Special Issue: AFOSR

Effects of particle size and oxide shell on
variable stiffness performance ofphase-changing materials

Journal of Composite Materials
2023, Vol. 57(4) 619–631© The Author(s) 2022
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DOI: 10.1177/00219983221129252
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Trevor L Buckner1 , Zachary J Farrell2, Amir M Nasab1 and Rebecca Kramer-Bottiglio1

Abstract
Phase-changing materials have recently found use as particulate ﬁllers to engender stiffness-changing behavior in composites. In the transition from solid to liquid particles, the composite modulus can be signiﬁcantly reduced, and then raised
again as the particles are allowed to solidify. While the general effect of stiffness tuning in phase-changing composites has
been demonstrated in a range of studies and applications, the most drastic stiffness ranges are provided by fusible metallic
alloys, which are commonly subject to the formation of oxide shells on their surfaces. In this work, we examine the effectthese oxide surfaces have on variable-stiffness performance, and investigate the corresponding role of particle size. In
particular, we study particles of a low-melting-point metallic alloy, Field’s Metal, and we also present a facile method of
manufacture that allows for control of particle size. We then manufacture a set of stiffness-changing composites using theseparticles and characterize mechanical performance, with the aim of controlling particle size to increase the total stiffness
ratio while resisting material failure.
Keywords
Multifunctional composites, phase-change, soft robotics, variable-stiffness

Introduction
Phase-changing materials provide a unique set of properties
that are made possible primarily via the reversible transition
from solid to liquid (or liquid to gas) and back. Typicallyderived from specialized metallic alloys, waxes, or thermoplastic polymers, solid-to-liquid phase-changing materials
are often used in manufacturing processes where internal
shape must be temporarily held and then easily extracted via
melting. For example, a pattern for investment casting mustremain solid during creation of a mold, but later is melted out
to leave behind the cavity for the product material.1,2More recently, a range of different phase-changing materials have become key tools for the development of mechanisms that utilize variable-stiffness or shape-change3–8
particularly in the ﬁeld of soft robotics, where on-demand
activation and suppression of rigid support structures is highly
sought after.9 In recent years, this strategy has brought about aseries of advancements toward the development of phasechanging composites – materials that have gained some addedfunctionality via the inclusion of an additive with phasechanging properties. In particular, soft materials such aselastomers, when modiﬁed via wax or fusible alloy microparticle ﬁllers, beneﬁt from substantial increases in rigidity inroom-temperature environments, yet retain access to the
ﬂexibility inherent in the matrix material via reversible heating
and melting of the internal phase-changing additive.10–12
While the general effect of stiffness tuning in phasechanging composites has been demonstrated in a range ofstudies and applications, the most drastic stiffness ranges areprovided by fusible metallic alloys, which are commonly
subject to the formation of oxide shells on their surfaces.
This oxide layer, often orders of magnitude more rigid thanthe underlying material, has a pronounced, detrimental
effect on the stiffness-changing performance of a given
composite. Motivated by this observation, we note that
controlling for particle size can offset this issue and as such1

School of Engineering and Applied Science, Yale University, New Haven,
CT, USA2
US Air Force Research Laboratory, Dayton, OH, USA
Corresponding author:
Rebecca Kramer-Bottiglio, School of Engineering and Applied Science,
Yale University, New Haven, CT 06520, USA.
Email: rebecca.kramer@yale.edu.
Correction (February 2023): Article updated with equation 8 and
Figure 5 since its original publication.

620
is a key design parameter for maximizing variable-stiffness
performance, although this brings with it other tradeoffs inmaterial properties.
Theoretical efforts applying Mori-Tanaka and Eschelby
equivalent inclusion theory can aid in the selection ofcomposite ﬁller dimensions to determine or design themechanical properties of standard composite materials withisotropic particles.13–15 However, few theoretical studies
have been able to capture the variation in mechanicalproperties of composites that result from phase-changing
particles,16 particularly in combination with a persistentoxide shell. Further, most previous studies of this nature
have considered a low packing fraction of particles, f < 0:2,
whereas recent variable-stiffness composites using fusible
metallic alloys10,12,17–19 often utilize higher concentrations,
0:3 ≤ f ≤ 0:5, where there are strong interactions between
particles. This study aims to ﬁll these gaps in knowledge via
the use of other modern core-shell modelling techniques,backed by experimental measurements.
In this work, we examine composites made from particles of Field’s metal (FM), a metallic alloy known for its
exceptionally low melting point (∼62°C) and nonhazardous composition. Generally, a variable-modulus
composite should be strong and fracture resistant, in addition to maximizing its modulus range. Therefore, ourstudy characterizes the impacts of particle size and oxide
formation on each of these dependent properties within theunique requirement space of variable-stiffness composites.
We developed a manufacturing method that allows for tightcontrol of particle size with high sphericity, in addition to afacile method of modifying the surface structure of FM
particles to improve mechanical bonding.

