Tunable flexible capacitive sensor for dynamic pressure monitoring

Abstract

Flexible capacitive pressure sensors have gained widespread application in health monitoring, robotics, and structural diagnostics. However, conventional designs that rely on flat or micropatterned dielectric layers typically offer high sensitivity only at low pressures and possess limited tunability, which makes them unsuitable for dynamic or harsh environments. In this study, we present a tunable capacitive pressure sensor fabricated via buckling-guided assembly and laser cutting, which transforms 2D precursors into 3D cage-like architectures. The sensor exhibits pressure-dependent sensitivity, characterized by low sensitivity under small loads and significantly enhanced sensitivity at higher loads due to nonlinear variations in electrode spacing. It achieves outstanding performance, including high durability over 6000 cycles, a low detection limit (~2 Pa), minimal hysteresis (~4%), and rapid response and recovery times (131/140 ms). Finite element analysis and experimental validation confirm the tunable mechanical response and accurate electromechanical behavior enabled by geometric design. The sensor also allows reversible tuning through lateral strain and liquid encapsulation, enhancing environmental robustness. Moreover, a compression-induced rotation mechanism further improves sensitivity by increasing electrode overlap during loading. Wind tunnel experiments validate the sensor’s performance under extreme conditions, demonstrating strong potential for practical applications.

Introduction

With the rapid advancement of intelligent connectivity and sensing technologies, flexible sensors have found extensive applications across various aspects of daily life and engineering fields, primarily due to their miniaturized design and adaptability. Among the various types of sensors, pressure sensors are in high demand in diverse domains such as health monitoring, integrated electronic devices, intelligent robotics, and smart Internet of Things systems^1–5. Recent progress in manufacturing and assembly techniques has facilitated pressure sensors to be used in non-invasive heart rate and pulse monitoring, skin health surveillance, implantable devices, electronic skin, and haptic feedback^6–12. Based on their operating principles, current flexible pressure sensors can be categorized into four categories: capacitive, resistive, piezoelectric, and triboelectric^13–17. Among them, capacitive pressure sensors have attracted significant attention in both research and practical applications owing to their high sensitivity, low power consumption, minimal hysteresis, and excellent long-term stability, as systematically summarized in recent review studies^18–20. Their typical structure comprises a fixed electrode plate, a movable electrode plate, and an intermediate dielectric layer. The electrical performance of such sensors is primarily determined by the effective facing area of the electrodes, the gap between the electrodes, and the relative dielectric constant of the dielectric layer^18,21,22. Current research predominantly focuses on optimizing the measurement range and sensitivity of capacitive sensors by designing specific microstructures (e.g., pyramidal, cylindrical, or porous) and tailoring the dielectric material (e.g., PDMS, TPU, MXene)^{10,11,23–32}.

In capacitive pressure sensors featuring microstructured dielectric layers, mechanical deformation typically induces simultaneous changes in both the dielectric constant and the contact area between the dielectric and the electrodes, thereby improving overall sensing performance. While this coupling effect enhances sensitivity, it also introduces pronounced nonlinearity in the variations of contact area and dielectric constant, complicating accurate theoretical prediction of capacitance changes. In contrast, most flexible capacitive pressure sensors with thin-film-like structures based on novel dielectric materials exhibit uniform geometric and material properties along the thickness direction. Consequently, these sensors demonstrate high sensitivity under low-pressure conditions but experience a marked decline in sensitivity at higher pressures, limiting their applicability in scenarios requiring tunable sensitivity across a broad pressure range. The fixed sensitivity profile also necessitates precise estimation of the expected loading conditions during the design phase, reducing the suitability of these sensors for applications characterized by long-term variability in pressure yet relatively stable short-term loads. Representative examples include structural health monitoring in civil engineering, wind pressure sensing in wind power systems, and moving load detection in road traffic^33–38. In contrast, pressure sensors exhibiting enhanced sensitivity in the high-pressure regime are particularly well suited for applications involving sustained or elevated loads, such as plantar pressure and gait analysis^39–41, grasping force sensing in robotics and prosthetics^42–45, and joint load monitoring in wearable systems^46–49, as well as pressure monitoring in subsea pipelines operating under high hydrostatic pressure conditions⁵⁰.

The development of tunable capacitive sensors has been explored through various design strategies, primarily focusing on material selection, structural design, and microstructural engineering. Flexible, highly deformable materials—such as liquid metals and soft elastomers—have been employed to create reconfigurable electrodes and dielectric layers. Advanced fabrication techniques, including three-dimensional (3D) printing, enable the construction of inflatable or architected dielectric structures with adaptable geometries. Electrically tunable mechanisms, incorporating dielectric elastomers or integrated MEMS components, offer active control over capacitance values. Additionally, microstructural engineering—via patterning, porosity modulation, or multilayer configurations—enhances sensitivity and functional diversity by tuning local electric fields and mechanical responses. Despite these advances, several persistent challenges complicate the practical deployment of tunable capacitive sensors. The reliance on soft or reconfigurable components often introduces susceptibility to environmental conditions, such as fluctuations in temperature, pressure, or humidity, which may lead to signal drift or long-term instability. Electrically driven systems, while offering dynamic tunability, typically require complex circuit integration and precise control schemes. Furthermore, microstructure-based approaches frequently involve intricate manufacturing processes—such as high-resolution patterning or multilayer assembly—that can limit throughput and hinder scalability. These interrelated factors collectively limit the robustness, reliability, and scalability of tunable capacitive sensing platforms in practical applications^51–58.

