Research Article
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Nonlinear Poisson’s Ratio for Modeling Hyperelastic
Capacitive Sensors
Elze Porte and Rebecca Kramer-Bottiglio*

Highly stretchable capacitive sensors are of great interest for soft robotic
control due to their ability to measure relatively large strains. These sensors
are often multilayered materials, with one or more of the layers made fromsilicones filled with functional particles. However, the models used to describe
the material behavior do not always account for the hyperelastic nature of thesilicones, the altered material properties due to fillers, and potential anisotropy
due to the layered structure. Large errors arise when predicting capacitanceusing widespread assumptions of linear elastic mechanics and isotropic
material properties. This study demonstrates how these modeling assumptions
are inadequate for predicting sensor performance, and compares alternativemodels based on empirical material mechanics. The Poisson’s ratio of multilayered hyperelastic capacitors is measured in both the width and thickness
directions by imaging the sensor dimensions during strain. The results indicatethat the sensors are anisotropic and have a strain-dependent Poisson’s ratio,
demonstrating the validity of the proposed model. Considering these properties
in capacitance models will lead to an improved ability to predict sensor
performance, especially at high strains.

1. Introduction
Elastomeric capacitive strain sensors are a promising technology
to measure the large strains associated with soft robotic systems.
They have been used for closed-loop control[1–5] and exhibit consistent performance over thousands of strain cycles.[6–8] Current
sensor models are inadequate for predicting electromechanicalperformance due to two fundamental deficiencies. First, the sensors are often made from hyperelastic silicones, but most models
are based on linear elastic mechanics.[8–14] Second, the conductive layers added to the silicone to create electrodes have different
properties than neat silicone, and can significantly affect theoverall material properties of the sensor.[6,9] To advance capacitive
sensor design and improve integration with soft robotic systems,
a greater understanding of the nonlinear mechanics and theinfluence of the conductive layers on capacitive sensor performance is required. In this work, we demonstrate that accounting
Dr. E. Porte, Prof. R. Kramer-Bottiglio
Department of Mechanical Engineering and Materials Science
9 Hillhouse Av
New Haven, CT 06511, USA
E-mail: rebecca.kramer@yale.edu
The ORCID identification number(s) for the author(s) of this article
can be found under https://doi.org/10.1002/admt.202001247.

DOI: 10.1002/admt.202001247

Adv. Mater. Technol. 2021, 2001247

for nonlinear and anisotropic material
properties improves our prediction of
capacitance by at least 15% compared tomodels based on traditional assumptions.
Capacitive sensors measure strain
based on changes in the geometry of thesensor during stretch. A very common
sensor design is the parallel-plate capacitor,[6,7,10,15–17] where thin electrode anddielectric layers are stacked. For ideal plate
capacitors, the capacitance C is related tothe sensor dimensions by
C = ∈0 ∈r

Lw
t

(1)

where ∈0 and ∈r are the free space and relative permittivity, respectively; L and w are
the length and width of the active area ofthe sensor, respectively; and t is the thickness of the dielectric layer. The dimensions of the sensor change during stretch,
resulting in a measurable change in capacitance. The changes in the different dimensions of the sensor are
related by the Poisson’s ratio ν, as schematically shown in 2D in
Figure 1a. When the Poisson’s ratio of the sensor is known, thecapacitance can be predicted for any change in L by expressing w
and t as a function of the strain in length direction εL.
The layered structure and the different material properties ofelectrodes and dielectric materials should be considered whendescribing the Poisson’s ratio. Electrode layers can be madefrom a composite of elastomer and conductive filler, such asgraphite[1,6] or carbon black.[7,15,18] The addition of fillers generally reduces the Poisson’s ratio of the composite compared toneat silicone[19–22] due to a mismatch between the Poisson’s ratioof the elastomer and of the filler material.[19,23] Similarly, a mismatch between the Poisson’s ratio of the electrode and the dielectric layers is expected to result in a Poisson’s ratio of the sensor
as a whole in between those of the separate layers.[24] In addition,
fillers in the silicone matrix can result in a strain-dependent Poisson’s ratio.[19,21,25,26] Nonlinear capacitance models for pressure
sensors highlight the importance of accurate material assumptions,[27,28] but, to the authors’ knowledge, no study provides
empirical evidence for a nonlinear capacitance model for stretch
sensors, and there are no studies investigating the potential
strain-dependent Poisson’s ratio of capacitive sensors.
The aim of this study is to use empirical material mechanics
to develop a hyperelastic capacitance model that accurately predicts the measured capacitance at large strains. We use silicone(DragonSkin10, Smooth-on, Inc) based sensors with expanded

