Additive manufacturing of channeled acoustic topological insulators^a)

Topological insulators have garnered considerable interest due to their unique edge state properties. The Chern number, a topological invariant, describes the sum of the Berry fluxes across the first Brillouin zone. At the interface between two materials, the difference between the Chern numbers determines the number of propagating interface states.¹ The significance of topological insulators stems from the existence of these interface states within the bandgap. The bulk material acts as an insulator, while waves propagate along the interface. As topologically protected edge states can offer immunity to backscattering from sharp corners or defects, considerable effort has been expended to explore systems known to produce these states.

Topological insulators first emerged with the discovery of the quantum Hall effect, which describes the quantization of energy for electrons in a plane subjected to a perpendicular magnetic field.² The quantization allows electrons at the edges to interact with the boundary and create chiral edge states characterized by their ability to propagate in only one direction.^2–5 Time (T)-symmetry must break for the chiral edge states to exist, which can occur through various means of dynamic modulation. Within condensed matter physics, this can include an external magnetic field or the intrinsic ferromagnetism.⁵ In anologs to the quantum Hall effect, classical wave systems break time symmetry through means such as circulated fluid flows,^6,7 active manipulation of material properties,⁸ or circularly polarized light.⁹ Quantum spin Hall states form through spin orbital coupling, which aligns the spin in relation to momentum, thus polarizing the edge and forming helical edge states. Unlike chiral edge states, helical edge states allow propagation in two directions and (T)-symmetry remains intact. Original efforts focused on the atomic structure of materials and found spin oribital coupling in telluride quantum wells,¹⁰ bismuth antimony alloys,^11,12 bismuth selenide, and bisthmuth telluride.^13–15 Efforts continue with machine learning to determine materials naturally exhibiting the desired crystal structure;¹⁶ however, analogues of the quantum spin Hall effect can be artificially created using other approaches breaking inversion symmetry with coupled resonators or mass spring dampers to induce coupling.^17–20 A more recent group of topological insulators rely on the quantum valley Hall effect, which is characterized by a break in inversion symmetry creating valleys in the Berry curvature with opposing chiralities.²¹ These systems produce semi-chiral modes at the interface between the original configuration and its mirror reflection. Quantum valley insulators originated with graphene systems breaking inversion symmetry for electronic applications,²² then expanded to optics,²³ photonics,²⁴ phononics,^25–28 and acoustics.^29,30

Acoustic topological insulators pose challenges both in design, as sound does not have an intrinsic spin, and in experimentation, as sound can experience large attenuation. To emulate a spin, designs emerged breaking inversion symmetry²⁹ and reflection symmetry³⁰ in quantum valley Hall topological insulators. These systems are easier to design yet weaken the topological protection as compared to quantum spin or quantum Hall analogous devices. The work of Fleury et al. provided insight to mimic intrinsic spin. Circulation of a fluid through a ring can introduce angular momentum bias and break nonreciprocity.³¹ This breakthrough led to a surge of acoustic topological insulators breaking time symmetry through various means, including a network of cavities with moving fluids^6,32–34 or a hexagonal lattice whose acoustic properties are modulated through time and space.³⁵ Later, quantum spin Hall systems created pseudospins through coupled ring resonators^17,36 or accidental degeneracies caused by alterations to the filling ratio.³⁷ Active acoustic topological insulators were introduced with a quantum spin Hall insulator directing sound propagation through manipulation of the fill ratio due to a controllable outer shell.³⁸ 

Additive manufacturing has greatly expanded the design possibilities for acoustic systems. Beyond the allowable geometric complexity, additive manufacturing reduces costs and production time through rapid prototyping. The ability to selectively fill the material permits a high strength-to-weight ratio and controllable porosity. As such, these techniques have been applied to absorb, shape, filter, and modulate sound. Topography in acoustic holograms,³⁹ snaking channels in some sound absorbing structures,^40,41 or coils in some negative index metamaterials,⁴² would be difficult or impossible to create without 3D printers. This is representative of a larger trend of increased complexity making the use of additive manufacturing more prominent for acoustic materials.

