Attenuation of surface acoustic waves (SAWs) by a disordered monolayer of polystyrene microspheres is investigated. Surface acoustic wave packets are generated by a pair of crossed laser pulses in a glass substrate coated with a thin aluminum film and detected via the diffraction of a probe laser beam. When a 170 μm-wide strip of micron-sized spheres is placed on the substrate between the excitation and detection spots, strong resonant attenuation of SAWs near 240 MHz is observed. The attenuation is caused by the interaction of SAWs with a contact resonance of the microspheres, as confirmed by acoustic dispersion measurements on the microsphere-coated area. Frequency-selective attenuation of SAWs by such a locally resonant metamaterial may lead to reconfigurable SAW devices and sensors, which can be easily manufactured via self-assembly techniques.

Granular media yield a range of unusual linear and non-linear acoustic phenomena of both fundamental and practical importance.1–3 Natural granular materials such as sand have long been known to exhibit strong broad-band attenuation of acoustic waves.4 More recently, studies of acoustic attenuation by model granular media comprised of monodisperse spherical particles offered insights into long-standing problems such as Anderson localization of elastic waves5 and Cosserat elasticity.6 Transmission of sound through ordered granular chains has been shown to yield narrowband spectral features such as phononic band gaps.7,8 In this work, we study a different aspect of acoustic attenuation by a granular material, namely, the resonant attenuation of surface acoustic waves (SAWs) traveling in a solid substrate induced by a monolayer of microparticles on the substrate surface.

Naturally occurring materials often have narrow optical absorption peaks owing to resonances of atoms or molecules at optical frequencies. By contrast, natural materials typically lack distinct acoustic resonances. Over the past fifteen years, artificial composite materials made of resonant unit cells9–11 have attracted growing attention. Such materials, referred to as locally resonant acoustic metamaterials,9 usually consist of periodic arrays of resonators, although the periodicity is not essential. Just as in optics, a random arrangement of resonant inclusions leads to macroscopic resonant properties and narrowband attenuation.9,12 In the realm of granular media, locally resonant behavior has been observed with fluid-saturated colloidal assemblies of nanospheres13 and in linear chains of macroscopic spheres with specially designed resonating elements.14 

A spherical particle adhered to a solid substrate yields a “mass-spring” axial contact resonance due to the Hertzian contact between the sphere and the substrate.15,16 Recently, it was found that an axial contact resonance of a monolayer of micron-sized spheres strongly interacts with Rayleigh surface waves in the substrate16 yielding a locally resonant metamaterial for SAWs.16 In the present work, we study the transmission of laser-generated SAWs through a thin strip of such a metamaterial. We find that despite a high degree of disorder in the granular monolayer, our metamaterial exhibits distinctly resonant attenuation, effectively blocking SAW transmission near the resonance frequency of ∼240 MHz, while yielding high transmission away from this frequency. Measurements of the acoustic dispersion inside the metamaterial confirm that the resonant attenuation is caused by the contact resonance of the microspheres.

The sample used in this study comprised a strip of microspheres adhered to a 1.5 mm thick soda-lime glass substrate coated with 100 nm of aluminum. A floating monolayer of 1.02 μm diameter monodisperse polystyrene spheres was self-assembled at an air/water interface using a modified Langmuir-Blodgett technique,17 and subsequently transferred onto the aluminum-coated side of the substrate. A micro-contact printing technique,18 using a poly(dimethylsiloxane) (PDMS) stamp, was employed to selectively remove the microspheres from the glass substrate using a lift-off process, leaving behind a 170-μm-wide strip of microspheres with bare substrate on either side of the strip, as shown in Fig. 1(a). A magnified image in Fig. 1(b) shows considerable disorder in the monolayer packing inside the strip.

FIG. 1.

(a) Photograph of the sample; excitation and probe spots on either side of the microsphere strip are visible due to partial transmission of light through the aluminum layer. (b) Magnified view of the spheres. (c) Schematic diagram of the measurement: SAWs are excited by a pair of crossed excitation pulses, travel through a microsphere-coated region and are detected via diffraction of the probe beam. The reference beam used for heterodyne detection is overlapped with the probe.

FIG. 1.

(a) Photograph of the sample; excitation and probe spots on either side of the microsphere strip are visible due to partial transmission of light through the aluminum layer. (b) Magnified view of the spheres. (c) Schematic diagram of the measurement: SAWs are excited by a pair of crossed excitation pulses, travel through a microsphere-coated region and are detected via diffraction of the probe beam. The reference beam used for heterodyne detection is overlapped with the probe.

