REVIEW
www.afm-journal.deLiquid Metal + x: A Review of Multiphase Composites
Containing Liquid Metal and Other (x) Fillers
Sophia Eristoﬀ, Amir Mohammadi Nasab, Xiaonan Huang,
and Rebecca Kramer-Bottiglio*
although there exists a Tm range depending
on processing and formulation[2,3] —liquid
metals boast low toxicity,[4] low viscosity,
exceptional electrical, and thermal conductivity, high deformability, self-healing
capability, near-zero vapor pressure, andvoltage-responsive interfacial energy.[5–7]
The interfacial energy of liquid metals is
dominated by surface gallium oxidation,[8]which forms a stable amorphous oxide
ﬁlm that has presented both processing
challenges and incredible shape stability.[9]Given such properties, it is unsurprisingthat liquid metals have emerged as a leading
candidate material for applications including stretchable conductors,[10] electrodes
and batteries,[11] sensors,[12–16] antennas,[17]transistors,[18,19]
memristors,[20]ensoftergy
harvesting
devices,[21–23]
actuators,[24] and integrated circuits.[25,26]
Incorporating liquid metal inclusions
into solid-phase materials generates liquid
metal composites, which have been shown
to impart many of the unique and advantageous properties of liquid metal tobulk solids. For example, Ford et al.[27]reported liquid metal inclusions in liquid crystal elastomer(LCE) actuators, which imparted conductivity without degrading the mechanical and shape-morphing properties of the LCE.
Markvicka et al.[28] proposed liquid metal droplets suspended
in soft elastomer, which, when coalesced via strain or pressure,formed stretchable elastomer conductors. Similarly, Thrasher
et al.[29] presented core–shell particles of liquid metal withsurface-bound acrylate ligands that polymerized together to create cross-linked particle networks, which when stretched, form
stretchable conductors with a suppressed electromechanical response. Bartlett et al.[30] showed the enhancement of thermal
conductivity in elastomers via liquid metal inclusions.
Recently, researchers have proposed increasingly complex liquid metal composites, often adding additional ﬁllers to either
amplify or multiply the functional properties. In this class ofcomposites, liquid metal can bridge between solid ﬁllers to reduce percolation thresholds,[31] forge inclusion alignments foranisotropy,[32] or mitigate the compliance mismatch between
rigid ﬁllers and soft polymers to reduce stress concentration
points that impact the extensibility of the matrix and longevity of
the composite.[33] The emergent beneﬁts of composites containing liquid and solid inclusions have summarily shown integrated

Multiphase mixtures containing both liquid metal and solid inclusions in asoft polymeric matrix can exhibit unique combinations of mechanical,
electrical, magnetic, and thermal properties. Gallium-based liquid metals haveexcellent electrical and thermal properties, and incorporating additional
conductive, magnetic, or other solid ﬁllers into liquid metal-embedded
elastomers can yield heightened electrical and thermal conductivities,
enhanced elasticity due to lowered percolation thresholds, and positive
piezoconductivity. This emerging class of liquid metal + x composites, wherex denotes any solid ﬁller type, has applications in stretchable electronics,
wearables, soft robotics, and energy harvesting and storage. In this review,the recent literature is consolidated on liquid metal + x composites and theirpotential to oﬀer uniquely ampliﬁed or multiplied bulk properties ishighlighted. The literature related to the materials and processing of liquid
metal + x composites is reviewed, through which it is found that theproperties of the resulting multiphase composites are sensitive to thesequence in which the distinct liquid metal and solid inclusions areincorporated into the continuous phase. This review further includes asummary of relevant predictive modeling approaches, as well as identiﬁes
grand challenges and opportunities to advance liquid metal + x composites.

1. Introduction
Gallium-based alloys that are liquid at room temperature, or liquid metals (LM), are a fascinating class of materials with highly
unique properties.[1] Most commonly, liquid metals refer to eutectic gallium indium (eGaIn; 75% Ga and 25% In by weight) or
Galinstan (68.5% Ga, 21.5% In, and 10% Sn by weight). Alongwith their sub-room temperature melting point—nominally
Tm = 15 °C and -19°C for eGaIn and Galinstan, respectively,
S. Eristoﬀ, A. M. Nasab, R. Kramer-Bottiglio
School of Engineering & Applied Science
Yale University
New Haven 06250, USA
E-mail: rebecca.kramer@yale.edu
X. Huang
Robotics Department
University of Michigan
Ann Arbor 48109, USA
The ORCID identiﬁcation number(s) for the author(s) of this article
can be found under https://doi.org/10.1002/adfm.202309529

DOI: 10.1002/adfm.202309529

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www.afm-journal.deties relative to liquid metal-only composites of the same liquid
metal loading percentage. In the following sections, we group
LM + x composites in the literature by their enhanced or added
mechanical, electrical, thermal, and magnetic properties. As a
versatile material with excellent electrical and thermal properties, adding liquid metal inclusions to existing composites has
been shown to enhance or enable electrical and/or thermal conductivity, often while also introducing favorable or maintained
mechanical and/or magnetic properties.

2.1. Mechanical Properties

Figure 1. Liquid metal + x composites, which are multiphase composites containing both liquid metal inclusions and other solid (x) ﬁllers. The
outer rings indicate which integrated properties are enhanced or enabledin the liquid metal + x composites based on their solid ﬁller. CNF and CNT
denote carbon nanoﬁbers and carbon nanotubes, respectively.

mechanical,[34–36] electrical,[37–39] magnetic,[40,41] thermal,[42,43]
self-healing,[44–47] rheological,[48] or tribological[49] properties insynergy with their liquid-solid-reinforced structures.[50]The purpose of this review is to highlight the immense potential for liquid metal (LM) + x multiphase composites, where“x” denotes any solid ﬁller (Figure 1). We note that the term
‘multiphase composite’ has been used in the literature to refer
to polymer composites containing two or more types of liquid
inclusions[51] or solid ﬁllers,[52] as well as inclusions of diﬀerent
sizes or shapes of the same material.[53] Here, we identify multiphase composites containing both liquid and solid inclusions
and further narrow our focus to those containing liquid metals.
This review complements other recent reviews that focus on composites containing liquid inclusions other than liquid metal,[54,55]
composites containing only liquid metal inclusions,[56–59] liquid metal particles,[60–63] and liquid metal ﬁbers.[64] We exclude
complex-phase capsules as composites on their own unless they
are embedded into host matrices. Thus, our aim is to report thecollective literature on “liquid metal + x” composites to elucidate
the potential to amplify or multiply bulk properties by adding liquid metal to existing ﬁller-matrix composite materials, as shownin Table 1. We further review the materials, processing, and modeling challenges associated with such multi-inclusion composites.

2. Integrated Properties
There are two ways to consider a LM + x multiphase composite: 1)
a solid ﬁller-matrix composite with added liquid metal, or 2) a liquid metal-matrix composite with added solid ﬁller. In Table 1, we
consider the literature based on the latter approach, thus grouping composites by x and listing their increased or added proper-

