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A liquid drop impacting a rigid substrate undergoes deformation and spreading due to normal reaction forces, which are counteracted by surface tension. On a non-wetting substrate, the drop subsequently retracts and takes off. Our recent work (Zhang et al., \textit{Phys. Rev. Lett.}, vol. 129, 2022, 104501) revealed two peaks in the temporal evolution of the normal force $F(t)$--one at impact and another at jump-off. The second peak coincides with a Worthington jet formation, which vanishes at high viscosities due to increased viscous dissipation affecting flow focusing. In this article, using experiments, direct numerical simulations, and scaling arguments, we characterize both the peak amplitude $F_1$ at impact and the one at take off ($F_2$) and elucidate their dependency on the control parameters: the Weber number $We$ (dimensionless impact velocity) and the Ohnesorge number $Oh$ (dimensionless viscosity). \VS{The first peak amplitude $F_1$ and the time $t_1$ to reach it depend on inertial timescales for low viscosity liquids, remaining nearly constant for viscosities up to 100 times that of water. For high viscosity liquids, we balance the rate of change in kinetic energy with viscous dissipation to obtain new scaling laws: $F_1/F_\rho \sim \sqrt{Oh}$ and $t_1/\tau_\rho \sim 1/\sqrt{Oh}$, where $F_\rho$ and $\tau_\rho$ are the inertial force and time scales, respectively, which are consistent with our data.} The time $t_2$ at which the amplitude $F_2$ appears is set by the inertio-capillary timescale $\tau_\gamma$, independent of both the viscosity and the impact velocity of the drop. However, these properties dictate the magnitude of this amplitude., Comment: version after first round of the review

The introduction of lubricant between fluid and substrate endows the Liquid-Infused Slippery Surfaces with excellent wetting properties: low contact angle, various liquids repellency, ice-phobic and self-healing. Droplets moving on such surfaces have been widely demonstrated to obey a Landau-Levich-Derjaguin (LLD) friction. Here, we show that this power law is surprisingly decreased with the droplet accelerates: in the rapid droplet regime, the slippery surfaces seem more slippery than LLD friction. Combining experimental and numerical techniques, we find that the meniscus surrounding the droplet exhibits an incompletely developed state. The Incompletely Developed Meniscus possesses shorter shear length and thicker shear thickness than the prediction of Bretherton model and therefore is responsible for the more slippery regime. With an extended Bretherton model, we not only provide an analytical description to the IDM behavior but also the friction when the Capillary Number of the moving droplet is larger than the Critical Capillary Number.

Large droplets emerging during dropwise condensation impair surface properties such as anti-fogging/frosting ability and heat transfer efficiency. How to spontaneously detach massive randomly distributed droplets with controlled sizes has remained a great challenge. Herein, we present a general solution called condensation droplet sieve, through fabricating microscale thin-walled lattice (TWL) structures coated with a superhydrophobic layer. Growing droplets were observed to jumped off this TWL surface with 100% probability once becoming slightly larger than the lattices. The maximum radius and residual volume of droplets were strictly confined to 16 {\mu}m and 3.2 nl/mm2 respectively, greatly surpassing the current state of the art. We reveal that this extremely efficient jumping is attributed to the large tolerance of coalescence mismatch and effective isolation of droplets between neighbouring lattices. Our work provides a new perspective for the design and fabrication of high-performance anti-dew materials.

A falling liquid drop, after impact on a rigid substrate, deforms and spreads, owing to the normal reaction force. Subsequently, if the substrate is non-wetting, the drop retracts and then jumps off. As we show here, not only is the impact itself associated with a distinct peak in the temporal evolution of the normal force, but also the jump-off, which was hitherto unknown. We characterize both peaks and elucidate how they relate to the different stages of the drop impact process. The time at which the second peak appears coincides with the formation of a Worthington jet, emerging through flow-focusing, and it is independent of the impact velocity. However, the magnitude of this peak is dictated by the drop's inertia and surface tension. We show that even low-velocity impacts can lead to a surprisingly high peak in the normal force, namely when a more pronounced singular Worthington jet occurs due to the collapse of an air cavity in the drop., Comment: Please find the supplemental movies here: https://youtube.com/playlist?list=PLf5C5HCrvhLGmlYTF1Gg2WviZ-Bkmy2qr

