Acoustically Trapped Colloidal Crystals That Are Reconfigurable in Real Time

Acoustically trapped colloidal crystals that are reconfigurable in real time

Mihai Caleap1 and Bruce W. Drinkwater

Faculty of Engineering, University of Bristol, Bristol BS8 1TR, United Kingdom

Edited* by John B. Pendry, Imperial College London, London, United Kingdom, and approved March 5, 2014 (received for review December 13, 2013) Photonic and phononic crystals are metamaterials with repeating tuned, they do not facilitate real-time reconfigurability and so unit cells that result in internal resonances leading to a range of cannot be used in an active mode. wave guiding and filtering properties and are opening up new In this article we report for the first time, to our knowledge, applications such as hyperlenses and superabsorbers. Here we the experimental realization of a 3D colloidal metamaterial that show the first, to our knowledge, 3D colloidal phononic crystal is reconfigurable in real time and demonstrate its ability to that is reconfigurable in real time and demonstrate its ability to rapidly alter its acoustic filtering characteristics. The reconfig- rapidly alter its frequency filtering characteristics. Our reconfigurable metamaterial is assembled from a low-density aqueous urable material is assembled from microspheres in aqueous solu- suspension of microspheres. Our metadevice generates a high- tion, trapped with acoustic radiation forces. The acoustic radiation amplitude megahertz-frequency (MHz) acoustic standing wave force is governed by an energy landscape, determined by an which results in acoustic radiation forces that lead to particles applied high-amplitude acoustic standing wave field, in which moving to, and becoming trapped in, a simple orthorhombic particles move swiftly to energy minima. This creates a colloidal lattice (although here we explore only simple tetragonal forms). crystal of several milliliters in volume with spheres arranged in an By varying the frequency of the acoustic standing wave in the orthorhombic lattice in which the acoustic wavelength is used to metadevice we are able to control the lattice geometry. We use control the lattice spacing. Transmission acoustic spectroscopy transmission acoustic spectroscopy to explore the reconfigur- shows that the new colloidal crystal behaves as a phononic meta- ability of the resulting metamaterial. These measurements reveal material and exhibits clear band-pass and band-stop frequencies

a complex and tunable distribution of band-gap and band-stop ENGINEERING which are adjusted in real time. phenomena over a wide range of frequencies.

