## On static chiral Milton-Briane-Willis continuum mechanics

#### Muamer Kadic, André Diatta, Tobias Frenzel, Sebastien Guenneau, and Martin Wegener

Recent static experiments on twist effects in chiral three-dimensional mechanical metamaterials have been discussed in the context of micropolar Eringen continuum mechanics, which is a generalization of Cauchy elasticity. For cubic symmetry, Eringen elasticity comprises nine additional parameters with respect to Cauchy elasticity, of which three directly influence chiral effects. Here, we discuss the behavior of the static case of an alternative generalization of Cauchy elasticity, the Milton-Briane-Willis equations. We show that in the homogeneous static cubic case only one additional parameter with respect to Cauchy elasticity results, which directly influences chiral effects. We show that the Milton-Briane-Willis equations qualitatively describe the experimentally observed chiral twist effects, too. We connect the behavior to a characteristic length scale.

https://arxiv.org/pdf/1901.02519.pdf

## Optical force rectifiers based on PT-symmetric metasurfaces

##### Rasoul Alaee, Burak Gurlek, Johan Christensen, and Muamer Kadic

We introduce here the concept of optical force rectifier based on parity-time symmetric metasurfaces. Directly linked to the properties of non-Hermitian systems engineered by balanced loss and gain constituents, we show that light can exert asymmetric pulling or pushing forces on metasurfaces depending on the direction of the impinging light. This generates a complete force rectification in the vicinity of the exceptional point. Our findings have the potential to spark the design of applications in optical manipulation where the forces, strictly speaking, act unidirectionally.

##### Phys. Rev. B 97, 195420 , 2018

## Experiments on Metamaterials with Negative Effective Static Compressibility

##### Jingyuan Qu, Alexander Gerber, Frederik Mayer, Muamer Kadic, and Martin Wegener

The volume of ordinary materials decreases in response to a pressure increase exerted by a surrounding gas or liquid, i.e., the material volume compressibility is positive. Recently, poroelastic metamaterial architectures have been suggested theoretically that allow for an unusual negative effective static volume compressibility—which appears to be forbidden for reasons of energy conservation at first sight. The challenge in the three-dimensional (3D) fabrication of these blueprints lies in the necessary many hollow 3D crosses sealed by thin membranes, which we realize in this work by using 3D laser microlithography combined with a serendipitous mechanism. By using optical-microscopy cross-correlation analysis, we determine an extraordinarily large negative metamaterial effective volume compressibility of κeff=−0.8% bar−1=−80 GPa−1 under pressure control.

Phys. Rev. X 7, 041060 ( 2017)

## Three-dimensional mechanical metamaterials with a twist

##### T. Frenzel, M. Kadic and M. Wegener

Rationally designed artificial materials enable mechanical properties that are inaccessible with ordinary materials. Pushing on an ordinary linearly elastic bar can cause it to be deformed in many ways. However, a twist, the counterpart of optical activity in the static case, is strictly zero. The unavailability of this degree of freedom hinders applications in terms of mode conversion and the realization of advanced mechanical designs using coordinate transformations. Here, we aim at realizing microstructured three-dimensional elastic chiral mechanical metamaterials that overcome this limitation. On overall millimeter-sized samples, we measure twists per axial strain exceeding 2°/%. Scaling up the number of unit cells for fixed sample dimensions, the twist is robust due to metamaterial stiffening, indicating a characteristic length scale and bringing the aforementioned applications into reach.

Science, 358, pp. 1072-1074 (2017)

## Mechanical metamaterials: When size matters

##### Muamer Kadic, Tobias Frenzel, Martin Wegener

That the unit cell of a metamaterial can't be considered vanishingly small like in ordinary crystals has long been deemed more burden than opportunity. The emergence of a characteristic length scale in metamaterial chains may change that trend.

