Elastocapillarity:Surface Tension and the Mechanics of Soft Solids

Robert W. StyleMathematical Institute, Oxford University, Oxford, UK

Anand JagotaDepartment of Chemical and Biomolecular Engineering and Bioengineering Program, Lehigh University, Bethlehem, USA

C.-Y. HuiDepartment of Mechanical and Aerospace Engineering, Cornell University, Ithaca, USA

Eric R. Dufresne∗

Department of Materials, ETH Zurich, CH-8093 Zurich, Switzerland(Dated: April 8, 2016)

It is widely appreciated that surface tension can dominate the behavior of liquids at small scales.Solids also have surface stresses of a similar magnitude, but they are usually overlooked. However,recent work has shown that these can play an central role in the mechanics of soft solids such as gels.Here, we review this emerging field. We outline the theory of surface stresses, from both mechanicaland thermodynamic perspectives, emphasizing the relationship between surface stress and surfaceenergy. We describe a wide range of phenomena at interfaces and contact lines where surfacestresses play an important role. We highlight how surface stresses causes dramatic departures fromclassic theories for wetting (Young-Dupre), adhesion (Johnson-Kendall-Roberts), and composites(Eshelby). A common thread is the importance of the ratio of surface stress to an elastic modulus,which defines a length scale below which surface stresses can dominate.

PACS numbers: Valid PACS appear here

I. INTRODUCTION

Soft solids are ubiquitous. They include gels, creams,foams, rubbers, pressure sensitive adhesives and muchof biological tissue. Soft solids have long been appliedin cosmetics, adhesives, sealants, and padding. Further-more, exciting new applications are developing in surgery,tissue engineering, flexible electronics and soft robotics(e.g. [1–8]), often utilising the fact that soft solids canexhibit mechanical phenomena that differ qualitativelyfrom hard engineering materials [9]. One key differenceis that surface stresses, which play a minor role in themechanics of stiff materials, can dominate the behaviorof soft solids.

The importance of surface stresses on the mechanicsof soft materials has only come to light in the last fewyears. For example, surface stresses stabilize the surfaceof a soft solid slab [10, 11], but break up soft solid fila-ments [12]. Surface stresses resist the deformation of fluidinclusions in a soft solid and stiffen fluid-solid composites[13]. Liquid droplets on soft substrates can violate theclassic Young-Dupre equilibrium [14–16]. Stiff solid par-ticles adhered to soft substrates do not obey the standardmodels for adhesive contacts [17–19].

The surface of a material has an energy penalty perunit area of surface, the surface energy, γ [20]. In liq-

∗eric.dufresne@mat.ethz.ch

uids, the surface energy penalty gives rise to a tensilesurface stress, Υ, that opposes surface stretching. Itacts to minimise the surface to volume ratio of the liquid,causing small droplets to bead up, and larger volumes ofliquid to have smooth, flat surfaces. Surface stress al-lows small dense objects to float at a liquid surface, andis the source of the Laplace pressure difference acrosscurved surfaces. Generally, Υ is a symmetric second or-der, two-dimensional tensor. However, in simple liquids,Υ is isotropic and thus can be represented by a scalarΥ. Another convenient property of simple liquids, as weshall see, is that γ = Υ. This has led to γ and Υ beingreferred to interchangeably as the surface tension [20, 21].

Surface energies and stresses in solids are different intwo key ways to their liquid counterparts. First, the sur-face stress and surface energy are not generally equal,γ 6= Υ, [22]. Therefore, one must use the term ‘surfacetension’ carefully. Secondly, solid surface stresses can beanisotropic and even compressive [21–26].

The relative importance of surface stress and bulk elas-ticity is a matter of length scale. Just as a fluid’s surfacetension creates a jump in hydrostatic pressure across acurved interface (the Young-Laplace equation), surfacestress causes a jump in the stresses across a solid inter-face. For a solid surface with mean curvature K andisotropic, uniform surface stress Υ, the jump in normalstress is simply ΥK. This stress jump drives local defor-mation in the bulk of the solid. For an elastic solid withYoung’s modulus E, stresses balance so that ΥK ∼ εE,where ε is the strain. Thus, we expect significant de-

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formations (ε ∼ 1) when 1/K ∼ Υ/E. This defines anelastocapillary length, Υ/E. Generally, at scales muchlarger than Υ/E, surface effects are negligible. At scalesmuch smaller than Υ/E, surface stresses dominate andone observes dramatic departures from classic elastic be-havior.

Elastocapillary phenomena at the continuum level arepossible when Υ/E is much larger than molecular scales.The surface energies of soft materials are typically oforder 10 − 100 mN/m [20]. Thus elastocapillarity canbe very important in soft polymeric materials like gels(E ∼ O(kPa)) and elastomers (E ∼ O(MPa)).

We briefly note that there is a large body of prior workthat has focussed on surface stresses in stiffer materialssuch as metals and other crystallographic solids (see pre-vious reviews: [26–29]). In this review, we will focus onrecent work that has evolved from new observations ofsurface stress effects in soft solids.

II. ELASTOCAPILLARY PHENOMENA INTWO-PHASE SYSTEMS

We begin this section by deriving the fundamentalboundary condition balancing bulk and surface stressesacross an interface. Then, we discuss the implications ofthis force balance in some examples involving two phasesseparated by an interface.

A. Surface stress as a boundary condition at aninterface

The surface stress tensor, Υ, is the 2D analogue ofthe more familiar 3D (Cauchy) stress tensor, σ. In 3D,the forces per unit area of a deformed surface (tractions)acting at a surface with normal n are given by σ · n.The tractions can be resolved into a normal component,n · σ · n, and a shear component, t · σ · n, where t isa unit vector tangential to the surface. Similarly, in 2D,the forces per unit length of a curve bounding a deformedsurface with boundary normal b are given by Υ · b (seeFigure 1). Surface stresses always act in the plane of thesurface and have a line-normal component, b ·Υ · b, andtangential component, t ·Υ · b, where t is the tangentvector to the curve.

