CN113808560A

CN113808560A - Ultra-thin broadband underwater sound absorber made of composite metamaterial based on impedance matching

Info

Publication number: CN113808560A
Application number: CN202110591992.XA
Authority: CN
Inventors: 沈平; 屈思超; 高楠
Original assignee: Hong Kong University of Science and Technology HKUST
Current assignee: Hong Kong University of Science and Technology HKUST
Priority date: 2020-06-12
Filing date: 2021-05-28
Publication date: 2021-12-17
Anticipated expiration: 2041-05-28
Also published as: CN113808560B

Abstract

An embodiment of the present invention provides an underwater sound absorber, which includes a plurality of Fabry-perot (fp) resonators. Each FP resonator is a rod made of a composite material, the plurality of FP resonators are spaced apart from each other by an air gap, and the cross-sectional dimension of each FP resonator is set small enough that the effective bulk modulus of each FP resonator is the same as its young's modulus.

Description

Ultra-thin broadband underwater sound absorber made of composite metamaterial based on impedance matching

Technical Field

The invention relates to a sound absorber, in particular to an ultrathin broadband underwater sound absorber made of a composite metamaterial based on impedance matching.

Background

Underwater sound waves play a central role in detection, measurement, underwater communication and the like in rivers and oceans. Today, underwater noise pollution is also becoming more and more severe in seashore and rivers near human civilization, which may harm the health of underwater living beings. Therefore, underwater sound absorption is important in detection (absorbed sound energy can be converted into electrical signals), stealth (by reducing surface acoustic reflections), and underwater noise management. However, for conventional sound absorbing materials, the thickness of the material required for the low frequency range is relatively large, if high absorption is to be achieved. Recent breakthrough research on acoustic metamaterials has enabled perfect broadband sound absorption in air at very thin thicknesses, which provides the potential for underwater sound absorption. In fact, the structure can produce a greater energy density by virtue of the resonance of the metamaterial, so that good absorption can always be achieved in a relatively narrow frequency band.

Disclosure of Invention

According to one aspect of the invention, there is provided an underwater sound absorber comprising a plurality of FP resonators. Each FP resonator is a rod made of a composite material, the plurality of FP resonators are spaced apart from each other by an air gap, and the cross-sectional dimension of each FP resonator is set small enough that the effective bulk modulus of each FP resonator is the same as its young's modulus.

According to an embodiment of the present disclosure, the composite material has a young's modulus E ═ B₀A, wherein B₀Is the bulk modulus of water, alpha is a constant greater than 1, and the density rho ═ alpha rho of the composite material₀Where ρ is₀Is the density of water.

According to an embodiment of the present disclosure, the composite material comprises a metal and a polymer, the volume fraction of the metal and the polymer being arranged such that the impedance of the composite material matches the impedance of water.

According to an embodiment of the disclosure, α > 1.

According to an embodiment of the present disclosure, the metal is tungsten.

According to an embodiment of the present disclosure, the polymer comprises polyurethane.

According to an embodiment of the present disclosure, the polymer includes a mixture of a urethane resin and a urethane rubber.

According to an embodiment of the present disclosure, the bottom of each FP resonator is supported by a rigid total reflection substrate.

According to the embodiment of the disclosure, the lateral dimension of each FP resonator is smaller than the wavelength corresponding to the resonance frequency of the FP resonator.

According to an embodiment of the present disclosure, the air gap has a dimension of about 50 microns.

Drawings

FIG. 1 is a schematic representation of a composite layer supported by a rigid fully reflective substrate.

Fig. 2 is a schematic diagram showing an incident wave, a reflected wave, and a transmitted wave on an interface of water and a composite material layer.

Fig. 3 is a diagram showing the reflected plane wave and the evanescent wave for different orders (m 1, 2).

Fig. 4 is a schematic diagram of a single FP resonator made of polyurethane-tungsten composite and an integrated unit composed of a plurality of FP resonators.

Fig. 5 is a schematic diagram of an absorption spectrum obtained by simulation.

Fig. 6 is a flow chart for forming an underwater sound absorber.

