Solid capillarity: when and how does surface tension deform soft solids?

Plastocapillarity: Partial and full Newtonian drop embedding into immiscible yield stress substrates

HYPOTHESIS

Recent advances have been made in elastocapillarity; reversible 3D deformations of solid substrates with low elastic moduli from the surface tension of deposited drops. This study explores permanent deformations caused by liquid drops on immiscible yield stress substrates. We hypothesize that the substrate's rheological properties play a major role in determining the shape and stability of the drop-substrate interface, and govern partial or full embedding into the substrate.

EXPERIMENTS

Substrate yield stress magnitudes are modified through altering the mixture ratios of petroleum jelly to paraffin oil. Water drops are deposited on substrates and deformation profiles of the deformed interface are quantified.

FINDINGS

Above a critical Bingham-Capillary number, which characterizes the ratio of yield stress magnitude to surface tension, deposited water drops deform the substrate surface permanently, but minimally. Below this value, drops become increasingly embedded as the substrate yield stress magnitude decreases, with larger indentation depths and increased circumferential ridge heights. With sufficiently low yield stress magnitudes, where surface tension forces dominate over yield stress forces, the plastically deformed ridges fully encapsulate the liquid drop surface, resulting in full drop embedding within the substrate. These results advance knowledge of interfacial wetting on soft yield stress substrates and has implications for binary fluids, functional materials, and new drug delivery systems.

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.

Enhancement of the Rate of Surface Reactions by the Elastocapillary Effect

We examined the effect of deformability of a solid substrate on the kinetics of a surface reaction that occurs between chemical species present in it and a liquid dispensed on it. In particular, we have dispensed aqueous solutions of gold and silver salt as sessile drops or as a liquid pool on a cross-linked film of poly(dimethylsiloxane) (PDMS). The PDMS surface contains organosilane (SiH), which reduces the salt, producing metallic nanoparticles at the solid-liquid interface. These experiments reveal that, for a sufficiently soft solid, the reaction proceeds about three times faster in the drop mode than in the pool mode. The reaction conditions in both cases remain exactly identical except that, for the drop, the vertical component of the liquid surface tension deforms the solid substrate at the three-phase contact line. We have estimated the solid-liquid and solid-air interfacial energy, which along with the surface energy of the liquid gives an estimate of excess free energy. This energy is found to be different for the drop and pool modes. By considering that this excess free energy decreases the activation energy barrier for the reaction, we have shown that the reaction rate constant in the drop mode should indeed exceed that in the pool mode by about three times.

A dive into the bath: embedded 3D bioprinting of freeform in vitro models

Designing functional, vascularized, human scale in vitro models with biomimetic architectures and multiple cell types is a highly promising strategy for both a better understanding of natural tissue/organ development stages to inspire regenerative medicine, and to test novel therapeutics on personalized microphysiological systems. Extrusion-based 3D bioprinting is an effective biofabrication technology to engineer living constructs with predefined geometries and cell patterns. However, bioprinting high-resolution multilayered structures with mechanically weak hydrogel bioinks is challenging. The advent of embedded 3D bioprinting systems in recent years offered new avenues to explore this technology for in vitro modeling. By providing a stable, cell-friendly and perfusable environment to hold the bioink during and after printing, it allows to recapitulate native tissues' architecture and function in a well-controlled manner. Besides enabling freeform bioprinting of constructs with complex spatial organization, support baths can further provide functional housing systems for their long-term in vitro maintenance and screening. This minireview summarizes the recent advances in this field and discuss the enormous potential of embedded 3D bioprinting technologies as alternatives for the automated fabrication of more biomimetic in vitro models.

Mechanical response of a thick poroelastic gel in contactless colloidal-probe rheology

When a rigid object approaches a soft material in a viscous fluid, hydrodynamic stresses arise in the lubricated contact region and deform the soft material. The elastic deformation modifies in turn the flow, hence generating a soft-lubrication coupling. Moreover, soft elastomers and gels are often porous. These materials may be filled with solvent or uncrosslinked polymer chains, and might be permeable to the surrounding fluid, which further complexifies the description. Here, we derive the point-force response of a semi-infinite and permeable poroelastic substrate. Then, we use this fundamental solution in order to address the specific poroelastic lubrication coupling associated with contactless colloidal-probe methods. In particular, we derive the conservative and dissipative components of the force associated with the oscillating vertical motion of a sphere close to the poroelastic substrate. Our results may be relevant for dynamic surface force apparatus and contactless colloidal-probe atomic force microscopy experiments on soft, living and/or fragile materials, such as swollen hydrogels and biological membranes.

