Piezoelectric cell structure and electronic device incorporating same

Abstract

The piezoelectric cell can have a stretching-dominant truss structure, made of a piezoelectric material and having a polarization direction coinciding with a truss direction of the truss structure.

Description

PIEZOELECTRIC CELL STRUCTURE AND ELECTRONIC DEVICE INCORPORATING SAME BACKGROUND

[0001] By tailoring their periodic or non-periodic underlying microarchitectures, metamaterials, as a category of rationally designed advanced materials (RDAMs) can exhibit distinctive and exotic multifunctional properties (e.g., negative optical refractive index, negative effective density, negative Poisson’s ratio (NPR), and shape reconfiguration) that surpass those found in natural or chemically synthesized substances. Multifunctional ferroelectric materials with inherent coupled structural, piezoelectric, and pyroelectric properties have the potential for use in actuators, sensors, catalysis, and energy harvesters. One type of effort to improve the thermo-electromechanical responses of ferroelectric materials is to focus on the engineering of material composition and underlying structures at the micro and / or nanoscale of bulk materials, involving templated grain growth, optimization of grain size, and domain engineering.SUMMARY

[0002] Rational design of microarchitecture of cellular ferroelectric metamaterials as a means to augment their multiphysical properties had yet to be explored.

[0003] The piezoelectric effect converts mechanical oscillations (du / dt where u is displacement and t is time) into electrical power, while the pyroelectric effect transforms temperature fluctuations (dT / dt where T is temperature) into electricity. Although porosity in ferroelectric ceramics is commonly regarded as a defect, the deliberate introduction of porosity can be beneficial for specific applications by significantly reducing the dielectric constant (K33) and volume-specific heat capacity (CE). For example, the piezoelectric voltage constant (933 = d33 / / <33) and the pyroelectric voltage sensitivity (Fv= P3 / (CE 33)), which are key in force and temperature sensors, exhibit higher values in porous ferroelectric materials compared to their fully dense counterparts. Herein, dss represents the longitudinal piezoelectric constant and P3 is the pyroelectric constant.

[0004] In addition to the relative density, pore topology engineering can significantly enhance ferroelectric properties. Indeed, it was found that, compared to ferroelectric foams with randompore architectures, those with aligned pores parallel to the polarization direction exhibit slightly smaller figures of merit, whereas those with porosity aligned perpendicular to the polarization direction demonstrate much higher values. For instance, the experimental 933 value of ferroelectric foams with perpendicular porosity is almost 2.1 and 3 times higher than foams with random and parallel porosity, respectively, at 0.4 relative density. However, insufficient polarization and inefficient stress transfer can cause the improvement of these figures of merit to come at the expense of the reduction of the piezoelectric constant dss compared to the constitutive bulk piezoelectric materials (e.g., around 30%, 25%, and 80% reduction in the d33value of piezoelectric foams with 0.4 relative density with random, parallel, and perpendicular pores have been reported, respectively). The trade-off between the decreased dss and the decreased dielectric constant can thus hinder the full potential of porous ferroelectric materials for reinforcing their multifunctionality. Furthermore, traditional fabrication methods such as burnt-out polymer spheres (BURPS), gel casting, replica templating, and freeze casting restrict the realization of ferroelectric metamaterials with truss, plate, or shell nano / microarchitecture.

[0005] Studying the mechanical behavior of intricately designed truss-based metamaterials reveals intriguing properties such as ultra-lightness, ultra-stiffness, structural multistability and extreme resilience. Based on the cell topology, lattice materials can be classified as bending-dominated or stretching-dominated. Their Young’s modulus (E) and yield strength (oy) follow a power law relationship with relative density as E / Es<x (p / ps)mand ay / ays<x (p / ps)n, where the subscript s denotes the corresponding properties of the solid material, and the exponents m and n are determined by lattice microarchitectures. Bending-dominated lattice with m = 2 and n = 1.5 bear loads through the bending of their struts, leading to uneven stress distribution and low material efficiency. In contrast, the struts of mechanically efficient stretching-dominated lattice predominantly deform via uniaxial compression or tension and exhibit linear scaling relationship with relative density, where m = 1 and n = 1.

[0006] Maxwell’s stability criterion can be applied to determine the deformation mode of lattice materials by assessing the rigidity of their pin-jointed configurations. Mathematically, this is expressed as M = b - 3j + 6, where M > 0 indicates a stretching-dominated lattice, and M < 0 indicates a bending-dominated lattice. Here, b represents the number of struts, and j represents the numberof frictionless joints in the 3D truss-based structure. On the other hand,progress in truss-based ferroelectric metamaterials has mainly remained limited to their diverse anisotropic piezoelectric behavior and actuation modes. Leveraging the advantages of additive manufacturing techniques, various architected ferroelectric materials can be investigated, such as gyroid and diamond skeletons, 2D diagonal lattices, and 3D SC lattices. However, a decrease in relative density is associated with weakened piezoelectricity. Although shell-based ferroelectric metamaterials can mitigate the influence of reduction of relative density on the decrease of direct piezoelectric constants, achieving further enhancement of piezoelectric constants through a decrease of relative density can be elusive.

[0007] It was found that the mechanical superiority of stretching-dominated lattice structures could be harnessed in the exploration of the feasibility of achieving simultaneous elevated piezoelectric and reduced dielectric constants in ferroelectric meta mate ria Is.

[0008] Programmable truss-based ferroelectric metamaterials can be achieved by adjusting their scaling ratios and relative densities and can lead to suitably high piezoelectric constants and ferroelectric figures of merit. A modified multiphysical asymptotic homogenization method, considering the effects of nonuniform local polarization electric fields, and analytical equations are proposed to elucidate the extraordinary characteristics of these ferroelectric meta mate ria Is. The rationally designed cellular ferroelectrics with intricate truss-based geometric features are 3D printed by a customized digital light processing (DLP) technology to deliver giant ferroelectric properties. In an innovative paradigm can be introduced for creating highly programmable and ultrasensitive lightweight multifunctional ferroelectrics for applications in self-powered vibration / thermal sensors, touchless control devices, wearable input devices, and precise trajectory recognition systems.