To address these challenges, this study presents a flexible capacitive pressure sensor fabricated through buckling-guided assembly and laser cutting technologies. In this approach, a two-dimensional (2D) precursor is compressed into a cage-like 3D configuration. The sensor exhibits a distinctive pressure-dependent sensitivity profile: it demonstrates relatively low sensitivity under low-pressure conditions and significantly enhanced sensitivity at higher pressures. This behavior is primarily attributed to the structural design, wherein the electrode gap changes gradually under small loads and steeply under larger ones. This distinct mechanism and the associated post-fabrication tunability differentiate our work from recent state-of-the-art platforms, such as the thermoforming-based sensors reported by Choi et al.⁵⁹. While their approach achieves high linearity through design-phase customization, our buckling-guided strategy offers in-situ range reconfigurability and enhanced sensitivity at high-load thresholds, thereby providing superior adaptability to dynamic and uncertain environments. Results from 6000 loading/unloading cycles confirm the sensor’s excellent long-term stability and reliability. Quantitative analysis further reveals minimal hysteresis (~4%), rapid response (131 ms), fast recovery (140 ms), and a low detection limit of approximately 2 Pa. By integrating experimental results with finite element analysis (FEA), this work demonstrates that the mechanical response of cage-like 3D structures can be flexibly and widely tuned through the design of 2D precursors, allowing customization for diverse pressure-sensing applications. Optimization of the local electrode architecture ensures parallelism between the upper and lower plates during deformation, allowing accurate prediction of the pressure–electrical response. Furthermore, the buckling-guided design supports reversible tuning of the electrode gap via lateral strain, facilitating real-time adjustment of sensitivity and detection range. The structure is also compatible with liquid encapsulation; embedding the 3D sensor in a glycerol-filled chamber preserves its flexibility and stretchability while enhancing mechanical protection, thus improving durability in harsh environments. Additionally, a compression-rotation-based enhancement mechanism is introduced, in which out-of-plane compression induces rotation of the top electrode, and strategically designed electrodes further modulate the effective facing area during compression, significantly boosting sensitivity. Wind tunnel experiments validate the sensor’s high accuracy, robustness, and applicability in demanding engineering scenarios, underscoring its potential for practical pressure-monitoring applications.

Results and discussions

Design, fabrication, and mechanical response

Figure 1a illustrates the fabrication process of the proposed 3D capacitive pressure sensor. As shown in the inset at the upper left of Fig. 1a, the capacitor consists of an upper cage-like structure and a lower circular electrode. The upper structure includes a flat platform composed of 25 μm polyimide (PI)/80 nm copper (Cu)/25 μm PI layers, and six curved supporting microstrips made of 80 nm Cu/25 μm PI. The bottom electrode is composed of 100 nm Cu and 50 μm PI. The entire structure is supported by an elastic substrate made of 1 mm-thick Dragon Skin (10 slow, Smooth-on, USA).

Fig. 1.

Design, fabrication, and mechanical response of the 3D capacitive pressure sensor. a Schematic of the buckling-guided assembly process. The main panel illustrates the design and sequential assembly steps, while the top right inset shows an optical image of the fabricated device, featuring a cage-like upper electrode and a circular lower electrode on an elastic substrate. Upon release of a 42% biaxial prestrain, the 2D precursor transforms into a 3D configuration. b FEA results (top and middle rows) alongside corresponding experimental images (bottom row) depicting the device deformation under increasing compressive strain. Scale bars, 1 mm

The 3D configuration is achieved via a buckling-guided assembly technique. Specifically, PI/Cu/PI multilayer 2D precursor structures are fabricated and patterned on a glass substrate using micro-nanofabrication and laser cutting techniques, and subsequently transferred onto a pre-stretched cruciform silicone elastomer substrate with a biaxial prestrain (ε_pre = 42%) (Fig. 1a, bottom left). By bonding the precursor at predefined locations using commercial adhesive (Super Glue, Gorilla, USA) and subsequently releasing the prestrain, the structure undergoes controlled lateral buckling, transforming into the final 3D geometry (Fig. 1a, center). The resulting stator (lower electrode) and rotor (upper electrode) together form a parallel-plate capacitor.

Out-of-plane compression leads to a change in the electrode gap, which in turn results in a variation of the device capacitance. Furthermore, the initial electrode gap can be tuned by applying a stretching strain to the elastic substrate using a custom-built stretching platform, thereby enabling the adjustment of the device’s sensing range and sensitivity. The stretching distance in this process is measured by a micrometer screw gauge with a precision of 0.01 mm.

