Fiber-based Miniature Strain Sensor with Fast Response and Low Hysteresis

Abstract

Flexible and stretchable strain sensors are in high demand in sports performance monitoring, structural health monitoring, and biomedical applications. However, existing stretchable soft sensors, primarily based on soft polymer materials, often suffer from drawbacks, including high hysteresis, low durability, and delayed response. To overcome these limitations, we introduced a stretchable miniature fiber sensor comprised of a stretchable core tightly coiled with parallel conductive wires. This fiber sensor is flexible and stretchable while exhibiting low hysteresis, a remarkable theoretical resolution of 0.015%, a response time of less than 30 milliseconds, and excellent stability after extensive cycling tests of over 16,000 cycles. To understand and predict the capacitive sensor response of the proposed sensor, an analytical expression was derived and proved to have good agreements with both experimental results and numerical simulation. The potential of the strain sensor as a wearable device is demonstrated by embedding it into belts, gloves, and knee protectors. Additionally, the sensor could extend its applications beyond wearable devices, as demonstrated by its integration into bladder and life safety rope monitoring systems. We envision our sensor can find applications in the field of sports performance evaluations, health care monitoring, and structural safety assessments.

Graphical Abstract

This paper presents a stretchable double coil sensor designed for wearables, structural health monitoring, and implantable devices. The sensor exhibits a theoretical resolution of 0.015%, rapid response times of 30 milliseconds, exceptional durability after over 16,000 cycles, and minimal hysteresis. Additionally, we introduce comprehensive analytical expressions for densely wound double coil capacitors to guide sensor designs and address existing gaps in the available theories on double coil capacitors.

1. Introduction

Flexible and stretchable electronic devices can find important applications in human-machine interfaces, structural health monitoring, and biomedical applications^[1–4]. In the realm of human-machine interfaces, the softness and flexibility of the electronic devices make them ideal wearable devices for continuously monitoring chemical, biomedical, and physiological signals from the human body. Among all the monitored parameters, the real-time strain data has particular significance since human bodies continuously experience various strain patterns such as joint bending, skin expansion, and respirations. In the realm of structural health monitoring, real-time strain data is crucial for equipment safety assessments and retirement criteria decisions. In addition, in biomedical applications, organ, and tissue go through extreme expansions and significant strain deformations and strains such that the real time strain monitoring of the dysfunctional organs, such as bladder monitoring, helps to improve the quality of life in patients. In these application scenarios, the sensors are required to possess flexibility and stretchability to adeptly follow the deformation patterns of the monitored objects. Additionally, the strain patterns in those applications are commonly random with nonmonotonic profiles and varying strain rates. To ensure the accuracy of the measurements, strain sensors are desired to have low hysteresis, short response, and high sensitivity ^[4,5]. In addition, these sensors must exhibit good dynamic durability to withstand prolonged usage in diverse and dynamic environments after deployment.

In recent years, resistive and capacitive sensors have been dominant options for stretchable strain sensor applications. Stretchable resistive sensors are either elastic material with conductive coating^[6,7] or soft supporting matrix with conductive fillers like semiconductor materials, carbon nanomaterials, and metal particles^[8–22]. Upon stretching, these sensors exhibit changes in resistance attributed to dimensional changes, disturbance of the conductive network, or intrinsic resistance changes within the conductive filler networks, such as tunneling effects, disconnection effects, or cracking ^[5]. However, resistive strain sensors usually have substantial hysteresis due to the high viscoelasticity of the supporting matrix and irreversible disturbance in the conductive network after stretching^[15]. On the other hand, capacitive sensors have less hysteresis comparing with resistive sensors, due to the fact that capacitance relies on the overlapped area and the separation between two electrodes rather than the conductance changes ^[5]. Several stretchable capacitive strain sensors have been proposed by wrapping or depositing conductive layer on elastic material^[23,24], doping conductive particles into soft matrix^[25–29], injecting liquid metal^[30,31], or multi-shell printing^[32]. However, those capacitive sensors have low sensitivities, with gauge factors $(GF=\delta (\Delta C/C_0)/\delta ϵ)$ close to unity. High sensitivity can be achieved in sensors with wrinkled gold film^[33] and silver nanofibers doped ionic hydrogels^[34], but those sensors possess either low durability or delayed response times. In addition, the existing capacitive sensors are not compatible with low-cost scalable fabrication. Consequently, finding a low-cost stretchable strain sensor with short response time, substantial durability and high sensitivity remains a challenge.

Lee, J. et al. recently introduced double coil capacitor sensors utilizing silver nanoparticle-doped elastomeric fibers and demonstrated high sensitivity of 12, low hysteresis and consistent responses under different strain rates. ^[29] However, they still lack the comprehensive theory on densely wound double coil structures, limiting their sensors to the sparse wound cases where the pitch lengths exceed the coil diameters. They showed that the increase of sensor’s sensitivity requires compromising its stretchability and pre-stretching. Here, we derived an analytical expression for densely wound double coil capacitors, and shows it is possible to simultaneously increase both the sensor’s stretchability and sensitivity and achieve high sensitivity at small strains without pre-stretching.

