Fluid droplets can be induced to move over rigid or flexible surfaces under external or body forces. We describe the effect of variations in material properties of a flexible substrate as a mechanism for motion. In this paper, we consider a droplet placed on a substrate with either a stiffness or surface energy gradient, and consider its potential for motion via coupling to elastic deformations of the substrate. In order to clarify the role of contact angles and to obtain a tractable model, we consider a two-dimensional droplet. The gradients in substrate material properties give rise to asymmetric solid deformation and to unequal contact angles, thereby producing a force on the droplet. We then use a dynamic viscoelastic model to predict the resulting dynamics of droplets. Numerical results quantifying the effect of the gradients establish that it is more feasible to induce droplet motion with a gradient in surface energy. The results show that the magnitude of elastic modulus gradient needed to induce droplet motion exceeds experimentally feasible limits in the production of soft solids and is therefore unlikely as a passive mechanism for cell motion. In both cases, of surface energy or elastic modulus, the threshold to initiate motion is achieved at lower mean values of the material properties.
We study equilibrium properties of polymer films and droplets on a solid substrate employing particle-based simulation techniques (Molecular Dynamics) and a continuum description. Parameter-passing techniques are explored that facilitate a detailed comparison of the two models. In particular, the liquid-vapor, solid-liquid and solid-vapor interface tensions, and the Derjaguin or disjoining pressure are determined by Molecular Dynamics simulations. This information is then introduced into continuum descriptions accounting for (i) the full curvature and (ii) a long-wave approximation of the curv…
Droplets deform soft substrates near their contact lines. Using confocal microscopy, we measure the deformation of silicone gel substrates due to glycerol and fluorinated-oil droplets for a range of droplet radii and substrate thicknesses. For all droplets, the substrate deformation takes a universal shape close to the contact line that depends on liquid composition, but is independent of droplet size and substrate thickness. This shape is determined by a balance of interfacial tensions at the contact line and provides a novel method for direct determination of the surface stresses of soft sub…
We investigated the electrostatic behavior of ferroelectric liquid droplets exposed to the pyroelectric field of a lithium niobate ferroelectric crystal substrate. The ferroelectric liquid is a nematic liquid crystal in which almost complete polar ordering of the molecular dipoles generates an internal macroscopic polarization locally collinear to the mean molecular long axis. Upon entering the ferroelectric phase by reducing the temperature from the nematic phase, the liquid crystal droplets become electromechanically unstable and disintegrate by the explosive emission of fluid jets. These je…
We demonstrate spontaneous bidirectional motion of droplets on liquid infused surfaces in the presence of a topographical gradient, in which the droplets can move either toward the denser or the sparser solid fraction area. Our analytical theory explains the origin of this bidirectional motion. Furthermore, using both lattice Boltzmann simulations and experiments, we show that the key factor determining the direction of motion is the wettability difference of the droplet on the solid surface and on the lubricant film. The bidirectional motion is shown for various combinations of droplets and l…
Gravitational settling of a droplet in air onto a soft substrate is a ubiquitous event relevant to many natural processes and engineering applications. We study this phenomenon by developing a three-phase lubrication model of droplet settling onto a solid substrate coated by a thin soft layer represented by a viscous film, an elastic compressible layer and an elastic sheet supported by a viscous film. By combining scaling analysis, analytical methods, and numerical simulations we elucidate how the resulting droplet dynamics is affected by the nature of the soft layer. We show that these soft l…
A range of technologies require the directed motion of nanoscale droplets on solid substrates. A way of realizing this effect is durotaxis, whereby a stiffness gradient of a substrate can induce directional motion without requiring an energy source. Here, we report on the results of extensive molecular dynamics investigations of droplets on a surface with varying stiffness. We find that durotaxis is enhanced by increasing the stiffness gradient and, also, by increased wettability of the substrate, in particular, when droplet size decreases. We anticipate that our study will provide further ins…
