Surface energy of strained amorphous solids

Nonmonotonic Motion of Sliding Droplets on Strained Soft Solids

Soft materials are ubiquitous in technological applications that require deformability, for instance, in flexible, water-repellent coatings. However, the wetting properties of prestrained soft materials are only beginning to be explored. Here we study the sliding dynamics of droplets on prestrained soft silicone gels, both in tension and in compression. Intriguingly, in compression we find a nonmonotonic strain dependence of the sliding speed: mild compressions decelerate the droplets, but stronger compressions lead again to faster droplet motion. Upon further compression, creases nucleate under the droplets until, finally, the entire surface undergoes the creasing instability, causing a "run-and-stop" motion. We quantitatively elucidate the speed modification for moderate prestrains by incremental viscoelasticity, while the acceleration for larger prestrains turns out to be linked to the solid pressure, presumably through a lubrication effect of expelled oligomers.

Bioinspired Slippery Surfaces for Liquid Manipulation from Tiny Droplet to Bulk Fluid

Slippery surfaces, which originate in nature with special wettability, have attracted considerable attention in both fundamental research and practical applications in a variety of fields due to their unique characteristics of superlow liquid friction and adhesion. Although research on bioinspired slippery surfaces is still in its infancy, it is a rapidly growing and enormously promising field. Herein, a systematic review of recent progress in bioinspired slippery surfaces, beginning with a brief introduction of several typical creatures with slippery property in nature, is presented. Subsequently,this review gives a detailed discussion on the basic concepts of the wetting, friction, and drag from micro- and macro-aspects and focuses on the underlying slippery mechanism. Next, the state-of-the-art developments in three categories of slippery surfaces of air-trapped, liquid-infused, and liquid-like slippery surfaces, including materials, design principles, and preparation methods, are summarized and the emerging applications are highlighted. Finally, the current challenges and future prospects of various slippery surfaces are addressed.

From a distance: Shuttleworth revisited

The Shuttleworth equation: a linear stress-strain relation ubiquitously used in modeling the behavior of soft surfaces. Its validity in the realm of materials subject to large deformation is a topic of current debate. Here, we allow for large deformation by deriving the constitutive behavior of the surface from the general framework of finite kinematics. We distinguish cases of finite and infinitesimal surface relaxation preceding an infinitesimal applied deformation. The Shuttleworth equation identifies as the Cauchy stress measure in the fully linearized setting. We show that both in finite and linearized cases, measured elastic constants depend on the utilized stress measure. In addition, we discuss the physical implications of our results and analyze the impact of surface relaxation on the estimation of surface elastic moduli in the light of two different test cases.

Wetting on silicone surfaces

Silicone is frequently used as a model system to investigate and tune wetting on soft materials. Silicone is biocompatible and shows excellent thermal, chemical, and UV stability. Moreover, the mechanical properties of the surface can be easily varied by several orders of magnitude in a controlled manner. Polydimethylsiloxane (PDMS) is a popular choice for coating applications such as lubrication, self-cleaning, and drag reduction, facilitated by low surface energy. Aiming to understand the underlying interactions and forces, motivated numerous and detailed investigations of the static and dynamic wetting behavior of drops on PDMS-based surfaces. Here, we recognize the three most prevalent PDMS surface variants, namely liquid-infused (SLIPS/LIS), elastomeric, and liquid-like (SOCAL) surfaces. To understand, optimize, and tune the wetting properties of these PDMS surfaces, we review and compare their similarities and differences by discussing (i) the chemical and molecular structure, and (ii) the static and dynamic wetting behavior. We also provide (iii) an overview of methods and techniques to characterize PDMS-based surfaces and their wetting behavior. The static and dynamic wetting ridge is given particular attention, as it dominates energy dissipation, adhesion, and friction of sliding drops and influences the durability of the surfaces. We also discuss special features such as cloaking and wetting-induced phase separation. Key challenges and opportunities of these three surface variants are outlined.

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.

