Introduction

Researchers have been inspired by electronic semiconductors, leading to the developments and studies of phononic semiconductor structures [1,2]. Phononic crystals (PnCs) represent a common approach to create bandgaps in the transmission of elastic, acoustic, or thermal waves [3][4][5]. For a desired range of operational frequencies, composite materials can mimic effective mediums via homogenization approximations for long wavelengths [2,6]. Periodic arrangements of elastically dissimilar materials in mesoscopic lattices and scatterers produce Bragg gaps [7] which was driven by Bragg scatteringinduced destructive interference [8]. PnCs can be designed in one-dimensional [9], two-dimensional [10], or even complex three-dimensional [11] forms to realize special applications such as bandpass filtering [12,13], beam collimation [14], and pulse synthesizers [15]. The introduction of asymmetry or anisotropy to simple PnCs can further lead to unique effects such as unidirectional transmission, gradient index, and Anderson localization [16] of acoustic waves. However, the increasing complexity of PnC designs poses challenges in fabrication.

Additive manufacturing (AM) has emerged as a practical solution for fabricating samples with complex geometries or arrangements, including non-conventional PnCs [17]. While costly metal/alloy AMs remain less accessible, polymer [18,19] printing techniques are commonly employed for PnC fabrication. These techniques include fused deposition modeling (FDM) for thermoplastics, or hydrogel composites [20,21], stereolithography (SLA) for UV laser/projection polymer curing [22,23], and inkjet AM for injection hybrid gel extrusion [24,25]. It is evident from these descriptions that printed materials exhibit distinct elastic or acoustic properties, necessitating careful consideration in composite selection to achieve sufficient impedance mismatching for PnC realization. However, existing literature primarily emphasizes printing accuracy in terms of dimensions [26], with a limited focus on the acoustic impedance printed materials.

Various additive manufactured polymers exhibit a wide range of physical properties, including elastic modulus and speed of sound, spanning from amorphous-liquid-like to crystalline-solid-like characteristics. These materials offer an alternative and tunable approach compared to conventional methods [27][28][29][30]. On the conventional PnC system, tuning the sound behaviors is feasible by shifting the operating frequency. For monochromatic wave systems, alternative approaches are needed to modify the geometrical configurations or effective properties of the PnCs. Different from electronic semiconductors or photonic crystals, due to the relatively large size of PnCs, more options of tuning mechanisms can be involved such as light irradiation [31], external magnetic fields [32], mechanical motions/ elastic deformations [10], and thermal or chemical energy transportations [33]. In existing literature, external optical and magnetic waves can tune small scale PnCs quickly, but lack penetration. Mechanical tunable PnCs were majorly achieved by the rotation of the asymmetric scatterers [10], which also cannot be scaled down due to finite size of motor or pumping systems. On the contrary, although thermal tunability responds not quick enough, there is no clear size limitation and penetration issue on the PnCs, which is considered as a decent mechanism to design tunable PnCs, especially in the underwater applications.

However, most existing thermal tunable PnCs exploit phase responsive hydrogels-prone to volumetric deformation during swelling-or liquid metals, which can embrittle and lose mechanical contact under cyclic use. To achieve thermal tunable PnC designs, the general consensus in the literature is to leverage the phase transformation of either the scatterers or lattice materials, as this induces sharp and significant changes in physical properties [34]. Previous studies have demonstrated the use of thermal-responsive hydrogels for underwater thermal tunable PnCs [31], where the hydrogels' lower critical solution point allows them to transition between two distinct states. Liquid metal gallium, which undergoes phase changes around room temperature, can provide similar effects. Both thermal-responsive hydrogels [35] and liquid metal gallium [36] are commonly employed to achieve strong variations in physical properties, particularly in sound speed and acoustic impedance. However, hydrogels face challenges due to large volumetric changes during phase transformations, which often result in inconsistent contact with metal or plastic scatterers during the reversible tuning process. On the other hand, liquid metal gallium struggles to maintain proper contact with polymers and can induce embrittlement in metals such as aluminum and copper [37], making it difficult to achieve stable, tunable PnC designs. These limitations hinder reliable operation in underwater or reconfigurable settings. Here, we propose a simple, robust alternative: by embedding additive manufactured polymer scatterers within a water/ice lattice, we leverage the well characterized impedance at the freezing point to switch Bragg gaps on and off without any moving parts or complex geometries. This strategy solves the stability and scalability issues of prior approaches, offers broad material selection, and achieves >20 dB tunability across multiple AM platforms.

