Direct measurement of strain-dependent solid surface stress

Unsteady wetting of soft solids

HYPOTHESIS

Spreading of liquids on soft solids often occurs intermittently, i.e., the liquid's wetting front switches between sticking and slipping. Studies of this so-called stick-slip wetting on soft solids mostly are confined within quasi-static or forced spreading conditions. In these situations, because the sticking duration is set much larger than the viscoelastic relaxation time of the solid, a ridge is persistently and fully developed at the wetting front as the soft solid yields to the liquid's surface tension. The sticking duration and spreading velocity, therefore, were shown to have little impact to the contact angle change required for stick-to-slip transitions. For unsteady wetting of soft solids, a commonly encountered but largely unexplored situation, we hypothesize that the stick-to-slip transition is controlled not only by a combination of sticking duration and the spreading velocity, but also by an increasing depinning threshold caused by the growing ridge at the wetting front.

EXPERIMENT

We performed unsteady wetting experiment on soft solids by letting water droplets spread freely on soft solid surfaces of various stiffness. We capture both the stick-slip spreading behavior and growing wetting ridges using synchronous high-speed imaging and high-speed interferometry. Recorded data of liquid spreading and solid deforming at the wetting front were analyzed to shed light on the relation between stick-slip characteristics and the growing wetting ridge.

FINDINGS

We find that intermittent wetting on a soft solid surface results from a competition between three key factors: liquid inertia, capillary force change during sticking, and growing pinning force caused by the solid's viscoelastic response. We theoretically formulate their quantitative contributions to predict how stick-to-slip transitions occur, i.e., how the contact angle change and sticking duration depend on the liquid's spreading velocity and the solid's viscoelastic characteristics. This provides a mechanistic understanding and methods to control unsteady wetting phenomena in diverse applications, from tissue engineering and fabrication of flexible electronics to biomedicine.

Deformation-dependent gel surface topography due to the elastocapillary and osmocapillary effects

Actively tuning surface topography is crucial for the design of smart surfaces with stimuli-responsive friction, wetting, and adhesion properties. This paper studies how elastocapillary deformation and osmocapillary phase separation can lead to rich deformation-dependent surface topography in polymeric gels. In a purely elastic material, stretching always flattens the surface due to the Poisson effect. We show that stretching can roughen the surface due to the elastocapillary and osmocapillary effects. The roughening can be tuned by the gel stiffness, the gel osmotic pressure, the deformation mode, and the initial amplitude of surface roughness. The rich deformation-dependent behavior of gel surface topography points to a new direction in designing smart surfaces.

Surface stress can initiate environment-assisted fracture in metals

Controlling environmental effects in surface plasticity/fracture of metals is of interest for areas as diverse as manufacturing processes, product performance, and structural safety. The key to controlling these effects is understanding the effect of adsorbates on surface energy (γ) and surface stress (f). While γ has been well studied, the role of surface stress has received much less attention. We characterize surface stress induced in metals by adsorption of organic monolayers. Linear alkanoic acids of varying chain length (3-18) are deposited by molecular self-assembly onto one side of an aluminum cantilever, several centimeters in length. The surface stress is estimated from in situ measurement of the cantilever deflection. We find that the organic adsorbates induce large surface stress of -4 to +30N/m. Furthermore, we show that f may be tuned by varying adsorbate-molecule chain length. The stress data explain beneficial embrittlement of metal surfaces by organic adsorbates in cutting and comminution processes, and point to a critical role, hitherto ignored, for f in environment assisted cracking (EAC) phenomena. Our results suggest opportunities for utilizing controlled environment-assisted fracture as an aid-fracture as a friend-to enhance material removal processes, apart from using surface stress itself as an experimental probe to explore various manifestations of EAC.

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.