Materials and methods
Particle manufacture
Field’s metal particles are manufactured via a process improved
from Buckner et al19 by modifying the thixotropic (shearthinning) mixing techniques from Yup et al20 and Eristoff etal.21 A 1000-mL beaker is ﬁlled with 200 mL of a thixotropic
aqueous mixture, in this case a solution of deionized water anda confectionery sodium alginate powder (Modernist Pantry)with a loading of 2 wt%. This mixture nominally has a viscosity of ∼2.6 pa∙s (similar to room-temperature honey), butthins to ∼0.2 pa∙s (similar to maple syrup) when subjected to ashear rate of 1000/s.21 The mixture is then warmed to approximately 70°C by placing the beaker inside of a heatedsilicone oil bath (95°C). Ingots of Field’s metal (Roto144 F,
RotoMetals, Inc.) with a total weight of 60 g are placed inside
the heated medium and allowed to melt over the course ofabout 10 min. Once fully melted, the Field’s metal is broken
apart using a homogenizer (Cole-Parmer LabGEN 850) with a
20 mm diameter saw-tooth generator attachment for 60 s,

Journal of Composite Materials 57(4)during which time the thixotropic mixture ﬂows freely. Afterhomogenization is stopped, the medium vitriﬁes in placealmost immediately, capturing the Field’s metal droplets insuspension where they are allowed to cool and solidify overtime with minimal aggregation or settling. The mixture canbe diluted in a large container by stirring in additional water,
at which point the FM particles are able to settle and collect atthe bottom of the container. The supernatant can be decantedoff, and the FM particles are extracted and rinsed, and allowed to fully dry in a vacuum chamber overnight(Figure 1(a)). Multiple diluting-decanting steps may be required for smaller particles, which will not fully settle until amajority of the sodium alginate mixture has been removed.
Because the particles become fully solid during the coolingstep, they can be handled directly without the need of any
additional stabilizers or encapsulants that might be needed for
liquid-core droplets.22 Further, in contrast to Yup et al,20 thismethod does not require any additional solvent or enzyme torelease the particles from the medium as it does not form apolymerized gel. Indeed, with sodium alginate powder andwater as the only required components, this process also
carries the beneﬁt of being inexpensive, simple to execute,and eco-friendly with sustainable by-products.

Particle size control
FM particle size is controlled by adjusting the homogenizer
mixing speed. At higher mixing velocities, the FM is broken
into smaller droplets (Figure 1(b)). Compared to previouslyreported methods,19 this technique tends to generate a morepredictable and less polydisperse spread of particle sizes(Figure 1(c)), which in turn results in a higher yield as large
unusable particles are not generated. Further, the highcounts of nanoscale shards broken off from fractured particles found in previous works19 do not appear to be present
after microscope inspection (Figure 1(d)), resulting in moreconsistent and predictable material behaviors.

Particle surface texturing
As the vitriﬁed thixotropic mixture inhibits aggregation andsettling of molten FM droplets, it also dampens the residual
shear from the homogenizing process and negates much of
the deformation that would occur due to gravity. Suspended
in this medium, the absence of external forces allows thesurface tension to dominate and pull the droplets into
spheres, while the gradual cooling of the mixture allows theparticles to solidify approximately uniformly. All of thiscontributes to particles that are not only highly spherical, butalso possess a very smooth surface.20 The surface texture
can be roughened as an additional processing step to improve bonding between the particles and the host matrix
material. After the particles have been collected and dried,they are heated to approximately 80°C, thereby melting theBuckner et al.

621

Figure 1. Manufacture and characterization of FM particles. (a) FM particle manufacturing process: (1) Sodium alginate is added into
deionized water and (2) mixed until homogenous. (3) FM ingots are added and heated. (4) Molten metal is homogenized at the desired
mixing rates. (5) Particles cool and solidify. (6) Alginate is rinsed from FM particles. (7) Particles are dried in a vacuum oven. (b) Meanparticle diameter dependent on homogenization speed. (c) Particle size distribution (by count, blue; by volume, red). (d) ScanningElectron Microscopy (SEM) images for each size, taken at the same scale.

FM particle cores while the oxide shell remains intact. At
this point, the particles are immediately submerged in an ice
bath, which causes the particles to shrivel slightly, resulting
in a wrinkled surface texture.