Simple method to measure rheological properties of soft surfaces by a micro-needle contact

We developed a simple method to investigate rheological properties of soft surfaces, such as polymeric liquids and colloidal suspensions, by capturing the images of a metal micro-needle inserted into the surface. At contact, a meniscus-like deformation is formed on the surface. By relating the shape of the deformation to the balance of applied forces, local elasticity and viscosity just inside the surface are obtained. With a facile setup and rapid measurement process, the present method can be implemented to variety of systems, for instance, drying sessile drops and small volume of liquid confined in a capillary.

Direct force measurement of microscopic droplets pulled along soft surfaces

When a droplet is placed on a soft surface, surface tension deforms the substrate, creating a capillary ridge. We study how the motion of the ridge dissipates energy in microscopic droplets. Using a micropipette based method, we are able to simultaneously image and measure forces on a microscopic droplet moving at a constant speed along a soft film supported on a rigid substrate. Changing the thickness of the thin film tunes the effective stiffness of the substrate. Thus we can control the ridge size without altering the surface chemistry. We find that the dissipation depends strongly on the film thickness, decreasing monotonically as effective stiffness increases. This monotonic trend is beyond the realm of small deformation theory, but can be explained with a simple scaling analysis.

Elastic deformation of soft substrates occurs upon wetting, yet it is challenging to follow its dynamics at a microscale. Khattak et al. show that the force required to pull a droplet along a soft surface decreases monotonically as the film thickness decreases and explain the phenomenon using a scaling analysis.

Micrometer-size double-helical structures from phospholipid-modified carbon nanotubes

Biomolecular self-assembly plays a key role in the life system. Herein, double-helical phospholipid-modified carbon nanotube structures were constructed via the self-assembly of phospholipids on carbon nanotubes. These micrometer size spring structures may find potential applications in biocompatible microrobots.

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.

Elastocapillary interaction between a long rectangular membrane and a liquid drop

We report elastocapillary interaction between a long rectangular membrane fixed along its central axis and a liquid drop dispensed at one of its ends. The introduction of the drop results in the elastocapillary-driven wrapping of the membrane along its width and a concomitant flow in the resulting conduit along its length. Depending upon the drop size (d) and capillary length scale (L_c), we identified general criteria for achieving complete wrapping of the membrane in the dry state from energy considerations. For small droplets satisfying d ≲ L_c, we find that the critical membrane length (W_c) required for complete wrapping is proportional to the elastocapillary length scale (L_ec). In the case of large droplets with d > L_c, the wrapping behavior depends on the ratio of membrane width to elastocapillary length scale (W/L_ec) and the ratio of capillary length scale to the elastocapillary length scale (L_c/L_ec). Our study suggests that the critical membrane width for complete wrapping is smaller in the wet state compared to that in the dry state, which can be attributed to the existence of a transmembrane pressure in the wet state. The effect of membrane thickness and width and drop volume on the length and cross-section of the wrapped conduit and attached width of the wrapped membrane is studied. For small droplets, the resulting elastocapillary flow exhibits an inertial regime at small times, followed by a Washburn regime at intermediate times, and finally an inertial regime, and for large droplets, only an inertial regime is observed throughout.

Droplet evaporation on soft solid substrates

Droplet evaporation on soft solid substrates is relevant to applications such as fabrication of microlenses and controlled particle deposition. Here, we develop a lubrication-theory-based model to advance fundamental understanding of the important limiting case of a planar droplet evaporating on a linear viscoelastic solid. A set of partial differential equations describing the time evolution of the liquid-air and liquid-solid interfaces is derived and solved with a finite-difference method. A disjoining-pressure/precursor-film approach is used to describe contact-line motion, and the one sided model is used to describe solvent evaporation. Parametric studies are conducted to investigate the effect of solid properties (thickness, viscosity, shear modulus, wettability) and evaporation rate on droplet dynamics. Our results indicate that softer substrates speed up droplet evaporation due to prolonged pinning of the contact line. Results from our model are able to qualitatively reproduce some key trends observed in experiments. Due to its systematic formulation, our model can readily be extended to more complex situations of interest such as evaporation of particle-laden droplets on soft solid substrates.

