## Hydrogel metamaterial design

Periodic repetition of geometry or material property such as Young's modulus in a structure can result in bandgaps within which the wave propagation is forbidden. In the case of a suspended periodic structure that has been excited horizontally from the upper fixed end. The wave propagation property is quantified by measuring its lower free end motion with a laser vibrometer (Fig. 1 (a)). Because the hydrogel is soft and the resulting beams are slender, large excitation amplitude may generate uncontrolled nonlinear vibration. Therefore, in this study, the hydrogels are only excited with low displacement amplitude to maintain the deformation in a linear regime. (Photos of the experimental setup can be found in the supplementary Figure S1.)

The variation of the local mass and stiffness in the periodic structure controls the bandgap characteristics. The concept can be easily demonstrated in a diatomic chain model, which is an infinite periodic discrete chain of two different springs and masses (Fig. 1 (a)). Here, the unit cell is defined as the periodically repeating segment of the diatomic chain. Using this model (supplementary document), one can derive the eigenvalues of this two-degrees-of-freedom system. Dispersion curves for this simplified structure can then be created by plotting the two resultant eigenvalues with respect to dimensionless wavenumber, *γ* (Fig. 1 (b)). The local minimum of the larger eigenvalue defines the upper bound of the bandgap, whereas the local maximum of the smaller eigenvalue defines the lower bound of the bandgap. Controlled variation of unit cell mass and stiffness ratio provides the foundation to tune bandgaps in periodic structures. In Fig. 1 (b), it can be observed that an increase in unit cell mass ratio (\({m_2}/{m_1}\)), *µ* results in a wider bandgap which occurs at a lower center frequency, whereas an increase in unit cell stiffness ratio (\({k_2}/{k_1}\)), *κ* produces a wider bandgap at a higher center frequency.

In this work, photo-responsive hydrogels are used to generate re-programmable periodic structures. Among various photo-chemical hydrogels, our recently developed PAAm-co-TPMLH hydrogel infused with NBA solution is selected in this study for its high photo-activation efficiency and decoupled activation and swelling behavior (62). The PAAm-co-TPMLH hydrogel is first prepared by copolymerizing TPMLH into a covelant crosslinked polyacrylamide hydrogel in dimethyl sulfoxide (DMSO)/water environment. The DMSO solvent is later removed by repeated water rinsing. To program the hydrogel, NBA is first infused into the hydrogel under basic conditions (pH 10.5). The hydrogel is then periodically activated using 365nm UV light and a pre-prepared photomask. After the photo-activation, the hydrogel is immersed in DI water for 4 hours to let it swell to its final periodic structure (Fig. 1 (c), Black box).

Since the photo-responsiveness of the hydrogel is reversible, its unit cell periodicity can be reconfigured through an erasing-reprogramming operation. It involves washing away reaction residuals from photo-activated NBA and recovering the photodissociated triphenylmethane cation (TPM+) back to its non-activated neutral state. The recombination of (TPM+) cation and hydroxide anion is carried out in a high pH environment. For efficient removal of reaction residual and other unwanted ions in the system, the hydrogel is washed in PEG water solution and NaOH water solution alternatively and repetitively for 4 times (Fig. 1 (c), Orange box). Following the washing step, NBA is re-infused into the hydrogel, making it ready for another round of photo-activation and shape morphing.

**Formation of Bandgap in Periodic Hydrogel Structure**

Figure 2 (a) compares the tip transmissibility (absolute tip displacement per base displacement) for both a homogeneous and a periodic hydrogel structure. Unlike the homogeneous hydrogel, bandgap formation can be observed for the periodic hydrogel structure (denoted by the red patch in Fig. 2 (a)). In this type of experiment, the bandgap can be physically interpreted as a band of frequency where the magnitude of tip displacement is much less than base displacement.

The experimental result for the periodic structure is further compared to a Finite Element Method (FEM) model in COMSOL. (Fig. 2 (b)) The FEM model performs linear frequency response analysis on a vertically suspended periodic beam which is axially deformed under gravitational load. The shaker excitation at the clamped end is emulated in the model with harmonic perturbation which is applied to one of the beam ends (yellow segment ends in the model's hydrogel representation provided in Fig. 1 (a)). The model assumes the hydrogel to be a linear elastic material undergoing small deformations when vibrating. Due to their low Young's modulus, hydrogels tend to statically deflect under their weight as they are suspended from the clamped end. The volumetric nature of gravitational loads would cause larger elongation in the unit cells closer to the clamped end when compared to the free end which in turn would slightly disrupt the structural periodicity. This factor is considered in the FEM model using the gravitational body load feature which considers the uneven distribution of the static deflection across the periodic structure. For this hydrogel-based periodic beam, the average diameter of the non-swollen and swollen unit cell segment is found to be 3.66 mm and 6.13 mm respectively, whereas the average length for these segments is 6.68 mm and 6.69 mm respectively. It is assumed that the entire hydrogel structure has uniform mass density, ρ = 1023.3 kg/m3, and Poisson's ratio, ν = 0.49 (low compressibility due to very high-water content). The Young's modulus for the non-swollen and swollen segment is found to be 7.50 ± 0.90 kPa and 4.72 ± 0.82 kPa respectively. Here, the uncertainty bounds are from the different values that are measured for each individual unit cell and multiple points on each unit of the same sample (supplementary document).