Nature Physics, 14, 8–9 (2018)

## Experimental Evidence for Sign Reversal of the Hall Coefficient in Three-Dimensional Metamaterials

##### Christian Kern, Muamer Kadic, and Martin Wegener

Effectively inverting the sign of material parameters is a striking possibility arising from the concept of metamaterials. Here, we show that the electrical properties of a p-type semiconductor can be mimicked by a metamaterial solely made of an n-type semiconductor. By fabricating and characterizing three-dimensional simple-cubic microlattices composed of interlocked hollow semiconducting tori, we demonstrate that sign and magnitude of the effective metamaterial Hall coefficient can be adjusted via a tori separation parameter—in agreement with previous theoretical and numerical predictions.

Phys. Rev. Lett. 118, 016601 (2017)

## An elasto-mechanical unfeelability cloak made of pentamode metamaterials

##### Tiemo Bückmann, Michael Thiel, Muamer Kadic, Robert Schittny, and M. Wegener

Metamaterial-based cloaks make objects different from their surrounding appear just like their surrounding. To date, cloaking has been demonstrated experimentally in many fields of research, including electrodynamics at microwave frequencies, optics, static electric conduction, acoustics, fluid dynamics, thermodynamics and quasi two-dimensional solid mechanics. However, cloaking in the seemingly simple case of three-dimensional solid mechanics is more demanding. Here, inspired by invisible core-shell nanoparticles in optics, we design an approximate elasto-mechanical core-shell ‘unfeelability’ cloak based on pentamode metamaterials. The resulting three-dimensional polymer microstructures with macroscopic overall volume are fabricated by rapid dip-in direct laser writing optical lithography. We quasi-statically deform cloak and control samples in the linear regime and map the displacement fields by autocorrelation-based analysis of recorded movies. The measured and the calculated displacement fields show very good cloaking performance. This means that one can elastically hide objects along these lines.

Nature Communications volume 5, Article number: 4130 (2014)

## On anisotropic versions of three-dimensional pentamode metamaterials

##### Muamer Kadic, Tiemo Bückmann, Robert Schittny and Martin Wegener

Pentamode materials are artificial solids with elastic properties that approximate those of isotropic liquids. The corresponding three-dimensional mechanical metamaterials or 'meta-liquids' have recently been fabricated. In contrast to normal liquids, anisotropic meta-liquids are also possible—a prerequisite for realizing many of the envisioned transformation-elastodynamics architectures. Here, we study several possibilities theoretically for introducing intentional anisotropy into three-dimensional pentamode metamaterials. In static continuum mechanics, the transition from anti-auxetic pentamode materials to auxetics is possible. Near this transition, in the dynamic case, approximately uniaxial versions of pentamode metamaterials deliver anisotropic longitudinal-wave phase velocities different by nearly a factor of 10 for realistically accessible microstructure parameters.

## Phonon band structures of three-dimensional pentamode metamaterials

##### Aude Martin, Muamer Kadic, Robert Schittny, Tiemo Bückmann, and Martin Wegener

Three-dimensional pentamode metamaterials are artificial solids that approximately behave like liquids, which have vanishing shear modulus. Pentamodes have recently become experimental reality. Here, we calculate their phonon band structures for various parameters. Consistent with static continuum mechanics, we find that compression and shear waves exhibit phase velocities that can realistically be different by more than one order of magnitude. Interestingly, we also find frequency intervals with more than two octaves bandwidth in which pure single-mode behavior is obtained. Herein, exclusively compression waves exist due to a complete three-dimensional band gap for shear waves and, hence, no coupling to shear modes is possible. Such single-mode behavior might, e.g., be interesting for transformation-elastodynamics architectures.

## Experiments on Transformation Thermodynamics: Molding the Flow of Heat

##### Robert Schittny, Muamer Kadic, Sebastien Guenneau, and Martin Wegener

## Three-dimensional labyrinthine acoustic metamaterials

##### Tobias Frenzel, Jan David Brehm, Tiemo Bückmann, Robert Schittny, Muamer Kadic and Martin Wegener

Building upon recent theoretical and experimental work on two-dimensional labyrinthine acoustic metamaterials, we design, fabricate, and characterize nearly isotropic three-dimensional airborne acoustic labyrinthine metamaterials. Our experiments on aluminum-based structures show phase and group velocities smaller than that of air by a factor of about 8 over a broad range of

frequencies from 1 to 4 kHz. This behavior is in agreement with three-dimensional band-structure calculations including the first and higher bands. The extracted imaginary parts of the phase velocity are 5–25 times smaller than the mentioned real parts. This ratio is better than for most optical metamaterials but still rather favors applications in terms of sub-wavelength broadband

acoustic absorbers.

