Liquid demixing in elastic networks: cavitation, permeation, or size selection?
Pierre Ronceray, Sheng Mao, Andrej Košmrlj, Mikko P. Haataja
LLiquid demixing in elastic networks: cavitation, permeation, or size selection?
Pierre Ronceray,
1, 2, ∗ Sheng Mao,
3, 4
Andrej Koˇsmrlj,
4, 5 and Mikko P. Haataja
4, 5, † Center for the Physics of Biological Function, Princeton University, Princeton, New Jersey 08544, USA Aix Marseille Univ, Universit´e de Toulon, CNRS, CPT,Turing Center for Living Systems, Marseille, France Department of Mechanics and Engineering Science, BIC-ESAT,College of Engineering, Peking University, Beijing 100871, People’s Republic of China Department of Mechanical and Aerospace Engineering,Princeton University, Princeton, New Jersey 08544, USA Princeton Institute for the Science and Technology of Materials (PRISM),Princeton University, Princeton, New Jersey 08544, USA
Demixing of multicomponent biomolecular sys-tems via liquid-liquid phase separation (LLPS)has emerged as a potentially unifying mechanismgoverning the formation of several membrane-less intracellular organelles (“condensates”) [1–7],both in the cytoplasm ( e.g. , stress granules) andin the nucleoplasm ( e.g. , nucleoli). While both invivo experiments [8] and studies of synthetic sys-tems [9, 10] demonstrate that LLPS is strongly af-fected by the presence of a macromolecular elasticnetwork, a fundamental understanding of the roleof such networks on LLPS is still lacking. Here weshow that, upon accounting for capillary forces re-sponsible for network expulsion, small-scale het-erogeneity of the network, and its nonlinear me-chanical properties, an intriguing picture of LLPSemerges. Specifically, we predict that, in additionto the experimentally observed cavitated droplets[8, 9] which fully exclude the network, two otherphases are thermodynamically possible: elasti-cally arrested, size-limited droplets at the net-work pore scale, and network-including macro-scopic droplets. In particular, pore size-limiteddroplets may emerge in chromatin networks, withimplications for structure and function of nucleo-plasmic condensates.
When LLPS occurs without mechanical constraints(Fig. 1A), the thermodynamically stable outcome of thedemixing is a macroscopic spherical droplet of the minor-ity liquid (red) embedded within the majority phase (yel-low). This ensures that the contact surface per dropletvolume between the two liquids is minimized, and resultsin a negligible, sub-extensive free energy penalty com-pared to the bulk phase-separated liquid. In the pres-ence of an elastic matrix hindering LLPS, in contrast,we distinguish and study here three distinct scenariosby which demixing can occur (Fig. 1B). Each scenarioresults in a specific free energy cost compared to thereference, unhindered case: (i)
The minority liquid cancreate a macroscopic cavity, which incurs a deformation ∗ [email protected] † [email protected] energy penalty E el associated with the elastic matrix. (ii) Alternatively, the minority liquid may form an ex-tensive number of microdroplets fitting within the poresof the network, which avoids elastic deformation but in-curs an extensive surface energy penalty E surf . (iii) Fi-nally, rather than fully excluding the network, the mi-nority droplet can permeate through it, resulting in afinite wetting energy E wet between the droplet and thenetwork. Below, we establish an equilibrium phase di-agram for LLPS within an elastic network by assessingthe relative thermodynamic stability of each scenario.Specifically, we introduce physically-based models foreach of the three energy penalties compared to the ref-erence situation of demixing in the absence of a network(Fig. 1A), and evaluate the stability of each phase byquantifying this free energy cost per volume of phase-separated minority liquid, an approach previously usedfruitfully in the context of block copolymer phase behav-ior [11]. We focus on identifying the thermodynamicallystable droplet configurations, and thus ignore all kineticprocesses (incl. nucleation, growth and coarsening). Inaddition, we assume that the droplets occupy a negli-gible volume fraction, and hence neglect all elastic andchemical inter-droplet interactions. We first employ sim-ple scaling arguments to establish a morphological phase FIG. 1. A. Liquid-liquid phase separation (LLPS) from aninitially mixed phase ( top ) results in a macroscopic droplet ofthe red minority liquid immersed in the yellow majority one( bottom ). B. When LLPS occurs with the initial mixed phaseimbibed in an elastic network ( top ), we identify three possibleoutcomes ( bottom ): (i) Cavitation. (ii)
Microscale droplets. (iii)
Permeation of the network into the minority phase. a r X i v : . [ c ond - m a t . s o f t ] F e b diagram by considering only the dominant term(s) of thefree energy for each phase. Motivated by existing the-oretical approaches [9, 10, 12, 13], we then develop acomprehensive theory of droplets constrained by elasticnetworks by accounting for capillary forces responsiblefor network expulsion from droplets, as well as hetero-geneities in the network structure and its nonlinear me-chanical properties. Both analytical and numerical ap-proaches are employed to confirm the salient features ofthe phase diagram and elucidate the nature of phase tran-sitions between the droplet phases.We begin by considering scenario (i) , where a macro-scopic droplet of size r → ∞ forms by creating a network-excluding cavity. This scenario was previously consideredfor in vitro oil-water mixtures in silicone gels [9, 10, 12]and in vivo droplets in the cell nucleus [8]. In orderto form a macroscopic cavity from an initial pore, largedeformations must occur in the network. Therefore, itis necessary to go beyond simple linear elasticity in thetreatment of the network mechanics. The simplest suchextension is a neo-Hookean (NH) constitutive relation,considered in Refs. [8–10], where the elastic energy E el ( r ) ∼ πr αG (1)scales as the volume of the cavity when r → ∞ . Here, G denotes the shear modulus of the network, while thenumerical coefficient α ∼ is a material parameter. Thissimple behavior reasonably describes a broad class of ar-tificial gels [14], and several mechanisms can lead to sucha volume scaling, such as detachment of cross-links orfracture at fixed hoop stress [15, 16]. For this scenario,the free energy per volume penalty compared to the ref-erence system without a network is thus∆ g ( i ) ∼ αG. (2)This constant free energy penalty results in a shift ofthe phase boundary to lower temperatures. Remarkably,this behavior was characterized and validated for in vitro systems [9], with a value α ≈ .