To account for the uncertainty in predicting the dynamic response of the periodic beam from the variation in Young's modulus of each non-swollen and swollen unit, the Monte Carlo simulation method is introduced. The fluctuations of the modulus value are within one standard deviation of the average Young's modulus. In the simulation, a total of 150 random combinations are created to define the variations in Young's modulus of the unit cells and the responses at the tip of the beam are obtained using the FEM model. In Fig. 2 (b), it can be seen that the random fluctuations in Young's modulus of the unit cells result in noticeable variations in the bandgap appearance. There are several combinations where the bandgap region is not as pronounced and there are combinations where bandgaps offer high attenuation. Despite these variations, we can observe an overall good agreement between the experimental results (solid black line) and model predictions (dotted red line). The mismatch between the model and the experiment is because the transition zone between the swollen and non-swollen region is gradual and smooth in the experimental sample, while it is simulated as a sharp transition between each unit in the FEM model, as shown in Fig. 2 (b).

**Hydrogel metamaterial bandgap formation reliability**

After verifying the bandgap formation of the periodically programmed hydrogel beam, we further explore the system repeatability and consistency after photo re-programming. A programmed periodic hydrogel sample with a length ratio of α = 1.26 is first to reset back to its non-activated state and then re-programmed with an identical periodicity. Here, α is defined as the length ratio between the swollen region and the non-swollen region. These values are measured after the sample is photo-activated and swollen in water. Figure 3 (a) shows the center frequencies and bandwidths of the bandgaps formed in the initially programmed periodic hydrogel beam and the re-programmed periodic beam with identical geometry. Despite the material defects and heterogeneities that commonly exist in hydrogel materials, it can be concluded that consistent bandgaps can be generated upon photo re-programming with the same photo pattern. Besides the bandgap repeatability of the same sample under different cycles of photo-actuation, we also explore the consistency of bandgap formation of different samples subject to the same photo-patterning. We test two separately prepared hydrogel samples that have the same chemical composition and are programmed into the same unit cell length ratio with the same photomask. Figure 3 (b) shows the comparison of the bandgap of the two separately prepared hydrogel periodic structures. Sample 1 is reused from the previous repeatability test but re-patterned to a different unit cell length ratio of α = 1.64. Sample 2 is newly prepared with the same unit cell length ratio of α = 1.64. The slight deviation in bandgap width between the two samples is likely due to the material defects and inhomogeneity that cause the shape variations between samples after being photo-activated and swollen in water.

**Hydrogel metamaterial bandgap tuning**

While the formation of the bandgap is realized and the reliability and re-programmability have been tested, we next explore the system's potential in bandgap tuning. Here, we focus on studying two control parameters: the photomask pattern that controls the length ratio of the swollen and non-swollen region and the TPMLH concentration that controls the swelling amount of the photo-activated region. A series of samples are prepared. All the samples contain the same amount of monomer (3 mol/L acrylamide (AAm)) and crosslinker (18 mmol/L N,N'-methylenebisacrylamide (Bis)). For quantitative comparisons, we define the following relevant parameters: diameter ratio, δ, Young's modulus ratio, ε, and length ratio, α of the swollen region to non-swollen region (Eq. 1):

\(\delta =\frac{{{d_{swollen}}}}{{{d_{non - swollen}}}}\), \(\varepsilon =\frac{{{E_{swollen}}}}{{{E_{non - swollen}}}}\), \(\alpha =\frac{{{L_{swollen}}}}{{{L_{non - swollen}}}}\) (1)