Materials and methods

Numerical simulations

The band structure was numerically calculated using the plane wave expansion method in the range between 0 and 0.5 with a step interval of 0.01 for the normalized Bloch vector k along Γ -X, X -M, and M -Γ directions. The dispersion relation ω = ω (k) is obtained from the wave equation. For sound wave propagating in inviscid inhomogeneous fluid the wave equation is written for acoustic pressure p (r, t),

Here K is the bulk elastic modulus and ρ is the mass density. Being periodic functions of coordinates these functions are expended in Fourier series

where G are the reciprocal lattice vectors. In a periodic structure a monochromatic solution of the wave equation ( 1) is represented by Bloch wave

Substitution of the Fourier series (2) and (3) into the wave equation (1) leads to a set of linear homogeneous equations for the Fourier coefficients p (G)

This infinite set of equations has a nontrivial solution if its determinant vanishes. Leaving a finite number of plane waves in the determinant equation

can be numerically solved over ω for a fixed value of the Bloch vector k, thus, giving the dispersion relation for sound. For elastic wave in a PnC a procedure of calculation of dispersion relation is similar but the wave equation ( 1) is replaced by the corresponding equation containing also the shear modulus. The details of calculations of dispersion of sound in PnC with solid and fluid constituents can be found in the review [38].

Numerical simulations of acoustic wave propagation behaviors were conducted using COMSOL Multiphysics 6. The material acoustic properties were experimentally obtained by monostatic time-of-flight. The densities were estimated by weight and volume measurements. The PnCs are considered as the arrays of 10 units × 10 units. The studied frequency range was 100 kHz to 400 kHz. The side walls are set to impedance-matched conditions with the water. The acoustic source has emission amplitude at 1 µPa, where the 0 dB reference line was also 1 µPa. In the simulation models, the ambient water areas were considered with impedance-matched boundary conditions to eliminate internal reflections from the outer boundaries.

Sample preparations

FDM samples were made by Snapmaker single extruder FDM printer using the ABS (acrylonitrile butadiene styrene) and PLA (polylactic acid) filament from Hatchbox with 0.4 mm nozzle. The designed 3D models were converted to G-Code by KISSlicer64 with a 100% infill rate setting and precision printing conditions for a slower printing process. Before the slicing processing, the infill direction (line type infill) was modified to a unidirectional orientation parallel to the wave propagation direction to decrease the acoustic attenuation. SLA samples were printed by a Formlab 2 printer with its own slicing function using identical settings. The resin liquids were obtained from Formlab. The agar sample was printed by Celllink BioX with a 5 ml syringe. During printing, the solution was held by the printer heat at 40 • C. The substrate of the printer was set to 5 • C during the printing. Agar is a polysaccharide vegetarian substitute for gelatin extracted from red algae, such as Gracilaria and Gelidium. Agar powder, 100% natural seaweed by ECO-TASTE, was used to make agar gels with 5 wt.% and 10 wt.% concentrations. For 5 wt.% agar gels, 5 g of agar powder was dissolved in 100 g de-ionized and degassed water, and the mixture was heated up to the boiling point and kept for 2 min with constant stirring. The mixture was then cooled down at room temperature for 24 h. We use the same six-well plates to mold the agar gel phantom.

