On the Use of Aerogel As a Soft Acoustic Metamaterial for Airborne Sound

On the use of aerogel as a soft acoustic metamaterial for airborne sound

Matthew D. Guild,∗ Victor M. Garc´ıa-Chocano, and JoséSánchez-Dehesa† Wave Phenomena Group, Departamento de Ingenier´ıa Electrónica, Universidad Politécnica de Valencia, Camino de vera s/n (Edificio 7F), E-46022 Valencia, Spain

Theodore P. Martin, David C. Calvo, and Gregory J. Orris U.S. Naval Research Laboratory, Code 7160, Washington DC 20375, USA (Dated: August 16, 2016) Soft acoustic metamaterials utilizing mesoporous structures have recently been proposed as a novel means for tuning the overall effective properties of the metamaterial and providing better coupling to the surrounding air. In this work, the use of silica aerogel is examined theoretically and experimentally as part of a compact soft acoustic metamaterial structure, which enables a wide range of exotic effective macroscopic properties to be demonstrated, including negative density, density- near-zero and non-resonant broadband slow sound propagation. Experimental data is obtained on the effective density and sound speed using an air-filled acoustic impedance tube for flexural metamaterial elements, which have previously only been investigated indirectly due to the large contrast in acoustic impedance compared to that of air. Experimental results are presented for silica aerogel arranged in parallel with either 1 or 2 acoustic ports, and are in very good agreement with the theoretical model.

PACS numbers: 43.20.+g, 43.28.+h, 43.58.+z Keywords: acoustic metamaterials, silica aerogels, negative dynamical density, slow sound

I. INTRODUCTION and exciting phenomena associated with effective properties that are near zero, particularly those associated with Acoustic metamaterials have received interest in re- extraordinary transmission, which can be achieved when cent years by enabling macroscopic physical characteris- either the effective density or wave speed approaches tics which cannot be obtained with traditional materials, zero[4, 5]. In the case of a density-near-zero material, such as negative, near-zero or anisotropic dynamic ef- the effective wave speed increases dramatically and leads fective fluid properties. Acoustic metamaterials are able to a quasi-static field within a given structure, which to achieve such previously unattainable exotic properties can exhibit a supercoupling effect through long narrow through the careful design of the microstructure, which channels[6–8]. At the opposite extreme, there are also in- create microscale dynamics that result in the desired teresting effects which arise as the effective acoustic wave macroscopic properties. The reader is addressed to the speed approaches zero, which is referred to as slow sound, recent reviews on this topic where one can find many of the analogue of slow light in optics. Previous demonstra- the exciting applications of acoustic metamaterials[1, 2]. tions have utilized resonant effects using either sonic crys- Until recently, such acoustic metamaterials have relied tals or detuned resonators[9–11], resulting in slow sound on materials which are much harder than the surround- that occurs over a relatively narrow bandwidth. A novel ing fluid medium, often treated as acoustically rigid or application of slow sound propagation was recently pro- nearly-rigid structures for airborne sound. Alternatively, posed for the improved design of acoustic absorbers by soft acoustic metamaterials utilizing mesoporous struc- Groby et al.[12], in which slow sound in large slits filled tures have been proposed as a novel means for the mold- with absorptive foam were used to significantly increase ing and tuning of the overall properties of the resulting the low frequency absorption in air. metamaterial, while simultaneously providing better cou- One of the fundamental aspects that gives a meta- pling with the acoustic environment around it [3]. Build- material its exotic macroscopic properties is the ho- ing upon this concept, the use of silica aerogel as part mogenization of the microstructure, which has recently of a compact and conformal soft acoustic metamaterial been explored for elastic and flexural metamaterial structure is examined theoretically and experimentally, components[13, 14]. This is particularly important be- arXiv:1509.08378v1 [cond-mat.mtrl-sci] 28 Sep 2015 yielding an interesting suite of useful yet exotic proper- cause the effective macroscopic properties of an acoustic ties. metamaterial can be significantly different than those of In addition to extremely large and/or negative dy- the constituent microstructural elements. When there is namic properties, there are a wide range of interesting open flow through the structure, such as a sonic crys- tal lattice[15, 16] or transmission-line arrangement of Helmholtz resonators [17], it is relatively straightforward ∗ [email protected]; Current address: NRC Re- to extract effective properties experimentally due to the search Associateship Program, U.S. Naval Research Laboratory, relatively low acoustic impedance. Code 7160, Washington DC 20375, USA A formidable challenge, however, arises when obtain- † [email protected] ing the effective properties of a metamaterial sample con- 2 taining elastic elements, which have acoustic impedances are made possible by its unique microstructure. One that are orders of magnitude greater than the surround- of the most common types of aerogels, silica aerogel, ing fluid and at frequencies well below those typically consists of a high-porosity frame made of fused silica used to obtain acoustic properties through direct time- nanoparticles. The most notable characteristic of sil- of-flight measurements. As a