Solid drops: large capillary deformations of immersed elastic rods

Experimental characterization of elastocapillary and osmocapillary effects on multi-scale gel surface topography

Surface topography significantly affects various surface properties of polymer gels. Unlike conventional materials where surface topography is largely a geometric property, the surface topography of a polymer gel is governed by the competition between capillary, elastic, and osmotic effects, which leads to complex stimuli-responsive effects. Elastocapillary deformation and osmocapillary phase separation are two phenomena that are known to flatten gel surface topography. Here we experimentally quantify how osmocapillary phase separation affects gel surface topography by fabricating ionogels with multi-scale topography and characterizing the swelling-dependent surface flattening. Our observation confirms the vital role of the osmocapillary length in governing the surface behavior of swollen ionogels. This study provides the first quantitative experimental verification of the osmocapillary phase separation and shows the insufficiency of the previous studies based on elastocapillary deformation alone.

Surface stress and shape relaxation of gelling droplets

Solidification is a heterogeneous transformation from liquid to solid, which usually combines transport, phase transition and mechanical strain. Predicting the shapes resulting from such a complex process is fascinating and has a wide range of implications from morphogenesis in biological tissues to industrial processes. For soft solids initially at equilibrium, elastic stresses, whether tensile or compressive, can be induced by heterogeneous volumetric deformations of the material. These stresses trigger surface instabilities leading to variations of curvature and shape of the solids. In this article, we study the shape evolution of elongated droplets of polymer and particle suspensions undergoing a solidification process caused by the inward diffusion of a gelling agent from the surface. We show experimentally and numerically that there appears a layer of gelled material growing at the surface. Due to volume contraction, this layer induces tensile stresses and drives a flow in the ungelled liquid core, resulting in the relaxation of the droplets toward spherical shapes. Over time, the thickness of this elastic membrane grows, hence the bending stiffness required to change its shape eventually balances the surface stresses, which arrests the relaxation process. These results provide general rules to understand the shape of solidifying materials combining both tension and bending driven deformations.

Elastocapillary menisci mediate interaction of neighboring structures at the surface of a compliant solid

Surface stress drives long-range elastocapillary interactions at the surface of compliant solids, where it has been observed to mediate interparticle interactions and to alter transport of liquid drops. We show that such an elastocapillary interaction arises between neighboring structures that are simply protrusions of the compliant solid. For compliant micropillars arranged in a square lattice with spacing p less than an interaction distance p^{*}, the distance of a pillar to its neighbors determines how much it deforms due to surface stress: Pillars that are close together tend to be rounder and flatter than those that are far apart. The interaction is mediated by the formation of an elastocapillary meniscus at the base of each pillar, which sets the interaction distance and causes neighboring structures to deform more than those that are relatively isolated. Neighboring pillars also displace toward each other to form clusters, leading to the emergence of pattern formation and ordered domains.

Active elastocapillarity in soft solids with negative surface tension

Active solids consume energy to allow for actuation, shape change, and wave propagation not possible in equilibrium. Whereas active interfaces have been realized across many experimental systems, control of three-dimensional (3D) bulk materials remains a challenge. Here, we develop continuum theory and microscopic simulations that describe a 3D soft solid whose boundary experiences active surface stresses. The competition between active boundary and elastic bulk yields a broad range of previously unexplored phenomena, which are demonstrations of so-called active elastocapillarity. In contrast to thin shells and vesicles, we discover that bulk 3D elasticity controls snap-through transitions between different anisotropic shapes. These transitions meet at a critical point, allowing a universal classification via Landau theory. In addition, the active surface modifies elastic wave propagation to allow zero, or even negative, group velocities. These phenomena offer robust principles for programming shape change and functionality into active solids, from robotic metamaterials down to shape-shifting nanoparticles.

A model for 3D deformation and reconstruction of contractile microtissues

Tissue morphogenesis and regeneration are essentially mechanical processes that involve coordination of cellular forces, production and structural remodeling of extracellular matrix (ECM), and cell migration. Discovering the principles of cell-ECM interactions and tissue-scale deformation in mechanically-loaded tissues is instrumental to the development of novel regenerative therapies. The combination of high-throughput three-dimensional (3D) culture systems and experimentally-validated computational models accelerate the study of these principles. In our previous work [E. Mailand, et al., Biophys. J., 2019, 117, 975-986], we showed that prominent surface stresses emerge in constrained fibroblast-populated collagen gels, driving the morphogenesis of fibrous microtissues. Here, we introduce an active material model that allows the embodiment of surface and bulk contractile stresses while maintaining the passive elasticity of the ECM in a 3D setting. Unlike existing models, the stresses are driven by mechanosensing and not by an externally applied signal. The mechanosensing component is incorporated in the model through a direct coupling of the local deformation state with the associated contractile force generation. Further, we propose a finite element implementation to account for large deformations, nonlinear active material response, and surface effects. Simulation results quantitatively capture complex shape changes during tissue formation and as a response to surgical disruption of tissue boundaries, allowing precise calibration of the parameters of the 3D model. The results of this study imply that the organization of the extracellular matrix in the bulk of the tissue may not be a major factor behind the morphogenesis of fibrous tissues at sub-millimeter length scales.

