as featured in - yale university · 2015. 8. 16. · registered charity number: 207890 as featured...

Registered charity number: 207890

www.softmatter.org

As featured in:

See Eric R. Dufresne et al., Soft Matter, 2015, 11, 672.

Highlighting Research from the Soft Matter Lab, Yale University,

New Haven, CT, USA. Prof. Eric R. Dufresne is the recipient of

the 2014 Soft Matter Lectureship award.

Title: Surface tension and the mechanics of liquid inclusions in

compliant solids

The standard models of solid mechanics were developed to

describe the behavior of stiff materials. Our recent experiments

have shown that the Johnson–Kendall–Roberts theory of

adhesion, Young–Dupre relation of wetting, and Eshelby theory

of composites each fail when elastic moduli drop below a

size-dependent scale. In all cases, surface tension drives new

phenomena. In this Soft Matter paper, we reformulate Eshelby’s

theory of composites to account for surface tension.

ARTICLESPUBLISHED ONLINE: 15 DECEMBER 2014 | DOI: 10.1038/NPHYS3181

Sti�ening solids with liquid inclusionsRobert W. Style1,2, Rostislav Boltyanskiy1, Benjamin Allen1, Katharine E. Jensen1, Henry P. Foote3,John S. Wettlaufer1,2,4 and Eric R. Dufresne1*

From bone and wood to concrete and carbon fibre, composites are ubiquitous natural and synthetic materials. Eshelby’sinclusion theory describes how macroscopic stress fields couple to isolated microscopic inclusions, allowing prediction ofa composite’s bulk mechanical properties from a knowledge of its microstructure. It has been extended to describe a widevariety of phenomena from solid fracture to cell adhesion. Here, we show experimentally and theoretically that Eshelby’stheory breaks down for small liquid inclusions in a soft solid. In this limit, an isolated droplet’s deformation is stronglysize-dependent, with the smallest droplets mimicking the behaviour of solid inclusions. Furthermore, in opposition to thepredictions of conventional composite theory, we find that finite concentrations of small liquid inclusions enhance the sti�nessof soft solids. A straightforward extension of Eshelby’s theory, accounting for the surface tension of the solid–liquid interface,explains our experimental observations. The counterintuitive sti�ening of solids by fluid inclusions is expected wheneverinclusion radii are smaller than an elastocapillary length, given by the ratio of the surface tension to Young’s modulus of thesolid matrix. These results suggest that surface tension can be a simple and e�ective mechanism to cloak the far-field elasticsignature of inclusions.

Composite materials can offer marked performanceimprovements over their individual components. Carbonfibre increases the strength and stiffness of polymer resins

as much as a hundredfold1. Densely packed gas bubbles in a liquidmatrix create a foam, which resists deformation like a solid2. Thefoundational theory of solid composites, due to Eshelby, describeshow isolated inclusions in a composite behave in response toapplied stresses3. Eshelby applied this result to predict the stiffnessof dilute solid composites3 and his theory has been extendedto finite concentrations, where neighbouring inclusions couplethrough their induced strain fields (for example, refs 4,5). Eshelby’stheory has been applied widely beyond composites, havinglong been used to understand the mechanics of fracture6,7 andplasticity8,9. More recently, it has been applied to understandingflow of sheared glasses10 and the interactions of cells with theextracellular matrix11,12.

Eshelby’s theory describes the matrix and inclusion as bulklinear-elastic solids, but does not account for the physics of theinterface, which generically includes excess surface free energy andsurface stress13–15. Surface energy is the reversible work per unitarea required to create new interfacial area by cutting. Surfacestress is the reversible work per unit area to create new interfacialarea by stretching. For liquids, surface energy and surface stressare identical, isotropic and strain-independent. For solids, surfacestress and energy are generally anisotropic and distinct, but can beisotropic for soft amorphous solids such as gels16,17. Cell membranesand other thin-walled vessels can exhibit large isotropic surfacestress with negligible surface energy18,19. In this manuscript, we usethe phrase surface tension, denoted by Υ , to denote an isotropicstrain-independent surface stress.

Recent work has underlined the importance of surface-tensioneffects in soft solids. These solid capillary effects include thesmoothing out of ripples and corners in soft solids20, and quali-tative changes to the phenomena of wetting21–26 and adhesion27–30.Furthermore, the competition of surface tension and elasticity

can select the wavelength of pearling and creasing instabilities31,32.Surface-tension effects typically appear in solids at length scales.L≡Υ/E, where E is Young’s modulus of the solid. In simple terms,this elastocapillary length represents the wavelength below whichsurface tension is capable of significantly deforming a solid21,27.Thus, it is reasonable to expect that when inclusions in an elasticbody have a characteristic size R<L, capillarity will become impor-tant and Eshelby’s theory will not apply. This has been suggestedby various theoretical studies (for example, refs 33,34) and recentexperiments on air bubbles embedded in emulsions35.

Here, we demonstrate the impact of surface tension on themechanical response of fluid inclusions in a soft solid matrix.We find that the deformation of isolated liquid inclusionsin a macroscopic stress field depends strongly on their size.Although large-droplet deformations are consistent with Eshelbytheory, droplets with radii below the elastocapillary scale deformsignificantly less than predicted. Furthermore, whereas finiteconcentrations of large droplets make a solid more compliant,droplets smaller than the elastocapillary scale make it stiffer. Ageneralization of Eshelby’s theory, accounting for surface tension,captures our experimental observations, and provides simpleanalytical results useful for the design of composites.

Stretching single inclusionsWe tested Eshelby’s inclusion theory in soft solids by observing themicroscopic deformation of droplets embedded in macroscopicallydeformed solids (Fig. 1a,b and Supplementary Section 1).We coatedthe soft solid on a thin, elastic sheet, and stretched it uniaxially,measuring the exact applied strain (ε∞x ,ε∞y ) by tracking fluorescentparticles attached to the surface of the sheet (SupplementaryFig. 1). The applied strain in the uniaxial stretch direction is ε∞xand ε∞y is the smaller, associated contraction that arises in theperpendicular, in-plane direction. The stretch lengthens the dropletsin the x-direction, and we imaged them at their equator frombelow with a ×60, NA 1.2, water objective. The droplets are ionic

1Yale University, New Haven, Connecticut 06520, USA. 2Mathematical Institute, University of Oxford, Oxford OX1 3LB, UK. 3University of North Carolina,Chapel Hill, North Carolina 27516, USA. 4Nordita, Royal Institute of Technology and Stockholm University, SE-10691 Stockholm, Sweden.*e-mail: [email protected]

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ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS3181

Ionic liquid

R = 2.5 μm

6.0%−1.3%

28.8%−6.9%

18.6%−4.3%

R = 6.4 μm R = 16.5 μm

w

Overlay

y∞∋

y∞∋

x∞∋

x∞∋

y∞∋

x∞∋ ,

Silicone

a b

Figure 1 | Stretching droplets embedded in soft solids. a, The sample is clamped and stretched in the x-direction. b, Example images of ionic-liquiddroplets in a soft, silicone solid E= 1.7 kPa. Larger droplets deform more at the same applied strain. Overlay shows small (blue), medium (red) and large(green) droplet images combined together for shape comparison.

liquid (1-ethyl-3-methylimidazolium dicyanamide, Ionic LiquidsTechnologies) and are completely immiscible in the silicone gel thatwe use for the solid phase. Silicone gels of two different stiffnesses(E= 1.7 kPa, E= 100 kPa) were prepared by mixing together baseand crosslinker at different ratios and curing at room temperaturefor 16 h, as described in Supplementary Section 2. Silicone gel isideal for these experiments as it behaves like a linear-elastic solidup to large strains. Supplementary Fig. 2 shows example rheologyfor the soft, E=1.7 kPa silicone.

The theory of Eshelby predicts that stretched inclusion shapesdepend only on the applied strain, and not on droplet size. Weconfirmed this result for droplets embedded in a stiff, 100 kPamatrix (Supplementary Fig. 3). However, micrometre-sized dropletsbehave quite differently in a compliant 1.7 kPa matrix (Fig. 1b).Here, small droplets are significantly less deformed than largedroplets under the same macroscopic strain.

Small liquid droplets seems stiffer than the surrounding solidmatrix. Figure 2a gives the aspect ratio, AR=`/w, of eight dropletsof different initial radii R at different stretches. As expected fora linear-elastic solid, the aspect ratio increases linearly with ε∞x .However, it also increases with R. In other words, large dropletsare deformed more by the stretch, and smaller droplets seem‘stiffer’. The dotted/continuous line shows Eshelby’s predictions:AR=(3+5ε∞x )/(3+5ε∞y ) for incompressible, spherical liquidinclusions, and AR=1 for rigid spherical inclusions3. We also plotEshelby’s prediction for an inclusion identical to the surroundingsolid, AR = (1 + ε∞x )/(1 + ε∞y ) as the dashed-dotted line—thisrepresents the bulk deformation of the solid matrix. The largestdroplet agrees well with the incompressible liquid limit. Smallerdroplets seem stiffer, with the smallest droplets approaching therigid inclusion limit.

The stiffening effect in small droplets seems to arise at a strain-independent length scale. Figure 2b shows the aspect ratio ofmany droplets as a function of their length, `, for six differentstrains. For each strain, the aspect ratio is insensitive to the sizeof large droplets (&30 µm). However, AR drops off sharply forsmaller droplets. It is interesting to note that a qualitatively similardependence of shape on size is seen for droplets in viscous shear orextensional flows36.

Composite sti�nessAccording to Eshelby’s classic result3, liquid inclusions, whichhave zero Young’s modulus, should reduce the stiffness of a solidcomposite. However, our data show that small, isolated dropletsresist deformation more strongly than one would expect fromEshelby’s theory. Now, we explore the impact of the increase inapparent stiffness of single droplets on the macroscopic stiffnessof a composite. We made soft composites out of silicone geland glycerol droplets by mixing silicone, glycerol (Sigma-Aldrich)

and a small quantity of surfactant (Gransurf 50C-HM, GrantIndustries) using a hand blender. Glycerol is used in place ofthe ionic liquid as it is cheap, non-toxic, and almost completelyimmiscible in silicone. We degassed the resulting emulsion ina vacuum, poured it into a mould and then cured it at 60 ◦Cfor two hours. This gives composites of droplets embedded insilicone with R=O(1 µm), at volume fractions φ from 4 to 20%(Supplementary Fig. 4). As explained in Supplementary Section 2band Fig. 5, we ignore composites with φ<4.4% to ensure that thestiffness of the continuous phase of the composite is unaffected bythe surfactant.

