Three-dimensional poroelastic metamaterials with extremely negative or positive effective static volume compressibility
Introduction
The static isothermal compressibility, , describes the change in the volume of a gas, liquid, or elastic solid in response to a change of the hydrostatic pressure exerted onto it under conditions of constant temperature , i.e.,
For porous elastic (poroelastic) structures, such as a sponge or an elastic foam, the effective or unjacketed [1] compressibility is obtained by replacing and . Here, the effective volume, , refers to the volume that is bordered by the overall porous structure, whereas the volume is the volume of the elastic solid itself, excluding the voids. The effective volume is what an observer sees. A fine porosity may even be below the diffraction limit. In a cubic box of volume , you can pack poroelastic cubes of effective volume These examples seem to indicate that the effective volume plays the same role as the ordinary volume. However, gas can move in and out of the pores, which are part of the effective volume — in sharp contrast to the ordinary volume of an elastic solid. Therefore, the associated effective compressibility does not underlie the same fundamental bounds for passive stable unconstrained materials as the ordinary compressibility [[2], [3], [4], [5], [6]]. In particular, the effective volume can increase in response to an increase in hydrostatic pressure , leading to the possibility of .
Several metamaterial architectures exhibiting such an unusual behavior have been suggested theoretically [[2], [5], [7]]. Lately, one of them [5] has also been realized experimentally [6] and has been measured. However, this metamaterial structure was rather complex. Within each cubic unit cell, it contained eight hollow three-dimensional (3D) crosses filled with air [6]. Each 3D cross was sealed by six thin membranes and connected by lever arms to the neighboring 3D crosses within the unit cell. The inner rotations of the 3D crosses around all three cubic axes led to frustrations for the lever arms, making the overall optimization quite complex.
In this paper, we present a significantly simplified unit cell design, which contains only a single hollow cube, sealed by six thin membranes, within each cubic unit cell. This unit cell is placed onto a simple-cubic translational lattice. The design process for 3D laser printing of hollow sealed volumes on the micrometer scale requires careful stress engineering of the structure for the chemical development process and is therefore discussed in some detail. We measure a negative effective compressibility that is six times larger than previous values. In addition, we also discuss and realize modified blueprints showing a large positive effective compressibility. We show that, conceptually, the effective positive compressibility can exceed the compressibility of air. Loosely speaking, this means that such poroelastic metamaterials would be easier to compress than air. However, the relative membrane thicknesses required to obtain such behavior are currently not in reach experimentally.
The poroelastic metamaterial structures discussed in this paper might find applications as actuators, e.g., in the spirit of programmable metamaterials [[8], [9], [10]], with hydrostatic air pressure serving as the stimulus rather than uniaxial pressure exerted by a stamp as in [10].
We note that the static (or quasi-static) case discussed in this paper is conceptually distinct from the case of time-harmonic modulation or acoustic or elastic waves, for which negative compressibility, the inverse of the bulk modulus in the isotropic case, can be accomplished by means of local resonances [[11], [12], [13], [14], [15]]. While resonances cannot be exploited in the static case, a large number of unusual static linear [[16], [17], [18], [19], [20], [21]] and nonlinear [[22], [23], [24], [25]] effective properties of mechanical metamaterials have recently been achieved by using other smart mechanisms (also, see the reviews [[26], [27], [28]]).
Section snippets
Metamaterial design
Fig. 1 (a) shows the novel blueprint for a metamaterial with negative effective compressibility. The artificial crystal has cubic symmetry. The geometrical parameters are indicated. Each cubic unit cell with lattice constant as shown in panel (b) contains one hollow cube. A cut-open version of the unit cell is illustrated in Fig. 1 (c). Six thin membranes seal the air within each cube. Hence, the number of moles inside, , is constant. We start from the same hydrostatic pressure in the
Sample fabrication
The design parameters discussed above are the target parameters for the 3D micro-fabrication. We use standard 3D laser nanolithography [[31], [32]] (Nanoscribe GmbH, Photonics Professional GT), a commercially available photoresist (Nanoscribe GmbH, IP-Dip), and mr-Dev 600 as the developer to wash out the remaining monomer after laser writing. Electron micrographs of selected fabricated polymer samples on glass substrate are depicted in Fig. 3 (a) and (b). The three-dimensional metamaterial
Measurements
Our measurements are performed under air pressure control and otherwise ambient conditions. We exert the controlled variable hydrostatic air pressure in a home-built sample cell. A glass window in this sample cell allows for in-situ optical imaging of the sample within the cell using a single microscope objective lens (LD Acroplan 20/0.40, Carl Zeiss) and a silicon charge-coupled-device (CCD) black/white camera (BFLYPGE-50H5M-C, Point Grey Research). Details and artifacts of this setup have
Conclusions
In conclusion, we have designed, fabricated by 3D laser nanoprinting, and characterized under hydrostatic air pressure control a novel three-dimensional metamaterial architecture leading to extremely negative or positive effective static volume compressibility. The underlying mechanism is based on the warping of the six thin faces (membranes) of one hollow sealed cube within each cubic unit cell. Connectors fixed to each membrane translate the warping into an isotropic expansion or contraction
Acknowledgments
We acknowledge support by the Helmholtz Program “Science and Technology of Nanosystems” (STN), the KIT “Virtual Materials Design” (VIRTMAT) project, the Karlsruhe School of Optics & Photonics (KSOP), and the Karlsruhe Institute of Technology Nanostructure Service Laboratory (NSL). M.K. acknowledges support by the Labex ACTION program (contract ANR-11-LABX-0001-01) and the French “Investissements d’Avenir” program, project ISITE-BFC (contract ANR-15-IDEX-03). We also acknowledge support by
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