Figure 1b depicts the deformation behavior of the 3D cage-like structure subjected to out-of-plane compression by a flat indenter (3 mm diameter) mounted on a solid analyzer (RSA-G2, TA Instruments, USA). The axial compressive strain is defined as

$${\epsilon }_z=h/h_0$$ 1

where h₀ and h denote the initial and compressed vertical heights of the upper structure, respectively. As the compressive strain increases linearly from 0 to 80%, the internal compressive stress rises monotonically, reaching 0.97 kPa at 80% strain. Accompanying this mechanical response, the capacitance undergoes a significant rise—from an initial value of 113.8 to 558.9 fF. Throughout the compression process, the deformation configurations predicted by FEA show excellent agreement with the experimental observations.

Geometric design and mechanical analysis

This section investigates the relationship between the mechanical response and geometric design of the cage-like structure. Nonlinear FEA was employed to inform key aspects of the device’s structural design, including the geometry of the 2D precursor, optimization of the PI thickness, and patterning strategies for the metal layer. The 2D precursor of the cage-like structure is depicted in Fig. 2a, with critical geometric parameters defined as follows: copper thickness t_Cu; polyimide thickness of the supporting base and strips t_PI; top protective polyimide layer thickness t_PI-Top; strip width W and central angle θ; radius of the central disk r; and distance from bonding sites to the center R.

Fig. 2.

Geometric design and mechanical analysis of the 3D cage-like structure. a Schematic of the 2D precursor design with labeled geometric parameters. b FEA of the strain distribution in the metal layer after assembly and under 80% compressive strain. Comparison of experimental and FEA results for pressure–displacement (c) and rotation angle–displacement (d) responses. e Normalized height (H/T) of the 3D structure as a function of the strip width-to-thickness ratio (W/T) and central angle θ. f Contour plot of equivalent stiffness (P/U) as a function of thickness (T) and central angle (θ) at a fixed width of W = 600 μm. g Contour plot of equivalent stiffness (P/U) as a function of strip width (W) and central angle (θ) at a fixed thickness of T = 27 μm. h Contour plot of equivalent stiffness (P/U) as a function of strip width (W) and thickness (T) for a fixed central angle of θ = 150°

The fundamental design parameters of the cage-like structure are summarized in Table 1.

Table 1.

Geometrical design parameters of the cage-like structure

R (mm)	r (mm)	W (µm)	θ (°)	tPI-Top (µm)	tCu (nm)	tPI (µm)
7.2	4.0	600	150	25	80	25

The reversibility of the mechanical response of the 3D cage-like structure is a crucial indicator of the device’s measurement repeatability. Figure 2b illustrates the strain distribution in the metal layer of the 3D cage-like structure both after buckling-guided assembly and under 80% out-of-plane compression. It can be observed that, due to the design of the thickened platform, the stiffness of the supporting strips is significantly lower than that of the platform, which helps maintain low strain levels in the central electrode plate. Consequently, deformation is primarily concentrated on the supporting strips, with the maximum principal strain remaining below the threshold for plastic deformation.

Furthermore, owing to the coupling of bending and torsional deformation, the strain along the width direction of the supporting strips exhibits an edge-dominated distribution—higher at both edges and lower in the center (Fig. S1 in the Supplementary Information). This observation suggests that placing copper interconnects near the center of the curved strips is an effective strategy to minimize strain on these conductive pathways.

Experimental and simulation observations of the compression process reveal that the upper electrode undergoes both vertical displacement and rotation. The rotation angle θ_R is defined as positive for clockwise rotation. Figure 2c, d presents the relationships between applied pressure (P), rotation angle (θ_R), and out-of-plane compression displacement (U), derived from both experimental data and FEA. The results show a near-linear mechanical response and a linear relationship between the compressive strain and rotation angle, with excellent agreement between experimental and simulated results. As shown in Fig. S2, minor, low-amplitude fluctuations observed in the force–displacement curve indicate the presence of a stick-slip phenomenon at the interface between the indenter and the upper electrode during compression. Importantly, the amplitude of these fluctuations relative to the full-scale output signal is less than 0.5%, confirming that while interfacial friction is detectable, it has a quantitatively minor effect and does not significantly influence the coupled out-of-plane compression and accompanying rotation of the structure.

The post-buckling height of the cage-like structure—which determines the electrode gap and thus the device’ s operational range—is closely related to the geometric parameters of the 2D precursor. As shown in Fig. 2e, the structural height increases with the width-to-thickness ratio of the supporting strips and exhibits a non-monotonic relationship with the central angle θ, initially decreasing and then increasing. The influence of the width-to-thickness ratio is more pronounced than that of θ, indicating that increasing strip width is an effective strategy for expanding the sensing range.