5. In the presence ofmacroscopic gradients in the network stiffness, Eq. (2)also implies that droplet growth is favored in softer re-gions of the system, where the phase-separated liquid hasa lower free energy [8, 10, 17, 18].While this model captures the macroscopic elastic re-sponse of the material, it does not account for small-scale heterogeneities. In both biological and artificialsystems considered here, the elastic network is consti-tuted by polymers with a finite pore size ξ characteriz-ing the size of interstices between polymers. Considernow scenario (ii) in Fig. 1B, in which microdroplets with r = ξ form within these pores without deforming the net-work. In this case, E el = 0, while due to their small size,the droplets incur a substantial surface energy penalty E surf = 4 πξ γ , where γ denotes the surface tension be-tween the two liquid phases. Per volume of the minority species, this result in a free energy penalty∆ g ( ii ) ∼ γξ (3)for scenario (ii) , compared to our reference system inabsence of elastic network. Comparing Eqs. (3) and(2) reveals that in such a porous network, the trade-offbetween elastic and surface energy is controlled by the elasto-capillary number [19]: h ≡ γξG . (4)When h > α , i.e. , for an elastically homogeneous net-work and large liquid-liquid surface tension, scenario (i) is thermodynamically favored, leading to the formation ofmacroscopic cavitated droplets. In contrast, when h < α ,pore-size-limited microdroplets corresponding to scenario (ii) are thermodynamically more stable (Fig. 2).In the scenarios considered thus far, the network isfully excluded from the droplets. We now consider sce-nario (iii) from Fig. 1B: the partial inclusion of the net-work in macroscopic droplets of the minority phase. Toassess the stability of this scenario, we introduce a wet-ting energy E wet , emanating from the minority phase per-meating through the network, as E wet = 4 πr − ϕ ) σ p , (5)where ϕ denotes the fraction of network expelled fromthe droplet compared to the undeformed state, and σ p denotes the permeation stress . Microscopically, σ p arisesfrom differential wetting energy per unit length of fila-ments constituting the network in contact with the twofluids [20] (see SI Sec. E). Eq. (5) translates this mi-croscopic wetting phenomenon into a macroscopic effect,which results in a stress discontinuity at the liquid-liquid FIG. 2. Putative phase diagram from simple scaling argu-ments, indicating the most stable state for liquid-liquid phaseseparation in an elastic network, as a function of the elasto-capillary number h (Eq. (4)) and the permeo-elastic number p (Eq. (7)). Note that only the dominant contribution to thefree energy is retained here, corresponding to Eqs. (2), (3)and (6), respectively for the cavitated ( i ), micro-droplets ( ii )and permeated ( iii ) phases. FIG. 3. Analysis of droplet phases within compressible NH networks.
A-B.
Elastic energy (blue, e el ), surface energy (orange, e surf ) and total free energy (green, ∆ g = e el + e surf ) per volume for a droplet as a function of pore size r , respectively forelasto-capillary numbers h = 4 (showing monotonic decay of ∆ g ) and h = 1 (showing a global minimum of ∆ g at r ∗ , red star).Dashed black line indicates the λ → ∞ cavitated limit. C. Equilibrium pore size r ∗ as a function of the elasto-capillary number h , for different Poisson ratios ν of the network. Star indicates the limit of stability of phase (ii) . D-E.
