he contact angle of a liquid droplet on a surface under partial wetting conditions differs for a nanoscopically rough or periodically corrugated surface from its value for a perfectly flat surface. Wenzel's relation attributes this difference simply to the geometric magnification of the surface area (by a factor $r_{\rm w}$), but the validity of this idea is controversial. We elucidate this problem by model calculations for a sinusoidal corrugation of the form $z_{\rm wall}(y) = Δ\cos(2πy/λ)$ , for a potential of short range $σ_{\rm w}$ acting from the wall on the fluid particles. When the vapor phase is an ideal gas, the change of the wall-vapor surface tension can be computed exactly, and corrections to Wenzel's equation are typically of order $σ_{\rm w}Δ/λ^2$. For fixed $r_{\rm w}$ and fixed $σ_{\rm w}$ the approach to Wenzel's result with increasing $λ$ may be nonmonotonic and this limit often is only reached for $λ/σ_{\rm w}>30$. For a non-additive binary mixture, density functional theory is used to work out the density profiles of both coexisting phases both for planar and corrugated walls, as well as the corresponding surface tensions. Again, deviations from Wenzel's results of similar magnitude as in the above ideal gas case are predicted. Finally, a crudely simplified description based on the interface Hamiltonian concept is used to interpret corresponding simulation results along similar lines. Wenzel's approach is found to generally hold when $λ/σ_{\rm w}\gg 1$, $Δ/λ<1$, and conditions avoiding proximity of wetting or filling transitions.
Understanding the critical condition and mechanism of the droplet wetting transition between Cassie-Baxter state and Wenzel state triggered by an external electric field is of considerable importance because of its numerous applications in industry and engineering. However, such a wetting transition on a patterned surface is still not fully understood, e.g., the effects of electro-wetting number, geometry of the patterned surfaces, and droplet volume on the transition have not been systematically investigated. In this paper, we propose a theoretical model for the Cassie-Baxter- Wenzel wetting …
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…
The free-energy barrier of filling a spherical cavity having an inner wall of various wettabiities is studied. The morphology and free energy of a lens-shaped droplet are determined from the minimum of the free energy. The effect of line tension on the free energy is also studied. Then, the equilibrium contact angle of the droplet is determined from the generalized Young's equation. By increasing the droplet volume within the spherical cavity, the droplet morphology changes from spherical with an equilibrium contact angle of $180^{\circ}$ to a lens with a convex meniscus, where the morphologic…
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 …
In this paper we have derived Young's, Wenzel's and Cassie-Baxter's equations using conceptual model rather than showing rigorous derivation to help the new-comers in this field. We then pointed out that if the substrate is initially hydorphilic then one can modify the surface morphology and make the substrate to become hydrophobic or superhydrophobic. But, if the substrate is initially hydrophobic then one can only make it superhydrophobic but not hydrophilic by modifying the surface morphology using the formalisms mentioned in this paper
When a liquid drops impinges a hydrophobic rough surface it can either bounce off the surface (fakir droplets) or be impaled and strongly stuck on it (Wenzel droplets). The analysis of drop impact and quasi static ''loading'' experiments on model microfabricated surfaces allows to clearly identify the forces hindering the impalement transitions. A simple semi-quantitative model is proposed to account for the observed relation between the surface topography and the robustness of fakir non-wetting states. Motivated by potential applications in microfluidics and in the fabrication of self cleanin…
When placed on rough hydrophobic surfaces, water droplets of diameter larger than a few millimeters can easily form pearls, as they are in the Cassie-Baxter state with air pockets trapped underneath the droplet. Intriguingly, a natural evaporating process can drive such a Fakir drop into a completely wetting (Wenzel) state. Our microscopic observations with simultaneous side and bottom views of evaporating droplets upon transparent hydrophobic microstructures elucidate the water-filling dynamics and the mechanism of this evaporation-triggered transition. For the present material the wetting tr…
We consider a three dimensional liquid drop sitting on a rough and chemically heterogeneous substrate. Using a novel minimization technique on the free energy of this system, a generalized Young's equation for the contact angle is found. In certain limits, the Cassie and Wenzel laws, and a new equivalent rule, applicable in general, are derived. We also propose an equation in the same spirit as these results but valid on a more `microscopic' level. Throughout we work under the presence of gravity and keep account of line tension terms.