Additionally, the device’ s sensitivity is primarily determined by the equivalent stiffness (P/U) of the structure. Figure 2f–h illustrates the influence of the geometric parameters of the strips on the effective stiffness of the structure. The results show that the effective stiffness of the cage-like structure increases significantly with decreasing central angle θ, and with increasing strip width W and thickness T (T = t_PI). These findings provide important, valuable design guidelines for tailoring the mechanical response of buckling-guided 3D cage-like structures to meet application-specific requirements.

Electrical characterization of the 3D capacitive pressure sensor

Based on the mechanical analysis, the intrinsic relationship between design parameters and the mechanical response of the cage-like structure has been elucidated. This section focuses on the electrical performance of the structure in relation to its deformation mechanism. For capacitive pressure sensors, the capacitance C is determined by the effective facing area between the electrodes A, the relative permittivity ε_r of the dielectric material, and the distance d between the top and bottom electrodes. The capacitance of a parallel-plate capacitor is expressed as:

$$C=\frac{{\epsilon }_0{\epsilon }_{r}A}{d}$$ 2

where ε₀ = 8.854187817 × 10⁻¹²F m⁻¹ is the vacuum permittivity. In this analysis, air is regarded as the dielectric with a relative permittivity ε_r = 1.0. Given that the deformation of the top and bottom electrodes is minimal, and the top electrode is intentionally designed to be slightly smaller than the bottom electrode so that the effective face-to-face overlap area remains constant, the parallel-plate capacitor approximation provides an adequate description of the sensor’s electrical behavior, as supported by the close agreement between experimental and simulated capacitance–pressure results. When the electrode overlap region undergoes more complex evolution, additional electrostatic effects may need to be considered.

To evaluate and optimize the sensor’s electrical performance, a high-precision experimental system was established, incorporating a precision capacitance meter (E4981A, Keysight, USA), a solid analyzer, and a computer for data acquisition and analysis. Figure 3a presents the experimental results of capacitance variation under applied pressure alongside FEA simulation predictions. The two datasets exhibit strong agreement, confirming that capacitance behavior derived from mechanical simulation accurately reflects the sensor’s electrical response (see SI Note S1 for calculation details), thus offering a reliable theoretical foundation for design and optimization.

Fig. 3.

Electrical characterization of the 3D capacitive pressure sensor. a Capacitance–pressure curves obtained from experimental measurements and FEA simulations. b Sensitivity curves derived from both experimental data and simulations. Contour plots showing the minimum pressure required to achieve a target sensitivity as functions of c thickness (T) and central angle (θ) at W = 600 μm, d strip width (W) and central angle (θ) at T = 27 μm, and e strip width (W) and thickness (T) at θ = 150°. f Relative capacitance changes under stepwise loading at pressures of 0, 0.1, 0.2, 0.3, and 0.5 kPa, with a 10 min holding period at 0.5 kPa. g Capacitance change–time curves during repeated loading–holding–unloading cycles at 0.2, 0.3, and 0.7 kPa. The enlarged views highlight the loading and unloading responses at 0.3 kPa. h Long-term durability performance showing relative capacitance change over 6000 loading/unloading cycles at 0.5 kPa

Sensitivity S, is a crucial metric for evaluating sensor performance, defined as:

$$S=\frac{\delta (\mathrm{\Delta }C/C_0)}{\delta P}$$ 3

where C₀ is the initial capacitance, (ΔC/C₀) is the relative capacitance change, and P is the applied pressure. Figure 3b presents both the experimentally measured sensitivity and FEA-based simulation results. The sensor exhibits an initial sensitivity of approximately 0.549 kPa⁻¹, which increases to 3.079 kPa⁻¹ at an applied pressure of 0.7 kPa. Overall, the sensitivity increases gradually in the low-pressure region and rises more sharply at higher pressures, indicating superior sensing performance under elevated loading conditions.

Figure 3c–e illustrates the influence of geometric parameters—central angle θ, strip width W, and PI layer thickness T—on the minimum pressure (P_S) required to achieve a target high sensitivity (S = 3.3 kPa⁻¹). It can be observed that P_S decreases significantly with decreasing θ, and increasing W and T. The gray regions in the contour plots denote parameter combinations where the sensor fails to meet the desired sensitivity threshold. These findings provide a design basis for tailoring the 2D precursor geometry to suit specific pressure ranges, ensuring the sensor maintains high resolution near the upper limit of its operating range while avoiding signal saturation or mechanical compression.

Beyond sensitivity, pressure sensors must also exhibit high stability, fast response, and reliable repeatability. Figure 3f shows the sensor’s capacitance response under stepwise loading at pressures of 0.1 kPa, 0.2 kPa, 0.3 kPa, and 0.5 kPa. The capacitance increases correspondingly with each loading step and remains stable during the constant-pressure holding periods. Notably, at 0.5 kPa, the signal is maintained for 10 min with negligible drift, demonstrating good stability under sustained loading.