The use of additive manufacturing in the fabrication of topological insulators is in its infancy and reported studies are sparse. A dynamic topological insulator using a 3D-printed outer shell to actively alter the fill density of the system was reported by Xia et al.³⁸ Higher-order topological states were achieved by 3D-printing unit cells with multiple channels.⁴³ Similarly, 3D topological insulators were 3D-printed to realize twisting three-dimensional pathways for airflow.^44,45 In the cited works, 3D-printing was used only as a means to fabricate complex geometry. Left unexplored is how additive manufacturing may be used to realize designs that minimize attenuation and permit reconfigurability.

In this paper, we numerically and experimentally explore a statically reconfigurable 3D-printed acoustic topological insulator designed to minimize sound losses as sound travels through air channels. We break inversion symmetry to create a semi-chiral state characterized by topological wave propagation along the interface of two acoustic lattices (which are mirror images of each other) that have opposing chiralities. The high attenuation rate of sound and approaches for addressing this concern in 3D printed structures are considered during design iterations.

We start with a hexagonal honeycomb lattice structure, then break inversion symmetry and invert half the lattice to produce two separate materials meeting at an interface. The system is depicted in Fig. 1(a), which illustrates the interior channels of the 3D-printed structure. If the following conditions are met, sound entering near the starred location will travel along the interface marked by the black line. First, the corresponding band structure for the bulk material must contain degenerate Dirac cones that separate when inversion symmetry is broken. Then, the valley Chern numbers for each material must be $±1/2$⁠, indicating the presence of semi-chiral modes at the interface between the two materials.⁴⁶ The following describes the procedure in greater detail and then discusses fabrication considerations.

To determine the band structure and possible Dirac cones, we must calculate the eigenvalues of the pressure across the first Brillouin zone. We use the governing equation describing wave propagation in a homogeneous fluid given by:⁴⁷ 

where p_t denotes the total pressure, ρ_c the density, and $k$ the wave number. The dipole domain source, q_d, and the monopole domain source, Q_m, are zero when calculating the eigenfrequencies. To obtain the band structure, we use the Floquet theory relating pressure at one edge of the unit cell [see Fig. 1(b)] to the pressure at the opposite edge. The following relation provides the Floquet periodic boundary condition:⁴⁷ 

where $rsrc$ represents the position of the first edge, $rdst$ the opposite, and k the wave vector.

The first Brillouin zone is the smallest repeating area containing one lattice point, as shown in Fig. 1(c). The shaded triangle illustrates the smallest representative area in reciprocal space needed to represent the band structure. The basis vectors $e1$ and $e2$ from Fig. 1(b) relate to the reciprocal vectors $e1*$ and $e2*$ from Fig. 1(c) using the following:⁴⁸ 

where $e3$ is a unit vector in the z-direction and the basis vectors $e1$ and $e2$ have length L, or the length of one side of the rhombus in Fig. 1(b). The geometry of a hexagon and the basis vectors in reciprocal space determine the edges of the Brillouin zone:⁴⁸ 

The pressure equation, Floquet periodic boundary conditions, and coordinates for the first Brillouin zone provide the information to evaluate the band structure using finite element analysis. The sixfold (⁠$C6v$⁠) symmetry of the hexagonal honeycomb lattice indicates the existence of Dirac cones at the K-point for the symmetric unit cell depicted in Fig. 2(a). We next break inversion symmetry by altering the heights of one chamber to produce the unit cell pictured in Fig. 2(b). In doing so, the $C6v$ symmetry becomes threefold (⁠$C3v$⁠) symmetry. Removing this symmetry separates the Dirac cones and satisfies the first condition, forming a bandgap. To evaluate if this gap is topologically protected, the Chern number is calculated using the Berry curvature. The Berry curvature can be found using the nodal pressures for each dispersion branch (i.e., eigenmode):

where A is the berry connection, and $Ωn(k)$ is the Berry curvature for the nth eigenvalue of wave vector k. The Berry connection is defined as,

where $Pn$ is an array of normalized nodal pressures of the nth eigenmode. The Berry curvature can be integrated local to the K and $K′$ points (⁠$K′$ being the opposite vertex to K in reciprocal space) to compute the valley Chern number. If the valley Chern number C_v equals $+1/2$ for the Dirac cone at K and $−1/2$ for the Dirac cone at $K′$⁠, then the material and its mirror reflection will have opposite topological character. At the boundary between the two topological materials, according to the bulk-edge correspondence principle, the total valley Chern number becomes $|∇Cvint|=1$⁠,⁴⁹ fulfilling the second condition and indicating that an interface mode exists within the bandgap.