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(a)

L0

L0(1+ξL)(1+ξw) = (1+ξL)-

W0

W0(1+ξw)

Poisson's ratio [-]

(c)

0.5

0.3

= -ξw/ξL

0.2
0.1
0

(b)

= -ln(1+ξw)/ln(1+ξL)

0.4

HenckyCauchy0

0.5(d) 0.8
Poisson's ratio [-]EIG-silicone

Silicone

1.5

0.5
0.4
sensor dielectric (width)
sensor dielectric (thickness)silicone (dielectric)
EIG-silicone (electrode)

0.2

0.5

(e) 0.8

ac ve areatransi on areas
tab areas (constrained, ξ = 0)

1

0.6

0
0

Silicone as
sensor dielectric

Poisson's ratio [-]Sensor

EIG-silicone as
sensor electrode

Strain [-]

0.6
0.5
0.4

Strain [-]1

1.5

thickness
width

0.2
0
0

standard
half length
half width
double thickness
0.5

width direction
thickness direction
fit standard thickness
fit double thickness

Strain [-]1

1.5

Figure 1. a) Schematic representation of Poisson’s ratio; b) schematic representation of the individual layers and sensor layout; c) comparison ofPoisson’s ratios of linear and hyperelastic models; d) Poisson’s ratios of the individual layers of the sensor, and the dielectric as part of the sensor;e) Poisson’s ratios of different sensor geometries including a linear fit to the data. The incompressibility condition (ν = 0.5) is indicated with a black
dotted line in (d) and (e) (n = 6 for all).

intercalated graphite (EIG)–silicone electrodes, as described inprevious work,[1,6] and schematically represented in Figure 1b.
First, we analyzed the Poisson’s ratio of silicone, comparing
a linear elastic and hyperelastic model. Second, we investigate the Poisson’s ratio of the individual layers of the sensor,
and as a complete sensor structure for different sensor dimensions. Variations in the sensor dimensions are all compared toour “standard” sensor, which has an initial active sensor area of
100 × 10 mm and dielectric thickness of approximately 0.6 mm.
Dimensions of all sensor geometries are given in Table S1 in the
Supporting Information. Third, we translated the measured Poisson’s ratio into a prediction of capacitance and compared the useof different models to the measured capacitance. Fourth, we predict the capacitance of sensors without direct measurements ofPoisson’s ratio, but based on the empirical model.

Adv. Mater. Technol. 2021, 2001247

2. Results and Discussion
There are two primary definitions of Poisson’s ratio: Cauchy’s
linear elastic ratio and Hencky’s hyperelastic ratio.[25] The definition of the linear elastic Poisson’s ratio isν w = −ε w /ε L(2)whereas the hyperelastic Poisson’s ratio isν w = − ln (1 + ε w )/ln (1 + ε L )(3)with νw being the Poisson’s ratio in width direction, and εw and εL
the strains in width and length direction, respectively. Replacing
εw in Equations (2) and (3) with the strain in thickness direction