Surface Tension and the Strain-Dependent Topography of Soft Solids

When stretched in one direction, most solids shrink in the transverse directions. In soft silicone gels, however, we observe that small-scale topographical features grow upon stretching. A quantitative analysis of the topography shows that this counterintuitive response is nearly linear, allowing us to tackle it through a small-strain analysis. We find that the surprising increase of small-scale topography with stretch is due to a delicate interplay of the bulk and surface responses to strain. Specifically, we find that surface tension changes as the material is deformed. This response is expected on general grounds for solid materials, but challenges the standard description of gel and elastomer surfaces.

A surface flattening method for characterizing the surface stress, drained Poisson's ratio and diffusivity of poroelastic gels

When a poroelastic gel is released from a patterned mold, surface stress drives deformation and solvent migration in the gel and flattens its surface profile in a time-dependent manner. Specifically, the gel behaves like an incompressible solid immediately after removal from the mold, and becomes compressible as the solvent is able to squeeze out of the polymer network. In this work, we use the finite element method (FEM) to simulate this transient surface flattening process. We assume that the surface stress is isotropic and constant, the polymer network is linearly elastic and isotropic, and that solvent flow obeys Darcy's law. The short-time and long-time surface profiles can be used to determine the surface stress and drained Poisson's ratio of the gel. Our analysis shows that the drained Poisson's ratio and the diffusivity of the gel can be obtained using interferometry and high-speed video microscopy, without mechanical measurement.

Capillary forces drive buckling, plastic deformation, and break-up of 3D printed beams

Capillary forces acting at the interfaces of soft materials lead to deformations over the scale of the elastocapillary length. When surface stresses exceed a material's yield stress, a plastocapillary effect is expected to arise, resulting in yielding and plastic deformation. Here, we explore the interfacial instabilities of 3D-printed fluid and elastic beams embedded within viscoelastic fluids and elastic solid support materials. Interfacial instabilities are driven by the immiscibility between the paired phases or their solvents. We find that the stability of an embedded structure is predicted from the balance between the yield stress of the elastic solid, τ_y, the apparent interfacial tension between the materials, γ', and the radius of the beam, r, such that τ_y > γ'/r. When the capillary forces are sufficiently large, we observe yielding and failure of the 3D printed beams. Furthermore, we observe new coiling and buckling instabilities emerging when elastic beams are embedded within viscous fluid support materials. The coiling behavior appear analogous to elastic rope coiling whereas the buckling instability follows the scaling behavior predicted from Euler-Bernoulli beam theory.

Elastic amplification of the Rayleigh-Taylor instability in solidifying melts

The concomitant mechanical deformation and solidification of melts are relevant to a broad range of phenomena. Examples include the preparation of cotton candy, the atomization of metals, the manufacture of glass fibers, and the formation of elongated structures in volcanic eruptions known as Pele's hair. Usually, solid-like deformations during solidification are neglected as the melt is much more malleable in its initial liquid-like form. Here we demonstrate how elastic deformations in the midst of solidification, i.e., while the melt responds as a very soft solid ([Formula: see text] Pa), can lead to the formation of previously unknown periodic structures. Namely, we generate an array of droplets on a thin layer of liquid elastomer melt coated on the outside of a rotating cylinder through the Rayleigh-Taylor instability. Then, as the melt cures and goes through its gelation point, the rotation speed is increased and the drops stretch into hairs. The ongoing solidification eventually hardens the material, permanently "freezing" these elastic deformations into a patterned solid. Using experiments, simulation, and theory, we demonstrate that the formation of our two-step patterns can be rationalized when combining the tools from fluid mechanics, elasticity, and statistics. Our study therefore provides a framework to analyze multistep pattern formation processes and harness them to assemble complex materials.