Adv. Funct. Mater. 2023, 2309529

The mechanical properties of polymers are generally impacted
by adding solid or liquid ﬁllers. For example, the addition ofsolid ﬁllers in a polymer network typically reduces the elasticity and embrittles the network, but may increase the strengthof the system.[33,98–100] The addition of liquid ﬁllers can increase the toughness of the network, as well as enhance energy absorption due to increased energy dissipation via the liquid
inclusions.[5,25,60,101] However, liquid inclusions at high concentrations in a polymer network may induce leakiness when subjected to pressure gradients.[68,87]
Together, both liquid and solid inclusions in a polymer network create complex interactions, and the speciﬁc composition
determines whether the liquid or solid inclusions are more dominant toward the bulk mechanical properties. Tutika et al.[68] studied the mechanical and functional tradeoﬀs in multiphase composites including both liquid and solid inclusions, yielding an
optimized composition of solid conductive particles and eGaIn
liquid metal in a silicone matrix with high compliance and thermal conductivity. In some cases, the unique interactions between
liquid and solid inclusions extend beyond mitigating the eﬀects
of one another and instead create unexpected outcomes. For example, in a work by Bury and Koh,[69] Galinstan and BTO particles were embedded in a polydimethylsiloxane (PDMS) matrix toyield a stretchable dielectric material, and increasing the Galinstan content was found to increase the composite’s modulus andviscosity—the opposite correlation from what is expected whentypically adding liquid ﬁller to a polymer.
Other examples of LM + x composites have demonstrated
enhanced or added mechanical properties such as increased
Young’s modulus,[68–70,76,78] shear hardening,[73] increased ultimate strain,[68] enhanced storage moduli,[69,71] and decreased
gauge factor.[87] For example, by mixing both Fe particles and liquid metal into a polymer matrix, the toughness and Young’s modulus of the bulk composite can be enhanced.[69,70,102] Incorporating carbon nanotubes (CNT),[73–75] carbon nanoﬁbers (CNF),[76]and graphene ﬂakes[78–80] with liquid metal into elastomers canalso enhance the toughness of the composite. In a work that
we ﬁnd particularly interesting, Saborio et al.[80] showed thatthe elastic modulus of a graphene-liquid metal composite initially increases as the loading content of graphene is raised to
1.4 wt.%, then subsequently decreases with further grapheneloading to 3 wt.%. Such non-monotonic dependency between
solid ﬁller content and elastic modulus is attributed to the ﬂow
of liquid metal into cavities, which reduces the friction between
the matrix and solid ﬁller, as depicted in Figure 2A (and further
investigated in other works[56,103] ). However, when the graphene2309529 (2 of 18)© 2023 Wiley-VCH GmbH

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Table 1. Liquid metal + x composites and their modiﬁed or multiplied properties relative to liquid metal-only composites of the same liquid metal loading
percentage.
x

Filler type

MatrixInvestigated properties

Reference

NdFeB

particles

Sylgard 184

Added magnetic properties

Zhang et al.[ 65]

NdFeB

particles

Metal-coatedparticles in liquid
metal

Increased thermal and electrical conductivity; added magnetic properties

Lu et al.[ 66]

NdFeB

particles

Sylgard 184

Added magnetic properties

Guan et al.[ 67]Feparticles

Gelest PDMS

Increased thermal conductivity, Young’s modulus, and ultimate strain

Tutika et al.[ 68]

Fe or BTO

particles

Gelest PDMS

Increased torsion storage modulus, Young’s modulus, and dielectric
permittivity

Bury and Koh[ 69]FeHO2particles

Sylgard 184

Added positive piezoconductivity; increased electrical conductivity and Young’s
modulus

Zhang et al.[ 70]

Carbonyl Fe

particles

Sylgard 184

Added positive piezoconductivity and magnetic properties; increased thermal
conductivity

Yun et al.[ 40]

Carbonyl Fe

particles

Sylgard 184

Increased thermal conductivity and storage modulus; added magnetic
properties

Li et al.[ 71]

Carbonyl Fe

particles

Sylgard 184

Added magnetic properties and metallic conductivity

Kim et al.[ 72]

Carbon (CNT)and Carbonyl
Fenanotubes
andparticles

PDMS-polyurethane
blend

Increased electrical conductivity; added electrical self-healing, magnetic
properties, and shear hardening

Ding et al.[ 73]

Carbon (CNT)

nanotubes

Sylgard 184

Increased electrical conductivity

Guan et al.[ 74]

Carbon (CNT)

nanotubes

Sylgard 184

Increased electrical conductivity

Li et al.[ 75]

Carbon (CNF)

nanoﬁbers

Sylgard 184

Increased Young’s modulus and tensile strength; added strain hardening andpositive piezoconductivity; decreased electrical conductivity

Zhang et al.[ 76]Graphite

ﬂakes

EcoFlex 00-50

Increased electrical and thermal conductivity

Bilodeau et al.[ 31]Graphite

ﬂakes

EcoFlex 00-30

Increased electrical and thermal conductivity

Li et al.[ 77]Graphene

nanoplatelets

Sylgard 184

Increased thermal conductivity and Young’s modulus

Sargolzaeiaval
et al.[ 78]Graphene

nanoplatelets

Sylgard 184

Increased thermal conductivity

Sargolzaeiaval
et al.[ 79]Graphene

ﬂakes

Sylgard 184

Increased electrical and thermal conductivity; non-monotonic Young’s modulus

Saborio et al.[ 80]

PolydopamineCoatedGraphene
Oxide

ﬂakes

Sylgard 184

Increased dielectric permittivity; decreased breakdown strength

Hu and Majidi[ 81]Agparticles

Silicone oil

Increased thermal conductivity

Uppal et al.[ 82]Ag

particles

Gelest PDMS

Increased thermal conductivity

Uppal et al.[ 83]Agparticles

Poly(styrene-bbutadiene-bstyrene)(SBS)

Increased electrical conductivity

Liu et al.[ 84]Agparticles

Gelest PDMS

Increased thermal conductivity and Young’s modulus

Tutika et al.[ 68]Agﬂakes andparticles

Ethylene vinyl
acetate (EVA)

Increased electrical conductivity

Wang et al.[ 85]

Ag

ﬂakes

Styrene-isoprene
block copolymer(SIS)

Increased electrical conductivity; non-smearing

Reis et al.[ 86]

Ag

ﬂakes

SIS

Increased electrical conductivity; decreased electromechanical gauge factor;
non-smearing

Lopes et al.[ 87]

Ag

ﬂakes

SIS

Increased electrical conductivity; non-smearing

Sgotti et al.[ 88]

Ag

ﬂakes

SIS

Increased electrical conductivity

Hajalilou et al.[ 89]Ag-coated Silica
Carbideparticles

Silicone oil

Increased thermal conductivity

Kong et al.[ 90]Ag-coated Fe2 O3particles

SIS

Increased electrical conductivity

Hajalilou et al.[ 89](Continued)

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Table 1. (Continued).
xFiller type

MatrixInvestigated properties

Reference

Ag-coated Ni

particles

SIS

Increased electrical conductivity

Hajalilou et al.[ 89]Ag-coated Ni

particles

EcoFlex 00-30

Increased electrical conductivity; added magnetic properties

Hoang et al.[ 91]Ni

particles

SIS

Increased electrical conductivity; decreased electromechanical gauge factor;
non-smearing

Hajalilou et al.[ 92]

Ni

ﬂakes

Gelest PDMS

Increased thermal conductivity and Young’s modulus

Tutika et al.[ 68]

Ni

ﬂakes

Carboxylated
polyurethane

Increased electrical conductivity; added electrical self-healing

Park et al.[ 93]

Ni

ﬂakes

EcoFlex 00-30

Added positive piezoconductivity; increased electrical conductivity and Young’s
modulus

Zhang et al.[ 70]Niparticles

Sylgard 184

Added positive piezoconductivity, magnetic properties; increased thermal
conductivity

Yun et al.[ 40]Nispike-shaped
particles

Sylgard 184, EcoFlex
00-30

Increased electrical conductivity; added positive piezoconductivity; increased
thermal conductivity

Yun et al.[ 94]Cuparticles

Thermoplastic
elastomer

Increased thermal and electrical conductivity; non-smearing

Chen et al.[ 95]Cuparticles

Polymer-coatedparticles in liquid
metal

Increased thermal conductivity

Zhang et al.[ 96]Cuparticles

EcoFlex 00-30

Increased thermal conductivity and Young’s modulus

Zhang et al.[ 97]Cuparticles

Gelest PDMS

Increased thermal conductivity and Young’s modulus

Tutika et al.[ 68]loading is increased beyond 3 wt.%, direct contact between ﬂakes
promotes van der Waals interactions, thus reverting the dependency between solid ﬁller content and elastic modulus back to a
positive correlation.
Another favorable mechanical property in LM + x composites
is non-smearing, which may be useful in scenarios where encapsulation of a material is not possible or favorable.[87] Frequently,
liquid metal composites leak as the liquid metal percolates out of
the network. However, the addition of solid metal ﬁller particles
can act as a strong anchor for liquid metal to prevent smearing
on the composite’s surfaces.[86–88] Liquid metals have been shown
to reactively wet or otherwise adhere to solid metals such as Ag,
Au, Cr, Cu, Fe, and Ni, and the inclusion of these ﬁller types incomposites can thus “hold” the liquid metal in place.[84,87,104–106]
For example, in Lopes et al.,[87] Ag nanoparticles were mixed
with eGaIn and styrene-isoprene block copolymer (SIS) to form
a conductive non-smearing composite. The addition of the silver
nanoparticles slightly enhances the electrical conductivity of thecomposite, but more favorably imparts a solid-like integrity seenin just solid-phase composites, which prevents smearing. Suchnon-smearing surfaces could lend toward applications in the actuator space, such as thermo-responsive actuators[88] or dielectric
elastomer actuators (DEAs)[86] (see Figure 2B), where smearingof conductive traces (as shown for the carbon black and carbon
grease traces in Figure 2B) can often lead to electrical shorting orother catastrophic failures.

