Dispersion assessment of three-dimensional Phononic Crystals using Laser Doppler Vibrometry
I. K. Tragazikis, D. A. Exarchos, P. T. Dalla, K. Dassios, T. E. Matikas, I. E. Psarobas
DDispersion assessment of three-dimensional Phononic Crystalsusing Laser Doppler Vibrometry
I. K. Tragazikis, D. A. Exarchos, P. T. Dalla, K. Dassios, and T. E. Matikas
Dept. of Materials Science & Engineering,University of Ioannina, 45110 Ioannina, Greece
I. E. Psarobas ∗ Section of Solid State Physics, National andKapodistrian University of Athens, 15784 Athens, Greece (Dated: March 6, 2019)The elastodynamic response of finite 3D phononic structures is analyzed bymeans of comparing experimental findings obtained through a laser Dopplervibrometry-based methodology and theoretical computations performed with theLayer-Multiple-Scattering method. The recorded frequency-gap spectrum of thephononic slabs exhibited a good agreement of theory to experiment. Along theselines, a newly developed technique, based on laser doppler vibrometry, has beenproposed and validated for the dispersion efficiency in 3D phononic metamaterials. a r X i v : . [ c ond - m a t . m t r l - s c i ] M a r INTRODUCTION
Wave propagation in inhomogeneous media is a problem of wide interest due to theimplications in technology and the scientific insight in understanding a large number ofphysical problems [1]. Classical wave transport in periodic media can provide the means tocontrol light (electromagnetic waves), sound (elastic waves) or both, with the development ofnovel materials, also known as classical spectral gap materials. Such a periodic arrangementof scatterers can obviously open up several directional spectral gaps. When for all directionsthe spectral gaps overlap so that there is a forbidden range of frequencies in which the wavescannot propagate in any direction, there is a special type of material that exhibits an absolutefrequency gap response. This paper deals with composite materials whose elastic propertiesvary periodically in space, and are also known as Phononic Crystals (PnC) [2]. PnCs on theother hand possess unique properties and exotic metamaterial features that can manifest ina wide range of frequencies from a macroscopic point of view with infrasound and seismicwaves, to mesoscopic systems with ultasound and up to hypersound and optomechanics aswell as nanoscale thermal devices [3]. All important physical phenomena associated withPnCs are completely scaled from the Hz to the THz regime and therefore conclusions andresults observed are independent of any limited frequency spectrum.Advanced materials with such properties aim to control the propagation of elastic waves(vibrations) in various technologically exploitable ways. Elastic wave transport in PnCs hasattracted an extraordinary amount of attention over the last decades [4], in fields rangingfrom frequency gap formation in 3D structures [5] and Anderson Localization of classicalwaves [6], to the study of structures of more exotic geometry [7, 8], negative modulus acousticmetamaterials [9–11] and even harvesting vibrations via PnC isolator modules [12].Omnidirectional frequency gaps do not appear easily in 3D solid PnCs. The reason is thatdirectional gaps corresponding to all degrees of freedom and for all directions of elastic wavesdo not necessarily overlap. However, it has been found that PnCs from non-overlappingscatterers of high density in a low density matrix (cermet topology) can function as absolutefrequency filters, regardless of directionality [5, 13]. There are various methods available forthe calculation of the elastic properties of PnCs [2], such as the traditional band-structuremethods, which mainly deal with periodic, infinite, and nondissipative structures. However,in an experiment, one deals with finite-size slabs and the measured quantities are, usually, thetransmission and reflection coefficients. Apart from that, realistic structures are dispersiveand exhibit losses. We recall that the usual band-structure calculation proceeds with agiven wave vector in order to compute the eigenfrequencies within a wide frequency rangetogether with the corresponding eigenmodes. On the contrary, on-shell methods proceeddifferently: the frequency is fixed and one obtains the eigenmodes of the crystal for thisfrequency. These methods are ideal when dealing with dispersive materials (with or withoutlosses). Moreover, on-shell methods are computationally more efficient than traditionalband-structure methods [14].Technological advancements, especially 3D-printing technology, have made easier to con-struct PnC slabs that mimic exactly the atom arrangement of crystalline matter so thatone can really benefit from the bulk properties of 3D phononic structures. Most of theexperiments so far have dealt with 2D phononic structures, in which cases theoreticalpredictions were adequately verified. In particular, the Layer-Multiple-Scattering (LMS)method [14, 15], a semi-analytical on-shell method with obvious advantages over pure nu-merical methods, has been examined for its accuracy in a 2D system formed by a monolayerof spheres [16]. In this paper, starting from modeling a 3D PnC within the framework ofLMS, we have constructed 3D PnC specimens of different symmetries and, following a thor-ough investigation of their crystallographic integrity, we were able to monitor their behaviorin the ultrasonic regime with Laser Doppler Vibrometry (LDV) [17], thus establishing anexperimental technique for observing the dispersion of 3D phononic metamaterials.