Stiff and compliant solids have opposing responses to liquidinclusions. We measured the composite Young’s modulus Ec bymacroscopic indentation (see Supplementary Section 2 for detailedprotocols). Figure 3a,b shows how composite stiffness changes withincreasing liquid content for composites with a stiffer solid matrixwith E∼100 kPa and a more compliant solid matrix with E∼3 kPa,respectively. The stiff-matrix composite becomes softer as the liq-uid content increases. This makes intuitive sense—as we replacea fraction of the solid by holes with no shear modulus, we see aproportional decrease to the stiffness. In fact, the data agrees withEshelby’s prediction for the stiffness of a solid containing dilute em-bedded monodisperse, incompressible droplets, Ec=E/(1+5φ/3)(ref. 3). The compliant-matrix composite shows the opposite trend:stiffening with liquid content. Stiffness increases by around a thirdwith a 20% increase in liquid content. The composites are elasticup to shear strains of ∼100%, and behave identically in subsequentcycles of indentation (see Supplementary Section 2c and Fig. 6).Thus, we find that the soft-matrix composite is unexpectedly stifferthan the pure soft solid, without a significant loss in strength.Conventional composite theory, such as Eshelby theory3, the lawof mixtures, and the Hashin–Shtrikman bounds4 uniformly predictdecreasing stiffness with increased fraction of liquid inclusions and,therefore, cannot describe this behaviour (for example, Fig. 3). Ina similar vein, recent experiments have shown that the stiffness ofan emulsion was unaffected when embedded bubbles were suffi-ciently small35.

Theory and discussionThe experimental data suggest that conventional composite theoryfails to describe our experiments because of the effect of surfacetension at the liquid/solid interface. Surface tension typically actsto smooth out interfaces and drive them towards a constantcurvature. In a solid, surface tension is typically overwhelmedby bulk elasticity. However, surface tension can cause significantdeformations in compliant solids (for example, ref. 37). In ourexperiments, surface tension acts to keep liquid inclusions spherical,opposing any applied stretch. Thus, surface tension can qualitativelyexplain the main features of our data. This echoes recent results

2 NATURE PHYSICS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturephysics




NATURE PHYSICS DOI: 10.1038/NPHYS3181 ARTICLES

0.000.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

1.0

1.1

1.2

1.3

1.4

1.5 Eshelby: inclusionsame as surrounding solid

Eshelby: incompressibleliquid inclusions

Eshelby: rigidinclusions

1.6

1.738 μm18 μm16 μm9 μm6 μm3 μm3 μm2 μm

1.8a

b

0.05 0.10 0.15 0.20 0.25 0.30

0 20 40 60 80

42.2%, −7.9%

33.9%, −6.6%

26.0%, −5.0%

17.5%, −2.6%

9.4%, −1.4%

5.6%, −1.5%

100

/w/w

/2 (μm)

x∞∋

, y∞∋

x∞∋

Figure 2 | Aspect ratio of stretched ionic-liquid droplets in a soft(E=1.7 kPa) silicone gel as a function of size and strain. a, Aspect ratios ofeight droplets of di�erent sizes increase linearly with applied strain. Thedashed–dotted line shows the change in aspect ratio of the solid under theapplied strain ε∞x ,ε∞y . Large droplets (R& 5µm) stretch more than the solid.Smaller droplets stretch less. The dotted (continuous) line shows theprediction from Eshelby theory for incompressible liquid (rigid) solidinclusions respectively3. b, Aspect ratio of droplets depends sensitively onsize. Di�erent colours correspond to di�erent applied strains. Dashedcurves show theoretical predictions using equations (2) and (3) withΥ =0.0036 N m−1.

on wetting and adhesion on compliant silicones, where capillaryaffects arose below a length scale of O(10 µm), similar to thatseen in Fig. 2b22,23,27. Here, we modify Eshelby theory to accountfor solid surface tension, and show that it accurately describesour data.

We consider an incompressible droplet embedded in a linear-elastic solid with a surface tension that acts on the droplet boundary.The solid’s displacements, u, obey the equation:

(1−2ν)∇2u+∇(∇ ·u)=0 (1)

where ν is Poisson’s ratio and we apply far-field strain boundaryconditions ε=ε∞. At the surface of the droplet, σ ·n=−pn+ΥKn,where σ is the stress tensor in the solid, n is the normal vector tothe deformed surface, p is the pressure in the droplet and K is thecurvature of the deformed surface. Note that we assume a surfacetension that is independent of surface strain. This is generally agood approximation for gels, although it is not true in general17,34.We derive analytic solutions to equation (1) (ref. 38) by extendingprevious work39. In the particular case of far-field, plane-stress

02.0

2.5

3.0

3.5

5

Eshelby theory

10 15 20

070

80

90

E c (k

Pa)

E c (k

Pa)

Eshelby theory

100

110Stiff

Soft

5 10 15 20

a

b

(%)φ

(%)φ

Figure 3 | Young’s modulus of soft composites as a function of liquidcontent. Glycerol droplets embedded in E∼ 100 kPa (a) and E∼3 kPa (b)silicone gels. Dashed curves show Eshelby’s predictions for incompressibleliquid droplets in an incompressible solid with Young’s modulus E.

boundary conditions (as in our experiment) εxx=ε∞x , εyy=ε∞y andσzz=0, the length and width of the stretched droplet are

`=2R[1+

5(2ε1−ε2)6+15 Υ

ER

](2)

and

w=2R[1+

5(2ε2−ε1)6+15 Υ

ER

](3)

where ε1 = (ε∞x + νε∞y )/(1− ν2) and ε2 = (νε∞x + ε∞y )/(1− ν2).In the limit Υ = 0, this reduces to Eshelby’s predictions. In thelimit Υ/ER� 1, surface tension dominates and the droplets stayspherical, as the elastic stresses become insufficient to deformthe droplet from its preferred shape. The dependence on theparameter Υ/ER indicates that surface-tension effects start toarise when the size of the droplets approaches the elastocapillarylength L=Υ/E (ref. 40). This is similar to previous experimentswhere solid capillarity becomes important: for example, contactmechanics results are altered when the size of the indenter is .L(refs 27–29), droplet contact angles change when drop radii are.L (refs 21,22), and thin fibres undergo instabilities when theirdiameters are .L (ref. 31).

Our theory agrees well with the isolated droplet data with onefitting parameter—the unknown surface tension Υ . In Fig. 2b, weplot the aspect ratio predicted by equations (2) and (3), usingν=1/2, E=1.7 kPa and Υ =0.0036Nm−1. The results agree withthe experiments up to large strains, suggesting that surface tension isindeed controlling droplet shape for small droplets. The agreementis surprisingly good as we use a linear-elastic theory which isonly strictly appropriate when ε . 10%, or equivalently when ε∞x ,ε∞y . 10% (ref. 38). Note that the value of the surface tension issmaller than we expected; we measured surface tension of an ionic






liquid in uncured silicone to be 0.025Nm−1 using the pendant dropmethod, and we might expect this value to be close to Υ . Thisdifference cannot be explained by measurement error, suggestingthat there is a significant change in the silicone/ionic-liquidinterface on crosslinking. This is not unprecedented—previousmeasurements have shown that there can be significant differencesbetween liquid and solid surface tensions of silicone22,24,41.

The individual droplet data collapses onto a single master curvewhen the ratio of the microscopic to macroscopic strains is plottedagainst the undeformed droplet radius (Fig. 4a). Using equations (2)and (3), we can obtain an estimate of the undeformed radii, R∗(R is unknown, as we did not track individual droplets from theirundeformed state for this large data set):

R∗=`

2−

A2(2ε1−ε2) (4)

where

A≡(`−w)(1+ν)3(ε∞x −ε∞y )

=10ER2

6ER+15Υ(5)

and the second equality comes from equations (2) and (3).The ratio 2A/R∗ compares the microscopic droplet strain to themacroscopic applied strain. Moreover, 2A/R∗ and R∗ dependonly on measured quantities, and nicely collapse the data overa factor of 70 in droplet size, and a range of strains from 5.6to 42.2% (Fig. 4a). There are two regimes: for droplets of sizeR∗ . 10 µm, (`−w)/R∗∝(ε∞x −ε∞y )R∗, whereas for larger droplets(`−w)/R∗∝(ε∞x −ε∞y ). From equation (5), we can interpret thisas the crossover from capillary- to elastic-dominated regimes as Rcrosses Υ/E. Note that although our theory effectively collapses thedata onto a universal curve, the data in the capillary regime seemsto have a stronger dependence on droplet size than predicted.

Our isolated droplet theory can be applied to predict compositestiffnesses3,42. Eshelby showed that the stiffness of a compositeconsisting of identical dilute inclusions can be calculated from theexcess energy of individual strained inclusions3; if the extra strainenergy due to the presence of a single inclusion in a uniaxiallystretched solid is W (σ∞,E,R,Υ ), where σ∞ is the applied stress,then the average strain energy density in a dilute composite is

E=12(σ∞)

2

E+φW43πR3

(6)

and Young’s modulus of the composite is Ec = (σ∞)

2/2E . We

can use equation (6) to predict the stiffness of a composite withmonodisperse, incompressible inclusions with surface tension38. Forthe particular case of an incompressible solid,

Ec=E1+ 5

2Υ

ER52Υ

ER (1−φ)+(1+53φ)

(7)

In the limit of small surface tension, or large droplets (R�Υ/E),this reduces to Eshelby’s result for liquid droplets in an elastic solid,which is Ec=E/(1+5φ/3). When surface tension dominates overelasticity (R�Υ/E), we obtain Ec=E/(1−φ), and the material isstiffened by the inclusions. This differs fromEshelby’s result for rigidparticles embedded in an elastic composite, Ec=E/(1−5φ/2)—although surface tension keeps the droplets spherical, they aredistinct from rigid particles because they have zero shear stress attheir surfaces. Equation (7) predicts that composites are stiffened bydroplets whenR<1.5Υ/E. Figure 4b shows how composite stiffnessdepends on liquid fraction, as predicted by equation (7) for differentvalues of Υ/ER. Intriguingly, we predict no change of the effectivemodulus of the composite when R= 1.5Υ/E. This suggests that

10−210−2

10−1

E i/E

Ei = E

E c/E

100

101

10−1

100

101a

b

c

102

0.5

1.0

1.5

2.0

2.5

10−1 100 101

Exact theory

102

/ERϒ

Ei = 2 /Rϒ

/ERϒ

= 0 (Eshelby)

100 101 102

R∗ (μm)

0 10 20 30 40 50 600.010.10.3

1

3

10100∞

(%)φ

42.2%, −7.9%33.9%, −6.6%26.0%, −5.0%17.5%, −2.6%9.4%, −1.4%5.6%, −1.5%

( −

w)/

(x∞

− y∞

)R∗

∋∋

, y∞∋

x∞∋

ϒ

Figure 4 | Theoretical predictions of composite behaviour. a, The datafrom Fig. 2b collapses onto a master curve when plotting the ratio of themicroscopic to macroscopic strains (equations (4) and (5)) against theestimated radius (equation (4)). The dashed curve shows the theoreticalprediction in equation (5). b, E�ective Young’s modulus of a compositeconsisting of monodisperse droplets embedded in a uniform solid, fromequation (7). The blue(black) data points are the composite sti�ness datafrom Fig. 3 scaled by E=3 kPa (E= 100 kPa) for the soft (sti�) compositesrespectively. c, Droplets with surface tension can be considered asequivalent elastic inclusions without surface tension. The red curve showshow the sti�ness of the equivalent elastic inclusion, Ei, depends on Υ/ER.For small Υ/ER, this agrees fairly well with the approximation Ei=2Υ/R,shown by the blue line.

surface tension can effectively cloak the far-field elastic signatureof inclusions43,44.