The crucial role of elasticity in regulating liquid-liquid phase separation in cells

Liquid-liquid phase separation has emerged as a fundamental mechanism underlying intracellular organization, with evidence for it being reported in numerous different systems. However, there is a growing concern regarding the lack of quantitative rigor in the techniques employed to study phase separation, and their ability to account for the complex nature of the cellular milieu, which affects key experimentally observable measures, such as the shape, size and transport dynamics of liquid droplets. Here, we bridge this gap by combining recent experimental data with theoretical predictions that capture the subtleties of nonlinear elasticity and fluid transport. We show that within a biologically accessible range of material parameters, phase separation is highly sensitive to elastic properties and can thus be used as a mechanical switch to rapidly transition between different states in cellular systems. Furthermore, we show that this active mechanically mediated mechanism can drive transport across cells at biologically relevant timescales and could play a crucial role in promoting spatial localization of condensates; whether cells exploit such mechanisms for transport of their constituents remains an open question.

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.

Capillary pressure mediated long-term dynamics of thin soft films

HYPOTHESIS

The conventional solid-solid contact is well studied in the literature. However, a number of practical applications, such as adhesive patches and biomimetic surfaces, require a much deeper understanding of soft contact where there is a distinct time-dependent adhesion behavior due to the dual-phase structure (solids and liquids). To understand this, currently existing solid-solid contact behavior is extrapolated to soft contact, wherein the size-effect of the gel film and the preload are typically neglected. When introducing the finite-size effect and preload, gels could experience distinctive long-term contact dynamics in contact with another material.

EXPERIMENTS

We reconstruct the evolving surface profile of the gel films intercalated between a glass sphere and glass slide using dual wavelength-reflection interference contrast microscopy. The macro-sized glass sphere compresses the gel. The indentation depth is comparable to the gel film thickness, wherein the conventional contact theories are inapplicable.

FINDINGS

The gel surface experiences two deformation stages. The natural preload and elastic force develop the contact area in the early state. In the later state, the viscous free molecules of the gel develop the ridge. We discover that the residual surface stress relaxes over 85 hr. Our findings on the long-term gel deformation provide a new perspective on soft adhesion, from developing soft adhesives to understanding biological tissues.

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.

Cell-Matrix Elastocapillary Interactions Drive Pressure-based Wetting of Cell Aggregates

Cell-matrix interfacial energies and the energies of matrix deformations may be comparable on cellular length-scales, yet how capillary effects influence tis sue shape and motion are unknown. In this work, we induce wetting (spreading and migration) of cell aggregates, as models of active droplets onto adhesive substrates of varying elasticity and correlate the dynamics of wetting to the balance of interfacial tensions. Upon wetting rigid substrates, cell-substrate tension drives outward expansion of the monolayer. By contrast, upon wetting compliant substrates, cell substrate tension is attenuated and aggregate capillary forces contribute to internal pressures that drive expansion. Thus, we show by experiments, data-driven modeling and computational simulations that myosin-driven 'active elasto-capillary' effects enable adaptation of wetting mechanisms to substrate rigidity and introduce a novel, pressure-based mechanism for guiding collective cell motion.

Epithelial Cell-Like Elasticity Modulates Actin-Dependent E-Cadherin Adhesion Organization

E-cadherin adhesions are essential for cell-to-cell cohesion and mechanical coupling between epithelial cells and reside in a microenvironment that comprises the adjoining epithelial cells. While E-cadherin has been shown to be a mechanosensor, it is unknown if E-cadherin adhesions can differentially sense stiffness within the range of that of epithelial cells. A survey of literature shows that epithelial cells' Young's moduli of elasticity lie predominantly in the sub-kPa to few-kPa range, with cancer cells often being softer than noncancerous ones. Here, we devised oriented E-cadherin-coated soft silicone substrates with sub-kPa or few-kPa elasticity but with similar viscous moduli and found that E-cadherin adhesions differentially organize depending on the magnitude of epithelial cell-like elasticity. Our results show that the actin cytoskeleton organizes E-cadherin adhesions in two ways─by supporting irregularly shaped adhesions at localized regions of high actin density and linear shaped adhesions at the end of linear actin bundles. Linearly shaped E-cadherin adhesions associated with radially oriented actin─but not irregularly shaped E-cadherin adhesions associated with circumferential actin foci─were much more numerous on 2.4 kPa E-cadherin substrates compared to 0.3 kPa E-cadherin substrates. However, the total amount of E-cadherin in both types of adhesions taken together was similar on the 0.3 and 2.4 kPa E-cadherin substrates across many cells. Our results show how the distribution of E-cadherin adhesions, supported by actin density and architecture, is modulated by epithelial cell-like elasticity and have significant implications for disease states like carcinomas characterized by altered epithelial cell elasticity.