To address the challenges and gaps in existing theories, we fabricated densely wound double coil sensor using a scalable manufacturing approach and systematically characterized it sensitivity, hysteresis, response time, and durability. More importantly, we derived an analytical expression of capacitance of densely wound double coil structures. It shows that depending on the winding density, the sensitivities under strains and pitch lengths are distinct from the sparsely wound cases. In conjunction with the currently available expressions on the sparsely wound cases, a comprehensive theoretical analysis of the double coil structure can be achieved and applied to a wider range of materials and structures. Finally, we demonstrated the sensor’s applications in sports performance monitoring (including joint bending, finger postures, and respiration patterns), structural health monitoring, and biomedical applications (shown in Figure 1a). The sensor was reported to have minimal hysteresis, high repeatability under varying strain rates and negligible degradations after over 16,000 cycles. It also shows a fast response time of 30 milliseconds, a gauge factor as high as 6.5, and a strain resolution of 0.015% without pre-stretching.

Figure 1. Design and fabrication of Double Coil Fiber.

a) Schematic representation of the multifunctional fiber strain sensors integrated into life safety ropes, gloves, clothing, and implantable applications. b) Schematic detailing the fabrication process of the double coil fiber. The double coil structure was wound onto the stretchable core by synchronized motion of the linear motor and rotator. Zoomed-in view highlights the structure of the insulated copper wires. c) Comparison photographs and d) microscopic images of the double coil fiber under 0% (left) and 100% stretching (right), demonstrating its flexibility and stretchability. The unstretched double coil fiber sensor has a 1 mm diameter stretchable core and 160 μm insulated copper wires with 350 μm/rad windings (scale bar, 200 μm). The winding angles α are highlighted in the picture with ${\alpha }_0\approx 85°$ and $\alpha \approx 80°.$ e) Photograph of a double coil sensor wound around a pencil (scale bar, 5mm).

2. Results and Discussion

2.1. Design and Fabrication of Double Coil Capacitive Strain Sensor

To demonstrate the double coil capacitive sensing idea, we fabricated the capacitive sensor by winding the insulted copper wire around a stretchable core, illustrated by the schematic view in Figure 1b. The elastic core which is driven simultaneously by a linear motor and a rotator (not shown in this figure) rotates with angular speed ${\omega }_R$ and translates with speed $v_L$. Two parallel conductive wires from the wire spools pass the clamp and attach to the dielectric elastic core. With the stress applied by the clamp, the two conductive wires are firmly wound around the elastic core. We define the pitch length ${\Lambda }_0$ as the separation between two consecutive turns and equal to the ratio between the translation speed and the rotational angular speed:

$${\Lambda }_0=\frac{\pi }{{\omega }_R}{v}_L$$ (1)

One example of the fabricated double coil fiber is shown in Figure 1c and 1d. The electrodes of the double coil capacitor are insulted copper wires (ELS, 34 AWG, uncoated diameter 160 μm) with 175 μm pitch length (350 μm/rad) and 1 mm elastic core. Image of the unstretched (left) and 100% stretched sensors (right) are shown in Figure 1c, along with the microscopic images in Figure 1d. Following the same definition conventions from J. et al.’s work ^[29], we define a winding angle, denoted as α, to quantify the density of the windings. Here, the winding angle is approximately 90°as the coil is densely wound.

With both ends of the coil structure glued to the elastic core, the double coil conductive wires follow the elongation and retraction of the elastic core. Notably, the mechanical properties of the sensors, illustrated in the strain-stress relations in Figure S1, are primarily dominant by the elastic core, where the coil structure exerts minimal impact on the overall mechanical properties. The mechanical hysteresis at different strain ranges is due to the viscoelasticity of the elastic core, which is typical for soft and stretchable strain sensors. However, the capacitance of the double coil fiber, which is a function of overlapped area and wire separations, is independent on the strain-stress relations and exhibits low hysteresis. Detailed hysteresis analysis will be presented in the experimental section. Two types of elastic cores were chosen in this work: the polyester (PES) elastic core was chosen for sensor characterizations and structural health monitoring applications as it has good stability and durability. The thermally drawn (Figure S2) thermal plastic elastomer (TPE, styrene and ethylene/butylene, SEBS) fibers with 1 mm diameter were used in implantable applications to extend the stretchability, minimize disruptions, and increase the sensitivities of the sensor. Although the electrodes of the sensor are stiff materials, the sensor has good flexibility (Figure 1 and Figure S2) and stretchability due to the dense coil structure. The densely wound sensor can undergo significant expansions while the conductive wires have no plastic deformations due to the dense winding and the large span of the coil.

2.2. Modeling of Double Coil Capacitive Sensor

To better understand the working principle of the densely wound double coil sensor and quantify the relative capacitance change as a function of strain, an analytical expression was derived. The analytical expression predictions were then compared with the Finite Element Analysis (FEA) simulations as well as experimental measurements to verify its accuracy.