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εt results in the Poisson’s ratio in thickness direction νt. Figure 1ccompares the two definitions by showing the linear and hyperelastic νw for silicone up to 150% strain. The hyperelastic analysis
results in a constant νw close to 0.5, while the linear elastic model
results in a strongly decreasing νw. The observed difference
between the two models is in line with the literature on hyperelastic behavior of elastomers.[25,29] Unless otherwise stated, we
will use the hyperelastic definition of Poisson’s ratio so the nearincompressibility of the silicone can be expressed as νw + νt ≈ 1.
The Poisson’s ratio of the EIG–silicone, silicone, and silicone
as sensor dielectric (Figure 1b) were measured to investigate
the isotropy and strain dependence of the sensor. Figure 1dshows νw and νt over 150% strain for the sensor dielectric, andνw for the individual layers, which are expected to be isotropic(νw = νt). The EIG–silicone failed at strains higher than 20%.
We found that the Poisson’s ratio of the sensor dielectric is
lower in width direction (≈0.30–0.39) than in thickness direction (≈0.56–0.62), and changes with strain in both directions.
The following paragraphs will first discuss the directionality ofthe sensor and subsequently the strain dependence.
The reduction of Poisson’s ratio of the sensor in the width
direction can be explained by the mismatch of the Poisson’s
ratios of the dielectric and electrode materials: the Poisson’s
ratio of the EIG-silicone (≈0.25) is much lower than the puresilicone (≈0.47) when tested separately (Figure 1d). The mismatch of Poisson’s ratios results in νw of the sensor dielectricin between those of the individual layers (≈0.30–0.39), whereasνt (≈0.56–0.64) increases because the dielectric is a nearly
incompressible material, where in this case νw + νt ≈ 0.94.
Other studies have reported a similar reduction in Poisson’s
ratio due to the addition of strain limiting[6,9] or auxetic layers
to the sensor.[8,30] The difference between the Poisson’s ratio
in width and thickness directions means the sensor cannot beassumed isotropic. Although nonlinear mechanics models (e.g.,
Neo-Hookean, Ogden) can be fit to the stress–strain curves,these models do not capture the anisotropic nature of the sensors and are therefore not suitable as sensor models.
A similar observation was made regarding the stiffness ofthe sensors; The EIG-silicone is much stiffer (≈0.8 MPa secant
modulus over 10% strain) than the dielectric silicone (≈0.2 MPa
secant modulus over 10% strain), resulting in a modulus of
the sensor as a whole in between those of the individual layers(≈0.3 MPa secant modulus over 10% strain). The average
stress–strain curves for the EIG-silicone, silicone, and sensor
are included in Figure S1 in the Supporting Information. The
stiffness of the sensors can affect the overall stiffness of asoft robotic system and should therefore be considered in thesystem design, but the measured capacitance is not dependent
on the sensor stiffness (Equation (1)).
The Poisson’s ratio of the sensor’s dielectric was not constant with strain (Figure 1d): νw increased from approximately
0.30 at 0% strain to 0.39 at 150% strain, and νt decreased fromapproximately 0.62 to 0.56 over the same strain interval. A
strain-dependent Poisson’s ratio has been reported for sensors
that include auxetic structures,[11] but has not been investigatedfor capacitive sensors in general. The addition of the electrodes
introduced a compressible component to the sensor, which
likely affected the overall material behavior and resulted in astrain-dependent Poisson’s ratio.