Elastic, wetting (red, e wet ) and free energy (green, ∆ g = e el + e wet ) per volume of a large droplet permeating through the network, as a function ofthe expelled volume fraction ϕ of the network, respectively for permeo-elastic numbers p = 4 (where cavitation is favored) and p = 1 (with global minimum at ϕ ∗ , red star). F. Equilibrium expelled volume fraction ϕ ∗ of the network as a function of thepermeo-elastic number p . Dashed lines indicate metastable states, with cavitation ( ϕ ∗ = 1) energetically favored. G. Phasediagram indicating the most stable phase in the ( p, h ) plane. Dotted lines indicate naive scaling results with α = 5 /
2, as inFig. 2. The shaded area in F,G indicate p < i.e. , a contractile droplet attracting the network. In A,B,D,E we take ν = 0 . G and the pore size ξ . interface through which the network permeates. In addi-tion to a bulk energy term (Eq. (5)), network wetting caninduce an effective change of liquid-liquid surface energy,in particular if filaments align with the interface. We donot consider such an effect in this article.Again, in the spirit of a simple scaling analysis, we firstneglect the network deformation in response to this stressand set ϕ = 0. The free energy per volume correspondingto this permeated scenario is thus∆ g ( iii ) ∼ σ p . (6)Comparing this expression with Eq. (2), we find that themost stable phase is controlled by a second dimensionlessquantity, namely the permeo-elastic number p ≡ σ p G , (7)which is a measure of the degree of network deforma-tion at the interface induced by the permeation stress.For p > α , scenario (i) is the most stable: the repul-sion between the network and the minority liquid is suffi-ciently strong to fully expel the network from the droplet,leading to cavitation. For p < α , the droplet permeatesthrough the network rather than excluding it, and sce-nario (iii) is preferred. Finally, when the elasto-capillarynumber h < α , the phase boundary between scenarios (ii) and (iii) is given by the line p = h . The results of this scaling analysis are summarized ina phase diagram in the ( p, h ) plane in Fig. 2, which pre-dicts the most stable demixed phase. These phase bound-aries depend only on the liquid and network properties,not on the degree of supersaturation: to assess whetherdemixing takes place or not, the free energy penalty ofthe most stable phase (Eq. 2, 3 or 6) should be addedto the demixing free energy per volume in the absenceof network. We note that for scenarios (i-ii) , the net-work hinders phase separation and stabilizes the mixedphase; for scenario (iii) , this depends on the sign of p :for σ p <
0, the network prefers the minority phase andfavors phase separation.We have so far considered only the dominant contri-bution to the free energy for each scenario – either E el , E surf or E wet . Network deformation will however occurin each of the three scenarios: in (ii) , microdroplets exerta pressure on the network, while in (iii) , a permeationstress σ p > (i-ii) for which the networkis fully excluded from the droplet, we consider a droplet FIG. 4. Numerical analysis of strain-stiffening materials withthe nonlinear contribution to the elastic energy density de-scribed by a term ∝ (( λ − /ε c ) (see SI Sec. D), where λ is the stretch and the parameter ε c controls the strengthof nonlinearity. Low (blue) and high (orange) values of ε c describe strong and weak nonlinearity, respectively. The ana-lytical solution for non-stiffening NH materials is also shown(green). A. Elastic energy per droplet volume as a functionof droplet size. B. Equilibrium droplet size r ∗ as a function ofthe elasto-capillary number h . For NH materials the cavita-tion transition is shown as a dotted line. C. Phase boundarybetween microdroplets (ii) and permeated (iii) phases. Cav-itation (i) is suppressed by the strain-stiffening. Dashed lineindicates naive scaling p = h . of radius r in a spherical cavity of initial radius ξ that cor-responds to the characteristic pore size of the network.When the elasto-capillary number h is large (Fig. 3A),the free energy per volume of the droplet ∆ g = e el + e surf decreases monotonically with droplet size r , indicatingthat cavitation (scenario i ) is thermodynamically fa-vored. At small h (Fig. 3B), in contrast, the free energyexhibits a global minimum at r ∗ (cid:38) ξ , and size-limited mi-crodroplets with radius r ∗ as per scenario (ii) are favored.For positive Poisson’s ratios ν , the radius r ∗ increasessharply with the elasto-capillary number h (Fig. 3C), butremains finite up to the limit of stability of microdroplets,indicating that the cavitation transition (i → ii) is weaklyfirst-order as surface tension is increased or, equivalently,as the shear modulus of the network is reduced. Inter-estingly, this transition becomes continuous for auxeticmaterials with ν < λ inside the droplet, we considera macroscopic phase separated droplet (thus neglectingthe surface energy e surf ) for which the free energy pervolume is a function of the fraction ϕ = 1 − λ − of thenetwork expelled from the droplet. When the permeo-elastic number p is large, the free energy exceeds thatof the cavitated case for all ϕ (Fig. 3D). In contrast, atsmall values of p (Fig. 3E) the global free energy densityminimum occurs at a finite value ϕ ∗ , and permeation isfavored. When p increases, the equilibrium expelled net-work fraction ϕ ∗ increases continuously up to the cavi-tation point, at which it experiences a compressibility-dependent jump (Fig. 3F), implying that the transitionis discontinuous. We summarize these results in a phasediagram for NH materials (Fig. 3G).While an NH constitutive law describes the defor-mation behavior of a broad class of materials at finite stretches, many biomolecular networks differ by exhibit-ing nonlinear strain-stiffening behavior [21–23] wherebythe (nominal) tensile stress grows faster than linearlywith the stretch – either as a power-law with exponent >