2.2. Electrical Properties
Imbuing soft polymers with conductive ﬁllers is a common
approach to stretchable conductors, which are essential components for emerging wearable and soft robotic technologies.

Adv. Funct. Mater. 2023, 2309529

Either conductive solid or liquid inclusions could be used
to achieve electrical conductivity in a polymer. Solid ﬁllers
such as silver or copper are commonly used due to theirhigh electrical conductivity, but often require high loading toachieve percolation.[85] The percolation threshold, or the volume fraction of conductive ﬁller required to achieve bulk
conductivity, is sensitive to the size and geometry of theparticles.[36] Spherical particles contribute to a very high percolation threshold since all of the particulate mass is concentrated into isolated spheres of low aspect ratio. Particles
with an amorphous geometry typically enable a lower percolation threshold. Generally, high loading of solid ﬁller leads
to bulk embrittlement, which is often unfavorable for conductive polymers. Conductive liquid ﬁllers, such as ionic ﬂuids
and liquid metals, can impart electrical conductivity to theirpolymer hosts without embrittlement. Liquid inclusions typically require lower loadings to achieve percolation when compared to their solid counterparts, as they are able to ﬂow
and form conductive pathways.[23,29] However, liquid inclusions
are often leaky, which can result in unreliable conductivity
and mechanical properties with compliance degrading at highloadings.[68,86–88]
The combination of both conductive liquid and conductive solid inclusions can reduce percolation thresholds andmitigate negative properties often seen with single-phasecomposites.[40,68,85] While ionic liquids have been used in electrically conductive multiphase composites,[107] the majority ofmultiphase electrically conductive polymer composites are manufactured by incorporating liquid metal and small amountsof conductive rigid inclusions with high aspect ratios such as
silver,[84–89] nickel,[70,91–94] graphite,[31,77] graphene,[80] and carbon nanotubes.[73–75] The synergy between both liquid metal
and the conductive solid (x) ﬁllers yields a balanced blend that

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www.afm-journal.deFigure 2. Examples of liquid metal (LM) + x composites and their enhanced or enabled mechanical, electrical, magnetic, and thermal properties.
A) Schematic of eﬀects of compression on a LM + x composite. Under compression, the LM ﬂows into the cavities formed around the graphene(x), enhancing the mechanical and electrical properties of the composite. Reproduced with permission.[80] Copyright 2019, Wiley-VCH. B) Images ofelectrodes used for dielectric elastomer actuators made using a non-smearing LM + x composite, and smearing carbon grease and carbon black.
Reproduced with permission.[86] Copyright 2021, Wiley-VCH. C) Schematic of the positive piezoconductivity achieved under deformation when using a LM + x composite. Reproduced with permission.[70] Copyright 2021, IOP. D) Images of a LM + x composite and its self-healing properties.
Post-healing, the material exhibits electrical self-healing under strain due to the ﬂow of LM into the healed zone. Reproduced with permission.[93]Copyright 2018, Wiley-VCH. E) Images of an inchworm-inspired magneto-responsive LM + x composite locomoting in response to a magnetic
ﬁeld. The x-axis units are mm. Reproduced with permission.[67] Copyright 2023, Elsevier. F) Schematic of the eﬀects of adding magneto-responsiveﬁllers to a LM-PDMS composite. Cross-sectional optical microscopy images show the composite without and with a magnetic ﬁeld applied during curing on the top and bottom, respectively. Reproduced with permission.[71] Copyright 2023, Elsevier. G) Finite elemental analysis (FEA) images of a LM + x composite (bottom) and its heat ﬂux transport compared to just pure LM (top). Reproduced with permission.[96] Copyright 2023,
Wiley-VCH.

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co-optimizes the composite’s bulk mechanical and electrical properties.
In LM + x composites, the liquid metal often acts as the secondary contributor to electrical conductivity. Solid ﬁllers are typically unable to achieve percolation alone, except at very high loadings. However, at high strains, liquid metal can percolate into thegaps between the solid ﬁllers and bridge an electrical network.
For example, Wang et al.[85] described a superelastic conductor
with an initial conductivity of 8331 S cm-1 and a maximum strainof 1000%, achieved using both Ag ﬂakes and eGaIn particles ina polymer matrix. When subject to strain, the eGaIn particles
rupture, maintaining the electrical connections between the Ag
ﬂakes. The eGaIn particles act as electrical anchors for the Ag
ﬂakes, but due to their liquid state, preserve the stretchability of
the composite. In Hajalilou et al.,[89] a variety of LM + x composites were probed and optimized to combine the best propertiesof liquid metal composites with the best properties of solid ﬁller
composites. A mixture of eGaIn and Ag ﬂakes yielded the highest
conductivity with optimized mechanical properties, as compared
to composites containing eGaIn and x = {Fe, Ni, Ag-coated Ni, or
Ag-coated Fe}.
Electrical resistivity usually increases by stretching a conductive material, as interconnects between electrically conductiveﬁllers reduce in magnitude.[108–110] However, in certain LM + xcomposite cases, added positive piezoconductivity results in electrical resistivity decreasing as tensile strain increases, which isa fascinating and under-explored phenomenon.[40,70,94] In Yun
et al.,[40] the electrical resistivity of a Fe-eGaIn-PDMS composite drops by three orders of magnitude during strain, exhibiting
positive piezoconductivity. The unique piezoconductivity properties of the Fe-eGaIn-PDMS composite are attributed to the deformation of eGaIn particles, and lack thereof of Fe particles, during strain. Under strain, the Fe particles may compress against
the PDMS matrix and eGaIn, leading to an overall reduction in
the thickness of the PDMS layer and a potential increase in FeeGaIn contacts, which in turn decreases resistivity, as shown in
Figure 2C. This positive piezoconductivity is also apparent inZhang et al.[70] and Yun et al.,[94] where the unusual positive
piezoconductivity property is used for a variety of sensing applications.
When tuning the electrical conductivity of a polymer composite, the relative permittivity (dielectric constant) can also be enhanced by increasing the volume fraction of liquid metal inclusions, regardless of the size of the inclusions.[22,111] However, enhancing the dielectric constant to desirable values often requires
high liquid metal loading (>50 vol.%), which can result in a drastic loss of material integrity and sometimes percolation.[69,81,112]
The combination of both solid and liquid metal ﬁllers can result in optimized dielectric permittivity with decreased breakdown strength, while still possessing rubber-like compliance andelasticity. In Hu et al.,[81] eGaIn and polydopamine (PDA)-coatedgraphene oxide ﬂakes were used in combination in a PDMS matrix to yield a composite with a high dielectric constant (≈10–57),
a low dissipation factor (≈0.01), and rubber-like compliance andelasticity. Bury and Koh[69] also found that by combining bothGalinstan and Fe rigid ﬁllers in a PDMS matrix, permittivity, andloss behavior could be enhanced while maintaining mechanicalintegrity. Furthermore, the ratio of Fe and Galinstan could be optimized for a speciﬁc sensing application, where both mechani-Adv. Funct. Mater. 2023, 2309529

cal and electrical properties could be precisely adjusted to yield
highly stretchable and responsive dielectric materials for use in
stretchable electronics and soft robotics.
Self-healing technologies have been extensively demonstrated
for the recovery of mechanically damaged parts,[44,47] but can also
be applied to maintaining electrical conductivity.[73,93,113,114] Electrical self-healing enables the healing and recovery of electricalconductivity in a composite structure after damage. Prior works
have shown that liquid metal encapsulated in solid shells can beincorporated into a matrix, and when damaged, will rupture andﬂow to restore the conductive pathways.[113,114] The combinationof both solid and liquid metal inclusions can enable reversible
electrical self-healing. For example, in Park et al.,[93] Ni ﬂakes
and eGaIn were added to a carboxylated polyurethane (CPU)matrix. CPU polymer is engineered to self-heal, and with justeGaIn, can restore electrical conductivity post-damage. However,
the eGaIn would eventually leak out of the composite over time,
preventing reversible self-healing. The addition of the Ni ﬂakes
enables reversible self-healing, as the Ni acts as an anchor forthe eGaIn, promoting optimal adhesion between the eGaIn and
CPU to prevent leaking over time (Figure 2D). In Ding et al.,[73]CNTs and gallium liquid metal were directly mixed into a PDMSpolyurethane (PU) blend. The PDMS-PU blend possessed natural self-healing properties, and when coupled with the synergistic
electrical properties of both the CNTs and liquid metal, could recover full electrical conductivity when damaged.