[0009] More specifically, it was found that for some categories of structures, increasing porosity increased piezoelectric effect instead of decreasing it. For instance, it was found that stretching-dominant structures, when polarized in a direction coinciding with a truss direction of the structure, could exhibit an enhancement, such as increased piezoelectric effect with increasing porosity. Stretching-dominant structures are structures in which the main type of deformation, when compressed, is stretching. This can be compared, for instance, to bendingdominant structures where the main type of deformation, when compressed, is bending, and which do not exhibit this enhancement. It was also found that this feature could continue to beexhibited when a cell of the structure is repeated, as a unit cell, along one or more axes of a periodic metastructure.

[0010] In accordance with one aspect, there is provided a unit cell having a stretchingdominant truss structure, made of a piezoelectric material and having a polarization direction coinciding with a truss direction of the truss structure.

[0011] Many further features and combinations thereof concerning the present improvements will appear to those skilled in the art following a reading of the instant disclosure.DESCRIPTION OF THE FIGURES

[0012] In the figures,

[0013] Fig. 1 A schematizes an example process of making a truss structure;

[0014] Fig. 1B schematizes an example process of imparting polarization to the truss structure of Fig. 1A;

[0015] Fig. 1C is a graph presenting the evolution of a polarization electric field overtime, in accordance with the example of Fig. 1 B;

[0016] Figs 2A, 2B and 2C are three example metastructures made of piezoelectric unit cells, including a 2X2X2 metastructure, a 4X4X2 metastructure, and a 8X8X2 metastructure, respectively;

[0017] Figs 3A, 3B and 3C are graphs presenting longitudinal piezoelectric charge constant c / 33 for ferroelectric lattice metamaterials having different densities, taking the examples of an octet truss structure, a tetrakaidecahedron truss structure, and foams, respectively;

[0018] Figs 4A and 4B schematize loading states of a truss for stretching and for bending, respectively;

[0019] Fig. 5 is a graph plotting relationships between c / 33 and inclination angle 0 for stretching-dominated cross-trusses and bending-dominant cross-trusses;

[0020] Figs 6A, 6B and 6C illustrates scaling of an octet truss structure along the X3 axis, with scaling ratios of 0.5, 1 and 1.5, respectively;

[0021] Fig. 7 is a graph plotting a 3D contour map showing the influence of relative density and scaling ratio on the c / 33 of stretching-dominated octet truss structure;

[0022] Figs 8A and 8B are graphs presenting a comparison in the relationships between c / 33 and density and scaling ratio, respectively, for different types of truss structures;

[0023] Fig. 9 presents a comparison of c / 33 and 933 across different types of lattices;

[0024] Fig. 10 presents experimental drop weight test results for an octet truss structure (pr= 0.2 and s = 0.2), tetrakaidecahedron structure (pr= 0.2 and s = 0.2), foam structure (pr= 0.5), and fully solid ferroelectric ceramics;

[0025] Figs 11 A, 11B and 11C illustrate scaling along different axes, including unsealed, scaling along xi-direction, and scaling along X3-direction, respectively;

[0026] Fig. 12 is a graph presenting simulation results (small dots) and experimental results (big dots) of c / 31 and c / 32 of octet truss and tetrakaidecahedron family lattices and foam;

[0027] Figs. 13A and 13B illustrate different loading states a ferroelectric octet truss compressed under xi-direction and X2-direction, respectively;

[0028] Fig. 14 depicts an alternate example of an octet truss structure having significant scaling;

[0029] Fig. 15A, 15B and 15C present voltage responses of the ferroelectric octet truss structure of Fig. 14 in different orientations;

[0030] Fig. 16 present voltage responses of different examples of tetrakaidecahedron-based ferroelectric truss structures;

[0031] Fig. 17 presents an example touchless sensor harnessing the pyroelectric properties of ferroelectric lattice metamaterials;

[0032] Figs 18A, 18B and 18C present the voltage response of the touchless sensor of Fig.17 for constant temperature, repeated presses by a cold finger and repeated presses by a normal finger, respectively;

[0033] Fig. 19 schematizes finite element simulation voltage responses of piezoelectric cells under different shear and compressive loads;

[0034] Fig. 20 presents a first example of a sensor optimized for sliding mode;

[0035] Fig. 21 presents a second example of a sensor optimized for press mode;

[0036] Fig. 22 presents an example sensor system including a sliding mode sensor worn on a first finger, a press mode sensor worn on a second finger, and a one-way grid panel;

[0037] Fig. 23 presents an example sensor system including a sliding mode sensor and a press mode sensor worn on an index finger;

[0038] Fig. 24 presents an example sensor system including two sliding mode sensors having different orientations worn on a same finger of a user and engaging a two-way grid panel; and

[0039] Fig. 25 presents output signals of the example sensor system of Fig. 24 when operated in a way to write the digit “5”.DETAILED DESCRIPTION

[0040] Fig. 1A presents an example production process for an example embodiment of a piezoelectric cell having a stretching-dominant truss structure, namely, in this example, an octet truss structure. Digital Light Processing (DLP) can be employed for the 3D printing of a green body. In this example, the green body can consist of 20 wt. % of photosensitive polymer Polyethylene Glycol Diacrylate (PEGDA) 250, 0.4 wt. % of photoinitiator diphenyl (2, 4, 6-trimethylbenzoyl) phosphine oxide (TPO), and 79.6 wt. % of ferroelectric ceramic particles BaTiO3. To reduce the viscosity of slurry and enhance the fluidity of the printing resin, the ferroelectric ceramic particles undergo surface treatment. The printed green body can then be debinded at 550 °C in a vacuum oven (01200-50, Zhengzhou Zylab Instruments Co., Ltd.) to remove the polymer binder, followed by sintering at a temperature of 1340 °C in an airatmosphere (M1500-12IT, Zhengzhou Zylab Instruments Co., Ltd.) to densify the ferroelectric ceramic particles, resulting in a 22% volume shrinkage that leads to the reduction of the as-designed 6 mm size of the octet truss ferroelectric ceramic to an as-built 4.7 mm size.

[0041] In Fig. 1A, the stretching-dominant truss structure takes the form of a single unit cell, whereas in other embodiments, the unit cell can be repeated along 1 , two, or three axes, for example, in a metastructure.