The schematic view of the double coil strain sensor is presented in Figure 2a with one turn of helical structure highlighted in red. The length of a single turn of the coil is:

$$l_t=\sqrt{{\left(\pi d_{\mathit{\text{coil}}0}\right)}^2+{\left(2{\Lambda }_0\right)}^2}=\sqrt{{\left(\pi \left(D_0+d\right)\right)}^2+{\left(2{\Lambda }_0\right)}^2}$$ (2)

Figure 2. Working mechanism of the double coil fiber.

a) Schematic representation of the double coil fiber with key geometric parameters marked. Inset: Schematic cross-section of consecutive electrodes with insulating coating. b) Numerical simulations showing electric field distributions in the double coil fiber, highlighting voltage distribution and electric field direction within one turn. The arrows represent electric field distributions, with the length of each arrow corresponding to the electric field strength in logarithmic scale. c) Measured lump capacitance of the double coil fiber as a function of the number of turns and linear regressions. d) Relative capacitance changes under different strains. The analytical expressions (solid line, equation 14) show good consistency with experimental data (dots), and numerical simulations (dash line) e) Capacitive and sensitivity response of the sensor with different pitch densities.

Where ${d_{\mathit{\text{coil}}}}_0$ is the major diameter of unstretched the coil structure, d is the diameter of the unstretched elastic core, $D_0$ is the diameter of the conductor, and the winding angle under unstretched condition is:

$${\alpha }_0=\text{arctan}\left(\frac{\pi \left(d+D_0\right)}{2{\Lambda }_0}\right)$$ (3)

With external tensile strain $ϵ$ applied to the elastic core, the double coil will not apply pressure onto the elastic core in the lateral direction, but it will follow the expansion of the core and straighten its windings while keeping the length of a single turn a constant value. The winding angle will decrease to:

$$\begin{gathered} \alpha =\text{arctan}\left(\frac{\pi d_{\mathit{\text{coil}}}}{2\Lambda }\right) \\ =\text{arctan}\left(\frac{\pi d_{\mathit{\text{coil}}}}{2\left(1+ϵ\right){\Lambda }_0}\right) \end{gathered}$$ (4)

Where $d_{\mathit{\text{coil}}}$ is the major diameter of the coil structure after stretching:

$$d_{\mathit{\text{coil}}}=\frac{\sqrt{l_t^2-{\left(2\Lambda \right)}^2}}{\pi }$$ (5)

When the winding angle (α) is less than 45°, it corresponds to a scenario of sparse winding. In this case, the electric field predominantly distributes between two facing turns that across the elastic core, and the capacitance arises from the spacings between these two facing turns^[29]. In contrast, for the sensor discussed here, the two electrodes are densely wound around the elastic core with the winding angle α significantly greater than 45° so that the electric field predominantly distributes between adjacent turns, resulting in turn-to-turn capacitance being dominant.

To further confirm our analysis on the electric field distributions, FEA models were built to analyze the electrical field distributions between the electrodes. The simulation results, presented in Figure 2b, illustrate that the electric fields distribute periodically (shown in Figure S3) and concentrate primarily between adjacent turns, and within two turns, the center of the circular shape exhibits the strongest amplitude of electric field. To further comprehend the relationship between the lump and turn-to-turn capacitance, we experimentally measured and plotted the total capacitance of the double coil structure against the number of turns, as shown in Figure 2c. As the number of turns exceeds 60, the total capacitance linearly increases with a slope of 358 fF/rad. The deviation from the linear relationships with less than 60 turns possibly comes from the significant impact of the stray field near the ends of the double coil structure. Therefore, the modeling of lump capacitance changes can be simplified to modeling the turn-to-turn capacitance changes. A circuit model and impedance measurements of the double capacitor structure presented in Figure S4(a) further confirm our conclusions:

$$\frac{\Delta C}{C_0}=\frac{C-C_0}{C_0}=\frac{C_{tt}-C_{tt0}}{C_{tt0}}$$ (6)

In the given expressions, $C_0$ and $C$ represent the initial lump capacitance and the lump capacitance after stretching, respectively. Likewise, $C_{tt0}$ and $C_{tt}$ denote the initial turn-to-turn capacitance and the turn-to-turn capacitance after stretching.