Nanoscale Elasto-Capillarity in the Graphene-Water System under Tension: Revisiting the Assumption of a Constant Wetting Angle

Wetting highly compliant surfaces can cause them to deform. Atomically thin materials, such as graphene, can have exceptionally small bending rigidities, leading to elasto-capillary lengths of a few nanometers. Using large-scale molecular dynamics (MD), we have studied the wetting and deformation of graphene due to nanometer-sized water droplets, focusing on the wetting angle near the vesicle transition. Recent continuum theories for wetting of flexible membranes reproduce our MD results qualitatively well. However, we find that when the curvature is large at the triple-phase contact line, the wetting angle increases with decreasing tension. This is in contrast to existing macroscopic theories but can be amended by allowing for a variable wetting angle.

Moving wetting ridges on ultrasoft gels

The surface mechanics of soft solids are important in many natural and technological applications. In this context, static and dynamic wetting of soft polymer gels has emerged as a versatile model system. Recent experimental observations have sparked controversial discussions of the underlying theoretical description, ranging from concentrated elastic forces over strain-dependent solid surface tensions to poroelastic deformations or the capillary extraction of liquid components in the gel. Here we present measurements of the shapes of moving wetting ridges with high spatiotemporal resolution, combining distinct wetting phases (water, FC-70, air) on different ultrasoft PDMS gels (∼100Pa). Comparing our experimental results to the asymptotic behavior of linear viscoelastocapillary theory in the vicinity of the ridge, we separate reliable measurements from potential resolution artifacts. Remarkably, we find that the commonly used elastocapillary scaling fails to collapse the ridge shapes, but, for small normal forces, yields a viable prediction of the dynamic ridge angles. We demonstrate that neither of the debated theoretical models delivers a quantitative description, while the capillary extraction of an oil skirt appears to be the most promising.

Direct Measurement of Solid-Liquid Interfacial Energy Using a Meniscus

Solid-liquid interactions are central to diverse processes. The interaction strength can be described by the solid-liquid interfacial free energy (γ_{SL}), a quantity that is difficult to measure. Here, we present the direct experimental measurement of γ_{SL} for a variety of solid materials, from nonpolar polymers to highly wetting metals. By attaching a thin solid film on top of a liquid meniscus, we create a solid-liquid interface. The interface determines the curvature of the meniscus, analysis of which yields γ_{SL} with an uncertainty of less than 10%. Measurement of classically challenging metal-water interfaces reveals γ_{SL}∼30-60  mJ/m^{2}, demonstrating quantitatively that water-metal adhesion is 80% stronger than the cohesion energy of bulk water, and experimentally verifying previous quantum chemical calculations.

Soft wetting with (a)symmetric Shuttleworth effect

The wetting of soft polymer substrates brings in multiple complexities when compared with the wetting on rigid substrates. The contact angle of the liquid is no longer governed by Young's Law, but is affected by the substrate's bulk and surface deformations. On top of that, elastic interfaces exhibit a surface energy that depends on how much they are stretched-a feature known as the Shuttleworth effect (or as surface-elasticity). Here, we present two models through which we explore the wetting of drops in the presence of a strong Shuttleworth effect. The first model is macroscopic in character and consistently accounts for large deformations via a neo-Hookean elasticity. The second model is based on a mesoscopic description of wetting, using a reduced description of the substrate's elasticity. While the second model is more empirical in terms of the elasticity, it enables a gradient dynamics formulation for soft wetting dynamics. We provide a detailed comparison between the equilibrium states predicted by the two models, from which we deduce robust features of soft wetting in the presence of a strong Shuttleworth effect. Specifically, we show that the (a)symmetry of the Shuttleworth effect between the 'dry' and 'wet' states governs horizontal deformations in the substrate. Our results are discussed in the light of recent experiments on the wettability of stretched substrates.

Temperature-dependent soft wetting on amorphous, uncrosslinked polymer surfaces

The wetting of polymer melts at high temperatures is studied by placing a glycerol drop on a poly(n-butyl methacrylate) film and measuring the wetting ridge. The height of the wetting ridge grows continuously over time. These wetting ridge growth rates can be explained by polymer chain dynamics occurring at the molecular level, determined using oscillatory shear rheology of the polymer melt. The shape of wetting ridge profile can be modeled using an equation previously used for elastomers, with a simple modification that incorporates the time-dependent storage modulus of the uncrosslinked melts.