[0042] Fig. 1 B presents an example process for polarizing a stretching-dominant truss structure in a manner to make it a piezoelectric cell. The electric polarization can activate the ferroelectric properties of sintered piezoceramics. The electric dipoles within the ceramic can initially be oriented in random directions, resulting in a lack of overall ferroelectricity. A high-voltage polarization can be applied, as illustrated in Fig. 1B, to align the dipoles along the electric field distributed within the underlying architecture of the ferroelectric material. This can lead to polarization of the piezoelectric material in a direction coinciding with the truss direction of the truss structure, and thus to the preservation of macroscopic ferroelectric properties even without the presence of an external electric field. When the electric field surpasses 1.5 kV / mm, the sintered bulk BaTiO3 ceramic becomes fully polarized, exhibiting a piezoelectric charge constant d33 of 270 pC / N, as measured by a d33 meter (PDK3-2000, PolyK, USA). The 3D printed ferroelectric samples can be polarized in an oil bath with a 3 kV / mm electric field for 30 minutes at room temperature, without encountering electric breakdown, in this specific example. This voltage level can be sufficiently high to fully activate the ferroelectric properties. The voltage can need to be maintained for a sufficient period, and the effect of time in the process is schematized in Fig. 1C, as an example.

[0043] The 3D fabrication approach can allow the creation of intricate ferroelectric architectures with precise structural features. As an example, a 3D printed and miniaturized intelligent Howrah bridge, constructed out of ferroelectric ceramics, demonstrates a voltage response to external impact loads due to its piezoelectric properties. Stretching-dominated octet truss and bending-dominated tetra ka id ecahedron were used as examples for primary investigation. Indeed, such structures can be repeated in a periodic metastructure, for instance. Figs 2A, 2B and 2C, for example, present three examples of metastructures using the octet truss as a unit cell, including a 2X2X2, a 4X4X2, and an 8X8X2, respectively. Inalternate embodiments, for comparison with conventional porous ferroelectric ceramics, foam structures that represent the porous ferroelectric ceramics produced by a BURPS method can be used. A ceramic fabrication platform can be used to 3D print ferroelectric metamaterials with alternative relative densities, numbers of unit cells, and topological structures. This fabrication technique can serve as a feasible tool for realizing high-performance ferroelectric metamaterials across various microarchitectures.

[0044] The longitudinal (c / 33) and transverse (-c / 31) piezoelectric coefficients can reduce as the porosity increases for porous ferroelectric ceramics. This decrease in coefficients may be due to a compromised polarization process efficiency by the increase of porosity, resulting in a diminished piezoelectric response, tied to the decline in remnant polarization. This poor polarization is intricately associated with the electric field distribution within the porous ferroelectrics with complex underlying architecture.

[0045] A rational design of the microarchitecture of ferroelectric metamaterials can unveil an unconventional behavior for c / 33. Figs 3A, 3B and 3C illustrate the c / 33 values of the octet truss, tetra ka id ecahedron, and five types of foams with varying relative density, respectively. Due to the structural disconnection at small pr, the relative densities for these 3D printed foams are kept above 0.3. Featuring well-developed mathematical foundations, the modified asymptotic homogenization considering the influence of polarization electric field may be implemented to determine the effective piezoelectric and pyroelectric properties for the octet truss and tetra ka id ecahedron with periodic boundary conditions, while a detailed finite element analysis may be conducted for multiphysical analysis of random ferroelectric foams. Considering the intricate polarization electric field, the local element coordinates undergo adjustments according to the electric field determined by polarization simulation. Remarkably, the c / 33 value of the octet truss displays an unprecedented rise as the relative density decreases, ascending from 285 ± 5 pC / N measured for an octet truss with pr= 0.5 to 347 ± 2 pC / N measured for an octet truss with at pr= 0.1. Conversely, the c / 33 values for the bending-dominated tetrakaidecahedron and bending-dominated foams exhibit a marginal decline when prdecreases.

[0046] In these examples, the piezoelectric properties of the ferroelectric lattice and foams are fully activated; beyond polarization electric fields (> 1.5 kV / mm), there may be nosubsequent enhancement in c / 33. The local polarization may predominantly align with the strut orientation in the lattice-based ferroelectric metamaterials. In addition, it is the stress transfer or loading state that governs the distinct behaviors of ferroelectric metamaterials depicted in Figs 3A to 3C. In the case of representative stretching-dominated cross truss, as presented in Fig. 4A, i.e., extracted from the octet truss, since the strut only experiences compressive or tensile deformation, the internal compressive force i is determined as follows upon an external compression F:F-| = F / sinO. (1)

[0047] However, for a bending-dominated cross truss extracted from a bending-dominated lattice, such as illustrated in Fig. 4B, the compressive force F1 is obtained by:Ft = FsinO. (2)

[0048] Since sinO < 1 , the internal force F1 within the metamaterial when subjected to a compressive force F is augmented for a stretching-dominated cross truss compared to a bending-dominated one, leading to an enhanced c / 33. The relationship between effective d33tretchin9and c / 3®ending, inclined angle 9, and d33 svalue of the constitutive ferroelectric material is given by:Stretching-dominated cross truss: c / 33tretchin9= c / 33 s / sin0; (3a)Bending-dominated cross truss: cf^endin9= d33 ssinO. (3b)

[0049] Verified by the asymptotic homogenization’s result, Eqs 3a and 3b demonstrate accuracy in predicting the effective c / 33 for both stretching and bending-dominated cross lattices as shown in Fig. 5. Since small 9 leads to an increased intersection area of the two lattice struts and thus perturbing the distribution of the force and polarization direction, the divergence between the homogenization (upper, dashed) and theoretical beam-model (upper, solid) results occurs when 9< 20°. Therefore, based on Eq. 3a, the stretching-dominated octet truss (O = 45°) displays a higher c / 33 value than the bulk material. Additionally, decreasing prtranslates to a reduced intersection region among beams. Consequently, this facilitates asuperior alignment of the force and polarization field with the beam direction, leading to an amplified c / 33. Although the beam incline angles of inside tetra ka id ecahedron and foams vary in space, their bending-dominated behaviors yield smaller c / 33 values compared to bulk materials, particularly evident at lower relative densities.

