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Are Liquid Metals Bulk Conductors?
Lina Sanchez-Botero, Dylan S. Shah, and Rebecca Kramer-Bottiglio*
liquid–solid GaIn mixtures, or biphasic
GaIn (bGaIn), can achieve a paste-likeconsistency without losing the electricalproperties of the liquid metal. Despite thisprogress in workability, the electrical properties of EGaIn and bGaIn are not fully
understood. Researchers have reported
a wide variation of the relative change inresistance results for EGaIn (Figure 1a)
and for composites of liquid metal[32,33](including biphasic materials and liquid
metal-embedded elastomers, or LMEEs,
shown in Figure 1b). Although somesamples seem to follow bulk-conductor
assumptions (Pouillet’s law), numerous
studies show resistances below the values
predicted by the model. Due to the wide
range of measurement techniques used inliquid metal research, it is often unclear
what differences are due to variations in
intrinsic conductivity versus uncorrected
errors caused by the experimental setup.
To illustrate the importance of measurement technique, consider the case of aclassic two-terminal measurement system.
These measurements are typically the easiest to carry out, but introduce significant
measurement error.[34] In this setup, a scientist or engineer will connect a multimeter to both ends of thesample using two wires (Figure 2a,b). The resistance reportedby the multimeter will necessarily include the resistance ofthe material of interest (EGaIn, bGaIn, etc.), in addition to the
combined “parasitic resistance” that includes the lead wires, thecontact resistance between the lead wires and the sample electrodes, and the resistance of any components (such as copper
terminals, conductive epoxies, oxides,[35] etc.) between the wires
and the material of interest. For higher resistance conductors(such as graphite-silicone conductive composites in sensors,which typically are in the range of several kΩ[36]) the parasitic
resistance is negligible. In contrast, if the sample resistance is
1 Ω compared to 0.1 Ω combined parasitic resistance (common
for LM circuits), the parasitic resistance represents a fixed 10%
dead-weight error.
Assuming reliable measurements can be obtained forstretchable electronics, what is the correct model for benchmarking the electromechanical behavior of liquid metal circuits? Figure 1 shows a significant range of behaviors reported
in the literature, yet many authors assume that the bulk conductor assumptions (Pouillet’s law[4,27,37]) are the appropriate
benchmark. Typically, liquid metal samples are containedin elastomer materials that reduce their cross-sectional area
when stretched, according to the material’s Poisson ratio. The

Stretchable electronics have potential in wide-reaching applications including
wearables, personal health monitoring, and soft robotics. Many recent
advances in stretchable electronics leverage liquid metals, particularly
eutectic gallium-indium (EGaIn). A variety of EGaIn electromechanical behaviors have been reported, ranging from bulk conductor responses to effectively
strain-insensitive responses. However, numerous measurement techniques
have been used throughout the literature, making it difficult to directlycompare the various proposed formulations. Here, the electromechanicalresponses of EGaIn found in the literature is reviewed and pure EGaIn is
investigated using three electrical resistance measurement techniques: four
point probe, two point probe, and Wheatstone bridge. The results indicate
substantial differences in measured electromechanical behavior between thethree methods, which can largely be accounted for by correcting for a fixed
offset corresponding to the resistances of various parts of the measurement
circuits. Yet, even accounting for several of these sources of experimental
error, the average relative change in resistance of EGaIn is found to be lower
than that predicted by the commonly used bulk conductor assumption,referred to as Pouillet’s law. Building upon recent theories proposed in theliterature, possible explanations for the discrepancies are discussed. Finally,suggestions are provided on experimental design to enable reproducible andinterpretable research.

1. Introduction
Room-temperature liquid metal alloys have been investigatedextensively for their potential applications in stretchable electronics as electronic skin (e-skin),[1–3] stretchable conductors,[4–6]electrodes,[7,8] sensors,[2,9,10] antennas,[10] transistors,[11] energyharvesting devices,[12–14] robots,[15–17] and flexible computational
systems.[16,18–20] However, liquid metals (LM) such as eutectic
gallium-indium (EGaIn) are intrinsically difficult to manipulate
because of their high surface tension, giving rise to challenges
in the patterning, scaling, and uniformity of EGaIn devices.
Researchers have sought to improve the workability of EGaIn by
embedding solid particles, such as exfoliated graphite (EIG),[21]carbon nanotubes,[22] magnetic particles,[23–26] highly crystalline
Ga2O3,[27] copper,[1,28] silver,[29–31] and quartz.[18] The resulting
L. Sanchez-Botero, D. S. Shah, R. Kramer-Bottiglio
School of Engineering & Applied Science
Yale University
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/adma.202109427.

DOI: 10.1002/adma.202109427

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www.advmat.deFigure 1. Compilation of reported results for relative change in resistance for a) EGaIn and b) EGaIn composites. We include biphasic GaIn (bGaIn)
pastes in addition to liquid metal embedded elastomers (LMEEs), since both achieve conductivity primarily through the inclusion of liquid metal, yet themechanics and rheology of both are different than pure eGaIn. The black, continuous line corresponds to bulk-conductor assumptions (Pouillet’s law).