How surface stress transforms surface profiles and adhesion of rough elastic bodies

The surface of soft solids carries a surface stress that tends to flatten surface profiles. For example, surface features on a soft solid, fabricated by moulding against a stiff-patterned substrate, tend to flatten upon removal from the mould. In this work, we derive a transfer function in an explicit form that, given any initial surface profile, shows how to compute the shape of the corresponding flattened profile. We provide analytical results for several applications including flattening of one-dimensional and two-dimensional periodic structures, qualitative changes to the surface roughness spectrum, and how that strongly influences adhesion.

Faraday waves in soft elastic solids

Recent experiments have observed the emergence of standing waves at the free surface of elastic bodies attached to a rigid oscillating substrate and subjected to critical values of forcing frequency and amplitude. This phenomenon, known as Faraday instability, is now well understood for viscous fluids but surprisingly eluded any theoretical explanation for soft solids. Here, we characterize Faraday waves in soft incompressible slabs using the Floquet theory to study the onset of harmonic and subharmonic resonance eigenmodes. We consider a ground state corresponding to a finite homogeneous deformation of the elastic slab. We transform the incremental boundary value problem into an algebraic eigenvalue problem characterized by the three dimensionless parameters, that characterize the interplay of gravity, capillary and elastic waves. Remarkably, we found that Faraday instability in soft solids is characterized by a harmonic resonance in the physical range of the material parameters. This seminal result is in contrast to the subharmonic resonance that is known to characterize viscous fluids, and opens the path for using Faraday waves for a precise and robust experimental method that is able to distinguish solid-like from fluid-like responses of soft matter at different scales.

Spinning elastic beads: a route for simultaneous measurements of the shear modulus and the interfacial energy of soft materials

Large deformations of soft elastic beads spinning at high angular velocity in a denser background fluid are investigated theoretically, numerically, and experimentally using millimeter-size polyacrylamide hydrogel particles introduced in a spinning drop tensiometer. We determine the equilibrium shapes of the beads from the competition between the centrifugal force and the restoring elastic and surface forces. Considering the beads as neo-Hookean up to large deformations, we show that their elastic modulus and interfacial energy constant can be simultaneously deduced from their equilibrium shape. Also, our results provide further support to the scenario in which interfacial energy and interfacial tension coincide for amorphous polymer gels.

Modeling of surface mechanical behaviors of soft elastic solids: theory and examples

Surfaces of soft solids can have significant surface stress, extensional modulus and bending stiffness. Previous theoretical studies have usually examined cases in which both the surface stress and bending stiffness are constant, assuming small deformation. In this work we consider a general formulation in which the surface can support large deformation and carry both surface stresses and surface bending moments. We demonstrate that the large deformation theory can be reduced to the classical linear theory (Shuttleworth equation). We obtain exact solutions for problems of an inflated cylindrical shell and bending of a plate with a finite thickness. Our analysis illustrates the different manners in which surface stiffening and surface bending stabilize these structures. We discuss how the complex surface constitutive behaviors affect the stress field of the bulk. Our calculation provides insights into effects of strain-dependent surface stress and surface bending in the large deformation regime, and can be used as a model to implement surface finite elements to study large deformation of complex structures.

A method for determining surface tension, viscosity, and elasticity of gels via ultrasonic levitation of gel drops

A method for obtaining the elasticity, surface tension, and viscosity of ultrasonically levitated gel drops is presented. The drops examined were made of agarose, a hydrogel. In contrast to previous studies where fluid properties are obtained using ultrasonic levitation of a liquid drop, herein the material studied was a gel which has a significant elasticity. The work presented herein is significant in that gels are of growing importance in biomedical applications and exhibit behaviors partially determined by their elasticities and surface tensions. Obtaining surface tension for these substances is important but challenging since measuring this quantity using the standard Wilhelmy plate or DuNuoy ring methods is not possible due to breakage of the gel. The experiments were conducted on agarose gels having elasticities ranging from 12.2 to 200.3 Pa. A method is described for obtaining elasticity, surface tension, and viscosity, and the method is experimentally demonstrated for surface tension and viscosity. For the range of elasticities explored, the measured surface tension ranged from 0.1 to 0.3 N/m, and the viscosity ranged from 0.0084 to 0.0204 Pa s. The measurements of surface tension are, to the authors' knowledge, the first obtained of a gel using ultrasonic levitation.