Revealing Capillarity in AFM Indentation of Cells by Nanodiamond-Based Nonlocal Deformation Sensing

Nanoindentation based on atomic force microscopy (AFM) can measure the elasticity of biomaterials and cells with high spatial resolution and sensitivity, but relating the data to quantitative mechanical properties depends on information on the local contact, which is unclear in most cases. Here, we demonstrate nonlocal deformation sensing on biorelevant soft matters upon AFM indentation by using nitrogen-vacancy centers in nanodiamonds, providing data for studying both the elasticity and capillarity without requiring detailed knowledge about the local contact. Using fixed HeLa cells for demonstration, we show that the apparent elastic moduli of the cells would have been overestimated if the capillarity was not considered. In addition, we observe that both the elastic moduli and the surface tensions are reduced after depolymerization of the actin cytoskeleton in cells. This work demonstrates that the nanodiamond sensing of nonlocal deformation with nanometer precision is particularly suitable for studying mechanics of soft biorelevant materials.

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.

Bowtie-Shaped Deformation Isotherm of Superhydrophobic Cylindrical Mesopores

Deformation of superhydrophobic cylindrical mesopores is studied during a cycle of forced water filling and spontaneous drying by in situ small-angle neutron scattering. A high-pressure setup is put forward to characterize the deformation of ordered mesoporous silanized silica up to 80 MPa. Strain isotherms of individual pores are deduced from the shift of the Bragg spectrum associated with the deformation of the hexagonal pore lattice. Due to their superhydrophobic nature, pore walls are not covered with a prewetting film. This peculiarity gives the ability to use a simple mechanical model to describe both filled and empty pore states without the pitfall of disjoining pressure effects. By fitting our experimental data with this model, we measure both the Young's modulus and the Poisson ratio of the nanometric silica wall. The measurement of this latter parameter constitutes a specificity offered by superhydrophobic nanopores with respect to hydrophilic ones.

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.

Calibrating surface hyperelastic constitutive models in soft solids

Soft solids such as silicone gels, with bulk shear modulus ranging from ∼10 to 1000kPa, exhibit strongly strain-dependent surface stresses. Moreover, unlike conventional stiffer materials, the effects of surface stress in these materials manifest at length scales of tens of micrometers rather than nanometers. However, the calibration of constitutive parameters for surface hyperelasticity has proved to be challenging. Using a reasonably general surface constitutive model, we explore the possibility of obtaining its parameters from force-twist, torque-twist, and force-extension (force-compression) responses of a soft cylinder held between two inert, rigid plates. The motivation behind using these responses is derived from the fact that the roles of the surface constitutive parameters, under suitably ideal conditions, are neatly separated from each other and the three responses easily yield values of the three parameters. Moreover, through large deformation finite-element simulations with coupled bulk and surface hyperelasticity, we delineate the extent to which deviation from the ideal conditions may be tolerated. Using an example with previously reported material parameters, we estimate that, for cylindrical specimens with a radius of the order of 100μm, the capability to measure forces and torques of the order of 1-100μN and 10^{3}-10^{5}μN-μm, respectively, will be required to determine the parameters accurately.

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.

Non-Destructive Imaging on Synthesised Nanoparticles

Our recently developed non-destructive imaging technique was applied for the characterisation of nanoparticles synthesised by X-ray radiolysis and the sol-gel method. The interfacial conditions between the nanoparticles and the substrates were observed by subtracting images taken by scanning electron microscopy at controlled electron acceleration voltages to allow backscattered electrons to be generated predominantly below and above the interfaces. The interfacial adhesion was found to be dependent on the solution pH used for the particle synthesis or particle suspension preparation, proving the change in the particle formation/deposition processes with pH as anticipated and agreed with the prediction based on the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory. We found that our imaging technique was useful for the characterisation of interfaces hidden by nanoparticles to reveal the formation/deposition mechanism and can be extended to the other types of interfaces.

