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{i.e.} the nucleation rate and the crystal growth speed, and show its sensitivity to the applied undercooling of the substrate. By combining the Johnson-Mehl-Avrami-Kolmogorov nucleation theory and scaling relations for the spreading, we can calculate the temporal evolution of the solid area fraction, which is in good agreement with our observations. We also show that for strong enough undercooling it is the rapid growth of the crystals which determines the eventual arrest of the spreading contact line.
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…
Spreading and solidification of liquid droplets are elementary processes of relevance for additive manufacturing. Here we investigate the effect of heat transfer on spreading of a thermoresponsive solution (Pluronic F127) that undergoes a sol-gel transition above a critical temperature $T_m$. By controlling the concentration of Pluronic F127 we systematically vary $T_m$, while also imposing a broad range of temperatures of the solid and the liquid. We subsequently monitor the spreading dynamics over several orders of magnitude in time and determine when solidification stops the spreading. It i…
The spreading under surface tension and gravity of a droplet of yield-stress fluid over a thin film of the same material is studied. The droplet converges to a final equilibrium shape once the driving stresses inside the droplet fall below the yield stress. Scaling laws are presented for the final radius and complemented with an asymptotic analysis for shallow droplets. Moreover, numerical simulations using the volume-of-fluid method and a regularized constitutive law, and experiments with an aqueous solution of Carbopol are presented.
When a droplet impacts a fabric mesh at a sufficiently high impact velocity, it not only spreads over the fabric but also penetrate its pores. To determine the influence of this liquid penetration of the fabric on droplet spreading on thin fabric meshes, we measured the droplet spreading ratio on fabric with and without an underlying substrate using a high-speed camera. For fabrics without a substrate, the droplet spreading ratio is reduced as the fabric penetration by the liquid reduces the droplet volume spreading on top of the fabric. Using entropic lattice Boltzmann simulations, we find th…
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, quantified through the Bond number. In particular, microgravity allows 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 experimental module provides a versatile platform to study a wide range of complex fluids thro…
We report an experimental investigation of spreading then shrinking of a surfactant droplet at the air-water interface with fingering instability appearing at the edge of the droplet stain. We find out that a droplet of a surfactant on the water shows three regimes of spreading, shrinking and resting at the appropriate parameters values. These regimes can be distinguished by measuring the mean square displacement of the droplet parts, fractal dimension of stain, or radius of fingerless part of the stain (to measure it, we define the angular box-counting dimension), versus time. The shrinking r…
Would a raindrop impacting on a coarse beach behave differently from that impacting on a desert of fine sand? We study this question by a series of model experiments, where the packing density of the granular target, the wettability of individual grains, the grain size, the impacting liquid, and the impact speed are varied. We find that by increasing the grain size and/or the wettability of individual grains the maximum droplet spreading undergoes a transition from a capillary regime towards a viscous regime, and splashing is suppressed. The liquid-grain mixing is discovered to be the underlyi…
The surfactant-driven spreading of droplets is an essential process in many applications ranging from coating flow technology to enhanced oil recovery. Despite the significant advancement in describing spreading processes in surfactant-laden droplets, including the exciting phenomena of superspreading, many features of the underlying mechanisms require further understanding. Here, we have carried out molecular dynamics simulations of a coarse-grained model with a force-field obtained from the statistical associating fluid theory to study droplets laden with common and superspreading surfactant…
The effect of freezing on contact line motion is a scientific challenge in the understanding of the solidification of capillary flows. In this letter, we experimentally investigate the spreading and freezing of a water droplet on a cold substrate. We demonstrate that solidification stops the spreading because the ice crystals catch up with the advancing contact line. Indeed, we observe the formation and growth of ice crystals along the substrate during the drop spreading, and show that their velocity equals the contact line velocity when the drop stops. Modelling the growth of the crystals, we…
Even a small fraction of nanoparticles in fluids affects the splashing behavior of a droplet upon impact on a smooth surface. Nanofluid drop impact onto a smooth sapphire substrate is experimentally investigated over wide ranges of Reynolds ($10^2<\mathrm{Re}<10^4$) and Weber ($50<\mathrm{We}<500$) numbers for three nanofluid mass concentrations (0.01%, 0.1%, 1%) using high-speed photography. Nanofluids are prepared by diluting a commercial Al$_2$O$_3$-water nanofluid in aqueous glycerol solutions without dispersants. In total, 30 samples are prepared and 1799 data points are acquired. Every s…
