Tiemo Bückmann

Tailored 3D Mechanical Metamaterials Made by Dip‐in Direct‐Laser‐Writing Optical Lithography

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 Poissons ratio and with sufficient overall size for direct mechanical characterization aim at demonstrating the new possibilities with respect to rationally designed effective materials.

On the practicability of pentamode mechanical metamaterials

Following the theoretical suggestion by Milton and Cherkaev in 1995, we fabricate pentamode metamaterials by dip-in direct-laser-writing optical lithography. Using finite element calculations and geometrical parameters corresponding to 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 1,000. This result opens new horizons for transformation acoustics.

Invisibility cloaking in a diffusive light scattering medium

To cast no shadow in a murky medium Cloaks can hide objects from view, but what about their shadows? Schittny et al. devised a cloak to remove even the shadow of an object embedded in a murky medium in front of a computer screen (see the Perspective by Smith). They engineered a core-shell structure within which an object could be hidden and tailored the optical properties of the cloak to match that of the medium. Light was routed around the cloak, leaving no trace of the hidden object. Science, this issue p. 427; see also p. 384 A core-shell structure provides a cloak for objects within a diffusive medium. [Also see Perspective by Smith] 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.

On three-dimensional dilational elastic metamaterials

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.

Three-dimensional labyrinthine acoustic metamaterials

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.

On anisotropic versions of three-dimensional pentamode metamaterials

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.

Elastic measurements on macroscopic three-dimensional pentamode metamaterials

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 Youngs 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.

Phonon band structures of three-dimensional pentamode metamaterials

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.

Mechanical cloak design by direct lattice transformation

Significance 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. Spatial coordinate transformations have helped simplifying mathematical issues and solving complex boundary-value problems in physics for decades already. More recently, material-parameter transformations have also become an intuitive and powerful engineering tool for designing inhomogeneous and anisotropic material distributions that perform wanted functions, e.g., invisibility cloaking. A necessary mathematical prerequisite for this approach to work is that the underlying equations are form invariant with respect to general coordinate transformations. Unfortunately, this condition is not fulfilled in elastic–solid mechanics for materials that can be described by ordinary elasticity tensors. Here, we introduce a different and simpler approach. We directly transform the lattice points of a 2D discrete lattice composed of a single constituent material, while keeping the properties of the elements connecting the lattice points the same. After showing that the approach works in various areas, we focus on elastic–solid mechanics. As a demanding example, we cloak a void in an effective elastic material with respect to static uniaxial compression. Corresponding numerical calculations and experiments on polymer structures made by 3D printing are presented. The cloaking quality is quantified by comparing the average relative SD of the strain vectors outside of the cloaked void with respect to the homogeneous reference lattice. Theory and experiment agree and exhibit very good cloaking performance.

Transient behavior of invisibility cloaks for diffusive light propagation

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.