1, or with a divergence at finite stretch. In the per-meated case, this nonlinearity limits the exclusion of thenetwork from the droplet, with moderate effects on thephase stability. In contrast, strain stiffening strongly af-fects phases (i-ii) where the network is fully excluded:the free energy of the cavity grows asymptotically fasterthan its volume, and the elastic penalty e el ( r ) diverges inthe limit of large droplets, as illustrated in Fig. 4A by nu-merical analysis of a minimal model for power-law strainstiffening materials (see SI Sec. D). As a consequence,effectively α → ∞ , and scenario (i) is suppressed: theglobal energy minimum always occurs at a finite dropletradius r ∗ , leading to size selection, as recently noted inthe context of the Gent model [24]. When the nonlin-earity is strong, the equilibrium droplet size is r ∗ (cid:38) ξ even at large capillary forces corresponding to h (cid:29) p (cid:38) h (Fig. 4C). When the nonlinearity is weak and emergesonly at large stretch, in contrast, microdroplets transitionfrom being linearly arrested with size r ∗ (cid:38) ξ at h (cid:46) r ∗ (cid:29) ξ at h (cid:38) h (Eq. (4)) andthe permeo-elastic number p (Eq. (7)), and constructeda phase diagram in the ( p, h ) plane (Figs. 2, 3G and 4C)that quantifies the equilibrium droplet size and networkdeformation behavior.Finally, we discuss the relevance of the predictedphases for experimental systems by providing the order-of-magnitude estimates for the relevant parameters, pre-sented in Table I. For fluorinated oil demixing in siliconegels (system I with h (cid:29) α ), consistently with experi-mental observations [9, 10, 12], only macroscopic phaseseparation appears to be relevant: these networks are toohomogeneous, and the surface tension too high, to per-mit microphase separation. We note that an independentstudy proposes that a combination of mesh size hetero-geneity, strongly heterogeneous nucleation at sparse loci, TABLE I. Order-of-magnitude estimates of the shear modulus G , network mesh size ξ , surface tension ξ and permeationstress σ p for three classes of experimental systems. We indicate the range of variation of the elasto-capillary number h and thepermeo-elastic number p , and conclude on the plausible scenarios for LLPS (most likely in bold). Details in SI Sec. F.System G ξ γ σ p h p ScenariosI Oil in silicone gel 10 − . Pa 2 − × − N m − − . Pa 20-700 1.1 - 6.5 (i) , (iii)
II Cytoplasmic cond. 10 −
100 Pa 50-150 nm 10 − N m − ± . − . ± − − . (ii), (iii) III Nuclear condensates 10 − Pa 7-20 nm 10 − − − N m − ± −
100 Pa 0 . ± . − (i), (ii), (iii) and network fracture under stress could lead to the coex-istence of microdroplets and cavitated droplets in thesesystems [16]. In contrast, for cytoplasmic condensates(system II), low surface tension, large mesh sizes andstiff filaments make permeation the most likely scenario,while cavitation appears to be ruled out by our theory:if droplets exclude the cytoskeleton, they are likely to besize-selected at the network mesh size. Finally, in thecontext of intracellular phase separation in the nucleo-plasm (III), all three scenarios are plausible. In particu-lar, we predict that mesh-size-selected microdroplets arepossible in chromatin for biologically relevant parame-ters. Interestingly, the chromatin mesh size is well be-low the optical resolution limit: if such microdropletsexist, they are likely not to have been fully characterizedyet. For instance, it was recently proposed that phase-separated condensates are involved in the activation andrepression of gene transcription [25–27]. Our work sug-gests that such condensates might be elastically limitedby the mechanisms presented herein.We note that our key theoretical predictions rely onseveral important assumptions. First, we have focused onthermodynamic equilibrium states, neglecting both thekinetic pathways leading to them such as droplet ripen-ing [10, 17, 18, 28] and merging [29] and, in the caseof biological systems, their inherently out-of-equilibriumnature. Second, we have ignored all elastic interactionsbetween the droplets, which is justifiable when the typi-cal droplet separations are much greater than their size.Third, we have neglected all visco-elastic effects in thenetwork: we thus considered systems over time scaleslong enough for phase separation to complete, yet shortenough for the network to retain its mechanical integrity.Exploring the effects of network-mediated droplet inter- actions and kinetic processes would provide additionalinsights into the behavior of elastically limited droplets,and is left for future work.Our study also suggests new ways to engineersize-controlled microdroplets through elastic limitation.These could be useful for nanofabrication, as well as toserve as crucibles for chemical reactions favored by phaseexchange: the very high surface-to-volume ratio wouldpermit fast exchange between the two phases. The multi-stage chemical reactions can be guided in structuredmulti-phase droplets, such as is the case with the ribo-some biogenesis in nucleoli [7], where the internal organi-zation of phases is dictated by their surface tensions [30].Finally, we note that while we have focused on the caseof droplets that (partially) expel the network, our theorypredicts that capillary forces are reversed when p < Acknowledgments.