Acoustic experimental setups

For the time-of-flight tests, all the samples were printed as disc geometry with 12 mm diameter and around 25 mm height. Totally 6 different additive manufactured materials from 3 distinguishable AM processes were experimentally tested acoustic impedance Z by expressing Z = ρc, where ρ represents density and c is the sound velocity. The sound velocity values were obtained by monostatic time-of-flight method, wherein the density values were estimated by obtained the ratio between samples' mass and volume. The samples included SLA clear resin, SLA flex resin, SLA draft resin, FDM ABS, FDM PLA, and inkjet agar gel. Water properties were also experimentally measured. The ice density was obtained from literature [39]. The speed of sound values of ice was also experimentally measured (5 times) with 25.4 mm width ice tubes by time-of-flight in around 0 • C water environment. The ice tubes were made by deionized water source with cooling rate around 2 • C h -1 in freezer. An Olympus Panametrics V301 1' diameter 0.5 MHz immersion transducer was used to generate a pulse and record signals reflected by the samples. The pulse source and time trigger were controlled internally by a JSR Ultrasonics DPR 500 Pulse/ Receiver, and the data was recorded by a Tektronix MDO 34. Aquaphor lotion was applied to the interface between the tested sample and the transducer surface as a coupling material. In the transmission tests of the PnC, the bistatic transducers were pre-aligned and immersed in refrigerated de-ionized water ambient as coupling with printed resin PnC. The ice/resin PnC was pre-prepared using a rectangular mold. The time domain transmission waveform was first recorded on the ice/resin. Along with the natural convection, the raised temperature of the water ambient causes the ice lattice to melt as a water lattice. Half and half PnC transmission were recorded when 5 water/PnC periods appeared. The water/resin PnC transmission with the identical setup after the water ambient is achieved around the room temperature. The frequency spectra were obtained by Fourier transformation of the temporal data.

Results and discussion

Tunable PnC design and numerical simulation

From conventional pulse time-of-flight measurements, figure 1(A) displays the speed of sound values (c) for water, ice, and SLA printed clear resin, revealing significant difference among these media. Figure 1(B) illustrates the corresponding acoustic impedance values calculated using the equation Z = ρc, where ρ represents density and c is the velocity. Notably, the density of ice and SLA printed resin results in highly comparable acoustic impedance values. Conversely, the low speed of sound value in liquid state water leads to a noticeable mismatch in calculated acoustic impedance compared to ice and SLA-printed resin. Based on these findings, a functional temperature-switchable 2D PnC was designed with a water/ice lattice and SLA-printed clear resin scatterers, as depicted in figures 1(C) and (D). The square lattice and circular scatterers have periodicities of 5 mm and diameters of 3 mm, respectively. The designed PnC achieves a filling fraction of approximately 29%. At room temperature, there is a 50% impedance mismatch between the scatterer material (SLA resin) and the lattice material (water). However, at freezing temperature, the impedance values become nearly matched, with only a 0.7% difference between the scatterer material (SLA resin) and lattice material (solid-state ice).

Using the plane wave expansion approximation, band structures of the ice/resin and water/resin PnCs were calculated along the 0 • (ΓX), 45 • (ΓM), and 90 • (XM) directions, as shown in figures 2(A) and (B). As expected, along the 0 • (ΓX) direction, both PnCs exhibit a linear 1st band, a 2nd band with a negative slope, and overlapping 3rd and 4th bands. In the water/resin PnC (figure 2(B)), the first band gap occurs around 145 kHz, with additional band gaps around 300 kHz alongside the 3rd/4th bands. Conversely, in the ice/resin PnC (figure 2(A)), the band gaps are closed due to impedance matching between ice and SLA printed resin, particularly affecting the 2nd and 3rd band gaps. The width of the 1st band gap is significantly reduced to only 1.3 kHz in the infinite long PnC (figure 2(A)), which may be negligible in practical finite length PnCs compared to the water/resin PnC (figure 2(B)). Additionally, neither PnC exhibits a complete band gap on the band structure, owing to the relatively small impedance mismatch compared to conventional immersion PnCs, such as water/metal PnCs. However, despite this limitation, the partial gap along the 0 • (ΓX) direction is sufficient for demonstrating the temperature-switchable phase-dependent bandpass filter. transmission frequency around 295 kHz in the water/resin PnC is not observed due to insufficient numbers of periods in simulation, the combined '2nd band gap' still offers more than -12 dB difference compared to the ice/resin PnC.

Figures D1, E1, F1 and G1 depict acoustic pressure distribution maps at 100 kHz, 150 kHz, 200 kHz, and 300 kHz, respectively, of the water/resin PnC. Sound waves are transmitted from the left side and received on the right side in all cases. At the 1st and 2nd transmission bands (figures 2(D1) and (F1)), there is sufficient transmission of sound waves through the proposed water/resin PnC without significant decay. In the gap frequency, the 1st (figure 2(E1)) and 2nd (figure 2(G1)) band gaps exhibit clear sound wave decay along the number of periods they propagate through. Conversely, in the ice/resin PnC case (figures D2, E2, F2 and G2), all selected frequencies demonstrate significant transmissions compared to the water/resin case. Notably, the wavelength presented in the ice/resin PnC experiences significant broadening effects compared to the water/resin case due to the significantly higher speed of sound in ice.