The role of crosslinking density in surface stress and surface energy of soft solids

Surface stress and surface energy are two fundamental parameters that determine the surface properties of any material. While it is commonly believed that the surface stress and surface energy of liquids are identical, the relationship between the two parameters in soft polymeric gels remains debatable. In this work, we measured the surface stress and surface energy of soft silicone gels with varying weight ratios of crosslinkers in soft wetting experiments. Above a critical density, k₀, the surface stress was found to increase significantly with crosslinking density while the surface energy remained unchanged. In this regime, we can estimate a non-zero surface elastic modulus that also increases with the ratio of crosslinkers. By comparing the surface mechanics of the soft gels with their bulk rheology, the surface properties near the critical density k₀ were found to be closely related to the underlying percolation transition of the polymer networks.

Elasticity-to-Capillarity Transition in Soft Substrate Deformation

Whereas capillarity controls fluid dynamics at submillimeter scale and elasticity determines the mechanics of rigid solids, their coupling governs elastocapillary deformations on soft solids. Here, we directly probed the deformations on soft substrates induced by sessile nanodroplets. The wetting ridge created around the contact line and the dimple formed underneath the nanodroplet were imaged with a high spatial resolution using atomic force microscopy. The ridge height nonmonotonically depends on the substrate stiffness, and the dimple depth nonlinearly depends on the droplet size. The capillarity of the substrate overcomes the elasticity of the substrate in dominating the deformations when the elastocapillary length is approximately larger than the droplet contact radius, showing an experimental observation of the elasticity-to-capillarity transition. This study provides an experimental approach to investigate nanoscale elastocapillarity, and the insights have the potential to kick-off future work on the fundamentals of solid mechanics.

Gradient-dynamics model for liquid drops on elastic substrates

The wetting of soft elastic substrates exhibits many features that have no counterpart on rigid surfaces. Modelling the detailed elastocapillary interactions is challenging, and has so far been limited to single contact lines or single drops. Here we propose a reduced long-wave model that captures the main qualitative features of statics and dynamics of soft wetting, but which can be applied to ensembles of droplets. The model has the form of a gradient dynamics on an underlying free energy that reflects capillarity, wettability and compressional elasticity. With the model we first recover the double transition in the equilibrium contact angles that occurs when increasing substrate softness from ideally rigid towards very soft (i.e., liquid). Second, the spreading of single drops of partially and completely wetting liquids is considered showing that known dependencies of the dynamic contact angle on contact line velocity are well reproduced. Finally, we go beyond the single droplet picture and consider the coarsening for a two-drop system as well as for a large ensemble of drops. It is shown that the dominant coarsening mode changes with substrate softness in a nontrivial way.

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.

Toward the Specificity of Bare Nanomaterial Surfaces for Protein Corona Formation

Aiming at creating smart nanomaterials for biomedical applications, nanotechnology aspires to develop a new generation of nanomaterials with the ability to recognize different biological components in a complex environment. It is common opinion that nanomaterials must be coated with organic or inorganic layers as a mandatory prerequisite for applications in biological systems. Thus, it is the nanomaterial surface coating that predominantly controls the nanomaterial fate in the biological environment. In the last decades, interdisciplinary studies involving not only life sciences, but all branches of scientific research, provided hints for obtaining uncoated inorganic materials able to interact with biological systems with high complexity and selectivity. Herein, the fragmentary literature on the interactions between bare abiotic materials and biological components is reviewed. Moreover, the most relevant examples of selective binding and the conceptualization of the general principles behind recognition mechanisms were provided. Nanoparticle features, such as crystalline facets, density and distribution of surface chemical groups, and surface roughness and topography were encompassed for deepening the comprehension of the general concept of recognition patterns.

