Anomalous contact angle hysteresis of a captive bubble: advancing contact line pinning

Life beyond Fritz: On the Detachment of Electrolytic Bubbles

We present an experimental study on detachment characteristics of hydrogen bubbles during electrolysis. Using a transparent (Pt or Ni) electrode enables us to directly observe the bubble contact line and bubble size. Based on these quantities we determine other parameters such as the contact angle and volume through solutions of the Young-Laplace equation. We observe bubbles without ("pinned bubbles") and with ("spreading bubbles") contact line spreading and find that the latter mode becomes more prevalent if the concentration of HClO4 is ≥0.1 M. The departure radius for spreading bubbles is found to drastically exceed the value predicted by the well-known formula of W. Fritz [Phys. Z. 1935, 36, 379-384] for this case. We show that this is related to the contact line hysteresis, which leads to pinning of the contact line after an initial spreading phase at the receding contact angle. The departure mode is then similar to a pinned bubble and occurs once the contact angle reaches the advancing contact angle of the surface. A prediction for the departure radius based on these findings is found to be consistent with the experimental data.

A new methodology for measuring solid/liquid interfacial energy

HYPOTHESIS

The interfacial energy γ_sl between a solid and a liquid designates the affinity between these two phases, and in turn, the macroscopic wettability of the surface by the fluid. This property is needed for precise control of fluid-transport phenomena that affect the operation/quality of commercial devices/products. Although several indirect or theoretical approaches can quantify the solid/liquid interfacial energy, no direct experimental procedure exists to measure this property for realistic (i.e. rough) surfaces. Makkonen hypothesized that the frictional resistance force per unit contact-line length is equal to the interfacial energy on smooth surfaces, which, however, are rarely found in practice. Consequently, the hypothesis that Makkonen's assumption may also hold for rough surfaces (which are far more common in practice) arises naturally. If so, a reliable and simple experimental methodology of obtaining γ_sl for rough surfaces can be put forth. This is accomplished by performing dynamic contact-angle experiments on rough surfaces that quantify the relationship between the frictional resistance force per unit contact-line length acting on an advancing liquid (F_p,a) and the surface roughness in wetting configurations.

EXPERIMENT

We perform static and advancing contact-line experiments with aqueous and organic liquids on different hydrophilic surfaces (Al, Cu, Si) with varying Wenzel roughnesses in the range 1-2. These parameters are combined with the liquid's known surface tension to determine F_p,a.

FINDINGS

F_p,a rises linearly with the surface roughness. Analysis based on existing theories of wetting and contact-angle hysteresis reveals that the slope of F_p,a vs.Wenzel roughness is equal to the solid/liquid interfacial energy, which is thus determined experimentally with the present measurements. Interfacial energies obtained with this experimental approach are within 12% of theoretically predicted values for several solid/liquid pairs, thereby validating this methodology.

Implementing Contact Angle Hysteresis in Moving Mesh-Based Two-Phase Flow Numerical Simulations

Contact angle hysteresis is a common phenomenon in nature, which also plays an important role in industrial applications. A numerical model based on the moving mesh two-phase flow method is presented for modeling contact angle hysteresis. The implementation includes a displacement-based penalty method and a state variable method. The pinning, moving, and repinning of the contact lines can be simulated. This method is robust considering both two-dimensional and three-dimensional geometries. To further demonstrate the performance of this method, a fluid-solid interaction model with a cylinder fluctuating on a water surface considering contact angle hysteresis is demonstrated.

Size-dependent behavior and failure of young's equation for wetting of two-component nanodroplets

HYPOTHESIS

For macroscopic systems, the interfacial properties are size-independent and Young's equation is generally valid for smooth substrates. For nanoscale systems, however, size-dependence and failure of Young's equation may emerge.

EXPERIMENTS

The wetting behavior of a nanodroplet containing two miscible liquids on a smooth substrate is explored by many-body dissipative particle dynamics simulations. The size-dependent surface tension of nanofilms is investigated as well.

FINDINGS

It is found that Young's equation is valid for nanodroplets of pure fluids but fails for two-component nanodroplets. The actual contact angle is always larger than the Young's contact angle, and their difference is getting smaller as the composition approaches pure fluids or the compatibility of the mixture is increased. The failure of Young's equation is closely associated with the size-dependent behavior in two-component nanodroplets and nanofilms. As the nanodroplet size is increased, the actual contact angle is found to decline but approaches a constant expected in macroscopic systems. Similarly, as the nanofilm thickness is increased, surface tension decreases and reaches its macroscopic value. The change of surface tension is attributed to the size-dependent surface composition, which is responsible for the failure of Young's equation.

