Active elastocapillarity in soft solids with negative surface tension
AActive elastocapillarity
Jack Binysh, Thomas R. Wilks, and Anton Souslov Department of Physics, University of Bath, Claverton Down, Bath, BA2 7AY, UK School of Chemistry, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK (Dated: January 12, 2021)Active solids consume energy to allow for actuation, shape change, and wave propagation notpossible in equilibrium. For two-dimensional active surfaces, powerful design principles exist thatrealise this phenomenology across systems and length scales. However, control of three-dimensionalbulk solids remains a challenge. Here, we develop both a continuum theory and microscopic simula-tions that describe an active surface wrapped around a passive soft solid. The competition betweenactive surface stresses and bulk elasticity leads to a broad range of previously unexplored phenom-ena, which we dub active elastocapillarity . In passive materials, positive surface tension rounds outcorners and drives every shape towards a sphere. By contrast, activity can send the surface tensionnegative, which results in a diversity of stable shapes selected by elasticity. We discover that inthese reconfigurable objects, material nonlinearity controls reversible switching and snap-throughtransitions between anisotropic shapes, as confirmed by a particle-based numerical model. Thesetransition lines meet at a critical point, which allows for a classification of shapes based on uni-versality. Even for stable surfaces, a signature of activity arises in the negative group velocity ofsurface Rayleigh waves. These phenomena offer insights into living cellular membranes and un-derpin universal design principles across scales from robotic metamaterials down to shape-shiftingnanoparticles.
Active metamaterials locally consume energy to powerfunctionality not possible in thermal equilibrium. Ex-amples range from mechanical actuation and shapechange [1–8] to overdamped wave propagation [9, 10].For two-dimensional (2D) active surfaces, powerful de-sign principles exist for the distribution and controlof active elements in order to achieve a target be-haviour [7, 8, 11, 12]. Achieving the same level of controlin bulk 3D materials remains a challenge.Embedding stress-generating, active elements into apassive soft solid allows for large deformations and ac-tuation in systems from the microscale [1, 3] to themacroscale [13]. In thermal equilibrium, the shape andstructure of a soft solid is determined not only by 3Dbulk elasticity (as in traditional stiff materials), but alsoby surface stresses on its 2D boundary. This compe-tition, termed (passive) elastocapillarity [14, 15], hasbeen used to stiffen composites [16], self-assemble micro-objects [17], and drive the coiling of nanoparticle he-lices [18]. These phenomena all originate in the minimi-sation of surface area due to the isotropic surface stresstensor Υ pij = γ p δ ij , where δ ij is the Kronecker delta.Because passive elastocapillary solids are in equilibrium,their surface tension γ p is constrained to be positive.Here, we ask what happens when the surface of thesolid is driven out of equilibrium, and the surface stressacquires an additional active term. We focus on the sim-plest active contribution: an isotropic dilational stressΥ aij ≡ γ a δ ij , representing a propensity for the surface tospontaneously grow its area, Fig. 1 a . The destabilisingeffects of active forces become apparent if γ a is negativeand bigger in absolute value than γ p , sending γ a + γ p < γ ≡ γ a + γ p [19].Such dilational active stresses will arise when surfaceelements swell or push against one another, with exam-ple mechanisms shown in Figs. 1 b–d . In Fig. 1 b , com-plementary polymers are continuously inserted into thesurface of an elastic nanoparticle, providing a supply ofexcess area. This area can only be accomodated by de-formations away from the equilibrium spherical shape [1].A biological context for active surfaces, shown in Fig. 1 c ,are cellular membranes [12, 20, 21]. These living sur-faces are coupled to an elastic cytoskeleton [21], can growtheir area due to exocytosis, and have been measured topossess an effective negative surface tension [20]. Fig-ure 1 d shows an active mechanical metamaterial builtfrom macroscopic surface actuators coupled to an elasticbulk. This design extends the phenomenology of self-folding origami [7, 8] to three-dimensional materials.Figures 1 e–h illustrate a selection of the active elas-tocapillary phenomena that arise due to spontaneousgrowth of surface area. Below we focus on exact solu-tions for deformations of a sphere (Fig. 1 e ) as well assurface waves and instabilities (Fig. 1 f ). First, we give anexample that typifies the phenomenology of active elasto-capillarity: the sharpening of a cone to a cusp (Fig. 1 g ).Passive elastocapillarity smooths out a vertical cone oftip angle θ [22–24]. By contrast, active elastocapillaritywill sharpen this feature to a power-law cusp, see detailsin Methods A– C. The solid’s boundary is described bythe curve | z ( ρ ) | = θ − ρ / , converging sharply to theorigin as a function of cylindrical coordinate ρ . Suchsharpening encourages crack proliferation through stressconcentration, Fig. 1 h .We now proceed to the general framework of our con-tinuum description. Active elastocapillarity is defined by a r X i v : . [ c ond - m a t . s o f t ] J a n bae c dfh nm m mm g FIG. 1:
Active elastocapillary phenomena . a . Activesurface stresses (red arrows), powered by a fuel source (bluegradient) localised near the surface of a soft solid, competeagainst bulk elasticity (orange). b – d . Mechanisms leading toactive surface stresses, from the nanoscale to the macroscale: b . Insertion of complementary polymers into a nanoparticlesurface causes steric crowding [1]. c . Living cellular mem-branes coupled to an elastic cytoskeleton [20]. d . An activemetamaterial in which mechanical actuators are embeddedin the surface of an elastic medium [8, 9]. e – h . Continuumphenomenology: e . Switching between spheres and complexshapes of higher surface area. f . Negative group velocity ofelastocapillary Rayleigh waves. g . Sharpening of corners andedges, with a cone of tip angle θ deforming into a cusp of theform | z ( ρ ) | = θ − ρ / . h . Promotion of crack propagation. two intrinsic length scales. The first, so-called elasto-capillary length l γ ≡ | γ | /µ , measures the ratio of effec-tive surface tension | γ | to shear elastic modulus µ of thesolid. Intuitively, at length scales larger than elastocapil-lary length l γ , 3D elasticity stabilises the surface. Whathappens at ever smaller scales? In passive elastocapil- larity, stability is provided by positive surface tension.By contrast, within active elastocapillarity, the destabi-lizing contribution of negative surface tension γ must beregularized by higher-gradient surface stresses. This in-troduces the second, bendoelastic , length l κ ≡ ( κ/µ ) / coming, for example, from a surface with bending mod-ulus κ [25]. Besides passive surface contributions, thisbendoelastic length can arise from the length dependenceof the active stress γ a . In the Methods D, we discuss analternative stabilisation mechanism via elastic dispersionand contrast these effects with viscous dissipation.An object’s shape results from the competition be-tween bulk elasticity and boundary conditions containingactive surface stresses. We solve the equations of linearelastodynamics