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The anisotropic, strain dependent behavior was observed
for all tested sensors geometries. Figure 1e shows the averagemeasured Poisson’s ratio over strain and a linear fit of thedata in both width and thickness direction for three additional
sensor designs: a “half-length,” “half-width,” and “double
thickness” sensor. The results from the standard sensor are
repeated from Figure 1d to provide a clear comparison. The
standard deviations of the measurements are not shownfor clarity, but are included in Figure S2 in the Supporting
Information.
The measurements of the standard, half-width, and halflength sensors show almost identical Poisson’s ratios. The halfwidth sensor shows a slightly lower average νt compared to thestandard and half-length sensors, which we hypothesize is dueto the sensitivity of the measurement method in thickness tosmall vibrations and twists in the sample that are more prone
to appear in the smaller samples. As a result, both the halflength and half-width sensors showed a much larger range inthe measured νt compared to the other sensors. The Poisson’s
ratio is assumed to be the same for sensors with the same
thickness, since νw was identical for all sensors with the same
dielectric thickness, and the small reduction in νt for the halfwidth sensor likely resulted from the sensitivity in the experimental method.
The potential effect of unwanted twisting in the sensor on
the measured capacitance was further investigated by stretching
the sensors in a twisted state, with the top end of the sample
rotated 90° with respect to the bottom. We did not observe any
differences between the measured capacitance of the twisted
and straight sensors (measurements are included in Figure S3
in the Supporting Information). Small twisting motions of thesensors are therefore not further accounted for in the current
work. In general, we expect the additional strains and the associated change in capacitance that result from a twisting motion
to be relatively small in sensors with a much larger length thantheir width and thickness.
The double thickness sensor shows a slightly higher νw andlower νt compared to the other sensors. Qualitatively, this is linewith the composite rule of mixtures, which predicts the Poisson’s ratio of a composite material νcomp based on the volume
ratio ϕ of the materials (νcomp = ν1ϕ1 + ν2ϕ2).[31–33] Because thedimensions of the dielectric and electric layers are the same inlength and width, only the thickness of the layers affects thevolume ratio. The rule of mixtures, however, is based on linear
elastic assumptions and assumes isotropy, and did therefore
not provide a quantitative prediction of the Poisson’s ratios thatmatched the experimental data for the sensor composites.
The insights into the Poisson’s ratio of the sensor can beused to translate the material properties into an accurate
capacitance prediction. Several capacitance models are considered in predicting the measured capacitance of the sensors. The most commonly used model assumes linear solidmechanics[8–14,18]
C = C0(1 + ε L ) (1 − ν w ε L )(1 − ν t ε L )(4)with C0 being the initial capacitance at 0% strain. Taking into
account the nonlinear mechanics and the anisotropy, while

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keeping the assumption on the incompressibility of the dielectric layer, the capacitance–strain relationship is predicted by[6,7]C = C0(1 + ε L )(1 + ε L )−ν
2(1−ν )
= C 0 (1 + ε L )( 1 + ε L )− ν
w

w

t

(5)

When the sensor is assumed as an isotropic incompressible
material (νw = νt = 0.5),[7,9,10,12–14,17] both the linear- and hyperelastic models reduce to
C = C 0 (1 + ε L )(6)Using the strain-dependent empirical data on νw and νt in
Equation (5) results in the empirical model presented in thisstudy.
It is well documented that capacitance and strain are linearly related for elastomeric capacitive sensors.[6,7,10,15] At highstrains, inaccurate measurements of the capacitance may occur
due to an increasing resistance in the electrodes in combination with the chosen measurement frequency,[15,34] resulting ina seemingly nonlinear relationship. The nonlinearity is, however, a measurement artifact, and if it occurred in the current
work the linear part of the data was extrapolated to 150%. The
measured Poisson’s ratios at 5% strain were used as constantνw and νt in the theoretical models.
Figure 2a shows the linear fit (extrapolated from 100%) of
the standard sensor compared to the predictions based on themodels from Equations (4)–(6) and the empirical model basedon the measured Poisson’s ratio. The only model that accurately
predicts the capacitance at 150% strain is the empirical model.
The assumption of an isotropic incompressible material underestimates the capacitance by almost 15% at 150% strain, overestimating the strain by about 30%. Similarly, the incompressible hyperelastic model with constant strain underestimates
the strain by about 30%. Although the linear elastic model
performs similarly to the other models up to about 30% strain,
it overestimates the capacitance at large strains, because thestrain dependent Poisson’s ratio (Figure 1c) was not accountedfor. It is therefore essential to consider the strain-dependent
Poisson’s ratio for an accurate prediction of the capacitance.