Close modal

Measurements were performed using the laser-induced transient grating (TG) technique19 modified to spatially separate the excitation and detection of SAWs and to enable a fine control of the SAW wavelength. The basic setup was similar to that used in earlier work.16,20 SAWs were generated by crossed laser pulses (wavelength 515 nm, pulse duration 60 ps, total energy at the sample 0.7 μJ) entering the sample through the transparent substrate and forming an interference pattern of period L on the Al film. The excitation spot (190 μm in diameter at 1/e2 intensity level) can be clearly seen in Fig. 1(a). The absorption of laser light in the Al film led to rapid thermal expansion that generated counter-propagating SAW wave packets at wavelength L. The detection of SAWs was done via diffraction of a probe laser beam (wavelength 532 nm) focused to a spot of 90 μm in diameter at a distance 440 μm from the excitation spot. The probe beam was gated by an electro-optic modulator to produce rectangular pulses of 68 μs in duration; on the time scale of the measurements the probe could be considered continuous with a power of 190 mW. A reference beam used for optical heterodyne detection of the diffracted signal21,22 was overlapped with the probe at the sample. The period of the interference pattern defining the SAW wavelength is set by the period of the transmission diffraction grating (phase mask) used to produce both excitation and probe/reference beam pairs.22 Thus the SAW wavelength can be varied over a discrete set of values by switching the phase mask patterns fabricated on the same substrate. For this work, we introduced a fine control of the SAW wavelength by rotating the phase mask.23 As will be shown below, this fine control is essential for mapping the frequency dependence of the SAW attenuation.

Figure 2(a) shows signal waveforms measured at different acoustic wavelengths without the microsphere strip (this was done by translating the sample to move the microsphere strip out of the space between the excitation and probe spots). Each waveform consists of a high-frequency wave packet whose arrival time corresponds to surface-skimming longitudinal wave in the substrate, and a higher-amplitude, low-frequency wave packet arriving at ∼140 ns, in agreement with a Rayleigh SAW velocity of 3177 m/s measured on Al-coated soda-lime glass.23 The envelope of the SAW wave packets is determined by the Gaussian laser intensity profile in the excitation spot, with the duration of the wave packet given by the ratio of the excitation spot size to the SAW velocity. Normally, the detection of acoustic oscillations via diffraction of the probe beam requires a standing acoustic wave, which acts as a diffraction grating with an oscillating diffraction efficiency. However, optical heterodyne detection yields oscillation in the signal from a traveling acoustic wave, thus making it possible to separate excitation and probe spots in the TG experiment.22 

FIG. 2.

SAW wave packets detected by the laser probe (a) without microspheres and (b) with a microsphere strip between excitation and probe spots; (c) corresponding Fourier transform peaks without spheres (solid lines) and with spheres (dashed lines). The dashed line in the middle represents a spectrum multiplied by a factor of 10.

FIG. 2.

SAW wave packets detected by the laser probe (a) without microspheres and (b) with a microsphere strip between excitation and probe spots; (c) corresponding Fourier transform peaks without spheres (solid lines) and with spheres (dashed lines). The dashed line in the middle represents a spectrum multiplied by a factor of 10.

Close modal

Figure 2(b) shows signal waveforms obtained with the microsphere strip placed between the excitation and probe spots as shown in Fig. 1(a). One can see that microspheres resulted in moderate attenuation of SAW wave packets at 130 and 422 MHz whereas the SAW wave packet at 235 MHz was completely blocked. By contrast, surface-skimming longitudinal waves are not affected by the presence of the spheres. To quantify the attenuation, we calculated the Fourier transform (FT) of the signals, as shown in Fig. 2(c), and determined the ratio of spectral peaks measured with and without the microspheres. Figure 3 shows the dependence of the attenuation on the SAW frequency, which reveals a deep dip in transmission around 240 MHz. Varying the frequency in small steps necessary to trace the attenuation peak was made possible by the fine control of the TG period mentioned above.23 The maximum attenuation could not be measured as the transmitted SAW signal was below the noise level; the filled circles in Fig. 3 show the upper bounds for the transmission at about −43 dB. This corresponds to an attenuation length of 17 μm at the 1/e intensity level, which amounts to only 1.3 SAW wavelengths (on the substrate without microspheres).

FIG. 3.

Measured SAW attenuation vs. frequency (open symbols). Solid symbols show upper bounds (transmitted SAWs were below the noise level). Dashed curve has been calculated based on the imaginary part of the wave vector obtained from Eq. (1) with the damping term. The shaded region corresponds to the acoustic bandgap, i.e., the frequency range yielding a complex wave vector in the model without damping.

FIG. 3.