We study spreading dynamics of nematic liquid crystal droplets within the framework of the long-wave approximation. A fourth order nonlinear parabolic partial differential equation governing the free surface evolution is derived. The influence of elastic distortion energy and of imposed anchoring variations at the substrate are explored through linear stability analysis and scaling arguments, which yield useful insight and predictions for the behaviour of spreading droplets. This behaviour is captured by fully nonlinear time-dependent simulations of three dimensional droplets spreading in the …
From biological tissues to layers of paint, macroscopic non-porous materials with the capacity to swell when brought in contact with an appropriate solvent are ubiquitous. Here, we study experimentally and theoretically one of the conceptually simplest of such systems, the swelling of a thin hydrogel layer by a single water drop. Using a bespoke experimental setup, we observe fast absorption leading to a radially spreading axisymmetric blister. Employing a linear poroelastic framework and thin-layer scalings, we develop a non-linear one-dimensional diffusion equation for the evolution of the b…
We report a novel surface-tension driven instability observed for droplets spreading on a compliant substrate. When a droplet is released on the surface of an agar gel, it forms arms/cracks when the ratio of surface tension gradient to gel strength is sufficiently large. We explore a range of gel strengths and droplet surface tensions and find that the onset of the instability and the number of arms depend on the ratio of surface tension to gel strength. However, the arm length grows with an apparently universal law L ~ t^{3/4}.
We present results on the leveling of polymer microdroplets on thin films prepared from the same material. In particular, we explore the crossover from a droplet spreading on an infinitesimally thin film (Tanner's law regime) to that of a droplet leveling on a film thicker than the droplet itself. In both regimes, the droplet's excess surface area decreases towards the equilibrium configuration of a flat liquid film, but with a different power law in time. Additionally, the characteristic leveling time depends on molecular properties, the size of the droplet, and the thickness of the underlyin…
We use a three-dimensional lattice Boltzmann model to investigate the spreading of mesoscale droplets on homogeneous and heterogeneous surfaces. On a homogeneous substrate the base radius of the droplet grows with time as $t^{0.28}$ for a range of viscosities and surface tensions. The time evolutions collapse onto a single curve as a function of a dimensionless time. On a surface comprising of alternate hydrophobic and hydrophilic stripes the wetting velocity is anisotropic and the equilibrium shape of the droplet reflects the wetting properties of the underlying substrate.
The interplay between phase separation and wetting of multicomponent mixtures is ubiquitous in nature and technology and recently gained significant attention across scientific disciplines, due to the discovery of biomolecular condensates. It is well understood that sessile droplets, undergoing phase separation in a static wetting configuration, exhibit microdroplet nucleation at their contact lines, forming an oil ring during later stages. However, very little is known about the dynamic counterpart, when phase separation occurs in a non-equilibrium wetting configuration, i.e., spreading dropl…
The influence of controlled vibrations on the granular rheology is investigated in a specifically designed experiment. We study experimentally a thin granular film spreading under the action of horizontal vibrations. A nonlinear diffusion equation is derived theoretically that describes the evolution of the deposit shape. A self-similar parabolic shape (the``granular droplet'') and a spreading dynamics are predicted that both agree quantitatively with the experimental results. The theoretical analysis is used to extract effective dynamic friction coefficients between the base and the granular …
We present a study of the spreading of liquid droplets on a solid substrate at very small scales. We focus on the regime where effective wetting energy (binding potential) and surface tension effects significantly influence steady and spreading droplets. In particular, we focus on strong packing and layering effects in the liquid near the substrate due to underlying density oscillations in the fluid caused by attractive substrate-liquid interactions. We show that such phenomena can be described by a thin-film (or long-wave or lubrication) model including an oscillatory Derjaguin (or disjoining…
The spreading of polymer droplets is studied using molecular dynamics simulations. To study the dynamics of both the precursor foot and the bulk droplet, large drops of ~200,000 monomers are simulated using a bead-spring model for polymers of chain length 10, 20, and 40 monomers per chain. We compare spreading on flat and atomistic surfaces, chain length effects, and different applications of the Langevin and dissipative particle dynamics thermostats. We find diffusive behavior for the precursor foot and good agreement with the molecular kinetic model of droplet spreading using both flat and a…