with a stress-matching condition at thesurface [26]: − σ nn = − γ − κ ∇ ⊥ ) δH (1)for slow variations in initial curvature H , where σ nn is thecomponent of the 3D elastic stress tensor normal to thesurface, ∇ ⊥ is the covariant (surface) Laplacian, and δH is the variation of mean curvature, see Fig. 2 a . In Meth-ods B, we include a derivation of Eq. (1) and details ofour solutions. Significantly, our approach accounts for3D-elastic coupling between active surface elements, de-scribing geometries inaccessible via a phenomenologicalfree energy restricted to two dimensions, c.f. Refs. [20, 21]and Methods E.For any shape of size R , the solutions that we find mustbe characterised by the lengthscale triplet ( l γ , l κ , R ). Wethen define two independent dimensionless ratios as thesurface tension and bending modulus rescaled by the size R : ˜ γ R ≡ γ/ ( µR ) = sign( γ ) l γ /R and ˜ κ R ≡ κ/ ( µR ) =( l κ /R ) . When size R is too large to be relevant (such asfor an infinite half-space), the solutions depend only onthe quantity ˜ κ γ = κµ / | γ | = ( l κ /l γ ) . Here ˜ κ γ describesthe ratio of stabilising elasticity ( µ and κ ) to destabilis-ing activity | γ | . We conclude that under overall rescaling,the phenomenology remains unchanged and that contin-uum elastocapillarity remains valid across any scale, fromnanoparticles to macroscopic metamaterials.Spheres minimise area at fixed volume, and positivesurface tension drives every initial shape towards that of asphere. By contrast, negative surface tension drives tran-sitions away from a sphere into a variety of shapes. Ourexact results demonstrate how to select between theseshapes using the elasticity of the underlying solid. Thephase diagram in Fig. 2 b shows that for low active driv-ing, | ˜ γ R | is small and spherical shapes are stable. As ac-tivity increases, spheres spontaneously destabilise. Thethreshold activity for instability, encoded in | ˜ γ R | , and theangular wavenumber l of the dominant unstable mode areboth determined by the balance of surface and 3D mod-uli ˜ κ R ( ∼ κ/µ ). When the bending modulus dominatesover 3D elasticity (large ˜ κ R ), the instability is pushedtowards bulk shape change. At the largest ˜ κ R , we find a b cd e f FIG. 2:
Shape instability and wave propagation within active elastocapillarity . a. Schematic of an elastic solidwith shear modulus µ and surface bending rigidity κ deformed by a negative surface tension −| γ | , which leads to an excessmean curvature H → H + δH . The stress-matching condition, Eq. (1), results in either restoring forces or surface instabilityand shape change. b. Phase diagram for the stability of an active elastocapillary sphere of radius R , controlled by rescaledsurface tension | ˜ γ R | [ ≡ | γ | / ( µR ), corresponding to activity] and bending modulus ˜ κ R [ ≡ κ/ ( µR )]. The border between thewhite and the coloured regions corresponds to activity strength at which the sphere goes unstable. The colours indicate theazimuthal mode number l , which is selected by the competition between shear and bending moduli. Insets show unstable modesat l = 2 , , and 4. c. Dispersion ˜ ω R ≡ ωR (cid:112) ρ/µ , with ρ bulk density, of spherical oscillations corresponding to points markedby square, triangle, and circle in b . A single mode is driven unstable (Im ˜ ω R >
0, dashed line) as the threshold ˜ γ R is crossed. d–e. The limit R → ∞ (equivalently, a zoom into the limit l → ∞ from b ) asymptotically describes a half-space, and thephase diagram collapses to become one-dimensional ( e ). The instability is controlled by the bending modulus ˜ κ γ ≡ κµ / | γ | ,given by the ratio of stabilising elasticity ( µ and κ ) to destabilising activity ( γ ). Each value of ˜ κ γ in e (coloured bars) givesa curve ˜ γ R ∼ ˜ κ / R in d . At ˜ κ ∗ γ = 1 /
27 [equivalently, | γ ∗ | = 3( κµ ) / ], the half-space destabilises at rescaled wavenumber˜ q ∗ γ ≡ q | γ | /µ = 3 ( e , green region). Even below threshold active driving a vestige of surface activity can be measured via thenegative group velocity of surface elastocapillary waves ( e , grey region), which occurs for ˜ κ γ (cid:46) . κ ∗ γ and ˜ q γ (cid:38) . q ∗ γ . f. Surfacewave dispersions ˜ ω γ ≡ ω (cid:112) ργ /µ corresponding to coloured lines in parts d–e , showing first a region of negative group velocity( d ˜ ω γ /d ˜ q γ < ω γ > κ γ decreases. shapes with uniaxial anisotropy, which are inaccessible,for example, via instabilities at fixed surface area [27].The opposite limit of small ˜ κ R allows instead for controlover fine structure and surface texture. These unsta-ble modes originate in the dispersion relation, shown inFig. 2 c , in which only a single mode is selected by neg-ative surface tension [with corresponding points markedby square, triangle, and circle in Fig. 2 b ].For a flat object, energy injection will soften surfacewaves and drive surface instabilities, Fig. 2 d–e . The two-dimensional phase diagram of a sphere collapses to theone-dimensional Fig. 2 e , described by ˜ κ γ and the planarwavenumber ˜ q γ ≡ ql γ . In Fig. 2 f , we show the planar dispersion relation ˜ ω γ (˜ q γ ), in which the frequency ω isrescaled by both elastocapillary length (cid:96) γ and transversespeed of sound c T ≡ (cid:112) µ/ρ . For weak active driving(large ˜ κ γ ) the half-space is stable, and plane waves stiffenat high wavenumbers. As active driving increases (˜ κ γ decreases), energy injection softens high wavenumbers,leading first to negative group velocity d ˜ ω γ /d ˜ q γ <
0, andthen to full-blown surface instability. Intuitively, this be-haviour stems from an effective shift of the shear modulusby negative surface tension, µ → µ − | γ | q (see Meth-ods C). This rescaling causes µ , restoring elastic forces,and phase velocities all to vanish on a scale set by l γ .The instability thresholds for wavenumber q ∗ = ( µ/κ ) / and active driving | γ ∗ | = 3( κµ ) / can both be tunedusing the surface modulus κ and 3D shear modulus µ .In other words, by selecting the material parameters ofthe passive solid, we can select the first mode that goesunstable once activity is turned on.Linear analysis can select only mode number, not modeamplitude. We now explore the role of nonlinear bulkelasticity in selecting shape, focusing on the l = 2 modeof uniaxial deformations. The amplitude of this uniax-ial strain can be approximated by a homogenous defor-mation with principal stretch factor λ , see Fig. 3 a . Weconsider the minimal Mooney-Rivlin model of nonlinearelasticity, which is often used for rubber [28] and poly-mer gels (see Methods F for a discussion of alternatemodels). An effective energy ˜ F ≡ F/µR for this far-from-equilibrium solid is given by (see Methods F):˜ F = ˜ F NH + ˜ F MR + ˜ F bend + ˜ γ R ˜ A. (2)For the active surface contribution ˜ γ R ˜ A , we take the area˜ A of a uniaxial ellipsoid (Fig. 3 a ). This is balanced by aHelfrich bending energy ˜ F bend [29] and equilibrium bulkelasticity, composed of the neo-Hookean contribution˜ F NH ≡ (1 − ˜ α ) (cid:0) λ − + λ (cid:1) / F MR ≡ ˜ α (cid:0) λ − + 2 λ (cid:1) /
2. Crucially, this model in-cludes a single parameter ˜ α to continuously tune mate-rial nonlinearity, which ranges between the neo-Hookeanlimit of only geometric nonlinearity, ˜ α = 0, and maximalnonlinearity, ˜ α = 1.Minimising Eq. (2) yields the phase diagram shown inFig. 3 b . The discontinuous shape transitions indicatedby solid lines correspond to bistable configurations in theeffective energy ˜ F ( λ ) (Fig. 3 c ). Increasing active drivingdestabilises spheres, but now the elastic nonlinearity ˜ α controls whether the resulting shape is a prolate ellip-soid (a ‘worm’) or an oblate one (a ‘pancake’). In theneo-Hookean limit ˜ α →