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Appendix A: Mathematical framework: modeling liquid droplets in an elastic network
We first discuss the framework we employ to assess the stability of each of the three scenarios considered in the maintext: (i) cavitation, (ii) microdroplets, and (iii) permeation. Throughout this article, we consider a single sphericaldroplet of phase-separated liquid, in an infinite elastic medium representing the network. We thus neglect mechanicalinteractions between droplets, mediated by the network; this assumption is valid if the separation between dropletsis much larger than their size ( i.e. when the volume fraction of phase-separated droplets is small). The stability ofeach scenario is measured by the difference ∆ g of free energy per droplet volume, compared to an infinite droplet ofphase separated liquid in the absence of an elastic network. This penalty is captured in three distinct terms: elasticenergy stored in the network, liquid-liquid surface tension, and wetting energy. The latter two have closed forms asa function of the droplet size and inner stretch. The mathematically non-trivial aspect thus lies in the evaluation ofthe elastic energy resulting from the network deformation induced by the droplet.We characterize the elastic medium by its stored energy function W ( λ , λ , λ ) (which we leave unspecified for now),where the λ i ’s correspond to the three principal stretches. This function corresponds to the elastic energy density inthe undeformed material coordinates. We consider a droplet of size r d in a spherically symmetric infinite medium.We write the equilibrium deformation r = r ( R ), such that a point at distance R from the droplet center in the initialundeformed state is displaced to radius r ( R ) in the deformed state. In this geometry, the principal stretches are theradial stretch λ ( R ) = drdR ≡ s and the hoop stretch λ ( R ) = λ ( R ) = r/R ≡ t .We distinguish two geometries, depending on whether the network is excluded from the droplet (scenarios (i-ii) )or included (scenario (iii) ): • excluded network (Fig. 5A): the medium is modeled as an infinite material with an initial spherical pore of radius ξ in the reference configuration (left). A droplet of radius r d = r ( ξ ) stretches this pore by a factor λ d = r d /ξ compared to this reference configuration (right). The elastic energy stored in the network outside the droplet isthus: E el , out = 4 π (cid:90) ∞ ξ W (cid:18) drdR , rR , rR (cid:19) R dR (A1)Introducing u = R/ξ , the radial stretch s = dr/dR and the hoop stretch t = r/R , we obtain the following formfor the elastic energy per unit volume of the droplet: E el , out v d = 3 λ d (cid:90) ∞ W ( s ( u ) , t ( u ) , t ( u )) u du ≡ f out ( λ d ) (A2)where v d = πr d is the droplet volume. Eq. (A2) should be minimized over the deformation field t ( u ), withboundary condition t ( u = 1) = λ d . • included (permeated) network (Fig. 5B): the medium is modeled as an intact infinite material, and the poresare considered to be infinitesimally small. The droplet of radius r d is placed at the center, and imposes a stressdiscontinuity at its surface. The material inside the droplet is isotropically and homogeneously deformed withstretch λ i . The material outside the droplet is deformed in a similar way as previously, and hence the totalelastic energy reads E el = E el , in + E el , out = 43 π (cid:18) r d λ i (cid:19) W ( λ i , λ i , λ i ) + 4 π (cid:90) ∞ r d /λ i W (cid:18) drdR , rR , rR (cid:19) R dR (A3) FIG. 5. Geometries of droplets considered here. A. Network exclusion, starting from a pore of size ξ stretched by a factor λ d . B. Permeation of the droplet through the network, with network stretch λ i inside the droplet. which, divided by the droplet volume, is: E el v d = 1 λ i W ( λ i , λ i , λ i ) + f out ( λ i ) (A4)where f out was defined in Eq. (A2). In this geometry, the fraction of the network excluded from the droplet is ϕ = 1 − λ − i , so that the wetting energy reads E wet = v d σ p λ − i .We finally recapitulate our definition of the free energy for each of the three phases considered in this article. • Cavitation (i) : the only contribution to the free energy is the elastic penalty, in the infinite-stretch limit ofEq. (A2): ∆ g ( i ) = lim λ d →∞ f out ( λ d ) . (A5) • Microdroplets (ii) : we combine the elastic energy with network exclusion (Eq. (A2)) with the surface tension.The free energy is found by minimizing over the pore stretch ( i.e. over the droplet radius):∆ g ( ii ) = min λ d (cid:20) γλ d ξ + f out ( λ d ) (cid:21) (A6)where γ is the surface tension. Note that the minimization does not always yield a finite value for λ d . • Permeation (iii) : we combine the elastic energy with network inclusion (Eq. (A4)) with the wetting energy. Thefree energy is found by minimizing over the pore stretch ( i.e. over the excluded fraction of the network):∆ g ( iii ) = min λ i (cid:20) σ p + W ( λ i , λ i , λ i ) λ i + f out ( λ i ) (cid:21) (A7)where σ p is the permeation stress.The mathematically non-trivial part, in all three scenarios, is the evaluation of the outer elastic energy density f out ( λ ).We combine two approaches, depending on the class of materials considered, i.e. on the functional form of W . In thecase of neo-Hookean materials, we consider slightly compressible systems, which allows us to solve for the deformationfield analytically, as discussed in Sec. B (corresponding to the results presented in Fig. 3 of the main text). Forstrain-stiffening materials (Fig. 4 of the main text), such an analytical approach is not possible, and we resort to anumerical estimation of f out , as presented in Sec. D. In all cases, the free energy minimization over the value of λ inEqs. (A6) and (A7) is then performed numerically. Appendix B: Analytical treatment of slightly compressible neo-Hookean materials