Escaping of the liquid molecules from their liquid bulk into the vapour phase at the vapour-liquid interface is controlled by the vapour diffusion process, which nevertheless hardly senses the macroscopic shape of this interface. Here, deformed sessile drops due to gravity and surface tension with various interfacial profiles are realised by tilting flat substrates. The symmetry broken of the sessile drop geometry leads to a different evaporation behavior compared to a drop with a symmetric cap on a horizontal substrate. Rather than the vapour-diffusion mechanism, heat-diffusion regime is defined here to calculate the local evaporation flux along the deformed drop interface. A local heat resistance, characterised by the liquid layer thickness perpendicular to the substrate, is proposed to relate the local evaporation flux. We find that the drops with and without deformation evaporate with a minimum flux at the drop apex, while up to a maximum one with a significantly larger but finite value at the contact line. Counterintuitively, the deviation from the symmetric shape due to the deformation on a slope, surprisingly enhances the total evaporation rate; and the smaller contact angle, the more significant enhancement. Larger tilt quickens the overall evaporation process and induces a more heterogeneous distribution of evaporative flux under gravity. Interestingly, with this concept of heat flux, an intrinsic heat resistance is conceivable around the contact line, which naturally removes the singularity of the evaporation flux showing in the vapour-diffusion model. The detailed non-uniform evaporation flux suggests ways to control the self-assembly, microstructures of deposit with engineering applications particularly in three dimensional printing where drying on slopes is inevitable., Comment: 12 pages, 7 figures

In this paper, we systematically investigate the wetting behavior of a liquid ring in a cylindrical capillary tube. We obtain analytical solutions of the axisymmetric Young-Laplace equation for arbitrary contact angles. We find that, for specific values of the contact angle and the volume of the liquid ring, two solutions of the Young-Laplace equation exist, but only the one with the lower value of the total interfacial energy corresponds to a stable configuration. The transition to an unstable configuration is characterized by specific critical parameters such as the liquid volume, throat diameter etc. Beyond the stable regime, the liquid ring transforms into a plug. Based on numerical simulations, we also discuss the transition of an axisymmetric ring into non-axisymmetric configurations. Such a transition can be induced by the Rayleigh-Plateau instability of a slender liquid ring. The results are presented in terms of a map showing the different liquid morphologies as a function of the parameters governing the problem., Comment: 31 pages, 10 figures

Superrepellency is a favorable non-wetting situation featured by a dramatically reduced solid/liquid contact region with extremely low adhesion. However, drop impact often brings out a notable extension of the contact region associated with rather enhanced water affinity, such renders irreversible breakdowns of superhydrophobicity. Here, we report an alternative outcome, a repeated Cassie-Wenzel-Cassie (CWC) wetting state transition in the microscale occurs when a drop impacts a two-tier superhydrophobic surface, which exhibits a striking contrast to the conventional perspective. Influences of material parameters on the impact dynamics are quantified. We demonstrate that self-cleaning and dropwise condensation significantly benefit from this outcome - dirt particles or small droplets in deep textures can be taken away through the transition. The results reported in this study allows us to promote the strategy to design functional superrepellency materials., Comment: 20 pages, 5 figures

An analytical model is proposed for the Young-Laplace equation of two-dimensional (2D) drops under gravity. Inspired by the pioneering work of Landau & Lifshitz (1987), we derive analytical expressions of the profile of drops on flat surfaces, for arbitrary contact angles and drop volume. We then extend our theory for drops on inclined surfaces and reveal that the contact line plays a key role on the wetting state of the drops: (1) when the contact line is completely pinning, the advancing and receding contact angles and the shape of the drop can be uniquely determined by the predefined droplet volume, sliding angle and contact area, which does not rely on the Young contact angle; (2) when the drop has a movable contact line, it would achieve a wetting state with a minimum free energy resulting from the competition between the surface tension and gravity. Our theory is in excellent agreement with numerical results., Comment: 11 pages, 7 figures

When a droplet (which size is characterized by R) is confined between two parallel planes, its morphology will change accordingly to either varying the volume of the droplet or the separation (characterized by h) between the planes. We are aiming at investigating how such a geometric confinement affects the wetting behaviours of a droplet. Our focus lies on two distinguished regimes: (1) a pancake shape in a Hele-Shaw cell when the droplet is highly compressed (i.e. h/R << 1), in which particular attention is paid on nonwetting and wetting cases, respectively; and (2) a liquid Hertzian contact rendered by a slight confinement (i.e. h/(2R) --> 1) in a nonwetting case. To realize this aim, we first develop strict analytical expressions of the shape of the droplet which are available for arbitrary contact angles between the liquid and the solid planes, but in which the elliptic integrals indicate that these solutions are implicit. By employing asymptotic methods, we are able to give expressions of relevant geometrical and physical parameters (the Laplace pressure, droplet volume, surface energy and external force) in terms of sole functions of R and h in an explicit manner. Comparisons suggest that over a large range of h/R, our asymptotic results quantitatively agree well with the numerical solutions of the analytical expressions. This systematic study of the parameter space allows a comprehensive understanding of the geometric confinement on wetting, especially a wide existence of logarithmic behaviours in a liquid Hertzian contact, to be identified., Comment: 21 pages, 9 figures