2.3. Magnetic Properties
Magnetorheological elastomers (MREs) are magneto-responsive
due to the inclusion of magnetic particles within the elastomer matrix.[115–118] Typically, MREs are synthesized by dispersing solid magneto-responsive ﬁllers, such as ferromagnetic
Fe,[119–121] Ni,[122–124] or NdFeB[125–127] particles, in a polymer matrix prior to curing in the presence or absence of a magnetic ﬁeld.
Based on the curing conditions, loading concentration, and typeof magnetic particle used, MREs may possess anisotropic physical properties, which can enable ﬁeld-dependant mechanical or
physical properties.[115,118,128,129] However, due to the solid natureof most magnetic ﬁllers, MREs are often very stiﬀ and tough, withlittle compliance or stretchability.[130,131]
Integration of liquid inclusions into MREs can yield composites with not only retained magnetic properties but also enhanced or introduced mechanical and thermal functionalities.
Most magneto-responsive LM + x composites make use of either carbonyl Fe[40,71–73] or NdFeb[65–67] particles due to their optimal ferromagnetic properties. Ni particles have been used in
MREs[91] but at a lower frequency because the magnetic permeability of Ni is relatively low compared to Fe and NdFeB.[122]In one example, Zhang et al.[65] used a NdFeB-liquid metalPDMS composite to create a miniature, untethered, magnetic
robot with dual-energy transmission mode capabilities, which locomotes in response to a low-frequency magnetic ﬁeld waveform.
When subjected to a fast-changing magnetic ﬁeld, the robot isable to generate enough current to power an LED light. In another example, Guan et al.[67] used the same materials (NdFeBeGaIn-PDMS) to synthesize a biomimetic soft inchworm withmagneto-responsivity (Figure 2E). The inclusion of both rigid

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NdFeB with liquid eGaIn yielded enhanced mechanical properties often seen in liquid metal composites, such as higher ﬂexibility and cross-linking density, decreased swelling, and enhanced
tear resistance.[67]A unique property that arises from liquid metal + ferromagnetic particle composites is that the electrical conductivity can betuned by controlling the direction and strength of a magnetic ﬁeld
generated by an electromagnet.[40,71,72] The magnetic particles in
a LM + x composite tend to align in the direction of the magnetic
ﬁeld, which reduces the spacing between the conductive particles. As a result, when the material is subjected to a magnetic
ﬁeld perpendicular to the resistance measurement direction, theelectrical resistance decreases. By monitoring the change in resistance, MREs with liquid metal can be used as sensors to measuremagnetic ﬂux density and pressures.[40]2.4. Thermal Properties
Eﬃcient thermal management is essential for electronic devices,
and in the case of wearable computing devices and soft robots,
stretchability and deformability are demanded as well. Dispersing high contents of rigid inclusions can enhance thermal conductivity in polymer composite materials but also results in reduced stretchability.[132–134] Liquid metals exhibit excellent thermal conductivity (relative to other liquid alternatives, such as water, ionic liquids, and hydrocarbons[135] ), in addition to superior
electrical properties, and can be dispersed into polymer composites to enhance thermal conductivity while maintaining mechanical compliance.[30,136] For example, Bartlett et al.[30] reported
a liquid metal elastomer composite with unprecedented metallike thermal conductivity in a highly stretchable (𝜖 >600%) state.
However, composites containing only liquid metal inclusions athigh loading concentrations exhibit low compliance and leakiness, thus preventing reliability.
If solid inclusions with high thermal conductivities such as
Fe,[40,68,71] graphite,[31,77] graphene,[78–80] Ag,[68,82,83,90] Ni,[40,68] orCu[68,95,97] are also added to the elastomer, in addition to liquid metal, the thermal conductivity of the multiphase composite will increase relative to the liquid metal-only composite at thesame loading percentage. However, the maximum possible volume fraction of the inclusions decreases when rigid inclusions
are involved, due to the stiﬀness of the rigid inclusions, which
prevents mixing and uniform formation of a multiphase composite. Therefore, the addition of solid ﬁllers can also reduce thehighest achievable thermal conductivity.[68,137]
Thermal interface materials (TIMs), which are used for thethermal management of high heat ﬂux devices, are required topossess high thermal conductivity with low Young’s modulus.
However, these two properties typically oppose one another.[97,138]
Zhang et al.[97] proposed a LM + x composite mixture, wherein aco-doped TIM consisting of liquid metal and Cu in a silicone matrix yielded a thermal conductivity of 3.94 WmK-1 and an elastic
modulus of 699 kPa. The performance of this speciﬁc TIM wasbetter in comparison to that of the single-phase TIMs, due to thesynergistic eﬀects of the highly thermally conductive Cu and theliquid metal ﬁllers, which have very little impact on the mechanical properties of the matrix material. In Sargolzaeiaval et al.,[78]eGaIn and graphene nanoplatelets (GNPs) in a PDMS matrix

Adv. Funct. Mater. 2023, 2309529

worked together to increase the thermal conductivity of the composite. The aﬃnity between the eGaIn and the GNPs createdmore paths for phonon transport without increasing the density of the barriers. The thermal conductivity correspondingly increased, as conductive paths formed not only through the GNPs
and eGaIn droplets but also through the eGaIn islands that aggregated on the surface of the GNPs. Furthermore, the low loadingof rigid GNP ﬁller (<2.2 wt.%) into the composite made it possible to possess high thermal conductivities without substantially
impacting the material’s ﬂexibility.
Since liquid metals are often denser than the polymer matrices
they are embedded in, they tend to settle during curing and prevent uniform heat transfer throughout the bulk composite. However, in some instances, the incorporation of both liquid metal
and magneto-responsive particles can aid in introducing multifunctionalities and enhancing the thermal conductivity of a material, promoting a more uniform distribution of heat. For example, in Li et al.,[71] PDMS and carbonyl Fe particles were mixed
with a liquid metal to form a multiphase composite. The thermal conductivity of just the liquid metal-PDMS composite was
0.371 WmK-1 , but increased to 0.440 WmK-1 with the addition of
the carbonyl Fe particles. The thermal conductivity is enhancedby the addition of carbonyl Fe because the particles form distributed chains in the PDMS matrix when subject to a magnetic
ﬁeld while curing, as shown in Figure 2F, increasing more uniform distribution of thermally conductive particles throughout
the matrix. In Yun et al.,[40] irregularly shaped Ni particles withsurface spikes were integrated into a liquid metal-PDMS matrix.
Due to the irregular surface area of the Ni particles, more contacts were formed between the particles and liquid metal, enhancing the thermal conductivity of the composite. Furthermore,because the Ni particles exhibit ferromagnetic properties, the enhanced thermal conductivity could be coupled with the magnetoactive properties to achieve an intelligent magneto-responsive
heating pad.
We note that in the vast majority of LM + x composites, a polymer typically serves as the matrix material encasing both the liquid and solid ﬁller. Here, we highlight some inverted instancesthat continue to meet our deﬁnition of LM + x composites, where
the liquid metal serves as the matrix material and hosts multimaterial ﬁllers. For example, in Lu et al.,[66] NdFeB particles arecoated with Ag before being incorporated into an eGaIn matrix.
Due to the corrosive behavior of gallium-based materials, the Ag
acted as a protective barrier to preserve the NdFeB and its inherent magnetic properties. Furthermore, by using a composite
material, rather than pure eGaIn, dynamic leakage-free properties were achieved via enhanced viscosity, while retaining excellent heat-storage performance, good thermal stability, low thermal resistance, and high thermal conductivity. In Zhang et al.,[96]polyphenol-coated Cu particles were dispersed in an eGaIn matrix to yield eﬀective thermal management components for CPU
chips, as demonstrated in Figure 2G. Since eGaIn has excellent
thermal dissipation properties but low viscosity, the polyphenolcoated Cu particles enhanced the ﬂow properties to prevent surface spreading. Furthermore, the polyphenol-based supramolecular networks served as an eﬀective barrier between the Cu andeGaIn, reducing the alloying interactions and preventing the formation of intermetallic compounds, eﬀectively enhancing thestability of the composite.