Results
Effects of particle size
Matrix-inclusion bonding. The quality of bonding between thehost matrix and the particle inclusions is key to achieving
several desirable properties in a variable-stiffness composite. Generally, secure bonding corresponds to higher
material strength,15 and poor bonding has been shown tocause early fracture in previous examples of FM composites.23 While there exist many methods to achieve highbonding in Field’s metal or other core-shell particles, including chemical, heat, and mechanical treatments,24 theyoften disrupt the eutectic balance of the alloy or increaseoxide growth which inhibits the stiffness change performance. As such, this section remains strictly focused onmechanical bonding strength due to particle size, which will
be applicable to variable stiffness ﬁllers regardless ofmaterial.

In an earlier work, we noted that a debonding event canbe detected by monitoring the ﬂexural modulus of thecomposite as it is stretched. As the rigid particles embedded
within an elastomeric matrix begin to detach from the matrix
material, they no longer contribute to the material stiffness,
which will manifest as a sudden drop in elastic modulus at a
certain level of strain.10 We used this observable drop in
modulus as a measure of the onset of debonding, by which
we qualitatively compared the bonding effectiveness of FM
ﬁllers dependent on particle diameter. The test was performed by axially stretching composite specimens of eachparticle inclusion size (5 of each) at a loading of 30 vol%,using Dragon Skin 10 Medium (Smooth-On) as the elastomeric matrix material. Specimens were cut out from a ﬁlm
of cured FM-Dragon Skin, resulting in rectangular coupons(100 mm × 8 mm × 0.6 mm). Fabric reinforcement tabs
(20 mm × 8 mm) were adhered to the front and back at bothends of each specimen, providing an inextensible region for
secure gripping in the testing clamps, and resulting in astretchable region of 60 mm × 8 mm. Specimens were
loaded at rate of 120 mm/min. We mark the debonding eventfor each specimen as the point of tightest curvature at the“elbow” of the resulting stress-strain curves (Figure 2(A)).
The results show that particles of smaller diameter are able
to maintain the bond at increased levels of stress and strain

622

Journal of Composite Materials 57(4)Figure 2. Particle size versus particle-matrix bonding strength. (a) Dragon Skin-FM specimens strained beyond the ﬁller debonding
point, marked with a circle. Includes neat Dragon Skin, at full scale as well as scaled to 70% stress to approximate 30% void content. (b)Mean debonding stress as a function of particle diameter, and expected debonding stress from model given a particle/matrix surface
adhesion energy of 1.6 J/m2. (c) Cyclic strain data for a specimen with 30 vol% of 100 μm particles, compared with FEA simulation data. (d)Simulated cube (30 vol%, 105 μm diameter particles).

(Figure 2(B)), which we attribute to the larger total surface
area involved at the material interfaces that consequently
increases the total energy absorbed per volume before
debonding occurs.25 These results indicate that in the absence of chemical binders or adhesives, the use of smaller
particles can contribute to a tougher variable-stiffness
composite.
Strain cycling the material several times to 20% strain, asafe level below the debonding point, results in a largely
repeatable stress-strain curve. However, once the debonding
event has occurred, the damage appears to remain in subsequent strain cycles (Figure 2(C)). Re-heating the specimen to melt the particles is not sufﬁcient to restore thematrix-ﬁller bonding, likely due to the presence of the thinoxide shell on the particle surface. Instead, it is necessary toheat and then apply a large amount of strain to disrupt theoxide shell and allow the melted material to reﬂow andreform the mechanical bond onto the microtexture of thematrix.10

We conﬁrmed that the modulus change after the “elbow”
point is indeed due to debonding by modeling the behaviorof bonded and non-bonded ﬁller materials with FiniteElement Analysis (FEA, SolidWorks 2020). In thesetests, we simulated a set of soft cubes ﬁlled with 105 μm
solid particles at 30 vol% and subjected them to small
displacements (Figure 2(D)). If the surfaces are fully
bonded, the measured stresses indicate a high elastic
modulus in agreement with our experimental data
(Figure 2(C)). If the simulated particles are not bonded, theeffective modulus closely matches experimental data for
specimens that have been manually stretched beyond thedebonding point before testing. Further, we note that theunbonded simulation has very similar results to an additional simulation wherein all particles are replaced by voids.
That is, the addition of ﬁller particle material that fails tobond to the matrix will behave similarly to voids and in-fact
reduce the composite modulus to below that of the neatmatrix material. Further, if bonding is poor, reducing

Buckner et al.