Using photomasks of different window sizes, the hydrogel beam can be reconfigured to exhibit different unit cell length ratios. Since the repeatability and sample-to-sample consistency have been proved, instead of re-patterning a single sample three times, three separate samples of the same chemical composition are prepared (3 mol/L AAm, 18 mmol/L Bis, and 45mM TPMLH) and programmed to three different unit cell length ratios. The photomasks are prepared using aluminum foils with 7 window openings cut-out at a 10mm spacing. The widths of the window openings are 1mm, 1.5mm, and 2mm, respectively. (Note: The length ratio of the final swollen hydrogel is not equal to the photomask pattern ratio, majorly due to swelling of the activated segments as well as light scattering through-thickness) The tip transmissibility results of the three samples are shown in Fig. 4 (a). It can be observed that the bandgap shrinks, and the center frequency shifts to a lower value as the unit cell length ratio increases. Similar trends can be observed in the FEM simulation results (Fig. 4 (b) and (c) respectively). The FEM model again uses the Monte Carlo method to account for random variation in Young's modulus on a unit-cell-to-unit-cell basis. For a given sample, these random combinations yield fluctuations in bandgap width and center frequency predictions. These bandgap fluctuations in the FEM model can be perceived as the effect of the difference in system sensitivity to gravitational loads with a stiffer or a softer modulus combination. A vertically hanged hydrogel will experience non-uniform axial stretch under gravity influence. The softer the hydrogel is, the greater axial stretch non-uniformity will it experience, which in turn would further jeopardize the structural periodicity and reduce the bandgap quality (narrowed bandgap). For quantitative comparison, the values of the system parameters including unit cell length ratio, diameter ratio, and modulus ratio of the swollen region to the non-swollen region of each sample as well as the values of the bandgap center frequency and bandgap width are listed in Table 1 Since the samples are made with the same chemical composition, the change of photomask patterns leads to only the change of length ratio α, but not the diameter ratio δ and modulus ratio ε. The reduction in the bandgap width and attenuation can be interpreted as that the periodic beam shape converges to a homogeneous beam shape as the non-activated zone length shrinks and the unit cell length ratio increases and a homogeneous structure is not expected to provide any bandgap. Details will be quantitatively discussed in the next section.

We next investigate the influence of TPMLH concentration on the bandgap performance of the periodic hydrogel. Here, three samples that have a TPMLH concentration of 38 mM, 41.5 mM, and 45 mM are prepared and tested. They are activated under the same photomask. The window openings on the photomask are 1.5mm wide and 10mm spacing from each other. The tip transmissibility results of the three samples are shown in Fig. 5 (a). It can be observed that the center frequency of the bandgap shifts to the left and the bandgap widens as the TPMLH concentration increases. Similar trends can be observed in the FEM simulation results (Fig. 5 (b) and (c)). For quantitative comparison, the values of the system parameters including unit cell length ratio, diameter ratio, and modulus ratio of the swollen part to the non-swollen part of each sample as well as the values of the bandgap center frequency and bandgap width are listed in Table 2 With a higher TPMLH concentration, the hydrogel swells more after photo-activation and the swollen hydrogel becomes softer. As a result, the diameter ratio δ and length ratio α increase, while the modulus ratio ε decreases as the TPMLH concentration increases. It is observed that the length ratio increase as TPMLH is not as large as the diameter ratio increase. The reason is thought to be that the light scattering near the boundary is more significant when the TPMLH concentration is low, which results in an effectively longer activated region. Consequently, it is the significant increase in diameter, a significant decrease of Young's modulus, and a moderate increase of length that leads to a wider bandgap and lower center frequency.

**Comparisons with diatomic chain model**

In the experimental and simulation results, we noticed that the increase in unit cell length ratio (α) can shrink the bandgap while shifting it to a lower center frequency, whereas the bandgap widens and shifts to a lower center frequency with an increase in TPMLH concentration. These trends can be interpreted using a diatomic chain model. The variations in unit cell length ratio and TPMLH concentration ultimately change the unit cell mass and stiffness ratios which are fundamentally responsible for bandgap formation in periodic structures. For a structure under bending vibration, the equivalent mass and stiffness ratios can be given by:

\(\mu ={\delta ^2}\alpha\), \(\kappa =\frac{{{\delta ^4}}}{{\varepsilon {\alpha ^3}}}\) (2)

To analyze the experimental trends using the diatomic chain model, we use Eq. 2 to solve for the mass(*µ*) and stiffness ratios(*κ*) of the total six hydrogel samples. As shown in Fig. 6 (a) and (b), higher unit cell length ratios result in a higher mass ratio but a lower stiffness ratio which ultimately gives a narrower bandgap at a lower center frequency. On the other hand, it can be seen that an increase in TPMLH concentration results in higher mass and stiffness ratios, which ultimately gives a wider bandgap at a lower center frequency. However, for the amount of increase in mass and stiffness ratios due to TPMLH variation, the diatomic chain model predicts a negligible reduction in bandgap center frequency. The bandgap shifting trends observed with the diatomic chain model are identical to the experimental trends. Diatomic chain model is an oversimplification of the structure on hand as it discretizes a continuous system and neglects more complex factors such as static axial deflection, hence diatomic model bandgap predictions carry little quantitative relevance. However, the trend agreement concludes that the experimental variations in the bandgap width and center frequency are ultimately a consequence of the change in unit cell mass and stiffness ratios which quantify periodicity in periodic structures.