A solid analyzer was used to apply pressure incrementally at a rate of 4 mm/s. Complete loading–holding–unloading cycles were performed twice at three pressures: 0.2 kPa, 0.3 kPa, and 0.7 kPa. The results are shown in Fig. 3g. The capacitance response exhibits clear stepwise transitions upon loading and unloading. At 0.3 kPa (as shown in the enlarged view of Fig. 3g), the sensor exhibits a loading response time of approximately 131 ms and a recovery time of approximately 140 ms. The details of the measurement setup and real-time data acquisition are provided in Supplementary Note S2. These dynamic characteristics are governed by the mechanical viscoelasticity of the elastomeric substrate and the structural relaxation of the 3D cage architecture during deformation.

Although the sensor displays a slight degree of hysteresis (Fig. S3), the hysteresis is limited to approximately 4%⁶⁰. We also compare our device with state-of-the-art linear sensors, such as the thermoformed 3D sensors reported by Choi et al.⁵⁹, which achieve high linearity (R² = 0.999) and negligible hysteresis (<0.5%) using rigid thermoplastics. In contrast, our sensor utilizes a viscoelastic elastomer substrate to enable unique strain-tuning capabilities. While this introduces a slightly higher hysteresis (~4%), it is a necessary trade-off to achieve reversible range adaptability. Furthermore, unlike linear sensors that offer uniform resolution, our device exhibits a compression-induced nonlinear sensitivity, where the sensitivity increases significantly (by ~5.6 times) at higher pressures. This progressive sensitivity profile provides enhanced resolution and signal-to-noise ratio specifically at high-load thresholds, making it uniquely suitable for detecting peak loads and dynamic gusts in wind monitoring applications. To evaluate long-term reliability, the sensor was subjected to 6,000 loading/unloading cycles at a frequency of 0.5 Hz under 0.5 kPa (Fig. 3h). The device maintained stable performance throughout the test, demonstrating high repeatability under prolonged cyclic loading.

Finally, the detection limit of the device was assessed by placing a lightweight kraft paper sheet (0.01 g, ~ 2 Pa) onto the unloaded sensor. This minimal pressure induced a clear and measurable capacitance change (Fig. S4), confirming the sensor’s ultra-low detection limit of around 2 Pa.

Strain tuning and electrode design for performance optimization

After device fabrication, the sensitivity and working range of the proposed sensor can be further tuned by applying a tensile strain ε_post, to the elastic substrate, which adjusts the initial gap between the top and bottom electrodes. Here, the applied post-strain ε_post is defined as the normalized change in the distance between the two bonding sites of the cage-like structure, given by ε_post = (l−l₀)/l₀. Specifically, l₀ denotes the distance between the bonding sites when the substrate prestrain is fully released, and the device has completed 3D assembly, whereas l represents the distance between the bonding sites after an additional tensile strain is applied to the substrate (Fig. S5). Figure 4a, b illustrates the strain-tuning setup: biaxial strain is applied to the cross-shaped elastic substrate using a tensile platform, with the stretching displacement precisely controlled by a micrometer, and custom fixtures fix the four straight strips to ensure stable and uniform strain beneath the device (Fig. S6). Two types of fixtures are used: one for calibration following performance adjustment (Fig. 4a) and another for mounting the device during application (Fig. 4b). The deformation of the cage-like structure under 15% applied strain is shown in Fig. 4a, with FEA simulation results demonstrating close agreement with experimental optical images. Figure 4b shows the sensor mounted onto a curved surface, showcasing the design’s adaptability and stability under real-world conditions.

Fig. 4.

Strain tuning and electrode design for performance optimization of the 3D capacitive pressure sensor. Experimental setup for applying biaxial tensile strain to the elastic substrate: a calibration fixture with 15% applied strain showing FEA simulation (left) and corresponding optical image (right); b mounting fixture used to attach the device onto irregular surfaces. Scale bars, 10 mm. c Experimental results with FEA simulations for relative capacitance change under out-of-plane compression at various applied post-strains ε_post. d Initial sensitivity as a function of the applied post-strain ε_post. e Workflow diagram for the sensor parameter tuning process

Figure 4c presents both experimental and FEA simulation results of the electromechanical response of the sensor under out-of-plane compression at different values of the applied post-strain ε_post (10–30%). As the applied post-strain ε_post increases, the sensor’s measurement range narrows, while the sensitivity improves. This is further quantified in Fig. 4d, where the sensitivity in the 0 - 0.1 kPa range rises with increasing ε_post, reaching a maximum of 1.687 kPa⁻¹ at 30% strain.

The strain-tunability of the device allows rapid optimization of operating parameters to suit varying measurement environments. Figure 4e outlines the sensor parameter adjustment workflow tailored to specific measurement needs. First, based on the intended working conditions, the approximate range and sensitivity are determined. Then, based on the geometric design diagrams shown in Fig. 3c–e, the dimensions of the 2D precursor are determined for sensor fabrication, which is then mounted onto the target surface for initial measurements. If the response meets expectations, data collection proceeds. If not, guided by the trends shown in Fig. 4c, d, lateral strain is applied to adjust sensitivity and range. The final configuration is then fixed using the mounting fixture. Through iterative adjustment, the optimal strain setting for the application is determined, enabling accurate and reliable pressure sensing.