This theory for composite stiffness is consistent with ourexperimental data. Figure 4b includes the data from Fig. 3,normalized by E = 3 kPa and E = 100 kPa for the softer andstiffer composites respectively. The soft-matrix composite resultsare modelled well by the surface-tension-dominated theory. Thestiff-matrix composite results are modelled well by the theorywith little, or no, surface-tension effects. Using rough estimates





NATURE PHYSICS DOI: 10.1038/NPHYS3181 ARTICLESof Υ ∼14mNm−1 (Supplementary Section 2b) and R∼ 1 µm, weindeed expect surface tension to dominate for the soft-matrixcomposite (Υ/ER>1), and to be small for the stiff-matrix composite(Υ/ER< 1). Note that the experimental data for the soft-matrixcomposite is consistently stiffer than the upper limit of our theory.We suspect that this is due to formation of chain-like structures ofdroplets (Supplementary Fig. 4). Our theory is strictly valid only inthe limit of isolated droplets.

We can greatly simplify the above results to give a simple physicalpicture of the effect of surface tension in soft composites. Thestiffness of a composite of incompressible elastic inclusions withYoung’smodulus Ei in a solid ofmodulus E, according to Eshelby3, is

Ec=E1+ 2

3EiE(

23 −

5φ3

) EiE +(1+

53φ)

(8)

If we equate equations (7) and (8), we find that embedded dropletsare equivalent to elastic inclusions42 with stiffness

Ei=E24 Υ

ER

10+9 Υ

ER

(9)

This recovers the result of ref. 35 (derived using equivalentinclusion dipoles) that was successfully used to describe bubblesin soft emulsions. When Υ/ER�1, the droplets behave likeinclusions with Young’s modulus Ei = 12Υ/5R. This value isclose to the droplet Laplace pressure Ei = 2Υ/R, which is oftentaken as its stiffness for describing the composite stiffness ofemulsions and gels45. In the capillary-dominated regime,Υ/ER�1,the effective Young’s modulus of the inclusions saturates atEi=8E/3. Thus the droplets cannot have an arbitrarily increasingeffective stiffness as they get smaller, as the common Ei = 2Υ/Ransatz suggests. By replacing capillary-dominated inclusions withequivalent elastic inclusions described by equation (9) (for example,refs 35,42), one can use established composite theory such asMori–Tanaka homogenization5,35 or self-consistent methods46 topredict denser composite stiffnesses. Futurework can also generalizethe Hashin–Shtrikman bounds on composite moduli4 to includesurface tension—either using equivalent elastic inclusions, or byusing the thin-layer analogy for interface area introduced for thecase of surface-strain-dependent surface stresses15,47.

Our experimental and theoretical results show that surfacetension can be important for soft composites consisting of aliquid phase embedded in a continuous solid phase. We expectthat surface tension will be important for solid/solid compositeswhenever R. 100Υ/E1, 100Υ/E2, where E1, E2 are the stiffnessesof the two solids. For compliant materials such as gels withE =O(kPa), capillarity needs to be addressed at scales of up toO(100 µm) (refs 31,35). For stiffer materials, such as elastomers,with E =O(MPa), capillarity needs to be addressed at scales ofup to O(100 nm). Capillary effects should negligible in structuralmaterials, such as glass and ceramics, with E=O(GPa).

We expect that our results should be of use in understanding themechanical properties of soft tissues, especially in soft connectivetissues. For example, the cortical tension of fibroblasts mayhave a larger impact on the bulk mechanical properties of acollagenous tissue than the fibroblasts’ elastic moduli48,49. Ourresults complement new approaches to measuring mechanicalforces within three dimensional tissues by quantification of thedeformation of embedded liquid droplets50.

Our theoretical results include simple analytic expressions forindividual droplet deformation and for the properties of the bulkcomposite that can be readily applied to the design of newmaterials.They suggest that surface tension provides a relatively simple andeffective means to cloak the elastic signature of inclusions insoft materials.

Received 15 July 2014; accepted 10 November 2014;published online 15 December 2014; corrected online8 January 2015

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22. Style, R. W., Che, Y., Wettlaufer, J. S., Wilen, L. A. & Dufresne, E. R. Universaldeformation of soft substrates near a contact line and the direct measurementof solid surface stresses. Phys. Rev. Lett. 110, 066103 (2013).

23. Style, R. W. et al. Patterning droplets with durotaxis. Proc. Natl Acad. Sci. USA110, 12541–12544 (2013).

24. Nadermann, N., Hui, C-Y. & Jagota, A. Solid surface tension measured by aliquid drop under a solid film. Proc. Natl Acad. Sci. USA 110,10541–10545 (2013).

25. Karpitschka, S., Das, S., Andreotti, B. & Snoeijer, J. Dynamic contact angle of asoft solid. Preprint at http://arXiv.org/abs/1406.5547v1 (2014).

26. Bostwick, J. B., Shearer, M. & Daniels, K. E. Elastocapillary deformations onpartially-wetting substrates: Rival contact-line models. Soft Matter 10,7361–7369 (2014).

27. Style, R. W., Hyland, C., Boltyanskiy, R., Wettlaufer, J. S. & Dufresne, E. R.Surface tension and contact with soft elastic solids. Nature Commun. 4,2728 (2013).

28. Salez, T., Benzaquen, M. & Raphael, E. From adhesion to wetting of a softparticle. Soft Matter 9, 10699–10704 (2013).

29. Xu, X., Jagota, A. & Hui, C-Y. Effects of surface tension on the adhesivecontact of a rigid sphere to a compliant substrate. Soft Matter 10,4625–4632 (2014).

30. Cao, Z., Stevens, M. J. & Dobrynin, A. V. Adhesion and wetting ofnanoparticles on soft surfaces.Macromolecules 47, 3203–3209 (2014).

31. Mora, S., Phou, T., Fromental, J-M., Pismen, L. M. & Pomeau, Y. Capillaritydriven instability of a soft solid. Phys. Rev. Lett. 105, 214301 (2010).




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32. Chen, D., Cai, S., Suo, Z. & Hayward, R. C. Surface energy as a barrier tocreasing of elastomer films: An elastic analogy to classical nucleation. Phys. Rev.Lett. 109, 038001 (2012).

33. Yang, F. Size-dependent effective modulus of elastic composite materials:Spherical nanocavities at dilute concentrations. J. Appl. Phys. 95,3516–3520 (2004).

34. Sharma, P. & Ganti, S. Size-dependent eshelbyõs tensor for embeddednano-inclusions incorporating surface/interface energies. J. Appl. Mech. 71,663–671 (2004).

35. Ducloue, L., Pitois, O., Goyon, J., Chateau, X. & Ovarlez, G. Coupling ofelasticity to capillarity in soft aerated materials. Soft Matter 10,5093–5098 (2014).

36. Cox, R. G. The deformation of a drop in a general time-dependent fluid flow.J. Fluid Mech. 37, 601–623 (1969).

37. Chakrabarti, A. & Chaudhury, M. K. Direct measurement of the surfacetension of a soft elastic hydrogel: Exploration of elastocapillary instability inadhesion. Langmuir 29, 6926–6935 (2013).

38. Style, R. W., Wettlaufer, J. S. & Dufresne, E. R. Surface tension and themechanics of liquid inclusions in compliant solids. Soft Matter (in the press).Preprint at http://arXiv.org/abs/1409.1998 (2014).

39. Duan, H. L., Wang, J., Huang, Z. P. & Karihaloo, B. L. Eshelby formalism fornano-inhomogeneities. Proc. R. Soc. A 461, 3335–3353 (2005).

40. Wang, J., Duan, H., Huang, Z. & Karihaloo, B. A scaling law for properties ofnano-structured materials. Proc. R. Soc. A 462, 1355–1363 (2006).

41. Park, S. J. et al. Visualization of asymmetric wetting ridges on soft solids withX-ray microscopy. Nature Commun. 5, 4369 (2014).

42. Duan, H. L., Yi, X., Huang, Z. P. & Wang, J. A unified scheme for prediction ofeffective moduli of multiphase composites with interface effects. part i:Theoretical framework.Mech. Mater. 39, 81–93 (2007).

43. Milton, G. W., Briane, M. &Willis, J. R. On cloaking for elasticity and physicalequations with a transformation invariant form. New J. Phys. 8, 248 (2006).

44. Bückmann, T., Thiel, M., Kadic, M., Schittny, R. & Wegener, M. Anelasto-mechanical unfeelability cloak made of pentamode metamaterials.Nature Commun. 5, 4130 (2014).

45. Dickinson, E. Emulsion gels: The structuring of soft solids withprotein-stabilized oil droplets. Food Hydrocolloids 28, 224–241 (2012).

46. Hill, R. A self-consistent mechanics of composite materials. J. Mech. Phys.Solids 13, 213–222 (1965).

47. Brisard, S., Dormieux, L. & Kondo, D. Hashin–Shtrikman bounds on the bulkmodulus of a nanocomposite with spherical inclusions and interface effects.Comput. Mater. Sci. 48, 589–596 (2010).

48. Brown, R., Prajapati, R., McGrouther, D., Yannas, I. & Eastwood, M. Tensionalhomeostasis in dermal fibroblasts: Mechanical responses to mechanical loadingin three-dimensional substrates. J. Cell. Physiol. 175, 323–332 (1998).

49. Zemel, A., Bischofs, I. & Safran, S. Active elasticity of gels with contractile cells.Phys. Rev. Lett. 97, 128103 (2006).

50. Campàs, O. et al. Quantifying cell-generated mechanical forces within livingembryonic tissues. Nature Methods 11, 183–189 (2014).

AcknowledgementsWe thank J. Fernandez-Garcia for the ionic liquids, and T. Kodger and R. Diebold foradvice in preparing silicone. We also thank J. Singer, M. Rooks, F. Spaepen,S. Mukhopadhyay, P. Howell and A. Goriely for helpful conversations. We gratefullyacknowledge funding from the National Science Foundation (CBET-1236086) to E.R.D.,the Yale University Bateman Interdepartmental Postdoctoral Fellowship to R.W.S., andthe John Simon Guggenheim Foundation, the Swedish Research Council and a RoyalSociety Wolfson Research Merit Award to J.S.W.