[0050] Based on Eq. 3, a design strategy for low relative density ferroelectric metamaterials can be aimed at achieving exceptionally high piezoelectric charge constants. Illustrated in Figs 6A, 6B and 6C, scaling the octet truss along the polarization direction with a scaling ratio of s allows adjusting the truss’s inclined angle, thereby enabling the attainment of the desired c / 33 value. As shown in Fig. 7, decreasing the relative density prand scaling ratio s corresponds to an augmentation in c / 33, observed in both homogenization and experiment results. For example, the c / 33 value of the octet truss with pr= 0.3 and s = 1 stands at 304 ± 3 pC / N, yet it is increased to 849 ± 11 pC / N for pr= 0.1 and s = 0.2.

[0051] In alternate embodiments, exemplified in Figs 8A and 8B, other stretching-dominated lattices can be used, e.g., BCC with Z = 8 and SC with Z = 6 that is stretching-dominated only along the three normal directions. Here, Z denotes the number of trusses connected to a single joint. Since its 0 is insensitive to the scaling ratio, the SC maintains an almost constant c / 33 with varying s and pr. Due to its high inclined angle (0 = 90°) of the four vertical beams, BCC shows a smaller c / 33 than that of the octet truss. At smaller s values, a minor decrease in the c / 33 value of SC ferroelectrics is observed, induced by a greater distribution of loads among the vertical beams. By contrast, bending-dominated lattices, e.g., tetrakaidecahedron (Z = 4), cross truss (Z = 4), and honeycomb (Z = 3), can exhibit distinctive trends where a decrease in both prand s results in a reduced c / 33 across these ferroelectric metastructures (not shown).

[0052] In some experiments, limitations of the resorted DLP 3D printer restricted the maximum achievable c / 33 value of the octet truss family to 849 ± 11 pC / N, observed for pr= 0.1 and s = 0.2. Nevertheless, the utilization of advanced 3D printing techniques, such as two-photon lithography (TPP), holds the promise of further amplifying this value by fabricating ferroelectric metamaterials with smaller relative densities and scaling ratios. The maximum measured c / 33 can surpass the corresponding values for BaTiCh ceramics achieved through material processing and sintering optimization, such as microwave sintering and grain and domain engineering. Combining the architected material design strategy with materialcomposition / manufacturing-enabled enhancement of piezoelectric properties can further escalate the ranges of attainable c / 33 in next generation of ferroelectric materials. For example, the c / 33 value of solid BaTiCh ceramics can reach a maximum of 788 pC / N using the templated grain growth method; by utilizing this material as feedstock for 3D printing of octet truss ferroelectric materials (with pr= 0.1 , s = 0.2), the c / 33 value can be further enhanced by approximately 3.3 times to reach ~2600 pC / N.

[0053] The piezoelectric voltage constant, denoted by 933, is linked to the electric field produced per unit of applied mechanical stress and is calculated as the ratio of the piezoelectric charge coefficient (c / 33) to the dielectric constant (K33). Unlike c / 33, K33 values for both the octet truss and tetra ka id ecahedron family demonstrate a declining trend by decreasing relative density (pr) and scaling ratio (s). Indeed, the decrease in prresults in fewer materials to generate electric charges under a given external electric field. In addition, the decrease of s value implies a reduction in the local electric field experienced by inclined beams, resulting in lower electrical charge generated along x^-direction per unit volume, i.e., K33. This decrease is proportional to the cosine value of the beam inclined angle. Hence, through the scaling operation, we can attain K33 values significantly smaller than ferroelectric foams with the same pr. For instance, the mean K33 values for 3D printed octet truss (s = 0.2) and tetrakaidecahedron (s = 0.2) with pr= 0.5 are 1.49 nC / m / V and 3.68 nC / m / V, respectively, whereas 6.21 nC / m / V for the foam and 20.36 nC / m / V for the solid BaTiCh piezoceramic, measured by an LCR meter (SR 715, Stanford Research Systems, USA) at 1 kHz.

[0054] The underlying topology, scaling ratio, and relative density play pivotal roles in determining the piezoelectric voltage constant. A notable outcome of the significant reduction in K33 is the amplification of 933 with the decrease of prand s. For instance, in ferroelectric foam structures, the mean 933 rises from 0.0412 Vm / N to 0.139 Vm / N as prdecreases from 0.5 to 0.3, whereas 933 is 0.013 Vm / N for solid ferroelectric materials. Similarly, for octet truss and tetrakaidecahedron structures with prof 0.3, the mean 933 value increases from 0.093 Vm / N and 0.070 Vm / N to 0.968 Vm / N and 0.352 Vm / N, respectively, as the scaling ratio diminishes from 1 to 0.2.

[0055] In bending-dominated tetrakaidecahedron structures, a decrease in prand s leads to a moderate decline in c / 33 whereas a more substantial reduction in K33 noticeably boosts 933.For instance, when prdecreases from 0.5 to 0.3, the average c / 33 of tetrakaidecahedron (s = 1) drops from 256 pC / N to 238 pC / N, accompanied by a decrease in K33 from 6.19 nC / m / V to 3.39 nC / m / V. Consequently, 933 increases from 0.041 Vm / N to 0.070 Vm / N. Notably, in conventionally manufactured porous ferroelectric ceramics, high porosity leads to a considerable decrease in c / 33, offsetting the benefits gained from the decreased K33. Nevertheless, c / 33 still maintains a reasonable value despite a small relative density in a 3D printed bending-dominated lattices and foams. Conversely, for stretching-dominated octet truss structures, c / 33 demonstrates an increasing trend with the decrease ofprand s, while K33 shows the opposite trend, allowing us to achieve ultrahigh 933 values in architected ferroelectric metamaterials. For instance, the mean 933 increases from 0.0319 Vm / N to 11.098 Vm / N for octet truss when prand s decreases from 0.5 and 2 to 0.1 and 0.2, respectively.

[0056] High values of c / 33 and 933 are essential for various piezoelectric applications as sensors, actuators and energy harvesters. However, achieving materials with both high c / 33 and 933 simultaneously has posed a longstanding challenge, as an increase in c / 33 is typically accompanied by a decrease in 933.