resulting relation between resistance change (R/R0) and engineering strain ε (defined as ΔL/L) becomes R = (1 + ε )2, whichR0
is the most commonly-used form of Pouillet’s law. The relativechange in resistance is conveniently a quadratic, and it is often
assumed that this is the appropriate model to compare novel
formulations against. However, many assumptions underlie theuse of this model, which are often unaccounted for, such as:
Poisson’s ratio, experimental design (the parasitic resistances
discussed above), sample morphology, liquid metal amount
per sample, and substrate adhesion. Recent efforts have sought
to model liquid metal circuits with higher fidelity, including
Zolfaghari et al.’s introduction of a parameter for tortuosity in
LMEEs.[37] However, as Zolfaghari pointed out, a pure liquid
metal trace has a tortuosity of zero, and should therefore follow
Pouillet’s law. Thus, the variations observed in Figure 1a remain
largely unexplained.
In this progress report, we seek to evaluate the claim that
EGaIn behaves like a bulk conductor. After reviewing theexperimental literature on liquid metal, we test the hypothesis that obtained resistances are strongly dependent on thetesting procedure. We evaluate EGaIn traces using three conventional methods used to measure electrical resistance—fourpoint probe (4PP), two point probe (2PP), and Wheatstonebridge (WB). Our results show substantial differences between
the three measurement methods. We investigate the cause of
the discrepancies, and provide empirical evidence that a significant portion of the difference between the three measurement methods can be accounted for with a single constant“parasitic resistance” which can be easily measured. However,none of our results suggest that EGaIn traces behave like abulk conductor. Therefore, we then derive Pouillet’s law frominitial assumptions and discuss the advantages and limitationsof this model. Seeking to determine whether biphasic liquid
metal materials are significantly different than EGaIn, we thenrepeat the tests on samples made with a biphasic material consisting of highly crystalline Ga2O3 particles mixed with EGaIn.
Finally, we provide a set of best practices and suggest avenues
for future research on the electromechanical response of liquid
metals and related materials. We hope that this discussion
encourages researchers to carefully remove as many sourcesof experimental error as possible to improve the accuracy andAdv. Mater. 2022, 2109427

reproducibility of future research results, thereby increasing
our understanding of the behaviors of stretchable electronics.

2. Electromechanical Response of Liquid Metals
Most studies that use liquid metal interconnects report resistance versus strain values significantly below that predicted byPouillet’s law (Figure 1). However, several papers also showneat EGaIn and EGaIn composites that behave similarly to bulk
conductors. In this section, we review the liquid metal literature, highlighting the diversity of compositions, manufacturingmethods, and measurement techniques used. We will focus
on studies that report electromechanical results. For further
reading, we refer readers to recent reviews on liquid metals,[38]stretchable electronics,[39] and wearable electronics.[40]Some studies report that EGaIn behaves in agreement with
Pouillet’s law. For example, Zhu et al. reported stretchable
conductive fibers with a triangular cross-sectional area and acore-shell structure, which consist of an EGaIn core and an elastomeric shell of Poly[styrene-b-(ethylene-co-butylene)-b-styrene](SEBS).[4] The cross-sectional area of the fibers was tuned byvarying the spinning rates: fibers with an inner diameter of≈360 μm were obtained at a spinning rate of 1000 m min−1,
while larger fibers with an inner diameter of ≈670 μm wereobtained at 100 m min−1. A four point probe (4PP) method anda custom deformation stage were used to characterize the electromechanical properties of these conductive fibers. The larger
diameter fibers only followed Pouillet’s law at strain values up
to ε < 100% (Figure 1a, yellow asterisks), exhibiting lower resistance change than the model from 100% to 700% strain (defined
as change in length divided by initial length, or ΔL/L0 = ε). It
is unclear whether the data corresponds to one sample or to
the average of multiple samples, and thus it is difficult to tell
how consistent these electromechanical trends are. It should
be noted that Zhu et al. reported subtraction of the parasitic
resistance value as part of their methodology. However, it is not
immediately clear how fibers were clamped or placed in theircustom deformation stage. Since liquid metal traces increase
their resistance when pressure is applied,[3,41] knowledge of theclamping mechanism and parasitic resistance calculation arenecessary for proper interpretation of the reported results.

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www.advmat.deFigure 2. Summary of measurement methods used for electrical resistance measurements. Schematics are shown for a,b) two probe point (2PP) c,d)four-point probe (4PP), and e,f) Wheatstone bridge (WB). Each method indicates, in red, the unwanted “parasitic” resistances that may introduce
error into the measurements of the liquid metal samples. g) Photos of the setup used for the 2PP and 4PP methods, including the multimeter (top)and tensile-testing stage (bottom). h) Photos of the WB circuit (top) and the WB circuit next to the tensile stage (bottom).

Other reports suggest that EGaIn deviates substantiallyfrom bulk conductor assumptions. Boley et al. printed tracesof EGaIn onto a moving PDMS substrate.[5] The electromechanical properties of the specimens were measured with the
2PP method (digital multimeter, Fluke 87V) using a custom
deformation stage. In this work, the resulting relative change
in resistance follows a quadratic function with R/R0 ≈ 3 at 100%
strain (Figure 1a, orange cross), which is 25% lower than thevalue predicted by Pouillet’s law. During the electromechanicaltesting, the clamps were positioned so that the added pressure
would not interfere with the liquid metal traces, and the samples were measured for five cycles. The data presented shows

Adv. Mater. 2022, 2109427

five data points per measured strain; however, it is unclear if
these values correspond to those five stretching cycles or to
five samples. The lack of reporting prevents the assessment ofreproducibility and margins of error.
By patterning serpentine or spiral microchannels filledwith EGaIn, Park et al. demonstrated strain sensors that wereselectively responsive to strain in particular directions (stretch
along x or y direction, or pressure along the z axis).[41] Figure 1a(green triangles) shows the electromechanical response of asingle channel during uniaxial strain. Using a linear approximation of Pouillet’s law that is valid for small deformations(ε < 200%), Park showed one sample with reasonable agreement2109427 (3 of 12)© 2022 Wiley-VCH GmbH

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www.advmat.deFigure 3. Schematics of the liquid metal samples. a) Exploded view of the sample components. b) Equivalent electrical circuit. c) Dimensions of thesamples in mm.