THEORY
The layered multiple-scattering theory provides a framework for a unified description ofwave propagation in three-dimensional periodic structures, finite slabs of layered structures,systems with impurities, namely isolated impurities, impurity aggregates, or randomly dis-tributed impurities [1]. In particular, the LMS method [14] is well-documented for the elas-todynamic response of PnCs with spherical and non-spherical inclusions [18]. The method,based on an ab initio multiple scattering theory [1], constitutes a powerful tool for an ac-curate description of the elastic (acoustic) response of composite structures comprised of anumber of different layers having the same 2D periodicity in the xy -plane (parallel to thelayers). LMS provides the complex band structure of the infinite crystal associated with a TABLE I. Mechanical properties of materials used, at room temperature.
Materials Density Longitudinal Shear Acoustic Young Shearspeed speed Impedance Modulus Modulus g/cm m/s m/s Kg/sm GPa GPaAir 1 . × −
343 - 4 × − − -Paraffin 0.9 2040 800 1.84 1.62 0.58Stainless steel 7.78 5760 3160 44.8 200 77.9Aluminum 2.7 6320 3130 17.06 70.76 26.45Cement paste 1.97 3680 1990 7.25 17.5 6.8 given crystallographic plane and also the transmission, reflection, and absorption coefficientsof an elastic wave incident at any angle on a slab of the crystal, parallel to a given plane, offinite thickness. An advantage of the method is that it does not require periodicity in the z -direction (perpendicular to the layers). In order to calculate the complex frequency bandstructure of the above crystal associated with the elastic field in the manner described inRef. [14], periodic boundary conditions are imposed initially and then, for a given angularfrequency ω and reduced wave vector k (cid:107) , we obtain the eigenmodes of the elastic field bydetermining k z . The reduced wave vector k (cid:107) (parallel to the crystallographic plane of stack-ing) and ω are given conserved quantities. k z follows from the definition of the wave vector k = [ k (cid:107) , k z ( ω, k (cid:107) )] of a generalized Bloch wave.The accuracy of the computations performed herein by the LMS code [14] is determined bythe cutoff values of the angular momentum number (cid:96) max coming from the spherical wave mul-tipole expansion and the number g max of reciprocal lattice vectors g used in the plane waveexpansion, necessary to incorporate Ewald summation techniques for faster convergence.In the following analysis, a cubic stacking, viewed as a succession of (001) crystallographicplanes, has been considered for both the hexagonal close-packing and the body-centeredcubic arrangements. For (cid:96) max = 7 and g max = 45 we have established a convergence ofless than 0.01% and no numerical instabilities were observed. Finally, all proper attenua-tion and realistic losses (experimentally measured) were taken into account, as the methodincorporates into the calculations many complex dispersion behaviors [19]. FIG. 1. PnC assembly. The white material on the right is the paraffin used.FIG. 2. The vacuum oven with the PnC inside.