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www.afm-journal.de3. Materials and Processing
In this section, we review a variety of synthesis techniques for
LM + x composites. For completeness, we focus not only on thesynthesis of existing LM + x composites, but also on processing
methods using liquid metals (absent of x) to comment on future
opportunities for the fabrication of LM + x composites.

3.1. Liquid Metal Emulsions
In most cases, the fabrication process of a LM + x composite begins with the preparation of an emulsion, where the uncured
polymer matrix is the continuous phase and the liquid metalis the dispersed phase. Thereafter, other ﬁllers can be incorporated prior to polymerization (curing) of the continuous phaseto obtain a solid structure containing the encapsulated liquid
metal and solid components. Several emulsion processes exist toachieve the mixture of two or more immiscible or partially miscible liquid phases.[139,140] Mechanical shear is utilized in the emulsiﬁcation process to break the large particles of the dispersed
phase into smaller polydispersed particles distributed throughout the continuous phase. The shear force required for breaking up the dispersed phase can be provided by manual stirring,
automated shear homogenization, sonication, or application of
acoustic waves.[141,142] Phase separation is another emulsiﬁcation
method for creating liquid inclusions in a mixture of two partiallyor entirely miscible ﬂuids. In the phase separation method, themiscibility in a saturated mixture of ﬂuids can be reduced by applying an external stimulus to form droplets in the mixture.[143]In the emulsiﬁcation process, there is a critical capillary num𝜏, below which the droplet dispersion is suppressed
ber, Ca = 2𝜎∕dwith surface tension, where 𝜏, 𝜎, and d are shear stress, surface energy, and droplet diameter, respectively. Shear stress obtained from the mechanical agitation causes spherical to ellipsoidal shape deformation in the droplets of the distributed phase,breaking them up into smaller particles, while surface tension
opposes this shape deformation.[144] The emulsiﬁcation process
also depends on the viscosity of the ﬂuids; the viscosity ratio ofthe dispersed phase to the continuous phase controls the abilityof a droplet to split in the continuous phase. At a ﬁxed capillary
number, droplet breakups happen if the viscosity ratio is within a
speciﬁc range of low and high bounds.[145] In other words, higher
capillary numbers are required for mixtures with very low orvery high viscosity ratios, which corresponds to higher applied
shear stress.
Using a third component as an emulsifying agent is sometimes necessary or useful to facilitate the emulsiﬁcation process and stability of the emulsion. There are diﬀerent stabilizing
mechanisms for emulsions depending on the system, however,two methods are most commonly used: 1) decreasing interfacial
energy, or using surfactants as an emulsifying agent to reducethe interfacial tension generated between the dispersed phaseand the continuous phase; and 2) mechanical protection (Pickering emulsion; discussed further in Section 3.3), or incorporating solid particles into the emulsion to be adsorbed and trappedat the interfaces of the dispersed and continuous phases, creating a mechanically strong ﬁlm to prevent droplet coalescence.
When using a surfactant as an emulsiﬁer, the phase in which

Adv. Funct. Mater. 2023, 2309529

the surfactant is more soluble becomes the continuous phase(Brancroft’s rule).[146] Therefore, if the surfactant is not carefully
chosen, the emulsion could invert and the intended dispersed
phase could emerge as the continuous phase. Surfactants to aid
in the emulsiﬁcation of liquid metals have been well studied, as
liquid metals possess a notably high surface tension and are thusdiﬃcult to incorporate into diﬀerent matrix materials. For example, Handschuch-Wang et al.[147] investigated the eﬀects of a variety of surfactants, such as Tween-80 and sodium dodecyl sulfate, on gallium-based liquid metals, with the goal of preventing
droplet coalescence. Diﬀerent surfactants were found to successfully adsorb onto the surface of the liquid metal droplets basedon the chemical environments, preventing liquid metal dropletsfrom agglomerating. Sometimes, dispersants can also be added
to aid in the formation of a stable, yet emulsiﬁer or surfactantfree, emulsion. For example, both Park et al.[93] and Wang et al.[85]utilized acrylic acid as a dispersant to evenly distribute ﬁller ﬂakes
and eGaIn in a polymer network.

3.2. Sequence of Material Incorporation
The literature on multiphase composites containing distinct solidand liquid metal inclusions indicates that the sequence in which
the diﬀerent inclusions are incorporated into the continuous
phase plays a vital role in the formation of inclusions and the bulk
properties of the resulting composite.[137] For example, Bilodeau
et al.[31] developed a composite using silicone elastomer as thecontinuous phase, and liquid metal and solid expanded intercalated graphite (EIG) ﬂakes as the dispersed phases. The liquid
metal and silicone elastomer were mixed ﬁrst, thus breaking up
the liquid inclusions into appropriate sizes via shear mixing. Subsequently, EIG suspended in a carrier solvent of cyclohexane wasadded to the mixture. However, if the EIG had been added to thesilicone elastomer ﬁrst, the viscosity of the mixture would have
dropped dramatically due to the incorporation of the solvent, preventing the appropriate breakup of the liquid metal droplets into
microscale inclusions.
In some works, the solid ﬁller is introduced into the external matrix prior to liquid metal incorporation.[75,86–88] In Lopes
et al.,[87] styrene-isoprene block copolymer (SIS) dissolved intoluene was ﬁrst mixed with Ag ﬂakes to disperse the rigid ﬁllers.
Once evenly dispersed, the eGaIn liquid metal was thoroughly
mixed into the composite to form a biphasic elastomer ink forstretchable electronics. Interestingly, when the eGaIn was incorporated into the SIS-Ag mixture, the composite yielded not only
eGaIn droplets with a gallium oxide shell and Ag ﬂakes in an
SIS matrix, but also AgIn2 microparticles on the surface of theoxide-coated eGaIn droplets. In another work, Yun et al.[40] placed
PDMS, Fe powder, and eGaIn into a container in the sequence
stated and shear mixed them to fabricate a composite with distinct dispersed inclusions of Fe particulates and LM inclusions in
the bulk PDMS matrix, as shown in Figure 3A. In Zhang et al.[70]eGaIn and solid ﬁllers consisting of either iron or nickel were directly mixed into pre-cure silicones in a one-pot emulsion process
to form a uniform stretchable conductive composite, suggestingthat simultaneous mixing may achieve similar outcomes to premixing the solid ﬁller and matrix before adding liquid metal insome cases.

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www.afm-journal.deFigure 3. Materials and processing methods for LM + x composites. A) Schematic and scanning electron microscopy (SEM) images with energy dispersive X-ray spectroscopy (EDS) elemental mappings of a LM + x composite. Reproduced with permission.[40] Copyright 2019, Springer Nature. B)
Synthesis process of LM nanodroplets using sonication techniques, which are then collected and incorporated into an elastomeric matrix with additional ﬁller. SEM images with EDS elemental mappings are shown to exhibit the dispersion of the ﬁllers in the elastomer matrix. Reproduced withpermission.[80] Copyright 2019, Wiley-VCH. C) SEM images of LM encapsulated in a polymer shell using an in situ polymerization process. When embedded in an epoxy matrix and fractured, the cross-sectional SEM image shows the location of the damaged area and liquid metal release (denoted
as Ga–In). Additional micro-CT data, with a schematic superimposed, shows the LM-encapsulated microcapsules and their release of liquid metal into
the crack plane of a healed specimen. Reproduced with permission.[113] Copyright 2012, Wiley-VCH. D) Images of an emulsion of LM and a monomer
pre- and post-sonication, with a schematic showing the resultant Pickering emulsion formed post-sonication, where organophilic particles stabilize
the surface of eGaIn particles. Reproduced with permission.[148] Copyright 2019, American Chemical Society. E) Schematic of the synthesis processof graphene-coated liquid metal droplets. Optical microscopy and SEM images of the particles show the graphene is evenly dispersed throughout thesurface of the eGaIn droplets. Reproduced with permission.[149] Copyright 2018, Wiley-VCH.