Coupled flow and deformation fields due to a line load on a poroelastic half space: effect of surface stress and surface bending

In the past decade, many experiments have indicated that the surfaces of soft elastic solids can resist deformation by surface stresses. A common soft elastic solid is a hydrogel which consists of a polymer network swollen in water. Although experiments suggest that solvent flow in gels can be affected by surface stress, there is no theoretical analysis on this subject. Here we study the solvent flow near a line load acting on a linear poroelastic half space. The surface of this half space resists deformation by a constant, isotropic surface stress. It can also resist deformation by surface bending. The time-dependent displacement, stress and flow fields are determined using transform methods. Our solution indicates that the stress field underneath the line load is completely regularized by surface bending-it is bounded and continuous. For small surface bending stiffness, the line force is balanced by surface stresses; these forces form what is commonly known as 'Neumann's triangle'. We show that surface stress reduces local pore pressure and inhibits solvent flow. We use our line load solution to simulate the relaxation of the peak which is formed by applying and then removing a line force on the poroelastic half space.

The elastic Rayleigh drop

Bioprinting technologies rely on the formation of soft gel drops for printing tissue scaffolds and the dynamics of these drops can affect the process. A model is developed to describe the oscillations of a spherical gel drop with finite shear modulus, whose interface is held by surface tension. The governing elastodynamic equations are derived and a solution is constructed using displacement potentials decomposed into a spherical harmonic basis. The resulting nonlinear characteristic equation depends upon two dimensionless numbers, elastocapillary and compressibility, and admits two types of solutions, (i) spheroidal (or shape change) modes and (ii) torsional (rotational) modes. The torsional modes are unaffected by capillarity, whereas the frequency of shape oscillations depend upon both the elastocapillary and compressibility numbers. Two asymptotic dispersion relationships are derived and the limiting cases of the inviscid Rayleigh drop and elastic globe are recovered. For a fixed polar wavenumber, there exists an infinity of radial modes that each transition from an elasticity wave to a capillary wave upon increasing the elastocapillary number. At the transition, there is a qualitative change in the deformation field and a set of recirculation vortices develop at the free surface. Two special modes that concern volume oscillations and translational motion are characterized. A new instability is documented that reflects the balance between surface tension and compressibility effects due to the elasticity of the drop.

Elastocapillary Transition in Gel Drop Oscillations

We report experimental observations of surface oscillations in an ultrasoft agarose gel drop. Ultrasonic levitation is used to excite shape oscillations in the gel drop and we report the natural frequency of the drop as it depends upon a nondimensional elastocapillary number, which we define as the ratio of the elastocapillary length to drop size. Our experiments span a wide range of experimental parameters and we recover the appropriate scaling laws in the elastic and capillary wave limits. The crossover between these two limits is observed and agrees well with a proposed frequency relationship.

The shape of hanging elastic cylinders

Deformations of heavy elastic cylinders with their axis in the direction of earth's gravity field are investigated. The specimens, made of polyacrylamide hydrogels, are attached from their top circular cross section to a rigid plate. An equilibrium configuration results from the interplay between gravity that tends to deform the cylinders downwards under their own weight, and elasticity that resists these distortions. The corresponding steady state exhibits fascinating shapes which are measured with lab-based micro-tomography. For any given initial radius to height ratio, the deformed cylinders are no longer axially symmetric beyond a critical value of a control parameter that depends on the volume force, the height and the elastic modulus: self-similar wrinkling hierarchies develop, and dimples appear at the bottom surface of the shallowest samples. We show that these patterns are the consequences of elastic instabilities.

A surface with stress, extensional elasticity, and bending stiffness

We demonstrate that the surface of a commonly used polydimethylsiloxane formulation (PDMS, Sylgard 184) treated by ultraviolet ozonolysis (UVO) has significant surface stress, considerable extensional elasticity (the "Shuttleworth Effect"), and surface bending elasticity. For soft solids, phenomena such as wetting, contact, surface flattening, and stiffening by liquid inclusions are often governed by their surface, which is usually represented by a liquid-like constant surface stress. Whether the surfaces of soft solids can have more complex constitutive response is actively debated. We studied the deformation of three surface-patterned materials systems: untreated polydimethylsiloxane (PDMS), an organogel, and patterned PDMS with surface treatment by UVO. The last of these three, we found, has complex surface elasticity. This is analogous to the situation for liquids in which the presence of a second phase at the interface yields Gibbs elasticity. Our finding is of broad applicability because in soft solids the behavior of the surface can often dominate bulk deformation.

Effects of strain-dependent surface stress on the adhesive contact of a rigid sphere to a compliant substrate

Recent experiments have reported that the surface stress of soft elastic solids can increase rapidly with surface strain. For example, when a small hard sphere in adhesive contact with a soft silicone gel is slowly retracted from its rest position, it was found that the retraction force versus displacement relation cannot be explained either by the Johnson-Kendall-Roberts (JKR) theory or a recent indentation theory based on an isotropic surface stress that is independent of surface strain. In this paper, we address this problem using a finite element method to simulate the retraction process. Our numerical model does not have the restrictions of the aforementioned theories; that is, it can handle large nonlinear elastic deformation as well as a surface-strain-dependent surface stress. Our simulation is in good agreement with experimental force versus displacement data with no fitting parameters. Therefore, our results lend further support to the claim that significant strain-dependent surface stresses can occur in simple soft elastic gels. However, significant challenges remain in the reconciliation of theory and experiments, particularly regarding the geometry of the contact and substrate deformation.