Plateau-Rayleigh instability in a soft viscoelastic material

A soft cylindrical interface endowed with surface tension can be unstable to wavy undulations. This is known as the Plateau-Rayleigh instability (PRI) and for solids the instability is governed by the competition between elasticity and capillarity. A dynamic stability analysis is performed for the cases of a soft (i) cylinder and (ii) cylindrical cavity assuming the material is viscoelastic with power-law rheology. The governing equations are made time-independent through the Laplace transform from which a solution is constructed using displacement potentials. The dispersion relationships are then derived, which depend upon the dimensionless elastocapillary number, solid Deborah number, and compressibility number, and the static stability limit, critical disturbance, and maximum growth rate are computed. This dynamic analysis recovers previous literature results in the appropriate limits. Elasticity stabilizes and compressibility destabilizes the PRI. For an incompressible material, viscoelasticity does not affect stability but does decrease the growth rate and shift the critical wavenumber to lower values. The critical wavenumber shows a more complex dependence upon compressibility for the cylinder but exhibits a predictable trend for the cylindrical cavity.

A thin-film model for droplet spreading on soft solid substrates

The spreading of droplets on soft solid substrates is relevant to applications such as tumor biophysics and controlled droplet condensation and evaporation. In this paper, we apply lubrication theory to advance fundamental understanding of the important limiting case of spreading of a planar droplet on a linear viscoelastic solid. The contact-line region is described by a disjoining-pressure/precursor-film approach, and nonlinear evolution equations describing how the liquid-air and liquid-solid interfaces evolve in space and time are derived and solved numerically. Parametric studies are conducted to investigate the effects of solid thickness, viscosity, shear modulus, and wettability on droplet spreading. Softer substrates are found to speed up spreading for perfectly wetting droplets but slow down spreading for partially wetting droplets. For perfectly wetting droplets, faster spreading is a result of more liquid being pumped toward the contact line due to a larger liquid-film thickness there arising from the repulsive component of the disjoining pressure. In contrast, slower spreading of partially wetting droplets is a result of less liquid being pumped toward the contact line due to a smaller liquid-film thickness there arising from the attractive component of the disjoining pressure. The model predictions for partially wetting droplets are qualitatively consistent with experimental observations, and allow us to disentangle the effects of substrate deformability and wettability on droplet spreading. Due to its systematic formulation, our model can readily be extended to more complex situations involving multiple droplets, substrate inclination, and droplet phase changes.

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.

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.

Effect of Polymer Volume Fraction on Fracture Initiation in Soft Gels at Small Length Scales

The influence of polymer volume fraction, ϕ_v on fracture initiation via puncture is studied in self-assembled triblock copolymer gels. Spherically tipped indenters of radii varying over a wide range were used to characterize puncture at length scales on the same order of magnitude as the elasto-capillary length (∼μm) and significantly below the elasto-fracture length (∼mm) for ϕ_v = 0.12-0.53. Critical energy release rate, G_c for ϕ_v = 0.12-0.30 was found to be in agreement with the predicted scaling from the classical Lake-Thomas model modified for gel fracture via the failure mechanism of chain pull-out and plastic yielding of micelles (G_c ∼ ϕ_v^2.2). Interestingly, we demonstrate that fracture initiation energy, Γ_o, from puncture scales as Γ_o ∼ ϕ_v, thus, indicating the role played by different fundamental mechanisms governing fracture initiation in soft gels. Additionally, gels with ϕ_v = 0.53 show deviation from experimental scalings for G_c and Γ_o, likely due to a change in micellar morphology leading to anomalous fracture behavior.

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.

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.

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.

Hierarchically Patterned Elastomeric and Thermoplastic Polymer Films through Nanoimprinting and Ultraviolet Light Exposure

The surface relief structure of polymer films over large areas can be controlled by combining nanoscale imprinting and microscale ultraviolet-ozone (UVO) radiation, resulting in hierarchical structured surfaces. First, nanoscale patterns were formed by nanoimprinting elastomer [poly(dimethylsiloxane) (PDMS)] films with a pattern on a digital video disk. Micron-scale patterns were then superimposed on the nanoimprinted PDMS films by exposing them to ultraviolet radiation in oxygen (UVO) through a transmission electron microscopy grid mask having variable microscale patterning. UVO exposure leads to conversion and densification of PDMS to SiO x , leading to micron height relief features that follow a linear scaling relation with pattern dimension. Further, the pattern scopes are shown to collapse into a master curve by normalized feature values. Interestingly, these relief structures preserve the nanoscale features. In this paper, the influence of the self-limiting PDMS densification, wall stress at the boundary of micro-depression, and UVO exposure energy is studied in control of the micro-depression scale. This simple two-step imprinting process involving both nanoimprinting and UV radiation allows for facile fabrication of the dimension adjustable micro-nano hierarchically structures not only on elastomer films but also on thermoplastic polymer films. Coarse-grained molecular dynamics simulations were performed to correlate the surface tension and elastic properties of polymeric materials to the deformation of the pattern structure.