Detailed Description

Compared with traditional underwater absorbers (such as Alberich coatings, porous materials and local resonance absorbers), the present invention uses the resonance of longitudinal elastic waves instead of shear waves to dissipate acoustic energy. By decoupling the different resonators and properly integrating the designed resonance profiles, a flat underwater sound absorption spectrum was obtained and verified by COMSOL simulation. Furthermore, by mixing tungsten powder with polyurethane to synthesize an artificial composite material having both a large mass density and a low young's modulus, the preliminary objective of matching the composite material with water resistance is achieved while significantly reducing the thickness of the absorber to a degree not previously achieved.

In the water according to the inventionIn the lower sound absorber, the rods made of composite material are separated from each other by a minute air gap, so that the pressure relief boundary condition is applied. The free boundary conditions of the pressure relief on the lateral surface of the FP resonator and the transverse dimension much smaller than the relevant wavelength in water are used to adjust the geometry of the bar, so as to obtain a Young's modulus E of the composite material instead of the effective bulk modulus B_effThe FP resonator of (1). By using composite materials to make FP resonators, such as metal to polymer composites, the young's modulus of the mixture can be "tuned" by mixing different proportions of metal powder in the polymer matrix. By synthesizing an artificial composite material with a large mass density and a low young's modulus, impedance matching of the composite material with water can be achieved while significantly reducing the thickness of the absorber.

The operation of the underwater sound absorber according to the embodiment of the present invention will be described in detail.

1. Causal constraints of acoustic absorbers

For any passive material (passive material), their response to an extraneous incident wave must satisfy causal criteria. In other words, for a sound wave, its reflected sound pressure p_r(t) is the direct reflection p of the incident sound pressure at that moment_r(t) superposition of the earlier time incident acoustic response, i.e. p_i(t- τ), where τ > 0. Therefore, the temperature of the molten metal is controlled,

causal properties of the material response not only can mathematically derive the well-known Kramers-Kronig relationship (which relates the real and imaginary parts of the response function of the material, e.g., mass density and bulk modulus in the acoustic case, permeability and permittivity in the electromagnetic case), but also the causal constraints that relate the sample thickness d to the absorption spectrum a (λ):

in the formula, λ (═ 2 π c)₀/ω) is the wavelength of the sound wave,

denotes the bulk modulus of water, B_effRepresenting the effective bulk modulus of the sound absorbing material. It is assumed here that the sample is supported by rigid reflective walls. Equation (2) reveals that the integral is mainly contributed by the long wavelength part (i.e. the low frequency part), which also explains the fact that a larger sample thickness is required for broadband low frequency absorption.

Once the target absorption spectrum A (λ) and the acoustic wave propagation medium (i.e., water in this example) are given, the minimum thickness of the absorber is only determined by its effective bulk modulus B_effAnd (6) determining. For sound in air, it is difficult to find a material with a bulk modulus smaller than that of air to achieve a reduction in d_minThe purpose of (1). However, for underwater sound waves, since the bulk modulus of water is in the order of GPa, it is entirely possible to find materials softer than water to implement ultra-thin absorbers. Therefore, by artificial design B_eff＜B₀The thickness of the material can be reduced under the cause and effect limit, which is the design basis of the invention. If the thickness of the absorber is equal to the causal limit thickness d_minAnd if the sound absorber is consistent, the sound absorber reaches the optimal state. In fact, in the usual case, the actual thickness of the absorber far exceeds the minimum thickness predicted by equation (2). However, an optimal sound absorbing structure may be achieved by using special design techniques, which will be discussed in detail later.

By means of B_eff＜B₀The possibility of obtaining a smaller sample thickness can also be explained from the point of view of acoustic energy density. Effective bulk modulus is less than that of water (B)_eff＜B₀) The speed of sound in the material is also less than the speed of sound in water (c < c)₀). Since the refractive index is defined by the speed of sound ratio, i.e., n-c 0/c, the wave vector in the composite material can be defined as k-nk₀Wherein k is₀Wave vector in water. Further, due to the density of states and n²And is proportional, so the state density and the energy density in the composite material are both higher. In other words, the same can be achieved within a smaller composite sample thicknessThe resonant frequency.