Are elastic moduli of biological cells depth dependent or not? Another explanation using a contact mechanics model with surface tension

Atomic force microscopy (AFM) has become the most commonly used tool to measure the mechanical properties of biological cells. In AFM indentation experiments, the Hertz and Sneddon models of contact mechanics are usually adopted to extract the elastic modulus by analyzing the load-indent depth curves for spherical and conical tips, respectively. However, the effects of surface tension, neglected in existing contact models, become more significant in indentation responses due to the lower elastic moduli of living cells. Here, we present two simple yet robust relations between load and indent depth considering surface tension effects for spherical and conical indentations, through dimensional analysis and finite element simulations. When the indent depth is smaller than the intrinsic length defined as the ratio of surface tension to elastic modulus, the elastic modulus obtained by classical contact mechanics theories would be overestimated. Contrary to the majority of reported results, we find that the elastic modulus of a cell could be independent of indent depths if surface tension is taken into account. Our model seems to be in agreement with experimental data available. A comprehensive comparison will be done in the future.

Extracting the surface tension of soft gels from elastocapillary wave behavior

Mechanically-excited waves appear as surface patterns on soft agarose gels. We experimentally quantify the dispersion relationship for these waves over a range of shear modulus in the transition zone where the surface energy (capillarity) is comparable to the elastic energy of the solid. Rayleigh waves and capillary-gravity waves are recovered as limiting cases. Gravitational forces appear as a pre-stress through the self-weight of the gel and are important. We show the experimental data fits well to a proposed dispersion relationship which differs from that typically used in studies of capillary to elastic wave crossover. We use this combined theoretical and experimental analysis to develop a new technique for measuring the surface tension of soft materials, which has been historically difficult to measure directly.

The mechanical equilibrium of soft solids with surface elasticity

Recent experiments have shown that surface stresses in soft materials can have a significant strain-dependence. Here we explore the implications of this surface elasticity to show how, and when, we expect it to arise. We develop the appropriate boundary condition, showing that it simplifies significantly in certain cases. We show that surface elasticity's main role is to stiffen a solid surface's response to in-plane tractions, in particular at length-scales smaller than a characteristic elastocapillary length. We also investigate how surface elasticity affects the Green's-function problem of a line force on a flat, incompressible, linear-elastic substrate. There are significant changes to this solution, especially in that the well-known displacement singularity is regularised. This raises interesting implications for soft phenomena like wetting contact lines, adhesion and friction. Finally, we discuss open questions, future directions, and close ties with existing fields of research.

Impact of Beads and Drops on a Repellent Solid Surface: A Unified Description

We investigate freely expanding sheets formed by ultrasoft gel beads, and liquid and viscoelastic drops, produced by the impact of the bead or drop on a silicon wafer covered with a thin layer of liquid nitrogen that suppresses viscous dissipation thanks to an inverse Leidenfrost effect. Our experiments show a unified behavior for the impact dynamics that holds for solids, liquids, and viscoelastic fluids and that we rationalize by properly taking into account elastocapillary effects. In this framework, the classical impact dynamics of solids and liquids, as far as viscous dissipation is negligible, appears as the asymptotic limits of a universal theoretical description. A novel material-dependent characteristic velocity that includes both capillary and bulk elasticity emerges from this unified description of the physics of impact.

Plateau-Rayleigh instability in solids is a simple phase separation

A long elastic cylinder, with radius a and shear-modulus μ, becomes unstable given sufficient surface tension γ. We show this instability can be simply understood by considering the energy, E(λ), of such a cylinder subject to a homogenous longitudinal stretch λ. Although E(λ) has a unique minimum, if surface tension is sufficient [Γ≡γ/(aμ)>sqrt[32]] it loses convexity in a finite region. We use a Maxwell construction to show that, if stretched into this region, the cylinder will phase-separate into two segments with different stretches λ_{1} and λ_{2}. Our model thus explains why the instability has infinite wavelength and allows us to calculate the instability's subcritical hysteresis loop (as a function of imposed stretch), showing that instability proceeds with constant amplitude and at constant (positive) tension as the cylinder is stretched between λ_{1} and λ_{2}. We use full nonlinear finite-element calculations to verify these predictions and to characterize the interface between the two phases. Near Γ=sqrt[32] the length of such an interface diverges, introducing a new length scale and allowing us to construct a one-dimensional effective theory. This treatment yields an analytic expression for the interface itself, revealing that its characteristic length grows as l_{wall}∼a/sqrt[Γ-sqrt[32]].