Droplets Sit and Slide Anisotropically on Soft, Stretched Substrates

Anisotropically wetting substrates enable useful control of droplet behavior across a range of applications. Usually, these involve chemically or physically patterning the substrate surface, or applying gradients in properties like temperature or electrical field. Here, we show that a flat, stretched, uniform soft substrate also exhibits asymmetric wetting, both in terms of how droplets slide and in their static shape. Droplet dynamics are strongly affected by stretch: glycerol droplets on silicone substrates with a 23% stretch slide 67% faster in the direction parallel to the applied stretch than in the perpendicular direction. Contrary to classical wetting theory, static droplets in equilibrium appear elongated, oriented parallel to the stretch direction. Both effects arise from droplet-induced deformations of the substrate near the contact line.

Contact lines on stretched soft solids: modelling anisotropic surface stresses

We present fully analytical solutions for the deformation of a stretched soft substrate due to the static wetting of a large liquid droplet, and compare our solutions to recently published experiments (Xu et al. 2018 Soft Matter 14, 916–920 (doi:10.1039/C7SM02431B)). Following a Green’s function approach, we extend the surface-stress regularized Flamant–Cerruti problem to account for uniaxial pre-strains of the substrate. Surface profiles, including the heights and opening angles of wetting ridges, are provided for linearized and finite kinematics. We fit experimental wetting ridge shapes as a function of applied strain using two free parameters, the surface Lamé coefficients. In comparison with experiments, we find that observed opening angles are more accurately captured using finite kinematics, especially with increasing levels of applied pre-strain. These fits qualitatively agree with the results of Xu et al ., but revise values of the surface elastic constants.

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.

Capillary-driven indentation of a microparticle into a soft, oil-coated substrate

Small scale contact between a soft, liquid-coated layer and a stiff surface is common in many situations, from synovial fluid on articular cartilage to adhesives in humid environments. Moreover, many model studies on soft adhesive contacts are conducted with soft silicone elastomers, which possess uncrosslinked liquid molecules (i.e. silicone oil) when the modulus is low. We investigate how the thickness of a silicone oil layer on a soft substrate relates to the indentation depth of glass microspheres in contact with crosslinked PDMS, which have a modulus of <10 kPa. The particles indent into the underlying substrate more as a function of decreasing oil layer thickness. This is due to the presence of the liquid layer at the surface that causes capillary forces to pull down on the particle. A simple model that balances the capillary force of the oil layer and the minimal particle-substrate adhesion with the elastic and surface tension forces from the substrate is proposed to predict the particle indentation depth.

Nonlinear theory of wetting on deformable substrates

The spreading of a liquid over a solid material is a key process in a wide range of applications. While this phenomenon is well understood when the solid is undeformable, its "soft" counterpart is still misunderstood and no consensus has been reached with regard to the physical mechanisms ruling the spreading of liquid drops over soft deformable materials. In this work we provide a theoretical framework, based on the nonlinear theory of discontinuities, to describe the behavior of a triple line on a soft material. We show that the contact line motion is opposed both by nonlinear localized capillary and visco-elastic forces. We give an explicit analytic formula relating the dynamic contact angle of a moving drop to its velocity for arbitrary rheology. We then specialize this formula to the experimentally relevant case of elastomers with the Chasset-Thirion (power-law) type of rheologies. The theoretical prediction is in very good agreement with experimental data, without any adjustable parameters. We then show that the nonlinear force balance presented in this work can also be used to recover classical models of wetting. Finally we provide predictions for the dynamic behavior of the yet largely unexplored case of a viscous drop spreading over a soft visco-elastic material and predict the emergence of a new form of apparent hysteresis.

The Influence of Nanoparticle Shape on Protein Corona Formation

Nanoparticles have become an important utility in many areas of medical treatment such as targeted drug and treatment delivery as well as imaging and diagnostics. These advances require a complete understanding of nanoparticles' fate once placed in the body. Upon exposure to blood, proteins adsorb onto the nanoparticles surface and form a protein corona, which determines the particles' biological fate. This study reports on the protein corona formation from blood serum and plasma on spherical and rod-shaped nanoparticles. These two types of mesoporous silica nanoparticles have identical chemistry, porosity, surface potential, and size in the y-dimension, one being a sphere and the other a rod shape. The results show a significantly larger amount of protein attaching from both plasma and serum on the rod-like particles compared to the spheres. Interrogation of the protein corona by liquid chromatography-mass spectrometry reveals shape-dependent differences in the adsorption of immunoglobulins and albumin proteins from both plasma and serum. This study points to the need for taking nanoparticle shape into consideration because it can have a significant impact on the fate and therapeutic potential of nanoparticles when placed in the body.