623

particle size will also increase crack propagation since
existing cracks do not have to deviate as far from particle toparticle to continue spreading.26 The Dragon Skin 10 matrix
material was modeled using the Ogden model parameters:
α1=1.62, α2=5.18, α3=-13.42, μ1=80,031.09 Pa,
μ2=1049.90 Pa, μ3=32.89 Pa,27 and the Field’s Metal
particles were modeled as a linear elastic material withparameters: Poisson’s ratio=0.33, elastic modulus=9.2
GPa.19Our experimental results also closely match analyticaldebonding models. For an FM composite with a known
particle size, the total energy absorbed up until the debonding point is measured as the area under the stressstrain curve, γtotal . Given a FM volume loading of f ¼ 0:3,
we know that some portion of the absorbed energy is due tostretching the Dragon Skin in the remaining 70% of thevolume, γsilicone . This is measured as the area under the
stress-strain curve of a neat Dragon Skin 10 specimen to
the same strain point, multiplied by 1  f. Note in
Figure 2(A) that scaling the stress of the neat specimen to
70% causes it to follow the trajectory of the other curves.
The remaining absorbed energy, γdebond , is due to particlematrix interfaces. Therefore, given the particle size andloading volume, we can calculate the total particle-matrix
interfacial surface area Ainterf ace to determine the per-areaadhesion energy γsurf ace.
γdebond ¼ γtotal  ð1  fÞγneatγsurf ace ¼ γdebond Ainterf ace(1)

As expected, γsurf ace is fairly consistent across particle
sizes, ranging from 1.30–1.88 J=m2 . Using the debonding
equation provided by Lauke25 and Nicholson28 which assumes the particles are rigid compared to the matrix material, we can see that a surface energy of 1.6 J=m2 provides
predicted debonding stress points that match our experimental elbow points very closely (Figure 2(B))
rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1
16γsurf ace Em ð1 þ vm Þ(2)
σ debond ¼Þ
3ð1  vmdwhere σ debond is the predicted debonding stress, vm and Em
are the Poisson’s ratio (0.5, estimate) and Young’s modulus(186 kPa10) of the matrix material, and d is the average
particle size. The 48 μm particle size does not quite follow
the curve, but we attribute this to the wider particle polydispersity of those samples (Figure 1(C)) and small inconsistencies in the manual manufacturing process.
We noted in Section 2.3 that the highly smooth surface of
the FM particles (Figure 3(A)) may have a negative impact
on particle-matrix bonding, although this can be improved
further by treating the particle surface so as to modify its
texture, creating additional physical features on the surface
to provide robust mechanical connection. We achieved

surface roughening by heating the particles, and thensubmerging them in an ice bath to induce wrinkling of theoxide shell as the particle shrivels (Figure 3(B)). After thisprocess, the FM particles lose their highly-reﬂective shiny
surface and appear a dull gray to the naked eye.
Using these roughened particles of the 11 μm size, we
repeated the strain experiment above and compared it to theuntreated particles. Figure 3(C) shows that the debonding
onset strain is indeed extended by approximately 16% andthe debonding stress is increased by 11%, corresponding toa predicted adhesion energy of 2.08 J=m2 (Equation (2)),
which is a 30% increase compared to smooth particles. At
the same time, the composite elastic modulus appears to beunaffected. Further modiﬁcation of the surface structure toinclude deeper crevices and folds can likely improve thisbond, and may be achievable by a similarly facile
procedure.
Composite Uniformity and Particle Settling. Depending on thecure time of the matrix material, ﬁller particles may settle
at the bottom of the composite. This naturally leads tononhomogeneous and unpredictable material properties29
due to particle agglomeration or the emergence bilayerinteractions,30 and so is generally undesirable. The densityof FM results in obvious settling when the particles arelarge (>100 μm diameter). In this work, we observed thecross-section of Dragon Skin 10 with 30 vol% FM ofdifferent particle sizes. While the 100 μm particles do settle
rapidly to the bottom of the composite (Figure 4(A)),
specimens using smaller sizes begin to appear welldistributed (Figures 4(B)-(D)). We also note that handmixing of composites was signiﬁcantly easier withdecreasing particle size, which may have aided in the even
distribution.
We further note that at 50 vol% loading, a number ofvoids begin to appear throughout the matrix19 which can be
detrimental to the strength of a variable-stiffness composite.
The presence of voids with increasing ﬁller content is aknown phenomenon.31 Following similar reports,32,33 wesuspect that as the composite is mixed by hand, small
pockets of air become entrapped in the resin. Althoughspecimens undergo a vacuum process to evacuate thecaptured air, this appears to become less successful withincreasing particle settling. Perhaps, this is because in addition to the resin already being highly viscous, particle
sedimentation is known to create aggregates which further
impede the ﬂow of trapped air through the resin medium,34
or cause uneven ﬁlling over the specimen height.29 To
measure void content, we created an additional set ofcomposite samples using a 2-part epoxy (detailed below in
Section 3.1.3) with a slightly lower viscosity and higher
curing time than Dragon Skin 10, which exacerbates settling. We estimated the void content in the resulting
specimens by comparing their measured weight against the624

Journal of Composite Materials 57(4)Figure 3. Improved particle bonding via “wrinkling” surface treatment. (a) Neat particles. (b) Particles from the same batch after being
heated then quickly submerged in ice water to cause shriveling. (c) Debonding strain is improved by 16%, and debonding stress is
improved by 11% over the untreated particles.