0, the preferred shape is a com-pressed pancake. Intuitively, this corresponds to max-imising surface area ˜ A without any elastic effects (Fig. 3 c ,left panel). For sufficiently large ˜ α , the preferred shapeis instead an elongated worm (Fig. 3 c , middle), reflect-ing the bias towards uniaxial elongation over compressionencoded in the Mooney-Rivlin theory [28]. These wormscan undergo a second snap-through transition (Fig. 3 c ,right), morphing to pancakes as active driving | ˜ γ R | isfurther increased, or as the nonlinearity ˜ α is tuned.In Figs. 3 d–e , we realise these continuum predictionsin a microscopic model of active elastocapillarity. We nu-merically simulate the relaxation of a spherical mesh ofbulk nonlinear springs, coupled to surface springs exert-ing active stresses, as shown in Fig. 3 d (see Methods Gfor details). At a critical active stress, the meshed soliddestabilises, exhibiting worms (Fig. 3 d , top), pancakes,and snap-through transitions (Fig. 3 d , bottom) depend-ing on spring-level nonlinearity, in agreement with ana-lytical predictions. In Fig. 3 e we compare the continuum theory Eq. (2), with numerical data for the stretch fac-tor λ (see Methods G). Significantly, the quantitativetheory-simulation agreement near the transition pointstowards universality.The three transition lines in Fig. 3 b meet at a criticalpoint. This critical point is a direct consequence of thesymmetry of the initial shape. In contrast to a sphere, aninitial uniaxial anisotropy λ causes the critical point tosplit, Fig. 3 f . For example, the phase diagram shown inFig. 3 f , top right, demonstrates how an initially elongatedshape with λ > f , bottom right). In activity-anisotropy space, acut through the critical point reveals an Ising-like transi-tion (Fig. 3 f , bottom left), with −| ˜ γ R | playing the role oftemperature, and λ an external field. This observationmotivates a universal characterization of shape transi-tions near the critical point using Landau theory.A general nonlinear elasticity introduces an infinite setof materials parameters, making inaccessible an exact so-lution like the one we obtained in the linear regime. How-ever, near the critical point, the complete behaviour ofthe active solid can be understood using symmetry-basedarguments. For any initial shape, the Landau expansionguarantees that the effective free energy has the form˜ F ( (cid:15) ) = r(cid:15) − w(cid:15) + u(cid:15) − h(cid:15), (3)where the linearised strain (cid:15) (= λ −
1) plays the role oforder parameter, the control parameter r ∼ ∆˜ γ R probesthe distance to linear instability, w ∼ ∆˜ α is the lowest-order nonlinear term, and u > h ( ∼ λ −
1) captures the effects of ei-ther shape anisotropy or external uniaxial stresses, and isabsent for cubes, spheres, and other spherical tops (i.e.,shapes with an isotropic moment-of-inertia tensor). Asa result, a critical point is generically expected for thesesymmetric shapes, with three weakly discontinuous tran-sitions emanating from it, as in Fig. 3 b . Although Lan-dau theory breaks down at higher strains, this criticalpoint controls the entire phase-diagram shape. We de-rive expressions for parameters r , w , u , and h within theMooney-Rivlin model Eq. (2) in the Methods F. How-ever, we emphasise that the form of Eq. (3) is fully con-strained by symmetry, presenting a universal classifica-tion across all elastocapillary materials and shapes.Active elastocapillarity couples field theories of differ-ent dimensionalities towards new materials design prin-ciples. The instabilities we have uncovered, from snap-through to smooth deformations, suggest active elasto-capillarity as a portable mechanism to achieve complexreconfigurable shapes. At the macroscale, we envisionthe design of soft robotic arms composed of an elasticbackbone covered in simple actuators. Scaling activeelastocapillarity down to soft nanoparticles, for which ab de f c Theory > ** < * FIG. 3:
Nonlinear active elastocapillarity selects mode amplitude. a.
For large amplitudes, we describe a homogenousdeformation by stretch factor λ , with λ > λ <
1, an oblate (pancake-like) one. b. Worm/pancake phase diagram. Beyond linear elasticity, sphere destabilisation due to active driving | ˜ γ R | results in eithera worm or a pancake, each stabilised by nonlinear terms. In the Mooney-Rivlin theory, the selected shape (mode amplitude)depends on material nonlinearity, parameterised by ˜ α . In the neo-Hookean limit, ˜ α →
0, pancakes are favoured. As ˜ α increases, material nonlinearity favours extension over compression, resulting in worms. These worms suffer a second ‘snap-through’ transition into pancakes at higher active driving, even at fixed nonlinearity. These three discontinuous transition linesmeet at a critical point. c. Landscapes for the effective energy describing discontinuous transitions along the arrows markedin b . d–e. Simulations of a minimal ball-and-spring model demonstrate continuum theory predictions, see Methods G fordetails. A spherical mesh of bulk nonlinear springs is coupled to surface springs exerting active stresses, with an energetic costto bending deformations of surface plaquettes (inset). At a critical active driving the meshed solid destabilises, with resultingshape tuneable via spring-level nonlinearity. For nonlinearity larger than the critical value (˜ α > ˜ α ∗ , top), the mesh forms aworm-like structure, which elongates as active driving is further increased. For ˜ α (cid:38) ˜ α ∗ , the mesh destabilises first to a worm,but then snaps through to a pancake upon increased active driving. e. Theory-simulation comparison for the dependence ofstretch factor λ on active driving | ˜ γ R | . For numerical data, orange squares and blue triangles correspond to data shown in d .Theoretical fit shows all of the (meta)-stable minima for the Mooney-Rivlin continuum theory Eq. (2) with parameters matchedto the orange squares (˜ α > ˜ α ∗ ). f. If the initial shape is anisotropic, parametrised by initial stretch factor λ , the critical pointfrom part b splits. For both worm ( λ >
1, top right) and pancake ( λ <
1, bottom right), the initial anisotropy can eithergrow smoothly or snap through a transition line depending on path in parameter space. A cut in | ˜ γ R | - λ space through thecritical point (bottom left) gives an Ising-like transition described by the Landau theory (3). no reliable shape-control mechanism exists, may proveuseful for applications ranging from drug delivery to self-assembly of photonic crystals. Acknowledgments
J.B. and A.S. acknowledge the support of the Engineer-ing and Physical Sciences Research Council (EPSRC)through New Investigator Award No. EP/T000961/1.J.B. and A.S. acknowledge illuminating discussionsthroughout the virtual 2020 KITP program on “Sym-metry, Thermodynamics and Topology in Active Mat-ter”, which was supported in part by the National Sci-ence Foundation under Grant No. NSF PHY-1748958. This work was performed in part at the Aspen Center forPhysics, which is supported by National Science Founda-tion grant PHY-1607611. The participation of A.S. at theAspen Center for Physics was supported by the SimonsFoundation. T.R.W. thanks the University of Birming-ham for funding and support. [1] Hua, Z. et al.