between theory and experiment up to ε = 100%, for a singlemicrochannel trace geometry (Figure 3a in ref. [41]). While subsequent serpentine samples were qualitatively explained to be
expected to have a non-Pouillet response, the presented models
did not account for the directionality of the electromechanicalresponse (as observed in Figure 10 in ref. [41]). For these experiments, the samples were clamped in a region not containing
any liquid metal, stretched using a universal testing machine(Instron 5544A), and measured using a custom 4PP resistancemeasuring circuit.
Gozen et al. introduced open elastomer molds with micronscale channels that were filled with EGaIn by a pressuretransfer process (Figure 1a, blue diamonds).[6] The electromechanical properties of EGaIn specimens were measured to a
maximum strain of 40% for a total of 50 cycles. The resistancevalues fluctuated within the first few cycles, followed by steadyvalues of R/R0 ≈ 1.18 at 40% strain, compared to R/R0 ≈ 2 predicted by Pouillet’s law. The axial loading was applied using amanual, lead-screw driven linear stage (Velmex UniSlide A15),
and the resistance measurements were acquired with an LCR
meter (B&K Precision 889B).
Similarly to neat EGaIn, some works report EGaIn composites with Pouillet-like behavior. Ozutemiz et al. presented
a circuit fabrication method utilizing EGaIn’s selective wetting of copper/chromium (Cu/Cr) traces.[19] In this fabrication
method, the 120 nm Cu/Cr films are sputter deposited onto a
PDMS substrate, then the desired circuit is patterned using alaser, followed by deposition of an EGaIn layer of several hundred μm. Although this manufacturing method uses very thinfilms of Cu/Cr ( 120nm), the contact of these elements withEGaIn may induce the formation of intermetallic compounds.
The relative change in resistance of single traces was measured
with a universal testing machine (Instron 5969) in conjunction
with a voltage divider circuit. For this review, we extracted thedata reported by Ozutemiz et al. corresponding to a sampleof Cu/Cr-EGaIn embedded in PDMS and cut into a dogbone
shape (ASTM D412) without HCl treatment (Figure 1b, purplex’s). One of the samples exhibited good agreement with Pouillet’s law in the wide range of applied strains, while the otherdeviated significantly from the theory, exhibiting larger valuesof R/R0. In the latter example, the authors observed rupturingof the elastomer and void formation at the interface after the
sample was stretched.
Yet, the majority of reported EGaIn composites suggest relative strain-insensitivity. Pan et al. sputtered thin films of Cr
and Cu onto a PDMS substrate, followed by the deposition of
EGaIn[20], similar to the method proposed by Ozutemiz et al.[19]Adv. Mater. 2022, 2109427

Pan reported deviation of EGaIn’s electromechanical properties from Pouillet’s law with R/R0 ≈ 2 at 100% strain (Figure 1b,
blue squares), which is 50% lower than the value predicted byPouillet’s law.
Many composites made by embedding irregularly-connected
droplets of EGaIn in silicone exhibit relatively strain-insensitive
responses.[21,31,42–44] For example, our group recently reported
one such liquid metal embedded elastomer (LMEE) that showed
resistance far below Pouillet’s law (Figure 1b, pink).[21] EGaIn
was vigorously shear-mixed into uncured silicone to embedliquid metal microparticles into the elastomer matrix. Then, agraphite-cyclohexane suspension was added, and the composite
was cast into sheets with relatively uniform thickness. After
curing, the sheets could be stretched to induce microcracks andsinter the microparticles, creating a continuously-connected
network of liquid metal and graphite embedded in the silicone. The electromechanical properties of the composite were
measured by a custom 4PP method (a constant-current source
paired with a B&K Precision multimeter) exhibiting R/R0 ≈ 1.5
at 100% strain.
Ford et al. demostrated a shape-morphing LMEE by embedding LM microparticles into a liquid crystal elastomer (LCE)
matrix (Figure 1b, blue triangles).[42] EGaIn was shear mixed
with uncured LCE to form and embed LM microparticles.
During synthesis, a native gallium oxide layer formed aroundthe LM-microparticles, requiring mechanical sintering to
induce electrical conductivity. The final LM-LCE composites
exhibited high electrical conductivity and high thermal conductivity. To induce reversible shape morphing, the voltage
was applied to the LM-LCE composite to enable electrical joule
heating and activate the shape-memory effect in the LCE. The
electromechanical properties of the LM-LCE composite were
measured with a USB data acquisition module (USB-6002, NI)
that collected data from both the Instron 5969 materials tester
and a voltage divider, exhibiting R/R0 ≈ 1.08 at 60% strain.
Haque et al. presented liquid metal elastomeric thin films(LET), which were made by mixing EGaIn with Styrene–Isoprene–Styrene tri-block copolymer and polybutadiene.[8] The
electro-mechanical characterization of the LETs was performed by bonding them to a stretchable supporting layer(3M 4910 VHB Double-Sided Tape, 1mm thickness) andattaching copper tape electrodes. LETs were mechanicallystretched in an Instron 5944 mechanical testing machine, andresistance was measured using a Keithley 2460 source meterwith capability for 2PP and 4PP modes. The resulting relativechange in resistance at 100% strain was reported to be ≈2.5,
lower than the value predicted by Pouillet’s law.