result, previous works in ica aerogel is its extremely low static density which is air have either been restricted to theoretical and nu- directly related to the very high porosity of the struc- merical evaluation[4, 7, 18–20] or limited to an indirect ture, making it much closer to that of air compared with comparison of the metamaterial properties using experi- any other type of elastic solid. Due to the nanoscale mental results for the reflected and transmitted pressure pore size, however, the air is locked in place by vis- field[5, 21]. In this work we make the significant step cous effects producing a higher acoustic density than of experimentally extracting the effective dynamic prop- that compared to typical porous media used in acous- erties (density and sound speed) of these flexural meta- tic applications[26]. Furthermore, the small cross-section material elements, which has to the authors knowledge connecting the fused nanoparticles results in a very low never been accomplished previously for such a metama- elastic stiffness, compared with a rigid silica structure terial structure in air. of the same porosity[27]. This combination gives a rela- It is expected that the elasticity of the materials defin- tively low acoustic impedance (for an elastic solid), and ing the metamaterial structure might play a fundamental in particular yields an exceptionally low flexural wave role in order to understand the phenomena observed in speed, making it ideal for use as a subwavelength flexu- sound transmission and reflectance through the channels ral element for airborne sound. defined by the structure. In fact, the role of the elastic When the wavelength is much larger than the mi- properties is paramount for the case of structures embed- crostructure, negative effective properties are achieved ded in water, as it has been recently demonstrated [22]. via control of the microstructure arrangement and the Although some recent work has begun to incorporate the resulting dynamics. This feature in the microstructure elastic effects into the metamaterial structure, many of design is typically achieved with two main types of ar- these designs continue to have the primary dynamic ele- rangements: either as a mass-spring system, or in a ment consisting of mass-spring resonators which are af- transmission-line consisting of mass and stiffness ele- fixed to an elastic plate as structural support[19, 20]. ments. The mass-spring systems, which demonstrate ex- Alternatively, soft acoustic metamaterials represent a treme effective mass and stiffness in the vicinity of the paradigm shift beyond this framework by creating the mass-spring resonance, are therefore referred to as lo- dynamics from the structure itself. It is important to cally resonant acoustic metamaterials (LRAMs) [5, 28– emphasize that such a soft acoustic metamaterial, real- 32]. Although such mass and spring elements can be ized with the unique properties of aerogels, can be tai- arranged in a compact configuration and are relatively lored to obtain a wide spectrum of desirable exotic prop- simple and robust, the resulting extreme effective properties in a single versatile subwavelength acoustic meta- erties are inherently narrowband and subject to appre- material element. In this work, theoretical and experi- ciable loss due to the close proximity of the mass-spring mental results for a compact metamaterial configuration resonance[8, 33]. Alternatively, acoustic metamaterials are presented, enabling a thin, conformal configuration have been proposed using thin elastic plates as a means to be realized. In particular, these structures represent for operating as a positive stiffness element in acous- a soft acoustic metamaterial, which are realized using tic transmission-line arrangements [18], which has re- the flexural resonance of the zeroth-order anti-symmetric cently been applied to acoustic metamaterial leaky-wave Lamb-wave mode in silica aerogel disks. In addition to antennas[34]. While the unit cell of such an arrangement its sub-wavelength thickness, extreme effective properties is much smaller than a wavelength, the entire configura- are demonstrated across a broad range of the operating tion requires many elements in series and can result in a bandwidth, with distinct regions exhibiting negative den- very long structure relative to the wavelength. sity, density-near-zero, and ultra-low sound speeds. In this work we have employed hydrophobic silica aerogel which has a static density of 107 kg/m3 and an (opti- cal) refractive index of 1.03. This silica aerogel has a high II. BACKGROUND ON AEROGELS resistance for water and moisture due to its hydropho- bicity, a feature allowing the disks to be fabricated using water jet cutting techniques. Their properties are stable While the exceptional thermal properties of aero- in any climate, a feature contributing the high reliability gel have led to a revolution in thermal insulation for the analysis provided below. applications, utilization of the unique acoustic char- acteristics have been minimal, primarily relating to marginal improvements of existing concepts such as quarter-wavelength impedance matching or ultrasonic III. THEORETICAL FORMULATION absorbers[23–25]. However, aerogels offer several unique features that enable it to function as a subwavelength In many acoustic metamaterial configurations, includ- flexural element in a soft acoustic metamaterial, which ing LRAMs and acoustic transmission line arrangements, 3 thin elastic disks and membranes are used as sub- As observed from Eq. (1), the resonance frequency de- wavelength stiffness elements. This ubiquitous imple- notes a critical point in the effective acoustic properties, mentation arises from the fact that the time-harmonic