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…
Super-antiwettability is an extreme situation of wetting where liquids stay at the tops of rough surfaces, in the so-called Cassie state1. Owing to the dramatic reduction of solid/liquid contact, it has many applications, such as antifouling2,3, droplet manipulation4,5, and self-cleaning6-9. However, super-antiwettability is often destroyed by impalement transitions caused by environmental disturbances10-16 while inverse transitions without energy input have never been observed12,17-21. Here we show through controlled experiments that there is a "monostable" region in the phase space of the re…
The effect of pitch of the pillars and impact velocity are studied for the impact dynamics of a microliter water droplet on a micropillared hydrophobic surface. The results are presented qualitatively by the high-speed photography and quantitatively by the temporal variation of wetted diameter and droplet height. A characterization of the transient quantitative results is a novel aspect of our work. Three distinct regimes, namely, non-bouncing, complete bouncing and partial bouncing are presented. A critical pitch as well as impact velocity exists for the transition from one regime to another.…
In this paper, the self-assembly of condensed droplets on smooth and constrained surfaces under saturated vapor atmosphere of ethanol and methanol binary system is reported. Hexagonally ordered array of pores are obtained on smooth surfaces with saturated vapors of binary liquids without the assistance of any additives. The results show that the addition of small amount of ethanol to methanol plays a role very similar to that of surface active agents in inducing the formation of regular droplet array. The effect of constraints on self-assembled droplet pattern such as movement of contact line …
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…
We study the size and shape of the final deposit obtained when a drop with colloidal particles has dried on a super-hydrophobic surface made of micro-posts. As expected, most of the particles lie inside a circular area, which radius roughly corresponds to the Laplace pressure threshold for liquid impalement inside the structure (Cassie-Wenzel transition), inducing a coffee-stain deposit due to contact-line pinning. Less expected is the observation of tiny deposits on top of posts in the area external to the main ring, despite the low macroscopic liquid/solid friction. Experiments are carried o…
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…
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…
The effects of line tension on the morphology of a sessile droplet placed on top of a convex spherical substrate are studied. The morphology of the droplet is determined from the global minimum of the Helmholtz free energy. The contact angle between the droplet and the spherical substrate is expressed by the generalized Young's formula. When the line tension is positive and large, the contact angle jumps discontinuously to $180^{\circ}$, the circular contact line shrinks towards the top of the substrate, and the droplet detaches from the substrate, forming a spherical droplet if the substrate …
Dynamic Wettability Modulation of Textured, Soft and LIS Interfaces Using Electrowetting
School of Mechanical Sciences, Indian Institute of Technology (IIT) Goa（印度理工学院果阿分校机械科学学院）；Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur（印度理工学院卡拉格普尔分校机械工程系）
The morphological phase transition between a sessile and lenticular shapes of a droplet placed in a nanochannel is observed upon increasing the droplet volume. The phase diagram for this system is discussed within the macro- and mesoscopic approaches. On the mesoscopic level, the van der Waals forces are taken into account via the effective interface potential acting between the channel walls and the droplet. We discuss the contact angle dependence on the droplet volume and the distance between the walls; this angle turns out to be smaller than the macroscopic Young's angle. The droplet presen…
When a drop of water is placed on a rough surface, there are two possible extreme regimes of wetting: the one called Cassie-Baxter (CB) with air pockets trapped underneath the droplet and the one characterized by the homogeneous wetting of the surface, called the Wenzel (W) state. A way to investigate the transition between these two states is by means of evaporation experiments, in which the droplet starts in a CB state and, as its volume decreases, penetrates the surface's grooves, reaching a W state. Here we present a theoretical model based on the global interfacial energies for CB and W s…
The study of wetting phenomena is of great interest due to the multifaceted technological applications of hydrophobic and hydrophilic surfaces. The theoretical approaches proposed by Wenzel and later by Cassie and Baxter to describe the behaviour of a droplet of water on a rough solid were extensively used and improved to characterize the apparent contact angle of a droplet. However, the equilibrium hypothesis implied in these models means that they are not always predictive of experimental contact angles due to strong metastabilities typically occurring on heterogeneous surfaces. A predictive…
A generalized Young's equation, which takes into account two corrections to the line tension by the curvature dependence of the liquid-vapor surface tension and by the contact angle dependence of the intrinsic line tension, is derived from the thermodynamic free-energy minimization. The correction from the curvature dependence can be qualitatively estimated using Tolman's formula. The correction from the contact angle dependence can be estimated for nanometer-scale droplets for which the analytical formula for the intrinsic line tension determined from the van der Waals interaction is availabl…
In this work we consider two possible wetting states for a droplet when placed on a substrate: the Fakir configuration of a Cassie-Baxter (CB) state with a droplet residing on top of roughness grooves and the Wenzel (W) state characterized by the homogeneous wetting of the surface. We extend a theoretical model based on the global interfacial energies for both states to study the wetting behavior of simple and double reentrant surfaces. Due to the minimization of the energies associated to each wetting state, we predict the thermodynamic wetting state of the droplet for a given surface and obt…