An analytical model of the turn-to turn capacitance is derived by referring to the model of the self-capacitance and parasitic capacitance of inductors ^[35–40]. The cross-section of one pair of adjacent turns is presented in the inset of Figure 2a. The capacitance between two adjacent turns can be treated as the capacitance of the insulating layer $C_i$ in series with the capacitance of the gap between the electrodes $C_g$ (shown in Figure S4). Considering an elementary cell of the insulating layer, the capacitance is expressed as:

$$d{c}_{io}=\frac{r{\epsilon }_{ri}{\epsilon }_{0}d\theta dl}{dr}$$ (7)

where ${\epsilon }_{ri}$ and ${\epsilon }_0$ are the relative permittivity of the coating material and free space permittivity respectively. The unit capacitance per angle is given by integrating equation (5) over radius r from conductor radius $r_c=\frac{D_c}{2}$ to coating radius $r_0=\frac{D_0}{2}$ and over length l from 0 to the length of one turn $s_0:$

$$\frac{d{c}_{io}}{d\theta }={\epsilon }_{ri}{\epsilon }_0{\int }_0^{l_t}dl{\int }_{r_c}^{r_0}\frac{r}{dr}=\frac{{\epsilon }_{ri}{\epsilon }_0{l}_t}{\text{ln}\left(\frac{r_0}{r_c}\right)}$$ (8)

According to the FEA simulation results of the electric field distribution in Figure 2b, the electric field is non-uniformly distributed within the gap of two adjacent turns due to the curvature of the electrode. For simplicity, the capacitance at the gap between the electrodes is analyzed under the assumption that the electric field follows the shortest possible distance between the electrodes and the total capacitance between the curvature can be divided into elements of capacitance in parallel with each other, which is a reasonable assumption referring to the electric field directions presented by arrows in Figure 2b. The distance between the electrodes is a function of the angle θ, where the separation is:

$$x\left(\theta \right)=\Lambda -D_{0}cos\theta ={\Lambda }_0*\left(1+ϵ\right)-D_{0}cos\theta$$ (9)

Therefore, for a unit angle dθ, the capacitance is:

$$\frac{d{C}_{g0}}{d\theta }=\frac{{\epsilon }_0{\epsilon }_{rg}{l}_t}{2}\frac{1}{\frac{\Lambda }{D_0}-\text{cos}\theta }$$ (10)

where ${\Lambda }_0$ is the spacing between the center of two adjacent turns before stretching, and ${\epsilon }_{rg}$ is the relative permittivity of the gap at the angle $\theta .$ The capacitance between two electrodes can be equivalently treated as three capacitance in series (Figure S4b): $C_{i0},C_{g0},$ and $C_{i0},$ so that the total capacitance for one turn $d{C}_{\mathit{\text{unit}}}$ is:

$$\frac{1}{d{C}_{\mathit{\text{unit}}}}=\frac{1}{d{C}_{i0}}+\frac{1}{d{C}_{g0}}+\frac{1}{d{C}_{i0}}$$ (11)

$$\frac{d{C}_{\mathit{\text{unit}}}}{d\theta }=\frac{{\epsilon }_0{l}_t}{2}\frac{1}{\frac{1}{{\epsilon }_{ri}}\text{ln}\left(\frac{r_0}{r_c}\right)+\frac{1}{{\epsilon }_{rg}}\left(\frac{\Lambda }{D_0}-\text{cos}\theta \right)}$$ (12)

The turn-to-turn capacitance for one turn is the integration over -π/2 to π/2. As shown in Figure 2b, the electric field is mainly distributed in the air gap between the turns, and only the stray field at the two ends of the double coil structure crosses through the elastic core. For all the cases we discuss here, the turns are stacking tightly, the effects of the stray electric field near the two ends of the double coil structure are ignored. For the turn-to-turn capacitance between two turns (highlighted in Figure 2a), the total capacitance of the capacitor is calculated by integrating the capacitance across all the angles $\theta$. As an approximation, we assume that the dielectric material between the electrodes is air with a relative permittivity taken as:

$${\epsilon }_{rg}=1$$ (13)

The total turn-to-turn capacitance between one pitch of the coil is calculated as:

$$\begin{gathered} C_{tt}={\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{C}_{\mathit{\text{unit}}}d\theta \\ ={\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\frac{{\epsilon }_0{l}_t}{2}\frac{1}{\frac{1}{{\epsilon }_{ri}}\text{ln}\left(\frac{r_0}{r_c}\right)+\frac{1}{{\epsilon }_{rg}}\left(\frac{\Lambda }{D_0}-cos\theta \right)}d\theta \\ =\frac{4b}{\sqrt{a^2-1}}\text{arctan}\left(\frac{\sqrt{a+1}}{\sqrt{a-1}}\right) \end{gathered}$$ (14)

where $a=\frac{1}{{\epsilon }_{ri}}\text{ln}\left(\frac{r_0}{r_c}\right)+\frac{\Lambda }{D_0},$ and $b=\frac{{\epsilon }_0{l}_t}{2}.$

Substituting the turn-to-turn capacitance expressions from Equation (14) into the calculations for relative capacitance change in Equation (6), we calculated the relative capacitance changes under different strains.

The analytical capacitance expressions are plotted alongside FEA simulations and experimental measurements in Figure 2d. The analytical expressions show excellent agreement with the FEA simulations and the experimental results in the tested range, justifying our analysis. In the context of dense windings, the sensor has the best sensitivity at small strain and then the gauge factor decreases after stretching. For the sensor design we presented here, the maximum GF is up to 6.5 in the tiny strain range, while the averaged GF is 5.1 under 0%~5% strain, 2.2 under 5%~15% and 0.9 under 15%~30% strain. This exceeds the maximum achievable unity GF of the parallel-plate structure capacitive sensor^[5,33,41] and demonstrates higher sensitivity than the reported capacitive sensor at small strains^[23–29].