Experimental characteristics

Following verification from numerical simulations, experimental characterizations of the SLA-printed clear resin 10 by 10 periods PnC were conducted using a bistatic immersion configuration in deionized water (figure 3(A)). In the water ambient, the printed resin scatterers functioned as in the water-/resin PnC case. For the ice/resin PnC case, the resin scatterers were pre-frozen in ice using a rectangular mold, as depicted in figure 3(B). The molded ice/resin PnC had parallel surfaces for attaching the ultrasonic emitter and detector. The molded ice/resin PnC was still tested in the deionized water ambient to ensure sufficient coupling between the ice surfaces and ultrasound transducers. However, the deionized water used in the ice/resin PnC case was pre-refrigerated to about 2 • C to minimize the melting process. Furthermore, frequency spectra were obtained by Fourier transformation of the time domain broadband pulse transmission signal, which is significantly faster than frequency domain sine wave sweeps. This approach ensured minimal degradation of the ice/resin PnC during the measurement period. For direct comparison, the water/resin PnC was also tested using the identical configuration at room temperature.

During the transmission test of the ice/resin PnC, without additional temperature control on the refrigerated water ambient under room temperature, the ice on the PnC completely melted within 1.8 h. As the ice was naturally replaced with water fraction, the transmission spectrum exhibited distinguishable transmission and gap frequency ranges from the 5 periods of water/resin PnC presented (figure 3(C)). In figure 3(D), the experimental frequency spectra were summarized, including ice/resin PnC, water/resin PnC, and half/half PnC, with the pure deionized water transmission serving as a normalizing reference. Similar to the numerical simulation results, the ice/resin PnC (blue line) exhibits uniform material transmission spectra without showing any band gap-like dips in the studied frequency range, thereby verifying the experimentally obtained acoustic impedance matching parameters between ice and SLA printed clear resin discussed in the previous section. The applied ultrasound transducer had a frequency of 100 kHz, approaching the low end of the effective band.

Hence, the normalized ice/resin PnC still exhibits slight fluctuations due to the relatively low signal-to-noise ratio.

As indicated by the red curve in figure 3(D), without the backfilled ice in the PnC, the water/resin PnC shows an appreciable drop in transmission reduction around 145 kHz and 300 kHz, consistent with the numerical simulation shown in figure 2(C). The difference in the transmission frequency band between the ice/resin PnC and water/resin PnC is -15 dB and -32 dB at 145 kHz and 295 kHz, respectively. Comparing the numerically simulated and experimentally measured transmission amplitude levels at room and freezing temperatures, the ice/resin PnC exhibits a lower overall transmission level with respect to the water/resin PnC, which may be attributed to additional reflection loss at ice surfaces or/and unexpected ice attenuation due to weak contact at ice/resin interfaces or/and massive grain morphology in ice. Furthermore, as indicated by the band structure (figure 2(B)), the 3rd and 4th overlapping transmission bands are located between the 2nd and 3rd band gaps with a low slope. From the experimental transmission results, around 300 kHz, a sub-state mode appeared in the band gap, which has the potential to be the presence of 3rd and 4th transmission bands. On the green curve, the spectra of half/half PnC also depict the presence of this sub-state mode in the gap around 300 kHz. Besides the 2nd gap, the half/half PnC exhibits transmission spectra with weak gap amplitude reduction with respect to the water/resin PnC, owing to the less effective water/resin PnC periods. It is also noteworthy that the spectra of the water/resin PnC show a slight red shift and broadening in the gaps with respect to the half/half crystal, which is potentially caused by the different operating temperatures inducing speed of sound differences at room temperature and refrigerated deionized water.