[0057] As depicted in Fig. 9, piezoceramics may exhibit a high c / 33 value (100 to 400 pC / N) but lack competitive 933 (0.01 to 0.13 Vm / N) due to their high dielectric constant. Introducing a second phase, such as polymer for composite ferroelectrics or voids for porous ferroelectrics, presents an effective strategy to enhance mechanical flexibility and as a result c / 33, albeit at the cost of reduced c / 33. For instance, c / 33 of ferrocomposite decreases from 438 pC / N to 219 pC / N accompanied by an increase in 933 from 0.065 Vm / N to 0.157 Vm / N. Similarly, ferroelectric polymers like polyvinylidene fluoride (PVDF) offer a balanced compromise between c / 33 (16 to 62 pC / N) and 933 (0.3 to 0.5 Vm / N). Tailoring the chemical constituents of molecular-based ferroelectric materials may allow to achieve relatively high values of both c / 33 and 933. For instance, a hybrid perovskite ferroelectric, [MesNC^CI]-CdBrC , can achieve c / 33 and 933 values of 440 pC / N and 6.215 Vm / N, respectively. By finely tuning the microarchitecture of ferroelectric metamaterials, our present work can simultaneously achieve ultrahigh c / 33 (849 pC / N) and 933 (11.098 Vm / N) even when utilizing relatively weak ferroelectric ceramics (i.e., BaTiCh). Moreover, the architected material design approach presented here for piezoelectric materials is independent of their constitutive solidbase materials and has no theoretical boundary for both c / 33 and 933 where a simultaneous reduction of prand s in stretching-dominated lattice piezoelectric materials may lead to infinitely high values.

[0058] Simulations can be performed to compare octet truss (pr= 0.2 and s = 0.2), tetra ka id ecahedron (pr= 0.2, s = 0.2), foam (pr=0.5), and solid piezoelectric materials. To evenly distribute the force, two thin plates can be incorporated at the top and bottom of otherwise cellular materials to experience the compression force. The octet truss can exhibit the highest normalized voltage of 1 , surpassing that of tetrakaidecahedron, foam, and fully solid ferroelectric materials by factors of 5, 77, and 200, respectively, owing to its ultrahigh 933 value.

[0059] As schematized in Fig. 10, each of these four structures can be 3D printed and then subjected to a drop-weight test using a steel ball dropped from a 5 cm height. Upon impact by a small ball (diameter of 1 mm), the voltage responses for foam and solid materials are undetectable while 5.9 V and 1.6 V responses are found for octet truss and tetrakaidecahedron, respectively. Subsequently, when impacted by a ball with a diameter of 3 mm, the octet truss yielded a average voltage response of 101 V, significantly greater than the other three designs. This observation aligns with the superiority of the octet truss results found by numerical simulation of the drop weight test results and by experimental measurement of 933 value (i.e. 2.753 Vm / N for octet truss). In addition to high sensitivity, ferroelectric metamaterials can also demonstrate a fast response time of around 0.2 ms.

[0060] To validate the energy harvesting performance of the rationally designed ferroelectric lattices, the voltage, current, and power density across varying load resistance for ferroelectric octet truss and foam can be measured and compared. For the measurement, the diameter of the dropped steel ball is first kept at 3 mm. With the increase of load resistance, the voltage increases while the current gradually decreases, and the power density peaks at 2 MO and 200 kQ, respectively, with corresponding values of 5.09 mW / cm3and 0.06 mW / cm3for octet truss and foam ferroelectrics. The advantage observed for generated power density persists across alternative drop weights. This is attributed to the elevated figure of merit for piezoelectric energy harvesting (FOM33), which evaluates electrical energy density undermechanical stress and is calculated as C / |3 / K^3. Specifically, the experimentally measured FOM33 values for octet truss and foam are 1762 x 1 o12m2 / N and 10 x 1 o12m2 / N, respectively.

[0061] As presented in Figs 11A, 11 B and 11C, to expand the design space for acquiring diverse transverse piezoelectric constants (c / 31 and c / 32), scaling operation can be performed along both xi-direction and / or X3-direction, with a scaling ratio referred to as si or S3, respectively. Alternative combinations of these two scaling ratios (ranging from 0.2 to 2) and relative density (ranging from 0.05 to 0.5) can be explored for ferroelectric octet truss and tetrakaidecahedron.

[0062] The (c / 31, c / 32) values for a total of 1453 distinctive topologies of ferroelectric lattices and foams are presented in Fig. 12. Additional octet trusses with prof 0.1 and 0.2, and (si, S3) of (0.2, 1), (0.4, 1), (0.6, 1), and (0.8, 1), can be fabricated. Interestingly, (c / 31 , c / 32) of the designed ferroelectric lattices exhibit a broad range of values across all four phases, expanding the range of transverse piezoelectric anisotropies. More specifically, most (c / 31, c / 32) of the tetrakaidecahedron family fall within the first phase while those of the octet truss family are primarily found in the third phase. For instance, without scaling operation (si= S3 = 1), the experimental c / 31 and c / 32 values for the tetrakaidecahedron structures range from 57 pC / N to 12 pC / N, while for the octet truss structures, vary from -5 pC / N to -22 pC / N as the relative density changes from 0.1 to 0.3. It is noted that by engineering the scaling operation, we can achieve ultrahigh values forthe magnitude of c / 31 and c / 32. For example, the mean experimental c / 31 value of octet truss with (pr, Si, S3) of (0.1, 0.2, 1) reaches -659 pC / N, 6.00 and 2.44 times higher than the magnitude of c / 31 (-110 pC / N) and c / 33 (270 pC / N) values of the solid ferroelectric materials, respectively. Similarly, for tetrakaidecahedron, we can also achieve an ultrahigh theoretical c / 31 value of 313 pC / N with geometrical parameters (pr, Si, S3) adjusted to (0.05, 0.2, 2). Opposed to lattice ferroelectrics, the (c / 31, c / 32) of ferroelectric foams is located in phase III (- , -), and decreases rapidly in terms of magnitude by decreasing relative density. Although diverse combinations of c / 31 and c / 32 can also be achieved in by spinodal ferroelectrics, their magnitude is limited to the c / 31 and c / 32 value of the constitutive solid ferroelectric material.

