correction factors should increase [275, 277]. These works further proposed that these end corrections are functions of both the pore diameter and thickness through empirical. The scientific backing behind this dependence is largely unknown. Thus, how to derive end correction factors remains unknown. Also, there are few published works on modeling lattices through effective methods, and only one work each on modeling lattice absorption using cavity resonance and the sonic black-hole method. The potentials of these methods remain uncertain, and their reliability could benefit from further exploration. If analytical modeling can be better comprehended, there is a promising outlook for enhancing the design potentials of sound-absorbing lattices through structural manipulation.

Another challenge is associated with a lack of systematic understanding of lattice geometry on sound absorption. Unlike mechanical properties, where lattice stiffness and strength follow Gibson-Ashby relationships, there is no such systematic classification for sound absorption properties. This lack of classification applies not only across different structures but also within the same structure, where the influence of relative density, cell size, and the number of cells along sound incidence is largely unknown. For instance, as shown in [272], even
<figure><img src="image_39.png" /><figcaption>**Figure 30.** (A) Overview on the different types of lattice structures, revealing structures with configurations such as truss, TPMS, plate, and hybrid combinations. Figure is reproduced and adapted with permission from [263–269]. (B) Schematic of three distinct types of sound absorption coefficient curves. Reproduced from [263]. CC BY 4.0. Reproduced with permission from [264]. CC BY-NC-ND 4.0. Reprinted from [265], Copyright (2023), with permission from Elsevier. [266] John Wiley & Sons. [Copyright © 2023 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim]. Reproduced from [267]. CC BY 4.0. Reprinted with permission from [268]. Copyright (2019) American Chemical Society. Reprinted from [269], Copyright (2019), with permission from Elsevier.</figcaption></figure>

for the same truss arrangement, at different relative densities, sound absorption behavior is drastically different. More studies are imperative to comprehensively characterize sound absorption properties based on lattice design. However, there are also challenges associated with undertaking such studies. As mentioned earlier, there are distinct types of lattice structures (figure 30), making it potentially difficult to quantify properties across different types of lattices. Moreover, beyond homogeneous unit cells, the sound absorption properties of lattices associated with the inhomogeneous assembly of units, configurations such as heterogeneous, fractal, hierarchical, and functionally grading, are also largely unknown.

## Advances in science and technology to meet challenges

Advancements in science and technology is crucial to addressing the challenges associated with 3D printed lattice structures for sound absorption. Firstly, there is a need for more sophisticated analytical methods to enhance the modeling of lattice sound absorption properties. For example, in MLHR-type structures, this entails developing a method to determine end correction factors beyond empirical means obtained through extensive experiments. This may necessitate

collaboration among experts from diverse fields, particularly involving physicists and fluid mechanics specialists, to delve into the physics underlying extended fluid vibration beyond the pore. Additionally, the development of other analytical acoustics models, such as cavity resonance, effective properties, and slow-sound effects, requires more experimental work to further refine the models. At the moment, there is an increasing number of studies adopting finite element analysis (FEA) using COMSOL Multiphysics to predict sound absorption behavior and observe sound dissipation through lattices (figure 31(B)) [274, 278]. Success have been demonstrated thus far. However, achieving high-fidelity modeling through COMSOL often demands substantial computational resources. The development of improved FEA methods or algorithms specifically for acoustics modeling could significantly enhance the efficiency of modeling lattice structures. Lastly, there is a critical need for more experimental data. Given the diverse array of lattice structures, each with its unique sound absorption mechanism, a substantial dataset is essential for gaining a deeper understanding of the structural-property relationship between lattice structure and acoustic properties. This empirical data will contribute significantly to refining and validating analytical and FEA models, ultimately advancing our comprehension of the intricate interplay between lattice design and acoustic performance.
<figure><img src="image_40.png" /><figcaption>**Figure 31.** (A) Illustration of the discretizing and modeling of a lattice into an MLHR. [274] (2022), reprinted by permission of the publisher (Taylor & Francis Ltd, www.tandfonline.com). (B) Illustration of FEA modeling of sound dissipation through different lattice structure arrangements. Reproduced from [277] with permission from the Royal Society of Chemistry.</figcaption></figure>

**Concluding remarks**

Overall, the integration of lattice structures, enabled by additive manufacturing, presents a promising avenue for addressing these challenges. The three-dimensional arrangements of interconnected struts, shells, and plates not only offer superior acoustic properties but also exhibit a broad design flexibility that opens possibilities for multifunctionality. However, amidst the progress, challenges such as complexities in analytical modeling complexities, especially regarding end correction factors in MLHR models, demand collaborative efforts across disciplines to unravel the intricacies of airflow mechanisms within lattice structures. Another significant challenge is associated with the lack of systematic study on lattice structures for sound absorption, lacking a classification. This knowledge gap hinders comprehensive understanding and necessitates further exploration, requiring studies in the diverse lattice types and inhomogeneous unit assemblies. The need for a substantial experimental dataset to comprehend the structural-property relationship between different lattice designs and acoustic properties remains crucial. The journey toward comprehensive solutions requires ongoing research, innovation, and a synthesis of expertise across various scientific and engineering disciplines.

**Acknowledgments**

We acknowledge the funding by the MOE AcRF Tier 1 Grant (Project No. WBS A-8002418-00-00).

**Data availability statement**

All data that support the findings of this study are included within the article (and any supplementary files).

**ORCID iDs**

Yabin Jin https://orcid.org/0000-0002-6991-8827

Zhongming Gu https://orcid.org/0000-0003-0865-540X

Qian Wu https://orcid.org/0000-0002-4252-1109

Pedro David García https://orcid.org/0000-0002-3422-178X

Yi Chen https://orcid.org/0000-0002-6614-976X

Muamer Kadic https://orcid.org/0000-0002-4692-5696

Yongquan Liu https://orcid.org/0000-0003-0627-9885

M Lázaro Navarro https://orcid.org/0000-0003-4949-8295

J-P Groby https://orcid.org/0000-0002-9308-8261

S Félix https://orcid.org/0000-0002-4951-9634
L M Garcia-Raffi https://orcid.org/0000-0003-3985-8453

Xiaoying Zhuang https://orcid.org/0000-0002-7150-296X

I Mahmoud Hussein https://orcid.org/0000-0002-6583-1425

Yan Pennec https://orcid.org/0000-0002-4719-9913

Feiyan Cai https://orcid.org/0000-0002-1652-7996

Xinwei Li https://orcid.org/0000-0002-1702-8670

Wei Zhai https://orcid.org/0000-0003-2307-5243

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