Finally, some works demonstrate that mixing liquid metal withsolid ﬁllers (such as metallic particles or ﬂakes) prior to addition
to the matrix can aid in the formation of a more uniform composite material with enhanced properties. For example, in Li et al.,[77]graphite ﬂakes and liquid metal were mixed thoroughly prior toaddition into a silicone rubber matrix to yield liquid metal-coatedgraphite ﬂakes. Both the liquid metal-coated graphite ﬂakes andindividual liquid metal droplets dispersed throughout the matrix work synergistically to form a thermally conductive pathway,
which is not present in composites of the same composition but
without the pre-mixing step. Uppal et al.[82] showed that whenpre-mixing Ag ﬂakes with liquid metal prior to incorporating insilicone oil, which is much less viscous than that of the pre-curesilicone used in Li et al.,[77] thermal properties were not enhanced

Adv. Funct. Mater. 2023, 2309529

because the ﬁller and liquid metal clumped more frequently. Furthermore, they showed that as the viscosity of the silicone oil increased, the clumping between the solid ﬁller and liquid metal
decreased and dispersion increased, which agrees with Li et al.[77]Tutika et al.[68] also showed that if solid metal particles (such as
Fe or Cu) are mixed with eGaIn prior to shear mixing with silicone elastomer, the liquid metal encapsulates most of the solid
particles. This encapsulation of solid particles is likely due tothe formation of a gallium-oxide coating around them, which
increases the aﬃnity between the particles and eGaIn. Related,
Kanetkar et al.[150] found that pre-mixing solid particles with liquid metal led to air pocket formation, which could negatively impact the bulk composite’s thermal conductivity or positively impact density.[151]2309529 (9 of 18)© 2023 Wiley-VCH GmbH

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www.afm-journal.de

3.3. Liquid Metal Core or Shell Capsules
While liquid metal and other ﬁllers are often embedded into soft
matrices via simpler mixing and emulsion processes, these ﬁllers
can also be introduced by pre-synthesizing capsules. By fabricating liquid metal capsules prior to addition into a matrix, one may
prevent the coalescence of liquid metal droplets in the composite,enable a higher degree of control over the size and morphologyof the droplets, and potentially amplify desirable properties.
Gallium-based liquid metal emulsiﬁed in a solvent may itself be considered to form core-shell capsules, due to the inherent formation of stabilizing gallium oxide at the surface of each
droplet.[26,152–159] Saborio et al.[80] synthesized liquid metal capsules by sonicating a mixture of bulk eGaIn in chloroform tobreak up the liquid metal into droplets, with gallium oxide enclosing the eGaIn particles. Thereafter, the eGaIn capsules andgraphene were dispersed in a silicone matrix to yield ﬂexible
and stretchable electrically conductive composites (Figure 3B).
For a higher degree of control over the size and morphologyof the liquid metal suspensions, prior works have also employed stabilizers.[156,157,160] In Lear et al.,[156] thiols were added
to a mixture of eGaIn and ethanol, with the thiolated emulsion
yielding smaller and more monodisperse liquid metal capsules.
Finally, microﬂuidic techniques can also be used to generatemonodisperse droplets of liquid metal.[161,162] Lin et al.[163] utilized a capillary-based microﬂuidic approach to synthesize liquid metal microspheres with excellent electrical conductivity andphotothermal properties. By modulating the continuous medium
in which the liquid metal capsules were synthesized, one canchange the size of the synthesized liquid metal spheres. For an
extensive review of the intersection between microﬂuidic techniques and liquid metals, we refer the reader to Zhu et al.[13]Other existing processes that can be used to synthesize capsules include interfacial polymerization,[164–171] in situ
polymerization,[113,114,172–176] and Pickering emulsions.[177–184] Interfacial polymerization and in situ polymerization techniques
are both frequently used to synthesize single core-shell architectures because these methods typically optimize the ratio of coreliquid to shell material to maximize the encapsulation of liquid,and some reactions involve a hybrid of both techniques.[185] Ina relevant example of in situ polymerization, Blaiszik et al.[113]encapsulated eGaIn in a polymeric shell using an in-situ ureaformaldehyde micro-encapsulation technique. The in situ process yielded ellipsoidal capsules with major-axis diameters ranging from 3–200 μm. Thereafter, the liquid metal capsules wereincorporated into an epoxy matrix, and upon mechanical damage to the cured matrix, the liquid metal capsules would fracture, enabling the ﬂow of the liquid metal (Figure 3C). While
formaldehyde-based in situ polymerization is frequently employed in synthesizing microcapsules, we note that safety concerns associated with formaldehyde,[186] especially for wearable and biological applications, have motivated the synthesisof melamine-urea microcapsules via interfacial polymerization
without the use of formaldehyde.[187]In a work employing Pickering emulsions, liquid metal particles were utilized as the Pickering stabilizer, where eGaIn particles self-organized on a porous surface during phase separation
to form conductive pathways when mechanically sintered.[188] Alternatively, the liquid metal can be stabilized by Pickering sta-Adv. Funct. Mater. 2023, 2309529

bilizers, such as organophilic particles, which can enable stableliquid metal emulsiﬁcation in a hydrogel matrix,[148] as shown in
Figure 3D. Finally, while not strictly a Pickering emulsion, Chenet al.[149] coated liquid metal droplets by dropping them into abeaker of graphene sheets and rolling them in the beaker until the droplets were suﬃciently coated (Figure 3E), which represents a system of smaller functional particles coating the surfaceof liquid metal droplets.
In the context of LM + x composites, some works synthesize capsules with solid ‘x’ ﬁller cores that are coated with liquid metal shells before mixing into a matrix.[91,95] For example, in He et al.,[189] polypropylene foam particles were coatedwith liquid metal by stirring both materials together. The process yielded foam particles with a liquid metal coating of approximately 50 μm. In Hoang et al.,[91] Ni particles were ﬁrst coatedwith eGaIn by mixing both components. Thereafter, the eGaIncoated Ni particles were dispersed into a silicone matrix. Sometimes, to enhance the wettability between the liquid metal andthe solid ﬁller material, the ﬁller can be surface-functionalized.
In Chen et al.,[95] Cu particles were ﬁrst surface-functionalizedwith silanes prior to coating with eGaIn to promote wetting anduniform coating. In other cases, the rigid ﬁllers can be coatedwith a polymer prior to being mixed into a liquid metal matrix.
For example, in Zhang et al.,[96] Cu particles were coated with tannic acid, which is a polyphenol polymer, prior to mixing into an
eGaIn matrix. The polyphenol polymer coating prevented alloying interactions between the eGaIn and Cu particles and delayed
the formation of intermetallic compounds, which proved favorable for thermal applications.
3.4. Inclusion Settling and Composite Homogeneity
Settling of the inclusions may impact the homogeneity of all
LM + x composite formulations. Liquid metals typically possessa high density, on the order of seven times that of the elastomers
they are frequently encapsulated in [190]. Because of the large
diﬀerence in densities between the liquid metal and the matrix,
settling of the dense liquid metals during curing often occurs,
leading to non-uniform samples. The settling phenomena foundin liquid metal composites can often have detrimental eﬀects,
causing a lowered dielectric constant along with lowered electrical and thermal conductivity. Several works have aimed to mitigate the segregation and settling of liquid metals in elastomer
composites.[190,191] One approach is to mix low-density materials,
such as hollow glass microspheres, with the liquid metal prior to
its addition into an elastomer matrix.[191] Mixing up to 50 vol.%of the low-density microspheres into the liquid metal can reducethe density to almost half of the neat liquid metal, which prevents settling during curing. In another approach, Xue et al.[190]
mixed pre-synthesized PDMS beads with liquid metal and precure PDMS to prevent settling and segregation. The PDMS beads
and liquid metal inclusions work synergistically to jam and prevent the inclusions from settling.