Consider Eq. (A1), written in terms of arbitrary inner and outer radii R min and R max : E el , out = (cid:82) R max R min πR W ( λ , λ , λ ) dR . In mechanical equilibrium, E el , out is a minimum. Thus, the equilibrium deformation r = r ( R ) can be obtained from a variational principle as δE el , out δr ( R ) = 8 πR ∂W∂λ − π ddR (cid:18) R ∂W∂λ (cid:19) = 0 , (B1)or ddR (cid:0) R W (cid:1) − RW = 0 , (B2)where W i ≡ ∂W/∂λ i . [Note that here we assume that the system is compressible. In an incompressible system, thedeformation is explicitly determined from J = drdR (cid:0) rR (cid:1) = 1 ↔ drdR = (cid:0) rR (cid:1) − .] It is straightforward to show that dW /dR = W r (cid:48)(cid:48) ( R ) + 2 W ( r (cid:48) ( R ) /R − r/R ), where W j ≡ ∂ W/∂λ ∂λ j . Upon introducing the hoop and radialstretches as t = r ( R ) /R and s ( t ) = dr/dR , respectively, it can be shown that r (cid:48)(cid:48) ( R ) = ds/dR = ds/dt ( s − t ) /R and dW /dR = W [ ds/dt ( s − t ) /R ] + 2 W [ s − t ] /R . Thus, Eq. (B2) becomes W dsdt = − (cid:18) W − W s − t + W (cid:19) . (B3)Let us next focus on the following simple form for the stored energy function W , corresponding to a slightlycompressible neo-Hookean network [34]: W ( λ , λ , λ ) = G (cid:104) λ + λ + λ − − λ λ λ −
1) + β ( λ λ λ − (cid:105) , (B4)with G and ν = (1 − β − ) / W , Eq. (B3) becomes (cid:0) βt (cid:1) dsdt = − (cid:0) βst (cid:1) . (B5)The exact solution of Eq. (B5) is given by [34] s ( t ) = C − Ψ( t ) (cid:112) βt , (B6)where C denotes an integration constant, and d Ψ( t ) dt = 2 (cid:112) βt ↔ Ψ( t ) = (cid:90) tt dτ (cid:112) βτ . (B7)Now, consider the case where we have an initial pore of radius ξ embedded within an infinite elastic, neo-Hookean ma-trix, and the pore walls are subjected to a constant pressure p . Far from the cavity, the matrix remains deformation-free, and hence lim t → s ( t ) = 1. From the exact solution we immediately obtain s I ( t ) = √ β − (cid:82) t dτ √ βτ (cid:112) βt . (B8)Now, consider subjecting the boundary of the pore to a stretch λ such that r ( ξ ) = λξ . The corresponding radialstretch is given by s I ( λ ) ≡ ∆ I = √ β − (cid:82) λ dτ √ βτ (cid:112) βλ . (B9)Now, the pressure p required to sustain the deformation is given by p ( λ, β ) G = − W Gλ = 1 − ∆ I λ − β (cid:0) ∆ I λ − (cid:1) . (B10)We obtain the stored elastic energy as the total work of pressure forces from the undeformed state: E el , out ( λ ) = 4 πξ (cid:90) λ p ( λ (cid:48) , β ) λ (cid:48) dλ (cid:48) (B11)Using the formal calculus software SymPy [35] to expand the integral in Eq. (B9) in powers of β − ( i.e. a weaklycompressible expansion), we finally obtain the following expression for the elastic energy per droplet volume f out asa function of the droplet stretch λ = r/ξ : f out ( λ ) G = + 52 − λ − λ + 32 λ + β − (cid:20) −
340 + 65 λ − λ + 65 λ − λ (cid:21) + β − (cid:20) − λ + 980 λ + 980 λ − λ + 148 λ (cid:21) + β − (cid:20) − λ − λ − λ − λ + 14325 λ − λ (cid:21) + β − (cid:20) − λ + 453328 λ + 1640 λ + 1640 λ + 453328 λ − λ + 214352 λ (cid:21) + β − (cid:20) − λ − λ − λ − λ − λ − λ + 221989 λ − λ (cid:21) + O (cid:0) β − (cid:1) (B12)0which we use to assess the stability of each phase, as described in Sec. A. For ν > β > β − ). In particular, we can read out the λ → ∞ limit, corresponding to the cavitated free energy (Eq. (A5)): α ≡ ∆ g ( i ) G = 1 G lim λ →∞ f out ( λ ) = 52 − β + 148 β − β + 214352 β − β + 9951200 β + O (cid:0) β − (cid:1) (B13)Note that we also have α = p ∗ /G , where p ∗ is the cavitation pressure. As expected, in the limit β → ∞ , p ∗ /G → / ν = 1 / β = 3) or ν = 1 / β = 2), Eq. (B13) yields p ∗ /G ≈ .
48 and 2 .
47, respectively. Finite compressibility thusreduces the critical cavitation pressure, albeit to a rather small degree. We also note that for the special case β = 1(corresponding to ν = 0), p ∗ /G = 2 −√ / √ π Γ (5 / ≈ .
44, where Γ( x ) denotes the Euler gamma function, whilethe series approximation in Eq. (B13) yields p ∗ /G ≈ .
44, in excellent agreement with the exact result. Cavitationpressures for several representative compressibilities are listed in
Table 1 . β ∞ ν α = p ∗ ( β ) /G Appendix C: Limit of metastability of microdroplets in the neo-Hookean model
We investigate here the nature of the equilibrium transition between microdroplets (scenario ii ) and cavitation(scenario i ), which is controlled by the elasto-capillary number h . To this aim, it is useful to consider the free energyper volume of a droplet of size r = λξ as a function of its stretch λ , in the large λ limit that can be read out fromEq. (B12): 1 G ∆ g ( λ ) = α ( β ) + h − λ + A ( β ) λ + O ( λ − ) (C1)where again β = 1 / (1 − ν ) is the compressibility parameter, and A ( β ) is the coefficient of the inverse cubic term.Interestingly, the leading order in the expansion changes sign when h = 3 (independently of β ), and around thisvalue the free energy is thus dominated by higher-order terms. Depending on the compressibility, we identify twoqualitatively distinct behaviors when varying h , as shown in Fig. 6: • for ν > i.e. for usual materials, we have the following sequence of regimes: – h <
3: ∆ g ( λ ) exhibits a single minimum at a finite λ ∗ = r ∗ /ξ , corresponding to the microdroplets scenario.Near equilibrium, droplets larger than r ∗ would shrink (“anti-ripen”) so as to reach the equilibrium size. – < h < h c : microdroplets are the global free energy minimum, but there is a local maximum at λ > λ ∗ .As a result, λ = ∞ is a local minimum of free energy, and cavitated droplets are metastable. – h c < h < h † : the global minimum of free energy is at λ = ∞ , and cavitation is the stable scenario; however,a local minimum exists at r † , corresponding to metastable microdroplets. – h > h † : the free energy is monotonically decreasing as a function of λ , cavitation is stable and there existsno metastable state.The transition between scenarios ( i ) and ( ii ) governed by h is thus first-order. However, plotting in Fig. 7 thevalues of h c and h † over the physical range of Poisson’s ratio values ν , we note that the range of metastabilitycorresponding to this first-order transition is very narrow, and restricted to values 3 < h < .