Liquid dispensing and writing in the extremely small size regime are important for applications in many current technologies, such as micro/nano fabrication, biological/chemical patterning and analysis, and drug discovery. Most of current dispensing/writing methods can be sorted into a category of liquid flowing through tiny tubes or nozzles that requires inputting an impulse energy, which leads to complex procedures, expensive equipment and narrow material applicability, especially for biomaterials. Here, we report a method that may lead to a new category: liquid flows over the tapered surface of a pin with longitudinal nano grooves on the surface to uninterruptedly perform droplet dispensing and direct writing. The dispensed droplet diameters were controllable from several microns down to 150 nm, and the written line heights were as low as 5 nm. The mechanism underlying automatic liquid storage on conical surface and spontaneous liquid transport through nano grooves is revealed and well modeled by a simple relationship. Furthermore, the nano-grooved pins are much simpler and cheaper in fabrication than nanoscale tubes and nozzles, and pins have much depressed clogging problems that are typically troublesome for tubes and nozzles. Our new strategy may constitute a basis for creating liquid dispensing/writing technologies that are simultaneously smaller, simpler, faster and applicable for more types of materials., Comment: 31 pages in total. Maintext: 10 pages, 4 figures. Supplementary information: 21 pages, 8 figures, 3 tables

Superantiwettability, including superhydrophobicity, is an enhanced effect of surface ruggedness via the Cassie-Baxter wetting state, and has many applications such as antifouling, drop manipulation, and self-cleaning. However, superantiwettability is easily broken due to Cassie-Baxter to Wenzel wetting state transition caused by various environmental disturbances. Since all observed reverse transitions required energy inputs, it was believed that the Cassie-Baxter state couldn't be monostable. Here we show that there is a regime in the phase space of the receding contact angle and ruggedness parameters in which a Wenzel state can automatically transit into the Cassie-Baxter one without an external energy input, namely the Cassie-Baxter state in this regime is monostable. We further find a simple criterion that predicts very well experimentally observed Wenzel to Cassie-Baxter transitions for different liquids placed on various pillar-structured substrates. These results can be used as a guide for designing and engineering durable superantiwetting surfaces., Comment: 17 pages, 4 figures

Although realizing wetting transitions of droplets spontaneously on solid rough surfaces is quite challenging, it is becoming a key research topic in many practical applications which require highly efficient removal of liquid. We report wetting transitions of condensed droplets occurring spontaneously on pillared surfaces with two-tier roughness owing to excellent superhydrophobicity. The phenomenon results from further decreased Laplace pressure on the top side of the individual droplet when its size becomes comparable to the scale of the micropillars, which leads to a surprising robust spontaneous wetting transition, from valleys to tops of the pillars. A simple scaling law is derived theoretically, which demonstrates that the critical size of the droplet is determined by the space of the micropillars. For this reason, highly efficient removal of water benefits greatly from smaller micropillar space. Furthermore, three wetting transition modes exist, in which the in situ wetting behaviors are in good agreement with our quantitative theoretical analysis., Comment: 20 pages, 9 figures

Dynamic behavior of micro water droplet condensed on a lotus leaf with two-tier roughness is studied. Under laboratory environment, the contact angle of the micro droplet on single micro papilla increases smoothly from 80 deg to 160 deg during the growth of condensed water. The best-known "self-clean" phenomenon, will be lost. A striking observation is the out-of-plane jumping relay of condensed droplets triggered by falling droplets, as well as its sustained speed obtained in continuous jumping relays, enhance the automatic removal of dropwise condensation without the help from any external force. The surface tension energy dissipation is the main reason controlling the critical size of jumping droplet and its onset velocity of rebounding., Comment: 14 pages, 8 figures

A new type of water droplet transportation on microstructured hydrophobic surface is proposed and investigated experimentally and theoretically - water droplet could be driven by scale effect which is different from the traditional methods. Gradient microstructured hydrophobic surface is fabricated in which the area fraction is kept constant, but the scales of the micro-pillars are monotonic changed. When additional water or horizontal vibration is applied, the original water droplet could move unidirectionally to the direction from the small scale to the large scale to decrease its total surface energy. A new mechanism based on line tension model could be used to explain this phenomenon. It is also found that dynamic contact angle decreases with increasing the scale of the micro-pillars along the moving direction. These new findings will deepen our understanding of the relationship between topology and wetting properties, and could be very helpful to design liquid droplet transportation device in microfluidic systems., Comment: 14 pages, 7 figures