Author contributionsR.W.S., R.B., B.A., K.E.J., H.P.F. and E.R.D. designed experiments. R.W.S., B.A. and R.B.acquired data. R.W.S. and E.R.D. analysed data. R.W.S., J.S.W. and E.R.D. developed thetheory. R.W.S., J.S.W. and E.R.D. wrote the paper.

Additional informationSupplementary information is available in the online version of the paper. Reprints andpermissions information is available online at www.nature.com/reprints.Correspondence and requests for materials should be addressed to E.R.D.

Competing financial interestsThe authors declare no competing financial interests.




http://arXiv.org/abs/1409.1998


http://www.nature.com/reprints


In the text following equation 7, the expression describing the limit where stiffening occurs was incorrect and should have read: surface tension dominates over elasticity (R << Υ/E). This has now been corrected in the online versions of the Article.

Stiffening solids with liquid inclusionsRobert W. Style, Rostislav Boltyanskiy, Benjamin Allen, Katharine E. Jensen, Henry P. Foote, John S. Wettlaufer and Eric R. Dufresne

Nature Physics 11, 82–87 (2015); published online 15 December 2014; corrected after print 8 January 2015.

CORRIGENDUM


Surface tension and the mechanics of liquidinclusions in compliant solids

Robert W. Style,ab John S. Wettlauferabc and Eric R. Dufresne*a

Eshelby's theory of inclusions has wide-reaching implications across the mechanics of materials and

structures including the theories of composites, fracture, and plasticity. However, it does not include the

effects of surface stress, which has recently been shown to control many processes in soft materials

such as gels, elastomers and biological tissue. To extend Eshelby's theory of inclusions to soft materials,

we consider liquid inclusions within an isotropic, compressible, linear-elastic solid. We solve for the

displacement and stress fields around individual stretched inclusions, accounting for the bulk elasticity of

the solid and the surface tension (i.e. isotropic strain-independent surface stress) of the solid–liquid

interface. Surface tension significantly alters the inclusion's shape and stiffness as well as its near- and

far-field stress fields. These phenomena depend strongly on the ratio of the inclusion radius, R, to an

elastocapillary length, L. Surface tension is significant whenever inclusions are smaller than 100L. While

Eshelby theory predicts that liquid inclusions generically reduce the stiffness of an elastic solid, our

results show that liquid inclusions can actually stiffen a solid when R < 3L/2. Intriguingly, surface tension

cloaks the far-field signature of liquid inclusions when R ¼ 3L/2. These results are have far-reaching

applications from measuring local stresses in biological tissue, to determining the failure strength of soft

composites.

I. Introduction

Eshelby's theory of inclusions1 provides a fundamental resultunderpinning a wide swath of phenomena in compositemechanics,2–5 fracture mechanics,6,7 dislocation theory,8 plas-ticity9,10 and even seismology.11 The theory describes how aninclusion of one elastic material deforms when it is embeddedin an elastic host matrix. At an individual inclusion level, itpredicts how the inclusion will deform in response to far-eldstresses applied to the matrix. It also allows the prediction ofthe macroscopic material properties of a composite from aknowledge of its microstructure.

Eshelby's theory does not include the effects of surfacestresses at the inclusion/matrix boundary. However, recentwork has suggested that surface stresses need to be accountedfor in so materials. This has been suggested both by theoret-ical models of nanoscale inclusions,12–14 and by recent experi-ments which have shown that surface tension (isotropic, strain-independent surface stress) can also signicantly affect sosolids at micron and even millimetric scales. For example, solidcapillarity limits the resolution of lithographic features,15–18

drives pearling and creasing instabilities,19–22 causes the Young–

Dupre relation to break down for sessile droplets,23–28 and leadsto a failure of the Johnson–Kendall–Roberts theory of adhe-sion.29–33 Of particular relevance are our recent experimentsembedding droplets in so solids, where we found that Eshel-by's predictions could not describe the response of inclusionsbelow a critical, micron-scale elastocapillary length.34 A similarbreak down was also seen in recent experiments that embeddedbubbles in so, elastic foams.35

To apply Eshelby's theory to a broad-class of mechanicalphenomena in so materials, we need to reformulate it toaccount for surface tension. Here, we derive analytic expres-sions for the deformation of individual inclusions, the defor-mation and stress elds around the inclusions, and the elasticmoduli of so composites. Our approach builds upon previoustheoretical works that have: focused on strain-dependentsurface stresses14,36–39 (which are relevant to nanoinclusions instiffer materials, but not for soer materials such as gels40), onlyconsidered isotropic loadings,12 used incorrect boundaryconditions13 (cf.41), or only considered incompressible solidsand employed a dipole approximation to calculate compositeproperties.42

II. Stretching individual inclusions

We begin by considering how surface tension affects Eshelby'ssolution for the deformation of individual inclusions embeddedin elastic solids subjected to far-eld stresses.1 We consider an

aYale University, New Haven, CT 06520, USA. E-mail: [email protected] Institute, University of Oxford, Oxford, OX1 3LB, UKcNordita, Royal Institute of Technology and Stockholm University, SE-10691

Stockholm, Sweden

Cite this: Soft Matter, 2015, 11, 672

Received 31st October 2014Accepted 1st December 2014

DOI: 10.1039/c4sm02413c

www.rsc.org/softmatter

672 | Soft Matter, 2015, 11, 672–679 This journal is © The Royal Society of Chemistry 2015

Soft Matter

PAPER

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isolated, incompressible, spherical droplet of radius Rembedded in a linear-elastic solid that is deformed by aconstant uniaxial far-eld stress, as shown in Fig. 1. Thedisplacement eld in the solid satises

(1 � 2n) V2u + V(V$u) ¼ 0, (1)

where n is Poisson's ratio of the solid.For far-eld boundary conditions, the stress in the solid s is

given by the applied uniaxial stress szz ¼ sN, sxx ¼ syy ¼ 0 incartesian coordinates. Stress and strain are related by

3ij ¼ 1

E

�ð1þ nÞsij � ndijskk

�; (2)

where dij is the Kronecker delta, and E is Young's modulus.Thus, the far-eld boundary conditions can also be written 3zz ¼3zz

N ¼ szzN/E, 3xx ¼ 3yy ¼ �n3zz

N. At the surface of the dropletthe elastic stress satises a generalised Young–Laplace equa-tion, which states that the difference in normal stress across aninterface depends on its surface stress, Y, and curvature K(equal to twice the mean curvature, or the sum of the principalcurvatures) via

s$n ¼ �pn + YK n (3)

(e.g. ref. 20 and 23). Here n is the normal to the deformeddroplet surface, s$n is the normal stress on the solid side, and pis the pressure in the droplet. The assumption that the surfacestress is simply an isotropic, strain-independent, surfacetension is appropriate for many so materials including gelsand elastomers.40 Expressions for n and K in terms of surfacedisplacements are given in Appendix A – these are differentfrom the expressions used in ref. 13 which ignored inclusiondeformation and assumed that K ¼ 2Y/R.14

We exploit symmetry and solve the problem in sphericalpolar co-ordinates by adopting as an ansatz the solution

ur ¼ F rþ Gr2þ P 2ðcos qÞ

��12nA r3 þ 2Brþ 2

ð5� 4nÞCr2

� 3Dr4

�;

and

uq ¼ dP 2ðcosqÞdq

��ð7� 4nÞA r3 þBrþ 2

ð1� 2nÞCr2

þDr4

�: (4)

as described by ref. 14. The surface displacements in the radialand q directions (q is the polar angle from the z-axis) are ur anduq respectively, P 2 is the Legendre polynomial of order 2, and Athrough G will be determined from the boundary conditions.

Applying the far-eld strain condition, we nd that A ¼ 0, F¼ (1� 2n)3zz

N/3 andB ¼ (1 + n)3zzN/3. Droplet incompressibility

requires thatðSu$n dS ¼

ð2p0

ðp0R2ur sinqdq df ¼0, where S is

the boundary of the stretched droplet, and the area integral isevaluated using results from differential geometry summarisedin Appendix A. This gives G ¼ �(1 � 2n)R33zz

N/3. Finally, byapplying the boundary condition (3) using eqn (2) to covertstresses to strains and displacements we obtain

C ¼5R3ð1þ nÞ

�R

L�ð1þ nÞ

�

6

�R

Lð7� 5nÞþ�

17� 2n�19n2��3zzN (5)

and

D ¼R5ð1þ nÞ

�R

L��1þ nþ 2n2

��R

Lð7� 5nÞ þ �

17� 2n�19n2� 3zz

N: (6)

Here Lh Y/E is the elastocapillary length, a material property ofthe solid–liquid interface. For perturbations of wavelengthmuch smaller than L, l� L, surface deformations are primarilyopposed by surface tension, whereas for l [ L, bulk elasticitysuppresses deformation of the surface (e.g. ref. 19, 25, 43 and44). With the expressions for A � G, eqn (4) gives us the exactdisplacement solutions. These also allow calculation of stressesin the solid: we convert displacements to strains (e.g. ref. 45)and then use eqn (2) to nd the non-zero stress components srr,srq, sqq and sff.

While these results are for uniaxial stress, they are readilyextended to provide the solution for general far-eld stresses. Inthe appropriate coordinate frame, the far-eld stress matrix isdiagonalisable so the only non-zero far-eld stresses are s1, s2and s3. Then, from linearity of the governing equations, we cancalculate the resulting displacements by simply summing thesolutions for uniaxial far-eld stresses s1, s2 and s3.

A. Inclusion shape

While Eshelby's results are scale-free, surface tensionmakes theresponse of a liquid inclusion strongly size-dependent. Wederive the equations describing the inclusion shapes inAppendix B. For large droplets, R [ L, the uid dropletdeforms more than the surrounding solid. In this limit, thedroplet shape only depends on sN/E ¼ 3zz

N and n, in agreementwith Eshelby's theory.1 However, as R approaches L, highinterfacial curvatures are suppressed by surface tension. For R� L, ur(R,q) ¼ 0 and the droplets remain spherical. This isvisualized in Fig. 2 for a uniaxially-stressed solid where the far-eld stress and strain are sN¼ 0.3E and 3zz

N¼ 0.3, respectively.

Fig. 1 Schematic diagram for uniaxial stretching of a single, incom-pressible droplet embedded in a linear-elastic solid. ‘ is the length ofthe deformed droplet in the stretch direction.

This journal is © The Royal Society of Chemistry 2015 Soft Matter, 2015, 11, 672–679 | 673

Paper Soft Matter

Here, the radial and polar displacements in the top two rows aremeasured relative to the far-eld displacement eld: ur

N¼ F r +2P2(cos q)Br and uq

N ¼ P20(cos q)Br.Changes in droplet shape are captured with an effective

droplet strain 3d ¼ (‘ � 2R)/R ¼ 2ur(R,0)/R, where ‘ is the long-axis of the droplet. For an incompressible solid, eqn (4) gives

3d ¼

0B@ 203zz

N

6þ 15L

R

1CA: (7)

This is plotted in Fig. 3(a). In both extremes of droplet size,the droplet shape is independent of size. In the capillary-dominated regime (R� L) the droplet stays spherical (3d¼ 0). Inthe large-droplet limit (R [ L), surface tension does not play arole, and Eshelby's results are recovered (3d ¼ 103xx

N/3). Thereis a smooth cross-over between these limits in the vicinity of R�L. Surface tension makes signicant changes to droplet shapefor droplet radii up to about 100L.

