The Rise of (Chiral) 3D Mechanical Metamaterials

materials

Article The Rise of (Chiral) 3D Mechanical Metamaterials

1, 1, 1 1,2, Janet Reinbold †, Tobias Frenzel †, Alexander Münchinger and Martin Wegener * 1 Institute of Applied Physics, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany; [email protected] (T.F.) 2 Institute of Nanotechnology, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany * Correspondence: [email protected] These authors have contributed equally. † Received: 10 October 2019; Accepted: 24 October 2019; Published: 28 October 2019

Abstract: On the occasion of this special issue, we start by briefly outlining some of the history and future perspectives of the field of 3D metamaterials in general and 3D mechanical metamaterials in particular. Next, in the spirit of a specific example, we present our original numerical as well as experimental results on the phenomenon of acoustical activity, the mechanical counterpart of optical activity. We consider a three-dimensional chiral cubic mechanical metamaterial architecture that is different from the one that we have investigated in recent early experiments. We find even larger linear-polarization rotation angles per metamaterial crystal lattice constant than previously and a slower decrease of the effects towards the bulk limit.

Keywords: mechanical metamaterials; chirality; acoustical activity

1. Introduction Consider the three-dimensional (3D) micro-lattice shown in Figure1 as an example. When giving talks to a broad audience, an ever-reoccurring question is whether one should see such an artificial crystal as a “structure” or as a “material”. The true answer is that both viewpoints are permissible. However, treating such rationally designed lattices as material or “metamaterial” has the advantage that the properties of the lattice can be mapped onto simple effective-medium parameters, such as, for example, the effective Young’s modulus in mechanics. Working further with these effective-medium parameters to design systems eases the treatment compared to looking at an entire system as a micro- or nanostructure. By analogy, computer chips are successfully designed by using electric conductivities etc., whereas the design would likely be impossible if the entire computer chip needed to be treated on the level of individual atoms. The “meta” in metamaterials emphasizes that these effective-medium properties go beyond the properties of the ingredient materials, qualitatively or quantitatively [1]. Sometimes, they can even go beyond what nature has to offer. However, one should be cautious when referring to nature. Nature offers a multitude of amazing microstructured materials, ranging from 3D photonic crystals in butterflies, beagles, and plants to mechanical materials such as wood, bone, or skin. It should be said that the understanding of the notion of “metamaterials” has changed significantly over the last 20 years [1]. Originally, metamaterials were solely associated with man-made lattices with centimeter-scale periods leading to effective negative refractive indices in electromagnetism at GHz frequencies. In fact, early definitions explicitly restricted metamaterials to electromagnetic waves [1]. Miniaturization of the unit cells by orders of magnitude brought the field to optical and even to visible frequencies [2]. This ground sometimes made the publication of early work on mechanical metamaterials difficult. One of us distinctly remembers reviewers’ comments such as “ ... the authors misuse the term

Materials 2019, 12, 3527; doi:10.3390/ma12213527 www.mdpi.com/journal/materials Materials 2019, 12, 3527 2 of 10 metamaterials ... ”, “ ... metamaterials require multiple different components. The structure discussed here has only one ... ”, “ ... metamaterials must address waves, whereas the authors speak about static properties ... ”, “ ... the static case lacks a characteristic length scale ... ” or “ ... the idea of mechanical metamaterials is not new. Many examples have been published decades ago ... ” etc. We know from personal discussions that others in the field of mechanical metamaterials faced very similar negative reactions early on. To some extent, these reactions were probably based on the fact that the notion “metamaterial” was a cash cow for funding in the early years of this century after the notion had been coined. For some, the idea of metamaterials just somehow sounded interesting and fashionable. Having said that, we should humbly admit that the idea of artificial composite materials with unusual properties is not new. The 2002 textbook by Graeme W. Milton “The Theory of Composites” [3] summarizes the history as well as the underlying physics and mathematics on more than 700 pages. The word “metamaterial” does not appear once in this book though. In mechanics, for example, Roderik Lakes published early work on auxetics [4]. Milton and Cherkaev introduced the concept of pentamodes (or “anti-auxetics”) [5], from which any linearly elastic material describable by Cauchy elasticity can be constructed conceptually. However, experimental advances in nanofabrication and in 3D printing on the macro- and on the micro-scale have spurred interest tremendously and have in turn stimulated novel designs and design approaches [1]. Mechanical metamaterials offer opportunities yet beyond those of optical metamaterials. In the linear regime, mechanics is richer than optics. This statement has been proven mathematically [6] by arranging the equations of mechanics and electromagnetism into the same form. Intuitively, mechanics always comprises transverse and longitudinal excitations at the same time. In contrast, longitudinal waves can appear in optics in only a few rare exceptions. In the nonlinear mechanical regime, nonlinear geometric effects can lead to huge nonlinearities and multi-stable behavior even if the constituents behave purely linearly elastic [7,8]. In contrast, nonlinear effects in optics are almost always very small corrections of the linear properties. Recent reviews on mechanical metamaterials in general can also be found in [9–11]. Nevertheless, some effects that have been known for decades in optics have not been observed in mechanics until recently. Ultrasound acoustical activity emphasized in this paper is one such example. More broadly, some of us have speculated previously [1] that the futures of 3D metamaterials and 3D printing may be closely linked: In 2D graphical printing (ink-jet or laser printing), one routinely uses the concept of “dithering” to generate thousands of apparent or effective colors from just four ink cartridges (black, cyan, magenta, and yellow). Looking through a microscope, one notices that the 2D printer can actually only print dot patterns, with each dot originating from one of these four cartridges. From a distance, the human eye cannot resolve this substructure anymore and the patterns appear as different homogeneous effective colors. The analogue of the “dithering” substructure in 3D material printing is the metamaterial unit cell. By tailoring the unit cell structure, thousands of different effective material properties can be achieved from just a few material cartridges of the 3D printer. The 3D micro-architectures discussed below are even composed of only a single polymer material and voids within.