Spreading on viscoelastic solids: are contact angles selected by Neumann's law?

The spreading of liquid drops on soft substrates is extremely slow, owing to strong viscoelastic dissipation inside the solid. A detailed understanding of the spreading dynamics has remained elusive, partly owing to the difficulty in quantifying the strong viscoelastic deformations below the contact line that determine the shape of moving wetting ridges. Here we present direct experimental visualisations of the dynamic wetting ridge using shadowgraphic imaging, complemented with measurements of the liquid contact angle. It is observed that the wetting ridge exhibits a rotation that follows exactly the dynamic liquid contact angle - as was previously hypothesized [Karpitschka et al., Nat. Commun., 2015, 6, 7891]. This experimentally proves that, despite the contact line motion, the wetting ridge is still governed by Neumann's law. Furthermore, our experiments suggest that moving contact lines lead to a variable surface tension of the substrate. We therefore set up a new theory that incorporates the influence of surface strain, for the first time including the so-called Shuttleworth effect into the dynamical theory for soft wetting. It includes a detailed analysis of the boundary conditions at the contact line, complemented by a dissipation analysis, which shows, again, the validity of Neumann's balance.

Distinguishing deformation mechanisms in elastocapillary experiments

Soft materials are known to deform due to a variety of mechanisms, including capillarity, buoyancy, and swelling. In this paper, we present experiments on polyvinylsiloxane gel threads partially-immersed in three liquids with different solubility, wettability, and swellability. Our results demonstrate that deformations due to capillarity, buoyancy, and swelling can be of similar magnitude as such threads come to static equilibrium. To account for all three effects being present in a single system, we derive a model capable of explaining the observed data and use it to determine the force law at the three-phase contact line. The results show that the measured forces are consistent with the expected Young-Dupré equation, and do not require the inclusion of a tangential contact line force.

Elastocapillary Ridge as a Noninteger Disclination

Understanding the interfacial properties of solids with their environment is a crucial problem in fundamental science and applications. Elastomers have challenged the scientific community in this respect, and a satisfying description is still missing. Here, we argue that the interfacial properties of elastomers, such as their wettability, can be understood with a nonlinear elastic model with the assumption of a strain-independent surface energy. We show that our model captures accurately available data on elastomer wettability and discuss its implications.

Wetting of Polymer Brushes by Polymeric Nanodroplets

End-anchoring polymers to a solid surface to form so-called polymer brushes is a versatile method to prepare robust functional coatings. We show, using molecular dynamics simulations, that these coatings display rich wetting behavior. Depending on the interaction between the brushes and the polymeric droplets as well as on the self-affinity of the brush, we can distinguish between three wetting states: mixing, complete wetting, and partial wetting. We find that transitions between these states are largely captured by enthalpic arguments, while deviations to these can be attributed to the negative excess interfacial entropy for the brush droplet system. Interestingly, we observe that the contact angle strongly increases when the softness of the brush is increased, which is opposite to the case of drops on soft elastomers. Hence, the Young to Neumann transition owing to softness is not universal but depends on the nature of the substrate.

Peeling an elastic film from a soft viscoelastic adhesive: experiments and scaling laws

The functionality of adhesives relies on their response under the application of a load. Yet, it has remained a challenge to quantitatively relate the macroscopic dynamics of peeling to the dissipative processes inside the adhesive layer. Here we investigate the peeling of a reversible adhesive made of a polymer gel, measuring the relationship between the peeling force, the peeling velocity, and the geometry of the interface at small-scale. Experiments are compared to a theory based on the linear viscoelastic response of the adhesive, augmented with an elastocapillary regularization approach. This theory, fully quantitative in the limit of small surface deformations, demonstrates the emergence of a "wetting" angle at the contact line and exhibits scaling laws for peeling which are in good agreement with the experimental results. Our findings provide a new strategy for design of reversible adhesives, by quantitatively combining wetting, geometry and dissipation.