and even though Eq. (9) provides the exact solution, it displacement of an elastic disk is proportional to the flex- must be solved numerically. To better understand the ural stiffness in the quasi-static (low frequency) limit, and relationship between the material properties and dimen- therefore yields an analogous inductive behavior equiva- sions of the plate and the resonance frequencies, an ap- lent to that of a mechanical spring. In general, however, proximate analytic expression is sought. To proceed, it the specific acoustic impedance of a lossless elastic plate will be assumed that the flexural wavenumbers δ1 and can be expressed in terms of the mass per area Mplate δ2 are sufficiently high that the large argument approx- and compliance Cplate as[35] imations can be used for the Bessel and modified Bessel 1 functions can be used, namely, [39] Zplate = jωMplate + jωCplate J1(x) π I1(x) ω 2 ≈ tan(x − ), ≈ 1. (10) = jωM 1 − res ≡ jω M (ω), (1) J0(x) 4 I0(x) plate ω eff −1/2 Even in the case of large wavelengths relative to the where ωres = (MplateCplate) is the angular resonance thickness of the plate (and therefore in the low frequency frequency of the plate. Written in this form, it is ap- limit in terms of the acoustic waves in the surrounding parent that the stiffness-controlled response of the elas- fluid), the flexural modes occur at frequencies where the tic plate can equivalently be treated as a frequency- wavelengths are on the order of the diameter of the disk dependent effective mass, Meff , which is negative for or less, suggesting that the assumption above is reason- ω <ωres. able for the case of flexural acoustic metamaterial ele- For canonical shapes and idealized boundary condi- ments being investigated in this work. Note that be- tions, analytic expressions have been developed to de- cause the ratio of the modified Bessel functions given in scribe the flexural wave motion of elastic solids. For the Eq. (10) is approximately equal to unity, the resonance flexural motion of thick plates, effects from shear defor- frequency under these conditions only depends on δ1. mation and rotational inertia become important, which Application of the approximate expressions in Eq. (10) can be formulated using Mindlin theory[36]. For flexural to Eq. (9) yields waves in a thick circular elastic disk, the modal displacement, w of the plate is given by[37, 38] n tan(δ1) ≈ 0, (11) h r r i wn(r, θ) = A1Jn(δ1 ) + A2In(δ2 ) cos(nθ), (2) a a for which δ1 =mπ with m being a non-zero integer. Equa- h i tion (3) can therefore be written as 2 1 4 p 2 −4 δ1 = λ (R + S) + (R − S) + 4λ , (3) 2 √ 1 h i 1 4 h −2 i 2 δ2 = λ4 p(R − S)2 + 4λ−4 − (R + S) , (4) λ (R + S) + 2λ 1 + ∆ = (mπ) , (12) 2 2 2 1 h2 where R = , (5) 12 a 2 2 1 4 2 3 2 2 D 1 h ∆ = λ (R − S) = (kph) − 1 , (13) S = = , (6) 4 144 (1 − ν)κ2 µh(κa)2 6(1 − ν)κ2 a 4 2 ρha ω with kp = ω/cp denoting the compressional plate wave λ4 = , (7) D number with plate wave speed, 3 Eh s D = , (8) E 12(1 − ν2) c = . (14) p ρ(1 − ν2) where Jn and In denote the Bessel function and modified Bessel function of the first kind, a is the plate radius, h Typical values of compressional plate wave speeds are is the plate thickness, ρ is the mass density, E is Young’s usually orders of magnitude higher than flexural wave modulus, µ is the shear modulus, ν is Poisson’s ratio, √ speeds, and thus the corresponding wavenumbers are ω is the angular frequency and κ = π/ 12 is the shear much lower. As a result, one expects that k h 1 and correction factor. Note that for axisymmetric loading, p likewise ∆ 1, in which case Eq. (12) can be simplified such as that encountered in an acoustic impedance tube, to give an expression for the flexural resonance frequency the only non-zero mode is n=0. of the mth mode, Due to the clamped boundary conditions corresponding to w(a) = w0(a) = 0, the characteristic equation for 1 cph 1 hp i the flexural resonance frequencies is given by f (m) = √ 1 + 2(mπ)2(R+S)−1 . res 4π 3 a2 R+S J1(δ1)I0(δ2) + J0(δ1)I1(δ2) = 0. (9) (15) 4

In addition to the flexural resonance frequency, the particular solution to the displacement is necessary to calculate the effective acoustic properties of the flexural disk. Assuming a time harmonic pressure P applied across the face of the disk and assuming clamped edges, the displacement becomes r r P I1(δ2)J0(δ1 a ) + J1(δ1)I0(δ2 a ) w(r)= 2 − 1 . (16) ρhω I1(δ2)J0(δ1) + J1(δ1)I0(δ2) Although Eq. (16) gives an expression for the displacement at any given radius r, it is actually the ensemble of the displacement over all the points on the surface which will be measured via the reflected or transmitted acoustic waves at some distance from the disk. Therefore, a more useful quantity is the spatial average of the displacement, which can be obtained from Eq. (16) such that

" 1 1 # 2( + )J1(δ1)I1(δ2) P δ1 δ2 wavg = 2 − 1 , (17) ρhω I1(δ2)J0(δ1) + J1(δ1)I0(δ2) from which one can obtain the average acoustic impedance for a thick clamped circular plate