Measured SAW attenuation vs. frequency (open symbols). Solid symbols show upper bounds (transmitted SAWs were below the noise level). Dashed curve has been calculated based on the imaginary part of the wave vector obtained from Eq. (1) with the damping term. The shaded region corresponds to the acoustic bandgap, i.e., the frequency range yielding a complex wave vector in the model without damping.

Close modal

To elucidate the origin of the sharp peak in the attenuation, we measured dispersion of SAWs inside the microsphere-coated area by overlapping the excitation and probe spots inside the microsphere strip. Figure 4(a) shows an example of an acoustic waveform measured inside the strip, and Fig. 4(b) the corresponding FT spectrum. The high-frequency peak at 500 MHz corresponds to the surface-skimming longitudinal wave; two low-frequency peaks between 200 and 300 MHz are surface modes resulting from the splitting of the Rayleigh wave dispersion due to the interaction with the microspheres.16 This splitting can be clearly seen in Fig. 4(c) showing the measured acoustic dispersion i.e., the positions of the frequency peaks vs acoustic wave vector. As discussed in Ref. 16, the Rayleigh SAW undergoes avoided crossing and hybridization with the axial Hertzian contact resonance of the spheres. The lower dispersion branch starts as a Rayleigh SAW at low frequencies and turns into the contact resonance mode on approaching the contact resonance frequency, whereas the upper branch, in the classic avoided-crossing picture would start as a contact resonance mode at k = 0 and turn into predominantly Rayleigh SAW at high frequencies. However, the upper branch does not extend all the way to k = 0 because it becomes overdamped upon crossing the “transverse threshold” ω = cTk due to the radiation of bulk waves into the substrate.16,24

FIG. 4.

(a) Typical signal waveform inside the strip (acoustic wave vector 0.546 μm−1) and (b) its FT spectrum (solid line) and a fitted spectrum consisting of a linear combination of two Lorentzians (dashed line) with complex frequencies that are calculated from the dispersion equation with a damping term. (c) Measured (symbols) and calculated (red curves) surface acoustic wave dispersion inside the strip. The solid black line is the calculated Rayleigh SAW dispersion without microspheres. The horizontal dashed-dotted line corresponds to the microsphere contact resonance frequency f0. The dashed line corresponds to the bulk transverse velocity of the substrate. The shaded stripe shows the band gap.

FIG. 4.

(a) Typical signal waveform inside the strip (acoustic wave vector 0.546 μm−1) and (b) its FT spectrum (solid line) and a fitted spectrum consisting of a linear combination of two Lorentzians (dashed line) with complex frequencies that are calculated from the dispersion equation with a damping term. (c) Measured (symbols) and calculated (red curves) surface acoustic wave dispersion inside the strip. The solid black line is the calculated Rayleigh SAW dispersion without microspheres. The horizontal dashed-dotted line corresponds to the microsphere contact resonance frequency f0. The dashed line corresponds to the bulk transverse velocity of the substrate. The shaded stripe shows the band gap.

Close modal

The hybridized dispersion is described by the following equation relating the angular frequency ω and wave vector k16 

(ω2ω021)[(2ω2k2cT2)24(1ω2k2cT2)1/2(1ω2k2cL2)1/2]=mNρω4cT4k3(1ω2k2cL2)1/2,
(1)

where m is the mass of a microsphere, N the average number of microspheres per unit area, ω0 the contact resonance frequency, ρ the substrate density, cL and cT longitudinal and transverse acoustic velocities of the substrate material, respectively. Solid curves in Fig. 4(c) show dispersion calculated using Eq. (1), with the contact resonance frequency ω0 being the only fitting parameter.25 Calculations with f0 = 2πω0 = 236 MHz can be seen to provide a good fit to the experimental data.

In the model without dissipation16 used to derive Eq. (1), SAWs with frequencies inside the bandgap region separating the lower and upper branches are attenuated, while outside the bandgap the only losses are associated with reflections at the boundaries of the microsphere strip. In reality, however, Fig. 3 shows significant attenuation outside the bandgap, which we attribute to the broadening of the contact resonance.16 For a more realistic description of the attenuation, we added a damping term to the contact resonance model. The derivation of the SAW dispersion equation with damped oscillators is similar to the derivation for undamped oscillators in Ref. 16 and leads to a dispersion equation similar to Eq. (1), with the replacement of the term (ω2/ω021) in the left-hand side by (ω2/ω02+2iωγ/ω021), where γ is the damping constant. From the modified dispersion equation, we calculated the complex wave vector as a function of frequency. Attenuation by microspheres over a propagation distance of 170 μm was then calculated from the imaginary part of the wave vector (see Fig. 3). The calculated attenuation for γ/ω0 = 0.18, shown by a dashed curve in Fig. 3, yields reasonable agreement with the experimental data. We note that in the case of a Bragg bandgap of a phononic crystal, the maximum of attenuation is typically located in the middle of the bandgap,26 whereas in the local resonance case, the attenuation curve is close to a Lorentzian centered at the resonant frequency, as is well known from the Lorentz oscillator model in optics.27 In general, the attenuation caused by a local resonance should not be interpreted as a consequence of the bandgap, particularly when the line width exceeds that of the bandgap, as is the case in our experiment. As shown in Ref. 24, depending on the damping and oscillator strength, the avoided crossing may or may not form, whereas the attenuation peak at the resonant frequency is always present (this is also well known in optics where the locally resonant bandgap corresponds to a band of negative dielectric constant above the resonant frequency27).