reconfigurable acoustic assembly | acoustic metamaterials | Results aqueous suspension of microspheres The metadevice used to form the 3D reconfigurable metamaterial is summarized in Fig. 1. To generate the desired lattice rtificially engineered metamaterials have attracted signifi- pattern, the metadevice creates a high-amplitude acoustic Acant research interest due to their useful wave guiding and standing wave field using five planar piezoceramic transducers transmission properties. The interest in these materials stems from arranged as shown in Fig. 1A. The metadevice consists of three the possibility of gaining previously unheralded control over wave orthogonally arranged systems: the levitation system, arranged phenomena, for example, controlling the path of waves leads to vertically, which lifts the particles and holds them in horizontal incredibly efficient lenses (1) or invisibility cloaking (2, 3) and planes, and two orthogonal manipulation stages, which use the controlling their transmission and reflection leads to highly ef- counterpropagating wave method to produce a grid of acoustic ficient filters (4), diodes (5–7), or superabsorbers (8). traps in the horizontal plane. The lattice spacing can be con- Photonic and phononic crystals are periodic metamaterials trolled by the frequencies of the standing waves and the form typically created from multiple elementary repeating cells. They and location of the lattice by the phase differences (16). The exhibit a well-known series of band-pass and band-stop frequencies that have attracted significant attention for use as filters Significance and absorbers. Here we demonstrate a phononic crystal reconfigurable in real time, although the same principles could be used We have been working on metamaterials that are reconfig- to fabricate other metamaterials including photonic crystals, as urable in real time with a view to creating genuinely active well as the more elaborate configurations required for applica- metamaterials. Such materials will allow researchers to gain tions such as cloaking. unprecedented control over a range of optical and acoustic The frequencies of the band gaps in phononic crystals can be wave phenomena. To date, although numerous examples of tuned by changing the lattice geometry (9), and the width of the metamaterials exist, none is reconfigurable in three dimensions. gaps depends on the contrast between the densities and sound We have developed a method for creating three-dimensional velocities of the component materials. Most of the work to date colloidal crystals that are reconfigurable in real time. Our method has not involved any reconfigurability, for example, 2D phononic uses acoustic assembly to trap a suspension of microspheres in crystals with a periodicity millimeter order and above have been patterns resembling crystal lattices. We show that it is possible to assembled manually and used to explore their basic properties dynamically alter the geometry of the colloidal crystal and use (10, 11). A reconfigurable phononic crystal has been created in our metamaterial as an ultrasonic filter that can be tuned in 2D using optical tweezers but this is limited to a relatively small real time. scale (’ 1 × 108-m2 area) and to transparent or dielectric particles (12). Three-dimensional experimental realizations of phononic Author contributions: M.C. and B.W.D. designed research; M.C. performed research; M.C. crystals have either been manufactured with millimeter period- and B.W.D. analyzed data; and M.C. and B.W.D. wrote the paper. icity or as colloids (13, 14). Colloids represent a particularly at- The authors declare no conflict of interest. tractive option as they are known to self-assemble into simple 3D *This Direct Submission article had a prearranged editor. crystalline structures. Recent work on 3D hypersonic (GHz) col- Freely available online through the PNAS open access option. loidal crystals has led to experimental observation of the hyper- 1To whom correspondence should be addressed. E-mail: [email protected]. sonic Bragg gaps (15). However, whereas the choice of particle This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. shape, size, volume fraction, etc. allows the colloidal crystal to be 1073/pnas.1323048111/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1323048111 PNAS Early Edition | 1of5 Downloaded by guest on September 27, 2021 Fig. 1. Acoustic metadevice capable of manipulat- ing the acoustic space and controlling the propagation of waves. (A) Schematic view of the fabricated metadevice with piezoceramic transducers for generating standing acoustic waves and a pair of broadband transducers for acoustic transmission spectroscopy. Note a photograph of the device can be seen in SI Text, Fig. S4.(B) Acoustic manipulator with four transducers, positioned as two orthogonal pairs. Exciting weakly resonant waves in the piezoceramic plates, polystyrene microspheres suspended in a fluid are driven to the pressure nodes of the standing wave field. (C) Acoustic levitator with one transducer at the base of the device holds particles against gravity in horizontal planes inside a thin- wall transparent polyester tube; the free surface of the aqueous suspension above the levitator acts as a reflector to form standing waves. (D) Experimental apparatus for acoustic transmission spectroscopy. The ultrasonic pulse generated by a monolithic transducer is incident upon the colloidal crystal in the center of the device and the transmitted signal is collected by a 2D matrix array transducer whose center elements are collinear with the incident wave. The image in the background shows a typical acoustic scattering be- havior of a colloidal crystal lattice at an instant time. Arrows indicate direction of propagation of the waves.

resultant 3D standing wave exerts an acoustic radiation force on cited in-phase, the particles were further formed into ap gridffiffiffi in the particles (17). This force stems from a combination of the the horizontalpffiffiffi X Y plane with lattice spacing ax = λx= 2 and time-averaged acoustic pressure and inertial interaction between ay = λy= 2. Here the wavelengths in the X- and Y directions, λx the particles and the acoustic field (18). Under Rayleigh scat- and λy, were used to reconfigure the metamaterial (see the sec- tering conditions, the wavelength is larger than the particle size ond section of SI Text). Typical acoustic landscape is depicted ðλ >> aÞ and acoustic radiation force due to a standing wave is in Fig. S2, while the particle positions are shown in Fig. S3. governed by the spatial gradient of the energy landscape, U, i.e., The acoustic standing wave patterns were first characterized F = −∇U (17). Particles with densities greater than the fluid ex- by a Schlieren optical system which maps the time-averaged perience forces toward pressure nodes. However, whereas this intensity field distribution. Fig. 2 shows the intensity distri- model is helpful to understand the basic operating principles of butions formed when the manipulation transducers are ex- our metadevice, the Rayleigh limit is not strictly satisfied in our cited at fx = fy = 2:25 MHz, 3:75 MHz, and 5:25 MHz which current metamaterial design (see extended theory in the first lead to ax = ay = 465 μm, 279 μm, and 199 μm, respectively. The section of SI Text). Fig. S1 illustrates the acoustic radiation force acoustic radiation force associated with these distributions traps for a polystyrene sphere in a standing wave field. particles at the intensity minima which set the X- and Y-lattice The experiments were performed using 90-μm-diameter poly- dimensions to be equal, leading to simple tetragonal lattice styrene spheres in aqueous suspension contained within a central arrangements. Fig. 3 A–C shows views of the X Y plane of the cylindrical chamber of 0:33 mL in volume which becomes our metamaterial assembled using the acoustic field distributions of reconfigurable colloidal metamaterial (Methods). Under the ac- Fig. 2. It is apparent that the lattice spacing depends on the tion of the levitation stage but without a signal applied to the frequency used and it becomes more densely spaced as the manipulation transducers, the particles collect in horizontal frequency increases. Note that the concentration of particles planes one above the other separated by az = λz=2, where λz is was ϕ ’ 5% by volume, chosen such that at 5:25 MHz most of the wavelength in the Z direction, typically az = 215 μm when the traps contained a single polystyrene sphere. Fig. 3D shows excited at 3.45 MHz assuming the speed of sound in water, an X Z view of the reconfigurable metamaterial when the c0 = 1; 480 m=s. When the manipulation transducers were ex- levitation stage was excited at fz = 3:45 MHz, together with