Revisiting the supplementary relationship of dynamic contact angles measured by sessile-droplet and captive-bubble methods: Role of surface roughness

HYPOTHESIS

Quantitative characterization of surface wettability through contact angle (CA) measurement using the sessile droplet (SD) or captive bubble (CB) methods is often limited by the intrinsic wetting properties of the substrate. Situations may arise when an extreme surface wettability may preclude using one of the two methods for predicting the behaviors of droplets or bubbles on the surface. This warrants a relationship between the dynamic CAs measured via the SD and CB methods. While the two dynamic CAs (e.g., the advancing CA of SD and receding CA of CB) add up to 180° on a smooth surface, the simple geometric supplementary principle may not apply for rough surfaces.

EXPERIMENTS

We perform a systematic wettability characterization of solid substrates with varying degrees of roughness using the sessile-droplet and captive-bubble methods, and interpret the experimental observations using a theoretical model.

FINDINGS

The dynamic contact angles measured by the sessile-droplet and captive-bubble methods deviate from the supplementary principle as the surface roughness is increased. We present a theoretical explanation for this disparity and predict the values of the contact angles using prevalent thermodynamic models of wetting and contact-angle hysteresis on rough substrates. The theoretical prediction is in good agreement with the experimental observations.

Influence of surface treatment on PEDOT coatings: surface and electrochemical corrosion aspects of newly developed Ti alloy

Surface treatment of metallic materials prior to the application of polymer coatings plays an important role in providing improved surface features and enhanced corrosion protection. In the current investigation, we aimed to evaluate the effect of surface treatment of newly developed TiNbZr (TNZ) alloys on the surface characteristics, including the surface topography, morphology, hydrophobicity and adhesion strength of subsequent poly(3,4-ethylenedioxythiophene) (PEDOT) coatings. The surface morphology, chemical composition, and surface roughness of both treated and coated alloys were characterized by scanning electron microscopy, energy dispersive spectroscopy, and optical profilometry, respectively. The adhesion strength of the coating was measured using a micro scratch machine. Furthermore, we also evaluated the performance of electrochemically synthesized PEDOT coatings on surface-treated TNZ alloys in terms of the surface protective performance in simulated body fluid (SBF) and in vitro bioactivity in osteoblast MG63 cells. Surface analysis findings indicated that the nature of the PEDOT coating (surface morphology, topography, wettability and adhesion strength) was intensely altered, while the surface treatment performed before electrodeposition facilitated the overall performance of PEDOT coatings as implant coating materials. The obtained corrosion studies confirmed the enhanced corrosion protection performance of PEDOT coatings on treated TNZ substrates. In vitro cell culture studies validated the improved cell adhesion and proliferation rate, further highlighting the important role of surface treatment before electrodeposition.

Shapes of Splattered Drops

Drops that impact and stick to a surface (splattered drops) commonly show noncircular triple lines. Physical or chemical defects on the surface are known to pin the triple line in this static metastable state. We report an experimental study to relate the defect distribution on a surface to the triple-line microstructure of such drops. Triple lines of an ensemble of splattered drops have been imaged on a range of surfaces varying in wetting properties. Local contact angles have been calculated, and the microscale pinning force distribution has been estimated. We propose a novel method of estimating defect strength distribution from the pinning forces, using extreme value analysis. From this analysis, we show that pinning force distributions have finite upper and lower bounds. We show that most common surfaces show both hydrophobic and hydrophilic defects, but their strength distributions are asymmetric in relation to the surface's advancing and receding angles. In addition, we show that the range of microscopic pinning forces varies linearly with macroscopic contact angle hysteresis but, surprisingly, with a nonzero intercept. We explain the intercept by drawing an analogy to static and dynamic friction.

Extraordinarily Rapid Rise of Tiny Bubbles Sliding beneath Superhydrophobic Surfaces

Tiny bubbles readily stick onto substrates owing to contact angle hysteresis (CAH). Nevertheless, they can slide slowly on a tilted surface with ultralow CAH because capillarity is overcome by buoyancy. It is surprising to observe experimentally that bubbles of 3-15 μL (diameter 1.79-3.06 mm) slide beneath a tilted superhydrophobic surface at a vertical ascent rate faster than that of freely rising ones of high Reynold numbers ≈O(10²). As the tilting angle increases, the drag coefficient remains essentially the same as that of a freely rising bubble, but the frontal area of the flat bubble rises monotonically. Nonetheless, the frontal area of the sliding bubble always stays much smaller than that of a freely rising bubble. Consequently, the small drag force associated with the sliding bubbles is attributed to their substantially small frontal areas on superhydrophobic surfaces.

