Adjustable Magnetic-Control Design of a Metasurface for Sound Insulation

Magnetic field is one efficient approach to realize non-contact control of materials. Here, we propose a type of membrane-based metasurface consisting of several parts of magnetorheological films embedded with magnetic particles, which enable non-contact control of sound insulation through magnetic field. The abundant degree of freedoms in membrane-type metamaterials are explored by considering composite structures. Through numerical calculations, we have investigated the effect of magnetic field excitation and film composition on the acoustic properties and the performance for sound insulation. We find an optimized model for better adjustability, wider acoustic band, and more balanced performance.


INTRODUCTION
In recent years, artificial acoustic metamaterials have significantly expanded the horizon of acoustic materials, helical-structured metamaterials, PT-symmetric acoustics, acoustic cloaking, the acoustic Meta-Skin Insulator, etc, leading to many novel phenomena (Christensen et al., 2007;Li et al., 2009;Liang et al., 2010;Boechler et al., 2011;Zhang et al., 2011;Zhu et al., 2011;Chen et al., 2014a;Zhou et al., 2014;Zhu et al., 2014;Molerón and Daraio, 2015;Zhu et al., 2016a;Cummer et al., 2016;Ma and Sheng, 2016;Yang and Sheng, 2017;Tang et al., 2018;Tong et al., 2020;Zeng et al., 2021). The single layer of acoustic metamaterials, i.e., acoustic metasurfaces, can manipulate the phase and amplitude of sound in unprecedented ways and enable many interesting applications like anomalous refraction and reflection, radiation control, acoustic holography, acoustic absorption, etc (Lee et al., 2010;Liang and Li, 2012;Mei et al., 2012;Zhao et al., 2013a;Zhao et al., 2013b;Li et al., 2013;Li et al., 2014;Tang et al., 2014;Xie et al., 2014;Kaina et al., 2015;Xiao et al., 2015;Yuan et al., 2015;Gao et al., 2016;Assouar et al., 2018). Mechanisms of exotic elastic metamaterials have also been investigated (Akl and Baz, 2012;Chen et al., 2014b;Bergamini et al., 2014;Wang et al., 2014;Zhu et al., 2016b;Chen et al., 2017a). In this work, we focus on membranetype metamaterials as they have excellent sound isolation and absorption properties with regard to low-frequency noise ranging from 50 to 1000 Hz (Yang et al., 2010). Their average transmission loss may exceed 40 dB. Moreover, they are light in weight and easy to prepare. Membrane-type materials use resonance as a means to absorb sound waves at some frequencies (Du et al., 2001). Negative effective parameters have also been achieved by membrane-type materials (Yang et al., 2013). Recently, external static magnetic force has been applied to harness the geometric nonlinearity of membrane structures so as to tune the transmission spectrum both theoretically and experimentally (Chen et al., 2017b). However, the abundant degrees of freedoms in the structure design of the membrane-type metasurfaces for adjustable magnetic control have not been fully explored yet.