Extraction of Silicone Uncrosslinked Chains at Air-Water-Polydimethylsiloxane Triple Lines

Silicone elastomers such as polydimethylsiloxane (PDMS) are convenient materials routinely used in laboratories that combine ease of preparation, flexibility, transparency, and gas permeability. However, these elastomers are known to contain a small fraction of uncrosslinked low-molecular-weight oligomers, the effects of which are not completely understood, particularly when used in contact with liquids. Here, we show that triple lines involving air, water, and PDMS elastomers are responsible for the contamination of water-air interfaces by uncrosslinked silicone oligomers through a capillarity-induced extraction mechanism. We investigate both the case of static and moving contact lines and study various geometries ranging from partially immersed PDMS plates to water droplets or air bubbles deposited on PDMS plates, all involving air-water-elastomer triple lines. We demonstrate experimentally that the contamination timescale is strikingly shorter for moving contact lines than in the static case. Eventually, we propose a simple poroelastic model capturing the main features of contamination observed in experiments.

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.

Tailored topography: a novel fabrication technique using an elasticity gradient

A facile methodology to create a wrinkled surface with a tailored topography is presented herein. The dependency of the elasticity of poly(dimethyl)siloxane (PDMS) on the curing temperature has been exploited to obtain a substrate with an elasticity gradient. The temperature gradient across the length of PDMS is created by a novel set-up consisting of a metal and insulator connected to a heater and the highest usable (no degradation of PDMS) temperature gradient is used. The time-dependent temperature distributions along the substrate are measured and the underlying physics of the dependence of the PDMS elasticity on the curing temperature is addressed. The PDMS substrate with the elasticity gradient is first stretched and subsequently oxidized by oxygen plasma. Upon relaxation, an ordered wrinkled surface with continuously varying wavelength and amplitude along the length of PDMS is obtained. The extent of hydrophobicity recovery of this plasma oxidized PDMS with varying elasticity has been studied. The change in the wavelength and amplitude of the regular patterns on the substrate can be controlled by varying operational parameters like applied pre-strain, plasma power and the heater temperature. It has been found that the spatial distributions of the topography and the hydrophobicity collectively decide the resultant wettability of the substrate. Such surfaces with gradients in the substructure dimensions demonstrate different wetting characteristics that may lead to a wide gamut of applications including droplet movement, cell adhesion and proliferation, diffraction grating etc.

The contact mechanics challenge: tribology meets soft matter

In the fall of 2015, Martin Müser suggested a Contact Mechanics Challenge for the Tribology community. The challenge was an ambitious effort to compare a wide variety of theoretical and computational contact-mechanics approaches, and involved researchers voluntarily tackling the same hypothetical contact problem. The result is an impressive collection of innovative approaches - including a surprise experimental effort - that highlight the continuing importance of surface contact mechanics and the challenges of solving these large-scale problems. Here, we describe how the Contact Mechanics Challenge also reveals exciting opportunities for the Soft Matter community to engage intensely with classical and emerging problems in tribology, surface science, and contact mechanics.

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.

Auxiliary soft beam for the amplification of the elasto-capillary coiling: Towards stretchable electronics

A flexible fiber carrying a liquid drop may coil inside the drop thereby creating a drop-on-fiber system with an ultra-extensible behavior. During compression, the excess fiber is spooled inside the droplet and capillary forces keep the system taut. During subsequent elongation, the fiber is gradually released and if a large number of spools is uncoiled a high stretchability is achieved. This mechanical behaviour is of interest for stretchable connectors but information, may it be electronic or photonic, usually travels through stiff functional materials. These high Young's moduli, leading to large bending rigidity, prevent in-drop coiling. Here we overcome this limitation by attaching a beam of soft elastomer to the functional fiber, thereby creating a composite system which exhibits in-drop coiling and carries information while being ultra-extensible. We present a simple model to explain the underlying mechanics of the addition of the soft beam and we show how it favors in-drop coiling. We illustrate the method with a two-centimeter long micronic PEDOT:PSS conductive fiber joined to a PVS soft beam, showing that the system conveys electricity throughout a 1900% elongation.