2. Importance of Using impedance matching materials

FIG. 2 is a schematic diagram showing incident, reflected and transmitted waves at the interface of water and the composite layer, where p_i、p_rAnd p_tRespectively representing the acoustic pressure of the incident wave, the reflected wave and the transmitted wave. Considering the characteristic impedance as Z₀(＝ρ₀c₀) Has a water and characteristic impedance of Z_cThe interface between the sound absorbing composite of (a). If the sound wave is normally incident from the side of the aqueous medium, the reflected wave amplitude R can be given by:

if the impedance matching condition is satisfied, i.e. Z_c＝Z₀There is no reflection. Impedance matching is a prerequisite to achieve high acoustic absorption, since the sample is assumed to be supported by a fully reflective substrate. In other words, if the sound wave is reflected at the water-solid interface, the absorbed sound energy is much reduced.

Fabry-Perot resonator and its effective bulk modulus B_eff

In the foregoing, a relatively small effective bulk modulus B has been described_effOf importance in obtaining thin absorbers. In a general sense, B_effIs the effective bulk modulus of the absorber solid material. However, if attention is focused on the particular structure of the fabry-perot (FP) resonator, i.e. a narrow rod of fixed length with a reflective bottom, it can be found that B is_effMay be more broadly defined as the longitudinal modulus that controls the velocity of longitudinal waves in a solid rod. Based on the above point, for B_effThere is more explanation since the wave velocity along the length of the rod may depend on the boundary conditions of the side surfaces of the solid rod. That is, if the solid rod is wrapped in a rigid container, B_effMust correspond to the bulk modulus of the composite. However, if the cylindrical side surface of the rod is in contact with air so as to represent a pressure relief boundary condition (i.e., watch)Effective zero pressure modulation on the face) when B is present_effThe Young's modulus E of the solid. Why it is the young's modulus E, rather than the bulk modulus, that is important in achieving impedance matching with water. This is because E can be tuned by using composite materials in combination with their mass density, while the bulk modulus of solids always floats within a range of values of similar magnitude and is therefore not easily manipulated by man. Another important aspect is that if the FP-resonator is used as the basis for the underwater broadband sound absorber of the invention, only one type of wave (i.e. longitudinal waves in a solid body) is very important in the FP-resonator, which means that shear waves in the FP-resonator and their generation are excluded. Since the FP resonator is made of a composite material of solid nature, which always has a non-zero shear modulus, the shear waves therein cannot be automatically excluded. However, shear waves can be functionally excluded by making the cross-sectional dimensions of the composite rod much smaller than the wavelength of interest of the lowest FP resonator, so that when a wave is incident from water, there is only a slight phase change in the cross-section of the rod, even with oblique angles of incidence, and so oblique waves are quite different from normal waves. It is known that at normal incidence, longitudinal waves incident on the liquid-solid interface cannot generate shear waves on the solid side. Based on the above, it is possible to obtain a Young's modulus E of the composite material instead of the effective bulk modulus B if the geometry of the rod can be adjusted using the free boundary conditions of the pressure relief on the cylindrical lateral surface of the rod and the transverse dimensions much smaller than the relevant wavelength in water_effAnd wherein only longitudinal waves propagate in the solid material. In particular, shear waves can be eliminated by limiting the lateral dimension of the composite rod to less than the wavelength corresponding to the resonant frequency of the resonator formed by the composite rod so that shear modes are not excited. Furthermore, the composite rods are separated from each other by a slight air gap, so that the pressure relief boundary conditions apply. The size of the air gap needs to be as small as possible, being the distance at which objects naturally touch each other, about 50 microns.

When the transverse dimension a of the absorbing material unit is deep sub-wavelength, i.e. a < lambda, only the longitudinal mode can couple to the propagating incident wave. The reason is the following dispersion relation:

wherein k is₀Is the wave vector, k, of the incident wave_||，k_⊥Representing the lateral and longitudinal components of the wave vector, respectively. Since the transverse component is related to the transverse mode, i.e. k_||∝(2mπ/a)²Where m is 0, 1, 2, 3, so that if m is 0, then k is_⊥＝k₀And the lateral component of the sound field is uniform, which is exactly the normal incident longitudinal propagation mode. If m.gtoreq.1, then a < lambda, so

k_⊥Is a pure imaginary number (evanescent wave). Since any shear wave means a non-zero k_||So that it cannot couple with the propagating longitudinal wave. Fig. 3 shows the reflected plane wave and the evanescent wave for different orders (m 1, 2).