[0063] The transverse piezoelectric properties exhibited by ferroelectric lattices are related to their intrinsic micro architectures. To analyze the c / 31 and c / 32 of octet truss, compressive forcesalong xi-direction and X2-direction are applied as shown in Fig. 13A and 13B, respectively, where blue beams experience tension states while orange beams experience compression states, both contributing to charge generation for c / 31 and c / 32. Like the theoretical analysis of c / 33, the force acting on the inclined beams of a lattice dominated by stretching can be intensified, thereby amplifying the corresponding electric charge component along X3-direction, and thus resulting in exceptionally high c / 31 values for stretching-dominated octet truss ferroelectric structures with low relative density. The relationship between c / 31 and c / 32 with the two scaling ratios can be expressed as follows

[0064] Based on Eq. 4, the magnitude of c / 31 shows an increasing trend with decreasing si and increasing S3. Since c / 31 is inversely proportional to si, cfei / c / ss.s ratio can be increased by reducing si. Conversely, c / 32 features small values compared to c / 31.

[0065] For bending-dominated tetrakaidecahedron ferroelectrics with low relative density with scaling ratios of unity, the c / 31 and c / 32 can be given as:d3 / = ^ ( / = 1, 2) (5)

[0066] This relationship elucidates the positive values of c / 31 and c / 32 observed in tetrakaidecahedron structures with low relative density. The remarkably high value of the three piezoelectric constants and small dielectric constant of the ferroelectric lattices contribute significantly to enhancing the performance of the hydrophone devices.

[0067] The voltage responses of ferroelectric lattices and foam with three representative transverse piezoelectric anisotropies under uniaxial compressive impacts along three directions can be analyzed. Figs 15A, 15B and 15C present results obtained with an octet truss structure shown in Fig. 14. Specifically, the geometrical parameters (pr, Si, S3) for the octet truss and tetrakaidecahedron ferroelectric structures are (0.2, 0.2, 1) and (0.2, 1 , 1),respectively, while the relative density of the foam is 0.6. Due to the notably high absolute value of c / 31 (-335 pC / N), the voltage response of the octet truss loaded in xi-direction surpasses that loaded in X3-direction 3. Introducing porosity results in a smaller c / 31 and c / 32 for ferroelectric ceramics, thereby causing considerably smaller voltage responses along xi-direction and X2-direction loading compared to the X3-direction. Conversely, since all three piezoelectric constants share the same sign, the voltage response of the tetrakaidecahedron structure exhibits the same sign when compressed along all three orientations.

[0068] The diverse transverse piezoelectric anisotropies enable the design of integrated force directionality sensors. More specifically, a one-directional graded tetrakaidecahedron, comprised of two relative densities of 0.2 and 0.4 without scaling operation, demonstrates a programmable local voltage response. For pr= 0.2, c / 31 and c / 32 in tetrakaidecahedron are positive whereas they are negative for pr= 0.4. Consequently, under compression along xi-direction, the voltage signs of two local electrodes, i.e., I / 1 and 1 / 2 differ, while due to the same sign of c / 33 for both cells, the signs of I / 1 and 1 / 2 under X3-direction loading are the same. This functionality is verified through both simulation and experimentation. To further distinguish the force direction among X2-direction and X3-direction, a tetrakaidecahedron with (pr, si, S3) of (0.25, 0.2, 1) can be introduced as shown in in Fig. 16, in a two-directional graded design, whose c / 31 is positive while c / 32 is negative. By analyzing the voltage signs of these three local electrodes, we can determine the force orientation applied in this integrated sensor.

[0069] Apart from rf3i, c / 32, and c / 33, c / 42 and c / 5iare also considerable nonzero piezoelectric constants for shear-based piezoelectric devices, linking the shear stresses <74 and 05 to the electric charges £>2 and £>i, respectively. Taking into consideration large values of c / 3 / ( / = 1 , 2, 3), the influence of scaling operations (only along X3— d i rect io n) and relative density on c / 42 (equivalent to c / 51) for both ferroelectric octet truss and tetrakaidecahedron structures can be explored. Like the behavior of c / 33 in the octet truss, decreasing the scaling ratio increases c / 42. Furthermore, for small values of s, reducing the relative density can further enhance c / 42. For instance, the measured c / 42 of the octet truss (pr= 0.5) increases from 371 pC / N to 499 pC / N as the scaling ratio (s) decreases from 2 to 0.2. This value is further enhanced to 836 pC / N in the octet truss with pr= 0.1 and s = 0.2. Conversely, although the bending-dominated tetrakaidecahedron structure exhibits a weakened c / 33, which decreases with both reduced prand s, the d2value demonstrates a different trend concerning these two design factors. Notably, for small relative density and scaling ratio, CZ42 of ferroelectric tetrakaidecahedron structure even displays higher values compared to the solid structure. For example, the experimental values of CZ42 are 572 pC / N and 621 pC / N for the tetrakaidecahedron structure with (pr, s) of (0.2, 0.2) and (0.2, 0.1), respectively, which are 36% and 48% higher than those of solid BaTiOs ceramic (0(42,5 = 420 pC / N).

[0070] The behavior can be explained by the load state. For simplification, consider the cross truss: one end of the two beams is fixed while the other end is subjected to a downward force, F. If these two beams are under stretching-dominated conditions, the force component along the beam direction, F1, is:Ft = F / sinO (6)and thus the effective d42for this stretching-dominated cross truss is:d42= d33iS / sin0 (7)

[0071] Similarly, for a bending-dominated cross truss, the effective d42isd42= d33,ssin0 (8)

[0072] Comparing Eqs 7 and 8 with Eq. 3, both the effective d2and CZ33 for bending- and stretching-dominated trusses share the same value at small relative densities, as predicted by the beam model. Decreasing the scaling ratio results in a reduced inclined angle Q, which in turn increases d2for stretching-dominated cross trusses (upper boundary) and decreases d2for bending-dominated cross trusses (bottom boundary). For the octet truss with prof 0.02, the homogenization result approaches the upper boundary. For a bending-dominated tetrakaidecahedron structure (pr= 0.02), the homogenization result is also closer to the upper boundary. This can be attributed to the fact that, under the shear mode, a portion of the tetrakaidecahedron structure is subjected to stretching-dominated conditions, thereby exhibiting a combination of stretching and bending-dominated behaviors.

