Experiments on Metamaterials with Negative Effective Static Compressibility

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 ${\kappa }_{\mathrm{eff}}=-0.8\%{\mathrm{bar}}^{-1}=-80{\mathrm{GPa}}^{-1}$ under pressure control.

Popular Summary

Increasing the air pressure exerted on most ordinary materials leads to a decrease in volume—in other words, the object is compressed. Mathematically, physicists describe the object as having “positive volume compressibility.” Common sense suggests that negative compressibility—an object inflating in response to an increase in air pressure—is impossible under static conditions. Such a situation violates both stability and energy conservation. But recent theoretical arguments suggest that certain metamaterials could exhibit this unusual property. In this paper, we report on the fabrication and testing of a three-dimensional (3D) artificial material that exhibits such “forbidden” behavior in an isotropic and stable manner.

The challenge in our microfabrication lies in the manufacture of the necessary hollow 3D crosses, containing concealed volumes that are sealed by thin membranes. We demonstrate such concealed microvolumes, which leak only by gas permeation through the bulk of the thin polymer membranes on a time scale of 10 minutes. Using optical imaging, we directly measure an unusually large negative metamaterial effective compressibility equivalent to a relative effective volume increase of about 1% at 1 bar excess air pressure.

We demonstrate experimentally, for what we believe is the first time, a metamaterial with a negative effective compressibility under quasistatic conditions. This is a critical step toward some unusual applications, such as in artificial muscles and actuators.

Figure 1

(a) Blueprint of a (nonprimitive) unit cell of the three-dimensional (3D) metamaterial. This unit cell includes eight hollow 3D crosses. The inset shows a cut-open hollow 3D cross. The relevant geometrical parameters are indicated. (b) This unit cell is placed onto a simple-cubic translational lattice with lattice constant $a$, leading to the shown artificial crystal. Its effective volume $V_{\mathrm{eff}}$ is defined by the volume enclosed by the metamaterial’s six surface facets (light gray) and exhibits a negative effective static volume compressibility ${\kappa }_{\mathrm{eff}}$.

Figure 2

(a) Electron micrograph of a metamaterial sample composed of $2\times 2\times 2=8$ unit cells [see Fig. 1] fabricated by 3D optical laser lithography. (b) Electron micrograph of a detail of a control structure, in which we have intentionally introduced a hole into each hollow 3D cross, for otherwise identical geometrical parameters. (c) 3D isointensity surface obtained from laser scanning fluorescence microscopy of one unit cell. The shown cut through a part of these fluorescence data reveals that the interior of the hollow 3D crosses is actually hollow. To allow for a direct comparison, the blueprint [also see Fig. 1] is depicted below this measurement for a similar viewing direction.

Figure 3

Measured mean relative length change $\mathrm{\Delta }L/L$ vs excess pressure $\mathrm{\Delta }P=P-P_0>0$, with ambient pressure $P_0$. The observed behavior leads to an effective volume compressibility of ${\kappa }_{\mathrm{eff}}=-0.8\times {10}^{-7}{\mathrm{Pa}}^{-1}$. The blue dots are measured (full: pressure ramping up, open: pressure ramping down); the red solid curve is calculated numerically. The green dots correspond to measurements on a control sample, in which we have intentionally introduced a hole in each of the hollow 3D crosses [compare Fig. 2]. The black dots refer to a control experiment on a copper surface.

Figure 4

Measurements as blue dots in Fig. 3. However, at $t=0$, the pressure of the gas is stepwise increased from ambient pressure $P=P_0=1.0\times {10}^5\mathrm{Pa}$ to $P=4.8\times {10}^5\mathrm{Pa}$ and then kept constant. The resulting spatial mean $\mathrm{\Delta }L/L$ is measured vs time using image cross-correlation analysis. The blue dots correspond to air (mainly composed of oxygen ${\mathrm{O}}_2$ and nitrogen ${\mathrm{N}}_2$) as gas, and the green dots to carbon dioxide ${\mathrm{CO}}_2$ instead. In the second half of the cycle, the pressure is switched back down to $P=1.0\times {10}^5\mathrm{Pa}$. The solid curves are single-exponential fits to the experimental data. For air, the derived time constant is ${\tau }_{\mathrm{air}}=745\mathrm{s}$. For carbon dioxide, the exponential time constant is ${\tau }_{{\mathrm{CO}}_2}=26\mathrm{s}$.

Figure 5

Effective compressibility vs difference in membrane thickness, $\mathrm{\Delta }t$, with respect to the thickness of $t=1.725\mu \mathrm{m}$ (equivalent to $t/a=1.15\%$ for $a=150\mu \mathrm{m}$) in Fig. 3. The solid curve is calculated for otherwise identical parameters as the calculations in Fig. 3; the full dots are measured. The different colors correspond to different nominally identical samples. The vertical error bars are $\pm$ three times the standard deviation of the mean for ten measurements each, such as the one shown in Fig. 3, which corresponds to the single open triangle at $\mathrm{\Delta }t=0$. The horizontal error bars are estimated. The effective compressibility changes by about a factor of 3 upon changing the membrane thickness by $0.4\mu \mathrm{m}$.