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Rubber Chemistry and Technology , 59–61 (1996). [43] Boal, D. H. & Rao, M. Topology changes in fluid mem-branes. Physical Review A , 3037–3045 (1992). MethodsA. Feature sharpening
A positive surface tension minimises area, causingsharp edges and features to be rounded. Given a two-[24] or three- [30] dimensional cone, positive surface ten-sion blunts the conical tip to a smoothed cap. Here, weconsider instead the case of negative surface tension. Wefind that, by contrast, surface area maximisation sharp-ens the cone to a cusped structure.A neo-Hookean cylinder of radius ρ , subject to surfacetension γ experiences a stretch factor λ along its axis of[31] λ = − γ µρ + (cid:115) (cid:18) γ µρ (cid:19) . (S1)This result applies regardless of the sign of γ . For a neg-ative surface tension, the stretch factor λ is greater thanone ( λ > ρ λ , so that at fixed volume, there is a cor-responding radial contraction ρ → ρ / √ λ . Taking thelimiting case | γ | /µ (cid:29) ρ (a thin cylinder) of Eq. (S1), fora negative surface tension we have λ ∼ ( l γ /ρ ) .Considering a three-dimensional cone of angle (cid:15) alignedalong z , each cylindrical slice z → z + dz then experiencesan elongation λ ( z ) ∼ ( l γ /(cid:15)z ) . Integrating up these elon-gations we find that Z − Z ∼ l γ (cid:15) (cid:18) ρl γ (cid:19) (S2)where Z ( ρ ) is the height of the deformed cone as a func-tion of its deformed radius. The exponent in Eq. (S2) isless than 1, indicating a cusped structure, which becomesmore pronounced with decreasing conical angle (cid:15) . B. Linear instability and shape change in an activeelastocapillary droplet
In this section we derive the linearised spectrum ofa (three-dimensional) active elastocapillary sphere. Re-cently, Ref. [26] has given an analysis of the vibrationsof a passive elastocapillary sphere, extending classical re-sults for the purely elastic [32] and capillary cases [33].Here we instead consider the active case, in which thesurface terms consist of both a negative surface tension γ and bending modulus κ . The approach is to take a bulkansatz satisfying the equations of linear elastodynamicsand impose a stress-matching boundary condition. Solv-ing the resulting equation we obtain the dispersion.We first derive the boundary condition, balancing ac-tive stresses with restoring elasticity. Given a (two-dimensional) surface with surface tension γ and bending rigidity κ , a variation of the Helfrich surface free energy F = (cid:90) dA (cid:104) κ ( H − c ) + γ (cid:105) (S3)gives the shape equation for vesicles [29], P =2 κ (cid:2) ∇ H + 2( H − c )( H − K + c H ) (cid:3) − γH, (S4)where P is (minus) the normal stress, H is the meancurvature, K is the Gaussian curvature and c allows for apreferred nonzero mean curvature. We now expand H, K and P to first order about a spherical shape, consideringa normal perturbation ψ n , where n is the outwards unitnormal: H = H + δH + O ( ψ ) = − R + δH + O ( ψ ) ,K = K + δK + O ( ψ ) = 1 R + δK + O ( ψ ) ,P = P + δP + O ( ψ ) . (S5)(Note the use of the outward normal gives H = − /R .)Here, δH , δK are given by [34, 35] δH = 1 R ψ + 12 ∇ ψ,δK = − R ψ − R ∇ ψ. (S6)Substituting Eqs. (S6) and (S5) into the shape equa-tion Eq. (S4) yields the normal component of the stressmatching condition − σ nn = − (cid:20)(cid:18) γ + 2 κc (cid:18) R + c (cid:19)(cid:19) − κ ∇ (cid:21) δH, (S7)with the tangential component σ τn = 0, where τ denotesthe surface tangent. For an expansion in terms of spher-ical harmonics, we take ψ = N Y ml = N P ml (cos θ ) e imφ ,with P ml the associated Legendre polynomial, and N anormalisation factor [36]. Then ∇ ψ = − l ( l + 1) ψ andEq. (S7) simplifies to − σ nn = (cid:20) γ + 2 κc (cid:18) R + c (cid:19) + κR l ( l + 1) (cid:21) (2 − l ( l + 1)) R ψ. (S8)From Eqs. (S7), (S8) we see the effect of the bendingmodulus and spontaneous curvature is to shift γ as γ → γ + 2 κc (cid:18) R + c (cid:19) + κR l ( l + 1) . (S9)Two natural special cases of this result are c = 0 (nospontaneous curvature) and c = − /R (spontaneouscurvature matching the initial mean curvature H ). Inthe main text, we focus on the case c = 0, for whichEq. (S7) simplifies to Eq. (1). However, note that onlythe last term in Eq. (S9) depends on l and the effect of c can be completely reabsorbed into the effective surfacetension γ .Using the shift Eq. (S9) we obtain the dispersion re-lation for spheroidal modes. The relevant dimension-less variables come from sphere radius R and associatedtimescale τ R = R/ (cid:112) µ/ρ : l = qR, ˜ ω R = τ R ω, ˜ γ R = l γ R , ˜ κ R = κµR = (cid:18) l κ R (cid:19) . (S10) In terms of these dimensionless quantities, the dispersionis given by the solution of˜ ω R (cid:2) ω R − l (˜ γ R + ˜ κ R l ( l + 1)) + 2 l (1 + ˜ γ R + ˜ κ R l ( l + 1)) − l (4 + ˜ γ R + ˜ κ R l ( l + 1)) (cid:3) j l (˜ ω R ) − (cid:2) ˜ ω R + l (2 + ˜ γ R + ˜ κ R l ( l + 1)) (cid:0) − l − l (cid:1)(cid:3) j l +1 (˜ ω R ) = 0 , (S11)where j l is a spherical Bessel function of the l th kind.Figs. 2 b–c are found by solving Eq. (S11) numerically.Note that the dispersion Eq. (S11) has an infinite numberof branches, corresponding to the roots of j l — spheroidalmodes ω R ( s, l ) are indexed by a radial ‘quantum number’ s and polar wavenumber l , being degenerate with respectto the azimuthal wavenumber m . Only the s = 1 branchcouples to the instability described in the main text, andit is this branch that is shown in Figs. 2 b–c . C. Waves and instabilities in an activeelastocapillary half space