11 for all ν . • for ν < i.e. for auxetic materials, we observe a second-order transition between scenarios (i) and (ii) (rightpanels in Fig. 6), with a continuous divergence of the droplet radius as r ∗ ∼ (3 − h ) − / as h → ν > weakly first order , characterized by the proximityto a critical point at ν = 0, sharp increase of the droplet size near the transition (as shown in Fig 3C of the maintext), and very limited range of metastability.1 FIG. 6. Plots of G ∆ g ( λ ) for different values of h and ν . The metastability regimes corresponding to a first-order transitionare apparent for ν > h = h c in orange and the end of metastability h = h † in red). For ν ≤ h = 3.FIG. 7. Left: equilibrium transition line h c (solid blue line) and metastability limit h † (dashed orange line) as a function ofPoisson’s ratio ν . The shaded area indicates the region in which microdroplets can be metastable. Note the very limited rangeof h values represented here. Right: maximum radius for stable microdroplets (solid blue line) and metastable droplets (dashedorange line). At ν < Appendix D: Minimal model for strain-stiffening effects
The stored energy function W for neo-Hookean materials [14] in Eq. (B4) does not capture strain-stiffening effectsoccuring in macromolecular systems at large stretches [21, 36–39]. To account for such effects, we consider thefollowing modified stored energy function: W = G (cid:34) ( I − − J −
1) + β ( J − + (cid:18) I − ε c (cid:19) (cid:35) , (D1)where I = λ + λ + λ and J = λ λ λ , and where ε c denotes a characteristic strain at which stiffening effectsbecome significant. It should be noted that the last term in Eq. (D1) emerges as the leading order term in a polynomialexpansion of the classic Arruda-Boyce [38, 39] and Gent [37] models for large-stretch behavior of polymer systems.The above choice for W is both convenient and physically-based: (1) At infinitesimally small strains, W ∼ G (cid:104) (cid:15) ij (cid:15) ij + ν − ν (cid:15) ii (cid:15) jj (cid:105) , in accordance with linear elasticity theory, where β − = (1 − ν ) with ν denoting the Poisson’sratio and (cid:15) the linear strain. (2) By taking ε c → ∞ , we recover the (slightly) compressible neo-Hookean model inEq. (B4). (3) Asymptotically, W ∼ [( I − / (6 ε c )] ∼ (cid:2) ( λ chain − / (2 ε c ) (cid:3) , indicating a strong stiffening effect when I → (3 + 6 ε c ). Therefore, this form of W can be viewed as a minimal model for slightly compressible, strain-stiffeninghyperelastic materials. Specifically, by tuning the parameter β , we can vary the compressibility with β → ∞ corre-2sponding to a perfectly incompressible material, while by tuning ε c , we can vary the material response from weaklystiffening (large ε c ) to strongly stiffening (small ε c ).We employ numerical simulations to study the influence of strain-stiffening of the network on liquid-liquid phaseseparation. Specifically, to evaluate f out ( λ ) (as given by Eq. (A2)), we discretize the displacement field over an unevengrid, u = [1 , . , . , . , ..., . , . , u max = 30] (with regular spacing of the values of √ u ). We evaluate theintegral (cid:82) u max W ( s ( u ) , t ( u ) , t ( u )) u du using finite differences of the displacement field, and use the SciPy optimizationpackage [40] ( scipy.optimize.minimize ) to perform the multivariate minimization of the energy of the displacementfield, under the constraint r ( u = 1) = λ . The outcome of this optimization is insensitive to the details of thediscretization, and recovers the analytical solution presented in Sec. B in the case of neo-Hookean materials. We thenpipe the resulting function f out ( λ ) into the free energy minimization described in Eqs. (A6)-(A7). Appendix E: Further discussion of permeation stress σ p . We now discuss the microscopic origin of the permeation stress σ p , and how it could be measured in practice.This term stems from the difference of wettability between the two liquids and the network. Microscopically, we canmodel the filaments composing the network as cylinders of radius r f , corresponding to a liquid-solid interface areaper unit length of 2 πr f . Taking the filaments immersed in the majority liquid (liquid 1) as the reference of energy,the interfacial energy per unit length of a filament immersed in liquid 2 is thus 2 πr f ( γ S − γ S ) where γ S and γ S respectively correspond to the interfacial energy between the network and liquids 1 and 2 (note that these interfacialenergies are defined at the microscopic level, not at the network level). Denoting by ρ the volume fraction of thenetwork in its rest state (we typically consider cases where ρ (cid:28) ρ/r f . The difference of energy per unit volume between the network immersed in liquid 1 and in liquid 2 is thus: σ p = 2 r f ρ ( γ S − γ S ) . (E1)In the presence of strain in the network, its volume fraction may change: denoting by ϕ = 1 − / ( λ λ λ ) the fractionof the network that is expelled compared to the undeformed state (where the λ i ’s are principal stretches), the energyper volume associated with immersing the stretched network into the minority liquid is thus (1 − ϕ ) σ p , correspondingto Eq. 5 of the main text.At the liquid-liquid interface, the difference in surface energy results in a capillary force F c ∼ πr f ( γ S − γ S ) oneach filament going through the interface. At the network scale, this implies a stress discontinuity in the network:the network is being “sucked in” by the best-wetting liquid. Note that while Eq. (E1) relies on microscopic modelingof the network and applicability of the surface energy at the level of individual filaments, this stress discontinuitysuggests that σ p could also be measured experimentally, in a way that is independent from microscopic models.Indeed, consider a tube separating two chambers containing respectively liquids 1 and 2, with a cork of clampednetwork in the tube. Then σ p corresponds to the pressure difference one must impose between the two chambers, sothat the liquid-liquid interface remains steady within the network cork. This provides an experimentally viable wayto measure σ p . The existence and microscopic origin of this term was previously noted by de Gennes in the contextof non-deformable porous media [20].Note that in addition to the difference of liquid-solid surface energy, it is possible that the rest state of the networkchanges when immersed in liquid 2 – either swelling or shrinking – and thus that permeation induces a spontaneousstrain in the network. This qualitatively distinct effect has, in practice, consequences similar to the effect discussedabove, and thus simply results in a modification of σ p for our purposes. Appendix F: Estimation of physical parameters.