Surface S

Closedloop C

FIG. 1: Schematic diagram for deriving the equation for forceequilibrium at an interface.

The essential physics of elastocapillary coupling liesin a connection between the surface stress and the bulkstress at the interface. The governing interfacial equationcan be derived by considering the forces acting on thearea of surface in Figure 1. The condition for staticequilibrium is,∫

S

σ1 · n dS −∫S

σ2 · n dS +

∮C

Υ · b dl = 0. (1)

Here, the integrals in the deformed configuration andσ1,2 are the true (Cauchy) stresses in the deformed body.Note that we have ignored any forces acting on the sur-face due to external potentials (e.g. surface charges in anelectric field), and have assumed that the surface has nobending rigidity. The first two terms represent forces dueto bulk stresses acting on either side of the area of inter-face, S. The last term is the contribution to total forcedue to surface stress. Using the surface divergence theo-rem, we can convert this term to a surface integral, andthen remove the integrals (as S is arbitrary) to obtainthe generalisation of Laplace’s law:

(σ1 − σ2) · n = −∇s ·Υ, (2)

where, ∇s is the surface gradient operator [24, 25, 30].When the surface stress is isotropic, Υ = ΥI2, where I2 isthe 2D identity tensor, this boundary condition simplifiesto

(σ1 − σ2) · n = −ΥKn +∇sΥ, (3)

where curvature K = ∇s ·n. Note that the stresses due tothe first term on the right-hand side are always normal tothe interface. The stresses due to the second are alwaysin-plane, and equivalent to Marangoni stresses in a liquid[31]. If the surface stress is both isotropic and uniform,∇sΥ = 0, and

(σ1 − σ2) · n = −ΥKn. (4)

which reduces to the Young-Laplace law for fluid inter-faces (∆P = γK). The simplified boundary condition,Eq. (4) shows excellent agreement with a wide range ofexperimental observations in hydrogels and silicone gels(e.g. [12, 13, 16, 19, 32–35]). The implications for morecomplex surface stresses are just beginning to be explored[36].

B. Surface Stresses Smooth Slabs

The simplest manifestation of the competition betweensurface stress and elasticity is the smoothing of featureson solid slabs [11, 34, 37–39]. An example of this is thechange in shape of soft gels that that are released fromrigid, patterned molds. Figure 2 (inset) shows the pro-file of a gel before and after release from a mold witha square wave pattern. Upon release, sharp corners arerounded out and the overall amplitude of the pattern is

3

FIG. 2: Flattening of a solid surface by surface stress. Agelatin gel with Young’s modulus E = 32.5kPa is cured(stress-free) in a PDMS mould, then released and allowedto relax under the influence of surface stress.Height of thesurface profile before and after release for different stiffnessesand wavelengths. The x-axis is Young’s modulus times surfacewavelength, and the black curve shows the theoretical predic-tion [39]. Inset shows optical profilometry (theory) measure-ments of the surface profile of the mould: blue (green) andafter release: red (black).

reduced. Flattening is strong for softer gels and shorterwavelengths (Figure 2).

The flattening process is intimately linked to the elas-tocapillary length, Υ/E [11, 37]. Consider a simplifiedexperiment with a solid having an initially sinusoidal sur-face profile of wavelength λ and amplitude A [11]. Ac-cording to Eq. 4, the capillary stress normal to the sur-face is σΥ = ΥK, where K ∼ A/λ2 is the surface curva-ture. This stress drives flattening, and is opposed by theelastic stress. Complete flattening requires the strain inthe solid ε ∼ A/λ, so the elastic stress σel ∼ Eε ∼ EA/λ.Thus, flattening is significant when σΥ ∼ σel, or whenλ ∼ Υ/E. In short, elasticity will prevent deformationat longer length scales, and surface stress overwhelmselasticity and flattens the surface at smaller scales. Sim-ilar arguments show that sharp corners will round out toleave a smooth surface with a radius of curvature ∼ Υ/E[37, 38].

In a similar vein, surface stresses also suppress com-pressional instabilities of soft solids films. It is well knownthat confined layers of elastic solids become unstableto buckling, wrinkling or creasing when compressed orswollen (e.g. [40]). However the morphology of the sur-face instability, the critical compression threshold, andthe critical defect size for instability nucleation all de-pend sensitively on the relative size of the layer thickness,H, and elastocapillary length Υ/E (e.g. Figure 3a,b).For example, when Υ/HE � 1, the surface becomes

unstable to creases with wavelength ∝ H [10, 32, 41].However, when Υ/HE � 1, the surface tends to wrin-kle instead with wavelength� H, while the compressionneeded for the instability to occur increases significantlybeyond classical predictions [32, 41–44].

C. Surface Stresses Deform Rods

Some of the most dramatic effects of surfaces stressin solids are seen in objects with slender sections. Sur-face stresses deform long, thin cylindrical beams of softhydrogels when they are released from rigid moulds[12, 34]. Sufficiently thin cylinders develop undulationsalong their axis, as shown in Figure 3c [12]. This insta-bility is analogous to the famous Rayleigh-Plateau cap-illary instability [45, 46]. In that case, surface tensiondestablizes a liquid cylinder of radius R to surface per-turbations that have wavelength >∼ O(R) . In the case ofa soft gel, however, this only occurs when the cylinder di-ameter is less than Υ/E, as shown in Figure 3c,d. Addi-tionally, surface stresses will round out a slender object’scorners, and when it has an asymmetric cross-section, theresulting asymmetric surface forces can cause significantbending [34].

D. Surface Stresses Stiffen Inclusions andComposites

Surface stresses stabilize spherical fluid inclusions insoft solids. When a soft gel with a dilute concentrationof embedded liquid droplets [13] is strained macroscopi-cally, the embedded droplets also deform. However, theextent of deformation depends on the size of the droplets.Figure 4a shows three inclusions of different initial sizessubjected to increasing far-field strains. At the same im-posed strains, the smaller the inclusion, the more spher-ical it remains. For a given macroscopic deformation,the droplet strain varies smoothly with its radius, Fig-ure 4b. When Υ/RE � 1, the droplet shape does notdepend on its size, in agreement with Eshelby’s classi-cal theory of inclusions in an elastic matrix [47]. ForΥ/RE � 1, the droplet shape scales with its size, as sur-face stresses oppose droplet deformation, contradictingEshelby.