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Pastes containing liquid EGaIn with solid filler particles,
or what we will refer to as biphasic GaIn (bGaIn), have also
become a field of active research, as several biphasic materials
have been reported to be strain-insensitive. A liquid metal paste
consisting of EGaIn and SiO2 (quartz) particles was preparedby Chang et al. The SiO2 particles were dispersed in the EGaIn
using a ball-milling method, creating a paste with excellent
electrical and rheological properties. The electromechanicalproperties of the material were tested on traces of the paste
painted onto spandex fabric substrates. Data was gathered withan acquisition unit (34972A, Agilent) with capability for 2PP
and 4PP acquisition modes. The sample exhibited a relativechange in resistance similar to the theoretical value at smaller
strains (ε < 20%), but then deviated to lower values resulting in
R/R0 ≈ 3.1 at ε = 100%.[18]Several research groups used nickel particles as their filler
material. For example, Guo et al. manually mixed nickel microparticles with EGaIn at different concentration ratios that
resulted in a Ni-GaIn amalgam (Ni-GaIn) paste[25]. The Ni-GaIn
paste was printed on an Ecoflex substrate directly by a rollingbrush. Flexible coils with a diameter of 4 cm were prepared,
and their relative change in resistance during stretching was≈1.9 at 70% strain (Figure 1b, orange cross). The resistancemeasurement method was not disclosed. Daalkhaijav et al.
sonicated nickel particles into EGaIn, reporting R/R0 ≈ 2 at
100% using a 4PP measurement technique (Figure 1b, red circles).[26] By varying the nickel content, the size of nickel particles, and the sonication energy, different rheological properties could be obtained, although it is unclear how these processparameters would affect the electromechanical properties of thebiphasic material.
In our previous work, motivated by the assumption thatneat EGaIn is a bulk conductor, we developed a biphasic gallium indium (bGaIn) material to explore its electromechanicalproperties. To obtain this new bGaIn, EGaIn was sonicated inethanol, spray-printed onto silicon wafers, and then baked at
900 °C to promote the in-situ formation of β-Ga2O3 (Figure 1b,
blue line).[27] After this, the biphasic mixture was transferred
to different substrates, such as PDMS and VHB (VHB 4905,
3M), utilizing various methods, including transfer-printing
and manual scraping. Using a Wheatstone bridge to measure
the resistance of rectangular traces, relatively strain-insensitive
behavior of the samples was observed (R/R0 ≈ 1.06 at 100%
strain on PDMS). Nevertheless, more detailed study of the samples revealed that our own results varied with measurement
method, exhibiting higher relative change in resistance when
2PP and 4PP were used. This finding motivated us to compile
the present progress report.
The presented literature review raises the question: What is
the true electromechanical behavior of EGaIn? The wide rangeof measurement techniques, experimental parameters and lackof cross-validation between studies have made it difficult tointerpret the strain (in)sensitivity observed in EGaIn samples,
which should behave identically across studies, since EGaIn isa specific mixture of gallium and indium. As novel formulations of bGaIn and EGaIn composites emerge with the claimof strain-insensivity, to what degree are these materials truly
strain-insensitive, and when is their behavior similar to previously reported materials? When working with low-resistanceAdv. Mater. 2022, 2109427

conductors, it is crucial to carefully report and ideally standardize the techniques used for sample preparation and resistance measurement, to ensure a fair comparison with priorart. In the following sections, we scrutinize some of the mostcommonly used techniques for resistance measurement, before
returning to a deeper discussion of the relevant theory andother experimental design considerations.

3. The Importance of Measurement TechniqueIn this study, we investigate three commonly-used measurement methods found in the stretchable-electronics literature:
two-point probe (2PP),[5] four-point probe (4PP)[4,41] and Wheatstone bridge (WB)[27] (Figure 2).[34] In brief, the 2PP method isequivalent to the common multimeter setups used in many
research, educational, and home settings. The WB method canhave relatively high sensitivity, and is well understood. Both
2PP and WB methods are simple to implement, but relatively
error-prone. To reduce errors when measuring low resistancespecimens, some researchers use a 4PP method.
The 2PP method consists of two electrodes, one at either
end of the sample (Figure 2a). The resistance is measured bysupplying the circuit with a constant current source and measuring the voltage across the sample with the same probes. Consequently, the measured voltage is across both the sample (Rs)and resistance of the leads (Rw) (Figure 2b). If the values of thetwo lead wire resistances are of the same order of magnitude
as the specimen (Rw ≈ Rs), the reported resistance value will
be inaccurate.
The effects of the resistance from lead wires may be avoided
by using the four-point probe method (Figure 2c,d). In a 4PP
measurement, the current source circuit and the voltage
sensing circuit are separated. If the input impedance of thevoltage sensing circuit is sufficiently high, then the current
flowing through Rw is minimal. This means the voltage drop
across Rw can be neglected, and finally, the voltage sensing circuit only measures the resistance across the sample, as desired(Figure 2d). In other words, the resistance of the lead wires (Rw)
is ignored, and the reading is solely dependent on the sample’s
resistance (Rs). The 4PP method produces relatively accurate
measurements of samples with low resistance values. However, the 4PP method is not recommended for samples withhigh resistance values, since the assumption of zero current
flowing through the voltage-sensing circuit’s lead wires may
become invalid.
A Wheatstone bridge is another conventional method tomeasure static or dynamic electrical resistance (Figure 2e).
Here, we utilize a Wheatstone bridge with three fixed resistors
and one variable resistor (Rs, our samples). In a Wheatstonebridge, a voltage Vs is applied between points A and C, and eachhalf of the bridge (R1, R2 and R3, Rs) forms a voltage divider. If
the voltage measured between points B and D (Vout) is zero, thebridge is balanced; conversely, the bridge will be unbalanced
when a Vout ≠ 0. Any change in the sample resistance (Rs) due
to tensile deformation will unbalance the bridge, making it suitable for the measurement of dynamic electrical resistance. It
should be noted that our Wheatstone bridge shown in Figure 2f
has an additional resistor R0 that was used to limit the current

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www.advmat.deflowing through circuit. The derivation of the equation used in
the calculation of the resistance value for the liquid metal samples can be found in Supporting Information.
Similar to the 2PP method, lead wire resistance has a significant impact on the operation of the circuit (Figure 2f, indicatedby the red Rw resistors). The error is negligible if Rw ≪ Rs, but
if Rw ≳ Rs, this source of error can result in significant signal
degradation. For example, our two alligator leads have a total
resistance of 0.85 Ω as measured using the 4PP method, compared to sample resistances of the order of 0.5 Ω.
To illustrate the differences in the measurements methods,
we manufactured our samples with a dogbone-shaped design
filled with a rectangular conductive trace with two square end
electrodes (Figure 3). Dogbone designs have been shown to bereliable and mechanically robust in other literature. Indeed, theASTM standard for testing rubber in tension (ASTM D412) suggests using a dogbone design, and utilizing the gauge length(17 mm in our samples) as the unstretched length for strain
calculation (ΔL/L0 = ε). Samples were made by depositing
EGaIn onto Dragon Skin 10 (Smooth-On, Inc.) silicone films
while using a polyethylene terepthalate (PET) mask. To interface the liquid metal to the measurement circuits, copper tape
was attached to the ends of the sample using silver epoxy. Theinterfaces were then reinforced with fabric squares to prevent
stretching at the interfaces. Finally, the specimens were encapsulated with another layer of Dragon Skin 10 and cut into a
dog-bone shape using a CO2 laser (ULS 3.0, Universal Laser)(Figure 3a). The electrical resistance of liquid metal samples
tends to increase with increasing applied pressure, due to the