Although we only consider the uniaxial stress case above, theresults can be generalised to the more general case of triaxialfar-eld strains. For example, for an incompressible solid inplane stress conditions (s1, s2 s 0, s3

N ¼ 0, as in our recentexperiments34), we calculate the solution by the summationtechnique described earlier. This gives the length of the dropletin the 1-direction as

‘1 ¼ 2R

26641þ 5

�2s1

N � s2N�

E

�6þ 15

L

R

3775: (8)

We recently compared this result to experimental measure-ments of individual liquid inclusions in so, stretched solids.We found good agreement over a wide range of droplet sizes,substrate stiffnesses and applied strains.34

B. Stress focussing by inclusions

The macroscopic strength of composites can be reduced due tostress focusing by inclusions. According to the Tresca yieldcondition, the solid will yield when the shear stress exceeds acritical value sc. Fig. 2 (bottom row) shows the maximum shearstress, smax, for an incompressible solid with a uniaxial far-eldstress for various values of R/L. The maximum shear stress isgreatest at the tip of the inclusion, and the value there increasessignicantly as R is reduced below L. In fact at the inclusion tip

smax

�r ¼ R; q ¼ 0

�¼ smaxN

5

�2þ 9

L

R

6þ 15L

R

: (9)

This is plotted in Fig. 3(b). There is a signicant increase inshear-stress concentration as surface tension becomes more

Fig. 2 Examples of droplets embedded in an incompressible solid under uniaxial strain with 3zzN ¼ 0.3. Top: excess radial displacements (ur �

urN)/R caused by the presence of the inclusion. The elastic dipole around the inclusion changes sign as R/L increases. Middle: excess tangential

displacements (uq � uqN)/R q is the polar angle from the z-axis. Bottom: shear-stress concentration factor smax/smax

N. When surface tensiondominates, smax is significantly increased at the inclusion tip. The black arrows denote the stretch of the host material.


Soft Matter Paper

important with smax(R,0) increasing from 5smaxN/3 when R [

L, to 3smaxN when R � L.

These results suggest that surface tension could weaken aso composite when inclusions fall below a size �100L. Thisalso means that the applied strain at which yielding is expectedto occur is no longer independent of the size of the liquidinclusion, as would be predicted from Eshelby's results, butdepends on the parameter R/L. These results hint at thepotential role of surface tension for fracture mechanics in somaterials where a critical value is the crack-tip stress. Thecapillary-induced stress focussing seen here shows how surfacetension could potentially signicantly alter this value.46

C. Dipole signature of inclusions

At nite concentrations, inclusions interact at a distancethrough their far-eld stresses. This can be important fordetermining mechanical properties of dilute composites (e.g.ref. 42 and 47). The far-eld solutions are convenientlyexpressed by a multipole expansion.

Here, inclusions appear as force dipoles in the far-eld.From eqn (4), we nd the leading order terms in the inclusion-induced displacements (ur � ur

N, uq � uqN) are proportional to

1/r2. This corresponds to a force dipole in an elastic body

Pij ¼ Pzizj + Pedij, (10)

with z being the unit vector in the z-direction. The displacementelds due to the dipoles are48

ur ¼ð1þ nÞ�ð1� 2nÞðPþ 3PeÞ þ P 2ðcos qÞð5� 4nÞP�

12pEð1� nÞr2 ; (11)

and

uq ¼ ð1þ nÞð1� 2nÞ12pEr2ð1� nÞ

dP 2ðcos qÞdq

P: (12)

Thus, from comparison with eqn (4),

P ¼ 24CpE(1 � n)/(1 + n), (13)

and

Pe ¼ 4pEð1� nÞG � 2C ð1� 2nÞð1þ nÞð1� 2nÞ : (14)

The rst dipole, P, is a force dipole of two point forces on thez-axis which also act along the z-axis – i.e. parallel to the appliedfar-eld stress. The second term Pe is an isotropic centre ofexpansion.48 When n ¼ 1/2, the displacement eld due to Pevanishes, and P ¼ 8pCE.

Intriguingly, the dipole strength, P, can be positive or nega-tive. Fig. 3(c) shows the normalised dipole strength P/sNR3 of aliquid inclusion in incompressible solid with a uniaxial appliedstress. For large inclusions (R > 1.5L), P > 0 and the dipole is apair of outward pointing point forces. This increases soliddisplacements – consistent with a weak point in the solid. Forsmall inclusions (R < 1.5L), P < 0 and so the dipole opposes theapplied far-eld stress, acting like a stiff point in the solid. Thesign switch is clearly seen in the displacement elds of Fig. 2. AtR ¼ 1.5L, the inclusion has no effect on the far-eld elasticityeld and is effectively invisible (e.g., see ref. 49).

III. Soft composites

We have shown that the surface tension of a small liquid dropletin a so linear elastic solid resists deformation imposed by far-eld stretch. Therefore, we expect that the dispersion of smallliquid droplets within a solid can increase its apparent macro-scopic stiffness. We calculate the effective Young's modulus Ecof a composite containing a dilute quantity of monodispersedroplets by following Eshelby's original approach.1,13 First, wecalculate the excess energy W due to the presence of a singleinclusion when a solid is uniaxially stretched. Then, weconsider uniaxial stretching of a dilute composite of noninter-acting inclusions. If the applied stress is szz ¼ sN, the strainenergy density of the composite is

Fig. 3 Liquid inclusion characteristics as a function of size R/L forinclusions in an incompressible solid with an applied uniaxial far-fieldstress as shown in Fig. 1 (a) Droplet strain, 3d¼ (‘� 2R)/R divided by far-field strain 3N only depends on R/L. When R/L � 1, surface tensiondominates and there is no droplet deformation. When R/L [ 1,surface tension is negligible and the shape prediction is that of classicalEshelby theory – given by the dash-dotted line. The dashed line showsthe material stretch, (‘ � 2R)/R ¼ 3N. (b) The shear-stress concentra-tion factor at the inclusion tip (r ¼ R, q ¼ 0). This corresponds to thehighest shear stress in the solid around the inclusion (see Fig. 2, bottomrow). Dash-dotted lines show surface-tension dominated and Eshelbylimits: smax/smax

N ¼ 3, 5/3 respectively. (c) The far-field dipole causedaround the inclusion. Note that this dipole changes sign at R ¼ 1.5L,indicating the transition between inclusion stiffening and inclusionsoftening of the composite.


Paper Soft Matter

E ¼ (sN)2/(2E) + WF/(4pR3/3) ¼ (sN)2/(2Ec), (15)

where F is the volume fraction of inclusions. The secondequality comes from the relationship between the strain energydensity and the effective modulus of the material, allowingcalculation of Ec from W.

The excess energy due to the presence of a single elasticinclusion in a uniaxially-stressed solid is

W ¼ 1

2

ðVi

sij3ij � sij

N3ijN�dV

þ 1

2

ðVm

sij3ij � sij

N3ijN�dV þ YDS: (16)

Here we assume that the inclusion is an elastic solid forgenerality – the droplet is the limiting case of zero shear modulus.

The volumes of the elastic matrix outside of the inclusionand the inclusion Vm and Vi, respectively, the far-eld stresses/strains are sij

N and 3ijN respectively, and the change in surface

area of the droplet upon stretching is DS. Using the divergencetheorem, the stress boundary condition (3), and the fact that inthe far-eld, sij

N ¼ sij,

W ¼ 1

2

ðVm

sij

N3ij � sij3ijN�dV þ 1

2

ðS þ

nisij

Nuj � nisijujN�dS

� Y

2

ðSK uinidS þ YDS:

(17)

Integration on the matrix side of the droplet surface S isdenoted by S +. From eqn (2), the rst term is zero, so Wdepends only upon displacements and stresses at the dropletsurface. Using our earlier results (e.g. eqn (4)), along withsecond-order (in the displacement) versions of the expressionsfor n, K , dS and DS shown in Appendix A, we obtain W for thecase of a uniaxial far-eld stress sN:

W ¼ 2pR3sN2ð1� nÞ

�

�R

Lð1þ 13nÞ�

9� 2nþ 5n2 þ 16n3

��

Eð1þ nÞ�R

Lð7� 5nÞþ�

17� 2n� 19n2�� : (18)

Finally, from eqn (15),

Ec

E¼ 1þ

3ð1� nÞ�R

Lð1þ 13nÞ��

9� 2nþ 5n2 þ 16n3��

ð1þ nÞ�R

Lð7� 5nÞ þ ð17� 2n� 19n2Þ

� F

2664

3775

�1

(19)

For an incompressible solid n ¼ 1/2 and we have

Ec

E¼

1þ 5

2

L

R5

2

L

Rð1� FÞ þ

�1þ 5

3F

: (20)

Fig. 4 plots the results of eqn (20) and shows the dramaticinuence of capillarity on so composite stiffness. When surfacetension is negligible (R [ L), the composite becomes morecompliant as the density of droplets increases – in exact agree-ment with Eshelby's prediction of Ec/E ¼ (1 + 5F/3)�1 (dottedcurve), and qualitatively agreeing with other classical compositelaws (e.g. ref. 2 and 3). However Eshelby's predictions break downwhen R ( 100L. In fact, when R < 1.5L, increasing the density ofdroplets causes the solid to stiffen, consistent with the dipolesign-switching seen earlier. In the surface-tension dominatedlimit,R�L, thedroplets stay spherical, andwend themaximumachievable composite stiffness Ec¼ E/(1�F) (dash-dotted curve).Note that the droplets do not behave like rigid particles in thislimit, for which Ec¼ E/(1� 5F/2) (ref. 1) (dashed curve). Althoughthedroplets remainspherical due tocapillarity, there arenon-zerotangential displacements, unlike the case of rigid particles.

These results agree with experiments. Recently, we made socomposites of glycerol droplets embedded in so siliconesolids. In quantitative agreement with the theory, we saw stiff-ening of solids by droplets in compliant solids, and soening instiffer solids.34 In the dilute limit (F / 0), eqn (20) matcheswith recent theoretical predictions (derived using the dipoleapproximation for inclusions in incompressible solids) thatdescribe experimental measurements of shear moduli ofemulsions containing monodisperse bubbles.35,42

IV. Conclusions

We have modied Eshelby's inclusion theory to include surfacetension for liquid inclusions in a linear-elastic solid, giving boththe microscopic behaviour and the macroscopic effects ofinclusions in composites. We have shown that surface tensionstiffens small inclusions, and focusses shear stresses at theinclusion tips. Thus composites with small, capillary-domi-nated inclusions will be stiffer but may be weaker. This stress-concentration illustrates the potentially strong role of surface

Fig. 4 The stiffness of soft composites. Young's modulus ofcomposites of droplets embedded in linear-elastic solids, Ec as afunction of liquid content. The dotted curve shows Eshelby's predic-tion without surface tension. The dash-dotted curve shows thesurface-tension dominated limit, R/L � 1. The dashed curve showEshelby's prediction for rigid spheres embedded in an elastic solid.