Importantly, this tunability not only enables static adjustment during device preparation but also suggests pathways for future dynamic control. For example, dielectric elastomer actuators allow real-time modulation of substrate strain via applied voltage⁶¹, while hydrogel substrates provide strain control through swelling induced by changes in water content. By integrating such actively tunable substrates, the sensor could be adapted to a wider range of measurement conditions in a controlled and reversible manner^62–64.

In addition to tuning the 2D precursor geometry, modifying the electrode shape provides another route for enhancing performance. Under compressive loading, the electrodes not only approach each other vertically but also undergo rotation. When electrode shapes lack central symmetry, this relative rotation results in changes to the effective facing area, which contributes to increased capacitance variation and improved sensitivity (see SI Note S3 for calculation details).

To demonstrate this concept, Fig. 5a shows a sensor design employing semicircular top and bottom electrodes, initially offset by a 10° central angle. Figure 5b compares FEA-simulated deformation with experimental optical images, showing that with increasing out-of-plane compression (up to ε_z = 80%), angular overlap increases from 10° to 17.5° (θ_S = 10°–17.5°), thereby enlarging the effective facing area. In this deformation process, the capacitance enhancement originates from the combined effects of electrode gap reduction and compression-induced geometric evolution that increases the effective face-to-face overlap area. To ensure that the normalized capacitance change (ΔC/C₀) evolves smoothly and remains physically meaningful under this combined deformation mode, a finite initial overlap area is deliberately introduced in the electrode design, leading to good consistency between model predictions and experimental responses. Further FEA results confirm that this patterning does not significantly affect the mechanical response or platform flatness of the cage-like structure (Fig. S7). Therefore, the parallel-plate capacitor model remains valid, and force–displacement data from FEA can be reliably used to predict the sensor’s electrical behavior (see SI Note S4 for calculation details). Figure 5c shows the pressure-capacitance response results, with the semicircular electrode design achieving an initial sensitivity of 1.582 kPa⁻¹, significantly higher than that of circular electrode designs.

Fig. 5.

Electrode design for performance optimization of the 3D capacitive pressure sensor. a Assembly of a 2D precursor with semicircular electrodes featuring a 10° central angle overlap. b FEA simulations and corresponding optical images showing electrode rotation and overlap progression under compressive strain. Scale bar, 1 mm. c Relative capacitance–pressure response for semicircular electrode design. d, e Schematic illustrations and corresponding FEA-derived relative capacitance–pressure curves for alternative electrode geometries: d X-shaped, e crescent-moon-shaped, and f designs featuring abrupt changes in facing area

Building on this concept, several alternative electrode geometries were explored, including an X-shaped design (Fig. 5d), a crescent-moon shape (Fig. 5e), and a segmented geometry designed to induce abrupt changes in effective facing area during compression (Fig. 5f). The latter generates a segmented linear pressure–capacitance response, where distinct pressure intervals correspond to different sensitivity levels, significantly enhancing responsiveness across a broad pressure range.

In summary, combining geometric tuning of the 2D precursor with innovative electrode shape design enables substantial improvements in capacitive sensor performance. These strategies allow precise control over sensitivity and range, offering a versatile design framework for high-resolution pressure sensing in diverse application scenarios.

Structural configuration and electrical characterization of the liquid encapsulated sensor

To further improve the reliability and environmental adaptability of the flexible capacitive pressure sensor for practical applications, a liquid encapsulation strategy is proposed. This encapsulated design not only provides effective environmental protection but also preserves compatibility between the sensor’s electromechanical performance and its structural deformation.

The specific fabrication process proceeds as follows: First, as shown in Fig. S8, Ecoflex is injected into a mold containing a dome-shaped chamber and wiring channels (1 mm width, 0.5 mm height) to form a soft silicone cover. After forming the 3D cage-like structure and completing electrical connections, the Ecoflex cover is bonded to the elastomer substrate using Sli-poxy adhesive, as depicted in Fig. 6a. The bonded assembly is then injected with non-volatile glycerol through a syringe needle to fill the encapsulated cavity. The glycerol layer serves a dual purpose: it increases the effective dielectric constant between the electrodes (reaching ε = 40–50 at 25 °C), and it preserves the mechanical flexibility of the device.

Fig. 6.

Structural configuration and electrical characterization of the liquid encapsulated sensor. a Schematic illustration of the encapsulation process. b Simulated and optical images of the encapsulated device. c Capacitance–pressure response of the encapsulated sensor, with the corresponding fitted curve. d Sensitivity variation of the encapsulated device under out-of-plane compression. e Comparison of capacitance noise levels between sensors using air and glycerol as dielectric media

Figure 6b presents a simulated configuration and corresponding optical image of the encapsulated sensor. It can be observed that the liquid-encapsulated packaging does not interfere with the deformation of the cage-like structure, demonstrating the successful integration of structural stability and protective functionality. Performance testing of the encapsulated device (Fig. 6c) reveals a slight increase in the measurement range and a minor decrease in sensitivity, with the pressure–capacitance response curve exhibiting near-linear behavior.