FABRICATION OF PHONONIC CRYSTALS
For the construction of phononic slabs with appreciable gap width (as predicted theo-retically), different small-scale phononic structures with varying volume filling fraction wereused. Firstly, we have constructed slabs consisting of stainless steel spheres placed in aparaffin matrix at specific crystallographic lattice arrangements. Optimization of phononicstructures requires a significant difference in the values of acoustic impedance between thefiller and the matrix [4]. Stainless steel was chosen as the material for the inclusions dueto its high density which enables high speeds of longitudinal waves hence endowing highacoustic resistance. Paraffin was selected as matrix owing to its small acoustic impedance.The main properties of the materials used are given in Table I.Very high purity paraffin was used for the preparation of specimens with matrix trans- (a) (f)(e)(d) (c)(b)
FIG. 3. PnCs consisted of stainless steel spheres in paraffin matrix. The spheres have a 2 mm diameter (top row) and 3 mm on the bottom. From left to right, top layer along the (001) directionof the bcc crystal slab [(a) and (d)], top layer along the (001) direction of the hcp crystal slab [(b)and (e)]. On the right, the 3D crystal in two different arrangements. hcp and bcc in (c) and (f),respectively. (a) (b) FIG. 4. PnC of 3 mm diameter stainless steel spheres in cement paste. (a) is a bcc crystal, while(b) is of hcp arrangement parency which facilitated visualization of the geometry of the phononic structure. Due to itsvery low viscosity and low surface tension, paraffin is a very good solution for the manufac-ture of such samples. Stainless steel spherical inclusions of two different diameters, namely2 mm and 3 mm were used. Theoretical predictions (LMS) mandated the consideration oftwo crystal lattices of different symmetry, namely body centered cubic ( bcc ) and hexagonalclose packed ( hcp ). Then the spheres were prepared for stacking according to each indi-vidual lattice arrangement. Before being embedded in the matrix, inclusions were cleaned (a) (d)(c) (b) FIG. 5. IR thermogram of the bcc slab of 2 mm spheres in paraffin (on top), where (a) is the topPnC layer and (b) the 3rd inner layer from top. The bottom set corresponds to the hcp slab, where(c) is the top layer and (d) the 3rd inner layer from top. (a) (d)(c) (b) FIG. 6. IR thermogram of the hcp slab of 3 mm spheres in paraffin (on top), where (a) is the topPnC layer and (b) the 3rd inner layer from top. The bottom set corresponds to the same hcp slabin cement paste, where (c) is the top layer and (d) the 3rd inner layer from top. FIG. 7. Aluminum waveguide. At the center of the front face of the waveguide appears a smallcube of dimensions 10 × × mm . a b c e d FIG. 8. Experimental setup: (a) Tektronic TDS 1012B pulse generator, (b) RITEC RPR-4000high voltage pulse generator, (c) Control unit of the Laser Doppler Vibrometer, (d) the PnC slab,and (e) the 2D scanning laser head. in acetone and then in deionized water, in an ultrasonic bath, for removal of organic andother superficial residues originating from the production phases. Then, compressed air wasused to dry the spheres which were subsequently stored in airtight glass containers to iso-late inclusions from surrounding environment and avoid eventual further contamination; thecontainers were subjected to the same cleaning procedure. Attempts to create phononicsslabs without initially cleaning the spheres failed due to low stacking efficiency and creationof a large number of defects per level.For the construction of phononic slabs, plastic molds of internal dimensions of 28 × × mm were used which had been previously thoroughly cleaned. Stacking of the steelballs per level was done manually. For specimens with 2 mm beads, seven-layers thick,each layer/plane consisted of 14 beads in the x -direction and 14 beads in the y -direction.Specimens with 3 mm -diameter beads were constructed equally thick with the difference of FIG. 9. A detailed view of the PnC slab mounted on the waveguide. The red spot on the top ofthe crystal corresponds to the laser beam of the LDV. having 9 beads in the x -direction and 9 in the y -direction per stacking plane. The packingdensity of each cell in the hcp arrangement was ∼ ◦ C (paraffin melting point approximately 60 ◦ C) to allowreduction of paraffin viscosity and enable wetting of the steel sphere stacks. After this, theoven was allowed to cool down to room temperature, and vacuum process was released.Based on the aforementioned procedure, phononic slabs were available for ultrasonictesting. Figures 3 and 4 show the phononic slabs with inclusion diameters of 2 mm and3 mm , respectively.Additional PnCs of equivalent quality and 3 mm diameter inclusions were fabricated withcement paste as matrix material. Fabrication of these samples was much more demandingthan their paraffin-based counterparts, due to the higher viscosity of cement paste comparedto paraffin, making its impregnation much more difficult (Fig. 5).The quality of fabricated phononic slabs and the geometric arrangement of phononiccrystals in particular, was evaluated non-destructively by means of Infrared (IR) Lock-Inthermography. Infrared thermography (IRT) is the most appropriate technique for this task,as the thermal waves can penetrate both transparent materials (such as paraffin) as well asto opaque materials (such as the cement paste). As seen in Figures 6, 7, 8 and 9, the IRT-0 processing unit Halogen lampsPulse GeneratorLamp dimmerIR cameraPnC FIG. 10. Setup for imaging the PnC slab using lock-in IR thermography.FIG. 11. A chirp signal. In FFT on the left and in Time domain on the right. prone geometry of both the outer and inner layers of the spheres was found to be of adequatequality.