Soft Matter Paper

tension in the failure of so-solids, highlighting the relevanceof this work to emerging elds like fracture mechanics andplasticity in so materials (e.g. ref. 46, 50 and 51).

Inclusions with surface tension can be viewed, at leadingorder, as elastic dipoles in a solid. The sign of the dipolecaptures the stiffening behaviour due to capillarity. Treatinginclusions as dipoles also offers a simplied picture of inclu-sions that give the interactions between features in elasticbodies, and can streamline calculations of bulk compositeproperties via standard theories. The analytic theory presentedfor bulk composite stiffness, which incorporates the entireelastic eld around inclusions, validates the dipole approach byrecovering previous results for incompressible materials in thelimit of dilute composites.35,42

Our work is applicable to a wide variety of so materialproblems. Most obviously it can be directly applied tocomposites comprising so materials such as gels and elasto-mers. As a specic example, we have shown how surface tensioneffects allow elastic cloaking, with inclusions of size R ¼ 1.5Lbeingmechanically invisible. Our work also has interesting usesin mechanobiology, as biological tissue is predominantly so.For example, a recent study embedded droplets in biologicaltissue and observed their deformations to extract local aniso-tropic stresses.52 The coupling between microscopic andmacroscopic stress also plays an important role in the tensionalhomeostasis of so tissues.53,54 Although we have only consid-ered liquid inclusions here our analysis can be repeated formore general so composites with elastic inclusions in placeof liquid droplets. In that case, we expect that similarcapillary effects to those presented here will be seen whenever R( 100Y/Ei, 100Y/Em with Ei/Em being the inclusion/matrixstiffnesses respectively.

AppendixA. Differential geometry

To calculate the effect of surface tension on the shape of adroplet embedded in a so solid, we need expressions for thenormal to the droplet surface, its curvature, and surface area interms of the surface displacements. We consider an initiallyspherical droplet with the position of its surface given by x ¼(R,0,0), and apply a uniaxial stretch so that x /x0 ¼ (R + ur, uq,0). From axisymmetry, the ur, uq are independent of the angle f.

Wecalculate thenormal to thedroplet surface,n, by taking thecross-product of the surface tangent vectors, vx0/vq and vx0/vf,55

n ¼vx0

vq� vx0

vf��vx0

vq� vx0

vf

��; (21)

with

vx0 vq ¼�vur

vq� uq;Rþ ur þ vuq

vq; 0

; (22)

and

vx0/vf ¼ (0, 0, (R + ur)sin q + uq cos q). (23)

At leading order in u we nd

n ¼�1;

uq

R� 1

R

vur

vq; 0

: (24)

The droplet surface curvature, K , can be calculated fromdifferential geometry using the rst and second fundamentalforms:55

K ¼ efGf � 2ffFf þ gfEf

EfGf � Ff2

(25)

where

Ef ¼ vx0

vq$vx0

vq; Ff ¼ vx0

vq$vx0

vf; Gf ¼ vx0

vf$vx0

vf; (26)

and

ef ¼ n$v2x0

vq2; ff ¼ n$

v2x0

vq vf; gf ¼ n$

v2x0

vf2: (27)

Thus, at leading order in u,

K ¼ 2

R� 1

R2

�2ur þ cot q

vur

vqþ v2ur

vq2

: (28)

Using the results above, we also obtain the area ele-ment dS ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEfGf � Ff 2

pdqdf.55 At leading order in u,

dS ¼"R2 sin qþ R

�uq cos qþ 2ur sin qþ vuq

vqsin q

#dq dF;

(29)

and aer integration we obtain the droplet surface area

S ¼ 4pR2 þð2p0

ðp0

"R

�uq cos qþ 2ur sin qþ vuq

vqsin q

#dq dF

¼ 4pR2 þð2p0

ð2p0

2ur sin q dq dF:

(30)

B. The shape of a uniaxially stretched droplet

We determine the shape of a uniaxially-stretched droplet byusing the calculated expressions for A � G in eqn (4) to obtainthe surface displacements:

urðR; qÞR

¼ 53zzN

2

264

�1� n2

�½1þ 3 cosð2qÞ�ð7� 5nÞþ L

R

�17� 2n�19n2

�375 (31)

and

uqðR; qÞR

¼ � 153zzN

2

�1þ L

Rð1þ nÞ

��1� n2

�sinð2qÞ

ð7� 5nÞ þ L

R

�17� 2n� 19n2

�2664

3775: (32)


Paper Soft Matter

When L [ R, we see that radial displacements vanish, andthe inclusions remain spherical. In the opposite limit, R [ L,the inclusion shape becomes independent of its size, as pre-dicted by Eshelby's results.

Acknowledgements

We thank Peter Howell and Alain Goriely for helpful conversa-tions. We are grateful for funding from the National ScienceFoundation (CBET-1236086) to ERD, the Yale University Bate-man Interdepartmental Postdoctoral Fellowship to RWS andthe John Simon Guggenheim Foundation, the SwedishResearch Council, and a Royal Society Wolfson Research MeritAward to JSW.

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Soft Matter Paper

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Paper Soft Matter

Stiffening solids with liquid inclusions

Robert W. Style,1 Rostislav Boltyanskiy,1 Benjamin Allen,1 Katharine E.

Jensen,1 Henry Foote,2 John S. Wettlaufer,1, 3 and Eric R. Dufresne1, ∗

1Yale University, New Haven, CT 06520, USA2University of North Carolina - Chapel Hill, NC 27516, USA

3Mathematical Institute, University of Oxford, Oxford, OX1 3LB, UK(Dated: October 30, 2014)

I. SINGLE-DROPLET EXPERIMENTS

We image individual droplet deformation in a soft solidusing the setup shown schematically in Figure 1. Wecoat a very dilute composite of ionic liquid droplets insilicone gel on a thin stretchable sheet (Silicone Sheeting.005” NRV Gloss/Gloss, Speciality Manufacturing Inc.).Before coating the composite, we attach 100nm fluores-cent beads to the surface of the stretchable sheet, usingthe same process described in previous work [1, 2]. Wecoat the silicone in a two-step process to ensure that alldroplets are embedded in the centre of the layer of softsolid. First, we spin coat a layer of pure silicone on thesheet and cure it at room temperature for 12 hours. Sec-ondly, we mix a very dilute suspension of ionic liquiddroplets in silicone (∼ 5µl in 6g of silicone), and spincoat this on top of the previous layer. Note that no sur-factant is added. As the second layer cures slowly atroom temperature for 12 hours, the droplets sink downto the top of the first layer of silicone where they becomeembedded.We stretch the sample with a home made uniaxial

stretcher. We clamp the sample so that there is a free,rectangular area between the two clamped ends. Westretch the sample in the x−direction by pulling theclamped ends apart horizontally. There is no samplewrinkling due to the small initial gap between the plates(∼ 1cm) relative to the sample width (∼ 3 − 4cm). Weimage droplets in a small area around the centre of thesample, thus avoiding edge effects and ensuring that allmeasured droplets experience the same applied strain.

A. Measuring applied strain

Although we stretch the sample only in thex−direction, there is always a small amount of associatedcontraction in the y−direction. We accurately quantifythe applied strain by tracking the positions of the fluo-rescent beads on the surface of the stretchable sheeting.We image a chosen cluster of beads each time that weincrease the stretch applied to the sample. Then usingimage analysis, we track the positions of the beads be-fore and after stretch (x0 and x1 respectively) [2]. The

∗ [email protected]

FIG. 1. Detailed schematic of the single droplet stretchingexperiment. The sample is clamped and stretched in thex−direction, resulting in a large tensile strain ε∞x parallel tothe stretch, and a small compressive strain ε∞y in the perpen-dicular direction.

stretch is an affine transformation, so x1 = Mx0 + T.We determine M and T by a least-squares minimisa-tion procedure using the data from each of the points inthe stretched cluster. This mapping accurately capturesthe transformation for all points. From nonlinear elas-ticity [3], the right Cauchy-Green deformation tensor isC = MTM, which has eigenvalues λ2

1, λ22. The principal

strains are then ε∞x = λ1 − 1 and ε∞y = λ2 − 1.We check to ensure that there is no slippage of the sam-

ple by imaging the fluorescent beads both immediatelybefore and after imaging droplet deformations. Thesemeasurements show no detectable change in the appliedstrains, and suggest that our strain measurement tech-nique is accurate to ±0.1%.

B. Extracting droplet shape

We image droplets with bright field microscopy usingmonochromatic light coming through a green-light fil-ter. We use a 60x water objective (NA 1.2) to imagelarger droplets, adding a 1.5x Optivar lens to achieve90x for smaller droplets. We manually focus on the waist(widest point of the droplet), to obtain images such asthose shown in Figure 1 of the main paper. We extract

Stiening solids with liquid inclusionsStiffening solids with liquid inclusions

Robert W. Style,1 Rostislav Boltyanskiy,1 Benjamin Allen,1 Katharine E.

Jensen,1 Henry Foote,2 John S. Wettlaufer,1, 3 and Eric R. Dufresne1, ∗

1Yale University, New Haven, CT 06520, USA2University of North Carolina - Chapel Hill, NC 27516, USA

3Mathematical Institute, University of Oxford, Oxford, OX1 3LB, UK(Dated: October 30, 2014)

I. SINGLE-DROPLET EXPERIMENTS

We image individual droplet deformation in a soft solidusing the setup shown schematically in Figure 1. Wecoat a very dilute composite of ionic liquid droplets insilicone gel on a thin stretchable sheet (Silicone Sheeting.005” NRV Gloss/Gloss, Speciality Manufacturing Inc.).Before coating the composite, we attach 100nm fluores-cent beads to the surface of the stretchable sheet, usingthe same process described in previous work [1, 2]. Wecoat the silicone in a two-step process to ensure that alldroplets are embedded in the centre of the layer of softsolid. First, we spin coat a layer of pure silicone on thesheet and cure it at room temperature for 12 hours. Sec-ondly, we mix a very dilute suspension of ionic liquiddroplets in silicone (∼ 5µl in 6g of silicone), and spincoat this on top of the previous layer. Note that no sur-factant is added. As the second layer cures slowly atroom temperature for 12 hours, the droplets sink downto the top of the first layer of silicone where they becomeembedded.We stretch the sample with a home made uniaxial

stretcher. We clamp the sample so that there is a free,rectangular area between the two clamped ends. Westretch the sample in the x−direction by pulling theclamped ends apart horizontally. There is no samplewrinkling due to the small initial gap between the plates(∼ 1cm) relative to the sample width (∼ 3 − 4cm). Weimage droplets in a small area around the centre of thesample, thus avoiding edge effects and ensuring that allmeasured droplets experience the same applied strain.