jωMplate FIG. 1. (Color online) Normalized plots versus frequency for Zplate = h i =jωMeff , an aerogel disk (solid red line), an acoustic port with a rigid 1−2( 1 + 1 ) J1(δ1)I1(δ2) δ1 δ2 I1(δ2)J0(δ1)+J1(δ1)I0(δ2) baffle (dash-dotted black line) and an aerogel disk combined (18) in parallel with an acoustic port (dashed line) for (a) effective where Mplate = ρh is the acoustic mass of the plate. Al- density relative to that of a baffled acoustic port, and (b) though not as obvious as the form presented in Eq. (1), effective sound speed relative to that of the surrounding fluid the expression presented in Eq. (18) also yields a nega- medium. The shaded regions denote broadband regions of tive effective mass below the first flexural resonance of extreme effective properties, namely (a) negative density, and the elastic plate, which is illustrated by the red solid line (b) non-resonant slow sound propagation. An illustration of the parallel arrangement of an aerogel disk with an acoustic in Fig. 1(a) for the case of a circular silica aerogel disk. port is shown in the inset of (b). The resulting negative effective density extends over the entire range below resonance and approaches −∞ as the frequency goes to zero. arranged in parallel with an acoustic port (black dash- Such a highly dispersive and divergent behavior is not dotted line) to obtain the effective density denoted by the ideal, particularly for broadband applications or if any dashed line. The effective sound speed of the combined type of acoustic impedance matching is desired. How- parallel arrangement follows that of the acoustic port at ever, this effective mass can be readily modified by plac- low frequencies, slowly decreasing towards zero before ing a positive acoustic mass (such as an acoustic port con- increasing again, producing a broad non-resonant region sisting of an air-filled hole in the support ring) in parallel where slow sound propagation occurs (denoted by the with the negative dynamic mass of the plate. The result- shaded region), slower than even that expected for the ing frequency dependence on the effective mass density case of a acoustic port. As the frequency increases, the is presented in Fig. 1(a), in which a circular disk (red effective sound speed becomes dominated by the elastic solid line) is combined in parallel with an acoustic port disk in the vicinity of the flexural resonance of the elastic (black dash-dotted line) to obtain the effective density plate, which occurs at f = f . Near this resonance, a denoted by the dashed line. This parallel configuration res large increase in the effective sound speed is observed as of the aerogel disk and acoustic port is illustrated in the the effective density passes through zero, with the peak inset of Fig. 1(b). Arranged in such a manner, the acous- value limited by the losses in the system. tic port will short circuit the plate as it approaches extremely large values, and allow for the magnitude and bandwidth of the negative dynamic mass (denoted by the shaded region) to be controlled using the same plate IV. EXPERIMENTAL SETUP and only varying the size and number of ports. A similar trend is observed for the effective sound Previous approaches for the acoustic characterization speed, which is illustrated in Fig. 1(b). As in Fig. 1(a), of silica aerogels have been based on ultrasonic time-of- the results for a circular elastic disk (red solid line) is flight measurements, which have produced experimental 5

FIG. 2. (Color online) (a) Diagram of the experimental setup using an air-filled acoustic impedance tube, and (b) a photo of a silica aerogel sample examined in this work. data for the compressional and shear properties of sil- of the effective complex acoustic properties of the acous- ica aerogels [40–43]. Unfortunately, such time-of-flight tic metamaterial sample [16, 47, 48]. The method for the measurements do not allow for one to examine the silica extraction of these properties can be found in previous aerogel as part of the metamaterial structure, requiring works [48], based on the complex reflection and transmis- the use of acoustic waves with wavelengths which are sion pressure coefficients, which are given by much smaller than the thickness of the sample for ac- −1 curate time-of-flight characterization. Although most of 1 Ztot 1 Ztot Rtot = 1 + . (19) these previous investigations involved compressional and 2 Z0 2 Z0 shear waves, Rayleigh surface waves have also been examined, which were observed to be less than 50 m/s, significantly lower than the compressional wave speed[44]. jω −1 Likewise, as can be seen from Fig. 1, the effective prop- Ttot = 1 + Meff +Mport , (20) 2Z0 erties due to the flexural behavior of the aerogel disks at large wavelengths (low frequencies) relative to the size where Ztot is the total acoustical impedance seen at the of the disk are dramatically different than those of the face of the sample, Z0 is the acoustical impedance of air, static density and compressional wave speed obtained via and Meff and Mport are the effective acoustic mass of the time-of-flight measurements for silica aerogel [40]. aerogel disk and port, respectively. Alternatively, an experimental setup was needed that This inverse process of using spectral acoustic measure- could examine the effective acoustic behavior of the en- ments to obtain the complex-valued acoustic properties semble arrangement including the silica aerogel disk, at has presented significant challenges, including ambiguity frequencies which were sufficiently low to be within the in identifying unique solutions [47] and a high sensitiv- homogenization limit for use in the microstructure of an ity in the extracted properties to even low levels of noise acoustic metamaterial. The experimental investigation in the spectral data obtained from the microphones [48]. of the silica aerogel samples in this work was performed Although these challenges have recently been overcome using an air-filled acoustic impedance tube. The exper- for samples such as sonic crystals[16, 48] with a relatively imental setup is illustrated in Fig. 2(a), which shows a low acoustic impedance relative to the surrounding air, standard 4-microphone configuration [45] for the mea- acoustic metamaterials consisting of solid elastic struc- surement of acoustic properties of a given sample. The tures, even relatively soft ones such as silica aerogels, acoustic properties, namely the complex-valued acoustic with acoustic impedances hundreds of times larger