With respect to the overall transmission amplitude in numerical simulation results, it is notable that the experimental transmission amplitude was lower. The first potential cause could be that the numerical simulation involved in this study was 2D simulations, which should have higher in 3D experiments due to the lack of Z-axis wave spreading and acoustic energy loss. The second potential reason could be that, the additive manufactured PnC might have slight diameter deviation and non-uniform mechanical property distribution in real experiments, which also decrease the overall transmission through the PnC. Such uncertainties are absent in numerical simulation. Furthermore, the additive manufactured PnC rods have slight surface roughness which also results in more energy dissipation in the transmission. However, in simulation, the PnC rods are ideally smooth, which minimized such energy dissipation in the transmission.

Impedance measurements on various additive manufactured materials

The preceding calculations, simulations, and experiments have demonstrated the feasibility of using SLA-printed clear resin to achieve a thermally switchable phase transformable acoustic bandpass filter PnC with water/ice. In addition to SLAprinted clear resin, several other commonly used AM polymers in the research field have the potential to fabricate PnCs or metamaterials. This study selected SLA draft resin, SLA flex resin, FDM ABS, FDM PLA, and inkjet agar gel for experimental characterization of acoustic impedance values and numerical application in the design of PnCs. It is noteworthy that inkjet gelatin is also a common material; however, due to its instability in the water environment, it was not included in this study. Similar to the previously used SLA clear resin, the acoustic impedance values of the mentioned printed materials were determined using monostatic time-offlight tests (figure 4(A)). The density values were estimated by measured weight and volume. Figure 4(B) summarizes the impedance comparison. FDM-printed ABS and PLA exhibit slightly higher impedance compared to liquid-state water but significantly lower than solid-state ice. The impedance mismatches are approximately 25% and 21% with ABS/water and PLA/water, 45% with ABS/ice, and 53% with PLA/ice, respectively. Similarly, the harder draft resin and softer flex resin generally exhibit matched impedance in ice with differences of 7% and 6%, respectively. However, sufficient impedance mismatches still exist with water, approximately 52% and 45%, respectively. On the contrary, inkjet-extruded agar gel demonstrates a high similarity in impedance values to water with only a 2% difference, but it mismatches with ice by about 91%. This soft and water-like acoustic impedance in agar gel suggests the potential for a reversed design of the proposed temperature-switchable PnC.

Figures 4(C) and (D) depict the 0 • (ΓX) direction band structures and corresponding 10 × 10 crystal transmissions of the designed PnC with the substitution of clear resin for draft resin and flex resin, respectively. The overall behaviors of draft resin and flex resin PnCs were comparable to the detailed discussions of the clear resin PnC in the previous section. However, due to the greater mismatch between draft resin/flex resin and ice, the reduction in transmission in the 2nd gap regions (around 300 kHz) is more pronounced compared to the clear resin case (figure 2(C)), particularly in the flex resin/ice PnC. Similar to the clear resin/water PnC, the absence of the 3rd and 4th transmission bands are also observed in the numerical simulated transmission spectra of the draft resin and flex resin cases. In the flex resin/water PnC (figure 4(D)), compared to the clear resin PnC (figure 2(C)), the reduction in transmission in the 1st gap is less significant due to the noticeably lower (closer-to-water) impedance value in flex resin (figure 4(H), columns 1 and 2).

Figures 4(E) and (F) illustrate the band structures and corresponding 10 × 10 crystal transmissions of FDM-printed ABS and PLA PnCs. Generally, the ABS and PLA PnCs exhibit similar behaviors, showcasing 1st and additional gaps in both water lattice and ice lattice. Unlike the harder SLA printed resins, the impedance values of FDM ABS and PLA are closer to water rather than ice. Consequently, the switchable states are reversed, resulting in significant transmission reductions in the ice lattice-based PnCs. Beyond 250 kHz, multiple narrow gaps appear on the band structures of both PLA and ABS PnCs, inducing relatively low and widespread transmission on the frequency spectra. Conversely, the waterbased PnCs exhibit narrow gaps with less weakening effects on transmission.

Due to the PLA's lower acoustic impedance compared to ABS, the 1st gap in ABS/water PnC is more pronounced than in the PLA/water case, while the 1st gap in ABS/ice PnC is less effective than in the PLA/water case. These behaviors lead to a greater temperature switchable transmission difference in PLA PnC than in ABS PnC. Additionally, FDM-based AM techniques allow for easy adjustment of the infill fraction of the printing object. Lowering the infill fraction setting can decrease the effective density and elastic modulus, potentially achieving lower acoustic impedance values to approximately match water impedance for future PnC or metamaterial designs.