Droplets Capped with an Elastic Film Can Be Round, Elliptical, or Nearly Square

We present experiments that show that the partial wetting of droplets capped by taut elastic films is highly tunable. Adjusting the tension allows the contact angle and droplet morphology to be controlled. By exploiting these elastic boundaries, droplets can be made elliptical, with an adjustable aspect ratio, and can even be transformed into a nearly square shape. This system can be used to create tunable liquid lenses and, moreover, presents a unique approach to liquid patterning.

Dynamic Solid Surface Tension Causes Droplet Pinning and Depinning

The contact line of a liquid drop on a solid exerts a nanometrically sharp surface traction. This provides an unprecedented tool to study highly localized and dynamic surface deformations of soft polymer networks. One of the outstanding problems in this context is the stick-slip instability, observed above a critical velocity, during which the contact line periodically depins from its own wetting ridge. Time-resolved measurements of the solid deformation are challenging, and the mechanism of dynamical depinning has remained elusive. Here we present direct visualisations of the dynamic wetting ridge formed by water spreading on a PDMS gel. Unexpectedly, it is found that the opening angle of the wetting ridge increases with speed, which cannot be attributed to bulk rheology, but points to a dynamical increase of the solid's surface tensions. From this we derive the criterion for depinning that is confirmed experimentally. Our findings reveal a deep connection between stick-slip processes and newly identified dynamical surface effects.

Microscopic details of a fluid/thin film triple line

In recent years, there has been a considerable interest in the mechanics of soft objects meeting fluid interfaces (elasto-capillary interactions). In this work we experimentally examine the case of a fluid resting on a thin film of rigid material which, in turn, is resting on a fluid substrate. To simplify complexity, we adapt the experiment to a one-dimensional contact geometry and examine the behaviour of polystyrene and polycarbonate films directly with confocal microscopy. We find that the fluid meets the film in a manner consistent with the Young-Dupré equation when the film is thick, but transitions to what appears similar to a Neumann-like balance when the thickness is decreased. However, on closer investigation we find that the true contact angle is always given by the Young construction. The apparent paradox is a result of macroscopically measured angles not being directly related to true microscopic contact angles when curvature is present. We model the effect with an Euler-Bernoulli beam on a Winkler foundation as well as with an equivalent energy-based capillary model. Notably, the models highlight several important lengthscales and the complex interplay of tension, gravity, and bending in the problem.

Paradox of Contact Angle Selection on Stretched Soft Solids

The interfacial mechanics of soft elastic networks plays a central role in biological and technological contexts. Yet, effects of solid capillarity have remained controversial, primarily due to the strain-dependent surface energy. Here we derive the equations that govern the selection of contact angles of liquid drops on elastic surfaces from variational principles. It is found that the substrate's elasticity imposes a nontrivial condition that relates pinning, hysteresis, and contact line mobility to the so-called Shuttleworth effect. We experimentally validate our theory for droplets on a silicone gel, revealing an enhanced contact line mobility when stretching the substrate.

Surface Stresses and a Force Balance at a Contact Line

Results of the coarse-grained molecular dynamics simulations are used to show that the force balance analysis at the triple-phase contact line formed at an elastic substrate has to include a quartet of forces: three surface tensions (surface free energies) and an elastic force per unit length. In the case of the contact line formed by a droplet on an elastic substrate an elastic force is due to substrate deformation generated by formation of the wetting ridge. The magnitude of this force f_el is proportional to the product of the ridge height h and substrate shear modulus G. Similar elastic line force should be included in the force analysis at the triple-phase contact line of a solid particle in contact with an elastic substrate. For this contact problem elastic force obtained from contact angles and surface tensions is a sum of the elastic forces acting from the side of a solid particle and an elastic substrate. By considering only three line forces acting at the triple-phase contact line, one implicitly accounts the bulk stress contribution as a part of the resultant surface stresses. This "contamination" of the surface properties by a bulk contribution could lead to unphysically large values of the surface stresses in soft materials.

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.