Viscoelastic and Poroelastic Relaxations of Soft Solid Surfaces

Understanding surface mechanics of soft solids, such as soft polymeric gels, is crucial in many engineering processes, such as dynamic wetting and adhesive failure. In these situations, a combination of capillary and elastic forces drives the motion, which is balanced by dissipative mechanisms to determine the rate. While shear rheology (i.e., viscoelasticity) has long been assumed to dominate the dissipation, recent works have suggested that compressibility effects (i.e., poroelasticity) could play roles in swollen networks. We use fast interferometric imaging to quantify the relaxation of surface deformations due to a displaced contact line. By systematically measuring the profiles at different time and length scales, we experimentally observe a crossover from viscoelastic to poroelastic surface relaxations.

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.

Rotation of a submerged finite cylinder moving down a soft incline

A submerged finite cylinder moving under its own weight along a soft incline lifts off and slides at a steady velocity while also spinning. Here, we experimentally quantify the steady spinning of the cylinder and show theoretically that it is due to a combination of an elastohydrodynamic torque generated by flow in the variable gap, and the viscous friction on the edges of the finite-length cylinder. The relative influence of the latter depends on the aspect ratio of the cylinder, the angle of the incline, and the deformability of the substrate, which we express in terms of a single scaled compliance parameter. By independently varying these quantities, we show that our experimental results are consistent with a transition from an edge-effect dominated regime for short cylinders to a gap-dominated elastohydrodynamic regime when the cylinder is very long.

Graphene-based sensing of oxygen transport through pulmonary membranes

Lipid-protein complexes are the basis of pulmonary surfactants covering the respiratory surface and mediating gas exchange in lungs. Cardiolipin is a mitochondrial lipid overexpressed in mammalian lungs infected by bacterial pneumonia. In addition, increased oxygen supply (hyperoxia) is a pathological factor also critical in bacterial pneumonia. In this paper we fabricate a micrometer-size graphene-based sensor to measure oxygen permeation through pulmonary membranes. Combining oxygen sensing, X-ray scattering, and Atomic Force Microscopy, we show that mammalian pulmonary membranes suffer a structural transformation induced by cardiolipin. We observe that cardiolipin promotes the formation of periodic protein-free inter-membrane contacts with rhombohedral symmetry. Membrane contacts, or stalks, promote a significant increase in oxygen gas permeation which may bear significance for alveoli gas exchange imbalance in pneumonia.

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.

Soft electrowetting

Electrowetting is a commonly used tool to manipulate sessile drops on hydrophobic surfaces. By applying an external voltage over a liquid and a dielectric-coated surface, one achieves a reduction of the macroscopic contact angles for increasing voltage. The electrostatic forces all play out near the contact line, on a scale of the order of the thickness of the solid dielectric layer. Here we explore the case where the dielectric is a soft elastic layer, which deforms elastically under the effect of electrostatic and capillary forces. The wetting behaviour is quantified by measurements of the static and dynamic contact angles, complemented by confocal microscopy to reveal the elastic deformations. Even though the mechanics near the contact line is highly intricate, the macroscopic contact angles can be understood from global conservation laws in the spirit of Young-Lippmann. The key finding is that, while elasticity has no effect on the static electrowetting angle, the substrate's viscoelasticity completely dictates the spreading dynamics of electrowetting.