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.

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.

Dropwise Condensation on Soft Hydrophobic Coatings

Promoting dropwise condensation (DWC) could improve the efficiency of many industrial systems. Consequently, a lot of effort has been dedicated to finding durable materials that could sustainably promote DWC as well as finding routes to enhance the heat transfer rate during this phase change process. Motivated by previous reports of substrate softening increasing droplet nucleation rate, here we investigated how mechanical properties of a substrate impact relevant droplet-surface interactions and DWC heat transfer rate. Specifically, we experimentally quantified the effect of hydrophobic elastomer's shear modulus on droplet nucleation density and shedding radius. To quantify the impact of substrate softening on heat transfer through individual droplets, we combined analytical solution of elastomer deformation induced by droplets with finite element modeling of the heat transfer process. By substituting these experimentally and theoretically derived values into DWC heat transfer model, we quantified the compounding effect of the substrate's mechanical properties on the overall heat transfer rate. Our results show that softening of the substrates below a shear modulus of 500 kPa results in a significant reduction in the condensation heat transfer rate. This trend is primarily driven by additional thermal resistance of the liquid posed by depression of the soft substrate.

Direct measurement of strain-dependent solid surface stress

Surface stress, also known as surface tension, is a fundamental material property of any interface. However, measurements of solid surface stress in traditional engineering materials, such as metals and oxides, have proven to be very challenging. Consequently, our understanding relies heavily on untested theories, especially regarding the strain dependence of this property. Here, we take advantage of the high compliance and large elastic deformability of a soft polymer gel to directly measure solid surface stress as a function of strain. As anticipated by theoretical work for metals, we find that the surface stress depends on the strain via a surface modulus. Remarkably, the surface modulus of our soft gels is many times larger than the zero-strain surface tension. This suggests that surface stresses can play a dominant role in solid mechanics at larger length scales than previously anticipated.Solid surface stress is a fundamental property of solid interfaces. Here authors measure the solid surface stress of a gel, and show its dependence on surface strain through a surface modulus.

Withdrawing a solid from a bath: How much liquid is coated?

A solid withdrawn from a liquid bath entrains a film. In this review, after recalling the predictions and results for pure Newtonian liquids coated on simple solids, we analyze the deviations to this ideal case exploring successively three potential sources of complexity: the liquid-air interface, the bulk rheological properties of the liquid and the mechanical or chemical properties of the solid. For these different complexities, we show that significant effects on the film thickness are observed experimentally and we summarize the theoretical analysis presented in the literature, which attempt to rationalize these measurements.

Drop-on-coilable-fibre systems exhibit negative stiffness events and transitions in coiling morphology

We investigate the mechanics of elastic fibres carrying liquid droplets. In such systems, buckling may localize inside the drop cavity if the fibre is thin enough. This so-called drop-on-coilable-fibre system exhibits a surprising liquid-like response under compression and a solid-like response under tension. Here we analyze this unconventional behavior in further detail and find theoretical, numerical and experimental evidence of negative stiffness events. We find that the first and main negative stiffness regime owes its existence to the transfer of capillary-stored energy into mechanical curvature energy. The following negative stiffness events are associated with changes in the coiling morphology of the fibre. Eventually coiling becomes tightly locked into an ordered phase where liquid and solid deformations coexist.

Liquid droplets on a free-standing glassy membrane: Deformation through the glass transition

In this study, micro-droplets are placed on thin, glassy, free-standing films where the Laplace pressure of the droplet deforms the free-standing film, creating a bulge. The film's tension is modulated by changing temperature continuously from well below the glass transition into the melt state of the film. The contact angle of the liquid droplet with the planar film as well as the angle of the bulge with the film are measured and found to be consistent with the contact angles predicted by a force balance at the contact line.

Electro-elastocapillary Rayleigh-plateau instability in dielectric elastomer films

We demonstrate, using both finite element simulations and a linear stability analysis, the emergence of an electro-elastocapillary Rayleigh-plateau instability in dielectric elastomer (DE) films under 2D, plane strain conditions. When subject to an electric field, the DEs exhibit a buckling instability for small elastocapillary numbers. For larger elastocapillary numbers, the DEs instead exhibit the Rayleigh-plateau instability. The stability analysis demonstrates the critical effect of the electric field in causing the Rayleigh-plateau instability, which cannot be induced solely by surface tension in DE films. Overall, this work demonstrates the effects of geometry, boundary conditions, and multi-physical coupling on a new example of Rayleigh-plateau instability in soft solids.