EXPERIMENTAL INVESTIGATION OF PNC SLABS
For the phononic gaps experiments, a high voltage pulse generator RITEC (RPR-4000)able to reproduce pulses of frequencies ranging from 0 to 22 MHz, up to 1000 V peak-to-peak, was used. As the generator cannot produce chirp signals, a Tektronic TDS 1012Bpulse generator was used as amplifier. In addition, a 2D laser Doppler vibrometer (LDV),Polytec PCV-400, was used as a non-contact sensor for performing non-contact vibrationmeasurements of the surface of the samples with the aid of a laser beam. Vibration am-plitudes and frequencies are extracted from the Doppler shift of the reflected laser beamfrequency due to the motion of the surface. LDV is advantageous in that the laser beamcan be directed at the surface of interest as well as in that the vibration measurement doesnot impose extra weight loading on the target structure. The technique has been used as a1 FIG. 12. Time Domain chirp signal on the left and after passing through a filter (bottom left).FFT of chirp signal on the right and after passing a filter (bottom right).FIG. 13. Experimental result of pure paraffin. vibration sensor in aerospace, industry, research, for crack detection in metallic structures,civil and mechanical engineering industries [17, 20–25]. Contact piezoelectric crystal-basedultrasound sensors with central frequency of 200 KHz 1.5 MHz and spectrums distributedaround its maximum frequency were also used.To guide ultrasound waves on the surfaceof the PnC slab, an aluminum waveguide of dimensions of 150 × × mm , with an2 Transmittance F r e qu e n cy ( M H z ) Lossless AttenuatedSteel spheres of 2 mm diameter in paraffin. structure: bcc - layers thick along the [ ] directionwith crystal lattice constant a= 2.3094 mm FIG. 14. Calculated transmission spectrum of a longitudinal wave along the [001] direction of a bcc phononic slab of 2 mm in diameter steel spheres in a paraffin matrix - 8 layers thick. The redcurve denotes the case with loses. The bcc arrangement has a lattice constant a = 2 . mm . additional central cube of dimensions 10 × × mm in the center of one surface, wasused (Fig. 7).Figure 10 shows the experimental arrangement for lock-in IR thermography, comprisingof the IR camera, the lamps, the pulse generator for thermal stimulation of the PnC, andthe processing unit of the thermographic results. RESULTS AND DISCUSSION
The vibrated stimulation of the PnCs was performed using a chirp signal, i.e. a constantamplitude signal with increased frequency over time. The phononic slabs were subjected toa range of vibration frequencies and their response to these frequency ranges was measured.This way, the stop bands for a specific frequency range was found. Initially, a pure paraffinstructure with the same thickness as the PnC slabs, without stainless steel spheres, wasevaluated. Figure 11 shows the resulting spectrum of application of a chirp vibration inpure paraffin, for a wide range of frequencies from 200 KHz to 1.25 MHz. As observed inthis figure, the spectrum in the entire frequency range is continuous and does not display3
FIG. 15. Experimental results for a bcc
PnC slab with 8 layers and steel spheres of 2 mm diameterin paraffin. The frequency bandgap extends between 480 and 750 KHz.FIG. 16. Frequency band structure at the center k (cid:107) = of the Surface Brillouin Zone (SBZ) ofthe (001) surface of an hcp PnC (denoted as dispersion). The crystal consists of 2 mm in diameterstainless steel balls in paraffin. The transmission spectrum of a longitudinal wave along the samedirection from a slab of the crystal 8 layers thick is presented in red. FIG. 17. Experimental results of hcp
PnC slabs with 8 layers and steel spheres of 2 mm diameterin paraffin. frequency gaps.For the design, construction and evaluation of materials that will provide shields to thevibrations, the physics of the interaction of elastic waves was studied. LMS is the mostreliable and complete solution for modeling PnCs in 3 dimensions. It is also the theoreticalapproach closer to a real experiment, since it calculates the coefficient of passage of acoustic(elastic) waves from a finite tile of the crystal. In the following, theoretical results basedon the LMS method are presented and compared to experimental data . First, the effect ofapplication of a chirp signal (750 KHz - 4 MHz) with a 1.2 3.2 MHz high-precision filteris depicted in Fig. 12. In this figure, the two upper frames show the reference chirp signalon time domain and FFT while the two lower frames are the results of cut filter. The chirpsignal is sent in bursts of duration of 200 µs , so that each frequency is generated in a specificamount of time.Figure 13 shows the results for pure paraffin. Therein, no frequency gap is observed.Figure 14) shows the LMS-expected frequency gaps in a 8-layer thick bcc phononic slabwith 2 mm diameter spheres as they appear in the appropriate transmission spectrum. Inaddition, an extra calculation was performed to account for all measured losses. It is evidentthat the positions of silent zones are not affected by any losses present in the system. Thistype of behavior was expected, following the theoretical predictions presented in Ref. [19].5 FIG. 18. Dispersion and transmission spectrum of the same hcp