## Metamaterials beyond electromagnetism

##### Muamer Kadic, Tiemo Bückmann, Robert Schittny and Martin Wegener

Metamaterials are rationally designed man-made structures composed of functional building blocks that are densely packed into an effective (crystalline) material. While metamaterials are mostly associated with negative refractive indices and invisibility cloaking in electromagnetism or optics, the deceptively simple metamaterial concept also applies to rather different areas such as thermodynamics, classical mechanics (including elastostatics, acoustics, fluid dynamics and elastodynamics), and, in principle, also to quantum mechanics. We review the basic concepts, analogies and differences to electromagnetism, and give an overview on the current state of the art regarding theory and experiment—all from the viewpoint of an experimentalist. This review includes homogeneous metamaterials as well as intentionally inhomogeneous metamaterial architectures designed by coordinate-transformation-based approaches analogous to transformation optics. Examples are laminates, transient thermal cloaks, thermal concentrators and inverters, 'space-coiling' metamaterials, anisotropic acoustic metamaterials, acoustic free-space and carpet cloaks, cloaks for gravitational surface waves, auxetic mechanical metamaterials, pentamode metamaterials ('meta-liquids'), mechanical metamaterials with negative dynamic mass density, negative dynamic bulk modulus, or negative phase velocity, seismic metamaterials, cloaks for flexural waves in thin plates and three-dimensional elastostatic cloaks.

## Elastic measurements on macroscopic three-dimensional

pentamode metamaterials

##### Robert Schittny, Tiemo Bueckmann, Muamer Kadic, and Martin Wegener

Pentamode metamaterials approximate tailorable artificial liquids. Recently, microscopic versions of these intricate three-dimensional structures have been fabricated, but direct experimental characterization has not been possible yet. Here, using three-dimensional printing, we fabricate macroscopic polymer-based samples with many different combinations of the small connection diameter d and the lattice constant a. Direct measurements of the static shear modulus and the Young’s modulus reveal that both scale approximately according to (d=a)3, in good agreement with continuum-mechanics calculations. For the smallest accessible values of d/a= 1.5%, we find derived ratios of bulk modulus B to shear modulus G of B/G= 1000.

## On three-dimensional dilational elastic metamaterials

##### Tiemo Bückmann, Robert Schittny, Michael Thiel1, Muamer Kadic, Graeme W Milton and Martin Wegener

Dilational materials are stable, three-dimensional isotropic auxetics with an ultimate Poissonʼs ratio of −1. Inspired by previous theoretical work, we design a feasible blueprint for an artificial material, a metamaterial, which approaches the ideal of a dilational material. The main novelty of our work is that we also fabricate and characterize corresponding metamaterial samples. To reveal all modes in the design, we calculate the phonon band structures. On this basis, using cubic symmetry we can unambiguously retrieve all different non-zero elements of the rank-four effective metamaterial elasticity tensor from which all effective elastic metamaterial properties follow. While the elastic properties and the phase velocity remain anisotropic, the effective Poissonʼs ratio indeed becomes isotropic and approaches −1 in the limit of small internal connections. This finding is also supported by independent, static continuum-mechanics calculations. In static experiments on macroscopic polymer structures fabricated

by three-dimensional printing, we measure Poissonʼs ratios as low as −0.8 in good agreement with the theory. Microscopic samples are also presented.