2of5 | www.pnas.org/cgi/doi/10.1073/pnas.1323048111 Caleap and Drinkwater Downloaded by guest on September 27, 2021 Fig. 2. Time-averaged intensity field distributions in the acoustic metadevice. (A–C) Schlieren images of the acoustic landscape in the center of the device obtained with the X–Y acoustic manipulator excited at 2.25, 3.75, and 5.25 MHz, respectively. The images are in a grayscale where black represents zero intensity and the maximum intensity amplitude is white. The central circular region is 5 mm in diameter.

fx = fy = 3:75 MHz. Fig. S4 shows an optical view of the working created, such as missing or multiple particles at some nodes, metadevice, while Fig. S5 displays views of the central cylindrical result in an increase in the level of the observed transmission region of Fig. S4 with 90-μm–diameter polystyrene particles as- minima compared with simulation (which assumed perfect sym- sembled into a tetragonal lattice at two instant times. metry). This is in line with other findings (20) where crystal imper- The colloidal metamaterials were characterized by transmission fections caused deviations from the perfect symmetry case. However, acoustic spectroscopy (Methods), where the transmission magnitude it is clear that the measured minima in transmission are in reason- T was obtained by normalizing the Fourier-transformed spectra able agreement with simulations indicating that the reconfigurable of the signal transmitted through the cylindrically shaped meta- metamaterial is behaving as expected. material, A1ð fÞ, with respect to the signal through the same Fig. 5 shows the transmission spectra measured with a scat- chamber filled with deionized water, A0ð fÞ; T = jA1ð fÞ=A0ð fÞj. tering angle θsc varying from −68 to 68 in the X Y plane (Fig. With respect to the lattice geometry, the incident wavefront used for 5A) for the metamaterial patterns shown in Fig. 3 A–C. An ad- transmission lies in the (100) plane of the lattice. We measured the ditional measurement was made through a random suspension of