Figure 4. Particle distribution and settling. (a) Composite with a loading of 30 vol% FM in Dragon Skin 10. The largest particle size, mean
diameter of 100 μm, settles before the matrix is cured, leaving particles in only the lower 73% of the specimen on average. (b-d) Smaller
particles are more evenly distributed throughout the matrix. (e) FM-epoxy composite. At higher loadings, increasing particle diameter
causes sedimentation which contributes to the presence of voids.

Buckner et al.

625

Figure 5. Particle size vs. toughness and strength for FM-Epoxy composites. (A) Representative stress-strain curves of 30 vol% epoxyFM specimens subjected to 3-point bending until failure. (B) Mean toughness, i.e., energy absorbed per volume before specimen failure
as a function of particle diameter. (C) Mean ﬂexural strength as a function of particle diameter.

expected mass for the given specimen volume and FM
loading percent, and note a measurable increase in void
content as particle size and settling increases, while at 30 vol% the porosity appears approximately uniform regardless ofparticle size (Figure 4(E)).
Composite strength and toughness. As variable-stiffness
structures are often designed to perform move-and-holdoperations, the capacity to stiffen in place and subsequently support loads without fracture is especially important. Noting the improved bonding strength of smaller
FM particles, we next measured the material toughness andstrength. As is well-documented in the literature, the size ofadditive particles has a signiﬁcant effect on the durability of
the resulting composite. Generally, a reduction in size will
result in increased material toughness and increased ultimate strength.15,35–37 In addition to the improved bonding
characteristics from an increased interfacial surface area, theincreased toughness is also usually due to the larger numberof rigid particle sites available to arrest crack propagation
via deﬂection or pinning.37
Composite specimens were manufactured from a thermoset epoxy resin consisting of Epon-828 and Jeffamine
D-400 combined at the stoichiometric ratio (100:58 byweight), and then loaded with FM particle inclusions at
30 and 50 vol% for a range of particle sizes (5 specimenseach). Each specimen, of size 3 × 10 × 80 mm, was subjected to a 3-point-bending test until fracture using a universal testing machine (Instron 3345, Figure 5(A)), with asupport span of 48 mm at a loading rate of 2 mm/min.
We calculated the toughness of each specimen bymeasuring the area under the stress-strain curves, which
represents the total energy absorbed by the specimen (KJ/m3) before failure occurs. In this case, it appears that thelargest particles achieve a higher toughness by allowing forlarger strains (Figure 5(B)), which may be an artifact of thebilayer system that develops due to settling of the particles
as seen in Section 3.1.2, while the smallest particles raise thetoughness via increased ﬂexural strength (Figure 5(C)).
Particles of intermediate size suffer from both lower ultimate strength and ultimate strains, and so have the lowest
overall toughness. We also note that the more heavilyloaded 50 vol% specimens tended to fracture more easily, which we attribute to the formation of voids within thecomposite (Figure 4(E)). No specimen could match thetoughness of neat epoxy by a factor of almost 1/2, andincreasing the ﬁller volume percent increases this disparity
by further raising the stiffness and brittleness of the material.
Composite stiffness change and impact of oxide shell. Particle
size has signiﬁcant inﬂuence on the composite stiffnesschange. Part of the undesired stiffening due to smaller
particles might be explained by interfacial tension effects,
which can cause small liquid inclusions with diameters
below a characteristic length scale D ≤ γ=E to resist deformation, increasing their effective modulus.38 Given thesurface tension of liquid Field’s metal, γLG ¼ 417 mN=m,39
the surface energy of cured epoxy γSG ≈ 35:1 mN=m,40 andthe contact angle of Field’s metal droplets with our cured
epoxy, θ ¼ ∼ 128°,19 we can estimate the interfacial
tension between Field’s metal and our epoxyγSL ¼ γSG  γLG cosðθÞ ¼ 302 mN=m. Keeping in mind
that the oxide shell on the FM can skew these types ofinterfacial tension measurements,41 these values alongwith the modulus of the epoxy matrix material,
E ¼ 11:14 MPa at 75°C, give a critical diameter of
D ¼ 27 nm. Even using the smallest particle size studied inthis work (mean diameter of 11 μm), our smallest detectedparticle had a diameter of 371 nm, an order of magnitudelarger than the critical diameter (Figure 1(C)) for thisparticular matrix material.
Instead, we propose that the stiffening behavior in small
FM particles is primarily a result of the formation of anoxide shell surrounding each droplet during manufacture,
which remains solid while the FM core melts. This oxideshell is mostly composed of In2O3, followed by SnO2, and626