## Transient behavior of invisibility cloaks for diffusive light propagation

##### Robert Schittny, Andreas Niemeyer, Muamer Kadic, Tiemo Bückmann, Andreas Naber, and Martin Wegener

An ideal invisibility cloak makes any object within itself indistinguishable from its surrounding—for all colors, directions, and polarizations of light. Nearly ideal cloaks have recently been realized for turbid light-scattering media under continuous-wave illumination. Here, we ask whether these cloaks also work under pulsed illumination. Our time-resolved imaging experiments on simple core–shell cloaks show that they do not: they appear bright with respect to their surrounding at early times and dark at later times, leading to vanishing image contrast for time-averaged detection. Furthermore, we show that the same holds true for more complex cloaking architectures designed by spatial coordinate transformations. We discuss implications for diffuse optical tomography and possible applications in terms of high-end security features.

## Experiments on cloaking in optics, thermodynamics and mechanics

##### Muamer Kadic, Tiemo Bückmann, Robert Schittny, Martin Wegener

Spatial coordinate transformations can be used to transform boundaries, material parameters or discrete lattices. We discuss fundamental constraints in regard to cloaking and review our corresponding experiments in optics, thermodynamics and mechanics. For example, we emphasize three-dimensional broadband visible-frequency carpet cloaking, transient thermal cloaking, three-dimensional omnidirectional macroscopic broadband cloaking for diffuse light throughout the entire visible range, cloaking for flexural waves in thin plates and three-dimensional elasto-static core–shell cloaking using pentamode mechanical metamaterials.

## Parallel Hall effect from three-dimensional single-component metamaterials

##### Christian Kern, Muamer Kadic and Martin Wegener

We propose a class of three-dimensional metamaterial architectures composed of a single doped semiconductor (e.g., n-Si) in air or vacuum which lead to an unusual effective behavior of the classical Hall effect. Using an anisotropic structure, we numerically demonstrate a Hall voltage that is parallel—rather than orthogonal—to the external static magnetic-field vector (“parallel Hall effect”). The sign of this parallel Hall voltage can be determined by a structure parameter. Together with the previously demonstrated positive or negative orthogonal Hall voltage, we demonstrate four different sign combinations.

## Diffuse-light all-solid-state invisibility cloak

##### ROBERT SCHITTNY, ANDREAS NIEMEYER, MUAMER KADIC, TIEMO BÜCKMANN, ANDREAS NABER, AND MARTIN WEGENER

An ideal invisibility cloak makes arbitrary macroscopic objects within the cloak indistinguishable from its surrounding—for all directions, illumination patterns, polarizations, and colors of visible light. Recently, we have approached such an ideal cloak for the diffusive regime of light propagation using a core–shell geometry and a mixture of water and white wall paint as the surrounding. Here, we present an all-solid-state version based on polydimethylsiloxane doped with titania nanoparticles for the

surrounding/shell and on a high-reflectivity microporous ceramic for the core. By virtue of reduced effects of absorption, especially from the core, the cloaking performance and the overall light throughput are improved significantly.

## Vibrant times for mechanical metamaterials

##### Johan Christensen, Muamer Kadic, Oliver Kraft and Martin Wegener

Metamaterials are man-made designer matter that obtains its unusual effective properties by structure rather than chemistry. Building upon the success of electromagnetic and acoustic metamaterials, researchers working on mechanical metamaterials strive at obtaining extraordinary or extreme elasticity tensors and mass-density tensors to thereby mold static stress fields or the flow of longitudinal/transverse elastic vibrations in unprecedented ways. In this prospective paper, we focus on recent advances and remaining challenges in this emerging field. Examples are ultralight-weight, negative mass density, negative modulus, pentamode, anisotropic mass density, Origami, nonlinear, bistable, and reprogrammable mechanical metamaterials.

## Tailored 3D Mechanical Metamaterials Made by Dip-in Direct-Laser-Writing Optical Lithograph

##### Tiemo Bückmann, Nicolas Stenger, Muamer Kadic, Johannes Kaschke, Andreas Frölich, Tobias Kennerknecht, Christoph Eberl, Michael Thiel, and Martin Wegener

Dip-in direct-laser-writing (DLW) optical lithography allows fabricating complex three-dimensional microstructures without the height restrictions of regular DLW. Bow-tie elements assembled into mechanical metamaterials with positive/zero/negative Poisson's ratio and with sufficient overall size for direct mechanical characterization aim at demonstrating the new possibilities with respect to rationally designed effective materials.