Making liquid droplets move spontaneously on solid surfaces is a key challenge in lab-on-chip and heat exchanger technologies. The best-known mechanism, a wettability gradient, does not generally move droplets rapidly enough and cannot drive droplets smaller than a critical size. Here we report how a curvature gradient is particularly effective at accelerating small droplets, and works for both hydrophilic and hydrophobic surfaces. Experiments for water droplets on tapered surfaces with curvature radii in the sub-millimeter range show a maximum speed of 0.28 m/s, two orders of magnitude higher than obtained by wettability gradient. We show that the force exerted on a droplet scales as the surface curvature gradient. Using molecular dynamics simulations, we observe nanoscale droplets moving spontaneously at over 100 m/s on tapered surfaces., Comment: 18 pages, 6 figures. arXiv admin note: substantial text overlap with arXiv:1202.6582

How to make small liquid droplets move spontaneously and directionally on solid surfaces is a challenge in lab-on-chip technologies, DNA analysis, and heat exchangers. The best-known mechanism, a wettability gradient, does not move droplets rapidly enough for most purposes and cannot move droplets smaller than a critical size defined by the contact angle hysteresis. Here we report on a mechanism using curvature gradients, which we show is particularly effective at accelerating small droplets, and works for both hydrophilic and hydrophobic surfaces. Experiments for water droplets on glass cones in the sub-millimeter range show a maximum speed of 0.28 m/s, two orders of magnitude higher than obtained by wettability gradient. From simple considerations of droplet surface area change, we show that the force exerted on a droplet on a conical surface scales as the curvature gradient. This force therefore diverges for small droplets near the tip of a cone. We illustrate this using molecular dynamics simulations, and describe nanometer-scale droplets moving spontaneously at over 100 m/s on nano-cones., Comment: 18 pages, 7 figures

We report experimental observation of a kind of fast spontaneous movements of water drops on surfaces of cones with diameters from 0.1 to 1.5 mm. The observed maximum speed (0.22 m/s) under ambient conditions were at least two orders of magnitude higher than that resulting from any known single spontaneous movement mechanism, for example, Marangoni effect due to gradient of surface tension. We trapped even higher spontaneous movement speeds (up to 125 m/s) in virtual experiments for drops on nanoscale cones by using molecular dynamics simulations. The underlying mechanism is found to be universally effective - drops on any surface either hydrophilic or hydrophobic with varying mean curvature are subject to driving forces toward the gradient direction of the mean curvature. The larger the mean curvature of the surface and the lower the contact angle of the liquid are, the stronger the driving force will be. This discovery can lead to more effective techniques for transporting droplets., Comment: 11 pages, 4 figures (supplementary information: 5 pages, 2 figures)

Earlier works found out spontaneous directional motion of liquid droplets on hydrophilic conical surfaces, however, not hydrophobic case. Here we show that droplets on any surface may take place spontaneous directional motion without considering contact angle property. The driving force is found to be proportional to the curvature gradient of the surface. Fast motion can be lead at surfaces with small curvature radii. The above discovery can help to create more effective transportation technology of droplets, and better understand some observed natural phenomena., Comment: 18 pages, 7 figures

Here we report a surface curvature-induced directional movement phenomenon, based on molecular dynamics simulations, that a nanoscale water droplet at the outer surface of a graphene cone always spontaneously moves toward the larger end of the cone, and at the inner surface toward the smaller end. The analysis on the van der Waals interaction potential between a single water molecule and a curved graphene surface reveals that the curvature with its gradient does generate the driving force resulting in the above directional motion. Furthermore, we found that the direction of the above movement is independent of the wettability, namely is regardless of either hydrophobic or hydrophilic of the surface. However, the latter surface is in general leading to higher motion speed than the former. The above results provide a basis for a better understanding of many reported observations, and helping design of curved surfaces with desired directional surface water transportation., Comment: 11 pages, 5 figures

Thousands of plant and animal species have been observed to have superhydrophobic surfaces that lead to various novel behaviors [1-5]. These observations have inspired attempts to create artificial superhydrophobic surfaces, given such surfaces have multitudinous applications [6-13]. Superhydrophobicity is an enhanced effect of surface roughness and there are known relationships that correlate surface roughness and superhydrophobicity, based on the underlying physics. However, while these recognize the level of roughness they tell us little about the independent effect of its scale. Thus, they are not capable of explaining why naturally occurring such surfaces commonly have micron-submicron sizes. Here we report on the discovery of a new relation, its physical basis and its experimental verification. The results reveal that scaling-down roughness into the micro-submicron range is a unique and elegant strategy to not only achieve superhydrophobicity but also to increase its stability against environmental disturbances. This new relation takes into account the previously overlooked but key fact that the accumulated line energy arising from the numerous solid-water-air intersections that can be distributed in the apparent contact area when air packets are trapped at small scales on the surface can dramatically increase as the roughness scale shrinks. This term can in fact become the dominant contributor to the surface energy and so becomes crucial for accomplishing superhydrophobicity. These findings guide fabrication of stable super water-repellant surfaces., Comment: arXiv-admin note: This version has been removed because it is in violation of arXiv's copyright policy