We show, using molecular dynamics simulations, that a two-dimensional Lennard-Jones solid is subject to droplet fluctuations characterized by {\em non-affine} deviations from local crystallinity. The fraction of particles in these droplets increases as the mean density of the solid decreases and approaches $\approx 20$% of the total number in the vicinity of the fluid-solid phase boundary. We monitor the geometry, local equation of state, density correlations and van Hove functions of these droplets and show that some of these droplets are fluid-like and compact, while some are glassy and stri…
The contact line of a liquid drop on a solid exerts a nanometrically sharp surface traction. This provides an unprecedented tool to study highly localised 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 …
The complicated dynamics of the contact line of a moving droplet on a solid substrate often hamper the efficient modeling of microfluidic systems. In particular, the selection of the effective boundary conditions, specifying the contact line motion, is a controversial issue since the microscopic physics that gives rise to this displacement is still unknown. Here, a sharp interface, continuum-level, novel modeling approach, accounting for liquid/solid micro-scale interactions assembled in a disjoining pressure term, is presented. By following a unified conception (the model applies both to the …
The equilibrium contact angle of a droplet resting on a solid substrate can reveal essential properties of the solid's surface. However, when the motion of a droplet on a surface shows significant hysteresis, it is generally accepted that the solid's equilibrium properties cannot be determined. Here, we describe a method to measure surface tensions of soft solids with strong wetting hysteresis. With independent knowledge of the surface tension of the wetting fluid and the linear-elastic response of the solid, the solid deformations under the contact line and the contact angle of a single dropl…
From bone and wood to concrete and carbon fibre, composites are ubiquitous natural and engineering materials. Eshelby's inclusion theory describes how macroscopic stress fields couple to isolated microscopic inclusions, allowing prediction of a composite's bulk mechanical properties from a knowledge of its microstructure. It has been extended to describe a wide variety of phenomena from solid fracture to cell adhesion. Here, we show experimentally and theoretically that Eshelby's theory breaks down for small liquid inclusions in a soft solid. In this limit, an isolated droplet's deformation is…
A finite element model was developed to compute the fluid flow inside a sessile evaporating droplet on hydrophilic substrate in ambient conditions. The evaporation is assumed as quasi-steady and the flow is considered as axisymmetric with a pinned contact line. The Navier-Stokes equations in cylindrical coordinates were solved inside the droplet. Galerkin weight residual approach and velocity pressure formulation was used to discretise the governing equations. Six node triangular mesh and quadratic shape functions were used to obtain higher accuracy solutions. Radial velocity profiles in axial…
When a droplet is brought in contact with an undercooled surface, it wets the substrate and solidifies at the same time. The interplay between the phase transition effects and the contact-line motion, leading to its arrest, remains poorly understood. Here we reveal the early solidification patterns and dynamics of spreading hexadecane droplets. Total internal reflection (TIR) imaging is employed to temporally and spatially resolve the early solidification behaviour. With this, we determine the conditions leading to the contact-line arrest. We quantify the overall nucleation behaviour, \textit{…
Young's law fails on soft solid and liquid substrates where there are substantial deformations near the contact line. On liquid substrates, this is captured by Neumann's classic analysis, which provides a geometrical construction for minimising the interfacial free energy. On soft solids, the total free energy includes an additional contribution from elasticity. A linear-elastic model incorporating an out-of-plane restoring force due to solid surface tension was recently shown to accurately predict the equilibrium shape of a thin elastic film due to a large sessile droplet. Here, we extend thi…
Understanding flow patterns and coupled transport phenomena during evaporation of droplets loaded with colloidal particles is central to design technical applications such as organizing proteins/DNA on a solid surface. We review recent reports on evaporating sessile droplets of colloidal suspensions on a solid surface. Starting from the classical mechanism of formation of a ring-like deposit, we discuss the influence of several problem parameters. Notably, thermal or solutal Marangoni effect, particle size, particle concentration, particle shape, substrate wettability, pH of the suspension etc…