Additive manufacturing was chosen for its ease and effectiveness in fabricating the complex geometries required in the proposed lattice structure. Figure 3(a) depicts the exterior boundaries of the unit cell as fabricated. To allow for recongifurability of the interface locations, each unit cell joins to another using the clearance-fit clips pictured in Fig. 3(c). The arched design of the clip forces the inward prongs to deform slightly as they enter the indentations in the unit cell, thus joining the unit cells. Rubber o-rings prevent leaks at the connection. To securely join unit cells, the clips must deform enough to enter the indentations yet not break, and must be of sufficient width to deform the o-ring yet reach the indentations. The likeliest mode of failure comes from delamination between printed layers. Orienting the shape flat against the print surface with the same orientation displayed in Fig. 3(b) minimizes overhang locations. However, the printing planes lie normal to the height, which is the direction of largest stress when joining the unit cells and deforming the o-rings. As such, the thickness and heights must be designed to prevent layer delamination and seal the system. A similar device secures lids to the top opening. To measure the acoustic pressure, we place a microphone at the top of any one unit cell. When the sound is not actively measured, rigid lids instead seal the opening. Both instances require the white fasteners visible in Fig. 3(b) that slide around the protruding ledge to secure the lid or microphone. Together this creates a uniquely reconfigurable system, and with the addition of topological mirror interfaces, one that is capable of directing sound with minimal losses due to backscattering and intrinsic losses.

The unit cells considered are parameterized by the quantities displayed in Table I and labeled in Fig. 4(a). Only the asymmetric unit cell is fabricated. When symmetric, both chambers have heights $h1=h2=2.5cm$⁠, a chamber curvature $a=2cm$⁠, and a top opening diameter $d=12mm$⁠. The solid lines in Fig. 4 depict the low-frequency band structure for the symmetric unit cell found using COMSOL's Multiphysics acoustics module. A Dirac degeneracy forms at the K-point where two distinct modes share the same frequency of 1408 Hz. To break inversion symmetry the height of chamber 2, h₂, increases to 3 cm. These values were chosen using an iterative process to maximize the width of the bandgap marked as the red region in Fig. 4(b). The dashed lines in Fig. 4(b) display the dispersion curves for the asymmetric unit cell. Breaking the symmetry separates the Dirac cones to form a bandgap spanning 1350 Hz to 1650 Hz. Within this region, waves cannot propagate through the bulk material.

To demonstrate the existence of the interface and edge modes, we join two sets of five unit cells with opposite chiralities to form a strip of ten. The application of Floquet boundary conditions along the $e1$ direction creates a system of infinite length and finite width. Figure 5(a) displays the computationally predicted band structure for this strip, which reveals two degenerate edge modes and an interface mode crossing the bandgap (Due to the non-integer difference between the Chern numbers for the lattice structure and air [Chern number is zero], the edge modes are not guaranteed to exist or have topological protection, and so are not of interest herein). Figure 5(b) depicts the eigenshapes corresponding to the edge modes, which have the largest pressure magnitudes at the ends of the strip, and the interface mode, which shows the largest magnitude at the center of the strip. As the waves cannot propagate within the bulk material for frequencies within the bandgap, these modes remain isolated and sound must propagate along the $e1$ direction without dispersing.

We next predict the frequency response for a finite system. The lattices in Fig. 6 consist of two materials with opposite chirality (i.e., they are mirror-symmetric) meeting at the boundary indicated by the black arrows. These interfaces are configured such that two large chambers lie across from each other at the interface, rather than continuing the pattern seen in the bulk. When the system is excited at a frequency outside the bandgap (e.g., 400 Hz), the source wave spreads across the entire lattice as shown in Fig. 6(a). In contrast, when the system is excited by a 1590 Hz pressure condition at the source (modeled using a moving baffle), the predicted response is restricted to the interfaces, indicating no propagation into the bulk material will occur at this frequency. Figure 6(b) displays the steady state pressure when the interface is oriented along a straight line, clearly documenting response only along the interface. Figure 6(c) displays the response when the interface is oriented along a sharp right turn. Once again, the predicted results indicate no waves travel into the bulk, and furthermore, no wave scattering occurs at the sharp turn, demonstrating topological protection from backscattering.