acoustic transmission through the metamaterial patterns with the same polystyrene particles. The transmission map is plotted ENGINEERING parameters shown in Fig. 3 (see typical experimental transmitted in Fig. 5B as a function of both the frequency and the scattering signals in Fig. S6). An additional measurement was made through angle. The transmission patterns are symmetrical with respect to a random suspension of the same polystyrene particles. Fig. 4 the center of the array receiver, indicating good alignment, and illustrates the experimental transmission spectra demonstrating considerable variation is seen with both angle and frequency the reconfigurability of the system. suggesting anisotropy. Various marked differences can be seen AlsoshowninFig.4A–C are the results of a finite-difference between the different reconfigured metamaterials, particularly in time-domain model of the acoustic propagation through an ideal- the frequency ranges 5–6.5 MHz and 8.5–9.5 MHz, which were ized version of the cylindrically shaped metamaterial (see the first also seen in Fig. 4. Fig. S8 shows the effect of real-time recon- part of the third section of SI Text). A generally good correspondence figuration (SI Text). Here a 30-cycle sinusoid, centered on 6 between measured and modeled data is observed. By using an ef- MHz, is transmitted through the metamaterial and the lattice fective field approach (19), the effective mass density and bulk reconfigured between ax = ay = 199 μm and ax = ay = 279 μm. modulus of the corresponding acoustic metamaterial with random The amplitude of the transmitted signal is seen to change by microstructure was calculated (see the second part of the third a factor of 6 and the reconfiguration takes significantly less than section of SI Text). Predicted values for the acoustic transmission 0.1 s (Movie S1). magnitude for the random acoustic metamaterial confined within the central cylindrical chamber with ϕ ’ 5% volume fraction of Discussion polystyrene particles in water are displayed in Fig. 4D and show This realization of a 3D colloidal metamaterial that is reconfig- good agreement with experiment. Fig. S7 shows measured and urable in real time represents a new and versatile means by predicted values for the transmission magnitude T with ϕ =2.5% which wave propagation phenomena can be controlled. The use vol fraction of polystyrene particles in water. The agreement of acoustic assembly is attractive as it has advantages over other between experimental results and predictions using Eqs. S5–S8 is potential techniques, such as optical tweezers. Acoustic assembly excellent over the entire frequency range displayed in Fig. S7. is easily scalable with wavelength; here we use wavelengths of By comparing Fig. 4 A–C (lattice) with Fig. 4D (random), it hundreds of micrometers and create a metamaterial of milliliters can be seen that some of the most prominent band gaps occur in in volume. This could be scaled down to the order of micro- the vicinity of the resonant frequencies of individual particles, meters before acoustic absorption and scattering become limit- whereas others suggest a simultaneous occurrence of structure- ing (21). Conversely, there is no obvious upper limit on the directed and particle-resonance-induced phononic gaps. We be- wavelength. The ability to trap particles in an acoustic field lieve that small imperfections in the symmetry of lattice we have requires contrast between at least one of the key fluid and particle

Fig. 3. Assembled colloidal crystals with the acoustic metadevice. (A–C) Experimental micrographs of the X–Y plane of the assembled colloidal metamaterials using the same parameters as Fig. 2 A–C, respectively. (D) Side view of the acoustic levitator excited at 3.45 MHz with the acoustic manipulator excited at 3.75 MHz.

Caleap and Drinkwater PNAS Early Edition | 3of5 Downloaded by guest on September 27, 2021 In the more immediate future these concepts will lead to, for example, the construction of a variety of reconfigurable quasi- crystalline structures (22, 23) which would be difficult by self- assembly techniques. Indeed, the range of structures that can be constructed is limited only by the ability to produce the appro- priate acoustic energy landscape. Multielement array devices would further extend the versatility (24). Arrays with wide bandwidths could, in principle, be used to produce gradations in the lattice parameters which would lead to the ability to con- struct reconfigurable waveguides and acoustic cloaking devices. In addition, such arrays could actively control the detail of the lattice geometry to fabricate more perfectly symmetrical lattices than were possible here. Tunability of the band structure could also be achieved with constitutive media with mixed properties such as acousto-optic or acousto-magnetic properties. Methods Device Fabrication. The four manipulation transducers consist of 10 × 15 × 3-mm- thick piezoceramic plates of NCE51 material [a soft-doped lead zirconate titanate (PZT) material: Noliac group]. These PZT plates are fitted into a specially designed polymer chamber which is fabricated using fused deposition modeling, a method commonly used for prototyping and rapid manufacturing. The distance between opposing pairs of transducers is 38 mm. The levitation transducer is a piezoceramic disk of NCE51 material of 5 mm diameter and is 2 mm thick. This PZT disk is fitted into one end of a round tube of Mylar material (a fairly rigid, transparent, heat-shrinkable polyester film, Precision Paper Tube Co.). The tube defines the central cylindrical chamber, within which the metamaterial is constructed; it is 17 mm in height and has an ∼ 50-μm wall thickness.

Particle Suspension. Monodisperse ð90 ± 3Þ-μm -diameter polystyrene spheres

(Polysciences Europe) were diluted in a solution of deuterium oxide (D2O) 3 3 and water (H2O). The solvent densities (1:11 g=cm for D2O and 1 g=cm for 3 H2O) were matched to keep the solid particles (1:05 g=cm for polystyrene) approximately neutrally buoyant. A solvent ratio of ∼1:1 volume will match the particles’ density, however only over a narrow temperature range. The buoyancy was further improved once the fluid–particle mixture was released inside a thin-wall polyester tube acting as an acoustic levitator. The re- mainder of the metadevice chamber was filled with deionized water.