reduced cross-sectional area of the samples under mechanicalcompression. To avoid mechanical compression during tensile testing, four holes were also laser cut into the interface
fabric area, to align and secure the samples to the tensile stage(Figure 3c). The EGaIn sample had a semi-circular cross-section, as seen in the micro-computed tomography (microCT)
images shown in Figure S2a,b, Supporting Information.
We manufactured EGaIn samples, and characterized
their electromechanical response using the three methods(Figure 4). The measurements for 2PP and 4PP methods wereacquired using a digital multimeter (B&K Precision model
5492B), interfaced with a set of kelvin probes (Pomona electronics, model 6730) as test leads (Figure 2g). The Kelvin test
leads have a resistance of 0.1 Ω, as measured by connecting theclips to each other and using the 2PP method. We note thatthe contact resistance between the test leads and the electrodesof some setups may be non-negligible. However, we found
the resistance to vary negligibly, within measurement precision (≈0.001 Ω), with different orientations and contact areas
between the leads and electrodes.
Five specimens were tested to get a mean value of the material behavior. The EGaIn samples have a characteristic resis­
tance value of about 0.2 Ω, including 0.1 Ω from the electrodematerials. All dynamic electrical resistance measurements weremade with a custom tensile stage controlled using an Arduino
Uno (Arduino, AG) (Figure 2g,h).
Across all specimens and measurement methods, weobserved that the electrical resistance monotonically increased
with increasing strain. The primary reason for this behavior is

Figure 4. Electromechanical behavior of EGaIn. a) Resistance responses of the dogbone shaped samples under a tensile strain up to ε = 100% with thethree electrical resistance measurement methods and their corresponding b) corrected resistance data. c) Relative change in resistance R/R0 of EGaIn
under a tensile strain up to ε = 100% and d) corrected R/R0 data. The shaded areas represent one standard deviation for five samples.

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www.advmat.dethat, during elongation, the channels increased in length anddecreased in cross-sectional area. However, additional factors
are likely at play, as discussed later in this manuscript.
Although a similar rate of change is observed in the resistanceof EGaIn using the three measurement methods, (Figure 4a),the magnitude of the reported resistance differs between
methods. For example, the average initial resistance estimatedfor EGaIn with the 4PP method is 0.25 Ω, while the 2PP and
WB methods yielded 0.33 and 1.15 Ω, respectively (Figure 4a).
One important observation is that the 4PP and 2PP methods
yielded different initial resistances for EGaIn, despite the factthat 4PP and 2PP used the exact same experimental setup(Figure 2). As mentioned above, the 2PP test lead resistancevalue is 0.1 Ω, which is 40% of the average value of the EGaIn
samples. Larger parasitic resistances will result in a larger difference between 2PP and 4PP techniques. Put another way, the2PP technique is more susceptible to experimental errors introduced by the connection between the measurement device andthe sample. A similar discrepancy can be observed for the WB
method: the estimation of the initial average resistance value
using WB is ≈4.6 times larger than the value obtained with 4PP
for EGaIn, owing to the relatively high resistance values of thealligator clips (0.85 Ω) used in this configuration.
The resistance of the electrode materials (copper tape andsilver epoxy) was also estimated using the 4PP method. Bothelectrodes have a total resistance value of 0.1 Ω, which is ashigh as the test lead resistance of the 2PP method. Using thisinformation in consultation with the electrical diagrams of
Figure 2, the estimated resistance values for the samples witheach method were then corrected by subtracting the extra resistance values (Figure 4b). Specifically, data acquired with the 4PP
method was corrected for the electrodes’ resistance, while the2PP and WB methods were corrected for test lead wires andelectrode resistances. The resulting curves for 4PP and 2PP
methods overlap, with average initial resistances values of 0.15
and 0.14 Ω, respectively.
These results highlight the importance of understanding
the assumptions behind the measurement methods used in anexperiment. For example, sample electrode resistances should
be considered when performing 4PP resistance measurements,
and in some cases it would make sense to adjust the data prior
to publication and clearly report the correction factors used. In
addition, the corrected data acquired by the WB method are
higher than 2PP and 4PP methods at lower strain, but the difference between measurement techniques decreases at higher
strains. The initial average resistance for EGaIn, after correction, is estimated by the WB method to be 0.2 Ω. Although
this value is not significantly higher than those obtained withthe corrected data for 4PP and 2PP, the overlap is not perfect,
implying that unaccounted resistance from other parts of thecircuit configuration may still be affecting the data.
3.1. Pouillet’s Law
Typically, liquid metal samples are contained in homogeneous
elastomers that reduce their cross-sectional area during stretch.
It has been widely accepted that the relation between resistancechange and mechanical strain for liquid metal samples follows

Adv. Mater. 2022, 2109427

Pouillet’s law,[4,27,37] which describes the resistance of an ideal
homogeneous conductor with a uniform cross section as afunction of its resistivity and dimensions:L(1)A
where R is the sample’s resistance, L is the length, A is the
cross-sectional area of the sample, and ρ is the resistivity ofthe material. The resistivity ρ is the reciprocal of electrical conductivity σ, and for EGaIn σ = 3.4x106 S m−1 at 22 °C.[38,45,46] Ifchanges in length and cross-sectional area are denoted by ΔL
and ΔA respectively, the resulting relative resistance change ofthe samples becomes