2. Enhanced Acoustical Activity The architecture shown in Figure1 has been inspired by the one introduced in [ 12]. It is composed of helical wire bundles, each bundle with four individual wires or beams. The different bundles are connected via small cubic elements and are arranged on a simple-cubic translational lattice, leading to a 3D cubic chiral mechanical metamaterial. The constituent material (gray) shall be describable by ordinary linear Cauchy elasticity [13]. Chiral objects are distinct from their mirror image, hence chiral structures lack centrosymmetry, mirror symmetries, and rotation-reflection symmetries [14–16]. However, any effect of chirality of the mechanical metamaterial goes beyond the regime of Cauchy elasticity. In other words, chirality plays strictly no role in a Cauchy continuum [13]. In sharp contrast, Eringen micropolar elasticity [17,18] (sometimes also loosely referred to as Cosserat elasticity) and Materials 2019, 12, 3527 3 of 10 Materials 2019, 12, x FOR PEER REVIEW 3 of 10

Willisagreement elasticity with [19 previous] can describe experimental eﬀects of results chirality on 3D and mechanical have led to metamaterials good agreement [18 with] in the previous static regime.experimental Various results other on chiral 3D mechanical mechanical metamaterials lattices have [ re18cently] in the also static been regime. inves Varioustigated in other the chiral static regimemechanical [18,20 lattices]. have recently also been investigated in the static regime [18,20].

Figure 1. ((aa)) and and ( (bb)) are are different different views onto the blueprint of of one cubic unit cell of the 3D chiral mechanical metamaterial metamaterial lattice lattice considered considered here. here. The The indicated indicated geometrical geometrical parameters parameters are: a = are:250 푎µ m=, b250= 10µmµ, m푏, c== 1050 µmµm, ,푐 and = d50= µm6.25, andµm 푑. Parameters= 6.25 µm of. Parameters the constituent of the polymer constituent material polymer are: material Young’s are:modulus Young’s (or storage modulus modulus) (or storageE = 4.18 modulus)GPa, Poisson’s 퐸 = 4 ratio.18 GPaν =, 0.4 Poisson’s, and mass ratio density 휈 = ρ0.=4 , 1.15 andg / masscm3 density(cf. [21]). 휌 For = some1.15 g of/c them3 calculations,(cf. [21]). For we some have of addedthe calculations, an imaginary we have part (oradded loss an modulus) imaginary of 0.20 partGPa (or lossto the modulus) quoted real of 0 part.20 GPa of the to polymer the quoted Young’s real part modulus. of the Whenever polymer Young’s applicable, modulus. we will Whenever explicitly applicable,mention this we finite will imaginary explicitly part, mention which this describes finite imaginary damping of part, the whichelastic waves.describes damping of the elastic waves. Here, we rather focus on the dynamic or wave regime of chiral architectures. The phenomenon of “acousticalHere, we activity”rather focus (alternatively on the dynamic named or mechanical wave regime activity of chiral or elasticarchitectures activity)—the. The phenomenon mechanical ofcounterpart “acoustical of activity” optical activity—was(alternatively named predicted mechanical theoretically activity years or elastic ago [22 activity)]. The— notionthe mechanical “activity” counterpartstands for the of rotation optical of activity the linear—was polarization predicted axistheoretically of a transversely years ago polarized [22]. The optical notion or mechanical “activity” wave,stands withfor the the rotation rotation of angle the linear being polarization proportional axis to theof a propagation transversely distance polarized and optical independent or mechanical of the wave,orientation with the of therotation incident angle linear being polarization. proportional To to avoid the propagation confusion, distance we emphasize and independent that the material of the orientationitself is passive, of the meaning incident that linear neither polarization. external energy To avoid sources confusion, nor switchable we emphasize material that parameters the material are itselfrequired. is passive, This rotation meaning due that to neither acoustical external activity energy must sources not be nor confused switchable with material Faraday parameters rotation and are is required.due to the This fact rotation that the eigenmodesdue to acoustical correspond activity to must left-handed not be confused and right-handed with Faraday circular rotation polarization and is dueand to that the these fact eigenmodesthat the eigenmodes propagate correspond with different to left phase-handed velocities. and right Following-handed the circular original polarization theoretical andprediction that these [22], direct eigenmodes experiments propagate on quartz with crystals different at about phase1 GHz velocitiesfrequency. Following were published the original [23]. theoreticalQuartz is also prediction a paradigm [22], crystal direct for experiments obtaining optical on quartz activity. crystals Later, at di ff abouterent continuum1 GHz frequency descriptions were publishedof acoustical [23 activity]. Quartz were is compared also a paradigm [24,25], all crystal for the for case obtaining of infinitely optical extended, activity. that Later, is bulk, different crystals. continuumMore recently, descriptions we have of presentedacoustical an activity early experimentalwere compared demonstration [24,25], all of for acoustical the case activity of infinitely in 3D extended,mechanical that metamaterials is bulk, crystals. [21]. The effects observed here [21] have been much larger than the effects for ordinaryMore recently, crystals we [23 have]. More presented importantly, an early one experimental cannot change demonstration the optimum operationof acoustical frequency activity forin 3Dordinary mechanical crystals, metamaterials whereas it can [21 easily]. The beeffects tailored observed by choice here of [21 the] have lattice been constant much in larger the case than of t 3Dhe ecrystallineffects for metamaterials. ordinary crystals The [23 aim]. of More the present importantly, paper is one to add cannot a second change experimental the optimum demonstration operation frequencyon a different for chiralordinary metamaterial crystals, whereas lattice (cf. it can Figure easily1). be tailored by choice of the lattice constant in the caseIn Figureof 3D crystalline2, we show metamaterials. numerically T calculatedhe aim of th phonone present band paper structures is to add (i.e., a second eigenfrequencies experimental demonstrationversus wave number) on a different of metamaterial chiral metamaterial beams based lattice on the(cf. crystalFigure unit1). cell depicted in Figure1. The geometricalIn Figure and 2, polymer we show material numerically parameters calculated are given phonon in the band caption structures of Figure (i.e.,1. The eigenfrequencies calculations in versusFigure 2wave refer number) to metamaterial of metamaterial beams with beams a cross based section on the of crystalNx unitNy cellunit depicted cells (as in indicated). Figure 1. The × geometricalmetamaterial and beams polymer are infinitely material extended parameters along are the given propagation in the caption direction of Figure of the 1. waves The calculations (z-direction, wavein Figure number 2 referkz ).to Thesemetamaterial solutions beams have with been a obtained cross section by using of 푁 the푥 × eigenmode 푁푦 unit cells solver (as indicated MUMPS). in The the metamaterialsoftware package beams Comsol are infinitely and by extended using Bloch-periodic along the propagation boundary conditionsdirection of along the waves the z-direction. (푧-direction, wave number 푘푧). These solutions have been obtained by using the eigenmode solver MUMPS in the software package Comsol and by using Bloch-periodic boundary conditions along the 푧-direction.