Journal of Composite Materials 57(4)then Bi2O3,42 all of which are signiﬁcantly more rigid thanField’s metal. Examining a 5 mm × 5 mm planar specimenof Field’s metal under X-ray photoelectron spectroscopy(XPS), the presence of Bi2O3 was negligible, while we
detected SnO2 at 18 at%, which corresponds to approximately 10.6 wt%. We performed these measurements according to the approach laid out in by Farrell et al43 and Cantet al,44 using a magnesium anode, and then again using an
aluminum anode which has a higher inelastic mean free path
(IMFP) and thus will measure deeper into the oxide. The
measurements in both cases are nearly identical, from which
we assume that the In2O3 and SnO2 are somewhat mixed,
rather than stratiﬁed. The mean oxide thickness was measured to be approximately 4.5 nm.
We then estimated the effective modulus of composites
containing liquid-core-shell particles following the differential
effective medium theory (D-EMT) modiﬁed by Garboczi andBerryman,45 supplemented with details from Young,46 using
the material properties in Table 1. This modiﬁed model accounts for higher loadings and effects of particle interaction,
which is critical for the loading volumes that we explore in thiswork, and also has a method for converting core-shell particles
into an equivalent homogenous particle which allows foranalysis of a three-phase composite.
We have collected the resulting equations here forconvenience, though we direct the reader to Garboczi andBerryman45 for the derivations. First, we deﬁne three materials: 1 is the matrix material, 2 is the particle shell material, and 3 is the particle inner core material. νi , Gi, Ki , Ei
refer to the Poisson’s ratio, and the shear, bulk, and elastic
moduli of the respective material i. Also, let a and b refer tothe diameters of the outer particle shell and of the inner core,respectively. We then calculate the effective properties ofthe particles νef f , p , Gef f , p , Kef f , p , Eef f , p by replacing thecore-shell inclusions with an equivalent homogenous particle of equal size, where Gef f , p is found by solving thefollowing quadratic equation for the positive square root

 

 
A Gef f , p G2 þ 2B Gef f , p G2 þ C ¼ 0

Property

Value

(3)




 2 63zηβ þ 2ηα ηγ pð7=3Þ
A ¼ 8zð4  5ν2 Þηα pþ 252zηβ pð5=3Þ  50z 7  12ν2 þ 8ν22 ηβ pþ 4ð7  10ν2 Þηβ ηγB ¼2zð15ν2 Þηα pð10=3Þ þ2 63zηβ þ2ηα ηγ pð7=3Þ
252zηβ pð5=3Þ þ75zð3ν2 Þηβ ν2 pþð3=2Þð15ν2 7Þηβ ηγ
C ¼4zð5ν2  7Þηα pð10=3Þ  2 63zηβ þ 2ηα ηγ pð7=3Þþ 252zηβ pð5=3Þ þ 25z ν22  7 ηβ p  ð7 þ 5ν2 Þηβ ηγ

Reference

In2O3 with 10 wt% SnO2
Young’s modulus
116 GPa
Poisson’s Ratio
0.35
Liquid Field’s metal(approximated using liquid indium)
Bulk modulus
23.96 GPa
Poisson Ratio
0.5
Solid Field’s metal
Young’s modulus
9.2 GPa
Poisson’s Ratio
0.33
Epoxy @75 °C
Young’s modulus
11.0 MPa
Poisson’s Ratio
0.33

[57]
[57]

[58]
Estimate[19]
Estimate
This work
Estimate3

p ¼ ða=bÞz ¼ G3 =G2  1
ηα ¼ zð7  10ν2 Þð7 þ 5ν3 Þ þ 105ðν3  ν2 Þηβ ¼ zð7 þ 5ν3 Þ þ 35ð1  ν3 Þ
ηγ ¼ zð8  10ν2 Þ þ 15ð1  ν2 Þ

(4)

At this point, Kef f , p is found using
Kef f , p ¼ K2 þpðK3  K2 Þ
K3  K2
1 þ ð1  pÞ
4
K2 þ G2
3(5)

and the remaining values are found using the known
relationsEef f , p ¼

with the coefﬁcients given byð10=3Þ

Table 1. Material properties used in oxide-shell analytical model.