[0073] Decreasing pralso leads a reduction in the dielectric constants «n and K22. Therefore, the enhanced CZ42 and rf5i, along with the decreased «n and K22, contribute to the improved 942 (= CZ42 / K22) and 051 (= cW n), indicating increased sensitivity to a shear force. In a setup for the shear-mode based piezoelectric test, one side of the lattices along the polarization direction is fixed to a 3D printed polymeric block by glue, while the other side is attached to another block to bear the impact load imposed by a steel ball dropped from a height of 5 cm. Four ferroelectric structures are considered: tetrakaidecahedron and octet trusses (pr= 0.3, s = 0.5), random foam (pr= 0.6), and a solid block with overall dimensions of 6.3 x 6.3 x 6.3 mm. The voltage responses of both architected ferroelectric lattices are approximately 2.4 and 4.4 times higher the corresponding values for the ferroelectric foam and solid structures, respectively, under two different drop heights. Numerical simulation also confirms the superiority of the designed ferroelectric lattice for shear-mode piezoelectricity.

[0074] Besides the piezoelectric effect (electromechanical coupling), polarized ferroelectric materials can also exhibit the pyroelectric effect (thermoelectric coupling). Indeed, the spontaneous polarization or orientation of electric dipoles can change with temperature fluctuations, releasing charges on the surface of the ferroelectric material. The pyroelectric constant, P3, defined as the generated electric charge per unit area per unit temperature change, assesses the pyroelectricity of the ferroelectric material. The P3 values of ferroelectric octet truss and tetrakaidecahedron structures with alternative relative densities and scaling ratios vary, where a decrease in prand s results in a weakened pyroelectric constant. The pyroelectric constant can also be experimentally measured by a PolyK pyroelectric test system, with a heating rate of 2°C / min from 30°C to 50°C. Lower prcan mean less material is available to generate charges under given temperature fluctuations, and a decrease in s results in fewer charge components along the polarization direction. It is noted that for the scaling operation of unity, octet truss, tetrakaidecahedron structures, and random foam exhibit close values, approximately around 50 pC / m2 / K when pr= 0.3.

[0075] To evaluate the voltage sensitivity of ferroelectric materials under a given power input, the pyroelectric voltage figure of merit, Fv, is defined as Fv= P (CEK33) where CE represents the volume-specific heat capacity. Independent of scaling ratio and structural topology, Fvis mainly determined by the relative density of ferroelectric lattices. For instance, with pr= 0.3,Fvvalues for the ferroelectric octet truss are 17.9 x w3and 18.1 x w3m2 / C for s = 1 and s = 2, respectively. In contrast, when s = 1 , this value increases significantly from 10.7 x w3to 56.7 x 10'3m2 / C as prdecreases from 0.5 to 0.1. This is because although ps decreases with the reduced prand s, K33 exhibits a similar pattern. Furthermore, since CE is proportional to pr, decreasing relative density effectively contributes to a substantial increase in Fv.

[0076] Utilizing the pyroelectric effect, a contactless sensor can be embodied, an example of which is presented in Fig. 17. The contactless sensor has a frame is higher than a ferroelectric octet truss structure (pr= 0.4, s = 1 , and overall size of 9.4 x 9.4 x 2.3 mm) to prevent direct contact between the finger and the ferroelectric sensor. Simulation results are presented in Figs. 18A, 18B and 18C showing voltage responses at constant temperature, at repeated cooling, and repeated warming, respectively. More specifically, Figs. 18B and 18C display the response of the ferroelectric sensor at room temperature (24 °C) when repeatedly approached by a cooled finger versus a normal finger. The opposite voltage signals under cooling and heating conditions confirm that the generated voltage is due to the variation of temperature, i.e., pyroelectric effect. A 0.1 -second gap before activation was achieved and can be considered a suitably rapid response during operation in many applications.

[0077] The distinctive responses of ferroelectric materials to external mechanical loads can allow their application in input devices such as wearable input devices. As illustrated in Fig. 19, shear-mode piezoelectricity results in opposing positive and negative voltage signals, i.e., 1 / 1, between two surfaces when shear forces act along X2-direction in the positive and negative directions. In contrast, compression along X3-direction yields a positive voltage (I / 2) between surfaces in X3-direction. Capitalizing these features, a sensor with slide mode and / or press mode functionalities can be implemented, as depicted in Figs 20 and 21 , respectively. Fig. 22 presents an example system which exploits both such sensors.

[0078] Referring to Fig. 19 in greater detail, the upper and middle images show a same sensor having electrodes engaged with two opposite faces of stretching-dominant truss structure rectangular prism shape, along axis X2. An electrical meter, not shown, such as a voltmeter here, is connected in an electrical circuit to the two electrodes, in a manner to sense a voltage difference between the two electrodes. This configuration is well adapted to sense shear in the structure as illustrated. In alternate embodiments, alternate features of the electrical circuitcan be sensed by the electrical meter, such as amperage. In the lower image, a different sensor is shown, also having electrodes engaged with two opposite faces of the stretchingdominant truss structure rectangular shape, but here along the X3 axis. This configuration is well adapted to sense compression in the structure as illustrated. In other embodiments, there can be electrodes disposed against opposite faces in more than one orientation, and corresponding meters, so as to provide information about deformation in different axes, for instance.

[0079] In one example, a 2 x 2 x 2 octet truss with size of 4.8 x 4.8 x 4.8 mm and relative density of 0.4 is selected to fabricate the sensor core. A sensor specifically adapted for slide mode operation, such as shown in Fig. 20 for example, can include a pointed tip to amplify the effect, for example, whereas in a sensor specialized for compressive mode operation may omit such feature. In one example implementation, illustrated in Fig. 22, the sensors of Fig. 20 and of Fig. 21 can be implemented as rings to be worn on one or more fingers, and can be designed to collaborate with a special mat / grid panel. Other implementations are possible.