Figure 6d illustrates the sensitivity variation under out-of-plane compression. Compared to the unencapsulated sensor, the encapsulated device displays a reduced rate of sensitivity increase at higher pressures. This attenuation is likely due to the flexible encapsulation layer reaching its deformation threshold, thereby partially restricting motion of the internal 3D structure. In addition, the high-permittivity glycerol significantly raises the baseline capacitance, which helps suppress noise from small fluctuations in the capacitance signal, as illustrated in Fig. 6e. The interfacial liquid layer also functions as a lubricant, effectively mitigating minor stick-slip fluctuations and further enhancing the signal-to-noise ratio.

Demonstration of the cage-like pressure sensor for wind speed sensing

Accurate measurement of wind direction and wind speed is critically important in meteorological monitoring. Traditional devices for wind speed and pressure measurements often face several limitations, including sensitivity degradation due to temperature fluctuations, lagged dynamic responses, and high maintenance costs. In contrast, capacitive flexible pressure sensors offer several advantages—low cost, lightweight construction, simple fabrication, and robustness to temperature variations (Fig. S9)—making them highly promising for meteorological applications.

To evaluate the suitability of the developed cage-like capacitive pressure sensor for wind speed measurement, a series of wind tunnel experiments were conducted. The experimental setup is schematically illustrated in Fig. 7a (left). The sensor was mounted inside the test section of a wind tunnel, connected via silver wires to a high-precision capacitance meter located outside the observation window for real-time data acquisition. A green laser beam projected from below the chamber enabled flow field visualization. Figure 7a (right) shows the interior layout, including the sensor under test, a Pitot tube for reference wind speed measurement, wiring connections, and the laser system. Airflow was precisely controlled by motor speed adjustments, flowing horizontally along the direction indicated by the arrow, and directly impacting the sensor. To ensure accurate wind speed control and mitigate potential interference from the Pitot tube following sensor installation, the relationship between motor speed and wind velocity was calibrated in advance using a hot-wire anemometer (GM8903), providing a reliable reference for subsequent experiments.

Fig. 7.

Demonstration of the cage-like pressure sensor for wind speed sensing. a Optical images of the wind tunnel test setup. Left: overall configuration of the experiment. Right: close-up view of the observation section with airflow trajectory visualized using green laser illumination. b Relative capacitance changes of the sensor over four wind speed cycles at 17.5 m/s and 20.0 m/s. c Relative capacitance change curve across various wind speeds. d Relative capacitance changes at 25 m/s measured in three orientations relative to the sensor: perpendicular to the electrode plates, and two parallel directions. e Comparison of relative capacitance changes under flat and curved surface mountings (curvatures: 0, 0.059 mm⁻¹, and 0.086 mm⁻¹) over four wind speed cycles at 25 m/s

Figure 7b shows the real-time capacitance response under wind speeds of 17.5 m/s and 20.0 m/s across four test cycles. The sensor exhibits rapid response and high signal stability under constant wind conditions. Capacitance fluctuation amplitude increases with wind speed, indicating high sensitivity to wind dynamics. Further calibration (Fig. 7c) demonstrates a nonlinear increase in capacitance with wind speed, particularly in the high-speed region.

Directional sensitivity was also assessed at a constant wind speed of 25 m/s, with airflow applied from three orientations relative to the electrode plates (Fig. S10). As shown in Fig. 7d, the highest capacitance change occurs when the wind is perpendicular to the electrode plates, whereas airflow parallel to the plates produces minimal signal variation. This result highlights the sensor’s effective suppression of off-axis airflow interference and its capability for directional wind speed sensing.

A major advantage of this flexible sensor is its ability to conform to curved surfaces, unlike conventional rigid sensors that require flat mounting. The device was tested on tubes with curvatures κ of 0, 0.059 mm⁻¹, and 0.086 mm⁻¹ under repeated wind loading at 25 m/s. As shown in Fig. 7e, the capacitance response on curved surfaces remains comparable to that on flat substrates, with high signal stability despite minor fluctuations. The Pearson correlation coefficients between the flat group and the 0.059 and 0.086 curvature groups were r = 0.96643 and r = 0.94718 (both p < 0.001), respectively, indicating strong consistency across configurations.

The sensor’s vibration resistance was further evaluated using a high-speed camera (500 FPS) to capture structural deformation under stable wind loading (Fig. S11). Image analysis and synchronized data acquisition were used to characterize the vibration-induced response. Under a frontal wind load of 25 m/s (Fig. S12), the maximum peak displacement of the sensor’s center was only 0.056 mm—significantly smaller than the electrode gap (2.2 mm)—resulting in a capacitance change ratio (ΔC/C₀) of approximately 0.028, with negligible impact on accuracy. Under lateral wind loading (Fig. S13), vibration response along the main wind direction was significantly enhanced, with the maximum displacement reaching 0.4 mm. Owing to the strategic design where the lower electrode is slightly larger than the upper one, the effective facing area remains nearly constant, resulting in stable capacitance output under lateral vibration.