In this section, we derive the spectrum of the linearisedequations of motion for an active elastocapillary half-space. These results follow from the l → ∞ limit of sec-tion B, in particular Eq. (S11). However, an independentderivation in the planar case has the virtue of being muchsimpler than the spherical case, and we shall extend it tostudy the effects of viscosity and bulk dispersion in sec-tion D. Passive elastocapillary waves have been studiedfrom the perspective of a viscous fluid [37] or an elasticsolid. Here, we take the elastic solids perspective [38],and consider the active case, in which a negative surfacetension γ is regularized in the high wavenumber limit bya bending modulus κ . Our approach applies equally totwo- or three-dimensional materials.Before giving a detailed argument, basic scaling con-siderations capture the main phenomenology. Consider a slab of material of undeformed surface area A with adeformed surface height h ( x ). The energy stored in sur-face deformations is E s ∼ (cid:0) γ h (cid:48) ( x ) + κ h (cid:48)(cid:48) ( x ) (cid:1) A , andthe bulk energy E b ∼ µh (cid:48) ( x ) Al , where l is the depththat surface deformations penetrate into the bulk. As-suming l ∼ /q , the total energy per unit volume is then E/ ( lA ) ∼ ( µ + γ q + κ q ) h , and we see that γ acts as a q dependent shift to the shear modulus, µ ( q ) = µ + γ q . As q increases, for γ < µ ( q ) softens, with higher wavenum-bers feeling progressively weaker elastic restoring forces.At q ∼ /l γ , restoring elasticity vanishes entirely, with κ regularising high wavenumbers. We thus expect thethreshold wavenumber for instability to scale as 1 /l γ .We now give a detailed derivation of the spectrum.We consider a half space z ≤
0. The equations of linearelastodynamics in the bulk material are [39] ρ ¨ u i = ∂ j σ ij , (S12)where ρ is the density and u i is the displacement. Foran isotropic material, the stress tensor σ ij = Bδ ij u kk +2 µ (cid:0) u ij − d u kk δ ij (cid:1) , where u ij = ( ∂ i u j + ∂ j u i ) is thelinearized strain, and d is the spatial dimension. Equa-tion (S12) supports longitudinal and transverse waves, ofwavevector q and frequency ω , propagating along x anddecaying as z → −∞ [38, 39]: u L = ( q e x − iα L e z ) exp[ i ( qx − ωt ) + α L z ] , u T = ( iα T e x + q e z ) exp[ i ( qx − ωt ) + α T z ] . (S13)Here α T = (cid:112) q − ρω /µ and α L = (cid:112) q − ρω /M are0the inverse penetration depths along z , with the longitu-dinal modulus M = B +2 µ ( d − d . A general displacement u = u x e x + u z e z is written as u = (cid:2) A L ( q e x − iα L e z ) exp α L z + A T ( iα T e x + q e z ) exp α T z (cid:3) exp i ( qx − ωt ) . (S14)We now take the bulk ansatz Eq. (S14) and substituteit into a stress matching boundary condition at z = 0.The boundary has surface tension γ , bending modulus κ ,and effective free energy F = (cid:90) dx (cid:34) γ (cid:18) dhdx (cid:19) + κ (cid:18) d hdx (cid:19) (cid:35) , (S15) where h ( x ) is the height of the free surface above z = 0.Matching the z component of bulk displacement u z toheight h yields the stress matching condition σ zz | z =0 = γ d u z dz − κ d u z dz ,σ xx | z =0 = 0 . (S16)Substituting Eqs. (S14) into (S16) gives (cid:20) [ − iα L M + iq ( M − µ )] − iα L ( γ + κq ) q µqα T + ( γ + κq ) q α L q i ( α T + q ) , (cid:21) (cid:20) A L A T (cid:21) = 0 . (S17)At this point we take the incompressible limit: as B, M → ∞ , α L → q , and Eq. (S17) simplifies to (cid:20) [ − i (2 µq + ( γ + κq ) q ) 2 µqα T + ( γ + κq ) q q i ( α T + q ) (cid:21) (cid:20) A L A T (cid:21) = 0 . (S18)The dispersion is obtained from the solvability conditionthat the determinant of Eq. (S18) must vanish, ρω − µq α T q + α T − γq − κq = 0 . (S19)Equation (S19) reduces to the passive elastocapillarydispersion (in the incompressible limit) for γ > κ → κ, γ → l γ = | γ | µ (elastocapillary length) ,l κ = (cid:18) κµ (cid:19) (bendoelastic length) ,τ γ = l γ (cid:112) µ/ρ = (cid:115) ργ µ (elastocapillary time) , (S20) which we use to define nondimensionalised variables˜ q γ = l γ q, ˜ ω γ = τ γ ω, ˜ α γ = (cid:113) ˜ q γ − ˜ ω γ , ˜ κ γ = (cid:18) l κ l γ (cid:19) = κµ | γ | . (S21)In dimensionless form, Eq. (S19) is then˜ ω γ − q γ ˜ α γ ˜ q γ + ˜ α γ − sgn( γ )˜ q γ − ˜ κ γ ˜ q γ = 0 . (S22)Here, we focus on the case of negative surface tension,sgn( γ ) = −
1. The phase diagrams shown in Figs. 2 d–e , and the dispersions shown in Fig. 2 f , are found bysolving Eq. (S22) numerically. However, the threshold atwhich instability occurs can be found analytically: letting˜ ω γ = 0 in Eq. (S22) gives the condition˜ q γ ( − κ ˜ q γ + ˜ q γ −
2) = 0 . (S23)We require the cubic in Eq. (S23) to posses degenerateroots, as must be the case at instability. This gives val-ues for the wavenumber ˜ q ∗ γ and dimensionless bending1modulus ˜ κ ∗ γ at which instability sets in:˜ q ∗ γ = 3 , ˜ κ ∗ γ = (cid:18) (cid:19) . (S24)We may compare the structure of the above derivationto the intuitive argument presented at the beginning ofthis section. To the extent that α T ≈ q (strictly true inthe Rayleigh wave limit q → q -dependent shear modulus, µ ( q ) = µ + γ q .Further, referring to Eq. (S24), we find that the thresh-old wavenumber for instability occurs at q ∼ /l γ , asexpected. D. Modifications to active elastocapillarity: theeffect of viscosity and bulk dispersion
1. The effect of viscosity
In the main text, we have focused on inertial dynamics,but we may easily consider the effects of viscosity usinga Kelvin-Voigt viscoelastic shear modulus, µ ( ω ) = µ − iνω, (S25)in Eq. (S19). For simplicity, we focus on the overdampedlimit, neglecting inertia. In this limit, Eq. (S19) simplifiesto iνω − µ − γ q − κ q = 0 , (S26)or in dimensionless units i ˜ ω ν − q γ −
12 ˜ κ γ ˜ q γ = 0 , (S27)where ˜ ω ν = νω/µ . Unlike some other examples of activesolids [10], here there is no phase lag between drivingand response, so ˜ ω ν is always purely imaginary and wedo not see overdamped waves. However, our tuneableshape instability remains, with the critical wavenumberand bending modulus as in Eq. (S24).