The quasistationary spreading of a circular liquid drop on a solid substrate typically obeys the so-called Tanner law, with the instantaneous base radius R(t) growing with time as R ~ t^{1/10} -- an effect of the dominant role of capillary forces for a small-sized droplet. However, for droplets of nematic liquid crystals, a faster spreading law sets in at long times, so that R ~ t^alpha with alpha significantly larger than the Tanner exponent 1/10. In the framework of the thin film model (or lubrication approximation), we describe this "acceleration" as a transition to a qualitatively differen…
Nanoscale topographic features of solid surfaces can induce complex metastable behavior in colloidal and multiphase systems. Recent studies on single microparticle adsorption at liquid interfaces have reported a crossover from fast capillary driven dynamics to extremely slow kinetic regimes that can require up to several hours or days to attain thermodynamic equilibrium. The observed kinetic regime resembling physical ageing in glassy materials has been attributed to unobserved surface features with dimensions on the order of a few nanometers. In this work, we study the spontaneous spreading o…
We study the spreading of droplets in a near-critical phase-separated liquid mixture, using a combination of experiments, lubrication theory and finite-element numerical simulations. The classical Tanner's law describing the spreading of viscous droplets is robustly verified when the critical temperature is neared. Furthermore, the microscopic cut-off length scale emerging in this law is obtained as a single free parameter for each given temperature. In total-wetting conditions, this length is interpreted as the thickness of the thin precursor film present ahead of the apparent contact line. T…
The contact angle of a fluid droplet on an heterogeneous surface is analysed using the statistical dynamics of the spreading contact line. The statistical properties of the final droplet radius and contact angle are obtained through applications of depinning transitions of contact lines with non-local elasticity and features of pinning-depinning dynamics. Such properties not only depend on disorder strength and surface details, but also on the droplet volume and disorder correlation length. Deviations from Wenzel or Cassie/Baxter behaviour are particularly apparent in the case of small droplet…
The spreading of one- and two-component polymer nanodroplets is studied using molecular dynamics simulation in a cylindrical geometry. The droplets consist of polymer chains of length 10, 40, and 100 monomers per chain described by the bead-spring model spreading on a flat surface with a surface-coupled Langevin thermostat. Each droplet contains ~350,000 monomers. The dynamics of the individual components of each droplet are analyzed and compared to the dynamics of single component droplets for the spreading rates of the precursor foot and bulk croplet, the time evolution of the contact angle,…
The spreading of a cap-shaped spherical droplet of non-Newtonian power-law liquids on a completely wettable spherical substrate is theoretically studied. Both convex spherical substrates and concave spherical cavities with smooth or rough surfaces are considered. The droplet on a rough substrate is modeled by either the Wenzel or the Cassie-Baxter model. The two sources of driving force of spreading by the surface-tension and the line tension are considered. Also, the two channels of energy dissipation by the viscous dissipation within the bulk and the frictional dissipation at the contact lin…
The spreading of a cap-shaped spherical droplet on a completely wettable spherical substrate is studied. The non-equilibrium thermodynamic formulation is used to derive the thermodynamic driving force of spreading including the line-tension effect. Then the energy balance approach is adopted to derive the evolution equation of the spreading droplet. The time evolution of the contact angle $θ$ of a droplet obeys a power law $θ\sim t^{-α}$ with the exponent $α$, which is different from that derived from Tanner's law on a flat substrate. Furthermore, the line tension must be positive to promote c…
The spreading of a circular liquid drop on a solid substrate can be described by the time evolution of its base radius R(t). In complete wetting the quasistationary regime (far away from initial and final transients) typically obeys the so-called Tanner law, with R t^alpha_T, alpha_T=1/10. Late-time spreading may differ significantly from the Tanner law: in some cases the drop does not thin down to a molecular film and instead reaches an equilibrium pancake-like shape; in other situations, as revealed by recent experiments with spontaneously spreading nematic crystals, the growth of the base r…