R=ρ

R
1+2ν
= (1 + ε ) (2)R0Hence, a nonlinear relation between resistance response
and strain is predicted, where ν is the Poisson’s ratio. Incompressible materials, such as Sylgard 184 (commonly referred
to in the literature as simply PDMS), exhibit a Poisson’s ratioof 0.5. Most authors take the incompressibility assumption for
liquid metal materials, yielding the relation between resistancechange and engineering strain (defined as ΔL/L)
R
2= (1 + ε ) (3)R0where R0 is the unstretched resistance. Finally, we have arrived
at the form of Pouillet’s law that has seen widespread use in thestretchable electronics field and beyond.[4,27,37] Complete derivation of Equation (3) can be found in Supporting Information.
To test the validity of Pouillet’s law on our samples, we cansimply divide the resistance measurements by each sample’s
initial resistance (Figure 4c). The response of the EGaIn samples measured with 4PP and 2PP was nonlinear and gradually
increased with strain, with a higher change in resistance for the
4PP curve. On average, R/R0 increased to 1.83 when ε = 100%for 4PP method, and 1.62 for 2PP method. This is expected,
since the 4PP method has a smaller parasitic resistance. Conversely, the EGaIn samples measured with WB method exhibited an almost strain-insensitive resistance response, indicating a maximum R/R0 value of 1.17 at 100% strain. While allmeasurement methods reported the same absolute changes inresistance (Figure 4c), the relatively high parasitic resistance in
the WB circuit results in lower relative changes in resistance(Figure 4c).
Applying the previous corrections to the data (Figure 4d), we
again see a better agreement between the three measurement
methods. However, although the WB data overlaps with the corrected results for 4PP and 2PP methods, an underestimation isstill apparent after correction. In addition, a higher nonlinear
R/R0 response was observed for all three methods.
Although these results fit within the ranges published for
EGaIn samples (c.f. Figure 1), none of our recorded data—raw
or corrected—closely follows the trend predicted by Pouillet’slaw. This suggests that other factors (such as cross sectional
geometry, trace shape and regularity, and quantity of encapsulated liquid metal) may also affect the electromechanicalbehavior of EGaIn samples, resulting in significant deviations
from the predicted Pouillet’s law (Equation 3).

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Other authors have proposed several mechanisms for nonPouillet behavior. For example, Zhu et al.[4] reported different
responses of the relative change in resistance R/R0 for two
types of ultra stretchable fibers filled with EGaIn (as mentioned previously in the section “Electromechanical Responseof Liquid Metals”). The EGaIn core of both fibers had a triangular cross-section, but their nominal diameter was different.
The narrower fibers showed a similar change in resistance asthat predicted with Pouillet’s law. In contrast, the fibers witha larger diameter exhibited lower changes in resistance during
deformation, and tended to change their cross section fromtriangular to a circular shape as the fibers were stretched. The
authors generated theoretical curves for both cases; the triangular cross-section model matches Pouillet’s law, while the triangle-to-circle cross section model has a lower R/R0 response
at high strain levels. However, the equations used were notexplicitly stated, so we were unable to independently evaluate
the model.
In another work, Neuman et al.[47] used spray depositionof liquid metal onto silicone substrates and reported resistance values as a function of trace length and trace width.
They observed that thinner traces exhibited higher resistancevalues than thicker traces. However, by plotting the resistanceas a function of trace length divided by cross-sectional area,
the estimation of the EGaIn resistivity using Pouillet’s law(Equation (1)) was within 10% of the value reported in the literature, suggesting that Pouillet’s law is accurate at zero strain(unstretched). In the same work, different geometrical patternsof liquid metal samples (serpentine, spiral, etc.) exhibited different responses to strain. However, the authors did not provide
an analytical solution to predict response a priori, leaving manyquestions unanswered.
Seeking to explain recent non-Pouillet behavior observed
across several studies on LMEE, Zolfaghari et al. presented
an FEM-informed theory that could generate curves similar to
those found in the literature.[37] Zolfaghari suggested that the
tortuosity of the connected liquid metal bubbles dictated theelectromechanical properties of the final composite.[37] Specifically, FEM simulations showed that the electromechanicalproperties are influenced by LM droplet spacing and the shapeof the connected conductive pathway.
In a similar vein of thought, it has been suggested that specific morphologies and substrates may induce lower changes
in the electromechanical response of liquid metal samples.
Ma et al.[48] fabricated a liquid metal fiber mat that exhibited
an R/R0 of 1.4 after 100 cycles at ε = 1800%. The stretchability
and electrical stability were ascribed to the wrinkled structures
formed by the relaxation of prestretched substrates. Wrinkled
structures tended to dissipate the induced strain during large
deformations. Similarly, Park et al.[49] developed a deformablehydrogel with an R/R0 of 2.3 at a strain value of 1500%. In thiscase, the resistance response was ascribed to the hydrogenbond interaction between the naturally formed gallium oxide
in EGaIn and the hydroxyl groups of their substrate. They
also showed that the resistance of different loadings of EGaIn
responded differently under deformation, with the lowest
EGaIn loading showing a higher change in resistance.
In sum, our experimental results did not match the bulkconductor model (Pouillet’s law), yet we cannot definitively stateAdv. Mater. 2022, 2109427

which experimental procedures or fabrication steps have leadother authors to find bulk-conductor behavior on EGaIn traces.
Correcting for the parasitic resistance led to higher R/R0 values,
yet there were still significant gaps between our trendlines andPouillet’s law (Figure 4d). Several explanations have been proposed in the literature, including the importance of trace geometry and the tortuosity of the conductive paths through traces,in addition to the presence of nonuniform cross-sectional
shapes. However, to definitively provide a theoretical model toexplain the totality of reported literature, we still need additional
studies pairing theory with controlled experiments to isolate themarginal contribution of each of these proposed contributors to
the final system-level electromechanical behavior.

