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

Phase separation dynamics in wetting ridges of polymer surfaces swollen with oils of different viscosities

When drops are placed on a sufficiently soft surface, the drop surface tension drives an out of plane deformation around the contact line (i.e., a wetting ridge). For soft elastomeric surfaces that are swollen with a liquid, capillarity from a drop can induce a phase separation in the wetting ridge. Using confocal microscopy, we study the dynamics of phase separation at the wetting ridge of glycerol drops on silicone elastomers, which are swollen with silicone oils of varying viscosity (i.e., molecular weight). We show that the viscosity of the swelling oil plays a large role in the oil separation size and separation rate. For networks swollen to near their maximum swelling (i.e., saturated), lower viscosity oil separates more and separates faster at early times compared to larger viscosity oil. During late-stage wetting, the growth rate of the separation is a function of viscosity and swelling ratio, which can be described by a simple diffusive model and a defined wetting ridge geometry. In this late-stage wetting, the higher viscosity oil evidently grows faster, likely because it is further from reaching equilibrium. Interestingly, the separated oil phase region grows with a nearly constant, geometrically similar shape. Understanding how phase separation occurs on swollen substrates should provide information on how to control drop spreading, sliding, adhesion, or friction on such surfaces.

Pinning Forces on the Omniphobic Dry, Liquid-Infused, and Liquid-Attached Surfaces

Omniphobic coatings effectively repelling water, oils, and other liquids are of great interest and have a broad number of applications including self-cleaning, anti-icing surfaces, biofouling protection, selective filtration, etc. To create such coatings, one should minimize the pinning force that resists droplet motion and causes contact angle hysteresis. The minimization of the free surface energy by means of the chemical modification of the solid surface is not enough to obtain a nonsticky slippery omniphobic surface. One should minimize the contact between the solid and the droplet. Besides coating the surface with flat polymer films, among the major approaches to create omniphobic coatings, one can reveal "lotus effect" textured coatings, slippery liquid-infused porous surfaces (SLIPS), and slippery omniphobic covalently attached liquid (SOCAL) coatings. It is possible to turn one surface type into other by texturizing, impregnating with liquids, or grafting flexible liquid-like polymer chains. There are a number of models describing the pinning force on surfaces, but the transitions between states with different wetting regimes remain poorly understood. At the same time, such studies can significantly broaden existing ideas about the physics of wetting, help to design coatings, and also contribute to the development of generalized models of the pinning force. Here we review the existing pinning force (contact angle hysteresis) models on various omniphobic substrates. Also, we discuss the current studies of the pinning force in the transitions between different wetting regimes.

Wetting on silicone surfaces

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

Understanding the Role of Loss Modulus of Viscoelastic Substrates in the Evaporation Dynamics of Sessile Drops

Viscoelastic properties of soft substrates play a crucial role in the evaporation dynamics of sessile drops. Recent studies have revealed that the modification of the viscoelastic properties of substrates changes the dynamics of the three-phase contact line, consequently affecting the evaporation behavior of sessile drops. Notably, these modifications occur without any noticeable changes to the substrate's wetting characteristics or surface topography. However, the individual role of storage (G') and loss (G″) moduli of substrates on drop evaporation dynamics remains unexplored. In this study, we investigate the evaporation dynamics of water drops on two groups of poly(dimethylsiloxane)-based viscoelastic substrates possessing either identical G' with varying G″ or identical G″ with varying G'. Our study reveals that on a substrate with constant shear modulus (G'), a reduction of an order of magnitude in loss modulus shifts the evaporation process from the constant contact radius mode to the constant contact angle mode. We hypothesize that this observed shift in behavior stems from the varying viscoelastic dissipation influenced by the plateau modulus and characteristic relaxation time of polymer gels. Our hypothesis is further supported from the observation that the evaporation process persists on the substrate with constant loss modulus (G″). Our study advances the current understanding of drop evaporation on soft substrates that may find potential applications involving soft composites, biological entities, tissue engineering, and wearable electronics.

Complexity in Wetting Dynamics

The spreading dynamics of a droplet of pure liquid deposited on a rigid, nonsoluble substrate has been extensively investigated. In a purely hydrodynamic description, the dynamics of the contact line is determined by a balance between the energy associated with the capillary driving force and the energy dissipated by the viscous shear in the liquid. This balance is expressed by the Cox-Voinov law, which relates the spreading velocity to the contact angle. More recently, complex situations have been examined in which dissipation and/or the driving force may be strongly modified, leading to sometimes spectacular changes in wetting dynamics. We review recent examples of effects at the origin of deviations from the hydrodynamic model, which may involve physical or chemical modifications of the substrate or of the wetting liquid, occurring at scales ranging from the molecular to the mesoscopic.

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.