Surface tension and a self-consistent theory of soft composite solids with elastic inclusions

The importance of surface tension effects is being recognized in the context of soft composite solids, where they are found to significantly affect the mechanical properties, such as the elastic response to an external stress. It has recently been discovered that Eshelby's inclusion theory breaks down when the inclusion size approaches the elastocapillary length L≡γ/E, where γ is the inclusion/host surface tension and E is the host Young's modulus. Extending our recent results for liquid inclusions, here we model the elastic behavior of a non-dilute distribution of isotropic elastic spherical inclusions in a soft isotropic elastic matrix, subject to a prescribed infinitesimal far-field loading. Within our framework, the composite stiffness is uniquely determined by the elastocapillary length L, the spherical inclusion radius R, and the stiffness contrast parameter C, which is the ratio of the inclusion to the matrix stiffness. We compare the results with those from the case of liquid inclusions, and we derive an analytical expression for elastic cloaking of the composite by the inclusions. Remarkably, we find that the composite stiffness is influenced significantly by surface tension even for inclusions two orders of magnitude more stiff than the host matrix. Finally, we show how to simultaneously determine the surface tension and the inclusion stiffness using two independent constraints provided by global and local measurements.

Interaction of Droplets Separated by an Elastic Film

The Laplace pressure of a droplet placed on one side of an elastic thin film can cause significant deformation in the form of a bulge on its opposite side. Here, we show that this deformation can be detected by other droplets suspended on the opposite side of the film, leading to interaction between droplets separated by the solid (but deformable) film. The interaction is repulsive when the drops have a large overlap and attractive when they have a small overlap. Thus, if two identical droplets are placed right on top of each other (one on either side of the thin film), they tend to repel each other, eventually reaching an equilibrium configuration where there is a small overlap. This observation can be explained by analyzing the energy landscape of the droplets interacting via an elastically deformed film. We further demonstrate this idea by designing a pattern comprising a big central drop with satellite droplets. This phenomenon can lead to techniques for directed motion of droplets confined to one side of a thin elastic membrane by manipulations on the other side.

A closed form large deformation solution of plate bending with surface effects

We study the effect of surface stress on the pure bending of a finite thickness plate under large deformation. The surface is assumed to be isotropic and its stress consists of a part that can be interpreted as a residual stress and a part that stiffens as the surface increases its area. Our results show that residual surface stress and surface stiffness can both increase the overall bending stiffness but through different mechanisms. For sufficiently large residual surface tension, we discover a new type of instability - the bending moment reaches a maximum at a critical curvature. Effects of surface stress on different stress components in the bulk of the plate are discussed and the possibility of self-bending due to asymmetry of the surface properties is also explored. The results of our calculations provide insights into surface stress effects in the large deformation regime and can be used as a test for implementation of finite element methods for surface elasticity.

Off-equilibrium surface tension in miscible fluids

The interfacial tension between immiscible fluids is responsible for a wealth of every-day phenomena, from the spherical shape of small drops and bubbles to the ability to walk on water of many insects. More than a century ago, physicist and mathematician D. Korteweg postulated the existence of an effective interface tension for miscible fluids, whenever a composition gradient exists, as encountered, e.g., in many flow geometries. In this mini-review, we discuss experimental work performed in the last decades that demonstrates the existence of a positive effective interface tension in a variety of systems, from molecular, near-critical liquids to complex fluids such as polymer solutions and colloidal suspensions. The various experimental strategies that have been deployed are discussed, together with their advantages and limitations. Finally, some of the key theoretical questions still open are outlined.

Finite-wavelength surface-tension-driven instabilities in soft solids, including instability in a cylindrical channel through an elastic solid

We deploy linear stability analysis to find the threshold wavelength (λ) and surface tension (γ) of Rayleigh-Plateau type "peristaltic" instabilities in incompressible neo-Hookean solids in a range of cylindrical geometries with radius R_{0}. First we consider a solid cylinder, and recover the well-known, infinite-wavelength instability for γ≥6μR_{0}, where μ is the solid's shear modulus. Second, we consider a volume-conserving (e.g., fluid filled and sealed) cylindrical cavity through an infinite solid, and demonstrate infinite-wavelength instability for γ≥2μR_{0}. Third, we consider a solid cylinder embedded in a different infinite solid, and find a finite-wavelength instability with λ∝R_{0}, at surface tension γ∝μR_{0}, where the constants depend on the two solids' modulus ratio. Finally, we consider an empty cylindrical channel (or filled with expellable fluid) through an infinite solid, and find an instability with finite wavelength, λ≈2R_{0}, for γ≥2.543...μR_{0}. Using finite-strain numerics, we show such a channel jumps at instability to a highly peristaltic state, likely precipitating it's blockage or failure. We argue that finite wavelengths are generic for elastocapillary instabilities, with the simple cylinder's infinite wavelength being the exception rather than the rule.