than impedance and wavenumber, were obtained as a function that of air present unique challenges using this technique. of input frequency from spectral measurements of the As a result of this high acoustical input impedance magnitude and phase of both the reflected and transmit- from the sample, the small but finite leakage from the ted acoustic pressure obtained from the 4 microphones. impedance tube becomes a significant source of error While this particular type of acoustic apparatus has in the measurement and must be accounted for. Typ- been utilized for many decades, such work has tradi- ically, this leakage has been observed as a result of im- tionally focused on simply measuring transmission loss proper sealing and mounting of the sample, with leakage through an absorptive sample [46]. In the last sev- of acoustic energy from the reflected side of the sam- eral years, this technique has been expanded to acoustic ple to the transmitted side. However, in the case of metamaterials, with a particular emphasis on extraction rigidly mounted high-impedance samples, this leakage is 6 primarily due to the microphones, which consist of thin V. RESULTS diaphragms and small acoustic ports for pressure equal- ization, with the main source of this leakage occurring on The experimental data for the reflectance are shown in the reflected side of the sample. As a result, two distinct Fig. 3(a)-(c), with the corresponding transmittance data differences arise for this case of microphone pressure leak- shown in Fig. 3(d)-(f), for the nominal aerogel sample age compared with that due to improper mounting: (1) plus with 1 and 2 acoustic ports, respectively. These the leakage occurs in the reflected pressure measurements measured values are relatively constant with frequency only, with negligible effects on the transmitted side, and except in the vicinity of 1500 Hz, where there is a rapid (2) for large impedances, the leakage should be indepen- decrease in reflectance with a corresponding increase in dent of the specific sample or its particular mounting in the transmittance, due to the flexural resonance of the the impedance tube. circular aerogel disk. This anomalous increase in the Under these conditions, the total acoustical impedance transmission of acoustic energy in the experimental data seen at the front of the sample can be written as far exceeds that expected based on quasi-static homoge- −1 nization theory for such a large impedance contrast with φplate φport φleak Z = + + , (21) the surrounding air, and corresponds to a region of ex- tot Z Z Z plate port leak traordinary transmission. where φ denotes the filling fraction of each component, Based on the measured data, the flexural resonance Zport is the impedance of the acoustic port including ther- frequency is obtained, and the Young’s modulus of the mosviscous effects [49], and Zleak is the pressure leakage aerogel sample can be obtained based on the theoretical impedance. expression for the resonance frequency given by Eq. (15). To determine the leakage impedance, one can consider The measured properties of the aerogel sample examined the measurement of a rigid or nearly-rigid sample within this work are tabulated in Table I. While these mea- out any acoustic ports, in which case the total impedance sured values are somewhat lower than some other similar is simply Ztot = Zleak/φleak. In the absence of the pres- aerogel samples[40], the results reported here fall within sure leakage, all the sound should be reflected and the the accepted range of measured aerogel properties[27]. reflectance should be unity; however, in the presence of From these measured values, an elastic model based the leakage, the measured reflection coefficient for a rigid on the theoretical formulation presented in Sec. III was or nearly-rigid sample, Rmeas, will exhibit a reduction calculated and compared to the data. Modeled results in the magnitude (and a phase difference as well). An obtained without accounting for the pressure leakage ef- expression for the leakage effects can be obtained using fects are denoted by the dashed line in Fig. 3 and Fig. 4. Eq. (19) in terms of the measured complex reflection co- While there is excellent agreement with the transmit- efficient Rmeas as tance shown in Fig. 3(d)-(f), the theoretical reflectance Z 2Z R shown in Fig. 3(a)-(c) predicts unity away from the flex- leak = 0 meas . (22) φ 1 − R ural resonance, a value which is not observed in the mea- leak meas surements. While this difference between the modeled Samples consisting of silica aerogel disks measuring and measured values represents less than a 10% error 2.43 cm in diameter and 1.1 cm thick were tested in a over most of the frequency band under investigation, this 3.5 cm diameter air-filled acoustic impedance tube, as leads to a significant variation in the resulting extracted pictured in Fig. 2(b). The acoustic impedance tube used effective acoustic properties, as illustrated in Fig. 4. This in this work consists of a circular tube having an inner is particularly the case below the flexural resonance fre- diameter of 3.5 cm. To mount the silica aerogel disks quency, for which the variation between the theoretical in the impedance tube, wooden rings were machined to model without accounting for the leakage differs by up support the aerogel disks and provide an acoustic baf- to an order of magnitude from that observed in the ex- fle. Due to the brittleness of the aerogel disks, the perimental data. holes were drilled into the wooden ring to create the Accounting for the leakage through the use of Eqs. (21) acoustic ports, each with a diameter of 1 mm. To ob- and (22), this observed difference in the reflectance can tain the acoustic characterization of the