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푁 × 푁 Figure 2. Calculated phonon phonon band band structures structures of of metamaterial metamaterial beams beams with with a across cross section section of of N푥x N푦y ⃗ ×⃗ unit cells in the 푥푦-plane and with 푁푧 = ∞ for wave vectors,→ 푘, along the 푧-direction,→ i.e., 푘 = unit cells in the xy-plane and with Nz = for wave vectors, k , along the z -direction, i.e., k = (0, 0, kz) (0,0, 푘푧) with wave number 푘푧. (a) 푁푥 =∞ 푁푦 = 2, (b) 푁푥 = 푁푦 = 4, and (c) bulk with 푁푥 = 푁푦 = with wave number kz.(a) Nx = Ny = 2,(b) Nx = Ny = 4, and (c) bulk with Nx = Ny = . The ∞. The transverse (or shear or flexural) bands are highlighted in red. Without chirality, the∞ two transverse (or shear or ﬂexural) bands are highlighted in red. Without chirality, the two transverse transverse bands would be degenerate due to the four-fold rotational symmetry of the metamaterial bands would be degenerate due to the four-fold rotational symmetry of the metamaterial crystal (see crystal (see dashed red curves). The blue bands correspond to longitudinal-like (or pressure-like) and dashed red curves). The blue bands correspond to longitudinal-like (or pressure-like) and the black the black bands to twist-like modes, respectively. The higher bands are not important in the context bands to twist-like modes, respectively. The higher bands are not important in the context of this paper ofand this are paper plotted and in lightare plotted gray for in clarity. light gray Parameters for clarity. have Parameters been given have in Figure been1 ;given the imaginary in Figure part 1; the of imaginary part of the polymer Young’s modulus is set to zero. In panel (a), the maximum splitting of the polymer Young’s modulus is set to zero. In panel (a), the maximum splitting of the red bands, ∆kz, the red bands, 훥푘푧, is about half of 휋/푎, corresponding to a rotation angle of about 45° per lattice is about half of π/a, corresponding to a rotation angle of about 45◦ per lattice constant, which approach constant, which approach the fundamental bound of 90° per lattice constant. the fundamental bound of 90◦ per lattice constant.