The fragmentation of liquid sheets produces a collection of droplets. The size distribution of the droplets has a considerable impact on the coating efficiency of sprays and the transport of contaminants. Although many processes commonly used particulate suspensions, the influence of the particles on the spreading dynamics of the sheet and its subsequent fragmentation has so far been considered negligible. In this paper, we consider experimentally a transient suspension sheet that expands radially. We characterize the influence of the particles on the dynamics of the liquid sheet and the fragm…
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 …
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 …
The coarsening kinetics for the beading instability for liquid contained in a groove with convex curved sides (for example between a pair of parallel touching cylinders) is considered as an open channel flow problem. In contrast to a V-shaped wedge or U-shaped microchannel, it is argued that droplet coarsening takes place by viscous hydrodynamic transport through a stable column of liquid that coexists with the droplets in the groove at a slightly positive Laplace pressure. With some simplifying assumptions, this leads to a t^(1/7) growth law for the characteristic droplet size as a function o…
The spreading of a cap-shaped spherical droplet of non-Newtonian power-law liquids on a flat and a spherical rough and textured substrate is theoretically studied in the capillary-controlled spreading regime. A droplet whose scale is much larger than that of the roughness of substrate is considered. The equilibrium contact angle on a rough substrate is modeled by the Wenzel and the Cassie-Baxter model. Only the viscous energy dissipation within the droplet volume is considered, and that within the texture of substrate by imbibition is neglected. Then, the energy balance approach is adopted to …
We study the time evolution of a sessile liquid droplet, which is initially put onto a solid surface in a non-equilibrium configuration and then evolves towards its equilibrium shape. We adapt here the standard approach to the dynamics of mechanical dissipative systems, in which the driving force, i.e. the gradient of the system's Lagrangian function, is balanced against the rate of the dissipation function. In our case the driving force is the loss of the droplet's free energy due to the increase of its base radius, while the dissipation occurs due to viscous flows in the core of the droplet …
Controlling the spatial distribution of liquid droplets on surfaces via surface energy patterning can be used to control material delivery to specified regions via selective liquid/solid wetting. While studies of the equilibrium shape of liquid droplets on heterogenous substrates exist, much less is known about the corresponding wetting kinetics. We make significant progress towards elucidating details of this topic by studying, via large-scale atomistic simulations, liquid nanodroplets spreading on chemically patterned surfaces. A model is presented for lines of polymer liquid (droplets) on s…
The spreading of a cap-shaped spherical droplet of non-Newtonian power-law liquids, both shear-thinning and shear-thinning liquids, that completely wet a spherical substrate is theoretically investigated in the capillary-controlled spreading regime. The crater-shaped droplet model with the wedge-shaped meniscus near the three-phase contact line is used to calculate the viscous dissipation near the contact line. Then the energy balance approach is adopted to derive the equation which governs the evolution of the contact line. The time evolution of the dynamic contact angle $θ$ of a droplet obey…
Electrowetting-on-dielectric (EWOD) is a powerful tool in many droplet-manipulation applications with a notorious weakness caused by contact-angle saturation (CAS), a phenomenon limiting the equilibrium contact angle of an EWOD-actuated droplet at high applied voltage. In this paper, we study the spreading behaviours of droplets on EWOD substrates with the range of applied voltage exceeding the saturation limit. We experimentally find that at the initial stage of spreading, the driving force at the contact line still follows the Young-Lippmann law even if the applied voltage is higher than the…
When sand flows out of a funnel onto a surface, a three dimensional pile that is stabilized by friction grows taller as it spreads. Here we investigate an idealized two dimensional analogue: spreading of a pile of monodisperse oil droplets at a boundary. In our system the droplets are buoyant, adhesive, and in contrast to sand, here friction is negligible. The buoyant droplets are added to the pile one-at-a-time. As the aggregate grows, it reaches a critical height and the 2D pile spreads out across the barrier. We find that, while granularity is important, the growth process is reminiscent of…
Even though electrowetting-on-dielectric (EWOD) is a useful strategy in a wide array of biological and engineering processes with numerous droplet-manipulation applications, there is still a lack of complete theoretical interpretation on the dynamics of electrowetting. In this paper, we present an effective theoretical model and use Onsager variational principle to successfully derive the governing equations of non-equilibrium electrowetting dynamics for a droplet in both overdamped and underdamped regimes. It is found that the spreading and retraction dynamics of a droplet on EWOD substrates …