Wetting hysteresis of nanodrops on nanorough surfaces

Nanodrops on smooth or patterned rough surfaces are explored by many-body dissipative particle dynamics to demonstrate the influence of surface roughness on droplet wetting. On a smooth surface, nanodrops exhibit the random motion and contact angle hysteresis is absent. The diffusivity decays as the intrinsic contact angle (θ_{Y}) decreases. On a rough surface, the contact line is pinned and the most stable contact angle (θ_{Y}^{'}) is acquired. The extent of contact angle hysteresis (Δθ) is determined by two approaches, which resemble the inflation-deflation method and inclined plane method for experiments. The hysteresis loop is acquired and both approaches yield consistent results. The influences of wettability and surface roughness on θ_{Y}^{'} and Δθ are examined. θ_{Y}^{'} deviates from that estimated by the Wenzel or Cassie-Baxter models. This consequence can be explained by the extent of impregnation, which varies with the groove position and wettability. Moreover, contact angle hysteresis depends more on the groove width than the depth.

Meniscus Shape and Wetting Competition of a Drop between a Cone and a Plane

The formation of a liquid bridge between a cone and a plane is related to dip-pen nanolithography. The meniscus shape and rupture process of a liquid meniscus between a cone and a plane are investigated by Surface Evolver, many-body dissipative particle dynamics, and macroscopic experiments. Dependent on the cone geometry, cone-plane separation, and wetting properties of cone and plane, three types of menisci can be observed before rupture and two types of wetting competition outcomes are seen after breakup. It is interesting to find that after rupture, the bulk of the liquid bridge volume is not necessarily retained by the cone which is more wettable. In fact, a sharp hydrophilic cone often loses wetting competition to a hydrophobic plane. To explain our findings, the "apparent" contact angle of the cone is introduced and the behavior of drop-on-cone/plane system is analogous to that of a liquid bridge between two parallel planes based on this concept.

Temperature Dependence of the Surface and Volume Hydrophilicity of Hydrophilic Polymer Brushes

The temperature-dependence of the volume and surface hydrophilicity of a series of water-swollen dense polymer brushes is measured by contact angle measurements in the captive bubble configuration, by ellipsometry, and by quartz crystal microbalance with dissipation monitoring (QCM-D). Thermoresponsive poly(N-isopropylacrylamide) (PNIPAM) and poly(di(methoxyethoxy)ethyl methacrylate) (PMEO2MA), strongly hydrophilic poly(N,N-dimethylacrylamide) (PDMA) and poly(oligo(ethylene glycol) methacrylate) (POEGMA), and weakly hydrophilic poly(2-hydroxyethyl methacrylate) (PHEMA) brushes were synthesized by surface-initiated atom-transfer radical polymerization (SI-ATRP). Conditions leading to reproducible measurements of the contact angle are first provided, giving access to the surface hydrophilicity. Volume hydrophilicity is quantified by measuring the swelling of the brushes, either by QCM-D or by ellipsometry. A model-free methodology is proposed to analyze the QCM-D data. Comparison between the acoustic and optical swelling coefficients shows that QCM-D is sensitive to the maximal thickness of swollen brushes, while ellipsometry provides an integral thickness. Diagrams of surface versus volume hydrophilicity of the brushes finally lead to identify two types of behavior: strongly water-swollen brushes exhibit a progressive decrease of volume hydrophilicity with temperature, while surface hydrophilicity changes moderately; weakly water-swollen brushes have a close-to-constant volume hydrophilicity, while surface hydrophilicity decreases with temperature. Thermoresponsive brushes abruptly switch from one behavior to the other, and do not exhibit an abrupt change of surface hydrophilicity across their collapse transition contrarily to a common erroneous belief. In general, there is no direct correlation between surface and volume hydrophilicity, because surface properties are dependent on the details of conformation and composition at the surface, whereas volume properties are averaged over a finite region within the brush.

Superhydrophilicity and spontaneous spreading on zwitterionic surfaces: carboxybetaine and sulfobetaine

Wetting behavior of zwitterionic surfaces fabricated by grafting sulfobetaine silane (SBSi) and carboxybetaine silane (CBSi) on glass slides.