9Kef f , p Gef f , p
3Kef f , p þ Gef f , p

3Kef f , p  2Gef f , pνef f , p ¼ 
2 3Kef f , p þ Gef f , p(6)

As particles are made smaller, the volume percent ofoxide material rapidly becomes signiﬁcant (Figure 6(a)) andbegins to stiffen the particles. Although the absolute increase in stiffness appears to be approximately the same
whether the particle core is solid or liquid, the relativechange in stiffness for liquid-core particles is quite drastic;
the predicted modulus of a heated 5 μm particle is ∼10×higher than that of a 50 μm particle.
We veriﬁed this equivalent particle model by measuring
the modulus of individual particles using force spectroscopy
via atomic force microscopy (AFM, Asylum Research,

Buckner et al.

627

Figure 6. Impact of FM particle oxide shell. (a) The volume percentage of oxide in particles increases sharply as particle size decreases.
(b) Elastic modulus of individual FM particles as measured via AFM force spectroscopy. (c) Elastic modulus of FM-epoxy composite at
75°C. The presence of the high-modulus oxide shell impacts the softened modulus and limits the overall stiffness range of the variablestiffness material. (d) Representative ﬂexural modulus plots of 30 vol% epoxy-FM specimens subjected to temperature sweep. (e) Mean
ﬂexural modulus range of each specimen type. The tops of the bars represent the “cold” modulus (25°C), and the bottom is the “hot”
modulus (70°C). (f) Mean variable stiffness ratio, Ecold/Ehot for each specimen type.

Cypher ES). While melted, individual particles express
some amount of hysteresis, losing some energy to plastic
deformation of the particle as the AFM probe tip is pressed
into the surface. Hence, we use the force-displacement
curve measured during retraction of the tip instead,which exhibits a more nearly elastic response. The AFM
probe was moved at a rate of 200 nm/s, indenting to a depthof 20 nm. The probe had a three-sided triangular proﬁlewith a rounded tip of radius of ∼7 nm. During testing, theforce curves indeed appeared to follow the response curveof a spherical indenter, which we used as an approximation
for the tip shape in the selected Hertz indentation model,47
where F is the force response, δ is the cantilever deﬂection, R is the probe radius, and E and ν are the particle specimen’s Young’s modulus and Poisson’s ratio
respectively (1).
F¼

4 ER1=2 δ3=2
3 ð1  ν2 Þ

(7)

While the Hertz model is not a perfect ﬁt, it sufﬁciently
approximates the value such that we can quantitatively
track the overall behavior. The experimental results
clearly follow the expected trend from the analyticalplot that small particles suffer from an increased
stiffness due to the increased inﬂuence of the oxide shell(Figure 6(b)).
We then used the three-phase D-EMT model from
Garboczi and Berryman45 to predict the modulus of FM
particles embedded in the same epoxy medium described in
section 3.1.3 (Figure 6(c)). The effective particles resulting
from equations (1)–(4) can be used to predict the composite
properties G, K
G ¼ G1 =ð1  fÞgk
K ¼ K1 ð1  fÞ




4
5 K1 þ 3 G1 Gef f , p  G1

g¼
8
3G1 K1 þ 9 G1 þ 2Gef f , p ðK1 þ 2G1 Þ




4
K1 þ G1 Kef f , p  K1
3


k¼
4
K Kef f , p þ G1
3

(8)

628

Journal of Composite Materials 57(4)and the remaining composite properties E, ν are found as
before using the known relations
E¼

9KG
3K þ G

ν¼

3K  2G
2ð3K þ GÞ(9)This model was validated by subjecting rectangular epoxy
composite specimens of size 3 × 12.5 × 30 mm (5 of eachcomposition) to a thermal sweep from 25°C to 75°C in a
10 mm dual cantilever beam clamp using a Dynamic Mechanical Analyzer (TA Instruments, DMA Q850). Specimens
were applied a sinusoidal force on the moving end of cantilever at a rate of 5 Hz, at a strain of 0.02%. Results show an
initial drop in ﬂexural modulus between 40 and 50°C, whichindicates the onset of glass transition of the epoxy. Up to thispoint, each specimen exhibits approximately equivalentbehavior, as particle size of solid ﬁllers is not expected tosigniﬁcantly affect the composite modulus.48,49 For example,
at 30 vol%, an initial ﬂexural modulus of ∼2.6 GPa at 20°C is
reduced to ∼60 MPa at 50°C due to the glass transition of thepolymer regardless of particle size (Figure 6(d)). However,after the onset of FM particle melting (62°C), it can be seenthat decreasing FM particle size has a detrimental effect,
reducing the drop in ﬂexural modulus that would be expectedfrom the phase change to liquid (Figures 6(e) and (f)).
Given this behavior, the allowable minimum ﬁller particle size may be limited by the desired lower bound forstiffness change. Alternatively, suppressing the formation ofan oxide shell (via acids,50 etc.), or use of an entirely
different phase-change inclusion material that does not forma shell (such as wax or other polymer) may be a path tomitigating this issue.