4. Measurements on bGaIn
Reflecting on what we have learned through this study, we raise
the question: does the introduction of solid particles, to makebGaIn, dramatically change the importance of measurement
method and interfacing on resistance measurements? As seenin Figure 1b, numerous previous studies on different bGaIn
formulations suggest that bGaIn is uniformly non-Pouillet. For
example, in our previous paper, we reported measurements
using a Wheatstone bridge, and found that our bGaIn was far
less resistive than the predictions from Pouillet’s law.[27] Here,
we use our bGaIn formulation as a case study on the influenceof solid particles on the conclusions derived above regarding
EGaIn (namely, that 4PP measurement technique reduces
error, and that there are still significant differences between
measurements and Pouillet’s law). While other biphasic formulations could, in principle, behave quite differently from ours,
this set of measurements should serve as a cautionary, representative tale and point of reference for other researchers.
We made five bGaIn specimens, finding remarkably similar
conclusions as we drew from the EGaIn samples. For example,
the bGaIn samples had an average initial resistance of 0.4 Ω
for bGaIn (including ≈0.1 Ω electrode resistance), which waslarger than EGaIn (0.25 Ω, including ≈0.1 Ω electrode resistance). The resistance of bGaIn samples as reported by 2PP ishigher (0.49 Ω), and WB is still higher (1.41 Ω, or 3.6 times
the 4PP value of 0.4), following a similar trend as the EGaIn
samples (Figure 5a,c). For 4PP and 2PP methods, a nonlinear
response is observed, with average R/R0 values at 100% strainof 1.72 and 1.61 respectively. In contrast, the WB method
yielded a relatively strain-insensitive response, with an average
R/R0 value of 1.24 at 100% strain. The electrode resistance wasalso 0.1 Ω for bGaIn samples, representing 30% of the averagevalue of the bGaIn samples. Using the same correction process used for EGaIn to correct the bGaIn data, we see reasonable overlap of the curves for 4PP and 2PP methods, withaverage initial resistances values of 0.3 and 0.29 Ω, respectively(Figure 5b,d).
These findings suggest that the introduction of solid particles
may change the electromechanical behavior, but biphasic mixtures do not eliminate the need to account for electrode resistance and use a suitable measurement technique (namely, a 4PP
method). Further, comparing Figures 4c and 5c, we note that
EGaIn and bGaIn initially appear equally “strain-insensitive.”

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www.advmat.deFigure 5. Electromechanical behavior of bGaIn. a) Resistance responses of the dogbone shaped samples under a tensile strain up to ε = 100% with thethree electrical resistance measurement methods and their corresponding b) corrected resistance data. c) Relative change in resistance R/R0 of bGaIn
under a tensile strain up to ε = 100% and d) corrected R/R0 data. The shaded areas represent one standard deviation for five samples.

However, after correcting the data to remove parasitic resistance caused by the different measurement techniques, we seethat bGaIn appears slightly more strain insensitive (comparing
Figures 4d and 5d).

5. Outlook
In summary, we have thoroughly examined the literature, andconducted experiments, to shed light on a central question:
are liquid metals bulk conductors? We found countervailingevidence, therefore failing to reject the null hypothesis. In ourinvestigation of Pouillet’s law, we found several factors that
may contribute to the electromechanical behavior of EGaIn,
but further study is needed to synthesize all of the hypotheses into a coherent theory. Additionally, in the course ofthis investigation, we uncovered several tips that should help
standardize the reporting of the electromechanical response of
stretchable circuits.
For example, we showed that when measuring low-resistance materials, the resistance of the test lead wires can introduce significant error if a 2PP or WB measurement method areused, suggesting the need for using a four-point probe method.
Furthermore, additional errors can be introduced if there is significant contact resistance at the interfaces between the sample
and its electrodes (often copper strips), and if the sample
design includes significant portions of liquid metal that are notstretched during sample elongation. The 4PP method could be
carried out using four independent copper wires connecting
to the traces (two on each side), eliminating the copper fromAdv. Mater. 2022, 2109427

the measurement. Independent measurements of these other“parasitic resistances” can be carried out and used to reduce—
but not fully eliminate—the measurement error. Despite thesesubtle experimental complexities, we emphasize that simply
using a four-point probe method will eliminate much experimental error and should generally be preferred over 2PP or
WB techniques.
Determining the relative effect of each influential factor on
the samples’ electromechanical response requires standardized
data and reproducible reporting. While we recommend using
the 4PP method and correcting for electrode resistance whenever possible, not all labs have access to a reliable 4PP measurement system. Hence, transparently reporting which method
was used as well as all (if any) corrections applied is essential.
Additionally, experiments with a perfect measurement system
will still have errors introduced during manufacturing, making
it important for researchers to report values obtained using
multiple samples and report a measure of variation, such as
standard deviation or confidence intervals. Such reporting willallow each result to be evaluated objectively, with appropriate
validations and reproductions carried out by other independent
researchers. Finally, we note the lack of standardized practices
to characterize stretchable electronics, despite the potential
sensitivity of measurements on numerous other experimental
procedures. For example, Park et al.[41] presented liquid metal
devices that exhibited change in electrical resistance underboth extension and compression. This result implies that theposition of the clamps used in the electromechanical characterization may also affect the resistance values of liquid
metal samples.

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Utilizing 4PP and standardized testing methods, we have a
few suggested experiments for future work investigating theelectromechanical behavior of liquid metal circuits. However,this list is by no means exhaustive, and we merely suggest
these experiments as examples. First, to investigate the effectof cross-sectional geometry, samples could be manufactured
with intentional variations in cross-section, such as traces withconstant circular cross-section versus traces with thinner andthicker sections, where injection methods may enable more
precise control over cross-section. Additionally, comparing thedifferent designs’ electromechanical response with their crosssections (rigorously quantified using, for example, micro computerized tomography imaging) could yield conclusions about
the effect of trace uniformity. Other experiments could test
whether the clamped end-regions effect the samples’ electromechanical response, for instance by varying the end-regions’volume. Larger volumes could flow into the main trace areaand prevent reductions in the cross-sectional area, thereby
reducing the relative resistance change. To measure the effectof Poisson’s ratio on the electromechanical response, samples
could be made in different substrates and observed opticallyduring stretch, yielding instantaneous Poisson’s ratios for the
stretching curve. While authors typically assume a constant
Poisson’s ratio, such as 0.5, we have found that silicones, such
as Dragon Skin, can have nonconstant Poisson’s ratio.[50] Aninteresting open problem complicating all of these proposed
studies is how to unify the theory for the electromechanical response of pure EGaIn traces, LMEEs, and bGaIn. For
example, should bGaIn be modeled more like EGaIn, or as a
tortuous LMEE-like conductive network? Does the introductionof solid particles induce sufficient oxide and microstructuring
to make bGaIn behave more like a tortuous path in an LMEE?
We hope these considerations can serve as a point of reference for future standardization efforts, thereby improving theinter-study comparability and furthering the theory and understanding of liquid metal circuits. By carrying out controlled
studies on the effects of different variables, such as sample
geometry, cross-sectional uniformity, and sample preparation
techniques, we hope that future advances can yield more reliable and predictable stretchable electronics for healthcare, soft
robotics, and wearable electronics.