A. Measuring applied strain

Although we stretch the sample only in thex−direction, there is always a small amount of associatedcontraction in the y−direction. We accurately quantifythe applied strain by tracking the positions of the fluo-rescent beads on the surface of the stretchable sheeting.We image a chosen cluster of beads each time that weincrease the stretch applied to the sample. Then usingimage analysis, we track the positions of the beads be-fore and after stretch (x0 and x1 respectively) [2]. The

∗ [email protected]

FIG. 1. Detailed schematic of the single droplet stretchingexperiment. The sample is clamped and stretched in thex−direction, resulting in a large tensile strain ε∞x parallel tothe stretch, and a small compressive strain ε∞y in the perpen-dicular direction.

stretch is an affine transformation, so x1 = Mx0 + T.We determine M and T by a least-squares minimisa-tion procedure using the data from each of the points inthe stretched cluster. This mapping accurately capturesthe transformation for all points. From nonlinear elas-ticity [3], the right Cauchy-Green deformation tensor isC = MTM, which has eigenvalues λ2

1, λ22. The principal

strains are then ε∞x = λ1 − 1 and ε∞y = λ2 − 1.We check to ensure that there is no slippage of the sam-

ple by imaging the fluorescent beads both immediatelybefore and after imaging droplet deformations. Thesemeasurements show no detectable change in the appliedstrains, and suggest that our strain measurement tech-nique is accurate to ±0.1%.

B. Extracting droplet shape

We image droplets with bright field microscopy usingmonochromatic light coming through a green-light fil-ter. We use a 60x water objective (NA 1.2) to imagelarger droplets, adding a 1.5x Optivar lens to achieve90x for smaller droplets. We manually focus on the waist(widest point of the droplet), to obtain images such asthose shown in Figure 1 of the main paper. We extract

SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHYS3181

NATURE PHYSICS | www.nature.com/naturephysics 1

© 2014 Macmillan Publishers Limited. All rights reserved.


2

droplet shapes using an ad-hoc MATLAB code that ap-plies a band-pass filter, then thresholds the image to findthe outline of the droplet. The code then fills in the areaof the droplet and uses MATLAB’s regionprops functionto find the ellipse that best fits the droplet shape. Thefitted ellipses always show excellent agreement with thedata, showing that droplets always remain elliptical. Fi-nally, we obtain the length and width of the droplet fromthe fitted shape.Some larger droplets are slightly too large to fit in the

field of view. For these droplets, we manually click atten points, widely spaced around the droplet perimeter,and fit an ellipse through the points. Again, these ellipsesshow excellent agreement with the outline of the droplets.

C. Preparation and chacterisation of silicone

We make soft silicones from pure components toavoid complications associated with additives that arecommon in commercial silicones. We combine siliconebase: vinyl-terminated polydimethylsiloxane (DMS-V31,Gelest Inc) with a crosslinker: trimethylsiloxane termi-nated (25-35% methylhydrosiloxane) - dimethylsiloxanecopolymer (HMS-301, Gelest Inc) in different ratios toobtain different stiffnesses. The reaction is catalysed bya platinum-divinyltetramethyldisiloxane complex in xy-lene (SIP6831.2, Gelest Inc). To make the silicone, weprepare two parts: Part A consists of the base with 0.05wt% of the catalyst. Part B consists of the base with 10wt% cross linker. We mix parts A and B together in aratio of 9:1 or 4:1 by weight to create softer/stiffer sil-icone solids respectively. The parts are mixed togetherthoroughly, degassed in a vacuum, and then either curedat room temperature, or at 60◦C.

Rheology data for the 9:1 mixture cured overnight atroom temperature is shown in Figure 2. The frequencysweep in Figure 2(A) shows that the material behaves asa solid under static loadings, as the shear modulus G′

tends to a finite value at low frequency, while the lossmodulus G′′ → 0 [2]. The strain sweep shows that thesolid behaves elastically up to high O(100%) strains, andthat elastic modulus is essentially independent of strain.Young’s modulus is related to shear modulus by E =2G′(1 + ν) [4], so for these silicones E = 3G′; previouswork has shown that silicones (and gels in general) areessentially incompressible, so Poisson’s ratio is ν = 1/2[2, 5]. Thus this sample has a Young modulus, E = 1.7kPa. The 4:1 sample has E = 100kPa. When the solidsare cured at 60◦C for two hours, their moduli are slightlystiffer. Young’s moduli are 6.2kPa and 208 kPa for the9:1 and 4:1 ratios, respectively.

D. Droplets in a stiffer solid

As a control experiment, we stretched ionic liquiddroplets embedded in stiffer silicone with E = 100kPa

10−2

10−1

100

101

102

100

101

102

103

Frequency [rad/s]

Mo

du

lus [P

a]

10−1

100

101

102

103

101

102

103

Strain [%]

Mo

du

lus [P

a]

G’

G’’

G’’

G’

(A)

(B)

FIG. 2. Rheology tests on soft silicone mixed 9:1 A:B, andcured at room temperature for 12 hours, performed on anARES-LS1 rheometer. (A) Frequency sweep at 0.5% strain.(B) Strain sweep at room temperature at 1rad/s.

0 10 20 30 40 501

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

l/2 [µm]

l/w

εx

∞=6.4%

εx

∞=23.8%

εx

∞=14.2%

FIG. 3. Aspect ratio of stretched ionic-liquid droplets in a stiff(E = 100kPa) silicone gel as a function of size and strain.Aspect ratio is essentially independent of applied strain, inagreement with Eshelby’s predictions [6]. Contrast with theresults in a soft solid (Figure 2B of the main paper).

(Figure 3). In this case, droplet shape is essentially in-dependent of size, in agreement with Eshelby’s predic-tions [6]. There is a little rounding of the very smallestdroplets, which may be due to surface tension effects, ordue to blurring of the droplet images as their size ap-proaches the diffraction limit of light. However, this issmall in comparison to the rounding of small dropletsseen in the softer silicone (Figure 2B of the main paper).Importantly, this verifies that the rounding effect in the

3

softer solid is not due to optical artifacts.

II. COMPOSITE EXPERIMENTS

We make composites by blending together premixed(curing) silicone, silicone surfactant and glycerol. Weuse 20g of silicone – mixed 9:1 A:B for the soft-matrixcomposites, and 4:1 A:B for the stiff-matrix composites.This is combined with 1g of silicone surfactant (Gransurf50C-HM, Grant Industries), and varying quantities ofglycerol. We blend together the ingredients with a handblender for approximately one minute, ensuring that theresulting emulsion is well-mixed. We degas the emulsionin a vacuum pump until all air bubbles are removed, pourit to completely fill a petri dish (10mm deep x 35mm indiameter, Falcon 353001, Corning Inc.) and immediatelyplace it in the oven for two hours. This procedure gaverepeatable measurements of the stiffness of the compos-ites.

A. Composite microstructure

We examine the microstructure of the composites bysmearing a thin film of the curing composite on a glassslide, curing it at 60◦C for two hours, and then imagingit with brightfield microscopy. Example micrographs areshown in Figure 4. We use a 60x water objective (NA1.2) with a 1.5x Optivar to image at 90x under green,monochromatic light. The example images show thatdroplets have a typical size of O(1µm), and are some-what polydisperse. Samples with higher glycerol contentswere difficult to image, as the high density of droplets in-creased light scattering and sample opacity. Figure 4(B)demonstrates how the embedded droplets appear to formchain-like structures at higher concentrations.

B. The influence of surfactant on matrix elasticity

It is important to ensure that the solid constituent ofthe composites has a constant Young’s modulus E thatis independent of glycerol content, φ. This may not betrue if the bulk surfactant concentration in the compos-ite, c, (i.e. the surfactant that is not adsorbed to thesurface of glycerol droplets) alters E, and if c depends onφ. In fact, Figure 5(A) shows that Young’s modulus ofsoft silicone (9:1 A:B) does depend on the bulk concen-tration of surfactant that is mixed in with the siliconeupon curing. The relationship is approximately linearwith dE/dc = 0.76kPa/wt%. This presumably occursbecause the surfactant dissolved in the silicone alters itscuring process. Thus we need to check that c does notchange with φ.We measure c as a function of composite φ by mea-

suring the surface tension of the composite’s uncuredsilicone phase, γ – which a strong function of c. We

(A)

(B)

FIG. 4. Brightfield micrographs showing the droplet sizesin typical soft-matrix composites for glycerol contents φ =0.5%, 4.4% in panels (A,B) respectively. This is the soft-matrix composite made with 9:1 A:B ratio in preparing thesilicone. Samples with higher volume fractions of glycerolwere hard to image due to light scattering causing sampleopacity.

make up emulsions using the same recipe as for the soft-matrix composites (using 9:1 A:B silicone), but withoutsilicone catalyst. We then gently centrifuge the sampleuntil the glycerol droplets have sedimented out of theemulsion, and pipette off the silicone phase. We measurethe surface tension of a pure glycerol droplet hangingin the silicone phase using the pendant droplet method

2 NATURE PHYSICS | www.nature.com/naturephysics

SUPPLEMENTARY INFORMATION DOI: 10.1038/NPHYS3181



2

droplet shapes using an ad-hoc MATLAB code that ap-plies a band-pass filter, then thresholds the image to findthe outline of the droplet. The code then fills in the areaof the droplet and uses MATLAB’s regionprops functionto find the ellipse that best fits the droplet shape. Thefitted ellipses always show excellent agreement with thedata, showing that droplets always remain elliptical. Fi-nally, we obtain the length and width of the droplet fromthe fitted shape.Some larger droplets are slightly too large to fit in the

field of view. For these droplets, we manually click atten points, widely spaced around the droplet perimeter,and fit an ellipse through the points. Again, these ellipsesshow excellent agreement with the outline of the droplets.

C. Preparation and chacterisation of silicone

We make soft silicones from pure components toavoid complications associated with additives that arecommon in commercial silicones. We combine siliconebase: vinyl-terminated polydimethylsiloxane (DMS-V31,Gelest Inc) with a crosslinker: trimethylsiloxane termi-nated (25-35% methylhydrosiloxane) - dimethylsiloxanecopolymer (HMS-301, Gelest Inc) in different ratios toobtain different stiffnesses. The reaction is catalysed bya platinum-divinyltetramethyldisiloxane complex in xy-lene (SIP6831.2, Gelest Inc). To make the silicone, weprepare two parts: Part A consists of the base with 0.05wt% of the catalyst. Part B consists of the base with 10wt% cross linker. We mix parts A and B together in aratio of 9:1 or 4:1 by weight to create softer/stiffer sil-icone solids respectively. The parts are mixed togetherthoroughly, degassed in a vacuum, and then either curedat room temperature, or at 60◦C.