4. Theory and ModelingIn this section, we review relevant literature that contributes
to the predictive modeling of LM + x composites. As LM + x

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www.afm-journal.demultiphase composites contain both liquid and solid inclusions,
we begin by discussing modeling approaches for composites containing each distinct ﬁller phase before reviewing combinatorial approaches.

4.1. Composites with Solid Inclusions
There are several theoretical models to estimate the eﬀect ofsolid inclusions on the mechanical properties of composites.
Mechanical properties of the inclusions and their volume fraction are the governing parameters in most of these models,
however, the geometry of the ﬁllers and their packing fraction have also been considered in some advanced models. The
simplest way to predict the mechanical properties of a composite such as elastic modulus, density, Poisson’s ratio, andstrength is through series and parallel models based on the ruleof mixtures. For elastic modulus, the series and parallel models consider that the ﬁllers and the matrix are under uniform
tension and deformation, which is a simpliﬁcation. In continuum mechanics, Eshelby’s theory[192] of solid composites expresses the responses of ellipsoidal inclusions in composites
to applied stresses, and the stiﬀness of solid composites withlow concentrations of inclusions can be predicted. Eshelby’s
theory has also been extended to higher concentrations of inclusions when the strain ﬁelds of neighboring inclusions affect each other.[193,194] On the other hand, the Halpin-Tsai empirical model equations have been used extensively to predictthe properties of composites reinforced with short ﬁbers considering the aspect ratio of the ﬁllers. The Halpin-Tsai model
reduces to the parallel and series models when the empirical parameter deﬁned in it becomes very large or very small,respectively. The Halpin-Tsai equations take a more intricate
form when the ﬁbers are randomly oriented in the composite
matrix.[195]4.2. Composites with Liquid Metal Inclusions
For composites with liquid inclusions, as predicted by Eshelby’s
classic theory, the stiﬀness of a composite should decrease byincorporating liquid inclusions, as the elastic modulus of liquid inclusions is zero. This theory has been shown to be effective for composites containing inclusions with radius R >𝛾/E, where 𝛾 is the surface tension and E is the elastic modulus of the matrix.[196] Eshelby’s theory further predicts that theelastic modulus of the composite Ec with volume fraction ϕ
E. However, this conventionalof liquid inclusions is Ec = 1+5𝜙∕3composite theory does not account for surface tension at theliquid–solid interface. L ≡ 𝛾/E is the elastocapillary length below which the surface tension at the liquid–solid interface cansigniﬁcantly oppose the deformation of the bulk composite, andthe conventional composite theories fail to predict the elastic
moduli of the composites with liquid inclusions at that scale(Figure 4A).[197] The reinforcing eﬀect of the liquid–solid interface has been shown for single liquid-ﬁlled microcapsules[35]and also dispersed liquid inclusions in elastomers.[198–203] Styleet al.[196] postulated that to include the surface tension eﬀect inthe Eshelby equation for predicting composite elastic modulus

Adv. Funct. Mater. 2023, 2309529

Ec , the liquid inclusions can be considered as soft elastic inclusions:E

1+ 2 iEc = E 2 5𝜙 E 3 E( 3 − 3 ) Ei + (1 + 5𝜙
)
3

(1)

where Ei is the eﬀective elastic modulus obtained from the properties of the dispersed and continuous phases as follows:
Ei = E

𝛾
24 ER

(2)

𝛾
10 + 9 ER

This equivalent elastic modulus can also be replaced inother established composite theories such as the Mori-Tanaka
method[194] to account for surface tension forces in structureswith higher contents of liquid inclusions since the Eshelby theory
just covers dilute composites.
Other force interactions at the liquid–solid interfaces in composites, such as cross-linking at the interface of silicone-ionicliquid[196] or oxide skin at the interface of eGaIn drops encapsulated in elastomer,[198] can also increase the eﬀective elastic modulus of the soft inclusions beyond the limit predicted by Equation (2). Bartlett et al.[198] reported that by dispersing eGaIn in
silicone elastomer, the value obtained from Equation (2) was Ei =129 kPa, but they found agreement between their experimental
data and Equation (1) when Ei = 320 kPa (Figure 4B). The difference in the elastic modulus of inclusions could be attributed
to the presence of stiﬀ oxide skin around the liquid metal inclusions, as studied by Lear et al.[156] and Buckner et al.,[202] and also
the interaction between adjacent inclusions, which was not predicted in Eshelby’s theory.

4.3. Composites with Liquid Metal + x Inclusions
To model the mechanical properties of composites containing
both solid and liquid inclusions, Mohammadi Nasab et al.[205]proposed that one can compute the composite property considering just one of the inclusions in the continuous phaseﬁrst, and then use that as the property of the continuous phasewhen considering the second ﬁller. Tutika et al.[68] proposed a
related approach to predict the thermal conductivity in multiphase composites using a variation of the Cheng-Vachon[206] andBruggeman[207] models, wherein the LM suspension in an elastomer is considered to be the continuous phase and Fe particles
are the discontinuous phase. Thus, the authors predict the multiphase composite’s thermal conductivity using a two-phase model(Figure 4C). Phan-Thien et al.[208] took the approach of a “diﬀerential scheme,” where the multiphase composite is sequentiallybuilt up through a series of incremental additions until the ﬁnal
volume fractions are reached. During each sequential modelingstep, the intermediate composite properties are used as the continuous phase and the incrementally added compounds as thedispersed phase, thus achieving a discretized implementation of
the two-phase model. In general, such sequential, embedded, or
discretized two-phase models oﬀer enough simplicity and explicitness to make them an attractive approach for the thermal andmechanical modeling of LM + x composites.

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www.afm-journal.deFigure 4. Predictive modeling of LM + x composites. A) Images of ionic-liquid droplets with diﬀerent radii in a soft matrix with E = 1.7 kPa, underdiﬀerent uniaxial stress in the x direction. Smaller droplets show less deformation under the same applied strain. Reproduced with permission.[196]Copyright 2015, Springer Nature. B) Tensile modulus versus volume fraction (ϕ) of liquid metal particles in a silicone elastomer. Diﬀerent regimesof inclusion stiﬀness are found using Equation (1), and the dashed line is the prediction from Equation (1) for liquid metal inclusions. Reproducedwith permission.[198] Copyright 2016, Wiley-VCH. C) Thermal conductivity as a function of Fe mass fraction in a LM + Fe composite, in comparison
to Bruggeman and Cheng-Vachon model predictions. The loading of LM is 30 vol.%. Reproduced with permission.[68] Copyright 2018, Wiley-VCH. D)
Eﬀective elastic modulus predicted by two-level double inclusion (DI) model for diﬀerent volume fractions and shell thickness to diameter ratio (t/D)
of LM inclusions in an elastomer. Comparisons to Eshelby’s model are shown using the properties of gallium-oxide and neat LM. Reproduced withpermission.[204] Copyright 2021, Elsevier.

Other approaches in the literature include a double inclusion
model capable of predicting the properties of polymer composites with core-shell liquid metal droplets, proposed by Chiew and
Malakooti.[204] In this double inclusion model (Figure 4D), theelasticity of the LM inclusions is actively tuned according to therelative shell thickness of gallium oxide surrounding each inclusion, where it is known that as the size of droplets decreases theoxide shell becomes more mechanically dominant.[156,202] Buryand Koh[69] employed a microcapacitor model in their work to
study the electrical behavior of multiphase composites. Therein,
the composite is modeled as a network of randomly distributed
microcapacitors in the host matrix. Finally, Chen et al.[95] modeled the thermal properties of multiphase composites using an
augmented Agari’s model,[209] which outputs a composite’s thermal conductivity as a function of the ﬁller properties. To conclude
this section, we note that theoretical approaches to predicting the
behavior of multiphase composites appear limited in the literature, thus positioning the derivation of models for composites
containing distinct solid and liquid inclusions as an opportunity
for future advancement.