aerogel samples, broadband noise is generated using an electromechani- cal driver at one end of the tube, and measured using 3 0.50 inch (1.27 cm) diameter G.R.A.S. condenser micro- fres (Hz) E (MPa) ρ (kg/m ) 1420 0.569 107 phones. The microphones are arranged in a standard 4-microphone configuration[45], allowing for the reflection and transmission pressure coefficients to be directly TABLE I. Measured values for the flexural resonance fre- determined using a transfer-matrix technique[46]. From quency, Young’s modulus E and density ρ for the silica aerogel this set of measurements, the complex impedance and examined in this work. The loss factor of the aerogel was ob- wavenumber were obtained for the range 300–2000 Hz, served to be 0.005 based on the reflectance and transmittance in a similar manner to previous experimental work on data presented in Fig. 3. The Poisson’s ratio was estimated acoustic metamaterial samples[16, 48]. to be 0.21 based on Gross et al.[40]. 7

FIG. 3. (Color online) Comparison of experimental data with the theoretical model presented in Sec. III for the acoustic reflectance (a)-(c), and transmittance (d)-(f). Experimental data is denoted by circles, and the theoretical results are presented with and without corrections due to the pressure leakage through the microphones, denoted by the solid and dashed lines, respectively. be correctly modeled, as shown by the solid lines in Fig. 3 creases. These effective properties, which arise from use and Fig. 4. In addition to the improved agreement in the of the flexural motion of the silica aerogel as a “hidden reflectance presented in Fig. 3(a)-(c), excellent agreement degree of freedom”[50] in the otherwise 1D planar ar- is also maintained with the transmittance in Fig. 3(d)- rangement of the acoustic impedance tube, lead to these (f). The theoretical model including pressure leakage and extreme effective properties which different greatly from the measured data of the extracted effective mass den- the static properties of silica aerogel. sity and sound speed are in excellent agreement, with the Likewise, the combination of the negative effective den- model capturing the correct magnitudes and frequency- sity of the silica aerogel disk with the positive effective dependence of these dynamic properties. Even with the density of one (or more) acoustic ports yields much more limitations of the experimental apparatus due to the fi- uniform and less dispersive regions of negative density, nite acoustical impedance of the microphones and the which are illustrated in Fig. 4(b) and (c) for the case of resulting acoustic pressure leakage, one is still able to 1 and 2 acoustic ports, respectively. As described above, observe an extremely large dynamic range of extracted the parallel arrangement of the silica aerogel disk and the acoustic properties, on the order of thousands of times acoustic ports leads to the effective properties being dom- that of the ambient air. inated by the acoustic port at lower frequencies, leading In addition to the agreement between the experimen- to this change in the effective density as a function of tal data and theoretical model, the results presented in frequency over this range below the flexural resonance Fig. 4 provide valuable information regarding the wide frequency. In the vicinity of the flexural resonance, how- range of effective properties which can be attained using ever, the effective properties are dominated by the silica the soft acoustic metamaterial arrangement illustrated in aerogel disk, and therefore the same density-near-zero re- Fig. 1. In particular, it can be observed that the overall gion is observed. trends as a function of frequency follow those predicted in Similarly, the effective sound speed is presented in Fig. 1(a) and (b) for silica aerogel disks with and with- Fig. 4(e) and (f) for the case of silica aerogel with 1 and 2 out parallel arrangements with acoustic ports, for the acoustic ports, respectively. In the vicinity of the flexural effective mass density and sound speed, respectively. In resonance, a similar spike in the effective sound speed is Fig. 4(a), the effective density for the data obtained for observed (corresponding to the effective density passing the silica aerogel disk is observed to be positive above the through zero) as was observed for the silica aerogel disk flexural resonance frequency (around 1500 Hz), passing without any acoustic ports. In addition to this, a dip in through zero, and then negative below the flexural reso- the effective sound speed is observed well below the flex- nance frequency and tending towards −∞ as frequency ural resonance, leading to a region of zero and near-zero decreases. Conversely, the effective sound speed shown effective wave speed in the acoustic metamaterial struc- in Fig. 4(d) increases significantly as the effective den- ture. Previous experimental investigations of slow sound sity passes through zero near the flexural resonance fre- have been observed over relatively narrow bands due to quency, and decreases towards zero as the frequency de- the resonant physical mechanisms employed, whereas the 8

FIG. 4. (Color online) Comparison of experimental data with the theoretical model presented in Sec. III normalized relative 3 to the properties of air (ρ0 = 1.21 kg/m , c0 = 343 m/s), for the eﬀective density (a)-(c), and eﬀective sound speed (d)-(f). Experimental data is denoted by circles, and the theoretical results are presented with and without corrections due to the pressure leakage through the microphones, denoted by the solid and dashed lines, respectively. The arrows denote extreme values, namely the peak negative density in (b) and (c), and near-zero slow sound propagation in (e) and (f).