To further understand the sensor’s sensitivity and its dependency on strain and the density of the coil, i.e. pitch length ${\Lambda }_0$, we plotted the relative capacitance changes as a function strain with different pitch lengths in Figure 2e. To quantify the sensitivity, we calculated the gauge factors of the sensor by taking the derivative of the relative capacitance changes with respect to strain and plotted the gauge factor in the inset plot. As it is shown by Figure 2d and 2e, the sensitivity decreases as the strain increases. We thus define the operative stretchability as the strain range that has higher sensitivity than a required sensitivity level. As an example, we marked the operative stretchability for S=3 in Figure 2e. For denser pitch length cases, the sensor tends to have higher sensitivity and larger operative stretchability for a given required sensitivity. Therefore, unlike sparse winding cases^[29], in the context of dense winding, the sensor can increase both operative stretchability and sensitivity simultaneously by increasing its pitch density.

2.3. Strain Sensor Characterization

Thanks to its large gauge factors, the sensor can sense miniature strains without pre-stretching. Figure 3a illustrates the relative capacitance changes of the sensor in response to a 0.5 Hz square-wave strain as low as 0.05%. The relative capacitance exhibits periodic changes following the applied strain changes. Note that the occasional spikes in the plot are attributed to system noise, with a standard deviation of 0.04 pF, corresponding to a theoretical relative capacitive sensitivity of 0.1% (3 σ). Consequently, when operating within the small strain range with a GF close to 6.5, the theoretical strain resolution of the sensor is 0.015%. Detailed insights into the theoretical resolution were explained in the experiment information. This unique feature, enabling the sensor to operate at high sensitivities at small strains and without pre-stretching, makes it an ideal candidate for biomedical and wearable applications. Particularly in cases where monitoring objects are delicate, the sensor’s high sensitivity without pre-stretching avoids imposing substantial tension on the object’s surface, protecting it against potential damage.

Figure 3. Performance Characterization of the Double Coil Fiber.

a) The relative capacitance changes under a 0.5 Hz square-wave with a 0.05% tiny strain. b) Relative capacitance changes over two rounds of stretch (solid line) and release cycles (dashed line). c) Sensor responses against cyclic strains ranging from 0.5 % to 25 %. d) Normalized capacitance changes in response to stretch and release strains under various strain rates. e) Sensor response time measurement involving stretching, holding, and releasing the sensor, illustrating the sensor’s quick response and recovery times. f) Durability assessment of the sensor through over 16,000 cyclic tests, with zoomed-in views of the sensor response during the initial and final 10 cycles.

The stability and durability of the sensor were also investigated through several tests. Two rounds of stretching and releasing patterns between 0% to over 25% strain were recorded with the corresponding relative capacitance changes in Figure 3b. A comparative analysis of the traces from these two stretching and releasing rounds reveal that the sensor exhibits minimal hysteresis and high repeatability across cycles. In contrast to the pronounced hysteresis observed in the stress-strain graph (Figure S1b), the sensor demonstrates minimal hysteresis and high repeatability in its capacitive response. This behavior is attributed to the fact that, despite the high viscoelasticity of the polyester core, the copper electrodes utilized in this sensor are rigid material and exhibit low viscoelastic properties. The high stability of the sensor can be further proved by the time evolution of the capacitance for an overnight test plotted in Figure S6, showing the sensor has stable capacitance readouts with negligible creeping effects. Due to its exceptional stability, the sensor exhibits consistent and distinguishable responses for repeated strain patterns over 0.5%, 3%, 10% 25% stretching, plotted in Figure 3c. In addition, the relative capacitance readouts of the sensor are independent of the stretching and releasing strain rates. As it is shown in Figure 3d, the sensor was stretched and released at 0.1 m/s, 0.01 m/s and 0.005 m/s, but the sensor exhibits consistent patterns of relative capacitance changes. In contrast to highly viscoelastic sensors, the sensor response is remarkably consistent, and the readout remains independent by the strain profile, proofing high reliability of the sensor. Another key advantage brought by the low viscoelasticity is the sensor’s fast response. As it is plotted in Figure 3e, the sensor response times were extracted by comparing the real time sensor location synchronized with the capacitance readout of the sensor. The stretching response time is less than 30 milliseconds (ms) (around 4%~23% stretching) and the releasing response time is around 100 milli-seconds. The delayed releasing response time is attributed to the delayed recovery of the coil structure due to the friction between the coil and elastic core. The overall response time is less than 1 second, which qualifies for most the wearable and mechanical monitoring applications. The durability of the sensor was assessed by the cycling tests plotted in Figure 3f. The repeatable relative capacitance is maintained upon repeated strain between 5% to 15% over 16,000 cycles, where the zoomed-in waveform of 10 cycles after 100 cycles and last 10 cycles show the minimum degradations of the sensor responses after extensive cycling tests.