Nanometer-precision non-local deformation reconstruction using nanodiamond sensing

Spatially resolved information about material deformation upon loading is critical to evaluating mechanical properties of materials, and to understanding mechano-response of live systems. Existing techniques may access local properties of materials at nanoscale, but not at locations away from the force-loading positions. Moreover, interpretation of the local measurement relies on correct modeling, the validation of which is not straightforward. Here we demonstrate an approach to evaluating non-local material deformation based on the integration of nanodiamond orientation sensing and atomic force microscopy nanoindentation. This approach features a 5 nm precision in the loading direction and a sub-hundred nanometer lateral resolution, high enough to disclose the surface/interface effects in the material deformation. The non-local deformation profile can validate the models needed for mechanical property determination. The non-local nanometer-precision sensing of deformation facilitates studying mechanical response of complex material systems ranging from impact transfer in nanocomposites to mechano-response of live systems.

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.

Probing Micro-Newton Forces on Solid/Liquid/Gas Interfaces Using Transmission Phase Shift

Many of the nature and life systems are driven by capillary interactions on solid/liquid/gas interfaces. Here, we present a profilometry technique called transmission phase shift for visualizing the liquid/gas interfaces in three dimensions with high resolution. Using this approach, we probe the change in tiny forces with particle radius at a solid/liquid/gas interface. We provide the first direct evidence that in the issues of floating versus sinking at small-scale, Archimedes' principle should be generalized to include the crucial role of surface tension and reveal the dominant regimes of floating particles based on the Bond number. Remarkably, the measured forces are in the range of micro-Newtons, suggesting that this terse methodology may guide the future design of a liquid microbalance and will be a universal tool for investigating capillarity and interface issues.

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.

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.

Depth-dependent hysteresis in adhesive elastic contacts at large surface roughness

Contact force-indentation depth measurements in contact experiments involving compliant materials, such as polymers and gels, show a hysteresis loop whose size depends on the maximum indentation depth. This depth-dependent hysteresis (DDH) is not explained by classical contact mechanics theories and was believed to be due to effects such as material viscoelasticity, plasticity, surface polymer interdigitation, and moisture. It has been observed that the DDH energy loss initially increases and then decreases with roughness. A mechanics model based on the occurrence of adhesion and roughness related small-scale instabilities was presented by one of the authors for explaining DDH. However, that model only applies in the regime of infinitesimally small surface roughness, and consequently it does not capture the decrease in energy loss with surface roughness at the large roughness regime. We present a new mechanics model that applies in the regime of large surface roughness based on the Maugis-Dugdale theory of adhesive elastic contacts and Nayak's theory of rough surfaces. The model captures the trend of decreasing energy loss with increasing roughness. It also captures the experimentally observed dependencies of energy loss on the maximum indentation depth, and material and surface properties.

Singular dynamics in the failure of soft adhesive contacts

We characterize the mechanical recovery of compliant silicone gels following adhesive contact failure. We establish broad, stable adhesive contacts between rigid microspheres and soft gels, then stretch the gels to large deformations by pulling quasi-statically on the contact. Eventually, the adhesive contact begins to fail, and ultimately slides to a final contact point on the bottom of the sphere. Immediately after detachment, the gel recoils quickly with a self-similar surface profile that evolves as a power law in time, suggesting that the adhesive detachment point is singular. The singular dynamics we observe are consistent with a relaxation process driven by surface stress and slowed by viscous flow through the porous, elastic network of the gel. Our results emphasize the importance of accounting for both the liquid and solid phases of gels in understanding their mechanics, especially under extreme deformation.

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.

Acceleration-induced pressure gradients and cavitation in soft biomaterials

The transient, dynamic response of soft materials to mechanical impact has become increasingly relevant due to the emergence of numerous biomedical applications, e.g., accurate assessment of blunt injuries to the human body. Despite these important implications, acceleration-induced pressure gradients in soft materials during impact and the corresponding material response, from small deformations to sudden bubble bursts, are not fully understood. Both through experiments and theoretical analyses, we empirically show, using collagen and agarose model systems, that the local pressure in a soft sample is proportional to the square of the sample depth in the impact direction. The critical acceleration that corresponds to bubble bursts increases with increasing gel stiffness. Bubble bursts are also highly sensitive to the initial bubble size, e.g., bubble bursts can occur only when the initial bubble diameter is smaller than a critical size (≈10 μm). Our study gives fundamental insight into the physics of injury mechanisms, from blunt trauma to cavitation-induced brain injury.