results presented in this work appear to be the first to ex- of a soft acoustic metamaterial. It has been shown perimentally demonstrate a broad region of non-resonant that the combination of the flexural motion of the sil- slow sound propagation. ica aerogel combined with the acoustic mass of one or Similar trends in the data described above were ob- more ports leads to a configuration with broadband nega- served for the silica aerogel disk combined with either 1 tive dynamic density, density-near-zero regions and non- or 2 acoustic ports. One interesting point of distinction resonant broadband slow sound propagation. The use that can be noted is that the addition of more acous- of silica aerogel as part of a soft acoustic metamaterial tic ports (and thus a larger, positive effective acoustic structure with subwavelength thickness was examined mass) results in an increase in the frequency at which theoretically and experimentally. Significant challenges the minimum of the negative density and region of slow were overcome to obtain direct measurements of the ef- sound occur. This stands in contrast with traditional res- fective density and sound speed for such high impedance onant acoustic metamaterials, such as those utilizing sim- metamaterial elements, which was achieved using an air- ple harmonic oscillators, for which the addition of mass filled acoustic impedance tube and correcting for the in- tends to decrease the resonant frequency and the corre- herent pressure leakage from the microphones. The ex- sponding frequencies of negative effective density. This perimental measurements were found to be in very good difference for the soft acoustic metamaterial arrangement agreement with the expected theoretical results. Unlike investigated here arises due to the parallel arrangement acoustic metamaterials utilizing mass-spring resonators, of the different components, compared with the tradi- the soft acoustic metamaterials described in this work tion series arrangement of mass-spring and transmission move beyond this framework by creating the dynamics line acoustic metamaterials. Although the soft acous- from the flexural motion of a soft elastic structure, while tic metamaterials examined in this work were limited to offering the tunability to achieve a wide range of desirable relatively simple lumped elements and canonical geome- exotic properties with a single subwavelength element. tries, this principle can be extended to a wide range of more elegant acoustic elements allowing for a vast range of tunable exotic properties in a compact, conformal de- ACKNOWLEDGEMENTS sign. This work was supported by the U.S. Office of Naval Research and by the Spanish Ministerio de VI. CONCLUSION Econom´ıa y Competitividad (MINECO) under grant number TEC2014-53088. The authors wish to acknowl- In conclusion, silica aerogel disks have been exam- edge Encarna G. Villora and Kiyoshi Shimamura for their ined theoretically and experimentally as building units help in aerogel fabrication and handling. 9

[1] M. Maldovan, Nature 503, 209 (2013). [26] L. Forest, V. Gibiat, and T. Woignier, J. Non-Cryst. [2] M. Kadic, T. Buckman, R. Schittny, and M. Wegener, Solids 225, 287 (1998). Rep. Prog. Phys. 76, 126501 (2013). [27] M. Gronauer and J. Fricke, Acta Acust. united Ac. 59, [3] T. Brunet, J. Leng, and O. Mondain-Monval, Science 177 (1986). 342, 323 (2013). [28] Z. Liu, X. Zhang, Y.Mao, Y. Y. Zhu, Z. Yang, C. Chan, [4] J. Christensen, L. Martin-Moreno, and F. J. Garcia- and P. Sheng, Science 289, 1734 (2000). Vidal, Phys. Rev. Lett. 101, 014301 (2008). [29] C. J. Naify, C.-M. Chang, G. McKnight, and S. Nutt, J. [5] J. J. Park, K. J. B. Lee, O. B. Wright, M. K. Jung, and Appl. Phys. 108, 114905 (2010). S. H. Lee, Phys. Rev. Lett. 110, 244302 (2013). [30] C. J. Naify, C.