Why is young's modulus, not bulk modulus, so important as longitudinal modulus? The reason is that even so-called "soft" rubber or polymer materials can have a bulk modulus in the GPa range. Therefore, it is impossible to obtain a small bulk modulus, which is advantageous for achieving impedance matching while obtaining a thin sound absorbing material, as described in the next section. The young's modulus is on the same order of magnitude as the shear modulus compared to the bulk modulus, so for soft materials the young's modulus can be one to four orders of magnitude smaller than the bulk modulus. Therefore, by using composite materials, such as metal and polymer composites, the Young's modulus of the mixture can be "tuned" by mixing different proportions of metal powder in the polymer matrix. At the critical volume fraction of metal (expressed as the "percolation" threshold), the metal powder can form an interconnectable metal network, and the shear/young's modulus of the composite will increase dramatically from that of the polymer matrix to that of the metal in the megapascal range. In other words, the young's modulus can be adjusted by the relative volume fractions of the metal powder and the polymer. Furthermore, as described in the following section, if a heavy mass density metal such as tungsten is used, it is possible to ensure a successful impedance matching with water while significantly reducing the wave velocity thereof.

4. Composite material mass density and Young modulus matched with water resistance

Since the longitudinal mode is the focus of the present invention, the longitudinal modulus M and the mass density ρ determine the characteristic impedance of the composite material

As mentioned in the previous section, the pressure relief boundary condition applies, i.e. the free boundary, if the rods made of composite material are separated from each other by a slight air gap. Thus, the effective bulk/longitudinal modulus is the same as the young's modulus (E ═ M ═ B)_eff). The characteristic impedance can be written as

At the same time, the impedance matching condition requires

Eρ＝B₀ρ₀. (6)

If the Young's modulus is written as follows, E ═ B₀A where α > 1 is a dimensionless quantity, B₀Is the bulk modulus of water, then to satisfy the impedance matching condition, the required density is determined as ρ ═ α ρ₀Where ρ is₀Is the density of water. The longitudinal wave velocity of the composite material with the density and the Young modulus is

Wherein c is₀Is the speed of sound in water. For a longitudinal wave of a given frequency, this means that the wavelength inside the impedance matching composite is reduced by a factor of a. When α > 1, an ultra-thin underwater absorber can be obtained by using a composite material matched to the water impedance. From the point of view of the causal limiting thickness (equation (2)), the parameter α will be the causal limiting thicknessFrom d_minReduced to d_min/α。

5. Design strategy for obtaining broadband absorption spectrum

5.1 Integrated Fabry-Perot resonator

A Fabry-perot (fp) resonator is a rod of length l and transverse subwavelength dimension a made of tungsten-polymer composite. Fig. 4 is a schematic diagram of a single FP resonator made of polyurethane-tungsten composite and an integrated unit composed of a plurality of FP resonators. The tungsten powder is selected to increase the mass density of the composite material. It is noted that the air gap separating adjacent FP resonator rods is critical in the design of the present invention. In the absence of an air gap, the longitudinal mode may couple with the transverse mode, as well as with an adjacent resonator. In addition, all the rods have a rigid backing, which acts as a reflective hard boundary for longitudinal waves. Thus, the acoustic displacement velocity v and pressure modulation p in a single FP resonator can be written as

Wherein E ═ B₀/α，ρ＝αρ₀And Z_c＝Z₀. The loss factor β is the viscoelastic loss coefficient of the composite. Speed of sound in composite materials

Specific velocity of sound in water

Slow. The surface impedance is defined by the ratio of p (z) and v (z) at z-0, which can be expressed as:

resonance frequency omega of ith resonator_iCan be obtained by the following formula:

where the resonance order m is 1, 2, 3. According to equation (10), one can obtain:

equation (11) gives the FP resonance frequency in the ith order resonator versus the rod length. Furthermore, for the resonant frequency of the ith resonator, the relationship of the rod length to wavelength can also be written as:

for simplicity, only first order resonances (m ═ 1) are considered in the following. The total impedance of the integrated resonator can be expressed as:

in the formula, alpha_i＝4dφΩ_iα/(πc₀N) is oscillation intensity, d, Ω_iAnd phi is the thickness, resonance frequency and length of the sample respectively_iThe porosity of the FP rod(s), N being the number of resonators. The porosity φ is defined as the ratio of the cross-sectional area of the composite rods to the entire cell. Since the air gaps can be very small, the porosity is close to 1, i.e.