Here we discuss how we obtain the experimental values of parameters presented in Table I of the main text, foreach of the three classes of systems considered. We focus on obtaining order-of-magnitude estimates for the twodimensionless parameters introduced in the main text: the elasto-capillary number h ≡ γ/ξG with γ the liquid-liquid surface tension, ξ the network pore size, and G the network shear modulus; and the permeo-elastic number p ≡ σ p /G with σ p the permeation stress. Note that values of σ p have not been reported in the literature, to the best ofour knowledge; for this reason, we employ Eq. (E1) with typical values for surface tensions to get order-of-magnitudeestimates of its range of variation. To this end, we substitute ρ ≈ r f /ξ , with r f denoting the radius of the filamentsconstituting the network. Below, we consider three distinct systems, one synthetic, and two biological ones. System I comprises the demixing of fluorinated oil embedded in a silicone gel, studied in great detail in Refs. [9,10, 12]. The elastic modulus G is in the range 1 . − ν = 0 . ξ through ξ ∼ ( k B T /G ) / with k B T = 4 × − J the thermal energy. Hence, ξ ∼ . − γ ≈ . − , as reported in Ref. [12]. We take a representative molecular radius r f ≈ . σ p ∼ − α ≈ . h for system I is h ∼ − (cid:29) α and p ∼ . − . (cid:39) α , where largervalues of h and p both correspond to softer gels. Our theory thus predicts that the relevant regime is predominantlycavitation (scenario i ), with permeation ( iii ) being marginally possible for very stiff gels. This is consistent with theexperimental observation of large, micron-sized droplets (while the mesh size is in the nanometer range) that fullyexclude the surrounding network, as characterized by coherent anti-Stokes Raman scattering [12]. System III generally encompasses liquid condensates found in the nucleus of eukaryotic cells and mechanicallyinteracting with the chromatin network, both native (such as nucleoli [7]) and biomimetic (such as CasDrop opto-genetically activated condensates [8]). Due to the broad class of systems considered and to the scarcity of availablequantitative data for physical parameters, we report only conservative ranges for our estimates. Following Ref. [8],we estimate the elastic modulus to be in the range G ∼ − ξ ∼ − γ ∼ − − − N m − .Indeed, such low values of surface tension have been reported for nucleolar proteins, γ ∼ × − N m − [7]. We takea radius r f = 1nm for DNA, and a volume fraction ρ ∼ . − . σ p ∼ ± − σ p depends on whether the nucleoplasm or the liquid condensate better wets the chromatin, which is notknown a priori ). No value of α has been reported to our knowledge, and so we take α ∼ .
5, corresponding to theneo-Hookean case, as a default. This results in a very broad range of possible values for dimensionless parameters, h ∼ − −
10 and p ∼ ± − −
10. In particular, all three scenarios appear to be plausible: cavitation ( i ) in softchromatin and for rather large values of the surface tension; nanodroplets confined at the mesh size ( ii ) if chromatin isstiffer and for low liquid-liquid surface tension; and finally permeation ( iii ) if the interfacial energy between chromatinand the condensate is low. Interestingly, only scenario (i) has been characterized yet: both nucleoli and engineeredcondensates form micron-sized droplets that have been shown to exclude the surrounding chromatin as they grow [8].However, it is possible that mesh-size-level droplets actually exist, but have not been characterized yet as they wouldbe significantly below optical resolution. System II , finally, encompasses cytoplasmic liquid condensates such as stress granules and P-bodies, which interactmechanically with cytoskeletal networks, in particular the actin cortex. The main changes compared to system IIare the properties of the elastic network. Reported values for the shear modulus of the cytoskeleton in intracellularconditions are similar in range to the nucleus, G ∼ − ξ ∼ − r f ∼ . γ ≈ µ N m − for cytoplasmic P-granules.The permeation stress is thus σ p ∼ . − h ∼ . − p ∼ ± − − .
2. Interestingly, this excludes cavitation ( i ): permeation ( iii ) is the predominant scenario, whilemicrodroplets ( iiii