Anti-smudge behavior of facilely fabricated liquid-infused surfaces with extremely low contact angle hysteresis property

Facile fabrication of hysteresis-free liquid-infused surfaces.

Air pocket stability and the imbibition pathway in droplet wetting

The stability of air pockets formed in grooves on a surface is relevant to contact angle hysteresis of droplet wetting and it is investigated by imbibition experiments and surface evolver (SE) simulations. Liquid drops of different wettabilities are placed atop a conical hole on a polymethyl methacrylate (PMMA) substrate. The stability of the air pocket depends on surface wettability. Four kinds of imbibition behaviors ranging from wetting to nonwetting are observed. The imbibition pathway for the kinetically unstable air pocket is observed by using the olive oil droplet. It involves an inward flow of a thin liquid film along the wall of the hole. The accumulation of liquid at the bottom leads to the rise of the air bubble. The energy-barrier profile associated with the imbibition pathway acquired by SE simulations is able to interpret the outcome of imbibition. The advancing and receding contact angles of various liquids on a PMMA substrate with drilled holes are also determined. Their wetting behaviors can be categorized into three types. Our experimental results for substrates with or without fluorination are in good agreement with the theory based on the stability of air pockets.

Plastron-Mediated Growth of Captive Bubbles on Superhydrophobic Surfaces

Captive bubbles on a superhydrophobic (SH) surface have been shown to increase in volume via injection of air through the surrounding plastron. The experimental contact diameter against volume trends were found to follow that predicted by the Surface Evolver simulation generally but corresponded with the simulated data at contact angle (CA) = 158° when the volume was 20 μL but that at CA = 170° when the volume was increased to 180 μL. In this regime, there was a simultaneous outward movement of the contact line as well as a small reduction in the slope that the liquid-air interface makes with the horizontal as air was injected. At volumes higher than 180 μL, air injection caused the diameter to reduce progressively until detachment. The inward movement of the contact line in this regime allowed the bubble body to undergo shape deformations to stay attached onto the substrate with larger volumes (300 μL) than predicted (220 μL at CA = 170°) using simulation. In experiments to investigate the effect of translating the SH surface, movement of captive bubbles was possible with 280 μL volume but not with 80 μL volume. This pointed to the possibility of transporting gas-phase samples on SH surfaces using larger captive bubble volumes.

Ultralow voltage irreversible electrowetting dynamics of an aqueous drop on a stainless steel surface

The electrowetting dynamics of a water drop on a stainless steel surface in air is investigated under ultralow voltages. The spreading behavior can be classified into three regimes. The drop expands slowly in regime I, but the spreading accelerates quite rapidly in regime II. The spreading becomes insignificant in regime III. The experimental results are compared to the equilibrium shapes acquired by Surface Evolver simulations. The good agreement between them indicates that the slow electrowetting dynamics can be considered to be a quasi-equilibrium process. The influences of the electric field and drop size on the spreading dynamics are examined. The variation of both the contact angle and base diameter with time in regimes II and III can be well described by the exponential change with a characteristic time, which grows with the drop volume but is inversely proportional to the electric field. A simple model based on the electromechnical mechanism is proposed to explain the spreading dynamics. The exponential change is attributed to ion migration from the bulk of the drop to the contact line. The experimental results agree well with the prediction of our simple theory.

Solute concentration-dependent contact angle hysteresis and evaporation stains

The presence of nonvolatile solutes in a liquid drop on a solid surface can affect the wetting properties. Depending on the surface-activity of the solutes, the extent of contact angle hysteresis (CAH) can vary with their concentration and the pattern of the evaporation stain is altered accordingly. In this work, four types of concentration-dependent CAH and evaporation stains are identified for a water drop containing polymeric additives on polycarbonate. For polymers without surface-activity such as dextran, advancing and receding contact angles (θa and θr) are independent of solute concentrations, and a concentrated stain is observed in the vicinity of the drop center after complete evaporation. For polymers with weak surface-activity such as poly(ethylene glycol) (PEG), both θa and θr are decreased by solute addition, and the stain pattern varies with increasing PEG concentration, including a concentrated stain and a mountain-like island. For polymers with intermediate surface-activity such as sodium polystyrenesulfonate (NaPSS), θa descends slightly, but θr decreases significantly after the addition of a substantial amount of NaPSS, and a ring-like stain pattern is observed. Moreover, the size of the ring stain can be controlled by NaPSS concentration. For polymers with strong surface-activity such as poly(vinylpyrrolidone) (PVP), θa remains essentially a constant, but θr is significantly lowered after the addition of a small amount of PVP, and the typical ring-like stain is seen.