Water droplet freezing is a common phenomenon in our daily life. In both natural scenarios and industrial production, different surface inclinations bring distinctive deformation and freezing dynamics to frozen droplets. We explore the freezing of pendent and sessile droplets at different Bond number regimes. The effect of gravity on the droplet freezing process is analyzed by considering droplet morphology, freezing front dynamics, and freezing time. It is found that gravity can significantly influence droplet freezing processes via shaping the initial droplet, resulting in the flattening or …
We predict and analyze the drying time of respiratory droplets from a COVID-19 infected subject, which is a crucial time to infect another subject. The drying of the droplet is predicted by diffusion-limited evaporation model for a sessile droplet placed on a partially-wetted surface with a pinned contact line. The variation of droplet volume, contact angle, ambient temperature, and humidity are considered. We analyze the chances of the survival of the viruses present in the droplet, based on the lifetime of the droplets in several conditions, and find that the chances of survival of the virus…
Viscous droplets impinging on soft substrates may exhibit several distinct behaviours including repeated bouncing, wetting, and hovering, i.e., spreading and retracting after impact without bouncing back or wetting. We experimentally study the conditions enabling these characteristic behaviours by systematically varying the substrate elasticity, impact velocity and the liquid viscosity. For each substrate elasticity, the transition to wetting is determined as the dependence of the Weber number We, which measures the droplet's kinetic energy against its surface energy, on the Ohnesorge number O…
The coalescence of droplets plays a crucial role in nature and modern technology. Various experimental and theoretical studies explored droplet dynamics in 3D and on 2D solid or liquid substrates. In this paper, we demonstrate the coalescence of isotropic droplets confined in thin quasi-2D liquids -- overheated smectic films. We observe the merging of micrometer-sized flat droplets using high-speed-imaging and analyse the shape transformations of the droplets on the timescale of milliseconds. Our studies reveal the scaling laws of the coalescence time, which exhibits a different dependence on …
We propose an approach to measure surface elastic constants of soft solids. Generally, this requires one to probe interfacial mechanics at around the elastocapillary length scale, which is typically microscopic. Deformations of microscopic droplets embedded in soft solids are particularly attractive, because they avoid intrinsic nonlinearities associated with previous experiments such as the equilibrium of contact lines and the relaxation of patterned surfaces. We derive analytical solutions for the shape of droplets under uniaxial deformation and for the radius of droplets upon hydrostatic in…
We study chemically driven running droplets on a partially wetting solid substrate by means of coupled evolution equations for the thickness profile of the droplets and the density profile of an adsorbate layer. Two models are introduced corresponding to two qualitatively different types of experiments described in the literature. In both cases an adsorption or desorption reaction underneath the droplets induces a wettability gradient on the substrate and provides the driving force for droplet motion. The difference lies in the behavior of the substrate behind the droplet. In case I the substr…
We experimentally investigated the splashing of dense suspension droplets impacting a solid surface, extending prior work to the regime where the viscosity of the suspending liquid becomes a significant parameter. The overall behavior can be described by a combination of two trends. The first one is that the splashing becomes favored when the kinetic energy of individual particles at the surface of a droplet overcomes the confinement produced by surface tension. This is expressed by a particle-based Weber number $We_p$. The second is that splashing is suppressed by increasing the viscosity of …
The state-of-the-art deep learning-based object recognition YOLO algorithm and object tracking DeepSORT algorithm are combined to analyze digital images from fluid dynamic simulations of multi-core emulsions and soft flowing crystals and to track moving droplets within these complex flows. The YOLO network was trained to recognize the droplets with synthetically prepared data, thereby bypassing the labor-intensive data acquisition process. In both applications, the trained YOLO + DeepSORT procedure performs with high accuracy on the real data from the fluid simulations, with low error levels i…
This paper investigates the properties of a three dimensional shear flow overpassing a hemispherical droplet resting on a plane wall. The exact solution is computed as a function of the viscosity ratio between the droplet and the surrounding fluid and generalizes the solution for the hemispherical no-slip bump given in an earlier paper by Price (1985). Several expressions including the torque and the force acting on the drop will be considered as well as the importance of the deformations on the surface for small Capillary numbers.
