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

∗ eric.dufresne@yale.edu

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

∗ eric.dufresne@yale.edu

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

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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

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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

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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.

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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.

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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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SUPPLEMENTARY INFORMATION DOI: 10.1038/NPHYS3181

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