## Pentamode Metamaterials with Independently Tailored Bulk Modulus and Mass Density

##### Muamer Kadic, Tiemo Bückmann, Robert Schittny, Peter Gumbsch, and Martin Wegener

We propose a class of linear elastic three-dimensional metamaterials for which the effective parameters bulk modulus and mass density can be adjusted independently over a large range—which is not possible for ordinary materials. First, we systematically evaluate the static mechanical properties and the phonon dispersion relations. We show that the two are quantitatively consistent in the long-wavelength limit. To demonstrate the feasibility, corresponding fabricated polymer microstructures are presented. Finally, we discuss calculations for laminates composed of alternating layers of two different metamaterials with equal bulk modulus yet different mass density. This lamination leads to metamaterials with effectively anisotropic uniaxial dynamic mass density tensors.

## Optically assisted trapping with high-permittivity dielectric rings: Towards optical aerosol filtration

##### Rasoul Alaee, Muamer Kadic, Carsten Rockstuhl and Ali Passian

Controlling the transport, trapping, and filtering of nanoparticles is important for many applications. By virtue of their weak response to gravity and their thermal motion, various physical mechanisms can be exploited for such operations on nanoparticles. However, the manipulation based on optical forces is potentially most appealing since it constitutes a highly deterministic approach. Plasmonic nanostructures have been suggested for this purpose, but they possess the disadvantages of locally generating heat and trapping the nanoparticles directly on the surface. Here, we propose the use of dielectric rings made of high permittivity materials for trapping nanoparticles. Thanks to their ability to strongly localize the field in space, nanoparticles can be trapped without contact. We use a semi-analytical method to study the ability of these rings to trap nanoparticles. The results are supported by full-wave simulations. Application of the trapping concept to nanoparticle filtration is suggested.

## Micro-Structured Two-Component 3D Metamaterials with Negative Thermal-Expansion Coefficient from Positive Constituents

##### Jingyuan Qu, Muamer Kadic, Andreas Naber, and Martin Wegener

Controlling the thermal expansion of materials is of great technological importance. Uncontrolled thermal expansion can lead to failure or irreversible destruction of structures and devices. In ordinary crystals, thermal expansion is governed by the asymmetry of the microscopic binding potential, which cannot be adjusted easily. In artificial crystals called metamaterials, thermal expansion can be controlled by structure. Here, following previous theoretical work, we fabricate three-dimensional (3D) two-component polymer microlattices by using gray-tone laser lithography. We perform cross-correlation analysis of optical microscopy images taken at different sample temperatures. The derived displacement-vector field reveals that the thermal expansion and resulting bending of the bi-material beams leads to a rotation of the 3D chiral crosses arranged onto a 3D checkerboard pattern within one metamaterial unit cell. These rotations can over-compensate the expansion and lead to an effectively negative thermal length-expansion coefficient – for all positive constituents – evidencing a striking level of thermal-expansion control.

## Tailored Buckling Microlattices as Reusable Light-Weight Shock Absorbers

##### Tobias Frenzel, Claudio Findeisen, Muamer Kadic, Peter Gumbsch, and Martin Wegener

Structures and materials absorbing mechanical (shock) energy commonly exploit either viscoelasticity or destructive modifications. Based on a class of uniaxial light-weight geometrically nonlinear mechanical microlattices and using buckling of inner elements, either a sequence of snap-ins followed by irreversible hysteretic – yet repeatable – self-recovery or multistability is achieved, enabling programmable behavior. Proof-of-principle experiments on three-dimensional polymer microstructures are presented.