For finite finite values values of ofN x푁푥= =N y푁,푦 we, we use use traction-free traction-free boundary boundary conditions conditions at the at metamaterial the metamaterial beam beamsurface surface along along the x- the and 푥 the- andy-direction. the 푦-direction. For N푁푥x = N푁푦y = ∞ (bulk(bulk case),case), we also use Bloch Bloch-periodic-periodic boundary boundary conditions conditions along along the the 푥x-- ∞ and the y푦-direction.-direction. This overall procedure is the same as in [21 [21], where it was used for a different different metamaterial architecture. To ease ease the the discussion, discussion, we have have colored colored the the bands bands in in FigureFigure2 2(see (see legend). legend). However, However, this this coloration has to be taken with a grain grain of of salt salt because because most most bands bands are are mixed mixed in in character. character. For For example, example, in thethe presencepresence of of chirality, chirality, the the longitudinal longitudinal pressure pressure bands bands (blue) (blue) and theand twist the bandstwist bands (black) (black) are mixed. are Thismixed. mixture This is mixture the immediate is the immediate dynamic counterpart dynamic of counterpart the (quasi-)static of the push-to-twist(quasi-)static conversion push-to-twist [18] (alsoconversion see [21 [18]).] Furthermore,(also see [21]) the. Furthermore, transverse polarizedthe transverse bands polarized or shear bands modes or or shear TA phonons modes or (red), TA whichphonons are (red) strictly, which speaking are s flexuraltrictly speaking modes for flexural beams modes with finite for beams cross section, with finite exhibit cross avoided section, crossings exhibit withavoided localized crossings (“optical”) with localized modes at (“optical”) higher frequencies modes at in higher the middle frequencies of the first in the Brillouin middle zone of the around first 1 π 1 휋 Brillouinkz . Thesezone around higher-frequency 푘푧 ≈ . T bandshese higher are plotted-frequency in light bands gray are because plotted they in arelight of gray lesser because importance they ≈ 2 a 2 푎 arein the of lesser context importance of this article. in the The context dashed of redthis curvearticle. shows The dashed the two red degenerate curve shows transverse the two bands degenerate in the transverseachiral reference bands case.in theHere, achiral each reference wire in case. the bundlesHere, each of fourwire (cf.in the Figure bundles1) is parallelof four (cf. to aFigure principal 1) is parallelcubic axis, to i.e.,a principal we have cubic eliminated axis, i.e., the we twisting have eliminate of the bundlesd the (nottwist depicted).ing of the Bybundles comparison (not depicted) with the. Byred comparison chiral case inwith Figure the 2red, we chiral see that case chirality in Figure leads 2, we to see a significant that chirality sti ffleadsening to of a significant the structure, stiffening thus to ofan the increase structure, of the thus phase to an velocity increase of theof the transverse phase velocity bands. of the transverse bands. The two red bands or chiral phonons [21 [21] in Figure 22 are are importantimportant inin thethe contextcontext ofof acousticalacoustical activity. ByBy thethe four-foldfour-fold rotational rotational symmetry symmetry along along a principal a principal cubic cubic axis axis (hence (hence also thealsoz -axis),the 푧-axis) these, thetwose bands two arebands degenerate are degenerate in absence in of absence chirality of (dashedchirality red (dashed curves). red In thecurves) presence. In the of chirality,presence at of a chirality,given fixed at angulara given frequencyfixed angularω, the frequency splitting 휔 of, thesethe splitting two bands of these in phonon two bands wave in number, phonon∆ kwavez(ω), number,determines 훥푘 the푧(휔 polarization-rotation), determines the polarization angle, ϕ,- viarotation the equation angle, 휑 [,21 via] the equation [21] 퐿 Lz푧 ϕ휑((ω휔))== ∆훥k푘푧((ω휔)) (1(1)) z 22

Here, 퐿푧 ≥ 0 is the propagation distance or length, and the factor 훥푘푧(휔) stems from the different phase velocities of left-handed and right-handed circular polarization (see above) [21]. The modulus of the rotation angle is fundamentally bounded by

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Here, Lz 0 is the propagation distance or length, and the factor ∆kz(ω) stems from the diﬀerent ≥ phase velocities of left-handed and right-handed circular polarization (see above) [21]. The modulus of the rotation angle is fundamentally bounded by

π Lz ϕ(ω) (2) ≤ 2 a that is, it cannot be larger than 90◦ per lattice constant a. This bound can be understood as follows. The two red bands correspond to circular polarization, which require three- or fourfold rotational symmetry → along the axis of the wave vector k = (0, 0, kz). As usual, the two wave numbers kz = π/a at the edge ± of the first Brillouin zone are equivalent. When replacing kz kz , a left-handed circular mode turns → − into a right-handed and vice versa. Therefore, the left- and the right-handed mode must be degenerate at the Brillouin-zone edge. Thus, the maximum possible wave number difference for a propagating wave in either the positive or the negative kz-direction at given frequency ω is ∆kz(ω) = π/a. From here, the above bound follows immediately. By comparison of Figure2 with the band structures of another previously discussed 3D metamaterial [21], we find that the anticipated rotation angles per lattice constant are larger in the present paper by about a factor of two for a given value of Nx = Ny. The difference is largest for the bulk case of Nx = Ny = Nz = . This aspect is beneficial for potential applications, in which one ∞ may want to convert one linear polarization into the orthogonal one (i.e., 90-degrees rotation) over a propagation distance as short as possible. The choice of the structure parameters shown in Figure1 and used in Figure2 as well as in the experiments to be described in the following section has resulted from a trade-off between experimental practicability, magnitude of acoustical activity, and bandwidth of acoustical activity. To help the reader appreciating the complexity of the behavior, we show in Figure3 a series of band structures for the bulk case of Nx = Ny = Nz . Here, as usual, the twist bands are completely absent due to the → ∞ three-dimensional Bloch-periodic boundary conditions. We fix the lattice constant a and vary the ratio c/a, i.e., we vary the size of the inner cube in Figure1. All other parameters are as in Figures1 and2. We depict values ranging from c/a = 0.1 to c/a = 0.8. While the behavior remains qualitatively the same when going from c/a = 0.2 (identical to Figure2) to c/a = 0.1 in Figure3, the red transverse bands develop into a bubble-like structure towards intermediate values of c/a = 0.5. Here, the two red bands are nearly degenerate of up to wave numbers of kz/a 1. This behavior means that little if any ≈ acoustical activity appears below a certain minimum frequency of 150 kHz. Together with an upper maximum frequency of the bands of about 210 kHz, acoustical activity is expected to appear only in a narrow frequency range in this case. For yet larger ratios towards c/a = 0.8, the band structures become again similar to the cases of c/a = 0.1 and 0.2. On this basis, we have selected c/a = 0.2 (cf. Figures1 and2) for our experiments. While many other geometrical parameters such as the b/a and the d/a ratio are not critical, it should be mentioned that it is important that the individual wires in the sets of wire bundles are not unintentionally connected on the way from one small cubic connection element to the next. Such connection would substantially reduce the twist of these beams, and hence the aimed-at effect (not depicted). This aspect has led us to stay away from yet smaller c/a ratios. Materials 2019, 12, x FOR PEER REVIEW 5 of 10