2.4. Wearable Application Demonstrations

The double coil sensor was integrated and embedded in several application scenarios to demonstrate its capability in sports performance monitoring, structure health monitoring and implantable applications. The stretchable strain sensor was stitched onto the index finger of commercially available gloves (Figure 4a). Thanks to the high sensitivity and stability of the strain sensor, it can monitor the bending gestures of the finger at different angles distinctively. (Figure 4a). By stitching the sensor to a stretchable running storage belt that goes around the waistline (Figure S7), the sensor can monitor the motion of respiration in real time (Figure 4b), with distinct signatures between different breathing patterns. Deep breathing usually involves more contraction and expansion of the lung, and a much larger relative capacitance change can be observed by the sensor stitched onto the running storage belt, where less amplitude of relative capacitance change can be observed in shallow breathing. By comparing the amplitude of the relative capacitive changes, shallow and deep breathing patterns can be distinguished. Like finger bending monitoring, the knee motion postures can be monitored when the stretchable sensor is integrated onto the knee protectors (Figure 4c). With high stretchability of the sensor, the sensor could follow the knee bending to nearly 180°, and with the fast response speed and high stability, the sensor can track the knee bending angles in real time. The fast response and high stretchability of the sensor allows the real time bending angle tracking of the joint. With a fusion of respiration monitoring and knee bending tracking, this sensor could be used for health monitoring and sports performance tracking. Shown in Figure 4d, a back-kicking exercise was recorded both by the breathing sensor and the knee sensor. The breathing data is shown in the top graph, where the breathing amplitude increases as the user exercises and more oxygen was demanded for the body motion. The knee data records the body motion where the user started with a static standing gesture, followed by periodic back-kicking motion, and then ended with another standing rest gesture. The breathing data shows the transition from shallow breathing to deep breathing after the aerobic movement. The combination of multiple wearable sensors could provide health monitoring data for the user and could potentially give instructions for the exercise and provide safety warnings.

Figure 4. Wearable demonstrations of the sensor.

a) Smart gloves demonstration: Capacitance changes of the embedded strain sensor with varying finger bending angles, with inset images displaying different finger gestures while wearing the smart gloves. b) Breathing monitoring: (i) Schematic illustration of human breathing. (ii) Sensor capacitance changes corresponding to both shallow and deep breathing patterns. c) (i) Knee bending pattern tracking: image of the strain sensor embedded in a knee support sleeve. (ii) Detailed view of the embedded strain sensor. (iii) Strain sensor readings change as the knee bends at different angles d) Sport Performance Monitoring: response of the strain sensors attached to the belt and knee during a back-kicking motion, demonstrating the sensor’s real-time monitoring capabilities in athletic movements.

2.4. Bladder Expansion Monitoring and Structural Health Monitoring

In addition, our double coil sensor has potential applications in implantable applications and structural health monitoring. Neurogenic bladder dysfunction is a prevalent issue impacting over 200 million people globally^[42], significantly compromising their quality of life. A stretchable strain technique that can effectively monitor the expansion of the bladder is highly desirable. However, the bladder loading process is random and involves several stages of hold, loading and unloading. Most the stretchable sensors suffer high viscoelasticity and exhibit high hysteresis between measurements which compromises the precision of the liquid loading level estimation of the bladder monitoring. The stretchable double coil fiber we developed here, however, has low hysteresis and high stability at a held strain and can be a solution to address current issues. To prove the capability of bladder monitoring, double coil fibers with thermoplastic elastomer (TPE) cores and silicone protected layers were fabricated for testing. TPE cores are adopted since it is soft and flexible and has same level of Young’s modulus with the soft sensor demonstrated for bladder expansion monitoring.^[43] The additional silicone layer was added to improve the sensor’s biocompatibility. The strain sensors were looped around a porcine bladder, with several sensor locations being tried so the sensor at the upper part of the bladder was picked as it had the highest sensitivity among the positions tested. The relative capacitance changes with two loading processes were plotted in Figure 5b, showing high repeatability of the sensor responses between the two loading processes. A polynomial curve fitting was applied to fit the measured relative capacitance changes ($\Delta C/C_0$) and volume of fluid ($V_{\mathit{\text{water}}})$injected into the bladder. Our specific relationship is expressed as: $\frac{\Delta C}{C_0}=4.834\times {10}^{-5}{V}_{\mathit{\text{water}}}^3-0.0044V_{\mathit{\text{water}}}^2+0.0106V_{\mathit{\text{water}}}-0.0229$. It’s important to note that this equation is tailored to the specific shape of the bladder and position of the sensor. However, through comprehensive calibrations and statistical analysis of a representative sample pool, a universal mathematical relationship can be derived. This broader relationship has the potential to serve as a predictive tool for estimating bladder fluid volume and monitoring the bladder health conditions. An additional loading and holding process was plotted in Figure 5b(ii). The sensor shows fast response as the loading and holding events alternating with minimum drifting observed during periods where there is no liquid loading. The stable and fast responses of sensor on monitoring the bladder liquid levels makes it a good candidate for implantable applications.