The stiffening of fluid inclusions by surface tension canhave a dramatic impact on the macroscopic behavior ofa fluid-solid composite. Intuition and classic compositetheory tell us that if you take a solid and fill a volumefraction φ of it with fluid-filled holes of radius R, its effec-tive modulus, Ec will as reduce φ increases. This worksrobustly when for Υ/RE � 1. However, the opposite isobserved when Υ/RE � 1. In this case, fluid inclusionsactually stiffen the composite solid (Figure 4c) [13, 48].

The reason that classic composite mechanics does notwork is that it omits Υ, which typically acts to keep in-clusions spherical. To address this, much recent work has

4

(a) (b)

GelSubstrate

L0

L

(c)(d)H

FIG. 3: a) In uniaxial compression of a soft layer (see schematic), the critical compression for instability depends only on theratio Υ/HE [32]. (b) Soft silicone layers of varying thickness are swollen by applying an electric field across them. As the ratioΥ/HE varies, there is a dramatic shift in the form of the surface instability [41]. c) Cylinders of soft agar gels are releasedin toluene and allowed to relax. From top to bottom, the cylinders have shear moduli, µ, reducing from 27 to 12Pa. d) Thecylinders become unstable (black points) when their diameters (2ρ0) are less than Υ/E – this theoretical prediction is given bythe curve. Unfilled points show stable cylinders [12]. All these soft solids are approximately incompressible so E ≈ 3µ.

0 10 20 30 40 50 600.5

1

1.5

2

2.5 ∞

0.1

0.01

100

10

1

3

0.3

γ = 0 (Eshelby)

Dro

plet

str

ain/

app

lied

stra

in(a) (b) (c)

FIG. 4: Surface stress affects the shape of small fluid inclusions in a stretched soft solid. a) Ionic-liquid droplets of differentinitial radii, R, are embedded in a soft silicone sheet and stretched under plane-stress conditions. The applied x, y strainsare ε∞x , ε

∞y respectively. Smaller droplets stay more spherical. Larger droplets obey classical elasticity theory. b) Droplet

stretch/applied stretch as a function of droplet radius for stretched ionic-liquid droplets in a soft silicone gel. Above theelastocapillary length (∼ 10µm), this is approximately constant (i.e. shape is independent of size). Below the elastocapillarylength, droplets stretch much less than the surrounding solid. The dashed curve is the theoretical prediction. c) The stiffnessof silicone/glycerol composites as a function of glycerol concentration. Blue circles: silicone with E = 3kPa, black diamonds:silicone with E = 100kPa. In the former, increasing glycerol volume fraction, φ, stiffens the composite. Curves: theoreticalpredictions for composites containing uniformly sized droplets [13].

focussed on augmenting composite mechanics to accountfor surface stresses (e.g. [48–54]). The augmented the-ory shows good agreement with experiments on emulsion[48] and silicone-gel [13] composites when Υ is taken asan isotropic, strain-independent surface stress (Figure4b). The theoretical work again again highlights the im-portance of Υ/E. For example, in incompressible elas-tic solids containing identical incompressible fluid ‘holes’,surface stress becomes important when R <∼ 100Υ/E,causing stiffening relative to classical predictions [54].

A practical simplification for composite calculations isapproximating soft inclusions as surface-stress-free, elas-tic inclusions. For example, in incompressible compos-ites with uniform, isotropic surface stress Υ, liquid in-

clusions effectively behave like elastic, surface-stress freeinclusions with modulus Eeff/E = 24/(9 + 10RE/Υ)[13, 48, 54]. Thus large inclusions have vanishing effec-tive modulus, small inclusions appear as elastic inclu-sions with constant stiffness 8/3 times stiffer than thesurrounding matrix, and inclusions with R = 3Υ/2E are‘cloaked’ with Eeff = E. Making this substitution allowsus to use the full power of classical composite mechanicsto predict the diverse behaviour of soft composites.

5

E. Surface Stresses Resist Fracture

Surface stresses will also affect the opening of a crackin a soft material. The movement of a crack tip througha solid is governed by the work of fracture Γ, the energyrequired per unit area of new crack. As this is a ma-terial parameter, we can form a second material lengthscale, an elasto-adhesive length, Γ/E, which character-izes the radius of curvature of a crack tip at propaga-tion (in linear-elastic fracture mechanics) [55]. Surfacestress will then induce a closing stress on the crack tipσ ∼ ΥK ∼ ΥE/Γ ∼ Eε. Thus a positive surface stresswill significantly blunt a crack tip when Υ is compara-ble to Γ. In this case crack growth will be retarded, aspointed out recently [56, 57].

III. ELASTOCAPILLARITY OF THREE-PHASESYSTEMS

So far, we have considered the impact of surfacestresses on the mechanics of a single two-phase interface.In this section, we will consider the structure of three-phase contact lines when two or more of the phases aresoft. First, we will consider the wetting of droplets onsoft solids. Then, we will consider adhesion of rigid par-ticles to soft solids.

A. Partial Wetting on Soft Substrates

There are two key results of classical wetting theory.First, Young [58] showed that a droplet’s contact angle, θ,on a rigid solid substrate was independent of any far-fieldboundary conditions (e.g. droplet size or the thicknessof a substrate), but varied for different combinations ofmaterials (Figure 5a). Minimising the interfacial en-ergy of the droplet/substrate system, one finds that θdepends only on the surface energies of the three inter-faces, γlv, γsv, γsl, through the Young-Dupre relation [59]:

γlv cos θ = γsv − γsl. (5)

Second, Neumann [60] considered a droplet resting on aliquid substrate with which it is immiscible, (Figure 5b).At the contact line, the angles between all three interfacesare independent of the far-field boundary conditions anddetermined by a vector balance of the surface stresses:

Υ12 · t12 + Υ13 · t13 + Υ23 · t23 = 0, (6)

where the surface stress tensor and the tangent vector ofthe interface between phases i and j are denoted by Υij

and tij respectively. Over the last 100 years, equations(5,6) have served as a basis for understanding the staticstructure and dynamics of fluids at small scales [20, 61].Recently, however, it has become apparent that wetting

on soft solids does not fall simply into either one of theselimits.