It is established that roughness and chemistry play a crucial role in the wetting properties of a substrate. Yet, few studies have analyzed systematically the effect of the non-uniformity in the distribution of texture and surface tension of substrates on its wetting properties. In this work we investigate this issue theoretically and numerically. We propose a continuous model that takes into account the total energy required to create interfaces of a droplet in two possible wetting states: Cassie-Baxter(CB) with air pockets trapped underneath the droplet; and the other characterized by the ho…
Wetting is a widespread phenomenon, most prominent in a number of cases, both in nature and technology. Droplets of pure water with initial radius ranging from 20 to 80 [Å] spreading on graphitic surfaces are studied by molecular dynamics simulations. The equilibrium contact angle is determined and the transition to the macroscopic limit is discussed using Young equation in its modified form. While the largest droplets are almost perfectly spherical, the profiles of the smallest ones are no more properly described by a circle. For the sake of accuracy, we employ a more general fitting procedur…
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…
The effects of line tension on the morphology of a lens-shaped droplet and bubble placed on the inner wall of a spherical cavity are studied. The contact angle between the lens-shaped droplet and the concave spherical substrate is expressed by the generalized Young's formula. The equator of the spherical substrate is found to play a crucial role. Neither a droplet with its contact line on the upper hemisphere of the substrate nor one with its contact line on the lower hemisphere can transform into each other continuously. On a hydrophobic substrate, the contact angle jumps discontinuously to $…
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 …
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…
The production of small fluid droplets relies on an instability of solutions to the Young-Laplace equation. We ask whether smaller droplets can be produced by changing the shape of the nozzle. At a given critical pressure, the circular nozzle actually produces the largest droplet. The droplet volume can be decreased by up to 18% using a triangular nozzle with stretched corners.
An efficient technique for the evaluation of the Gibbs adsorption of a liquid on a solid substrate is presented. The behavior of a water nanodroplet on a silicon surface is simulated with molecular dynamics. An external field with varying strength is applied on the system to tune the solid-liquid interfacial contact area. A linear dependence of droplet's volume on the contact area is observed. Our modified Young--Laplace equation is used to explain the influence of the Gibbs adsorption on the nanodroplet volume contraction. Fitting of the molecular dynamics results with these of an analytical …
To evaluate the role of common substrates in the transmission of respiratory viruses, uniformly distributed microdroplets of artificial saliva were generated using an advanced inkjet printing technology to replicate the aerosol droplets, and subsequently deposited on five substrates, including glass, PTFE, stainless steel, ABS, and melamine. The droplets were found to evaporate within a short timeframe, which is consistent with previous reports concerning the drying kinetics of picolitre droplets. Using fluorescence microscopy and atomic force microscopy, we found that the surface deposited mi…
Heat transfer via phase change is a major contributor to heat removal in numerous engineering applications. Thin films of liquid result in increased heat transfer due to a reduction of conduction resistance, in addition the pressure jump at the liquid-vapor interface also affects the rate and direction of rate of phase change. Because of these effects the morphology of the substrate surface is expected to affect the film shape, hence heat transfer, especially in thin films. In this study, the influence of surface characteristics on the rate of phase change from micron and sub-micron sized 2-D …
Growing tissues are highly dynamic, flowing on sufficiently long time-scales due to cell proliferation, migration and tissue remodeling. As a consequence, living tissues can be approximated as liquids. This means the shape of microtissues is governed by a surface stress state as in fluid droplets. Recent work showed that cells in the near-surface region of fibroblastic or osteoblastic microtissues contract with highly oriented actin filaments, thus making the surface stress state anisotropic, in contrast to what is expected for an isotropic fluid. Here, we extend the Young-Laplace law to inclu…
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…
The equilibrium properties of polymer droplets on a soft deformable surface are investigated by molecular dynamics simulations of a bead-spring model. The surface consists of a polymer brush with irreversibly end-tethered linear homopolymer chains onto a flat solid substrate. We tune the softness of the surface by varying the grafting density. Droplets are comprised of bead-spring polymers of various chain lengths. First, both systems, brush and polymer liquid, are studied independently in order to determine their static and dynamic properties. In particular, using a numerical implementation o…
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…
Modern pendant drop tensiometry relies on numerical solution of the Young-Laplace equation and allow to determine the surface tension from a single picture of a pendant drop with high precision. Most of these techniques solve the Young-Laplace equation many times over to find the material parameters that provide a fit to a supplied image of a real droplet. Here we introduce a machine learning approach to solve this problem in a computationally more efficient way. We train a deep neural network to determine the surface tension of a given droplet shape using a large training set of numerically g…