Hydrostatic pressure effect on micro air bubbles deposited on surfaces with a retreating tip

The effect of hydrostatic pressure on 6 μL air bubbles formed on micropillar structured PDMS and silicone surfaces using a 2 mm diameter stainless steel tip retreated at 1 mm/s was investigated. Dimensional analysis of the tip retraction process showed the experiments to be conducted in the condition where fluid inertial forces are comparable in magnitude with surface tension forces, while viscous forces were lower. Larger bubbles could be left behind on the structured PDMS surface. For hydrostatic pressures in excess of 20 mm H2O (196 Pa), the volume of bubble deposited was found to decrease progressively with pressure increase. The differences in width of the deposited bubbles (in contact with the substrate) were significant at any particular pressure but marginal in height. The attainable height before rupture reduced with pressure increase, thereby accounting for the reducing dispensed volume characteristic. On structured PDMS, the gaseous bridge width (in contact with the substrate) was invariant with tip retraction, while on silicone it was initially reducing before becoming invariant in the lead up to rupture. With silicone, hence, reductions in the contact width and height were both responsible for reduced volumes with pressure increase. Increased hydrostatic pressure was also found to restrict the growth in contact width on silicone during the stage when air was injected in through the tip. The ability to effect bubble size in such a simple manner may already be harnessed in nature and suggests possibilities in technological applications.

Growth of bubbles on a solid surface in response to a pressure reduction

A diffusion-controlled method is presented to study the growth of bubbles on a solid surface. The bubbles are nucleated spontaneously on a hydrophobic smooth surface in response to a sudden pressure reduction and then grow with an expanding contact line. The evolution of the bubbles in the early stage is found to grow with a constant bubble radius and a decreasing contact angle, while the bubbles continue their growth with a constant contact angle and an increasing bubble radius after the contact angle reaches its equilibrium value. A total variation of about 60° of the contact angle is observed during the growth of the bubbles with the size scale of 10-100 μm in radius. The growing process is described by the diffusion theory with the validation of the growth constant.

Superoleophobic textured copper surfaces fabricated by chemical etching/oxidation and surface fluorination

We report a convenient route to fabricate superoleophobic surfaces (abridged as SOS) on copper substrate by combining a two-step surface texturing process (first, the substrate is immersed in an aqueous solution of HNO3 and cetyltrimethyl ammonium bromide, and then in an aqueous solution of NaOH and (NH4)2S2O8) and succeeding surface fluorination with 1H,1H,2H,2H-perfluorodecanethiol (PFDT) or 1-decanethiol. The surface morphologies and compositions were characterized by field emission scanning electron microscopy and X-ray diffraction, respectively. The results showed that spherical micro-pits (SMP) with diameter of 50-100 μm were formed in the first step of surface texturing; in the second step, Cu(OH)2 or/and CuO with structures of nanorods/microflowers/microballs were formed thereon. The surface wettability was further assessed by optical contact angle meter by using water (surface tension of 72.1 mN m(-1) at 20°C), rapeseed oil (35.7 mN m(-1) at 20°C), and hexadecane (25.7 mN m(-1) at 20°C) as probe liquids. The results showed that, as the surface tension decreasing, stricter choosing of surface structures and surface chemistry are required to obtain SOS. Specifically, for hexadecane, which records the lowest surface tension, the ideal surface structures are a combination of densely distributed SMP and nanorods, and the surface chemistry should be tuned by grafted with low-surface-energy molecules of PFDT. Moreover, the stability of the so-fabricated sample was tested and the results showed that, under the testing conditions, superhydrophobicity and superoleophobicity may be deteriorated after wear/humidity resistance test. Such deterioration may be due to the loss of outermost PFDT layer or/and the destruction of the above-mentioned ideal surface structures. For UV and oxidation resistance, the sample remained stable for a period of 10 days.