MAGNETIC ACOUSTIC METAMATERIAL MODELS AND THEORY Theoretical Model for the Monolayer Magnetoacoustic Metamaterial
In the past decade, researchers have proposed to realize active acoustic metamaterials with magnetorheological films (Chen et al., 2017b;Domel et al., 2018). Such a metamaterial is composed of magnetorheological film, aluminum and lead mass block ( Figures 1A-D). For the magnetorheological film, the Young's modulus changes with magnetic field, thus the resonance frequency also varies accordingly, leading to the active control functionality.
To achieve better performances in sound insulation at low frequencies, a local resonance of the system at low frequencies is realized by attaching a mass block of lead to the magnetic film. The inertial force generated by the additional mass is added to the vibration equation of the film as an external incentive force. The equation describing free vibrations of the film structure with the added mass is (Sun et al., 2010) where ε r is the radial strain, ε θ is the circumferential strain, u and w are the radial and out-of-plane displacements of the membrane, respectively. Due to the axial symmetry, the balance equation in circumferential direction is automatically satisfied. Meanwhile, the relation between the bending angle and the deflection is introduced by sinθ − dw dr . The equilibrium equations are: The free vibration characteristic equation of the membrane-type acoustic metamaterial is where M and K denote the mass matrix and stiffness matrix, respectively, K f denotes the stiffness matrix for the pre-stressed material, ω is the natural angular frequency of the system, andq an eigenvector. When an incident plane wave propagates normal to the surface, it first interacts with the magnetic film and additional mass block. For this mass-weighted film, the surface-averaged Green function of Equation is (Chen et al., 2017b) Here 〈W〉 is the surface-averaged displacement of the membrane-type structure, δ P is the sound pressure variation through the structure, S is the relevant area, ρ is the local surface density, W i is the ith eigenmodes, and ω i is the relevant angular eigenfrequency. When the magnetic field is imposed, both the eigenmodes and the eigenfrequencies will be changed accordingly. The effect of a mass block can be analyzed by using the equivalent-lumped-element method (Du et al., 2001). The equivalent masses of the film and the mass block at the center of the circle are denoted M 1 and M 2 , and the equivalent elasticity coefficient of the film is denoted K 1 . The natural frequency of the system is then From Eq. 5, the natural frequency of the initial system is reduced by adding the mass block. By increasing the mass, the sound insulation effect of the model system at low frequencies becomes better. For this reason, we chose to use blocks of lead. The Young's modulus E of magnetorheological films varies with the volume fraction of ferromagnetic particles φ, the strength of the magnetic field H 0 , the vacuum permeability μ 0 , relative permeability μ m and constant D. An expression for E is given by the linear magnetic dipole model (Feng et al., 2015), From Eq. 8, it is clear that the Young's modulus of a magnetorheological film can be conveniently changed by adjusting the volume fraction of magnetic particles in the magnetorheological film, and hence the working frequency of the active acoustic material is altered accordingly. The unit cell of the monolayer active acoustic metamaterial comprises a magnetron film, aluminum frame, and lead mass block. The unit cell is exposed to air. At the upper boundary of the air layer, an incoming plane-wave is incident at a pressure of 1 Pa; at the lower boundary of the air layer, a set of outgoing plane waves are emitted. In this manner, the performance of the unit cell with regard to sound insulation can be calculated from the changes in sound pressure of these incoming and outgoing waves.
The acoustic properties of the metamaterials were numerically analyzed using the frequency mode of the acoustic-solid coupling module implemented in COMSOL Multiphysics 5.4. The structural size of the cell is as follows: film thickness d 1 mm, radius r 15 mm, frame thickness d 1 3 mm, inner diameter r 1 15 mm, r 2 17 mm, mass block radius r 3 6 mm, and thickness d 2 2 mm. The support frame of the unit cell is chosen as aluminum. The membrane-type film is made of magnetorheological material. And the mass block is made of lead. The material parameters of aluminum are: density ρ 2,700 kg/m 3 , Young's modulus E 70 GPa, and Poisson's ratio ν 0.33. The parameters for the magnetorheological material are: density ρ 2,100 kg/m 3 , Young's modulus E 6.5 MPa, and Poisson's ratio ν 0.45. Table 1 gives the dependence of the Young's modulus of the material on the magnetic field strength over the range of 0-300 kA/m for a 188 KPa pre-stressing of the magnetron film used in this study.

Composite Magnetic Control Metasurface Film
We note that there are abundant degrees of freedoms to be explored in the magnetic membrane-type metasurface design.
In this work, we consider a kind of composite magnetic membrane-type metasurface that are composed of multiple films instead of only one film. Compared to the previously investigated simple single-film structure, the multiple film design in principle can realize controllable broadband sound insulation by tuning the independent resonances of each film.
The composite magnetic membrane-type metamaterial is composed of one circular film and two annular films (Figures 2, 3), which are interposed by two aluminum spacers, yielding the sequence of film-1-aluminum-film-2aluminum-film-3. The whole structure is embedded in an aluminum frame. The structural size of the composite membrane-type metamaterial is the same as that of the previously investigated simple membrane-type metamaterial with only one film.
The acoustic properties of the metamaterials were numerically analyzed using the frequency mode of the acoustic-solid coupling module implemented in COMSOL Multiphysics 5.4. The structural size of the cell is as follows:  We get better sound insulation performance. The working bandwidth also turns wider, and hence sound insulation in the lowfrequency range is improved. By adding a magnetic field, controlling adjustments can be made in the peak value of sound insulation for the film material. Non-contact active control of noise is thereby realized, along with a broader frequency range and better performance for sound insulation of the film material.   Figure 5D shows the performance curves of the three-film membrane-type metamaterials.