Numerical solutions of high-speed microdroplet impact onto a smooth solid surface are computed, using the interFoam VoF solver of the OpenFOAM CFD package. Toward the solid surface, the liquid microdroplet is moving with an impinging gas flow, simulating the situation of ink droplets being deposited onto substrate with a collimated mist jet in the Optomec Aerosol Jet printing process. The computed values of maximum spread factor, for the range of parameters of practical interest to Aerosol Jet printing, were found in very good agreement with some of the correlation formulas proposed by previou…
When a liquid droplet impacts a hot solid surface, enough vapor may be generated under it as to prevent its contact with the solid. The minimum solid temperature for this so-called Leidenfrost effect to occur is termed the Leidenfrost temperature, or the dynamic Leidenfrost temperature when the droplet velocity is non-negligible. We observe the wetting/drying and the levitation dynamics of the droplet impacting on an (isothermal) smooth sapphire surface using high speed total internal reflection imaging, which enables us to observe the droplet base up to about 100 nm above the substrate surfac…
Emulsion droplets trapped in an ultrasonic levitator behave in two ways that solid spheres do not: (1) Individual droplets spin rapidly about an axis parallel to the trapping plane, and (2) coaxially spinning droplets form long chains aligned with their common axis of rotation. Acoustically-organized chains interact hydrodynamically, either to merge into longer chains or to form three-dimensional bundles of chains. Solid spheres, by contrast, form close-packed planar crystals drawn together by the sound-mediated secondary Bjerknes interaction. We demonstrate the chain-to-crystal transition wit…
Sessile droplets of aqueous poly(ethylene oxide) solution, with average molecular weight of 100 kDa, are monitored during evaporative drying at ambient conditions over a range of initial concentrations $c_0$. For all droplets with $c_0 \geq 3%$, central conical structures, which can be hollow and nearly 50% taller than the initial droplet, are formed during a growth stage. Although the formation of superficially similar structures has been explained for glass-forming polymers using a skin-buckling model which predicts the droplet to have constant surface area during the growth stage (L. Paucha…
We study the rheology of a suspension of soft deformable droplets subjected to a pressure-driven flow. Through computer simulations, we measure the apparent viscosity as a function of droplet concentration and pressure gradient, and provide evidence of a discontinuous shear thinning behaviour, which occurs at a concentration-dependent value of the forcing. We further show that this response is associated with a nonequilibrium transition between a `hard' (or less deformable) phase, which is nearly jammed and flows very slowly, and a `soft' (or more deformable) phase, which flows much more easil…
It is well known that jammed soft materials will flow if sheared above their yield stress - think mayonnaise spread on bread - but a complete microscopic description of this seemingly sim- ple process has yet to emerge. What remains elusive is a microscopic framework that explains the macroscopic flow, derived from a 3-D spatially resolved analysis of the dynamics of the droplets or particles that compose the soft material. By combining confocal-rheology experiments on compressed emulsions and numerical simulations, we unravel that the primary microscopic mechanisms for flow are strongly influ…
We simulate the response of a particle-stabilized emulsion droplet in an external force field, such as gravity, acting equally on all $N$ particles. We show that the field strength required for breakup (at fixed initial area fraction) decreases markedly with droplet size, because the forces act cumulatively, not individually, to detach the interfacial particles. The breakup mode involves the collective destabilization of a solidified particle raft occupying the lower part of the droplet, leading to a critical force per particle that scales approximately as $N^{-1/2}$.