휋 퐿푧 |휑(휔)| ≤ (2) 2 푎 that is, it cannot be larger than 90° per lattice constant 푎. This bound can be understood as follows. The two red bands correspond to circular polarization, which require three- or fourfold rotational ⃗ symmetry along the axis of the wave vector 푘 = (0,0, 푘푧). As usual, the two wave numbers 푘푧 = ± 휋/푎 at the edge of the first Brillouin zone are equivalent. When replacing 푘푧 → −푘푧, a left-handed circular mode turns into a right-handed and vice versa. Therefore, the left- and the right-handed mode must be degenerate at the Brillouin-zone edge. Thus, the maximum possible wave number difference for a propagating wave in either the positive or the negative 푘푧 -direction at given frequency 휔 is |훥푘푧(휔)| = 휋/푎. From here, the above bound follows immediately. By comparison of Figure 2 with the band structures of another previously discussed 3D metamaterial [21], we find that the anticipated rotation angles per lattice constant are larger in the present paper by about a factor of two for a given value of 푁푥 = 푁푦. The difference is largest for the bulk case of 푁푥 = 푁푦 = 푁푧 = ∞. This aspect is beneficial for potential applications, in which one may want to convert one linear polarization into the orthogonal one (i.e., 90-degrees rotation) over a propagation distance as short as possible. The choice of the structure parameters shown in Figure 1 and used in Figure 2 as well as in the experiments to be described in the following section has resulted from a trade-off between experimental practicability, magnitude of acoustical activity, and bandwidth of acoustical activity. To help the reader appreciating the complexity of the behavior, we show in Figure 3 a series of band structures for the bulk case of 푁푥 = 푁푦 = 푁푧 → ∞. Here, as usual, the twist bands are completely absent due to the three-dimensional Bloch-periodic boundary conditions. We fix the lattice constant 푎 and vary the ratio 푐/푎, i.e., we vary the size of the inner cube in Figure 1. All other parameters are as in Figures 1 and 2. We depict values ranging from 푐/푎 = 0.1 to 푐/푎 = 0.8. While the behavior remains qualitatively the same when going from 푐/푎 = 0.2 (identical to Figure 2) to 푐/푎 = 0.1 in Figure 3, the red transverse bands develop into a bubble-like structure towards intermediate values of 푐/푎 = 0.5. Here, the two red bands are nearly degenerate of up to wave numbers of 푘푧/푎 ≈ 1. This behavior means that little if any acoustical activity appears below a certain minimum frequency of 150 kHz. Together with an upper maximum frequency of the bands of about 210 kHz, acoustical Materialsactivity2019 is expected, 12, 3527 to appear only in a narrow frequency range in this case. For yet larger ratios6 of 10 towards 푐/푎 = 0.8, the band structures become again similar to the cases of 푐/푎 = 0.1 and 0.2.

Figure 3. Calculated bulk phonon band structures (i.e., 푁 = 푁 = 푁 → ∞) for the metamaterial Figure 3. Calculated bulk phonon band structures (i.e., N푥x = N푦y = Nz푧 ) for the metamaterial → ∞ deﬁneddefined inin FigureFigure1 .1. The The lattice lattice constant constanta and푎 and the the other other parameters parameters are the are same the same as in Figureas in Figure2c, where 2c, cwhere/a = 0.2푐/,푎 but = we0.2, vary but we the varyc/a ratio the (as푐/푎 indicated). ratio (as indicated). The bands The are bands colored are as colored in Figure as2 .in In Figure particular, 2. In theparticular, chiral transverse the chiral acoustical transverse bands acoustical are again bands highlighted are again highlightedin red. (a) c / ina = red.0.1 (,(a)b 푐)/c푎/ a== 00.2.1, ,( (bc) c푐/푎a ==0.3 0.,(2,d ()cc)/ 푐a/=푎 0.4= ,(0.e3), c/ (da) =푐/0.5푎 ,(=f) 0c./4a, = (e)0.6 푐/,(푎g )=c/ 0a.5=, 0.7 (f) ,푐 and/푎 (=h) 0c./6a, = (g)0.8 푐./푎 For = each 0.7,c / anda ratio, (h) an푐/푎 inset = 0 illustrates.8. For each the 푐/ corresponding푎 ratio, an inset metamaterial illustrates the unit corresponding cell. metamaterial unit cell.