5. Conclusion and Outlook
In this review, we consolidated the literature on multiphase composites containing liquid metal inclusions and other solid (x)Adv. Funct. Mater. 2023, 2309529

ﬁllers. We view this sub-class of liquid metal-inclusive multiphase composites as increasingly important. Liquid metals have
proven to be exciting, intriguing, and useful materials, as evidenced by the number of papers published annually using “liquid metal” as a keyword doubling over the last 10 years (Clarivate, Web of Science). With this explosion of ﬁndings on liquid
metal properties and potential applications, it is expected that wemight seek to improve or otherwise expand the capabilities ofother materials and composites by simply adding liquid metal.
This is how we arrived at our characterization of multiphase composites containing liquid metal as ‘liquid metal + x,’ which weview as analogous to the common approach of wrapping artiﬁcial intelligence (AI) around arbitrary models or simulations inassociated areas (e.g., x = healthcare, medicine, biology, cybersecurity, ﬁnance, business, manufacturing, and transportation) toyield previously undiscovered solutions (AI + x). Similarly, here
we propose that liquid metal can be integrated into many composite materials with functional ﬁllers (e.g., x = Ag, Cu, Fe, Ni,graphite, graphene, etc.) to yield previously untapped, unparalleled, or multiplied functional properties.
We identify two grand challenges that should be addressed toenable future progress in LM + x composites, pertaining to 1)
fabrication and 2) modeling. As discussed herein, the fabricationof LM + x composites, and multiphase composites overall, canbe either straightforward or highly complex, depending on the2309529 (12 of 18)© 2023 Wiley-VCH GmbH

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www.afm-journal.deFigure 5. Applications of LM + x composites. A) A stretchable and printable superelastic conductor based on a LM + silver composite. Images show
the composite in relaxed and stretched (up to 800%) states. Reproduced with permission.[85] Copyright 2018, Wiley-VCH. B) Schematic and imagesof a liquid metal coil for wireless charging via inductive coupling. Images show the ﬂexibility and charging capabilities of the coil. Reproduced withpermission.[212] Copyright 2019, Royal Society of Chemistry. C) Schematic and images of remotely assembled microwires using an external magnetic
ﬁeld. Gallium-coated iron particles align in response to a magnetic ﬁeld, eﬀectively electrically coupling the two copper tapes together. Reproduced withpermission.[72] Copyright 2022, American Chemical Society. D) Images of an untethered miniature magnetic soft robot locomoting in a tissue-covered
chamber. The robot can move to a target region and heat the area, performing localized ablation, in response to a magnetic ﬁeld. Reproduced withpermission.[65] Copyright 2022, Wiley-VCH.

speciﬁc material compatibilities. The literature indicates that successful sample fabrication is sensitive to the sequence in which
components are incorporated. Correspondingly, we found works
in which i) the liquid metal and matrix should be ﬁrst mixed before adding a solid ﬁller, ii) the solid ﬁller and matrix should beﬁrst mixed before adding the liquid metal inclusions, and iii) theliquid metal and solid ﬁllers should be ﬁrst mixed before incorporation into the matrix material. Furthermore, while the properties of LM are mostly constant, the properties of x and the matrix
may vary, making synthesis processes sensitive to speciﬁc material compatibilities and thus diﬃcult to generalize. The study
and discovery of guidelines to synthesize LM + x composites will
aid the community in the identiﬁcation of appropriate fabrication
processes for increasingly complex and multifunctional composites.
We also found in our literature review that the derivation ofaccessible predictive models for LM + x composites represents
a substantial challenge for the ﬁeld, and the few models that exist tend to be highly complex and bespoke. This is unsurprisingand coupled to the variability in fabrication approaches discussed
above. For example, Mohammadi Nasab et al.[205] proposed a sequential approach—that one could model a single-inclusion composite, and the resulting properties could be input as the contin-

Adv. Funct. Mater. 2023, 2309529

uous phase in the model again with a new dispersed phase—
which demands a certain processing order. It is unclear if such asequential modeling approach would apply to LM + x composites
fabricated using simultaneous mixing of all components or otherwise inverted sequencing. Deriving the underpinning mechanistic principles of multiphase composites will enable generalized,
predictive models of bulk thermal and mechanical material properties.
Overall, LM + x composites are becoming increasingly common in the literature, and their potential impact spans broad
applications including stretchable electronics, wearables, soft
robotics, thermal management, and energy harvesting and storage. Many technological advancements toward these applications can still be attained through fabrication and modeling breakthroughs. For example, although highly stretchable
conductors have already been achieved by ﬁlling elastomers
with conductive LM + x inclusions, secure, and eﬀective integration of rigid microelectronics with multilayer ﬂexible circuitry remains a challenge. Such integration challenges could
be solved via precise control of the sizes, shapes, and distribution patterns of the LM + x inclusions.[210] Maturationand generalization of the manufacturing processes used for
LM + x composites will enable translation to commercialization.

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Exciting products have already been proposed, including ultrastretchable conductive interconnects[85,107] (Figure 5A), wearable
wireless power transmitters[211,212] (Figure 5B), remotely assembled circuits[72] (Figure 5C), and sensors and actuators for soft
robots[65,86] (Figure 5D), and we foresee the continued proliferation of LM + x composites due to their uniquely enhanced, enabled, and multiplied properties.

Acknowledgements
S.E. and A.M.N. contributed equally to this work. S.E. was supported by aNASA Space Technology Graduate Research Fellowship under Grant no.
80NSSC21K1269. A.M.N. was supported by the National Science Foundation (NSF) under Grant no. EFMA-1830870. X.H. and R.K.B. were supported by the NSF under Grant no. IIS-1956027.

Conﬂict of Interest
The authors declare no conﬂict of interest.

Keywords
functional composites, gallium, liquid inclusions, multiphase inclusions,solid inclusions
Received: August 11, 2023
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Sophia Eristoﬀ is a graduate student in the department of Mechanical Engineering and Materials Science at Yale University and a NASA space technology graduate research fellow. She earned her B.S.
in materials science and engineering and biomedical engineering at Carnegie Mellon University. Her
interests primarily lie in creating soft materials that possess unique stimuli-responsive properties
to enable advantageous functionalities. Additional interests include polymer synthesis, composites
manufacturing, and potential applications in biotechnology.

Amir Mohammadi Nasab received his B.S. in aerospace engineering from Amirkabir University ofTechnology and his M.S. in mechanical engineering from Iran University of Science and Technology.
He completed his Ph.D. degree in mechanical engineering at the University of Nevada-Reno under thesupervision of Professor Wanliang Shan. Subsequently, he held a postdoctoral position in mechanical engineering and material science at Yale University in Professor Rebecca Kramer-Bottiglio’s lab.
His research contributed to the ﬁelds of soft robotics, adhesion mechanics, thermodynamics, andmultifunctional composites.

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Xiaonan Huang is an Assistant Professor in the Robotics Department at the University of Michigan,
Ann Arbor, specializing in autonomous dynamic soft robots for challenging environments. He earned
his Ph.D. and M.S. in Mechanical Engineering from Carnegie Mellon University in 2020 and 2016, anda B.S. in Mechanical Engineering and B.A. in Japanese from Harbin Institute of Technology in 2014 and2013. His notable publications include works in Science Robotics, Nature Communication, Nature
Materials, Nature Electronics, PNAS, Advanced Materials Technologies, Soft Robotics, and Robotics
and Automation Letters.

Rebecca Kramer-Bottiglio is the John J. Lee Associate Professor of Mechanical Engineering and Materials Science at Yale University. Focusing on the intersection of materials, manufacturing, and robotics,
her group is deriving new multifunctional materials that will allow next-generation robots to adapt
their properties, morphology, and control policy to changing tasks and environments. She is a recipient of the NSF Career award, NASA Early Career award, AFOSR Young Investigator award, ONR
Young Investigator award, and Presidential Early Career Award for Scientists and Engineers (PECASE).
She was named a National Academy of Engineering (NAE) Gilbreth Lecturer in 2022 and a National
Academy of Science (NAS) Kavli Fellow in 2023.

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