Surface tension and the Mori-Tanaka theory of non-dilute soft composite solids

Eshelby's theory is the foundation of composite mechanics, allowing calculation of the effective elastic moduli of composites from a knowledge of their microstructure. However, it ignores interfacial stress and only applies to very dilute composites-i.e. where any inclusions are widely spaced apart. Here, within the framework of the Mori-Tanaka multiphase approximation scheme, we extend Eshelby's theory to treat a composite with interfacial stress in the non-dilute limit. In particular, we calculate the elastic moduli of composites comprised of a compliant, elastic solid hosting a non-dilute distribution of identical liquid droplets. The composite stiffness depends strongly on the ratio of the droplet size, R, to an elastocapillary lengthscale, L. Interfacial tension substantially impacts the effective elastic moduli of the composite when [Formula: see text]. When R<3L/2 (R=3L/2) liquid inclusions stiffen (cloak the far-field signature of) the solid.

Interfacial tension and a three-phase generalized self-consistent theory of non-dilute soft composite solids

In the dilute limit Eshelby's inclusion theory captures the behavior of a wide range of systems and properties. However, because Eshelby's approach neglects interfacial stress, it breaks down in soft materials as the inclusion size approaches the elastocapillarity length L≡γ/E. Here, we use a three-phase generalized self-consistent method to calculate the elastic moduli of composites comprised of an isotropic, linear-elastic compliant solid hosting a spatially random monodisperse distribution of spherical liquid droplets. As opposed to similar approaches, we explicitly capture the liquid-solid interfacial stress when it is treated as an isotropic, strain-independent surface tension. Within this framework, the composite stiffness depends solely on the ratio of the elastocapillarity length L to the inclusion radius R. Independent of inclusion volume fraction, we find that the composite is stiffened by the inclusions whenever R < 3L/2. Over the same range of parameters, we compare our results with alternative approaches (dilute and Mori-Tanaka theories that include surface tension). Our framework can be easily extended to calculate the composite properties of more general soft materials where surface tension plays a role.

Adsorption of soft particles at fluid interfaces

Soft particles can be better emulsifiers than hard particles because they stretch at fluid interfaces. This deformation can increase adsorption energies by orders of magnitude relative to rigid particles. The deformation of a particle at an interface is governed by a competition of bulk elasticity and surface tension. When particles are partially wet by the two liquids, deformation is localized within a material-dependent distance L from the contact line. At the contact line, the particle morphology is given by a balance of surface tensions. When the particle radius R≪L, the particle adopts a lenticular shape identical to that of an adsorbed fluid droplet. Particle deformations can be elastic or plastic, depending on the relative values of the Young modulus, E, and yield stress, σp. When surface tensions favour complete spreading of the particles at the interface, plastic deformation can lead to unusual fried-egg morphologies. When deformable particles have surface properties that are very similar to one liquid phase, adsorption can be extremely sensitive to small changes of their affinity for the other liquid phase. These findings have implications for the adsorption of microgel particles at fluid interfaces and the performance of stimuli-responsive Pickering emulsions.

Softening of edges of solids by surface tension

Surface tension tends to minimize the area of interfaces between pieces of matter in different thermodynamic phases, be they in the solid or the liquid state. This can be relevant for the macroscopic shape of very soft solids and lead to a roughening of initially sharp edges. We calculate this effect for a Neo-Hookean elastic solid, with assumptions corresponding to actual experiments, namely the case where an initially sharp edge is rounded by the effect of surface tension felt when the fluid surrounding the soft solid (and so surface tension) is changed at the solid/liquid boundary. We consider two opposite limits where the analysis can be carried to the end, the one of a shallow angle and the one of a very sharp angle. Both cases yield a discontinuity of curvature in the state with surface tension although the initial state had a discontinuous slope.