3. Experiments To test the above prediction of large polarization rotation angles (cf. Figure2) for the architecture shown in Figure1, we have manufactured corresponding polymer samples by standard 3D laser microprinting. The target parameters have been deﬁned in Figure1. Concerning the fabrication details, we refer the reader to Refs. [18,21] and the early work on “dip-in” mode [26], which is now widely used for the making of microstructured polymer-based 3D mechanical metamaterials by 3D laser nanoprinting [27]. We depict example electron micrographs in Figure4. In contrast to our previous work [21], we have not added a plate at the sample top. We sinusoidally excite the samples at their bottom by a piezoelectric actuator at frequency f = ω/(2π), with an amplitude of some 10 nm along the y-direction. We stroboscopically illuminate the samples by short pulses of two light-emitting diodes (850 nm center wavelength, 1.5% duty cycle), the repetition rate of which is synchronized with the piezoelectric excitation. We process the obtained microscope images at the top of the sample and at the sample bottom by using image cross-correlation analysis [18,21]. In this fashion, we can detect displacement vectors at diﬀerent locations, →u →r , the moduli of which are much smaller than a pixel of the camera used for the recording and which are much smaller than the illuminating wavelength. At the sample bottom, we track the markers (cf. Figure4); at the sample top, we track the ends of the four rods of the inner four unit cells, that is, 16 rod ends. By comparing the measured time-dependent displacement vectors at the sample bottom and sample top at a given frequency, we extract the polarization rotation angle ϕ, that is, the strength of acoustical activity. We refer interested readers to a more detailed description of this measurement setup given in Ref. [21]. Materials 2019, 12, x FOR PEER REVIEW 6 of 10

On this basis, we have selected 푐/푎 = 0.2 (cf. Figures 1 and 2) for our experiments. While many other geometrical parameters such as the 푏/푎 and the 푑/푎 ratio are not critical, it should be mentioned that it is important that the individual wires in the sets of wire bundles are not unintentionally connected on the way from one small cubic connection element to the next. Such connection would substantially reduce the twist of these beams, and hence the aimed-at effect (not depicted). This aspect has led us to stay away from yet smaller 푐/푎 ratios.

Figure 4. SelectedFigure 4. oblique-viewSelected oblique electron-view electron micrographs micrographs of aof 3D a 3D chiral chiral cubic cubic polymer polymer metamaterial metamaterial sample sample manufactured by standard 3D laser micro-printing, following the blueprint illustrated in manufactured by standard 3D laser micro-printing, following the blueprint illustrated in Figure1. Figure 1. (a) Total view onto one metamaterial sample with 푁푥 × 푁푦 × 푁푧 = 5 × 5 × 12 unit cells (a) Total viewand the onto bottom one sample metamaterial holder. Here, sample we use with no plateNx atN they top.Nz (b=) Zoom5 5-in, 12showingunit cellsthe intricate and the bottom × × × × sample holder.interior Here, composed we useof sets no of plate twisted at rods. the top. (b) Zoom-in, showing the intricate interior composed of sets of twisted rods. We sinusoidally excite the samples at their bottom by a piezoelectric actuator at frequency 푓 = 휔/(2휋), with an amplitude of some 10 nm along the 푦-direction. We stroboscopically illuminate the 4. ResultsMaterials and Discussion 2019, 12, x FOR PEER REVIEW 7 of 10 samples by short pulses of two light-emitting diodes (850 nm center wavelength, 1.5% duty cycle), Example4.the Results repetition raw and data rate Discussion of for which three is synchronized selected excitation with the piezoelectric frequencies excitation. f are We depicted process the in obtained Figure 5. Here, microscope images at the top of the sample and at the sample bottom by using image cross-correlation Nx = Ny = 3Exampleand N rawz = data12 for. Wethree showselected the excitation individual frequenciesx- andf are depictedy-components in Figure 5 of. Here, the 푁 displacement = analysis [18,21]. In this fashion, we can detect displacement vectors at different locations, 푢⃗ (푟 ),푥 the 푁 = 3 and 푁 = 12. We show the individual 푥- and 푦-components of the displacement vector vector →u =moduli푦 ux, ofu ywhichversus푧 are much time smaller at the than sample a pixel bottomof the camera (left used column) for the recording as well and as atwhich the are sample top 푢⃗ = (푢 , 푢 ) versus time at the sample bottom (left column) as well as at the sample top (middle (middle column).much smaller푥 푦 than the illuminating wavelength. At the sample bottom, we track the markers (cf. column).Figure 4); at the sample top, we track the ends of the four rods of the inner four unit cells, that is, 16 rod ends. By comparing the measured time-dependent displacement vectors at the sample bottom and sample top at a given frequency, we extract the polarization rotation angle 휑, that is, the strength of acoustical activity. We refer interested readers to a more detailed description of this measurement setup given in Ref. [21].