The complexity arises because a droplet’s surface ten-sion can deform soft solid substrates. For instance, theout-of-plane component of a droplet’s surface tension,Υlv sin θ, pulls up on the surface. To estimate the mag-nitude of this deformation, let us consider a straight con-tact line on a semi-infinite, elastic substrate (Figure 5c).The line-force, Υlv sin θ, is spread out over an width ofthe order of the molecular size, a [62–66]. The tensiletraction applied to the substrate at the contact line canthus be approximated as Υlv/a [67]. This produces astrain under the contact line, ε ≈ Υlv/Ea. When thisparameter is very small, the substrate is effectively rigid,and Young-Dupre’s relation holds [67, 68].

On the other hand, when Υlv/E >∼ a, there are sig-nificant deformations at the contact line [16, 33, 69–71].In this case, we can determine the contact-line geom-etry by considering the force balance on a small testvolume around the contact line (Figure 5d). Equilib-rium requires force balance between the bulk and surfacestresses:∫

W1

σ1.n1dL+

∫W2

σ2.n2dL+

∫W3

σ3.n3dL

+Υ12.t12 + Υ13.t13 + Υ23.t23 = 0. (7)

Now shrink the size of the test volume. If the bulkstresses are bounded, or diverge more slowly than 1/r(which we expect), their contributions vanish while thosefrom the surface stresses remain finite. In other words,Equation (7) reduces to Neumann’s vector balance, Eq.(6), which requires the three interfaces to meet with fixedorientations (Figure 5b) [14, 16, 67, 72–74]. Note thatthis analysis ignores any long-range forces between theinterfaces [75, 76].

Thus, microscopic behaviour near the contact line de-pends critically on the parameter Υlv/Ea. Using thesurface tension and molecular dimensions of liquid water,this parameter is of order one for substrates with Young’smodulus of 100 MPa. On much stiffer solids, such asstructural materials, Young-Dupre’s law is recovered. Onmuch softer solids, such as gels, Neumann’s triangle de-scribes the contact line geometry. There is a smoothtransition between the two limits [15, 67, 73, 77, 78].This phenomenon allowed one of the first techniques forthe measurement of the absolute values of solid surfacestresses, by measuring the angles between the phases atthe contact line, and the liquid-vapour surface tension(e.g. [20]), and using equation (6) to calculate the solid-vapour and solid-liquid surface stresses.

While the shape near the contact is universal, theoverall shape of the droplet depends on its size (Figure6a). This is related to the fact that the surface stress-dominated regime near the contact line has a width ofroughly Υs/E, which has been confirmed experimentally(Figure 6) [16, 71]. For large droplets, R� Υlv/E, thewetting ridge is small compared to the droplet size, andthe apparent contact angle is unperturbed from its value

6

Solid

Liquid

Air Fluid 1

Fluid 2

Fluid 3

(a) (b)

O(a)

Rigid substrate Simple fluid substrate

(c) (d)

Soft solid substrates

FIG. 5: a) Young-Dupre’s law: the contact angle θ is determined by the three surface energies. b) Neumann’s triangle: theangles at the contact line between three simple fluids are determined by a force balance of the three surface stresses. c) Thenanoscale view of the contact line: the surface stress is spread out over a region of the order of a molecular diameter, a. Theresulting pressure pulls the underlying surface up into a ridge. d) The microscale view of a contact line on a soft solid, showingthe forces acting on a small volume (with size � a) around the contact line.

on a rigid substrate, [14]. Thus, large droplets behaveas if they were on rigid surfaces. On the other hand, forsmall droplets with R <∼ Υlv/E, the Laplace pressure inthe droplet can easily depress the underlying solid sur-face. In fact, for R � Υlv/E, the shape of the dropletis entirely determined by the three interfacial stresses, asit would for a liquid substrate, (Figure 5b). There is asmooth transition for the appararent contact angle fromthe Neumann to Young-Dupre limits with droplet size,as shown in Figure 6.

B. Wetting of Slender Objects

When a soft solid has no geometrically-imposed lengthscale, we have seen that liquid surface tension can sig-nificantly deform it at scales <∼ Υ/E. However, sur-face tension can also deform thin rods or sheets madefrom stiffer materials (cf the recent review [81]). In suchslender geometries, elastocapillary effects can be signif-icant at length scales much larger than Υ/E. A sessiledroplet of characteristic size R can signficantly deforma free plate or rod when R >∼

√Kb/γlv, where Kb is

the bending stiffness of the plate/rod (in Joules). The

length scale,√Kb/γlv, is also commonly referred to as

an elastocapillary length. To avoid confusion with Υ/E,we refer to it as a bendocapillary length, since it describesbending due to capillary forces. Note that

√Kb/γlv is

not a material parameter, like Υ/E, since it depends onthe cross-sectional dimensions of the rod or sheet. Fur-thermore, bendocapillary phenomena become more pro-nounced as the system gets larger, in stark contrast tothe elastocapillary deformations described in the previ-ous section, where capillary phenomena are more pro-nounced for smaller droplets. Bendocapillarity phenom-ena are a subject of intense research. Examples includecapillary origami, the bundling of wet hair and fibres, andclumping of micro-beams in micro-electromechanical de-vices [82–87]. An interesting new limit has also recentlybeen identified for the interactions of droplets with ex-tremely bendable sheets [88, 89].