2. Bulk dispersion
In the main text, we focused on a bending modulus asthe regularizing mechanism for the q → ∞ limit of ac-tive elastocapillarity. The lengthscale that κ introduces, l κ , comes from the surface physics. An alternate ad-ditional lengthscale, l µ , comes instead from stabilisinghigher-order gradients in the the bulk physics. In Fourierspace, we consider a q -dependent 3D shear modulus µ ( q ) = µ + µ q + ..., (S28) where l µ = (cid:112) µ /µ and µ >
0. We can assess theeffects of such higher-order terms using Eq. (S19), bysetting κ = 0 and sending µ → µ + µ q . The result is anew dispersion, ρω − µ ( q ) q α T q + α T − γq = 0 , (S29)where now α T = (cid:112) q − ρω /µ ( q ). In the absence ofsurface tension, Eq. (S29) admits Rayleigh waves ω ∼ (cid:112) µ ( q ) /ρq ∼ µ q ) as q → ∞ . By contrast, pure capillarywaves scale as ω ∼ γq . A power count thus indicatesthat bulk dispersive effects also regularize the large q limit. The asymptotic behaviour of Eq. (S29) as q → ∞ is ρω + | γ | q − ξ µ q = 0 , (S30)where ξ = 0 . ... is the ratio of Rayleigh to bulk wavevelocity [39]. In dimensionless form, Eq. (S30) is˜ ω γ + ˜ q γ − ξ ˜ µ γ ˜ q γ = 0 . (S31)where ˜ µ γ = ( l µ /l γ ) . Comparing Eq. (S30) to Eq. (S19),we see the effects of bulk dispersion on high wavenumbersare qualitatively similar to a bending modulus, but theexact scalings differ. High wavenumbers are stabilised as q with the bending modulus κ , but as q with bulk dis-persion µ . Instead of ˜ κ γ = ( l γ /l γ ) controlling the phaseplanes Fig. 2 d–e , we have ˜ µ γ = ( l µ /l γ ) . These dimen-sionless variables scale differently with their associatedlengthscales l κ and l µ . One consequence of this differ-ence would be a shifted scaling of the phase boundariesin Figs. 2 d–e . E. Explicit bulk elasticity versus an externalsurface potential
In this section, we contrast two approaches to mod-elling coupling between a surface and an elastic bulk.The first approach, taken in this work, explicitly usesboth bulk and surface free energy terms. The free energyof a nearly-planar surface is then F = (cid:90) d x (cid:104) Bu ii + µ ( u ik − δ ik u ll ) (cid:105) + (cid:90) d x (cid:104) γ ∇ h ) + κ (cid:0) ∇ h (cid:1) (cid:105) . (S32)Equation (S32) is translationally invariant (invariant un-der u i → u i + a, h → h + a ). One consequence of thisinvariance is that the spectrum of Eq. (S32) is gapless,tending to the Rayleigh wave dispersion ω ∼ q . The scal-ings of the relevant length scales with material parame-ters (see Eq. (S20)) are l γ = | γ | /µ and l κ = ( κ/µ ) .The second approach is to use a surface-only free en-ergy, with a phenomenological correction to account for2bulk coupling. Such an approach has been used to anal-yse how the cytoskeleton in red blood cells affects theirmembrane fluctuations [40], and has been proposed tomodel cytoskeletal wave propagation [21]. In this ap-proach, we add an external Hookean potential V h to asurface-only free energy: F = (cid:90) d x (cid:20) V h + γ ∇ h ) + κ (cid:0) ∇ h (cid:1) (cid:21) . (S33)Equation (S33) models the bulk elasticity as a fixed bedof springs at h = 0, attached to the surface but notcoupled to one another. Crucially, unlike Eq. (S32),Eq. (S33) is not translationally invariant. This differ-ence leads to qualitatively different predictions. For ex-ample, Eq. (S33) predicts a gapped spectrum, with azero-frequency gap ∼ √ V . Further, Eq. (S33) gives dis-tinct scalings of l γ , l κ with material parameters: l γ =( | γ | /V ) and l κ = ( κ/V ) . F. Nonlinear theory
1. Derivation of the Mooney-Rivlin energy
In this section we detail the derivation of Eq. (2), giv-ing expressions for ˜ F NH , ˜ F MR , ˜ γ R ˜ A and ˜ F bend for thecase of a uniaxial ellipsoid. We emphasise at the outsetthat the details of the results depend on the ellipsoidalgeometry we have chosen, but their structure does not.One may repeat these calculations for other starting ge-ometries, for example a cube, and obtain similar results.Elastic deformations are described by the deformationgradient tensor Λ [41]. For a three-dimensional material,Λ ≡ ∂ X ∂ x is a three-dimensional tensor which gives thelocal mapping of material points from the undeformedstate x to the deformed state X . In general, Λ dependson x , the position within the undeformed state. Herewe assume a homogeneous deformation, for which Λ isconstant.Given Λ, the three lowest-order rotational invariantswhich can appear in the elastic free energy density f elastic are I = Tr C , I = (Tr C ) − Tr( C T C ), I = Det C ,where C = Λ T Λ is the right Cauchy–Green deformationtensor [41]. I is the neo-Hookean term and I describesvolumetric deformations, i.e., I = 1 for incompressiblesolids as we consider here. I is the Mooney-Rivlin term,often used to phenomenologically account for materialnonlinearity in rubbers [28]. The elastic part of the freeenergy density can be written as f elastic = c I + c I + · · · (S34)We consider a uniaxial deformation, Λ =diag(1 / √ λ, / √ λ, λ ), and let µ = ( c + c ) ( µ is indeedthe linear elastic shear modulus [41]), α = ( c − c ). Equation. (S34) is then f elastic = µ − α (cid:18) λ + λ (cid:19) + α (cid:18) λ + 2 λ (cid:19) . (S35)For a uniaxial ellipsoid of radii ( R/ √ λ, R/ √ λ, Rλ ) thetotal elastic free energy is F elastic = 4 π µ (cid:20) − ˜ α (cid:18) λ + λ (cid:19) + ˜ α (cid:18) λ + 2 λ (cid:19)(cid:21) R , (S36)where we identify the first term in Eq. (S36) as F NH , andthe second as F MR .The surface energy is γA ellipsoid = 2 πγλ (cid:34) λ e ( λ ) arcsin e ( λ ) (cid:35) R , (S37)where e ( λ ) = √ − λ − is the eccentricity. For the bend-ing energy F bend we use the Helfrich form discussed insection B: F bend = 2 κ (cid:90) dA ( H − c ) . (S38)We now evaluate Eq. (S38) for the case of a uniaxial ellip-soid. Finite c does not qualitatively change the structureof our results, and we consider c = 0 for simplicity. Thearea element is dA = R √ (cid:18) λ + λ + (cid:18) λ − λ (cid:19) cos 2 v (cid:19) sin v d u d v, (S39)where u , v are the azimuthal and polar angles on theellipsoid. The mean curvature H is H = 3 + λ − ( λ −
1) cos 2 v √ Rλ (cid:0) λ + λ + (cid:0) λ − λ (cid:1) cos 2 v (cid:1) . (S40)Equation (S38) then simplifies to F bend = 4 πκ (cid:90) π d v (cid:0) λ − ( λ −
1) cos 2 v (cid:1) sin v √ λ (cid:0) λ + λ + (cid:0) λ − λ (cid:1) cos 2 v (cid:1) , (S41)which may be evaluated exactly; the result is F bend = 2 πκ (cid:32) λ + 3 λ tanh − (cid:0) √ − λ (cid:1) √ − λ + 7 (cid:33) . (S42)Combining Eqs. (S36), (S37) and (S42), all divided by µR , gives the total free energy Eq. (2).3 ab FIG. S1:
The structure of the worm/pancake phasediagram . a. Three weakly discontinuous transitions, de-scribed by the Landau theory Eq. (S45), meet at a criticalpoint given by Eq. (S44). b. As the bending modulus ˜ κ R increases, the critical point is driven to larger active driving | ˜ γ R | and lower material nonlinearity ˜ α , enlarging the worm-like region of parameter space.