Figure 2. a) Predicted capacitance based on different models compared to the measured capacitance. The light blue shaded area indicates the 95%
confidence interval of the linear fit to the experimental capacitance data. b) Comparison of different empirical models. Predicted capacitance using alinear fit to Poisson’s data for c) half-length, d), half-width, and e) double thickness sensors (n = 6 for all).

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The empirical model was further refined by predicting thecapacitance based on a linear fit of the measured Poisson’s
ratio, and by assuming near incompressibility of the dielectric (νw + νt = 0.94) where the fit of νw was used to calculate νt.
The comparison of these models, shown in Figure 2b, indicatesthat all models provide a close prediction of the capacitance.
Although the models that use both the measurements of νw
and νt are slightly more accurate, assuming incompressibility
can save valuable testing and analysis time. This final empiricalmodel can accurately predict the electromechanical performanceof the different sensor variations (Figure 2c–e). Although the prediction for the half-width sensor (Figure 2d) deviates more fromthe measured capacitance than is the case for the half-length anddouble thickness (Figure 2c,e), the prediction still falls within theconfidence interval. As expected from Equation (1), the reduction in length and width and increase in thickness for the sensor
variations resulted in an almost identical C0 (24–30 pF) at 0%
strain that is about half C0 of the standard sensor (50 pF). The
double thickness sensor (Figure 2e) has a slightly lower capacitance increase per unit strain than the other sensors, which canbe related to its different Poisson’s ratio.
The Gauge factor ((ΔC/C)/(ΔL/L)) is larger than 1 for all
sensor variations. For isotropic sensors with a nearly incompressible dielectric material (ν ≈ 0.5) the Gauge factor will not
exceed 1.[7,10] The increased sensitivity of the sensors in the current work is a direct consequence of the low Poisson’s ratio of
the electrodes, since a low Poisson’s ratio in width direction
and high Poisson’s ratio in thickness direction of the dielectricincreases the capacitance change per unit strain (Equation (5)).
The Poisson’s ratio mismatch between the sensor layers can be
exploited to increase sensor sensitivity by adding materials witha low Poisson’s ratio to the composite structure, such as auxetic
materials[8,30] and fabrics.[9] The Poisson’s ratio of particle
filled composites can generally be reduced by increasing thefiller content,[19,20,22,26] but there may be practical limits since
we observed embrittlement of the conductive composite in previous work that used a high filler content (>15 wt%).[35]The empirical Poisson’s ratio model was used to predict
the sensor performance without directly measuring Poisson’s ratio. Two additional sensor designs were chosen: sensors with approximately half the dielectric thickness of thestandard sensor to investigate if the model can be extrapolated
for thin dielectrics; and sensors that are a quarter the lengthand half the width of the standard sensor to check if the modelis accurate for much smaller active areas that deviate from thestandard sensor in more than one direction. The graphs in
Figure 3 show that, for both sensors, the predicted capacitanceclosely matches the experimental capacitance data.
The prediction of the quarter length sensors (Figure 3a) canbe improved by accounting for the deformations in the transition area between the gauge aura with uniform strain and thetab area with no strain (as defined in Figure 1b). In all sensors,
the strain was assumed uniform across both the transition andgauge area, since the transition area accounted for only a small
percentage of the total stretchable area (<10%). In the small sensors, the transition area is larger than 20%, and assuming thatthe average strain in this area is 0.5 times the strain in the gauge
area provided an even closer prediction of the measured capacitance (adapted model is included in the Supporting Information).

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Figure 3. Predictions of capacitance without direct measurements ofPoisson’s ratio for a) a quarter length, half-width sensor based on
the standard model and accounting for the transition area (Figure 1b)of the sensor, and b) a half-thickness sensor based on the standard
model and on an extrapolated model accounting for layer thickness(n = 4 for all).