Controllable Wetting Transitions on Photoswitchable Physical Gels

Softness plays a key role in the deformation of soft elastic substrates at the three-phase contact line, and the acting forces lead to the formation of a wetting ridge due to elastocapillarity. The change in wetting ridge and surface profiles at different softness has a great impact on the droplet behavior in different phenomena. Commonly used materials to study soft wetting are swollen polymeric gels or polymer brushes. These materials offer no possibility to change the softness on demand. Therefore, adjustable surfaces with tunable softness are highly sought-after to achieve on-demand transition between wetting states on soft surfaces. Here, we present a photorheological physical soft gel with adjustable stiffness based on the spiropyran photoswitch that shows the formation of wetting ridges upon droplet deposition. The presented photoswitchable gels allow the creation of reversibly switchable softness patterns with microscale resolution using UV light-switching of the spiropyran molecule. Gels with varying softness are analyzed, showing a decrease in the wetting ridge height at higher gel stiffness. Furthermore, wetting ridges before and after photoswitching are visualized using confocal microscopy, showing the transition in the wetting properties from soft wetting to liquid/liquid wetting.

Phase Separation in Wetting Ridges of Sliding Drops on Soft and Swollen Surfaces

Drops in contact with swollen, elastomeric substrates can induce a capillary mediated phase separation in wetting ridges. Using confocal microscopy, we visualize phase separation of oligomeric silicone oil from a cross-linked silicone network during steady-state sliding of water drops. We find an inverse relationship between the oil tip height and the drop sliding speed, which is rationalized by competing transport timescales of the oil molecules: separation rate versus drop-advection speed. Separation rates in highly swollen networks are as fast as diffusion in pure melts.

Spreading of droplets under various gravitational accelerations

We describe a setup to perform systematic studies on the spreading of droplets of complex fluids under microgravity conditions. Tweaking the gravitational acceleration under which droplets are deposited provides access to different regimes of the spreading dynamics, as quantified through the Bond number. In particular, microgravity allows us to form large droplets while remaining in the regime where surface tension effects and internal driving stresses are predominant over hydrostatic forces. The vip-drop² (visco-plastic droplets on the drop tower) experimental module provides a versatile platform to study a wide range of complex fluids through the deposition of axisymmetric droplets. The module offers the possibility to deposit droplets on a precursor layer, which can be composed of the same or a different fluid. Furthermore, it allows us to deposit four droplets simultaneously while conducting shadowgraphy on all of them and observing either the flow field (through particle image velocimetry) or the stress distribution inside the droplet in the case of stress birefringent fluids. It was developed for a drop tower catapult system, is designed to withstand a vertical acceleration of up to 30 times the Earth's gravitational acceleration in the downward direction, and is capable of operating remotely under microgravity conditions. We provide a detailed description of the module and an exemplary data analysis for droplets spreading on-ground and in microgravity.

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.

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.

Probing Liquid Drop Induced Deformation on Soft Solids Using Dual-Wavelength Reflection Interference Contrast Microscopy

A liquid drop resting on a soft solid deforms the surface at the three-phase contact line. The surface deformation, also called "wetting ridge", varies in size from nanoscales to microscales, depending on the elasticity and thickness of the soft layer. In this work, we probe how surface elasticity and coating thickness influences normal and tangential surface deformation profiles induced by a sessile liquid drop using dual-wavelength reflection interference contrast microscopy. Furthermore, we experimentally verify the appropriate characteristic length scale, which closely describes the ridge profiles on both thick and thin soft layers for two different surface elasticities.

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.

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.

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.

Model of spontaneous droplet transport on a soft viscoelastic substrate with nonuniform thickness

Dynamic wetting of droplets on soft solids has many industrial and biological applications which require an understanding of the underlying fluid transport mechanism. Here we study the case of a droplet on a viscoelastic substrate of variable thickness which is known to give rise to a spontaneous droplet transport. This phenomenon is known as droplet durotaxis and has been observed experimentally. Here we develop a model assuming a small linear gradient in substrate thickness to reveal the physical mechanism behind this transport phenomena. We show the variable thickness causes an asymmetric deformation along the drop contact line, which causes a variation in the contact angle. This generates a net driving force on the drop, causing it to move in the direction of higher thickness. The resulting drop velocity is determined by balancing the work done by the moving drop with the viscoelastic dissipation of the substrate (viscoelastic braking) and computed from a self-consistent model. We find our results to be in qualitative agreement to previously reported experimental findings.

Impact of substrate elasticity on contact angle saturation in electrowetting

The electrostatically assisted wettability enhancement of dielectric solid surfaces, commonly termed as electrowetting-on-dielectric (EWOD), facilitates many microfluidic applications due to simplicity and energy efficiency. The application of a voltage difference between a conductive droplet and an insulated electrode substrate, where the droplet sits, is enough for realizing a considerable contact angle change. The contact angle modification is fast and almost reversible; however it is limited by the well-known saturation phenomenon which sets in at sufficiently high voltages. In this work, we experimentally show and computationally support the effect of elasticity and thickness of the dielectric on the onset of contact angle saturation. We found that the effect of elasticity is important especially for dielectric thickness smaller than 10 μm and becomes negligible for thickness above 20 μm. We attribute our findings on the effect of the dielectric thickness on the electric field, as well as on the induced electric stresses distribution, in the vicinity of the three phase contact line. Electric field and electric stresses distribution are numerically computed and support our findings which are of significant importance for the design of soft materials based microfluidic devices.

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.