Results from experimental testing are presented next to confirm the existence of interface states. Using a Raise3D Pro2 printer (Irvine, CA) operating with a 0.4 mm nozzle size and a 0.16 mm layer height, we 3D-print the exterior unit cells using polylactic acid (PLA) and then assemble them to form the structure depicted in Fig. 7. A blue arrow marks the location of the interface between the two mirror-symmetric lattices. We excite the system using a PUI Audio (Mansfield, TX) AS08308CR-R 7 W Mouser speaker surrounded with damping material to absorb sound leaking from the speaker container and reduce the speaker's impact on ambient noise. A circuit with an I2S Adafruit (New York, NY) MEMS microphone and Arduino (Somerville, MA) MAKR Zero collects the data at a rate of 115.2 kHz. Each chamber is sealed with a solid lid unless we clip the microphone to the top to measure sound. Testing reveals that an excitation frequency of 1600 Hz (very close to the 1590 Hz required in the computational models) provides clear excitation of the interface states. After the system reaches a steady state we measure the pressure within each chamber using the movable microphone—this requires running the experiment separately for each microphone location. Figures 8(a) and 8(b) display the superposition of each of the steady-state sound pressure levels when the interface is straight and turns sharply left, respectively.

Similar to our computational results, the experimental results document sound traveling along the desired interfaces. Unlike the numerical results, however, the lattices exhibit appreciable losses. The straight interface loses approximately 1.5 dB per unit cell. This is comparable to the results of Ding et al.³³ observed in water-borne propagation along a topological interface. However, specific efforts were made to reach these levels of loss including design iteration and post-processing of fabricated parts, as described next.

The external unit cell first fabricated consisted of two cylindrical cavities with a cylindrical tube connecting the two, as shown in Fig. 9(a). This design exhibited severe losses and was not suitable for wave propagation out to appreciable distances from a source. To reduce the losses caused by sharp geometries, we introduced convexity, thus producing the smooth shape illustrated in Fig. 9(b). For the 0.4 mm nozzle size and 0.16 mm layer height, the work of Alsoufi and Elsayed suggests the surface roughness for these conditions would be about 20 μm.⁵⁰ With these surface parameters and a frequency of 1600 Hz, the attenuation evaluated by Whelan and Chambers suggest a negligible difference between losses due to a 3D-printed surface and a smooth surface.⁵¹ However, the remnants of surface supports and printer errors over curvilinear surfaces resulted in roughness significantly greater than that predicted. As such, in an effort to reduce losses, the interior of each exterior unit cell was sanded and painted [see Fig. 9(b)] before assembly to result in the final unit cell depicted in Fig. 9(c). To quantify the improvements, a chain of unit cells consisting of the main chambers and connecting passages was printed. Measurements of the pressure in the second chamber produced the frequency response graph provided in Fig. 10. The figure documents a measured peak pressure for the final design which is more than double that for the initial design at the operating frequency of 1600 Hz. While satisfactory, this reduction in attenuation presents an area for further improvement, which could be addressed using larger interior dimensions, or by active means such as amplification. Larger dimensions will certainly decrease losses at a given frequency due to both the greater volume-to-surface ratio and the lower expected interface frequencies, the resulting bandgap width decreases and thus, requires further study for optimized sound losses.

We have presented a 3D-printed, channeled structure for propagating sound along statically reconfigurable interfaces exploiting topological protection from backscattering. Unit cells are designed specifically to exploit the additive manufacturing process, to include inclusion of complex geometries for reducing propagation losses, consideration of the printing orientation for dimensional accuracy, and dimensional sizing for clearance fits. Computational models reveal the topological character of the introduced interfaces and the experimental results verify the existence of reconfigurable interface propagation. Measurement results document an acceptable, but still significant, attenuation rate of 1.5 dB per unit cell. Efforts exploited herein to minimize sound loss include geometric alterations and treatment of the material surface; however, this represents an area for future improvement. While the presented design is statically reconfigurable, future work will be aimed at actively altering the two chamber volumes in each unit cell, thus yielding dynamic reconfigurability of topological interfaces. This, in turn, may enable applications such as spatial acoustic filtering, spatial multiplexing/demultiplexing, or four-dimensional sound projection.

This material is based upon work supported by the National Science Foundation under Grant No. 1929849 and the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1650044.