-M. Chang, G. McKnight, F. Schuelen, and [6] R. Fleury, C. F. Sieck, M. R. Haberman, and A. Alù,J. S. Nutt, J. Appl. Phys. 109, 104902 (2011). Acoust. Soc. Am. 132, 1956 (2012). [31] G. Ma, M. Yang, Z. Yang, and P. Sheng, Appl. Phys. [7] R. Fleury and A. Alù,Phys. Rev. Lett. 111, 055501 Lett. 103, 011903 (2013). (2013). [32] Y. Chen, G. Huang, X. Zhou, G. Hu, and C. T. Sun, J. [8] R. Gracia-Salgado, V. M. Garc´ıa-Chocano, D. Torrent, Acoust. Soc. Am. 136, 2926 (2014). and J. Sanchez-Dehesa, Phys. Rev. B 88, 224305 (2013). [33] D. Torrent and J. Sánchez-Dehesa, New. J. Phys. 13, [9] A. Santillánand S. I. Bozhevolnyi, Phys. Rev. B 84, 093018 (2011). 064304 (2011). [34] C. J. Naify, C. N. Layman, T. P. Martin, M. Nicholas, [10] W. M. Robertson, C. Baker, and C. B. Bennett, Am. J. D. C. Calvo, and G. J. Orris, Appl. Phys. Lett. 102, Phys. 72, 255 (2004). 203508 (2013). [11] A. Cicek, O. A. Kaya, M. Yilmaz, and B. Ulug, J. Appl. [35] D. T. Blackstock, Fundamentals of Physical Acoustics Phys. 111, 013522 (2012). (John Wiley & Sons, New York, 2000), 1st ed., ISBN [12] J.-P. Groby, W. Huang, A. Lardeau, and Y. Aurégan,J. 0-471-31979-1. Appl. Phys. 117, 124903 (2015). [36] K. Graff, Wave Motion in Elastic Solids (Dover Publica- [13] D. Torrent, Y. Pennec, and B. Djafari-Rouhani, Phys. tions, 1975). Rev. B 90, 104110 (2014). [37] H. Lee and R. Singh, J. Sound Vib. 282, 313 (2005). [14] P. Li, S. Yao, and X. Zhou, J. Acoust. Soc. Am. 135, [38] C. D. Hettema, Ph.D. thesis, Naval Postgraduate School 1844 (2014). (1988). [15] F. Cervera, L. Sanchis, J. V. Sánchez-P’erez, [39] M. Abramowitz and I. A. Stegun, Handbook of Mathe- R. Mart’inez-Sala, C. Rubio, F. Meseguer, C. L’opez, matical Functions: With Formulas, Graphs and Mathe- D. Caballero, and J. Sánchez-D’ehesa, Phys. Rev. Lett. matical Tables (Dover, New York, 1972). 88, 023902 (2001). [40] J. Gross, G. Reichenauer, and J. Fricke, J. Phys. D: Appl. [16] M. D. Guild, V. M. Garcia-Chocano, W. Kan, and Phys. 21, 1447 (1988). J. Sánchez-Dehesa, J. Appl. Phys. 117, 114902 (2015). [41] T. E. G. Alvarez´ Arenas, F. R. M. de Espinosa, [17] N. Fang, D. Xi, J. Xu, M. Ambati, W. Srituravanich, M. Moner-Girona, E. Rodriguez, A. Roig, and E. Molins, C. Sun, and X. Zhang, Nature Mater. 5, 452 (2006). Appl. Phys. Lett. 81, 1198 (2002). [18] F. Bongard, H. Lissek, and J. R. Mosig, Phys. Rev. B [42] H. Lu, H. Luo, and N. Leventis, in Aerogels Handbook, 82, 094306 (2010). edited by M. A. Aegerter, N. Leventis, and M. M. Koebel [19] Y. Xiao, J. Wen, and X. Wen, J. Sound Vib. 331, 5408 (Springer, 2011), chap. 22. (2012). [43] Y. Xie and J. R. Beamish, Phys. Rev. B 57, 3406 (1998). [20] M. Oudich, X. Zhou, and M. B. Assouar, J. Appl. Phys. [44] J. A. Rogers and C. Case, Appl. Phys. Lett. 75, 865 116, 193509 (2014). (1999). [21] S. H. Lee, C. M. Park, Y. M. Seo, Z. G. Wang, and C. K. [45] Y. Salissou and R. Panneton, J. Acoust. Soc. Am. 128, Kim, Phys. Rev. Lett. 104, 054301 (2010). 2868 (2010). [22] A. Bozhko, V. M. Garc´ıa-Chocano, J. Sanchez-Dehesa, [46] B. H. Song and J. S. Bolton, J. Acoust. Soc. Am. 107, and A. Krokhin, Phys. Rev. B 91, 094303 (2015). 1131 (2000). [23] L. W. Hrubesh, J. Non-Cryst. Solids 225, 335 (1998). [47] V. Fokin, M. Ambati, C. Sun, and X. Zhang, Phys. Rev. [24] H. Nagahara and M. Hashimoto, in Aerogels Handbook, B 76, 144302 (2007). edited by M. A. Aegerter, N. Leventis, and M. M. Koebel [48] M. D. Guild, V. M. Garcia-Chocano, W. Kan, and (Springer, 2011), chap. 33. J. Sánchez-Dehesa, AIP Adv. 4, 124302 (2014). [25] V. Gibiat, O. Lefeuvre, T. Woignier, J. Pelous, and [49] L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. J. Phalippou, J. Non-Cryst. Solids 186, 244 (1995). Sanders, Fundamentals of Acoustics (John Wiley & Sons, New York, 2000), 4th ed. [50] G. W. Milton and J. R. Willis, Proc. R. Soc. A 463, 855 (2007).