Experimental Setup and Data Acquisition. Fabrication of the colloidal metamaterial. Each of the five piezoceramic transducers was driven using a continuous sine wave by a signal generator. The four X Y manipulation transducers (arranged in opposing pairs) require additional power amplification to provide voltages of 30 Vp−p. The manipulation transducers were excited at three resonance frequencies (2.25, 3.75, and 5.25 MHz, related to the thickness of these plates) to gain additional control, and each pair was synchronized to allow the relative phase of the excitations of the opposing transducers to be adjusted. Synchronization between channels was achieved using two arbitrary waveform generators (33220A, Agilent) providing four outputs allowing independent control of the amplitude, phase, and frequency of the transducer pairs. The levitation and manipulation stages are not synchronized together. The resonant nature of the levitation stage en- Fig. 4. Acoustic transmission spectroscopy showing reconfigurability of the sured that the signal direct from the generator (10 Vp−p at 3.45 MHz) was system. (A–C) Spectroscopy results for the colloidal metamaterials shown in sufficient to trap 90-μm-diameter polystyrene spheres against gravity. Fig. 3 A–C, respectively. (D) Transmission result for a random distribution of Transmission acoustic spectroscopy. The measurements of the transmission of the same concentration ðϕ ’ 5%Þ of polystyrene microspheres in water. the acoustic waves in the megahertz frequency range were performed in Note that both experimental and simulated results are shown. immersion. Two immersion transducers of nominal central frequency 15 MHz were fitted onto the metadevice in between two opposing manipulation transducers, along the Y direction. The distance between the two trans- acoustic properties: density, speed of sound, or compressibility. ducers is 35 mm. The excitation signal is a broadband pulse generated by a monolithic transducer (V313-SM, Panametrics) of 6:3 mm diameter. The However, this is not a significant restriction and allows most fluid– – receiver is a 2D matrix array probe transducer (CDC9663-1, Imasonic) and has solid as well as many liquid liquid combinations to be explored. 128 elements. The array has an element pitch of Δx = Δz = 300 μm and covers 2 Additionally, from a practical perspective, acoustic systems are a total active area ðLx × LzÞ of 3:85 × 3:55 mm . The transmitted field is av- relatively cheap and easy to integrate with other systems com- eraged over 10–15 successive acquisitions on individual array elements. The pared with their optical counterparts. measured time signals are then Fourier transformed to obtain the frequency The techniques used to assemble and reconfigure the 3D dependence of the complex transmission, so that both amplitude and phase are determined. Transmission spectra in Fig. 4 were measured on a single colloidal metamaterials described in this article could be ex- centrally located element, and angular variations in Fig. 5 were averaged tended to gain real-time control of highly efficient filters, diodes, over four elements in the Z direction (i.e., 1,150 μm). We note this averaging or superabsorbers with applications in both optics and acoustics. procedure reduces the depth of the transmission minima in Fig. 5 compared

4of5 | www.pnas.org/cgi/doi/10.1073/pnas.1323048111 Caleap and Drinkwater Downloaded by guest on September 27, 2021 Fig. 5. Acoustic transmission maps displaying angular-dependent scattering. (A) Schematic diagram of the experimental arrangement used to measure the acoustic transmission as a function of angle. Note that

the incident wavevector lies in the (100) plane of the ENGINEERING crystal lattice. The transmission result was averaged across four central elements in the Z direction (i.e., 1,150 μm). (B) Acoustic transmission loss plotted as a function of the wave frequency and the scattering angle for the metamaterial patterns shown in Fig. 3 A–C. Here, f0 ≡ fx = fy .

with the results in Fig. 4; however, the main features of the different met- reach a lattice location is governed by a combination of viscosity of the amaterial lattices are preserved (see SI Text, Fig. S6 for a detail of these suspending fluid and the acoustic pressure amplitude and distribution, results in the [100] direction, i.e., θsc = 0). as well as the various parameters that govern the acousto-phoretic contrast. As in practice, the upper limit on acoustic pressure is limited by Modulation Time. The modulation (or switching) time for the metadevice is the disrupting effects of acoustic streaming to below p0 ≈ 1 MPa. If we determined by the time taken for particles to move between different use this limiting pressure and take the dynamic viscosity of the D2O H2O patterns. If the acoustic radiation force on the polystyrene particles (SI mixture to be η ≈ 0:9 kPa · s the time taken for a 90-μm polystyrene par- Text,Eq.S1) is locally balanced by Stokes’ drag, the particle trajectories ticle to move from antinode to node in a 1D standing wave at 5:25 MHz can be simulated. This reveals that the time taken for a single particle to is ∼ 5 ms.