2. Landau theory coefficients for the Mooney-Rivlin model
In the main text, we argued that the behaviour of anactive elastocapillary sphere near the critical point λ = 1can be understood based only on symmetries, using theLandau expansion Eq. (3). Here, we derive the coeffi-cients r , w , u of Eq. (3) for the case of the Mooney-Rivlin free energy Eq. (2). Expanding Eqs. (S36), (S37) and (S42) in the strain (cid:15) (= λ −
1) we obtain
FµR = A (cid:15) + A (cid:15) + A (cid:15) + ...,A = 25 π (4˜ γ R + 24˜ κ R + 5) ,A = − π
105 (35˜ α + 52˜ γ R + 360˜ κ R + 35) ,A = 2105 π (105˜ α + 110˜ γ R + 1056˜ κ R + 70) . (S43)Equation. (S43) has a critical point at ˜ γ ∗ R , ˜ α ∗ , ˜ κ ∗ R , where˜ γ ∗ R = −
14 (5 + 24˜ κ ∗ R ) , ˜ α ∗ = 635 (5 − κ ∗ R ) . (S44)Expanding as ∆ γ = ˜ γ R − ˜ γ ∗ R , ∆ α = ˜ α − ˜ α ∗ , ∆ κ =˜ κ R − ˜ κ ∗ R , we obtain the structure of the free energy aboutthis critical point: FµR = r(cid:15) + w(cid:15) + u(cid:15) + ...,r = 85 π (∆ γ + 6∆ κ ) ,u = 1105 π ( − α − γ − κ ) ,w = 1105 π (504˜ κ ∗ R + 45) , (S45)where we omit terms like ∆ γ(cid:15) .In Fig. S1 we show phase diagrams obtained from min-imising the exact free energy Eq. (2). We can interprettheir structure in light of Eqs. (S44), (S45), with a fo-cus on the novel interplay between bulk elasticity andsurface effects. The Landau theory of Eq. (S45) corre-sponds to three weakly discontinuous transitions in the˜ α, | ˜ γ R | plane, meeting at a critical point (Fig. S1 a ). Thisstructure is unchanged by varying the bending modu-lus ˜ κ R . However, increasing the bending modulus drivesthe critical point to higher values of active driving | ˜ γ R | and lower material nonlinearity ˜ α (Fig S1 b ), enlargingthe region of phase space in which worms are favouredover pancakes. In this sense, both material nonlinearity˜ α and bending rigidity ˜ κ R conspire to produce worm-likestructures, as opposed to the more intuitively obviouspancake.
3. Varying the elastic strain energy: The Gent model
In the main text, we explored the effects of materialnonlinearity on the phase diagram of an active elastocap-illary sphere, using the Mooney-Rivlin model as a mini-mal example. Here, we discuss the effects of using a dif-ferent elastic strain energy. The expansion Eq. (S45) hasthe nonlinearity parameter ˜ α entering at cubic order in4 a c db FIG. S2:
Phase diagram of the Gent model . a–b. Cutsin ˜ κ R – | ˜ γ R | space for the neo-Hookean limit β = 0 ( a ), and ageneric nonzero β ( b ). The nonlinearity β does not changethe location of the critical point. c–d. Cuts in β – | ˜ γ R | spaceabove and below ˜ κ ∗ R , showing a wormlike region opening upwithout a critical point. (cid:15) , as is generically expected. The structure of the phasediagram Fig. 3 b stems directly from this cubic term, andas such should be preserved across different choices ofstrain energy. A singular example where this does notoccur is the Gent model [42], f Gent = − µ β log (1 − β ( I − . (S46)Here, β is a phenomenological parameter modelling finitechain extensibility, with the limit β → f Gent = 3 µ(cid:15) − µ(cid:15) + 14 (9 β + 4) µ(cid:15) + ..., (S47)in which β enters at quartic order. The result is that theGent model behaves essentially as a neo-Hookean solidfor all β .Equation (S44) gives the location of the critical pointwithin the Mooney-Rivlin model. It describes a line in ˜ α –˜ κ R – | ˜ γ R | space. This line intersects a generic coordinateplane to exhibit the critical point. In the Gent model,this line is parallel to the β axis, and so a cut in material ab rotation2 FIG. S3:
Two dimensional active elastocapillarity . a. In two dimensions, worms and pancakes are related by a π/ b. The phase diagram, in ˜ γ R only, predicts a con-tinuous transition from a sphere to an ellipse with increasedactive driving. nonlinearity space will not exhibit a critical point. How-ever, a generic cut, including the ˜ κ R – | ˜ γ R | plane, will. InFig. S2 a–b , we show the ˜ κ R – | ˜ γ R | plane of the Gent phasediagram, for β = 0 (the neo-Hookean limit) and a genericnonzero β . The phase plane contains a critical point,but upon varying β its location does not change, as isgenerically expected. Rather, the wormlike region of thephase plane simply narrows. The location of the criticalpoint can be found by setting ˜ α = 0 in Eq. (S44), giving˜ κ ∗ R = 5 /
8. In Figs. S2 c–d we show the β – | ˜ γ R | plane be-low and above ˜ κ ∗ R . The locus of the critical point runsparallel to these cuts, and we do not see a critical point inthese diagrams. Instead, at low bending modulus, pan-cakes are favoured for all β . At high bending modulus, awormlike region opens up, separated from pancakes by acurve running to β → ∞ .