Figure 5. Demonstration on implantable applications and structural health monitoring.

a) Photograph showing the elastic fiber sensor securely looped around a porcine bladder. b) Relative capacitance changes of the sensor looped around the bladder in response to (i) two rounds of water loading and (ii) alternating water loading and holding patterns. c) (i) Photo displaying the strain sensor braided inside a life safety rope. (ii) Image of the rope stretching lab test setup. d) Demonstration of the real time measurements on the rope strain. The predicted strain based on relative capacitance measurements is plotted in blue, while the reference strain measured by the video extensometer is plotted in red. The difference between the predicted strain and the true reference strain is plotted in the black trace.

Additionally, the sensor can be embedded to structures to monitor the operation conditions and structural health of the structure, and to provide remaining structural life estimation and safety warnings. Life safety ropes are commonly used for rock climbing to provide protection for drop-and-fall events, and the real time strain data during use and the plastic deformation after use are important parameters in condition monitoring and retirement criteria^[44,45]. The fast response time of the double coil fiber allows it to measure the drop-and-fall event with minimum delay, and the good durability of the sensor allows a one-time deployment of the sensor with continuous readouts. The sensor was embedded in the core of a life safety rope (Teufelberger) with the sensor’s two ends protruding out, which allows the sensor to follow the stretching behavior of the life safety rope. Two eyelet endings were fabricated in the rope to allow a good connection with the test fixtures. The ropes with stretchable sensor were tested under several stretch and hold steps with a tensile tester (Instron 5969). The first cycle of the testing with small strain range (0~14%) was used to calibrate the sensor and derive the function between the relative capacitance changes with the ground truth strain readings from Instron video extensometer. After the function between the relative capacitive changes against the strain was derived from the calibrations, the rope was stretched further. The predicted strain from the relative capacitance changes was plotted with the ground truth data in the same plot with the difference of the calibrated sensor measurements and the ground truth plotted by the black trace. The predicted strain exhibits the real time prediction with minimal lag, high stability at the holding stage and less than 0.5% discrepancy with the ground truth values, showing our sensor can provide accurate measurements of the life safety rope strains and information of the life safety rope conditions.

3. Discussion

In summary, we proposed a novel capacitive strain sensor with dense winding structures that can be scaled up for mass production. We proposed an analytical expression to model its relative capacitive changes against various strains, and the comparison between the analytical expressions, experimental measurements, and FEA simulations shows good agreement, which verifies our expressions. Even though the electrodes of the double coil sensor are much stiffer than the elastic core, the sensor is still stretchable due to the straightening effect of the helical structure. In addition, due to the dense winding and the circular cross-sectional area of the electrodes, the sensor was demonstrated to have a remarkable high Guage Factor (GF) of 6.5 without pre-stretching, which surpasses the geometric limitations typically associated with the planar capacitive sensors. Thanks to the characterizations, the sensor has high repeatability, low hysteresis, insensitivity to strain rates, rapid response, and robust durability. Leveraging these advantages, the sensor was integrated into several systems for demonstrations. The integration in the wearable applications shows its capabilities in real-time tracking of finger, joint, and breathing patterns with high stability. Furthermore, when braided into life safety ropes, the sensor demonstrates its capability to measure the rope strain in real time. The sensor’s applications also extend to implantable biomedical applications, as shown by our demonstrations on porcine bladder expansion monitoring.

This sensor exhibits promising potential across diverse domains, including sports performance monitoring, human-machine interface, structural health monitoring, and biomedical applications. With the proposed sensor geometry, the densely wound structure can be further extended to other materials and designs. By incorporating the analytical expression for the densely wound structure with the currently available expression for the sparsely case, a comprehensive theoretical analysis can serve as a valuable guidance for the future design of double coil capacitive sensors.

4. Experiment Section

Fabrication of the double coil fiber:

A pair of insulated copper wires (ELS, 34 AWG, uncoated diameter 160 μm were attached to the elastic core. The elastic core was held by the capstan attached to the rotator (ZYT520) and a translational motor (Montomatic motor generator, Electro-craft corporation). The translational motor was tuned to operate at the speed of 7.5 mm/min and the rotator was tuned to synchronize with the translational motor to produce the designed pitch. After the winding process, the two ends of the coil were glued to the elastic core with cyanoacrylate (Loctite, 4310) to allow the double coil structure to follow the elastic core expansions.

Elastic TPE (SEBS) fiber fabrication:

The thermal drawing process is shown in supplementary information. The preform diameter was around 1 inch and the length about 6 inches. The core of the preform is made of thermal plastic elastomer material, poly(styrene-(ethylene-co-butylene)-styrene) triblock copolymer (SEBS, Kraton G1657M), followed by a layer of eCOC and a sacrificial layer of PMMA (Rowland Technologies). The preform was drawn at the temperatures of 150 °C, 260 °C and 120 °C, and the final diameter of the fiber is around 1 mm. The PMMA sacrificial layer was etched by soaking in acetone (Sigma-Aldrich) and peeled off.