5.2 Style design strategy

Consider an ideal case where the resonance has a continuous distribution, with a modal density per unit frequency of D (ω), defined as:

in this case, equation (13) can be converted into an integral form,

after defining μ (ω) ═ α (ω) D (ω), one can obtain,

where δ (ω)²-Ω²) In the form of Dirac functions, or written as

By combining equation (14) and equation (17), a simple differential equation can be obtained

Wherein

According to the requirement Z (Ω) ═ Z₀The solution is

Therefore, the ideal resonance frequency distribution can be predicted by the power law.

COMSOL simulation results and related analysis

To validate the design strategy of the present disclosure, a numerical simulation study was conducted using the Comsol Multiphysics commercial finite element software.

6.1 discretization of the resonant frequency

Discretization of the resonance frequency is very important, since in practical cases an infinite number of resonance frequencies are not possible within a limited frequency range. Consider 9 FP resonators as a unit with length l_i(i ═ 1, 2.., 9) and the associated resonance frequency Ω_iThe formula (i ═ 1, 2., 9) can be given by formula (11). To calculate the 9 lengths of the FP resonators, the cut-off frequency f_cIs set to 5000Hz (omega)_c＝2πf_c) This is a relatively low frequency in water acoustics. Since the speed of sound in water is about 1500m/s, the wavelength is about 30cm at 5000 Hz. Next, frequency discretization is performed according to the following formula:

where N is 9 and phi is 0.81. The side length a of the square cross section is 5 mm. According to the relation a²/(b/3)²Phi, the size of each unit is

6.2 setting of relevant parameters and boundary conditions

The Comsol modeling is divided into two areas, which are defined and simulated by a pressure acoustic module and a solid mechanics module respectively. In the acoustic module, the medium is water, having a speed of sound c₀1500m/s and water mass density ρ₀＝1000kg/m³. In the solid mechanics module, the Young's modulus of the composite material is set to

Wherein the imaginary part introduces a loss factor of the composite material. Further, assume that the Poisson's ratio is 0.49998 and the density is 5 ρ₀. In the region of the pressure-acoustic module, the boundary conditions are: the two sides are periodic conditions, and the top is plane wave radiation conditions. At the interface of water and composite, the boundary condition is acoustic-elastic fixationBulk coupling conditions, where the pressure wave can propagate to/reflect from a solid, while the bottom of each rod is supported by rigid walls, corresponding to the solidary boundary in solid mechanics. The cylindrical side of the FP-bar is exposed to the air in the gap, set as a pressure relief/free boundary condition, which is crucial for the decoupling of wave propagation in different cubes/resonators. It should be noted that the acoustically rigid boundary in the present invention is optional. In case the gap between the FP rods is small enough, the effect of the gap boundary is negligible. Therefore, the technical effect of the present invention can be achieved without providing a rigid boundary.

6.3 Acoustic absorption Spectroscopy

The simulation results are shown in fig. 5, and it can be seen from the results that the absorption rate of the design is generally above 95%, and is very flat above the designed cut-off frequency. Local small oscillations in the absorption spectrum are due to only a limited number of first-order resonance frequencies, and to coupling between evanescent waves in different resonators. In this case, if the resonator length, i.e. the length of the longest FP rod, at the lowest cut-off frequency is taken to be 15mm, or about 1/20 wavelengths (corresponding to the lowest resonance frequency). The causal limit sample thickness calculated from equation (2) and the simulated absorption spectrum was 6.7 mm. The thickness of the design is very close to the limit minimum thickness if an average of 7.05mm of resonator length is used.

7. Design of experiments

7.1 preparation of Metal-Polymer composites

Having E-B as discussed in the previous section₀α ρ and ρ ═ α ρ₀A composite material of properties is defined as a soft impedance matching material. To experimentally achieve the desired properties, tungsten (W) and Polyurethane (PU) resin/rubber were selected as raw materials for composite preparation, the properties of which are shown in table 1. Tungsten has a high mass density and a large Young's modulus, while Polyurethane (PU) has a density close to that of water (1 g-cm)^-3) Young's modulus is lower than the bulk modulus of water (2.25 GPa). By mixing tungsten powder with PU and adjusting the volume ratio x, one chi can be expected to exist_cSo that the impedance matches condition Z_c＝Z₀Is satisfied.Because the tungsten powder is wrapped in the PU matrix, the X shape is only needed_cBelow the percolation threshold, the young's modulus of the composite is still of the same order of magnitude as that of PU. That is, E ═ B₀α ρ and ρ ═ α ρ₀The object of (. alpha. > 1) is thus achieved.