Conclusions
By utilizing a thixotropic medium in the manufacture stage,
we have shown that it becomes straightforward to repeatably produce FM particles at a speciﬁed size with highsphericity. The approach allows for tuning of variablestiffness composite properties to a degree that is previously not demonstrated. We also hypothesize that the
manufacturing technique presented in Section 2.1 might beuseful in other areas. For instance, repeatable control ofparticle dimensions would beneﬁt the development andoptimization of multi-material composites which havepreviously used low-melting-point metals alongside otherﬁllers, for example to enhance electrical conductivity.19,51
We also note that this technique is not limited to metals, andcan also be applied to other phase-changing materials suchas waxes or thermoplastics.

In seeking a durable variable-stiffness composite, thiswork shows that dimensional control of the phase-changing
inclusions is a key parameter for material performance. In
the case of FM, a small particle size can be chosen toenhance bonding between the inclusions and matrix material. This will also increase the composite ultimate
strength and toughness. Alternatively, allowing large particles to settle into a densely-packed FM layer reduces thecomposite strength and stiffness, but can beneﬁt from thenatural bilayer formation of neat matrix material above
which appears to act as a stabilizing “backbone”, absorbing
larger strains before failure. However, matching thetoughness of the neat matrix material may require supplemental additives such as rubber nanoparticles52 or further
bonding enhancements, such as the use of silanes.53
In terms of variable-stiffness, this work suggests thatmaximizing the modulus range requires larger particle sizes.
The presence of a stiff oxide layer becomes a signiﬁcant
impediment to the stiffness-changing ability of the FM
particles as particle dimensions are reduced, especiallybelow 50 μm. This impact can be accurately predicted using
known analytical models, applicable not only to FM but
other oxide-forming materials as well. Increasing the ﬁller
content volume also increases the overall stiffness change,
although this comes with other downsides, including reduced ease of manufacture and development of voids.
Based on these results, it would seem that for the greatest
variable stiffness range of an FM composite, a mean particle
size of ∼50 μm may be desired, as it does not suffer greatly
from the effects of the oxide shell, and can support higher
loadings of at least 50% while remaining easily mixable andhomogenous without succumbing to large numbers ofvoids. Much larger particles do not seem to appreciably
improve variable-stiffness performance, and risk settling
into a layered system at the cost of manufacturing difﬁculty
and high concentrations of voids. Much smaller particles
will increase the strength at the cost of reduced variablestiffness performance, although this may not be an issue forother materials that do not form a shell.
While this work has focused exclusively on the mechanical behaviors of composites with phase-changing
particles, there exist other aspects of variable-stiffness tobe studied and characterized. Of interest for thermallyresponsive phase-changing ﬁller materials are energy
costs, heating times, and hotspots, all of which can be reduced by increasing composite thermal conductivity.
Thermally-conductive particles such as FM in variablestiffness composites have already been demonstrated tohave a positive impact in this regard,23 and there is evidence
that smaller particles55 or a mixture of particle sizes56 may
be able to improve thermal conductivity further. Electricalconductivity in phase-changing composites is also an areaof interest, and studies focused on ﬁller shape54 have shownthat elongated particles can enable direct Joule heating,Buckner et al.
promoting improved integration into systems-level applications. These works, in conjunction with future explorations of different phase-changing ﬁller and matrix materials,
all serve to widen and clarify our understanding of theemerging use of inclusions to impart variable-stiffness
behavior. As we see phase-changing inclusions becoming
more prevalent for variable-stiffness applications, we are
optimistic that this work may supply important design
considerations and processes for the creation of robust,high-stiffness-change composites.
Declaration of conﬂicting interests
The author(s) declared no potential conﬂicts of interest with respect to the research, authorship, and/or publication of this article.

Funding
The author(s) disclosed receipt of the following ﬁnancial support
for the research, authorship, and/or publication of this article: This
work was supported by The National Aeronautics and Space
Administration (grant no. 80NSSC17K0553).

ORCID iD
Trevor L Buckner  https://orcid.org/0000-0002-6602-8511

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