Simulation of the Monolayer Magnetron Film
Comparing with the previous cases where the films have exactly the same Young's moduli, we find that when the three films' Young's moduli are different, a more stable sound insulation effect is achieved in a larger bandwidth. To be more specific, the working bandwidth for the regime with sound insulation effect of maximum −5 dB is found to be 3,980 Hz ( Figure 5D), which is much bigger than those observed in Figure 4. We find that the working bandwidth of sound insulation is greatly broadened and produces a better effect for sound insulation. Therefore, this multi-film design solves the issue of narrow working bandwidth for single-film magnetic control films.
By comparing Figures 6A-E with Figure 5D, we find that by tuning the Young's moduli in different films, a distinct modulation is achieved in the sound insulation effect. The bandwidths relating to the maximum −5 dB working range are, in sequential order, 3,810 Hz,i.e.,3,430 Hz,2,880 Hz,i.e.,3,550 Hz,2,580 Hz,i.e.,3,140 Hz,1,980 Hz,i.e.,3,480 Hz,and 4,140 Hz,i.e.,3,780 Hz, respectively, whereas that observed in Figure 5D is 3,980 Hz. Clearly, with a decreasing contrast in Young's moduli for different films, the working bandwidth of the three-film metasurface gradually diminishes.

Influence of the Magnetic Field for the Designed Three-Film Metasurface on the Performance for Sound Insulation
The magnetic field excitation is also a major influence factor on magnetically controlled films and, in particular, on the performance of such films for sound insulation.
For reference, we utilize the performance curves obtained under a 200-kA/m magnetic field for the three-film metasurface with Young's moduli of 2, 16, and 32 MPa ( Figure 5D). Retaining the same film conditions, we adjust the intensity of the magnetic field to obtain the performance curves for sound insulation ( Figures 7A,B), for which the Young's modulus were 0.02, 0.16, and 0.32 MPa under a 20-kA/m magnetic field, and 10, 80, and 160 MPa under a 890-kA/m magnetic field. In comparing Figure 5D with Figures 7A,B, the maximum −5-dB working bandwidth for sound insulation is 1,200 Hz, i.e., 4,401-6,501 Hz ( Figure  7A) and 1,7700 Hz, i.e., 10,500-28,200 Hz ( Figure 7B). Clearly, reducing the magnetic field intensity narrows the working frequency band and decreases the frequency of the peak; conversely, increasing the magnetic field intensity widens the working frequency band and raises the frequency of the peak. That is, by adjusting magnetic field strength, an adjustable working frequency band can be obtained, and this modulation is achieved through noncontact means. The results also show that this three-film metasurface is suitable for high frequency applications, whereas for low frequency, applying a mass block is better than implementing modulations in the magnetic field strength.

SUMMARY AND DISCUSSION
The acoustic characteristics and sound-insulation performance of multi-film composite metasurfaces with magnetic control are designed and analyzed. Compared with the single-film structure, multi-film magnetic metasurfaces have a wider working bandwidth and offer a better performance with regard to sound insulation. Together, they improve the performance of magnetron films and also broaden their range of application.
This work proves that magnetic field is one efficient approach to realize non-contact control of materials. The magnetic field is controllable and adjustable. And the adjustment effect is evident. We can get the desired changes by adjusting the magnetic field parameters.
We also have other ways to realize magnetic adjustment. For example, we can use Niobium metal to make materials. Nb has diamagnetism and superconductivity at a certain temperature. At the temperature, we can realize non-contact control of materials. Adjusting the magnetic field can control the parameters of the materials. Changing the temperature can also realize the switch of the magnetic control function. This idea has rich application prospect.
Our work shows that the use of film combination can greatly improve the overall performance of membrane-type metamaterials. This enhancement can also be considered in the other fields of acoustics and acoustic applications to achieve a better performance beyond a single film structure.

DATA AVAILABILITY STATEMENT
The raw data supporting the conclusion of this will be made available by the authors, without undue reservation.

AUTHOR CONTRIBUTIONS
PX conducted the simulations. PX, YL, and XL contributed to the writing. XL conceived and led the project.