Finite element method simulation:

Numerical simulations were built to explain the electric field distribution in the densely wound double coil fiber structure and its capacitance changes under different strains. The double coil fiber was simulated with COMSOL Multiphysics (version 6.0). The double coil fiber structure was modelled by a PES core with double coil structure following the parameters given by the experiment measurement. A model with 70 turns (35 turns for each conductor) was simulated, and the diameter of the copper wire is 160 μm with the pitch length of 350 μm/rad. The FEA model was meshed using the physics controlled free tetrahedra with 458366 vertices. The material of PE elastic core was chosen using the default material library, with the relative permittivity as 3.1. The copper wire was chosen from the library as well, and the electrical properties are as follows: electric conductivity σ= 5.998 $\times {10}^7$ [S/m], relative permittivity ${ϵ}_{rCu}=1$, reference resistivity $1.667\times {10}^{-8}$. As an approximation for the analytical calculations and FEA models, we assume that the double coil windings are equally spaced. For simplicity, the insulating layer was ignored in both FEA simulations and analytical expressions considering the insulating layer for 34 AWG wires is thin. For the simulation of capacitance under different strains, the double coil structure pitch was stretched uniformly respect to the applied strain and the diameter of the coil structure was assumed to decrease according to the function of: $\left(\sqrt{l_t^2-{\left(2\Lambda \right)}^2}\right)/\pi$, where $l_t$ is the length of one turn and Λ is the pitch length after stretching. The elastic total capacitance was calculated through the global evaluation parameter Maxwell capacitance.

Sensor characterizations:

(1) Impedance measurement: two electrodes from the same side of the sensor were connected to the Lcur and Hcur port of the impedance analyzer (Keysight 4990A) and the impedance between 1 kHz to 30 MHz was recorded. (2) LCR meter measurement: The electrodes from the tested sensor were connected to the LCR meter (BK precision 891) for the measurement. The LCR meter was set to parameter mode at 100 kHz, 0.5VRMS and fast speed mode. The LCR was connected to the PC and the real time capacitance data was recorded by LabVIEW (National Instrument). (3) Small strain range measurement: one end of the sensor was securely tightened to the stage and the other end of the sensor was attached to a stage driven by micrometer (LTA-HS motorized actuator). The micrometer was programmed to generate 0.5Hz square wave. (4) Theoretical strain sensitivity: we calculated capacitance sensitivity by calculating the 3 $\sigma$ variations of the capacitance when the sensor was held at a constant strain. Then, the theoretical strain sensitivity is calculated as: $S\left(\epsilon \right)=S(\Delta C/C_0)/GF$. (5) Hysteresis, strain rate, response time and stability test: the sensor was attached to the programmable linear motor (LinMot 015–5080) and the motor was programmed to the designed waveform. (6) Mechanical responses: Mechanical stress–strain tests and cyclic tests were measured using a dynamic mechanical analysis (TA Instrument, DMA Q800). The stiffness test was carried out using a dynamic mechanical analyzer installed with the dual cantilever module. The tested fiber length is 8 mm, which equals the distance between the fixed and movable clamps. The vibrating frequency varies between 0.1 Hz to 5 Hz with the amplitude of $500\mu m.$ In plot in Figure S2 b, plots for each of the fiber designs are measured over 3 samples.

(7) Relationship between number of turn and lump capacitance: The relationship plotted in Figure 2c is measured experimentally. We counted the total number of turns of the double coil structures and measured its capacitance after unwinding the coil.

Sensor Demonstrations:

(1) Wearable applications: The double coil sensor was stitched to the commercial gloves, running storage belt and knee protector, and the real time data was recorded by the customized LabVIEW script. The finger and the knee bent at different angles and then held the posture for a few seconds for data recordings. The belt with stretchable sensor was placed at the stomach of the subject. The subject was asked to take shallow and deep breath and the pattern of the capacitive changes are recorded in the same way as the joint movement. (2) Bladder demonstrations: The TPE (SEBS) core double coil fiber was packaged inside the silicone and then tightened around the porcine bladder. A silicone tubing was inserted into the top of the bladder. Water was pumped into and extracted the bladder through tubing (Pump Tubing, 3-Stop, Tygon^® S3^™ E-Lab) with peristaltic pump (ISMATEC ISM829B). The loading and extraction speed is ~0.63 ml/s. (4) Life safety rope demonstrations: the double coil fiber was braided into the core of the life safety rope (Teufelberger), with two ends of the fiber protruding through the cladding. The total length of the sample, including the eyelet section, is around 34 cm and the separations between the extruded position is 12.5 cm. The rope passed through the fixture of the tester (Instron 5969) and was stretched under programmed procedures as described in the previous section. The recorded relative capacitance changes were fitted with the reference strain from the video extensometer (Instron, AVE) by a customized MATLAB code using the 3rd order polynomial function. The prediction of strain in real time is achieved by the customized LabVIEW code shown in Figure S8.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.