Figure 5. MeasuredFigure 5. Measured displacement-vector displacement-vector components components (black (black and and green) green) versus time, time, taken taken at atthe the sample sample bottom (left column) and at the top of the sample (middle column), respectively for three bottom (leftdifferent column) frequencies and at f. theThe right top ofcolumn the sampleshows the (middle 푦-component column), versus respectivelythe 푥-component for for three the diﬀerent frequenciesbottom f. The (blue) right as column well as for shows the top the (red).y-component The sample versus is excited the at x its-component bottom by a piezoelectricfor the bottom (blue) as well astransducer for the top with (red). (a) 푓 = The 10 kHz sample, (b) 푓 is= excited20 kHz, and at (c its) 푓 bottom= 30 kHz by. The a metamaterial piezoelectric beam transducer has a with (a) f = 10crosskHz section,(b) off =푁푥20 × kHz푁푦 =, 3 and × 3 ( cunit) f cells= 30 andkHz a height. The of 푁 metamaterial푧 = 12 unit cells. beam From these has aexample cross section of data, we derive a polarization rotation angle of (a) 휑 = 2°, (b) 휑 = 32°, and (c) 휑 = 43°. Nx Ny = 3 3 unit cells and a height of Nz = 12 unit cells. From these example data, we derive a × × polarizationThe rotation right column angle shows of (a) ϕthe= same2◦,( bdata) ϕ represented= 32◦, and as (c )푦ϕ-component= 43◦. versus 푥-component for the sample bottom (blue) and the sample top (red). The extracted rotation angles 휑 are indicated, too. A summary of many experiments (open dots) similar to the one shown in Figure 5, plotted in

the form of rotation angle 휑 versus frequency 푓, with 푁푥 = 푁푦 and 푁푧 as parameters, is depicted in Figure 6. The experiments are compared with numerical results from metamaterial-phonon band- structure calculations (dashed curves) and with numerical results for the finite samples as in the experiment. The finite length of the structure unavoidably leads to reflections. However, due to time- reversal symmetry, the rotation of the polarization is reversed on its way back and has the same direction as the excitation when it reaches the bottom of the sample again [21]. Therefore, the finite length of the sample influences the amplitude of the measured wave, yet it leaves the polarization direction unaffected [21].

Materials 2019, 12, 3527 8 of 10

The right column shows the same data represented as y-component versus x-component for the sample bottom (blue) and the sample top (red). The extracted rotation angles ϕ are indicated, too. A summary of many experiments (open dots) similar to the one shown in Figure5, plotted in the form of rotation angle ϕ versus frequency f , with Nx = Ny and Nz as parameters, is depicted in Figure6. The experiments are compared with numerical results from metamaterial-phonon band-structure calculations (dashed curves) and with numerical results for the finite samples as in the experiment. The finite length of the structure unavoidably leads to reflections. However, due to time-reversal symmetry, the rotation of the polarization is reversed on its way back and has the same direction as the excitation when it reaches the bottom of the sample again [21]. Therefore, the finite length of the sample influences the amplitude of the measured wave, yet it leaves the polarization direction unaffected [21]. Materials 2019, 12, x FOR PEER REVIEW 8 of 10

FigureFigure 6. Derived from from data data like like those those shown shown in in Figure Figure 5,5, we we plot plot the the rotation rotation angle angle 휑ϕ == 휑ϕ((푓f)) versus 2 excitation frequency f for different beam cross sections, (푁 × 푁2 )푎 , and for different numbers of excitation frequency f for different beam cross sections, Nx 푥 Ny a 푦, and for different numbers of unit 푁 × cells,unit cells,Nz, along 푧, along the propagation the propagation direction. direction. The measured The measured results results are depicted are depicted as circles; as circles the solid; the curve solid hascurve been has obtained been obtained from numerical from numeric finite-elemental finite-element frequency-domain frequency-domain calculations calculations for finite-size for finite samples-size (assamples in the (as experiment, in the experiment, cf. Figure 5cf.), accountingFigure 5), accounting for a finite for imaginary a finite partimagina of thery Young’spart of the modulus Young’sE (cf.modulus Figure 1퐸); (cf. and Figure the dashed 1); and curves the dashed have been curves obtained have frombeen phononobtained band-structure from phonon calculationsband-structu (cf.re Figurecalculations2), assuming (cf. Figure zero 2), imaginary assuming part zero of imaginaryE. In the upperpart of right-hand퐸. In the upper side panel, right-hand we vary sideN panel,z at fixed we Nvaryx = 푁N푧y .at In fixed the lower 푁푥 = right-hand 푁푦. In the sidelower panel, right we-hand show side by panel, the dashed we show black by curve the dashed the expectation black curve for thethe bulkexpectation limit, obtained for the bulk from limit, phonon obtained band-structure from phonon calculations, band-structure again with calculations, zero imaginary again with part zero of E. Obviously,imaginary part the N ofx 퐸N. Obviously,y = 5 5 case the (blue) 푁푥 × is 푁 already푦 = 5 × very 5 case close (blue) to the is bulk already limit very (dashed close back to the curve). bulk × × Atlimit around (dashedf = back100 curve). kHz, we At obtain around a polarization푓 = 100 kHz rotation, we obtain as large a polarization as about 30 rotation◦ per lattice as large constant. as about 30° per lattice constant. We find the largest rotation angle per lattice constant of 30◦ for Nx = Ny = 2. at f = 100 kHz. The rotationWe angle find generallythe largest decreases rotation with angle increasing per latticeNx = constantNy. The of 3D 30° bulk for limit 푁 (=N x푁= N=y 2 at 푓) is= 푥 푦 → ∞ already 100 kHz approached. The rotation at aroundangle generallyNx = Ny decreases= 5, for which with weincreasing obtain about 푁푥 =20 푁◦푦polarization. The 3D bulk rotation limit ( angle푁푥 = per 푁 lattice→ ∞ ) constant is already at a frequency approached of f at= around100 kHz .푁 The = bulk 푁 = limit 5 , of forL / whicha weleads obtain to a finite about value 20° 푦 푥 푦 → ∞ ofpolarizationϕ because the rotation ratio of angle wavelength per lattice to lattice constant constant, at a λ frequency/a, remains of finite.푓 = 100 Cauchy kHz. elasticity, The bulk for limit which of 퐿/푎 → ∞ leads to a finite value of 휑 because the ratio of wavelength to lattice constant, 휆/푎, remains finite. Cauchy elasticity, for which the rotation effect would be zero, requires both, 퐿/푎 → ∞ and 휆/푎 → ∞, i.e., the large-sample limit and the static limit.