For the half-thickness sensors (Figure 3b), the capacitance
was predicted using both the Poisson’s ratio of the standard
thickness sensors and an extrapolated Poisson’s ratio toaccount for the reduction in dielectric thickness (the extrapolated Poisson’s ratio calculation is included in the Supporting
Information). Both predictions show close correspondence
to the experimental data, and the difference between the two
predictions is much smaller than between different models aspresented in Figure 2a. The half-thickness sensors were theonly samples where we observed buckling in the lateral direction when tensile strain was applied. Since the model provided
accurate predictions of the capacitance, we did not include
potential buckling effects in the model. Considering the straindependent hyperelastic Poisson’s ratio is more important thanaccounting for small variations due to sensor thickness.
Studies that directly compare a capacitance model to measured data show either over- or underestimates of predicted
capacitance.[10,12] The comparison of the different models in thecurrent work indicates that it is essential to include the hyperelastic nature of the sensors to accurately predict their capacitance at high strains. Isotropic sensors are an exception to this,since the hyperelastic and elastic models are identical. Althoughsome capacitive sensors with extremely thin electrodes may be
assumed isotropic,[10] the assumption should be carefully considered due to the Poisson’s mismatch between the conductive
and dielectric layers.

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The presented model is the first step to predict the Poisson’s
ratio of the sensor composite based on the Poisson’s ratio of theindividual layers. Currently, the model relies on the collected
empirical data specific to the sensors. Hyperelastic material
models, such as Neo-Hookean and Ogden assume isotropy,
and can therefore not be directly used to model the mechanicsof the sensors. Traditional composite models such as Voigt
and Reuss have been reported to be insufficient to describe
nonlinear material mechanics for fiber and particle composites,[36,37] and for materials with an incompressible matrix.[38]Although adapted composite models are presented in literature,[20,38] the experimental verification for a generalized theory
is limited. Further understanding of the material mechanics
of hyperelastic composite materials is required to accurately
predict the Poisson’s ratio of the sensor composite.

3. Conclusion
This work investigated the nonlinear and strain dependent
material properties of elastic capacitive sensors by using ameasured Poisson’s ratio to predict sensor capacitance. The
presented empirical model contributes to our understanding of
hyperelastic capacitive sensors and can be used in the sensor
design process to predict sensor sensitivity. Two main conclusions can be drawn. First, considering hyperelastic and anisotropic properties of the sensor is essential to accurately predict
capacitance. The presented model showed much higher accuracy than models based on linear elastic and isotropic assumptions. Anisotropy was caused by a mismatch in the Poisson’s
ratio of the dielectric and electrode layers of the sensor. Second,the capacitance was most accurately predicted when the strain
dependence of the Poisson’s ratio was considered. The sensors
showed an increasing Poisson’s ratio in width direction, anddecreasing Poisson’s ratio in thickness direction with strain.
The capacitance was also accurately described for sensors ofwhich the Poisson’s ratio was not directly measured, but thestrain-dependence of the Poisson’s ratio was included in thedescription.