1. Zhang X, Liu Z (2008) Superlenses to overcome the diffraction limit. Nat Mater 7(6): 13. Yang S, et al. (2002) Ultrasound tunneling through 3D phononic crystals. Phys Rev Lett 435–441. 88(10):104301. 2. García-Meca C, et al. (2013) Analogue transformations in physics and their application 14. Liu Z, et al. (2000) Locally resonant sonic materials. Science 289(5485):1734–1736. to acoustics. Sci Rep 3:2009. 15. Cheng W, Wang J, Jonas U, Fytas G, Stefanou N (2006) Observation and tuning of 3. Zhang S, Xia C, Fang N (2011) Broadband acoustic cloak for ultrasound waves. Phys hypersonic bandgaps in colloidal crystals. Nat Mater 5(10):830–836. Rev Lett 106(2):024301. 16. Courtney CRP, et al. (2011) Manipulation of particles in two dimensions using phase 4. Zhu H, Semperlotti F (2013) Metamaterial based embedded acoustic filters for structural controllable ultrasonic standing waves. Proc R Soc A Math Phys. Eng Sci 468:337–360. applications. AIP Adv 3(9):092121. 17. Gor’kov LP (1962) On the forces acting on a small particle in an acoustical field in an 5. Liang B, Guo XS, Tu J, Zhang D, Cheng JC (2010) An acoustic rectifier. Nat Mater 9(12): ideal fluid. Sov Phys Dokl 6:773–775. – 989 992. 18. King LV (1934) On the acoustic radiation pressure on spheres. Proc R Soc A Math Phys. 6. Liang B, Yuan B, Cheng JC (2009) Acoustic diode: Rectification of acoustic energy flux Eng Sci 147:212–240. in one-dimensional systems. Phys Rev Lett 103(10):104301. 19. Caleap M, Drinkwater BW, Wilcox PD (2012) Effective dynamic constitutive pa- 7. Li Y, Liang B, Gu Z, Zou X, Cheng J (2013) Unidirectional acoustic transmission through rameters of acoustic metamaterials with random microstructure. New J Phys 14: a prism with near-zero refractive index. Appl Phys Lett 103:053505. 033014. 8. Mei J, et al. (2012) Dark acoustic metamaterials as super absorbers for low-frequency 20. Psarobas I, Stefanou N, Modinos A (2000) Phononic crystals with planar defects. Phys sound. Nat Commun 3:756. – 9. Kushwaha MS, Halevi P, Dobrzynski L, Djafari-Rouhani B (1993) Acoustic band struc- Rev B 62:5536 5540. ture of periodic elastic composites. Phys Rev Lett 71(13):2022–2025. 21. Lee J, et al. (2011) Targeted cell immobilization by ultrasound microbeam. Biotechnol – 10. Montero de Espinosa F, Jiménez E, Torres M (1998) Ultrasonic band gap in a periodic Bioeng 108(7):1643 1650. two-dimensional composite. Phys Rev Lett 80:1208–1211. 22. Shechtman D, Blech I, Gratias D, Cahn J (1984) Metallic phase with long-range ori- 11. Vasseur JO, et al. (2001) Experimental and theoretical evidence for the existence of entational order and no translational symmetry. Phys Rev Lett 53:1951–1953. absolute acoustic band gaps in two-dimensional solid phononic crystals. Phys Rev Lett 23. Christiansen B, Alstrom P, Levinsen MT (1992) Ordered capillary-wave states: quasi- 86(14):3012–3015. crystals, hexagons, and radial waves. Phys Rev Lett 68:2157–2160. 12. Baumgartl J, Zvyagolskaya M, Bechinger C (2007) Tailoring of phononic band struc- 24. Courtney CRP, et al. (2013) Dexterous manipulation of microparticles using Bessel- tures in colloidal crystals. Phys Rev Lett 99(20):205503. function acoustic pressure fields. Appl Phys Lett 102:123508.

Caleap and Drinkwater PNAS Early Edition | 5of5 Downloaded by guest on September 27, 2021