What happens when a sheet carries tension T as well asbending rigidity? The tension in the sheet has contribu-

tions from the surface stress on the two sides, ∼ 2Υ, andthe bulk elastic stress ∼ εEh, where ε is the tensile strain.For very thin sheets, h � 2Υ/εE, the surface stressesdominate, and T ≈ 2Υ. For elastic sheets, the Foppl-von-Karman equations predict the response of the sheetto an applied load. For instance, a small drop placed ona sheet applies a line force due to liquid-vapor surfacetension and distributed Laplace pressure. By examiningthe magnitude of the forces due to bending and tensionof the sheet, we find a new length scale

√Kb/Υ. Like

Υ/E for a thick elastic solid, this length scale capturesthe intrinsic deformability of a slender elastic object. Be-cause Kb ∼ Eh3, it does not scale with overall systemsize, but only with the thickness of the sheet. For lateraldimensions much smaller than

√Kb/Υ, bending rigid-

ity resists applied forces. On length scales much largerthan

√Kb/Υ, the sheet’s surface stresses resists applied

forces.Thus, at length scales intermediate between

√Kb/Υ

and the droplet radius, R, the equilibrium configurationof the sheet near the contact line should be given bybalance of tensions: a sort of Neumann’s triangle, Fig-ure 5b, with the liquid-vapour surface tension balancingthe tensions on the wet and dry segments of the sheet.Neumann’s construction implies that that significant de-formations of the sheet occur when γlv >∼ Υ. Since theliquid surface tension is comparable in magnitude to thesurface stress, large deformations are generally expectedfor sheets in the thin limit. As an example, Figure 6eshows out-of-plane deflection of a silicone film stretchedacross an annular disc due to the Laplace pressure andsurface tension of a water droplet placed under it. Thedeformed shape can be represented accurately using platetheory [80]. At the macroscale, the overall shape at largerlength scales resembles a droplet at a fluid-fluid interface:it obeys Neumann’s triangle. Bending is influential onlyin a small region near the contact edge.

These simple scaling arguments results are supportedby recent theory and experiments with droplets placed onthin silicone sheets [80, 90, 91]. Thick thick/stiff sheetswith h � Υ/E are undeformed by droplets and thusfollow classical wetting behaviour, agreeing with Young-Dupre’s law. Thin sheets with h � Υ/E deform signifi-cantly as the Laplace pressure in the droplet causes the

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0 100 200R [µm]

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FIG. 6: a-c) Experimental data for droplets on soft, silicone substrates (E = 3kPa). a) Surface profiles underneath glyceroldroplets of different radii. The sharp peaks correspond to the position of the contact line. b,c) when the surface profilesare aligned (by translation and rotation) at the contact line, the local ridge geometry is always found to be the same for agiven liquid/substrate combination. (b: glycerol, c: fluorinated oil). d) The apparent contact angles θa of glycerol dropletsreduce smoothly as droplet radius approaches the elastocapillary length. The dashed curve shows the linear-elastic theoreticalprediction θa [79].e) Out-of-plane deflection of a stiff silicone film stretched across an annular disc by the Laplace pressure andsurface tension of a water droplet attached to it. The deformed shape can be represented accurately using the theory of plateswith in-plane tension [80].

film to bulge out, taking a shape identical to fluid-on-fluid wetting. Note that this transition does not dependon the droplet size, so the experiments can be performedwith macroscopic droplets. The contact angles of thebulging droplets can then be readily measured and usedto extract the surface stresses of the sheet, both insideand outside the droplet (e.g. Figure 6e [80].

C. Adhesion

In adhesion theory, as with wetting, there are two keyfundamental results. First, Hertz calculated the force-indentation relationship between two adhesion-less, elas-tic spheres [92]. This forms the basis of contact mechan-ics, and works very well for hard materials. Second, John-son, Kendall & Roberts (JKR, [17]) noticed that Hertz’stheory breaks down on soft materials – in particular itcannot explain why there is a significant pull-off force.This is due to adhesion between the surfaces, typicallyrepresented by a work of adhesion, W , the energy reduc-tion per unit area of adhered surface (caused by attrac-tive intermolecular forces). For the case of a rigid sphereadhering to a soft substrate, JKR showed that adhesionis important whenever the elasto-adhesion length W/Eapproaches or exceeds the particle radius, R. i.e. Hertz→ JKR when W/E >∼ R. JKR has been verified experi-mentally [93] and is widely acknowledged as the standardmodel for adhesive contact [94]. Notably, although it as-sumes small contact and linear elasticity, it is surprisinglyaccurate for contact radii, a, up to ≈ R/2 [95].

Recent experiments have shown that JKR theorybreaks down on very soft solids [19, 43]. For example,

load-free glass microspheres were found to indent signifi-cantly less into soft silicone gels than would be expected[19]). This breakdown is because JKR theory neglects therole of surface stress. To see this, consider the process ofplacing a load-free sphere on a soft substrate. We breakthe adhesion process into two steps. First, we deformthe substrate to its final geometry (Figure 7a). Second,we adhere the sphere (Figure 7b). JKR theory assumesthat in step 1, we only expend energy in elastically de-forming the substrate, and this energy comes from theadhesion energy released in step 2. However, as Figure7a illustrates, we have significantly stretched the surfacein step 1 – meaning that we have also expended energyon surface work.

When will surface stress become important? This canbe determined for load-free adhesion by evaluating thework done in stretching the surface, EΥ and the energyreleased upon adhesion EW . Figure 7c shows data forload-free, glass microspheres placed on soft silicone sub-strates. To good approximation, the indentation takesthe form of a spherical cap in a flat surface. Thus theextra surface area created (by stretching) in the first step∼ d2 (d is the indentation depth) so EΥ ∼ Υd2. Similarly,the adhesion occurs over the area of the spherical cap,and we find that EW ∼ WRd (R is the sphere radius).Surface stress effects are negligible when EΥ � EW , orequivalently when d�WR/Υ. From classical JKR the-ory, d ∼ (W 2R/E2)1/3, so the condition for JKR theoryto hold becomes ω � 1, where ω = (W/ER)2/3Υ/W[18, 19, 96–98]. This is illustrated in Figure 7d whichshows finite-element computations of zero-load, adhesivecontact with a rigid spherical indenter [96]. When ω = 0,the deformed shape displays the characteristic cusp of

8

d

surface stresspenalty fromstretching surface

elastic energypenalty fromdeforming substrate

adhesion energyreleased uponcontact

(b)(a)

−40 −20 0 20 40

0

10

20

30

r [µm]

z [

µm]

E = 3kPa

E = 85kPa

E = 500kPa

(c)

JKR YD

(d)

FIG. 7: Adhesion of load-free spheres on soft substrates. a,b) Schematic of the adhesion process of a sphere to a soft substrate.c) Experimental data for glass microspheres on silicones with three different stiffnesses, measured with confocal microscopy [19].d) Computed deformed shapes of a substrate in adhesive contact with a rigid sphere under zero external load [96]. When theadhesion parameter ω = 0, the deformed shape obeys JKR theory [17]. When ω � 1 the particle behaves as if at a fluid-fluidinterface with a contact angle given by the Young-Dupre Equation (5).