4. The two-dimensional case
In the main text, we have focused on nonlinear effectsin a three-dimensional context, but they apply equally toa two-dimensional active elastocapillary disk. A notabledifference is that in two dimensions, worms and pan-cakes are related by a π/ a .This symmetry will constrain the resulting Landau the-ory. The two-dimensional version of the nonlinear modelEq. (2) is˜ F = 12 (cid:18) λ + λ (cid:19) + 2˜ γ R (cid:18) λ + λ (cid:19) , (S48)which has the symmetry λ → /λ arising from π/ a ). Note that there is noMooney-Rivlin type invariant in two dimensions [41], and5 a t ij b k FIG. S4:
Microscopic ball-spring model . a. Schematicof the microscopic model at the surface of the meshed ball,showing vertices i , j , k... , edges ij , jk... , surface triangularplaquettes α , and volume tetrahedra t . b. Cut-through of themeshed ball used in simulation, showing pre-stressed surfacesprings (blue) and unstressed bulk springs (orange). our phase diagram becomes one dimensional, in ˜ γ R only.For simplicity we have neglected the bending modulus— its effect is only to shift the critical point to higheractive driving. We first expand Eq. (S48) in the strainas (cid:15) = λ −
1, for which the rotational symmetry reads (cid:15) → − (cid:15) + (cid:15) − ... Defining ∆ γ = ˜ γ R + 2, we have˜ F ( (cid:15) ) = const + 2∆ γ(cid:15) − γ(cid:15) + (1 + 2∆ γ ) (cid:15) + O ( (cid:15) ) . (S49)In Eq. (S49) rotation symmetry constrains r = − w , withboth the quadratic and cubic terms vanishing at the criti-cal point. We thus find a continuous transition at ˜ γ R = 2from a disk to an ellipse, as shown in Fig. S3 b . We canmake this symmetry more explicit by defining a new or-der parameter, φ = ln λ . The symmetry λ → /λ trans-lates to φ → − φ . In terms of φ , the nonlinear modelEq. (S48) reads˜ F = 2(cosh(2 φ ) + 2 γ R cosh( φ )) , (S50)which is manifestly even in φ . Expanding in small φ , wefind ˜ F = 2∆ γφ + 16 (6 + ∆ γ ) φ + O ( φ ) . (S51)Equation (S51) only contains even powers of φ , showingthe phase diagram to be exactly that of an Ising model. G. Numerics
In this section we describe the microscopic ball-springmodel and numerical methods used to realise the predic-tions of the continuum theory shown in Fig. 3.
1. Microscopic Model
We first construct a disordered tetrahedral meshing ofthe ball, as shown in Fig. S4. We label the vertices i , edges ij , triangles α and tetrahedra t . A microscopicenergy for deformations of this mesh is given by a springenergy F spring , a surface bending energy F bend , and anapproximate volume constraint F bulk : F = F spring + F bend + F bulk . (S52)For F spring , we place a spring along every edge ij of themesh. These springs have length r ij , and rest length r ij ,from which we define the extension λ ij = r ij /r ij . Thespring energy is then F spring = k m (cid:88) i>j (1 − α m ) (cid:0) λ − ij + λ ij (cid:1) + α m (cid:0) λ − ij + 2 λ ij (cid:1) , (S53)i.e. each spring acts as an incompressible Mooney-Rivlinsolid with microscopic neo-Hookean constant k m and ma-terial nonlinearity α m . To implement dilational surfacestresses, we pre-stress the springs on the surface of theball, initialising them at an extension λ m <
1. The bulksprings are initialised at their rest length, λ = 1. Wevary the macroscopic nonlinearity in material response ˜ α by varying α m for the bulk springs, keeping the surfacesprings at α m = 0. A bending energy F bend is given byRef. [43]: F bend = κ m (cid:88) α,β (1 − n α · n β ) , (S54)where the sum is over neighboring triangular plaquettes α, β on the surface of the ball, with n α the normal to pla-quette α . Finally, we approximately enforce incompress-ibility with an additional energetic penalty on volumechanges of the tetrahedra t of the mesh [41]: F bulk = B m (cid:88) t ( V t − V t ) , (S55)where V t is the current volume of a tetrahedron and V t is its rest volume.The microscopic energy Eq. (S52) contains k m , α m κ m , λ m and B m as microscopic parameters. We now describea mapping to the continuum shear modulus µ , bulk mod-ulus B , material nonlinearity ˜ α , bending rigidity κ andsurface tension γ . Given a typical mesh lengthscale a ,dimensional analysis gives µ ∼ k m a , (S56)˜ α ∼ α m , (S57) B ∼ B m a . (S58)For an analytical estimate of the relation between κ m and κ , we may calibrate using the continuum limit ofthe discrete bending energy Eq. (S54) for a sphere,64 πκ m / √ πκ gives the relation κ = 12 √ κ m . (S59)For an estimate of the mapping from λ m to | γ | , onecan show that the energy per unit area of a triangu-lar spring mesh of side length λ m a , composed of neo-Hookean springs, is given by | γ | = 2 √ a k m (cid:18) λ m (cid:19) . (S60)
2. Numerical Methods
The data in Fig. 3 are generated by numerically mini-mizing Eq. (S52) at fixed k m , α m , κ m , B m , with λ m pro-gressively decreasing from λ m = 1 (giving progressivelystronger dilational surface stresses). The final state ofeach minimization is then used as an initialisation condi-tion for the next. The minimizer used is the SciPy imple-mentation of BFGS algorithm, with mesh vertex coordi- nates as input, and gradient norm stopping threshold of10 − . For the data shown in Fig. 3, a ball of radius R = 1(which defines the arbitrary spatial unit) is meshed withtypical edge spacing a = 0 .
2. Microscopic parameters k m = 0 . B m = 5000, κ m = 2 . e correspond to microscopicnonlinearities α m = 0 (green triangles), 0 . . λ m to γ using Eq. (S60). To obtain the valuesof λ shown in Fig. 3 e , an ellipsoid is then least-squaresfit to the boundary vertices of the numerically relaxedmesh. The fit returns three ellipsoid axes, two of whichare of similar magnitude ( δλ/λ < . λ .Finally, we use the location of the critical point to fitthe continuum theory Eq. (2) to this data, with ˜ κ R , ˜ α as fitting parameters. The theoretical fit shown in Fig. 3corresponds to ˜ κ R = 0 .
3, ˜ α = ˜ α ∗ via Eq. (S44). An inde-pendent assessment of ˜ κ R may be made using Eqs. (S59–S58), from which we estimate κ ≈ . µ ≈ µ for the meshed sphere used insimulation places µ ≈ κ R ≈ ..