5. Conclusions After briefly reviewing the history and the perspectives of the field of 3D mechanical metamaterials, we have presented our original results on the design, calculation of phonon band structures, manufacturing by 3D laser microprinting, and ultrasound experimental characterization of one type of chiral 3D mechanical metamaterial exhibiting acoustical activity—the mechanical counterpart of optical activity. Compared to our recent early corresponding results on a different kind of chiral cubic 3D micro-lattice, we have found enhanced effects in terms of the rotation angle per lattice constant in the ultrasound frequency range 10– 100 kHz. The largest values found are around 45° rotation per lattice constant, in good agreement between theory and experiment. Notably, these values approach the fundamental bound of 90° rotation per lattice constant.

Materials 2019, 12, 3527 9 of 10 the rotation eﬀect would be zero, requires both, L/a and λ/a , i.e., the large-sample limit → ∞ → ∞ and the static limit.

5. Conclusions After briefly reviewing the history and the perspectives of the field of 3D mechanical metamaterials, we have presented our original results on the design, calculation of phonon band structures, manufacturing by 3D laser microprinting, and ultrasound experimental characterization of one type of chiral 3D mechanical metamaterial exhibiting acoustical activity—the mechanical counterpart of optical activity. Compared to our recent early corresponding results on a different kind of chiral cubic 3D micro-lattice, we have found enhanced effects in terms of the rotation angle per lattice constant in the ultrasound frequency range 10–100 kHz. The largest values found are around 45◦ rotation per lattice constant, in good agreement between theory and experiment. Notably, these values approach the fundamental bound of 90◦ rotation per lattice constant. Furthermore, the observed rotation angles decrease more slowly with increasing number of unit cells towards the bulk limit compared to our previous results. However, this advance comes at the price of a decreased robustness against fabrication errors, especially concerning the elongated twisted nearby yet non-touching rods, which form the heart of this chiral metamaterial. By decreasing the cubic lattice constant from the value of a = 250 µm considered here, our results can be scaled to lower or larger operation frequencies. This possibility of scaling is a major advantage of chiral metamaterials compared to chiral ordinary crystals. For ordinary crystals, the maximum effects are much smaller to begin with. More importantly, they appear at a certain frequency (typically on the order of some THz), which is given by nature and which cannot be changed. Therefore, sizable effects of acoustical activity are not available from natural crystals at kHz and MHz frequencies, whereas they are available from 3D chiral mechanical metamaterials. Thereby, these metamaterials provide new degrees of freedom to control the polarization of elastic waves. This enables applications such as mode conversion from one incident transverse propagation to the orthogonal one.

Author Contributions: T.F. and M.W. conceived the experiment; M.W. supervised the work; J.R. conducted the main experiments and performed numerical simulations with the help of T.F; J.R. analyzed all experimental and numerical data; T.F. took the scanning electron micrographs; A.M. contributed symmetry considerations; All authors discussed the results; M.W. wrote a first draft of the manuscript. All authors contributed to the final version of the manuscript. Funding: This research has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy via the Excellence Cluster “3D Matter Made to Order” (EXC-2082–390761711), which has also been supported by the Carl Zeiss Foundation through the “Carl-Zeiss-Focus@HEiKA”, by the State of Baden-Württemberg, and by the Karlsruhe Institute of Technology (KIT). Acknowledgments: We acknowledge support concerning the construction of the measurement setup by Johann Westhauser (Karlsruhe) and discussions with Muamer Kadic (Karlsruhe and Besancon). This research has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany´s Excellence Strategy via the Excellence Cluster “3D Matter Made to Order” (EXC-2082–390761711), which has also been supported by the Carl Zeiss Foundation through the “Carl-Zeiss-Focus@HEiKA”, by the State of Baden-Württemberg, and by the Karlsruhe Institute of Technology (KIT). We further acknowledge support by the Helmholtz program “Science and Technology of Nanosystems” (STN), by the associated KIT project “Virtual Materials Design” (VIRTMAT), and by the Karlsruhe School of Optics & Photonics (KSOP). Conflicts of Interest: The authors declare no conflict of interest.

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