Spreading of respiratory diseases, such as COVID-19, from contaminated surfaces is dependent on the drying time of the deposited droplets containing the virus. The evaporation rate depends on environmental conditions, such as ambient temperature and relative humidity and physical properties (e.g., droplet volume, contact angle and composition). The respiratory droplets contain salt (NaCl), protein (mucin), and surfactant (dipalmitoylphosphatidylcholine) in addition to water, which are expected to influence the evaporation in a big way. A diffusion-based theoretical model for estimating the dry…
The early stages of sintering of highly viscous droplets are very similar to the early stages of a viscous droplet spreading on a solid substrate. The flows in both problems are closely analogous to the displacements in a Hertzian elastic contact. We exploit that analogy to provide both a scaling argument and a calculation for the early growth of the contact radius $a$ with time, namely $a=(3 πγR^2 t/(32 η))^{1/3}$. (This result is complementary to the well-known Tanner law for spreading, $a \sim t^{1/10}$, which holds in the regime of low contact angles.) For viscoelastic fluids, the linear s…
Understanding the dynamics in the deposition of water droplets onto solid surfaces is of importance from both fundamental and practical viewpoints. While the deposition of a water droplet onto a heated surface is extensively studied, the characteristics of depositing a droplet onto a cold surface and the phenomena leading to such behavior remain elusive. Here we report the formation of a triple condensate halo observed during the deposition of a water droplet onto a cold surface, due to the interplay between droplet impact dynamics and vapor diffusion. Two subsequent condensation stages occur …
The spreading of liquid nanodroplets of different initial radii $R_{0}$ is studied using molecular dynamics simulation. Results for two distinct systems, Pb on Cu(111), which is non-wetting, and a coarse grained polymer model, which wets the surface, are presented for Pb droplets ranging in size from $\sim55 000$ to $220 000$ atoms and polymer droplets ranging in size from $\sim200 000$ to $780 000$ monomers. In both cases, a precursor foot precedes the spreading of the main droplet. This precursor foot spreads as $r_{f}^{2}(t)=2D_{eff}t$ with an effective diffusion constant that exhibits a dr…
This research found that when an acidic solution with a low surface tension spread on the surface of a glycerol solution mixed with milk, a star shaped pattern was spontaneously formed on the surface in the horizontal plane during the spreading process. We investigated the emergence of the star shaped pattern owing to an interfacial instability in experiments using glycerol solutions with several viscosities and 2-methoxyethanol aqueous solutions, which are acidic solutions, with several concentrations. This result demonstrated that the star shaped pattern emerged in the high concentration of …
The classical wetting models, such as the Wenzel and Cassie-Baxter have been extensively used to quantify the wettability of laser-textured surfaces. However, these models do not provide any description of the corresponding droplet dynamics. In this work, we propose a detailed continuum-level modelling to study the wetting dynamics of a water droplet on Ti-6Al-4V alloy. The calculations are performed for flat surfaces and surfaces with various reliefs. The calculated evolutions of the droplet spreading parameter for flat surfaces, surfaces with triangular reliefs and one with two different per…
Spreading of nano-droplets is an interesting and technologically relevant phenomenon where thermal fluctuations lead to unexpected deviations from well-known deterministic laws. Here, we apply the newly developed fluctuating non-ideal lattice Boltzmann method [Gross et al., J. Stat. Mech., P03030 (2011)] for the study of this issue. Confirming the predictions of Davidovich and coworkers [PRL 95, 244905 (2005)], we provide the first independent evidence for the existence of an asymptotic, self-similar noise-driven spreading regime in both two- and three-dimensional geometry. The cross over from…
In this article, we study the spreading of droplets of density-matched granular suspensions on the surface of a solid. Bidispersity of the particle size distribution enriches the conclusions drawn from monodisperse experiments by highlighting key elements of the wetting dynamics. In all cases, the relation between the dynamic contact angle and the velocity of the contact line follows a similar relation as that of a simple fluid, despite the complexity introduced by the presence of particles. We extract from this relation an apparent wetting viscosity of the suspensions that differs from that m…
Molecular dynamics simulations are used to study the spreading of binary polymer nanodroplets in a cylindrical geometry. The polymers, described by the bead-spring model, spread on a flat surface with a surface-coupled Langevin thermostat to mimic the effects of a corrugated surface. Each droplet consists of chains of length 10 or 100 monomers with ~350,000 monomers total. The qualitative features of the spreading dynamics are presented for differences in chain length, surface interaction strength, and composition. When the components of the droplet differ only in surface interaction strength,…