by a water bath (Cole–Parmer Polystat) kept at 6 °C. The EGaIn NPs
suspension was then poured into a 50 mL falcon tube and 5 mL ofethanol were further added.
Thin films of EGaIn NPs were prepared by spray-casting the solutionof EGaIn NPs onto four silicon wafers using a customized 3D printer(Monoprice Maker Select v2). The solution was spray-printed at aconstant rate (0.3 mL min−1) using compressed air (20psi) blown overa 20-gauge stainless steel needle. The x, y positions were determined tocover the area of the four silicon wafers following a serpentine pattern,with the z height set so there was a 55 mm gap between the syringe
needle and the printer bed. The printing bed speed was set to 5 mm s−1.
The weight of the film after five coatings was ≈180 mg for each siliconwafer. The resulting films were thermally treated at 900 °C for 30 min.
bGaIn powder was obtained after the films were scraped from the siliconwafers with a razor blade, transferred to a vial, and shaken vigorously
using a vortex mixer. The bGaIn powder was then mixed with EGaIn at asolid concentration of 50 wt%.
Electromechanical Characterization: The specimens were fabricatedwith the dimensions presented in Figure 3. First, a film of Dragon Skin
10 (DS10) was draw coated onto an acrylic sheet using a custom madedrawbar with 500 μm gap height. After the film was cured, a stencil was
cut out of 25 μm PET film using a UV laser (LPKF Protolaser U4) andplaced onto the DS10 films. EGaIn (or bGaIn) was then spread acrossthe mask. Then, copper tape was affixed on each end of the conductive
traces using silver epoxy (Part 8331-14G, MGchemicals). Once the silver
epoxy cured, square pieces of plain-weave fabric were impregnated withsilicone (DS10), placed on each end of the conductive traces, and left tocure for 1 h at 60 °C. The specimens were then encapsulated by drawcoating another layer of silicone (DS10), and left to cure overnight at
60 °C. At this point, the film containing the samples could be lifted
and removed from the acrylic sheet. Then, the samples were cut into adogbone shape using a laser cutter (VLS 3.50, Universal Laser Systems
Inc.) at 90% intensity and at 11% speed.
For electromechanical testing, the specimens were subjected touniaxial tensile loading at 15 mm min−1 using a customized tensile
stage controlled using an Arduino Uno. The dynamic resistancemeasurements were acquired using three methods: 1) four point probe,2) two point probe, and 3) Wheatstone bridge. The resistance and strain
values were recorded using Matlab 2021.

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

Acknowledgements
6. Experimental Section
Sample Preparation: Gallium (Ga, 99.99%) and indium (In, 99.99%)
were purchased from Rotometals, USA. Ethanol (200 proof) was
obtained from VWR. Silicon wafers were obtained from UniversityWafer,
Inc (75 mm diameter, boron-doped, <100> orientation, polished oneside). Dragon Skin 10, parts A and B were purchased from Smooth-On.
Eutectic gallium-indium (EGaIn) alloy was prepared by mixing
75.5 wt% Ga and 24.5 wt% In and heating at 200 °C on a hot plate
overnight. Biphasic gallium-indium (bGaIn) was prepared using a threestep process (similar to the procedure proposed by Liu et al.[27]): (1)EGaIn nanoparticles (EGaIn NPs) suspension preparation, (2) Spraycasting of EGaIn NPs onto silicon wafers and (3) Thermal sintering
process. The EGaIn NPs suspension was prepared by weighing
1.8 g of EGaIn into a 5 dram vial (Kimble) using a 3mL syringe witha 20-gauge needle. Then, Ethanol (11 mL) was slowly added, followedby sonication using a probe sonicator (Qsonica, Q700) coupled with a
1/4 inch diameter microtip (part number 4420) at 30% amplitude for 2 h.
During the sonication process, the sample temperature was controlled
Adv. Mater. 2022, 2109427

This material is based upon work supported by the NSF under grant no.
IIS-1954591. D.S. was supported by a NASA Space Technology Research
Fellowship (80NSSC17K0164).

Conflict of Interest
The authors declare no conflict of interest.

Keywords
eutectic gallium-indium, liquid metals, soft robotics, stretchable
electronics

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Received: November 19, 2021
Revised: February 1, 2022
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Lina Sanchez Botero received her B.S. in Engineering Physics and M.S. in Applied Physics fromthe National University of Colombia. She completed her Ph.D. degree at Cornell University under
the supervision of Professor Juan Hinestroza, in 2018. She is currently a Postdoctoral Associate
in Mechanical Engineering and Materials Science at Yale University in Rebecca Kramer-Bottiglio’sLab. Her current research interests include advanced manufacturing and materials development,
soft and wearable technologies, and human-machine interaction.

Dylan Shah received his B.S. in mechanical engineering and M.S. in Agricultural Engineeringfrom Iowa State University, in 2015 and 2016, respectively. He completed a Ph.D. degree at Yale
University under the supervision of Professor Rebecca Kramer-Bottiglio in 2022 and was a NASASpace Technology Research Fellow in collaboration with Massimo Vespignani at NASA Ames
Research Center. His research contributes to the fields of soft robotics, reconfigurable robotics,
and stretchable electronics.

Rebecca Kramer-Bottiglio is an associate professor at Yale University. She leads the Faboratory,
which innovates in materials, manufacturing, and robotics to develop new multifunctional materials that will allow next-generation robots to adapt their morphology and behavior to changing
tasks and environments. Prior to joining Yale, she held a faculty position at Purdue University.
She received her B.S. from the Johns Hopkins University, M.S. from UC Berkeley, and Ph.D. from
Harvard University.

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