Sensor Fabrication: Sensors were produced using an earlier reported
method.[6] The conductive EIG-silicone composite was rod-coated witha thickness of approximately 0.2 mm on a PET substrate. After curing,the dielectric silicone was coated on top of the EIG layer using a film
applicator (SH0340, TQC Sheen), and the resulting sheet was folded
onto itself just before full curing. The applicator settings were 450, 750,
and 1350 µm for the half-thickness, standard thickness, and double
thickness sensors, respectively. The final thickness is slightly lower
than the coated thickness due to slight spreading of the silicone before
curing. Sensors were cut to size with a laser cutter (VLS 2.30, Universal
Laser Systems).
Sensor Interfacing: The sensors were interfaced with an LCR meter(E4980AL, Keysight Technologies) using copper strips glued to thesensor tabs (Figure 1b). Conductive glue was prepared by mixing
Silpoxy (Smooth-On, Inc.) with the EIG mixture in the same ratio as
the electrode layers. The copper strips were attached to a tab that was
stationary during mechanical testing. Custom made attachments wereglued with Silpoxy to either side of the sensor tabs to securely attach thesensors to the materials testing system.
Precycling Sensors: Samples were repeatedly (10×) stretched using
a material testing system (Instron 3345) to 200% their original length
before electromechanical testing to exclude the Mullin’s effect from themeasurements. The length of the stretchable area of the samples after
the cyclic test was recorded as the new original length (L0).
Electromechanical Characterization: Sensors were stretched toapproximately 150% strain from L0, at a rate of 5 mm s−1 using thematerials testing system. For the half-length sensors, the rate was
reduced to 2.5 mm s−1. The capacitance of the sensors was recordedwith the LCR meter at an excitation frequency of 400 Hz. Simultaneously,
either the sensor width or thickness were captured with an imaging
system (Grasshopper 3, Point Grey Research). Images were captured at
approximately every 5% strain. The measured capacitance was adjusted
to represent only the capacitance of the stretchable area by subtracting
the capacitance of the stationary tab areas (Figure 1b) from the LCR
measurements. The capacitance of the tab areas was calculated as apercentage of the initial capacitance C0 using the relative size of the tab
areas reported in Table S1 in the Supporting Information.
Image Analysis: Images were analyzed using the contrast of the black
EIG layer against a white background for the width measurements andusing the contrast between the white dielectric and black EIG layers forthe thickness measurements. The width of the sensor was found by
taking the average width over a length of 40–45 mm for each image. The
thickness evolution of the dielectric was found by tracking the thickness
change of a specified area over two subsequent images. This moredetailed tracking method was required for the thickness measurements,
due to the relatively large variation in thickness compared to thevariation in width across the sensor.

4. Experimental Section
Dielectric Silicone: DragonSkin10 Slow (Smooth-on, Inc) was dyed
white for imaging purposes using Silc Pig (Smooth-on, Inc) silicone
dye. Part A was mixed with approximately 6 wt% white dye and stirred
manually to ensure even distribution of the dye in the final product.
Part B was added in equal ratio to the dyed Part A, and mixed in acentrifugal mixer (Thinky ARE-310) at 2000 rpm for 30 s and degassed
at 2200 rpm for 30 s.
Conductive Composite Silicone: Expanded intercalated graphite (EIG)
was prepared as conductive filler to DragonSkin10 Fast (Smooth-on, Inc)for the electrode layers of the sensors. Expandable graphite (5 g, Sigma
Aldrich) was expanded in a crucible heated to 800 °C, and subsequently
soaked in cyclohexane (500 mL). The mixture was sonicated (Q500 1/2″
tip, Qsonica) at 70% for 1 h and sieved through a 212 µm sieve. Excess
cyclohexane was decanted after settling of EIG particles for at least 1 h,
and the mixture was boiled down until approximately 3 wt% EIG wasachieved. Silicone was prepared with equal Parts A and B and mixed
as described for the dielectric silicone. The EIG mixture was manuallystirred into the silicone at 79 wt% to achieve approximately 10 wt% EIG
in the electrodes after evaporation of the cyclohexane during curing.

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Supporting Information
Supporting Information is available from the Wiley Online Library orfrom the author.

Acknowledgements
The authors would like to thank Thomas Sipple and our collaborators
at Otherlab, Karen Robinson, Tim Duggan, Conor O’Brien, and Jim
McBride for valuable discussions regarding capacitive sensors andthe applications of this work. This work was supported by a National
Aeronautics and Space Administration Small Business Technology
Transfer grant (80NSSC17C0030).

Conflict of interest
The authors declare no conflict of interest.

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Data Availability Statement
The data that support the findings of this study are available from thecorresponding author upon reasonable request.

Keywords
capacitance, composites, Poisson’s ratio, soft robots, strain sensors
Received: December 14, 2020
Revised: April 28, 2021
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