## Scattering problems in elastodynamics

##### Andre Diatta, Muamer Kadic, Martin Wegener, and Sebastien Guenneau

In electromagnetism, acoustics, and quantum mechanics, scattering problems can routinely be solved numerically by virtue of perfectly matched layers (PMLs) at simulation domain boundaries. Unfortunately, the same has not been possible for general elastodynamic wave problems in continuum mechanics. In this Rapid Communication, we introduce a corresponding scattered-field formulation for the Navier equation. We derive PMLs based on complex-valued coordinate transformations leading to Cosserat elasticity-tensor distributions not obeying the minor symmetries. These layers are shown to work in two dimensions, for all polarizations, and all directions. By adaptative choice of the decay length, the deep subwavelength PMLs can be used all the way to the quasistatic regime. As demanding examples, we study the effectiveness of cylindrical elastodynamic cloaks of the Cosserat type and approximations thereof.

## Hall-Effect Sign Inversion in a Realizable 3D Metamaterial

##### Muamer Kadic, Robert Schittny, Tiemo Bückmann, Christian Kern, and Martin Wegener

In 2009, Briane and Milton proved mathematically the existence of three-dimensional isotropic metamaterials with a classical Hall coefficient that is negative with respect to that of all of the metamaterial constituents. Here, we significantly simplify their blueprint towards an architecture composed of only a single-constituent material in vacuum or air, which can be seen as a special type of porosity. We show numerically that the sign of the Hall voltage is determined by a separation parameter between adjacent tori. This qualitative behavior is robust even for only a small number of metamaterial unit cells. The combination of simplification and robustness brings experimental verification of this striking sign inversion into reach. Furthermore, we provide a simple intuitive explanation of the underlying physical mechanism.

## Invisibility cloaking in a diffusive light scattering medium

##### Robert Schittny, Muamer Kadic, Tiemo Bückmann, Martin Wegener

In vacuum, air, and other surroundings that support ballistic light propagation according to Maxwell’s equations, invisibility cloaks that are macroscopic, three-dimensional, broadband, passive, and that work for all directions and polarizations of light are not consistent with the laws of physics. We show that the situation is different for surroundings leading to multiple light scattering, according to Fick’s diffusion equation. We have fabricated cylindrical and spherical invisibility cloaks made of thin shells of polydimethylsiloxane doped with melamine-resin microparticles. The shells surround a diffusively reflecting hollow core, in which arbitrary objects can be hidden. We find good cloaking performance in a water-based diffusive surrounding throughout the entire visible spectrum and for all illumination conditions and incident polarizations of light.

## Mechanical cloak design by direct lattice transformation

##### Tiemo Bückmann, Muamer Kadic, Robert Schittny, and M. Wegener

Calculating the behavior or function of a given material microstructure in detail can be difficult, but it is conceptually straightforward. The inverse problem is much harder. Herein, one searches for a microstructure that performs a specific targeted function. For example, one may want to guide a wave or a force field around some obstacle as though no obstacle were there. Such function can be represented by a coordinate transformation. Transformation optics maps arbitrary coordinate transformations onto concrete material-parameter distributions. Unfortunately, mapping this distribution onto a microstructure poses another inverse problem. Here, we suggest an alternative approach that directly maps a coordinate transformation onto a concrete one-component microstructure, and we apply the approach to the case of static elastic–solid mechanics.

## On the practicability of pentamode mechanical metamaterials

##### Muamer Kadic, Tiemo Bückmann, Nicolas Stenger, Michael Thiel, and Martin Wegener

Conceptually, all conceivable three-dimensional mechanical materials can be built from pentamode materials. Pentamodes also enable to implement three-dimensional transformation elastodynamics—the analogue of transformation optics. However, pentamodes have not been realized experimentally. Here, we investigate inasmuch the pentamode theoretical ideal suggested by Milton and Cherkaev in 1995 can be approximated by a metamaterial with current state-of-the-art lithography. Using numerical calculations calibrated by our fabricated three-dimensional microstructures, we find that the figure of merit, i.e., the ratio of bulk modulus to shear modulus, can realistically be made as large as about 1000.