Interface deformations due to counter-rotating vortices: Viscous versus elastic media

Capillary forces determine the shape of a liquid interface. Although often not considered, elastic solids with a free surface are also subjected to surface forces and these become important for materials of low Young's modulus. Here we consider two equivalent problems where a capillary free surface deforms due to vortices: (i) in a steady viscous flow [solved by Jeong and Moffatt, J. Fluid Mech. 241, 1 (1992)], and (ii) in an elastic medium. The equations of linear incompressible elasticity and viscous flow are strictly identical, and the two-dimensional problems that we consider are solved using complex variable methods. Despite the similarity, the kinematics of the free surface is very different for the viscous and elastic cases. We show for the present problem that these kinematics result in displacement and velocity fields of different topology. Unexpectedly, the resulting surface deflections are even of opposite sign.

Elastocapillarity can control the formation and the morphology of beads-on-string structures in solid fibers

Beads-on-string patterns have been experimentally observed in solid cylinders for a wide range of material properties and structural lengths, from millimetric soft gels to nanometric hard fibers. In this work, we combine theoretical analysis and numerical tools to investigate the formation and nonlinear dynamics of such beaded structures. We show that this morphological transition is driven by elastocapillarity, i.e., a complex interplay between the effects of surface tension and bulk elasticity. Unlike buckling or wrinkling, the presence of an axial elongation is found here to favor the onset of fiber beading, in agreement with existing experimental results on electrospun fibers, hydrogels, and nerves. Our results also prove that the applied stretch can be used in fabrication techniques to control the morphology of the emerging beads-on-string patterns. Such quantitative predictions open the way for several applications, from tissue engineering to the design of stretchable electronics and the microfabrication of functionalized surfaces.

Surface tension and the mechanics of liquid inclusions in compliant solids

Eshelby's theory of inclusions has wide-reaching implications across the mechanics of materials and structures including the theories of composites, fracture, and plasticity. However, it does not include the effects of surface stress, which has recently been shown to control many processes in soft materials such as gels, elastomers and biological tissue. To extend Eshelby's theory of inclusions to soft materials, we consider liquid inclusions within an isotropic, compressible, linear-elastic solid. We solve for the displacement and stress fields around individual stretched inclusions, accounting for the bulk elasticity of the solid and the surface tension (i.e. isotropic strain-independent surface stress) of the solid-liquid interface. Surface tension significantly alters the inclusion's shape and stiffness as well as its near- and far-field stress fields. These phenomena depend strongly on the ratio of the inclusion radius, R, to an elastocapillary length, L. Surface tension is significant whenever inclusions are smaller than 100L. While Eshelby theory predicts that liquid inclusions generically reduce the stiffness of an elastic solid, our results show that liquid inclusions can actually stiffen a solid when R<3L/2. Intriguingly, surface tension cloaks the far-field signature of liquid inclusions when R=3L/2. These results are have far-reaching applications from measuring local stresses in biological tissue, to determining the failure strength of soft composites.

Gravity driven instability in elastic solid layers

We demonstrate the instability of the free surface of a soft elastic solid facing downwards. Experiments are carried out using a gel of constant density ρ, shear modulus μ, put in a rigid cylindrical dish of depth h. When turned upside down, the free surface of the gel undergoes a normal outgoing acceleration g. It remains perfectly flat for ρgh/μ<α* with α*≃6, whereas a steady pattern spontaneously appears in the opposite case. This phenomenon results from the interplay between the gravitational energy and the elastic energy of deformation, which reduces the Rayleigh waves celerity and vanishes it at the threshold.

The energy release rate of a pressurized crack in soft elastic materials: effects of surface tension and large deformation

In this paper we present a theoretical study on how surface tension affects fracture of soft solids. In classical fracture theory, the resistance to fracture is partly attributed to the energy required to create new surfaces. Thus, the energy released to the crack tip must overcome the surface energy in order to propagate a crack. In soft materials, however, surface tension can cause significant deformation and can reduce the energy release rate for crack propagation by resisting the stretch of crack surfaces. We quantify this effect by studying the inflation of a penny-shaped crack in an infinite elastic body with applied pressure. To avoid numerical difficulty caused by singular fields near the crack tip, we derived an expression for the energy release rate which depends on the applied pressure, the surface tension, the inflated crack volume and the deformed crack area. This expression is evaluated using a newly developed finite element method with surface tension elements. Our calculation shows that, when the elasto-capillary number ω ≡ σ/Ea is sufficiently large, where σ is the isotropic surface tension, E is the small strain Young's modulus and a is the initial crack radius, both the energy release rate and the crack opening displacement of an incompressible neo-Hookean solid are significantly reduced by surface tension. For a sufficiently high elasto-capillary number, the energy release rate can be negative for applied pressure less than a critical amount, suggesting that surface tension can cause crack healing in soft elastic materials.