Multi-Analytical Framework to Assess the In Vitro Swallowability of Solid Oral Dosage Forms Targeting Patient Acceptability and Adherence

A lack of effective intervention in addressing patient non-adherence and the acceptability of solid oral dosage forms combined with the clinical consequences of swallowing problems in an ageing world population highlight the need for developing methods to study the swallowability of tablets. Due to the absence of suitable techniques, this study developed various in vitro analytical tools to assess physical properties governing the swallowing process of tablets by mimicking static and dynamic stages of time-independent oral transitioning events. Non-anatomical models with oral mucosa-mimicking surfaces were developed to assess the swallowability of tablets; an SLA 3D printed in vitro oral apparatus derived the coefficient of sliding friction and a friction sledge for a modified tensometer measured the shear adhesion profile. Film coat hydration and in vitro wettability was evaluated using a high-speed recording camera that provided quantitative measurements of micro-thickness changes, simulating static in vivo tablet–mucosa oral processing stages with artificial saliva. In order to ascertain the discriminatory power and validate the multianalytical framework, a range of commonly available tablet coating solutions and new compositions developed in our lab were comparatively evaluated according to a quantitative swallowability index that describes the mathematical relationship between the critical physical forces governing swallowability. This study showed that the absence of a film coat significantly impeded the ease of tablet gliding properties and formed chalky residues caused by immediate tablet surface erosion. Novel gelatin- and λ-carrageenan-based film coats exhibited an enhanced lubricity, lesser resistance to tangential motion, and reduced stickiness than polyvinyl alcohol (PVA)–PEG graft copolymer, hydroxypropyl methylcellulose (HPMC), and PVA-coated tablets; however, Opadry^® EZ possessed the lowest friction–adhesion profile at 1.53 a.u., with the lowest work of adhesion profile at 1.28 J/mm². For the first time, the in vitro analytical framework in this study provides a fast, cost-effective, and repeatable swallowability ranking method to screen the in vitro swallowability of solid oral medicines in an effort to aid formulators and the pharmaceutical industry to develop easy-to-swallow formulations.

Nanoscale Soft Wetting Observed in Co/Sapphire during Pulsed Laser Irradiation

Liquid drops on deformable soft substrates exhibit quite complicated wetting behavior as compared to those on rigid solid substrates. We report on a soft wetting behavior of Co nanoparticles (NPs) on a sapphire substrate during pulsed laser-induced dewetting (PLID). Co NPs produced by PLID wetted the sapphire substrate with a contact angle near 70°, which is in contrast to typical dewetting behavior of metal thin films exhibiting contact angles greater than 90°. In addition, a nanoscale γ-Al2O3 wetting ridge about 15 nm in size and a thin amorphous Al2O3 interlayer were observed around and beneath the Co NP, respectively. The observed soft wetting behavior strongly indicates that the sapphire substrate became soft and deformable during PLID. Moreover, the soft wetting was augmented under PLID in air due to the formation of a CoO shell, resulting in a smaller contact angle near 30°.

Contact line relaxation of sessile drops on PDMS surfaces: A methodological perspective

HYPOTHESIS

Characterization of contact angle hysteresis on soft surfaces is sensitive to the measurement protocol and might present adventitious time-dependencies. Contact line dynamics on solid surfaces is altered by the surface chemistry, surface roughness and/or surface elasticity. We observed a "slow" spontaneous relaxation of static water sessile drops placed on elastic surfaces. This unexpected drop motion reveals unresolved equilibrium configurations that may affect the observed values of contact angle hysteresis. Drop relaxation on deformable surfaces is partially governed by a viscoelastic dissipation located at the contact line.

EXPERIMENTS

In this work, we studied the natural relaxation of water drops formed on several smooth PDMS surfaces with different elastic moduli. We monitored in time the contact angle and contact radius of each drop. For varying the initial contact angle, we used the growing-shrinking drop method.

FINDINGS

We postulate that the so-called "braking effect", produced by the surface deformability, affects the contact line velocity and in consequence, the contact angle measurements. We conclude that the wetting properties of elastic surfaces should be properly examined with reliable values of contact angle measured after drop relaxation.

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.

The relationship between viscoelasticity and elasticity

Soft materials that are subjected to large deformations exhibit an extremely rich phenomenology, with properties lying in between those of simple fluids and those of elastic solids. In the continuum description of these systems, one typically follows either the route of solid mechanics (Lagrangian description) or the route of fluid mechanics (Eulerian description). The purpose of this review is to highlight the relationship between the theories of viscoelasticity and of elasticity, and to leverage this connection in contemporary soft matter problems. We review the principles governing models for viscoelastic liquids, for example solutions of flexible polymers. Such materials are characterized by a relaxation time λ, over which stresses relax. We recall the kinematics and elastic response of large deformations, and show which polymer models do (and which do not) correspond to a nonlinear elastic solid in the limit λ → ∞. With this insight, we split the work done by elastic stresses into reversible and dissipative parts, and establish the general form of the conservation law for the total energy. The elastic correspondence can offer an insightful tool for a broad class of problems; as an illustration, we show how the presence or absence of an elastic limit determines the fate of an elastic thread during capillary instability.

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.