JKR theory [17]. In the limit of large elastocapillarynumber the surface stress dominates and so the free sur-face is horizontal and meets the particle surface at anangle given by the Young-Dupre Equation (5).

In fact, the dimensionless parameter ω completely de-termines the adhesion behaviour, even when a force isapplied to the indenter [98]. When ω � 1 surface stressdominates elasticity (just as in the small droplet on a softsubstrate limit). In the limit ω → 0, JKR predicts thatthe pull-off force of a rigid spherical indenter from a soft,flat, adhesive substrate is Fpo = −3πWR/2. As ω →∞,this pull-off force can be reduced by more than 30% [98].Useful, semi-analytical expressions [98] and scaling laws[19, 97, 99] relating force to indentation can be found inrecent work and some results are provided in the Supple-mentary Information.

Surface stress also provides an alternative way to reg-ularize a conceptual difficulty with JKR theory, whichpredicts an infinite tensile stress at the contact line. Thissingularity effectively reduces JKR contact mechanics toan interface fracture problem [57], which can be regular-ized by the use of a cohesive zone model [100]. Alter-natively, recent theoretical work has predicted that forrigid indenters on soft substrates with isotropic surfacestresses with Υ = γ, the soft substrate meets the in-denter with a contact angle given by Young-Dupre’s law[19, 96, 101]. However, some molecular dynamics simu-lations disagree with this – and observe that the contactangle is dependent on parameters including indenter sizeand substrate stiffness [102]. Recent experimental workhas demonstrated an alternate mechanism to cut off theelastic stress singularity. Many of the soft solids whichdemonstrate strong elastocapillary phenomena consist ofelastic networks swollen by a solvent. When rigid par-ticles adhere to such a solid, the elastic singularity canbe avoided by phase separation of the solvent from theelastic network near the contact line [103, 104].

IV. SURFACE STRESS AND SURFACEENERGY IN SOFT MATERIALS

From a continuum perspective, the interface of twomaterials is a two-dimensional sheet. While it has nothickness, it can have mechanical properites that drivediverse phenomena. In this section, we review the rela-tion between two surface properties, the surface energy,γ, and surface stress, Υ. After highlighting key conceptsfor simple fluids, complex interfaces and simple solids, wediscuss the expected behaviour of soft solids like gels.

A. Simple liquid interfaces

A theoretical understanding of the surface stress ofsimple liquid interfaces has been complete for sometime [105, 106]. Simple liquids are isotropic and theirmolecules rearrange freely under thermal fluctuations.Thus, the increase in Helmholtz Free Energy (uponchanging the surface area A) is simply

dF = γdA

∣∣∣∣V,n,T

(8)

with a surface energy, γ, that is independent of fluid de-formation or shape. In equilibrium, the work done bysurface stress upon stretching the surfaces must equalthe change of surface free energy, dF = dW . If we as-sume that every point on the surface moves from x tox + ∆x, then the work done by the surface stress on anarea surrounded by the closed curve C is

dW =

∮C

∆x ·Υ · b dl, (9)

where b is the outward surface normal as in Figure 1.Since the interface is an isotropic fluid, the surface stress

9

must be isotropic Υ = ΥI2. Then dW = Υ∮c∆x ·b dl =

ΥdA. Comparing this with equation (8), we concludethat the surface stress and surface energy are numericallyequivalent, Υ = γ.

B. Simple liquid with insoluble surfactants

The next level of complexity is a fluid-fluid interfacedecorated by molecules or particles that are insolublein the adjoining fluids. Examples include phospholipids,particles, and polymers at interfaces. Because these ad-sorbed species are confined to the interface, their densitychanges as the surface is compressed or stretched.

For vanishing concentrations of adsorbed species, thesurface stress is isotropic and identical to that of thebare interface, Υo. As the density increases, the sur-face stress, Υ, decreases. Conventionally, these systemsare described by the surface pressure, Π = Υo −Υ [107],which is readily measured in a Langmuir trough.

More generally, surface stresses can have isotropic anddeviatoric contributions [107, 108]:

Υ = Tr (Υ) I2/3 + Υd. (10)

The first term is the isotropic tension that opposes in-crease in surface area. The second, deviatoric, part rep-resents in-plane shear forces. This has many possibleorigins and is quantified by the study of surface rheology[107, 109]. For static systems at low concentrations, thesurfactant molecules form a dilute, liquid-like layer onthe surface that cannot support shear, and so Υd = 0.However, at higher concentrations, surfactants can formsolid-like layers on surfaces with Υd 6= 0. Non-zero Υd

can also be found in dynamic, fluid-like interfaces at finiterates of shear deformation [108].

C. Simple solid interfaces

To uncover the connection between surface energy andsurface stress for a solid [110], we follow Cahn [111] anddistinguish between the reference or Lagrangian config-uration R and the current or deformed configuration C.The surface free energy F = γRAR = γCAC with areasAR and AC in the reference and current configurations,respectively, and free energy densities γR and γC definedper unit area of the reference and current configurations,respectively.