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.

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.

Surface energy of strained amorphous solids

Surface stress and surface energy are fundamental quantities which characterize the interface between two materials. Although these quantities are identical for interfaces involving only fluids, the Shuttleworth effect demonstrates that this is not the case for most interfaces involving solids, since their surface energies change with strain. Crystalline materials are known to have strain-dependent surface energies, but in amorphous materials, such as polymeric glasses and elastomers, the strain dependence is debated due to a dearth of direct measurements. Here, we utilize contact angle measurements on strained glassy and elastomeric solids to address this matter. We show conclusively that interfaces involving polymeric glasses exhibit strain-dependent surface energies, and give strong evidence for the absence of such a dependence for incompressible elastomers. The results provide fundamental insight into our understanding of the interfaces of amorphous solids and their interaction with contacting liquids.

Whether or not amorphous solids exhibit strain-dependent surface energies, like those of crystalline materials, is still a matter of debate. Here, Schulman et al. monitor the contact angle of droplets on strained polymeric glasses and elastomers, which directly probes energy variation at the interfaces.

Effect of large deformation and surface stiffening on the transmission of a line load on a neo-Hookean half space

A line force acting on a soft elastic solid, say due to the surface tension of a liquid drop, can cause significant deformation and the formation of a kink close to the point of force application. Analysis based on linearized elasticity theory shows that sufficiently close to its point of application, the force is borne entirely by the surface stress, not by the elasticity of the substrate; this local balance of three forces is called Neumann's triangle. However, it is not difficult to imagine realistic properties for which this force balance cannot be satisfied. For example, if the line force corresponds to surface tension of water, the numerical values of (unstretched) solid-vapor and solid-liquid surface stresses can easily be such that their sum is insufficient to balance the applied force. In such cases conventional (or naïve) Neumann's triangle of surface forces must break down. Here we study how force balance is rescued from the breakdown of naïve Neumann's triangle by a combination of (a) large hyperelastic deformations of the underlying bulk solid, and (b) increase in surface stress due to surface elasticity (surface stiffening). For a surface with constant surface stress (no surface stiffening), we show that the linearized theory remains accurate if the applied force is less than about 1.3 times the solid surface stress. For a surface in which the surface stress increases linearly with the surface stretch, we find that the Neumann's triangle construction works well as long as we replace the constant surface stress in the naïve Neumann triangle by the actual surface stress underneath the line load.

Surface elastic constants of a soft solid

Solid interfaces have intrinsic elasticity. However, in most experiments, this is obscured by bulk stresses. Through microscopic observations of the contact-line geometry of a partially wetting droplet on an anisotropically stretched substrate, we measure two surface-elastic constants that quantify the linear dependence of the surface stress of a soft polymer gel on its strain. With these two parameters, one can predict surface stresses for general deformations of the material in the linear-elastic limit.

Surface Stress and Surface Tension in Polymeric Networks

Understanding of how surface properties could change upon deformation is of paramount importance for controlling adhesion, friction, and lubrication of soft polymeric materials (i.e., networks and gels). Here, we use a combination of the theoretical calculations and coarse-grained molecular dynamics simulations to study surface stress dependence on deformation in films made of soft and rigid polymeric networks. Simulations have shown that films of polymeric networks could demonstrate surface properties of both polymer melts and elastic solids depending on their deformation. In particular, at small film deformations the film surface stress ϒ is equal to the surface tension obtained at zero film strains, γ₀, and surface properties of networks are similar to those of polymer melts. The surface stress begins to show a strain dependence when the film deformation ratio λ approaches its maximum possible value λ_max corresponding to fully stretched network strands without bond deformations. In the entire film deformation range the normalized surface stress ϒ(λ)/γ₀ is a universal function of the ratio λ/λ_max. Analysis of the simulation data at large film deformations points out that the significant increase in the surface stress can be ascribed to the onset of the bond deformation. In this deformation regime network films behave as elastic solids.