Hypothesis Emerging energy-related technologies deal with multiscale hierarchical structures, intricate surface morphology, non-axisymmetric interfaces, and complex contact lines where wetting is difficult to quantify with classical methods. We hypothesis that a universal description of wetting on multiscale surfaces can be developed by using integral geometry coupled to thermodynamic laws. The proposed approach separates the different hierarchy levels of physical description from the thermodynamic description, allowing for a universal description of wetting on multiscale surfaces. Theory and …
Droplet interface bilayers are a convenient model system to study the physio-chemical properties of phospholipid bilayers, the major component of the cell membrane. The mechanical response of these bilayers to various external mechanical stimuli is an active area of research due to implications for cellular viability and development of artificial cells. In this manuscript we characterize the separation mechanics of droplet interface bilayers under step strain using a combination of experiments and numerical modeling. Initially, we show that the bilayer surface energy can be obtained using prin…
We consider the equilibrium of liquid droplets sitting on thin elastic sheets that are subject to a boundary tension and/or are clamped at their edge. We use scaling arguments, together with a detailed analysis based on the Föppl-von-Kármán equations, to show that the presence of the droplet may significantly alter the stress locally if the tension in the dry sheet is weak compared to an intrinsic elasto-capillary tension scale $γ^{2/3}(Et)^{1/3}$ (with $γ$ the droplet surface tension, $t$ the sheet thickness and $E$ its Young modulus). Our detailed analysis suggests that some recent experimen…
Accurate estimation of surface wettability for various degrees of hydrophobicity becomes increasingly important in the molecular design of membrane. In this paper, we develop simple yet physically realistic model for estimating contact angle via hybridizing molecular dynamics and pseudocontinuum theory. Molecular dynamics simulations were carried out to compute the macro-scale contact angle between a water droplet and smooth walls from the nanoscale calculations. A macro-level droplet including countless degrees of freedom due to an infinite number of molecules is impossible to be studied dire…
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…
In this paper, the pinning and depinning mechanism of the contact line during droplet evaporation on chemically stripe-patterned surfaces is numerically investigated using a thermal multiphase lattice Boltzmann (LB) model with liquid-vapor phase change. A local force balance in the context of diffuse interfaces is introduced to explain the equilibrium states of droplets on chemically patterned surfaces. It is shown that, when the contact line is pinned on a hydrophobic-hydrophilic boundary, different contact angles can be interpreted as the variation of the length of the contact line occupied …
Migrating cell clusters as active droplets with asymmetric shape and contact angles
Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；Center for Systems Biology Dresden（德累斯顿系统生物学中心）；Cluster of Excellence Physics of Life, Technische Universität Dresden（德累斯顿工业大学生命物理卓越集群）；Departament de Física de la Matèria Condensada, Universitat de Barcelona（巴塞罗那大学凝聚态物理系）；Universitat de Barcelona Institute of Complex Systems (UBICS)（巴塞罗那大学复杂系统研究所）；Institució Catalana de Recerca i Estudis Avançats (ICREA)（加泰罗尼亚研究高级学院）
We use the Poisson-Boltzmann theory to predict contact angle saturation of aqueous droplets in electrowetting. Our theory predicts that injection of ions from the droplet into its surrounding medium is responsible for the deviation of the apparent contact angle from the Young-Lippmann equation for large applied voltages. The ion injection substantially decreases the Maxwell stress and increases the osmotic pressure at the interface between the two media, leading to saturation of the apparent contact angle. Moreover, we find that the contact angle does not saturate, but only has a broad minimum…
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 drople…
Soochow University（苏州大学）；Department of Physics, Technion – Israel Institute of Technology（以色列理工学院物理系）；Department of Physics, Massachusetts Institute of Technology（麻省理工学院物理系）
Soft particles can be better emulsifiers than hard particles because they stretch at fluid interfaces. This deformation can increase adsorption energies by orders of magnitude relative to rigid particles. The deformation of a particle at an interface is governed by a competition of bulk elasticity and surface tension. When particles are partially wet by the two liquids, deformation is localized within a material-dependent distance $L$ from the contact line. At the contact line, the particle morphology is given by a balance of surface tensions. When the particle radius $R \ll L$, the particle a…
In this paper, we report experimental results on the shape and motion of a mercury droplet, placed in a horizontally rotating cylinder in the rpm range 8-93, so that the Reynolds number of the drop 2500<Re<26000 and its capillary number 0.0002<Ca<0.0023. When contact angle variations can be neglected at low speeds (Re<8150), the velocity of the drop is much lower than that predicted by the Ho. Young Kim's [6] relation. This observed discrepancy is overcome by modifying Kim's relation by substituting the dissipation estimated from a boundary layer near the solid surface instead of bulk dissipat…
We present a lattice-Boltzmann method that can simulate the coupled hydrodynamics and electrostatics equations of motion of a two-phase fluid as a means to model electrowetting phenomena. Our method has the advantage of modelling the electrostatic fields within the lattice-Boltzmann algorithm itself, eliminating the need for a hybrid method. We validate our method by reproducing the static equilibrium configuration of a droplet subject to an applied voltage and show that the apparent contact angle of the drop depends on the voltage following the Young-Lippmann equation up to contact angles of …