[0080] In slide mode, as the sensor moves along a grid panel, the applied shear force due to the contact between the soft probe tip and beams of the grid generates a voltage whose polarity depends on the direction of the movement. In the press mode, compression of the sensor produces a voltage pulse. Such sensors can be integrated to a game controller, for instance, or to a pointing device such as a computer mouse. Alternately, such a sensor may be used for simulating keyboard functions, thereby enabling interaction with webpages.

[0081] Another embodiment is presented in Fig. 23 where a first sensor specialized for slide detection and a second sensor specialized for compression detection are mounted to a same finger. Different sliding directions of the finger over the slide mode sensor generate opposing voltages, corresponding to the "Up" and "Down" arrow keys. A single compression of the press mode sensor signifies the "Tab" function. A double activation of these piezoelectric sensors with slide and compression modes can be associated to other functions, such as the "Space" and "Enter" commands, respectively. Thus, by utilizing the thumb and the wearable sensors, users can efficiently perform inputs, searches, and web browsing.

[0082] In embodiments where a single slide mode sensor can detect two movement states (forward and backward) along one direction, employing two sensors of this type allows for the detection of four distinctive movement states across two orthogonal directions. An example of such a configuration, embodied here in combination with a two-way grid panel, is presented in Fig. 24. By analyzing the output voltages corresponding to finger movements over a two-way grid panel, handwritten letters can be recognized. Figure 25 illustrates the representative piezoelectric voltages for digit “5”, recorded from two channels representing the horizontal and vertical directions, respectively. Correspondence between finger movement and voltage generation can be found, which is essential for effective handwriting recognition. For example, the initial and final strokes correspond to downward and rightward movements, resulting in negative voltages for the vertical direction sensor and positive values for the horizontal direction sensor, respectively. To achieve real-time recognition of handwriting digits, each of them can be repeated 50 times and used to train a bidirectional long short-term memory (LSTM) network, which leverages information from both past and future contexts, enhancing the understanding and capturing of long-range dependencies. Experiments made with such two slide mode sensors demonstrated 99 % accuracy on the test dataset containing 100 cases.

[0083] To detect more movement directions, further slide mode ferroelectric metamaterials can be arranged in a circular pattern with a smaller angular separation (e.g., 12 sensors with 30° angular separation), to achieve a concentric ring panel designed to apply shear force through contact with the probe tip during finger movement. By analyzing the magnitudes of the output voltages from these twelve sensors, the movement direction can be determined. For instance, when moving along 0 = 90°, sensors l / i and 1 / 7 display voltages close to 0, while sensors 16 and l / io exhibit maximum and minimum values, respectively, indicated by the orange line. Increasing the number of sensors can further enhance the direction sensitivity of the wearable device, where N movement states can be determined with N slide mode sensors, desired for applications such as advanced gesture recognition, nuanced control in virtual and augmented reality, and precise input for robotic systems.

[0084] Accordingly, truss-based ferroelectric metamaterials with suitable piezoelectricity, diverse anisotropic piezoelectric properties, and suitable ferroelectric figures of merit can bedeveloped based on the selection of underlying microarchitecture. While introducing porosity typically reduces piezoelectricity in porous ferroelectrics, stretching-dominated truss-based lattices achieve remarkable improvements. At a relative density of 0.1 , the c / 31 (or c / 32), c / 33, c / 42 (or c / 51) values can reach 849, 659, and 836 pC / N, respectively — 3.14, 6, and 2 times higher than a solid counterpart. The bending-dominated ferroelectric tetrakaidecahedron demonstrates promising piezoelectric anisotropy with a positive combination of (c / 31 , c / 32, c / 33) values, resulting from distinctive local polarization directions and force distribution patterns. Additionally, the dielectric constants can decrease with reduced scaling ratio and relative density, leading to significantly enhanced ferroelectric figures of merit. For example, from octet truss family, the piezoelectric voltage constant, piezoelectric energy harvesting figure of merit, and pyroelectric voltage sensitivity can reach 11.098 Vm / N, 9422x10-12m2 / N, and 0.056.7 m2 / C, respectively. Accordingly, parameterizable ferroelectric lattice metamaterials can offer versatile applications, spanning energy-absorbing structures, ultrasensitive force / thermal sensors, wearable self-powered input devices, and precisely controlled microelectromechanical systems, to name some examples.

[0085] As can be understood, the examples described above and illustrated are intended to be exemplary only. The scope is indicated by the appended claims.

Claims

WHAT IS CLAIMED IS:1 . A piezoelectric cell having a stretching-dominant truss structure, made of a piezoelectric material and having a polarization direction coinciding with a truss direction of the truss structure.

2. The piezoelectric cell of claim 1 wherein the stretching-dominant truss structure has b struts, and j frictionless joints, where M = b - 3j + 6 and M > 0.5.

3. The piezoelectric cell of claim 1 or 2 wherein the stretching-dominant truss structure is an octet truss structure.

4. The piezoelectric cell of any one of claims 1 to 3 wherein the stretching-dominant truss structure is scaled by a scaling ratio s along 1 or 2 axes relative to a cubic virtual reference, the scaling ratio s being below 0.8.

5. The piezoelectric cell any one of claims 1 to 4 wherein the relative density of the stretching-dominant truss structure is below 0.4.

6. The piezoelectric cell of any one of claims 1 to 5 wherein the piezoelectric material is a ceramic.

7. The piezoelectric cell of any one of claims 1 to 4 periodically repeated a plurality of times along at least one orthogonal axes of a rectangular parallelepiped metastructure.

8. The piezoelectric cell of any one of claims 1 to 7 wherein the stretching-dominant truss structure has a rectangular prism shape, with three pairs of opposing faces, and at least one pair of electrodes engaged with respective faces of one of the three pairs of opposing faces.

9. The piezoelectric cell of claim 8 wherein an electrical meter is connected to one of the at least one pair of electrodes.

10. The piezoelectric cell of claim 8 or 9 wherein the truss structure is supported by a ring wearable around a finger of a user.

11. The piezoelectric cell of claim 10 further comprising a conical structure narrowing in a direction extending away from the truss structure and from the finger.

12. The piezoelectric cell of any one of claims 8 to 11 provided in combination with a mat having a periodically repeating structure.

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