Equating work performed on a system to its change infree energy (similar to the process in equations (8,9)), wefind

Υ =∂γR∂εs

. (11)

(or Υij = ∂γR/∂εsij in suffix notation) where εs is the

surface strain, which is assumed to be small. This is a2D version of the familiar connection of stress and free

energy used in 3D mechanics. Note that the 2D sur-face energy density being used is per unit area of thereference configuration. In most cases, it is more con-venient to write the surface energy per unit area of thecurrent configuration, γC , as that is what is constant inthe liquid-like limit. If the strain that connects the twoconfigurations is small, then AC = AR(1 + Tr(εs)), soγC = γR/ (1 + Tr(εs)). Then, equation (11) in termsof surface energy density in the deformed configurationbecomes

Υ = γCI2 +∂γC∂εs

. (12)

This relationship is known as Shuttleworth’s equation[22].

For small, elastic surface strains, we can derive a gen-eral form of the surface stress by Taylor-expanding γR interms of εsij (here we use suffix notation with the sum-mation convention). To leading order,

γR = γ0 +Bijεsij +

1

2Cijklε

sijε

skl. (13)

If we also assume that the material is isotropic then Bij =Υ0δij and Cijkl = λsδijδkl + µs(δikδjl + δilδjk). Withthese, equation (12) gives that, at leading order,

Υij = Υ0δij + λsεskkδij + 2µsεsij . (14)

This is the general linear-elastic form of the surface stress.λs and µs are the surface Lame constants. By analogywith bulk elasticity, µs is the surface shear modulus, andKs = λs + 2µs/3 is the surface bulk modulus, or Gibbselasticity. If the surface Lame constants are vanishinglysmall, the magnitude of surface stress is constant butneed not equal the surface free energy.

D. Soft solid interfaces

Now we put forward some simple hypotheses for theform of the surface stress and surface energy of a softsolids. We restrict our attention to polymer gels andelastomers, which have been the focus of experimentalstudies of elastocapillary phenomena. Gels consist ofcross-linked networks of polymer swollen by a solvent,while elastomers have no solvent.

In the limiting scenario of the ‘ideal gel,’ the surfacehas the same structure and composition as the bulk.Here, as a gel is deformed, solvent can move freely be-tween the surface and bulk. For a dilute gel, the bulkof the surface material is liquid, and thus the surfaceenergy is approximately that of the solvent: γC ≈ γl.In these conditions,the gel will have an isotropic surfacestress with magnitude that is identical to the surface en-ergy [112, 113]. This assumption appears to be consistentwith many experimental studies on soft hydrogels, as de-cribed in the earlier sections, (e.g. [12, 32, 34]).

Generally, however, the polymer network will be per-turbed by the presence of the surface. For example,

10

polymer chains could preferentially adsorb to the surfaceor the cross-linking density could vary near the surface.Let’s consider a simple extension of the ideal gel wherea gel with bulk Lame moduli λ1, µ1 has different moduliλ2, µ2 within a distance, h, from the surface. Here, h isassumed to be much smaller than other length scales inthe problem, so we can subsume the effects of the surfacelayer into a surface stress term. Assuming the surfacezone is still fully permeable to solvent, then there is aconstant contribution from the solvent’s surface tension,as in the ideal gel. However, upon stretching the bulksolid with a strain εij , the surface layer provides an ex-cess surface stress, so that

Υij = γlδij + (λ2 − λ1)hεskkδij + 2(µ2 − µ1)hεsij . (15)

This simple model is consistent with the general expres-sion for solid surface stress above (14). If the difference inmoduli is on the same order of magnitude as the modulusitself, and h is no larger than a few tens of nm, then weexpect the strain dependent terms to be quite small com-pared to γl for moderate strains. However, if the modulinear the surface are much larger than those in the bulk,the strain-dependent terms become significant. This maybe expected when polymer chains, or another componentof the system, absorb strongly to the interface.

This, perhaps, is the simplest example one could imag-ine for non-trivial surface rheology of a gel, i.e. Υ 6= γI2.It is very likely that the full menagerie of surface rheo-logical behavior such as seen in complex liquid systems(IV B) can occur on the interfaces of soft solids. Furtherprogress requires systematic measurements of the surfacestress in soft solids.

V. ANALYTICAL AND NUMERICALMETHODS

Several methods for analysis of deformation of softsolids that account for surface stress have been developedand used. These include:

1. Green’s functions for continuum analyses of small-strain elastic deformations that incorporate theboundary condition (eq. 4) described in SectionII A [67, 98, 114, 115]. These are useful for de-veloping analytical and semi-analytical solutions,as well as scaling relationships. Key results arethat the singularity of the stress and deformationfields at point or line loads on linear-elastic sur-faces are reduced significantly by the ability of sur-face stress to resist deformation. Green’s functiontechniques are closely related to transform methodswhich have proven useful in solving elastocapillar-ity problems. These include Fourier transforms (2dgeometries [33]), Hankel transforms (axisymmet-ric geometries [14, 74]), and Legendre transforms(spherical geometries [131]).

2. General purpose computational (e.g., finite ele-ment) methods [31, 34, 37, 116–118]. These meth-ods typically represent the role of surface stressin a modular way, say as a surface finite ele-ment, which allows surface stress effects to be com-bined with nearly any form of bulk mechanicalbehavior. In the Supplementary Material we de-scribe a 2-node surface finite element for use withnonlinear, implicit, static, finite-element simula-tions for including the influence of surface stress inplane stress/strain and axisymmetric models. Weprovide a user element file (usurf_2n_2d_axi.f)for use with the commercial finite element codeABAQUS(R) as well as an input file each for 2D(Rippled.inp) and axisymmetric (Hole_axi.inp)examples.

3. Molecular dynamics and density functional meth-ods. In molecular dynamics methods continuumproperties such as elastic moduli and surface stressemerge automatically in terms of the underlyinginter-particle potentials. They have been used suc-cessfully to study elastocapillary phenomena in-cluding contact mechanics [18, 97, 102] and wetting[119, 120]. They are best suited to relatively smalllength scales. An alternative approach is DensityFunctional Theory (DFT) [121]. This shares someof the features of molecular simulation in that itis a microscopic calculation based on inter-particlepotentials. However, the model is solved semi-analytically neglecting fluctuations and represent-ing densities by smooth functions. It is in the spiritof the classical Molecular Mechanics methods [106].It has been used, for example, to study the struc-ture near a wetting contact line [15].