Liquid Droplets Act as "Compass Needles" for the Stresses in a Deformable Membrane

We examine the shape of droplets atop deformable thin elastomeric films prepared with an anisotropic tension. As the droplets generate a deformation in the taut film through capillary forces, they assume a shape that is elongated along the high tension direction. By measuring the contact line profile, the tension in the membrane can be completely determined. Minimal theoretical arguments lead to predictions for the droplet shape and membrane deformation that are in excellent agreement with the data. On the whole, the results demonstrate that droplets can be used as probes to map out the stress field in a membrane.

Geometry-Driven Folding of a Floating Annular Sheet

Predicting the large-amplitude deformations of thin elastic sheets is difficult due to the complications of self contact, geometric nonlinearities, and a multitude of low-lying energy states. We study a simple two-dimensional setting where an annular polymer sheet floating on an air-water interface is subjected to different tensions on the inner and outer rims. The sheet folds and wrinkles into many distinct morphologies that break axisymmetry. These states can be understood within a recent geometric approach for determining the gross shape of extremely bendable yet inextensible sheets by extremizing an appropriate area functional. Our analysis explains the remarkable feature that the observed buckling transitions between wrinkled and folded shapes are insensitive to the bending rigidity of the sheet.

Elasto-electro-capillarity: drop equilibrium on a charged, elastic solid

We present a study here on elasto-electro-capillarity - for the first time, the matter of drop equilibrium on a soft (elastic and incompressible) and charged solid has been studied. Charges on the elastic solid induce an electric double layer or EDL at the solid-drop interface. Our analysis accounts for the electrostatic wetting contribution of the EDL in the overall energy balance. Our results reveal that (a) with an increase in "softness", the equilibrium solid-liquid contact angles show transition from the EDL-modified Young's law (rigid limit) to the EDL-modified Neumann's law (soft limit); (b) the EDL effects invariably enhance solid deformation and lower the apparent contact angle made by the drop with the undeformed solid; (c) the solid contact angles increase and the cusp made by the deformed solid undergoes enhanced rotation due to the EDL effects; and (d) the EDL effects are more prominent for the case where the solid-vapor surface energy exceeds the solid-liquid surface energy. The fact that the EDL effects invariably decrease the overall wetting energy of the system, thereby supporting a larger increase in the elastic energy associated with a larger solid deformation, explains all these findings and establishes that surface charges enhance the "softness" of a soft surface in the context of elastocapillarity.

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.

Droplet impact on soft viscoelastic surfaces

In this work, we experimentally investigate the impact of water droplets onto soft viscoelastic surfaces with a wide range of impact velocities. Several impact phenomena, which depend on the dynamic interaction between the droplets and viscoelastic surfaces, have been identified and analyzed. At low We, complete rebound is observed when the impact velocity is between a lower and an upper threshold, beyond which droplets are deposited on the surface after impact. At intermediate We, entrapment of an air bubble inside the impinging droplets is found on soft surfaces, while a bubble entrapment on the surface is observed on rigid surfaces. At high We, partial rebound is only identified on the most rigid surface at We≳92. Rebounding droplets behave similarly to elastic drops rebounding on superhydrophobic surfaces and the impact process is independent of surface viscoelasticity. Further, surface viscoelasticity does not influence drop spreading after impact-as the surfaces behave like rigid surfaces-but it does affect drop recoiling. Also, the postimpact drop oscillation on soft viscoelastic surfaces is influenced by dynamic wettability of these surfaces. Comparing sessile drop oscillation with a damped harmonic oscillator allows us to conclude that surface viscoelasticity affects the damping coefficient and liquid surface tension sets the spring constant of the system.

Tuning the Receding Contact Angle on Hydrogels by Addition of Particles

Control of the swelling, chemical functionalization, and adhesivity of hydrogels are finding new applications in a wide range of material systems. We investigate experimentally the effect of adsorbed particles on hydrogels on the depinning of contact lines. In our experiments, a water drop containing polystyrene microspheres is deposited on a swelling hydrogel, which leads to the drop absorption and particle deposition. Two regimes are observed: a decreasing drop height with a pinned contact line followed by a receding contact line. We show that increasing the particles concentration increases the duration of the first regime and significantly decreases the total absorption time. The adsorbed particles increase the pinning force at the contact line. Finally, we develop a method to measure the receding contact angle with the consideration of the hydrogel swelling.