Figure 4(G) illustrates the band structure and numerical simulation of transmission in the agar gel/water and agar/ice PnCs. As expected, due to the sufficient impedance matching condition between water and agar gel, the water/agar PnC exhibits a uniform material-like behavior with consistent and frequency-independent transmission amplitude. Conversely, the agar/ice PnC displays a 40 kHz wide gap associated with significant amplitude reduction on the frequency spectra.

Figure 4(I) presents the temperature switchable transmission difference in terms of absolute values between the room temperature state and freezing temperature state. Both SLAprinted resins and inkjet-printed agar can achieve over 20 dB with 10 periods of PnC, which is considered to be sufficiently tunable. However, AM products have limitations compared to conventional manufacturing, such as complex residual stress, non-uniform mechanical properties distributions, and surface/internal structure issues. Among the AM techniques studied, SLA offers relatively stable and consistent printing, while inkjet printing allows for printing soft gel or hydrogel. FDM products offer material selections and flexibility but may have surface roughness issues. Nonetheless, this study presents a functional temperature-tunable PnC design based on the material properties of polymer additive manufactured materials, offering the potential for breakthroughs in acoustic functional materials and metamaterials in future research.

As stated in the previous section, the design of this tunable PnC is purely relied on the material properties of additive manufactured polymers and water/ice, especially on acoustic impedance. The mismatch amount between the acoustic impedance between lattice material (water/ice) and scatterer materials (additive manufactured materials), leads to the presence/absence of the bandgap. For example, resin scatterers have strong enough impedance mismatch with water lattice which can offer bandgap in a PnC. However, it does not have clear enough impedance mismatch between the ice lattice, where effectively offer an acoustically homogenous material, even though in a PnC configuration, still leading to the absence of the bandgap. Hence, the tunability is not PnC design dependent.

As the simplest square lattice PnC indicated in this work, there is no special design needed to achieve the tunability, which has potential to be generalized on the complex PnC design in the existing literature [40], with the material combinations suggested in this work. The absolute position of the bandgaps should be relocated on the frequency window, but the temperature tunability should be properly maintained. Hence, the other PnC applications, such as lensing [33,41], collimation [29,42], pulse-decomposition [15], nonreciprocity [10,43], or even Anderson localization [16], can be further constructed with temperature tunability, which can offer more feasibility and flexibility in the underwater acoustic applications.

Conclusion

This study presents a design for a novel functional temperature-tunable PnC based on the material properties of polymer additive manufactured materials, which the band gap can be switched between On-Off states by heating/cooling between room temperature and water freezing point. This work novelly demonstrated the temperature tunable PnC structure using the phase transition between water and ice via both numerical simulation and practical experiments. In the existing literature, the common tunable material applied in PnC was temperature tunable hydrogels, which were not easily accessible and not cost-friendly. Through experimental characterizations of acoustic impedance values on some common polymer additive manufactured materials, it was observed that the additive manufactured materials were experimentally characterized to find the impedance mismatching/matching condition with respect to the water/ice. It was found that the SLA printed hard resins have almost matched impedance with ice, whereas the impedance is strongly mismatched with water. On the contrary, inkjet extruded agar gel has highly matched impedance with water but not ice. By using those additive manufactured materials to produce PnC scatterers, they can provide significant physical contrast to offer Bragg gap only in either water lattice or ice lattice, as the numerical simulation results showed. Both resin PnCs and agar PnCs demonstrated over 20 dB of tunability between the room temperature state (water lattice) and freezing temperature (ice lattice) in numerical simulation. Furthermore, experimental transmission tests were performed on a 10 by 10 SLA-printed clear resin PnC. The results aligned with numerical simulations, showing over 15 dB and 30 dB of tunability on the 1st and 2nd band gaps, respectively. This novel design approach of additive manufactured temperature tunable PnC offers novel tunable mechanism, material selections, and unlocked flexibility of the potentially future complex functional PnC designs. Moreover, this design unlocks flexibility for future complex PnC architectures, with potential applications in bandpass filtering, beam collimation, pulse synthesizers, acoustic lensing, non-reciprocal waveguiding, pulse decomposition, and even Anderson localization devices.