In supplementary information we provide some furtherdetails about the Green’s function for 3D and 2D prob-lems, and about a finite element implementation of asurface stress element in the commercial finite elementcode ABAQUSr [122].

VI. CONCLUSIONS

The research reviewed in this manuscript has estab-lished that interfaces in soft solids carry sufficient surfacestress to strongly influence and sometimes to dominatemechanical phenomena. Collectively, we use the termelastocapillarity to represent these phenomena. In manyof these phenomena, the elastocapillary length, Υ/E, de-fines the characteristic length scale over which surfacestress dominates over elasticity as the agent resisting(and sometimes driving) deformation. This length scalecan be much larger for soft solids such as gels and elas-tomers because the interactions that determine moduli(chain entropic elasticity [123]) are substantially weakerthan and disconnected from those that determine surface

11

energy and stress (near-neighbor intermolecular interac-tions [20, 124]).

In the simplest cases, surface stress is isotropic, ho-mogeneous, and independent of surface strain, and thissuffices to explain quantitatively many experiments. Ingeneral, surface stresses need not be any of these three.There is also an intimate connection between surfacestress Υ and surface free energy γ. This is captured bythe Shuttleworth relation, which relates surface stress tohow the surface free energy varies with surface strain.It can be used to develop prototypical 2D surface-stress/surface-strain relations. These are analogous toconstitutive relations in bulk elasticity that can be de-rived from a 3D energy density and how it depends onstrain. However, the idea of surface stress and more com-plex surface constitutive behaviour need not be limited tothose described under thermodynamic equilibrium. Weanticipate that soft solid interfaces will exhibit complexrheology, as observed at complex fluid interfaces. Notealso that we have assumed in this review that the inter-face has no bending rigidity (i.e. it cannot support a mo-ment). However there are many soft interfaces, such aslipid vesicles and cell membranes, where bending rigidityis important [125]. Thus a complete description of elas-tocapillarity may also need to incorporate this possibility[126].

Two-phase systems with a single interface representsthe simplest class of problems where the role of surfacestresses has been investigated. Phenomena such as theinstability and bending of rods, flattening of a structuredsurface, and stiffening of a solid by liquid inclusions aresome of the examples studied so far; there are certainlymany others to be studied. For example, Eshelby the-ory (which we have seen is strongly modified by surfacestress) is widely used beyond composite mechanics, forexample in plasticity [127], fracture mechanics [128] andthe cell mechanics [129, 130], so surface stress may playa role in these phenomena. In particular theory suggeststhat surface stress can strongly attenuate the energy re-lease to a crack tip and thus effectively increase resistanceto fracture, but this remains to be tested experimentally.Many soft materials exhibit plasticity, so we expect awhole range of ‘plastocapillary’ effects, but this area isin its infancy [131]. Similarly, much biological materialis soft, so there is almost certainly a range of biophysicalelastocapillary phenomena to be uncovered (e.g. [132]).

The influence of surface stress on three-phase systemshas been studied most in the context of wetting (two flu-ids and one solid), and contact (two solids and one fluid).The departures from classical wetting behaviour on softsurfaces have many interesting applications and pose awide range of questions for future research. One exampleis the ability to control droplet motion. Droplets spon-taneously slide along stiffness gradients towards softersurfaces – a process called droplet durotaxis that has par-allels with cellular durotaxis [79]. Droplets [79, 133] andeven solid objects [134, 135]) spontaneously slide towards

each other over homogeneous surfaces – driven by sub-strate deformations. Further examples include changes inevaporation, condensation and droplet-nucleation rateson soft substrates [136–138]; increases in the effectivenessof soft colloids as emulsifiers [131, 139]; the use of soft sur-faces as protection against icing [140]; control of the cof-fee ring effect [141]; likely effects on nanobubble and nan-odroplet formation on soft surfaces [142]; wetting of bio-logical materials; and the potential for adhesion betweensoft surfaces by capillary bridges [143, 144] – a strategyused by many insects [145]. There are many outstand-ing questions, both theoretical and experimental, still tobe tackled. For example, it has been known for sometime that contact lines typically move at slower speedson softer substrates. This ‘viscoelastic braking’ is caus-ing by dissipation in the deforming wetting ridge under amoving contact line [146]. It allows the material proper-ties of the substrate to dramatically affect speed, contactangle and smoothness of contact-line motion (stick-slipor not) [19, 35, 147, 148]. Indeed recent work has sug-gested that contact-line motion can potentially be used asa sensitive measure of substrate rheology [35]. A secondkey area is the measurement of surface stresses: detailedimaging of the wetting ridge shape has been shown to beone of the first techniques for measuring absolute valuesof surface stresses. There are also fundamental questionsthat have arisen from current research. For example, inthe case that surface stresses differ in magnitude fromtheir corresponding surface energies, there can be rathercounterintuitive effects on substrate strains near the con-tact line (e.g. [68, 90, 149]).

Futher discussion of the outstanding questions in elas-tocapillarity can be found in [150].

DISCLOSURE STATEMENT

The authors are not aware of any affiliations, mem-berships, funding, or financial holdings that might beperceived as affecting the objectivity of this review.

ACKNOWLEDGMENTS

We acknowledge vibrant discussions with all of themembers of our emerging community, especially thosewho participated in the Lorentz Center workshop on Cap-illarity of Soft Interfaces. We also acknowledge essentialintellectual contributions from long-term collaborators inthis topic, including John Wettlaufer and Manoj Chaud-hury. Work by ED and RWS was supported by the Na-tional Science Foundation (CBET-1236086). Work byAJ and CYH was supported by the U.S. Department ofEnergy (DOE), Office of Science, Basic Energy Sciences(BES), Division of Material Sciences and Engineering un-der Award (DE-FG02-07ER46463).

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