Rheology data for the 9:1 mixture cured overnight atroom temperature is shown in Figure 2. The frequencysweep in Figure 2(A) shows that the material behaves asa solid under static loadings, as the shear modulus G′

tends to a finite value at low frequency, while the lossmodulus G′′ → 0 [2]. The strain sweep shows that thesolid behaves elastically up to high O(100%) strains, andthat elastic modulus is essentially independent of strain.Young’s modulus is related to shear modulus by E =2G′(1 + ν) [4], so for these silicones E = 3G′; previouswork has shown that silicones (and gels in general) areessentially incompressible, so Poisson’s ratio is ν = 1/2[2, 5]. Thus this sample has a Young modulus, E = 1.7kPa. The 4:1 sample has E = 100kPa. When the solidsare cured at 60◦C for two hours, their moduli are slightlystiffer. Young’s moduli are 6.2kPa and 208 kPa for the9:1 and 4:1 ratios, respectively.

D. Droplets in a stiffer solid

As a control experiment, we stretched ionic liquiddroplets embedded in stiffer silicone with E = 100kPa

10−2

10−1

100

101

102

100

101

102

103

Frequency [rad/s]

Mo

du

lus [P

a]

10−1

100

101

102

103

101

102

103

Strain [%]

Mo

du

lus [P

a]

G’

G’’

G’’

G’

(A)

(B)

FIG. 2. Rheology tests on soft silicone mixed 9:1 A:B, andcured at room temperature for 12 hours, performed on anARES-LS1 rheometer. (A) Frequency sweep at 0.5% strain.(B) Strain sweep at room temperature at 1rad/s.

0 10 20 30 40 501

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

l/2 [µm]

l/w

εx

∞=6.4%

εx

∞=23.8%

εx

∞=14.2%

FIG. 3. Aspect ratio of stretched ionic-liquid droplets in a stiff(E = 100kPa) silicone gel as a function of size and strain.Aspect ratio is essentially independent of applied strain, inagreement with Eshelby’s predictions [6]. Contrast with theresults in a soft solid (Figure 2B of the main paper).

(Figure 3). In this case, droplet shape is essentially in-dependent of size, in agreement with Eshelby’s predic-tions [6]. There is a little rounding of the very smallestdroplets, which may be due to surface tension effects, ordue to blurring of the droplet images as their size ap-proaches the diffraction limit of light. However, this issmall in comparison to the rounding of small dropletsseen in the softer silicone (Figure 2B of the main paper).Importantly, this verifies that the rounding effect in the

3

softer solid is not due to optical artifacts.

II. COMPOSITE EXPERIMENTS

We make composites by blending together premixed(curing) silicone, silicone surfactant and glycerol. Weuse 20g of silicone – mixed 9:1 A:B for the soft-matrixcomposites, and 4:1 A:B for the stiff-matrix composites.This is combined with 1g of silicone surfactant (Gransurf50C-HM, Grant Industries), and varying quantities ofglycerol. We blend together the ingredients with a handblender for approximately one minute, ensuring that theresulting emulsion is well-mixed. We degas the emulsionin a vacuum pump until all air bubbles are removed, pourit to completely fill a petri dish (10mm deep x 35mm indiameter, Falcon 353001, Corning Inc.) and immediatelyplace it in the oven for two hours. This procedure gaverepeatable measurements of the stiffness of the compos-ites.

A. Composite microstructure

We examine the microstructure of the composites bysmearing a thin film of the curing composite on a glassslide, curing it at 60◦C for two hours, and then imagingit with brightfield microscopy. Example micrographs areshown in Figure 4. We use a 60x water objective (NA1.2) with a 1.5x Optivar to image at 90x under green,monochromatic light. The example images show thatdroplets have a typical size of O(1µm), and are some-what polydisperse. Samples with higher glycerol contentswere difficult to image, as the high density of droplets in-creased light scattering and sample opacity. Figure 4(B)demonstrates how the embedded droplets appear to formchain-like structures at higher concentrations.

B. The influence of surfactant on matrix elasticity

It is important to ensure that the solid constituent ofthe composites has a constant Young’s modulus E thatis independent of glycerol content, φ. This may not betrue if the bulk surfactant concentration in the compos-ite, c, (i.e. the surfactant that is not adsorbed to thesurface of glycerol droplets) alters E, and if c depends onφ. In fact, Figure 5(A) shows that Young’s modulus ofsoft silicone (9:1 A:B) does depend on the bulk concen-tration of surfactant that is mixed in with the siliconeupon curing. The relationship is approximately linearwith dE/dc = 0.76kPa/wt%. This presumably occursbecause the surfactant dissolved in the silicone alters itscuring process. Thus we need to check that c does notchange with φ.We measure c as a function of composite φ by mea-

suring the surface tension of the composite’s uncuredsilicone phase, γ – which a strong function of c. We

(A)

(B)

FIG. 4. Brightfield micrographs showing the droplet sizesin typical soft-matrix composites for glycerol contents φ =0.5%, 4.4% in panels (A,B) respectively. This is the soft-matrix composite made with 9:1 A:B ratio in preparing thesilicone. Samples with higher volume fractions of glycerolwere hard to image due to light scattering causing sampleopacity.

make up emulsions using the same recipe as for the soft-matrix composites (using 9:1 A:B silicone), but withoutsilicone catalyst. We then gently centrifuge the sampleuntil the glycerol droplets have sedimented out of theemulsion, and pipette off the silicone phase. We measurethe surface tension of a pure glycerol droplet hangingin the silicone phase using the pendant droplet method





4

(with an error of ∼ ±0.5mN/m). The results in Figure5(B) show that surface tension initially increases quicklywith increasing glycerol content of the emulsion, and thenplateaus for φ � 3%. Figure 5(C) shows how the glycerol-silicone surface tension depends on bulk surfactant con-centration, from pendant droplet experiments on silicone(9:1 A:B ratio, without silicone catalyst). By combiningthe results of Figure 5(B,C) we see that there is morebulk surfactant in the composites at low glycerol con-tents, but that it reduces, reaching a constant for largerglycerol contents.These results suggest that we restrict the experiments

to composites with φ > 3% in order to ensure that Eis constant for the silicone phase. For these composites,from Figure 5(B), we can estimate the solid-liquid surfacetension as Υ = 13.75 ± 0.75mN/m. We justify this esti-mate by noting that previous experimental observationsof glycerol on silanated silicone have shown that Υ ∼ γfor these materials [5].

C. Large-strain elasticity of composites

Wemeasured the stiffness of the composites from force-indentation profiles. We indented sample surfaces with

the flat end of cylindrical metal rod of radius a = 1.6mm,using an Instron E-1000 Stress Tester with a 5N load cell.The sample consisted of a filled (10mm deep × 35 mmdiameter) petri dish. We indented the sample by 5mm forthe soft-matrix composites, and 3mm for the stiff-matrixcomposites.Typical force-indentation profiles for the soft-matrix

composite are shown in Figure 6 for two different glyc-erol contents. In these experiments, we performed a se-ries of four indentations of increasing magnitude (yellow,red, cyan, blue) to investigate whether plasticity was oc-curring at large strains. Despite indenting to a depthof 5mm in a 10mm deep sample, we see no evidence ofplasticity, as all the profiles coincide. These profiles alsoshow how the force-displacement curve is initially linearfor up to 2mm of indentation, allowing us to accuratelydetermine its slope, and thus composite stiffness.From Boussinesq’s flat-punch solution, the sample in-

dentation, d, is related to the force applied by F =8aEcd/3. Here we have assumed that the composite isincompressible as both silicone gel and glycerol are ef-fectively incompressible (e.g. [2]). Thus we extract Ec

from the slope of the (initially-linear) force-displacementindentation profile (See Supplementary Figure S6 for ex-ample profiles).

[1] Y. Xu, W. C. Engl, E. R. Jerison, K. J. Wallenstein, C. Hy-land, L. A. Wilen, and E. R. Dufresne, Proc. Nat. Acad.Sci. 107, 14964 (2010).

[2] R. W. Style, R. Boltyanskiy, G. K. German, C. Hyland,C. W. MacMinn, A. F. Mertz, L. A. Wilen, Y. Xu, andE. R. Dufresne, Soft Matter 10, 4047 (2014).

[3] R. W. Ogden, Non-linear Elastic Deformations (Ellis Har-wood Ltd., 1997).

[4] L. D. Landau and E. M. Lifshitz, Course of TheoreticalPhysics, Volume 7: Theory of Elasticity, Third Edition(Pergamon Press, London, 1986).

[5] R. W. Style, Y. Che, J. S. Wettlaufer, L. A. Wilen, andE. R. Dufresne, Phys. Rev. Lett. 110, 066103 (2013).

[6] J. D. Eshelby, Proc. Roy. Soc. Lond. A 241, 376 (1957).[7] P.-G. de Gennes, F. Brochard-Wyart, and D. Quere, Cap-

illarity and Wetting Phenomena: Drops, Bubbles, Pearls,Waves (Springer, 2010).

5

10−1

100

101

5

10

15

20

25

30

c [wt%]

γ[mN/m]

0 5 10 15 209

10

11

12

13

14

15

φ [%]

γ[mN/m]

0 1 2 3 4 52

2.5

3

3.5

4

4.5

5

5.5

6

6.5

c [wt%]

E[kPa]

(A)

(B)

(C)

FIG. 5. Controlling the stiffness of the solid matrix in sili-cone/glycerol composites. (A) Young’s modulus of the soft sil-icone used for the soft-matrix composite, as a function of sur-factant concentration (no glycerol). E is measured with thesame indentation tests used for measuring composite stiffness.(B) Surface tension of glycerol in the uncured silicone phaseof the composite, as a function of composite glycerol content.We made emulsions using the same recipe as the soft-matrixcomposites but without the silicone catalyst. These sampleswere gently centrifuged down to separate glycerol dropletsfrom the bulk silicone phase, and the silicone was pipettedoff. Then we measured the surface tension of a large, pureglycerol droplet in the silicone by the pendant drop method[7]. (C) The surface tension of glycerol droplets in uncuredsilicone with different surfactant concentrations. We mixedthe silicone in the same ratio as used in the soft-matrix com-posites, but without catalyst, and added surfactant of differ-ent concentrations. Then we measured the surface tension ofa large, pure glycerol droplet in the silicone by the pendantdrop method.





6

0 1 2 3 4 50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Displacement [mm]

Forc

e [N

]

18.7% glycerol

4.4% glycerol

FIG. 6. Force-displacement profiles for indentation of com-posites with 4.4%, 18.7% glycerol. Profiles show four sequen-tial indentations of increasing magnitude (yellow, red, cyan,blue). All indentation profiles coincide, showing no evidenceof plasticity. Dashed lines highlight the initially linear be-haviour upon indentation, from which composite stiffness isextracted.





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