On the feasibility of pentamode mechanical metamaterials

Abstract: Conceptually, all conceivable three-dimensional mechanical materials can be built from pentamode materials. Pentamodes also enable to implement three-dimensional transformation acoustics - the analogue of transformation optics. However, pentamodes have not been realized experimentally to the best of our knowledge. 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 1,000.

• The paper reports a novel fabrication strategy using dip-in DLW that achieves a bulk/shear modulus ratio of up to 1,000 in pentamode metamaterials.
• Numerical simulations and experiments reveal that reducing connection region diameters preserves high pentamode behavior while ensuring structural stability.
• The study’s findings pave the way for advanced acoustic devices by demonstrating near-ideal material properties in a diamond-symmetric polymer lattice.

Feasibility of Pentamode Mechanical Metamaterials

The paper explores the practical realization of pentamode mechanical metamaterials, a subclass of metamaterials that permit the uncoupling of compression and shear waves by achieving a bulk modulus vastly exceeding the shear modulus. Initially conceptualized by Milton and Cherkaev in 1995, pentamode materials have thus far evaded experimental realization. They offer promising potential for applications in transformation acoustics, akin to transformation optics in directing light waves.

The researchers employ the current advancements in lithography to fabricate microstructures that approximate the theoretical pentamode ideal. Through numerical simulations and experimental methodologies, they devise structures that achieve a figure of merit—the ratio of bulk modulus ($B$) to shear modulus ($G$)—of up to 1,000. This ratio signifies a substantial leap toward realizing pentamaterial behavior. The materials utilized were polymers constituted into a diamond-symmetric lattice, leveraging dip-in direct-laser-writing (DLW) techniques for microfabrication.

Structurally, the ideal pentamode metamaterial consists of connected truncated cones forming a diamond-type crystal structure with a lattice constant that is critical for effective material behavior, particularly in the context of finite acoustic frequencies. The paper identifies that infinitely small connection points between cones would destabilize the structure under mechanical perturbation. To counteract this, they propose a finite overlap volume, trading off some modulus ratio for structural stability.

Notably, the experiments validate that reductions in the diameter of the connection regions can sustain high figure of merit values, pointing to future directions in minimizing these dimensions to achieve even higher fidelity pentamode properties. The paper also explores variations in the constituent material properties, such as Poisson's ratio and Young's modulus, and their relatively insignificant effect on the pentamode ratio under certain conditions.

The implications of these findings are substantial, as they mark a significant step toward practical implementation of three-dimensional transformation acoustics. This could lead to novel applications in areas requiring precise control over wave propagation, such as advanced acoustic metamaterials. However, challenges remain, particularly in reducing structural dimensions further and refining phonon band structure calculations for frequency-dependent behavior. Achieving a critical dimension with features scaling down from 10 µm to 1 µm could further escalate the figure of merit, opening new capabilities for metamaterials.

Overall, the paper demonstrates significant progress in creating mechanical metamaterials that closely approach the idealized properties of pentamodes, underscoring the power of contemporary fabrication techniques and numerical modeling to innovate within the field of nanophotonics and acoustics. Future research will undoubtedly focus on overcoming the technical hurdles that remain, particularly in terms of scaling, material characterization, and applying these structures to functional acoustic devices.