PnC and slab as in Fig. 16, alongthe same crystallographic direction. In this case the spheres are 3 mm in diameter. Figure 15 shows the experimental result of a slab of a bcc
PnC with 8 layers and 2 mm diameter balls. The experimental results for this case are in full agreement with the resultsof LMS method. In particular, a frequency gap in the range of 480 - 750 KHz was observed.Figure 16 shows the LMS results (directional frequency band structure of an infinitecrystal) of a hcp PnC with 8 layers and 2 mm spheres while Fig. 17 shows the experimentalresults of the same. The observed gap is in the range of 420 -850 KHz as it is supportedby theory. In addition, the hcp arrangement exhibits larger gap as compared to its bcc counterpart. The red lines in Fig. 16 show the fluctuation of a longitudinal wave incidentperpendicular to a tile of the crystal.The frequency of gaps are reduced significantly for the 3 mm inclusions. By examinationof Figures 18 and 19, which represent theory and experiment, respectively, frequency gapappears between 290 and 600 KHz while a range of 280 - 590 KHz is theoretically predicted.The results of a PnC with the cement paste matrix are given in Figures 20 and 21 andfeature an hcp phononic slab, 8 layers thick with steel spheres of 3 mm in diameter. In thiscase, the frequency gap appears in the range of 0.8 - 1.1 MHz. Agreement of theory andexperiment follows as observed in all cases so far. In particular, it is interesting to mentionthat that the presence of deaf and shear bands in the band structure of Fig 20 do not couple6 FIG. 19. Experimental results for a PnC slab, 8 layers thick, following the specifics presented inFig. 18.TABLE II. Comparison between theory and experiment. ∆ ω/ω G represents the gap width overmidgap frequency and denotes the gap efficiency, which can be scaled to a wide range of sizeparameters.PnC type Freq. Gap Freq. Gap Gap Width Gap Width ∆ ω/ω G ∆ ω/ω G (sphere diameter) Theory Exp. Theory Exp. Theory Exp.KHz KHz KHz KHz % %Paraffin hcp (2 mm ) 400-850 420-850 450 430 72 68Paraffin hcp (3 mm ) 280-580 280-590 300 310 69.8 71.9Paraffin bcc (2 mm ) 450-750 460-750 300 290 50 48Cement hcp (3 mm ) 800-1100 820-1050 300 230 31.6 25 with the external elastic field and therefore remain inactive. Fact which is clearly seen onthe theoretical transmission spectrum and the experimental results on Fig. 21.A comparative summary of all theoretically determined vs experimentally obtained fre-quency gap ranges, widths and gap width over midgap frequency is presented in Table 2.Therein, one can clearly see that the measured observables are consistent with the underlyingtheory.7 FIG. 20. Dispersion and transmission response of an hcp
PnC and a slab 8 layers thick, consistingof 3 mm steel spheres in cement paste, along the [001] direction. Inside the gap deaf and shearbands remain inactive for longitudinal normal incidence. CONCLUSION
In the present study, a nondestructive technique based on LDV was employed in order torecord the band-gap formation of 3D PnCs. Several types of PnC slabs of varying crystallo-graphic symmetry in paraffin and cement matrices were fabricated and their frequency gapspectrum was recorded experimentally using the LDV method. The results were comparedwith theoretical expectations and good agreement was found in all cases. The favorablecomparison renders the LDV technique an effective and promising tool for the non-contact,non-destructive assessment of PnCs. The developed methodology has the potential to beequally versatile in cases where classical evaluation methods, requiring contact sensors, isnot possible. Such cases may include very small regions, high temperature conditions as wellas nanostructured metamaterials, where the mass of a contact sensor can affect the experi-mental results. Finally, it should be noted that the experimental methodology developed inthis study does not impose any size or scale limitations concerning the evaluation of a PnCslab.8
FIG. 21. . Experimental results for 8 layers thick hcp
PnC slab consisting of 3 mm in diametersteel spheres embedded in cement paste matrix. ACKNOWLEDEMENTS
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