Conclusion

In summary, this study introduces a cage-shaped 3D flexible capacitive pressure sensor based on buckling-guided design and laser cutting. The sensor features a nonlinear sensitivity profile with significantly enhanced sensitivity at higher pressures. Experimental and FEA results confirm that its electromechanical response can be finely tuned by adjusting key geometric parameters and electrode design.

Post-fabrication strain engineering enables structural height modulation, allowing programmable control over sensitivity and pressure range. This post-fabrication configuration offers decisive advantages: it decouples fabrication from application, avoiding costly iterative redesigns; allows a single standardized device to be rapidly tuned on-site for new or uncertain measurement tasks; and ensures consistent performance across all tuned ranges, avoiding the variability introduced by modifying precursors or materials. The sensor also supports liquid encapsulation, which not only protects against environmental influences but also enables tunable dielectric properties through injectable liquids such as EMI-TFSI, EMI-BF₄, and PYR₁₄-TFSI^65–71.

With demonstrated stability and reliability in wind tunnel tests, the sensor shows strong potential for applications in intelligent wind monitoring and structural wind load evaluation. Its design strategy offers a versatile platform for future flexible electronics in areas such as smart infrastructure and environmental sensing. Importantly, future developments could enable active control of the sensor’s strain response through advanced substrate engineering, potentially expanding its applicability across a wider range of measurement conditions.

Methods

Fabrication of developed sensors

To fabricate the buckling-guided pressure sensor, a custom cross-shaped silicone elastomer substrate (Dragonskin 10 Slow, 1 mm thick) was first cast using an acrylic mold. Subsequently, copper electrodes were patterned on an ultra-thin copper-clad polyimide film (IF-2LD, PI as the base film) via a subtractive process. This process involved photolithography (photoresist: RD1215, exposure dose: 2100 mJ/cm²), chemical etching (etchant: APS), and resist stripping. After electrode formation, a circular protective layer was constructed on top by spin-coating a photosensitive polyimide (PSPI) and performing a second round of photolithography (exposure dose: 300 mJ/cm², developer: THMA, development time: 5 min), followed by nitrogen baking at 120 °C for 1 h and thermal curing. The resulting PI/Cu/PI tri-layer structure (total thickness ~75 μm) was then patterned into the desired 2D precursor using a laser cutter (power: 15 W, cutting accuracy: ±20 μm) on a glass substrate. The bottom electrode was prepared using the same process as the top electrode. To enable programmable buckling behavior, the elastomer substrate was biaxially pre-stretched to 100% strain using a custom-designed stretching apparatus. Based on FEA, this substrate pre-stretch (ε_sub = 100%) corresponds to an effective biaxial prestrain of approximately ε_pre ≈ 42% in the uniform strain region at the bonding sites. The bottom electrode (stator) and the 2D precursor (rotor) were then aligned and bonded at designated positions using an adhesive. Releasing the prestrain at a controlled rate of 0.03 s⁻¹ induced the buckling of the 2D precursor into a 3D configuration, forming a parallel-plate capacitor together with the bottom electrode. Finally, the upper and lower electrode leads were connected using silver wires and conductive silver paste, completing the sensor assembly.

Finite element analysis

Nonlinear FEA was employed to predict and optimize the mechanical response of the cage-like structure, using the commercial software ABAQUS for simulating the mechanical behavior of the pressure sensor. To model the deformation of the 3D structure, four-node shell elements (S4R) with composite layups were used for meshing, allowing for geometric nonlinearity to capture large deformations. A refined mesh consisting of approximately 20,000 elements was applied to ensure computational accuracy. For the simulation of the deformation of the 2D precursor adhered to the elastic substrate, eight-node brick elements with reduced integration (C3D8R) were employed with a fine mesh of approximately 160,000 elements. The substrate was modeled using a Mooney–Rivlin hyperelastic material model, with material parameters: elastic modulus E_sub = 197 kPa and Poisson’s ratio μ_sub = 0.49. The metallic and polyimide (PI) layers were modeled as linearly elastic materials, with parameters: E_Cu = 120 GPa, μ_Cu = 0.34; and E_PI = 9 GPa, μ_PI = 0.27.

Measurement

The measurement procedures for electrical and mechanical characterization of the sensors are described in detail in the main text. For all experimental measurements, multiple independently fabricated devices were tested under identical conditions. The corresponding sample size, mean values, and standard deviations are summarized in Supplementary Note S5.

Author contributions

Z.Z. and H.F. contributed equally to this work. Z.Z. and H.F. co-wrote the manuscript and jointly performed the numerical simulations. Z.Z. conducted all the experiments and prepared all the figures. H.Z., H.L., and G.L. assisted with the experimental procedures. The manuscript was written through the contributions of all authors. All authors have given approval to the final version of the manuscript.

Funding

This work is financially supported by the National Natural Science Foundation of China (12272342, 51988101).

Conflict of interest

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41378-026-01252-x.