While transmission measurements imply a real frequency and complex wave vector, dispersion measurements inside the microsphere strip imply a real wave vector (defined by the excitation pattern) and complex frequency. The dashed curve in Fig. 4(b) shows a fit by a linear combination of two Lorentzians using the complex frequencies calculated from Eq. (1) with the damping term γ/ω0 = 0.18 (amplitudes and phases are treated as free parameters). One can see that the calculated line widths qualitatively agree with the measured ones, even though the model overestimates the line width of the lower mode, which is dominated by the contact resonance vibrations of the spheres. We note that the calculated attenuation in Fig. 3 also overestimates the linewidth. Reducing the damping will reduce the line width but will cause a larger mismatch in the absolute values of the attenuation close to the resonance in Fig. 3. This indicates that our model is not entirely accurate. Indeed, we realize that the observed line width may not be due to the actual elastic damping in the Hertzian contact. Other significant factors would be sphere-to-sphere variations of the resonant frequency related e.g., to local roughness at contact points, which would lead to inhomogeneous broadening of the spectral peak, as well as randomness in the microsphere positions, which would lead to scattering losses.24 We consider the inclusion of the damping term in the contact spring model as a phenomenological device allowing us to account for the broadening of the resonance, even if not in a rigorous manner. Further progress in understanding the origin of the observed line width will require elucidating the role of the inhomogeneous broadening, which is very difficult to isolate based on linear attenuation measurements.28 Measuring the motion of a single sphere may help resolve this problem. Furthermore, our calculations did not take into account reflection losses (as well as scattering into bulk waves) at the boundaries of the strip. These losses are likely to be small far from the resonance frequency when the SAW inside the strip very much resembles the Rayleigh wave outside; however, they may not be negligible close to the resonance.

In conclusion, we have observed a strong peak in the attenuation of SAWs by a microsphere strip at the resonant frequency of the Hertzian contact between the microspheres and the substrate. Using the contact resonance of microspheres is not the only way of fabricating a locally resonant metamaterial for SAWs. For example, arrays of pillars or ridges also yield locally resonant behavior, and resonant attenuation of SAWs by ordered and disordered arrays of pillars has been observed.11,12 What makes a microgranular metamaterial special is that microspheres are only weakly connected to the substrate by van der Waals adhesion forces. They can easily be removed (for example, by the stamping technique used to fabricate the sample for this study, or even by high-amplitude SAWs29), which could be used to make a reconfigurable SAW filter. Furthermore, the resonant frequency can be controlled by affecting the stiffness of the Hertzian contact; for example, it could be tuned by temperature near the glass transition of the polymer used to make microspheres or as a layer between the microspheres and the substrate. It could also be tuned by any factor affecting the adhesion force; for example, the surface of the substrate or the microspheres could be functionalized to make the adhesion force sensitive to a certain chemical or biological agent. Finally, it would be interesting to see what happens at high SAW amplitudes: the nonlinearity of the Hertzian contact30 should lead to an amplitude dependence of the attenuation. At some amplitude, SAWs will cause the detachment of the particles,29 at which point the attenuation is expected to disappear. Thus we believe that the concept of using granular monolayers as SAW metamaterials opens multiple possibilities both for fundamental studies of wave phenomena and for practical applications.

The authors would like to thank Ryan Duncan for his help in the analysis of the results. The contribution by J.K.E., A.V.-F., K.A.N, and A.A.M. was supported by NSF Grant No. CHE-1111557. A.V.-F. appreciates support from CINVESTAV and CONACYT through normal, mixed and PNPC scholarships. A.K. and N.B acknowledge support by the NSF through Grant No. CMMI-1333858. M.H. was supported through the NSF Graduate Research Fellowship Program.

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Other parameters used in the model were set to the following values: ρ = 2440 kg/m3, cT = 3438 m/s, cL = 5711 m/s, m = 5.89 × 10−16 kg (the latter value calculated based on the polystyrene density 1060 kg/m3).

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