We propose a theory for the multi-ring pattern of the deposits that are formed when droplets of suspension are dried on a substrate. Assuming a standard model for the stick-slip motion of the contact line, we show that as droplets evaporate, many concentric rings of deposits are formed, but are taken over by a solid-circle pattern in the final stage of drying. An analytical expression is given to indicate when ring-pattern changes to solid-circle pattern during an evaporation process. The results are in qualitative agreement with existing experiments, and the other predictions on how the evapo…
On a sufficiently-soft substrate, a resting fluid droplet will cause significant deformation of the substrate. This deformation is driven by a combination of capillary forces at the contact line and the fluid pressure at the solid surface. These forces are balanced at the surface by the solid traction stress induced by the substrate deformation. Young's Law, which predicts the equilibrium contact angle of the droplet, also indicates an a priori radial force balance for rigid substrates, but not necessarily for soft substrates which deform under loading. It remains an open question whether the …
The transport of small quantities of liquid on a solid surface is inhibited by the resistance to motion caused by the contact between the liquid and the solid. To overcome such resistance, motion can be externally driven through gradients in electric fields, but these all inconveniently involve the input of external energy. Alternatively, gradients in physical shape and wettability - the conical shape of cactus spines to create self-propelled motion. However, such self-propelled motion to date has limited success in overcoming the inherent resistance to motion of the liquid contact with the so…
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 stretc…
Freezing or solidification of impacting droplets is omnipresent in nature and technology, be it a rain droplet falling on a supercooled surface, be it in inkjet printing where often molten wax is used, be it in added manufacturing or in metal production processes or in extreme ultraviolet lithography (EUV) for the chip production where molten tin is used to generate the EUV radiation. For many of these industrial applications, a detailed understanding of the solidification process is essential. Here, by adopting a totally new optical technique in the context of freezing, namely TIR (Total-Inte…
Liquid infused surfaces with partially wetting lubricants have recently been exploited for numerous intriguing applications, such as for droplet manipulation, droplet collection and spontaneous motion. When partially wetting lubricants are used, the pinning force is a key factor that can strongly affect droplet mobility. Here, we derive an analytical prediction for contact angle hysteresis {in the limit where the meniscus size is much smaller than the droplet}, and numerically study how it is controlled by the solid fraction, the lubricant wetting angles, and the various fluid surface tensions…
By solving the Young Laplace equation of capillary hydrostatics one can accurately determine equilibrium shapes of droplets on relatively smooth solid surfaces. The solution, however of the Young Laplace equation becomes tricky when a droplet is sitting on a geometrically patterned surface and multiple, and unknown a priori, three phase contact lines have not been accounted for, since air pockets are trapped beneath the liquid droplet. In this work, we propose an augmented Young-Laplace equation, in which a unified formulation for the liquid/vapor and liquid/solid interfaces is adopted, incorp…
Ice accumulation hinders the performance of, and poses safety threats for infrastructure both on the ground and in the air. Previously, rationally designed superhydrophobic surfaces have demonstrated some potential as a passive means to mitigate ice accretion; however, further studies on material solutions that reduce impalement and contact time for impacting supercooled droplets and can also repel droplets that freeze during surface contact are urgently needed. Here we demonstrate the collaborative effect of substrate flexibility and surface nanotexture on enhancing both icephobicity and the …
Eshelby's theory is the foundation of composite mechanics, allowing calculation of the effective elastic moduli of composites from a knowledge of their microstructure. However it ignores interfacial stress and only applies to very dilute composites -- i.e. where any inclusions are widely spaced apart. Here, within the framework of the Mori-Tanaka multiphase approximation scheme, we extend Eshelby's theory to treat a composite with interfacial stress in the non-dilute limit. In particular we calculate the elastic moduli of composites comprised of a compliant, elastic solid hosting a non-dilute …
Using experiments and simulations, we study the flow of soft particles through quasi-two-dimensional hoppers. The first experiment uses oil-in-water emulsion droplets in a thin sample chamber. Due to surfactants coating the droplets, they easily slide past each other, approximating soft frictionless disks. For these droplets, clogging at the hopper exit requires a narrow hopper opening only slightly larger than the droplet diameter. The second experiments use soft hydrogel particles in a thin sample chamber, where we vary gravity by changing the tilt angle of the chamber. For reduced gravity, …
Wettability is the affinity of a liquid for a solid surface. For energetic reasons, macroscopic drops of liquid are nearly spherical away from interfaces with solids, and any local deformations due to molecular-scale surface interactions are negligible. Studies of wetting phenomena, therefore, typically assume that a liquid on a surface adopts the shape of a spherical cap. The degree of wettability is then captured by the contact angle where the liquid-vapor interface meets the solid-liquid interface. As droplet volumes shrink to the scale of attoliters, however, surface interactions become si…
We present a multi-phase-field model to describe quantitatively nanowire growth by the vapor-liquid-solid (VLS) process. The free-energy functional of this model depends on three non-conserved order parameters that distinguish the vapor, liquid, and solid phases. The evolution equations for those order parameters describe basic kinetic processes including the rapid (quasi-instantaneous) equilibration of the liquid catalyst to a droplet shape with constant mean curvature, the slow incorporation of growth atoms at the droplet surface, and crystallization within the droplet. The standard constrai…