In the preparation of the sample, tungsten powder with the diameter of 45 microns, polyurethane resin and polyurethane rubber are selected as raw materials. The properties of W and PU are shown in Table 1. Because the Young's moduli of the polyurethane resin and the polyurethane rubber differ by three orders of magnitude, the polyurethane resin and the polyurethane rubber can be mixed in different proportions so that the Young's moduli are within a desired range. And then, uniformly mixing the polyurethane mixture and the tungsten powder through mechanical stirring, and finely adjusting the Young modulus by combining mass density to finally realize impedance matching between the composite material and water. After mixing tungsten powder and a polyurethane resin-rubber mixture according to different volume ratios, when the volume ratio of the tungsten powder is more than 30%, the tungsten powder is difficult to be uniformly mixed due to small proportion of high polymer materials, high viscosity and the influence of large surface active energy of the tungsten powder. When the volume ratio of tungsten is changed between 20% and 28%, the density of the composite material is about 4-5 g-cm^-3。

Table 1: material Properties of tungsten and polyurethane used in the experiment

Material	Density rho (g cm)^-3)	Young's modulus E (GPa)
			Tungsten powder (W)	19.35	411
Polyurethane resin	1.07	1.01
			Polyurethane rubber	1.09	6.83×10^-3

7.2 design of design

The preparation of the sample is mainly divided into the following steps, as shown in fig. 6. Step S1, mixing tungsten powder and polyurethane in a certain proportion. Step S2, pour the mixture into a mold. Step S3, a metal plate is attached to each FP rod using a waterproof adhesive. In step S4, the cured product is cured at room temperature for about 9 hours. And step S5, demolding.

Among them, the following points should be noted:

1. in order to make the demoulding more convenient, a soft material with lower Young modulus than that of the composite material is adopted to manufacture a mould, such as tin-catalyzed silica gel;

2. the purpose of the metal plate is to act as an acoustically hard border;

3. the thickness of the waterproof glue is less than 50% of the air gap between the FP rods.

The basic unit of the experimental sample was an array of 3 × 3 FP pillars. The overall planar dimensions of the sample were 1m × 1m, consisting of 16 identical subunits, each subunit consisting of a plurality of elementary units. The entire 1m × 1m sample can be composed by combining 16 subunits together.

It will be understood that the above embodiments are merely exemplary embodiments taken to illustrate the principles of the present invention, which is not limited thereto. It will be apparent to those skilled in the art that various changes and modifications can be made therein without departing from the spirit and scope of the invention, and these are to be considered as the scope of the disclosure.

Claims

1. An underwater sound absorber comprises a plurality of FP resonators,

wherein each FP resonator is a rod made of a composite material, the plurality of FP resonators are spaced apart from each other by an air gap, and the cross-sectional dimension of each FP resonator is set small enough so that the effective bulk modulus of each FP resonator is the same as its Young's modulus.

2. The underwater sound absorber of claim 1, wherein the young's modulus E ═ B of the composite material₀A, wherein B₀Is the bulk modulus of water, alpha is a constant greater than 1,

the density rho ═ alpha rho of the composite material₀Where ρ is₀Is the density of water.

3. The underwater sound absorber of claim 2, wherein the composite material includes a metal and a polymer, the volume fractions of the metal and the polymer being set such that the impedance of the composite material matches the impedance of water.

4. An underwater sound absorber as claimed in claim 3 wherein α > 1.

5. The underwater sound absorber of claim 3, wherein the metal is tungsten.

6. The underwater sound absorber of claim 5, wherein the polymer comprises polyurethane.

7. The underwater sound absorber of claim 6, wherein the polymer comprises a mixture of polyurethane resin and polyurethane rubber.

8. The underwater sound absorber of any one of claims 1 to 7, wherein the bottom of each FP resonator is supported by a rigid fully reflective substrate.

9. The underwater sound absorber of claim 1, wherein a lateral dimension of each FP resonator is less than a wavelength to which a resonant frequency of the FP resonator corresponds.

10. The underwater sound absorber of claim 1, wherein the air gap has a dimension of 50 microns.