CO2 adhesion on hydrated mineral surfaces

Hydrated mineral surfaces in the environment are generally hydrophilic but in certain cases can strongly adhere CO2, which is largely nonpolar. This adhesion can significantly alter the wettability characteristics of the mineral surface and consequently influence capillary/residual trapping and other multiphase flow processes in porous media. Here, the conditions influencing adhesion between CO2 and homogeneous mineral surfaces were studied using static pendant contact angle measurements and captive advancing/receding tests. The prevalence of adhesion was sensitive to both surface roughness and aqueous chemistry. Adhesion was most widely observed on phlogopite mica, silica, and calcite surfaces with roughness on the order of ~10 nm. The incidence of adhesion increased with ionic strength and CO2 partial pressure. Adhesion was very rarely observed on surfaces equilibrated with brines containing strong acid or base. In advancing/receding contact angle measurements, adhesion could increase the contact angle by a factor of 3. These results support an emerging understanding of adhesion of, nonpolar nonaqueous phase fluids on mineral surfaces influenced by the properties of the electrical double layer in the aqueous phase film and surface functional groups between the mineral and CO2.

Trapped liquid drop at the end of capillary

The liquid drop captured at the capillary end, which is observed in capillary valve and pendant drop technique, is investigated theoretically and experimentally. Because of contact line pinning of the lower meniscus, the lower contact angle is able to rise from the intrinsic contact angle (θ*) so that the external force acting on the drop can be balanced by the capillary force. In the absence of contact angle hysteresis (CAH), the upper contact angle remains at θ*. However, in the presence of CAH, the upper contact angle can descend to provide more capillary force. The coupling between the lower and upper contact angles determines the equilibrium shape of the captured drop. In a capillary valve, the pinned contact line can move across the edge as the pressure difference exceeds the valving pressure, which depends on the geometrical characteristic and wetting property of the valve opening. When CAH is considered, the valving pressure is elevated because the capillary force is enhanced by the receding contact angle. For a pendant drop under gravity, the maximal capillary force is achieved as the lower contact angle reaches 180° in the absence of CAH. However, in the presence of CAH, four regimes can be identified by three critical drop volumes. The lower contact angle can exceed 180°, and therefore the drop takes on the shape of a light bulb, which does not exist in the absence of CAH. The comparisons between Surface Evolver simulations and experiments are quite well.

Evaporation stains: suppressing the coffee-ring effect by contact angle hysteresis

A ring-shaped stain is frequently left on a substrate by a drying drop containing colloids as a result of contact line pinning and outward flow. In this work, however, different patterns are observed for drying drops containing small solutes or polymers on various hydrophilic substrates. Depending on the surface activity of solutes and the contact angle hysteresis (CAH) of substrates, the pattern of the evaporation stain varies, including a concentrated stain, a ringlike deposit, and a combined structure. For small surface-inactive solutes, the concentrated stain is formed on substrates with weak CAH, for example, copper sulfate solution on silica glass. On the contrary, a ringlike deposit is developed on substrates with strong CAH, for example, a copper sulfate solution on graphite. For surface-active solutes, however, the wetting property can be significantly altered and the ringlike stain is always visible, for example, Brij-35 solution on polycarbonate. For a mixture of surface-active and surface-inactive solutes, a combined pattern of a ringlike and concentrated stain can appear. For various polymer solutions on polycarbonate, similar results are observed. Concentrated stains are formed for weak CAH such as sodium polysulfonate, and ring-shaped patterns are developed for strong CAH such as poly(vinyl pyrrolidone). The stain pattern is actually determined by the competition between the time scales associated with contact line retreat and solute precipitation. The suppression of the coffee-ring effect can thus be acquired by the control of CAH.

Trapped liquid drop in a microchannel: multiple stable states

A liquid drop trapped in a microchannel, in which both contact angle (wettability) and opening angle (geometry) can vary with position, is investigated based on the minimization of free energy. The calculus of variation yields the Young-Laplace equation and its further integration leads to the general force balance. The equilibrium position of the trapped drop is determined by the balance between the area-mean capillary force and the area-mean hydrostatic pressure difference. Trapped liquid drops in truncated cones and hyperboloids are studied to elucidate our theory. As the volume of the drop trapped in the hydrophilic cones is increased, four regimes separated by three critical volumes are identified. The drop is either trapped at the narrow end or away from the cone top. The solution at the cone top satisfies the force balance by adjusting the upper contact angle, which is experimentally observed and verified by Surface Evolver (SE) simulations. Multiple stable states can exist in a particular regime. The hyperboloid tube in which the opening angle varies with position is also considered. As the gravitational strength is increased in hydrophilic hyperboloid, four regimes separated by three critical gravitational strengths are identified. The drop is either trapped near the neck or below the neck. Unlike hydrophilic cones, the drop stays near the neck of the hyperboloid due to varying opening angles. Multiple stable states are also observed. For both cone and hyperboloid, hydrophobic cases are studied as well and all theoretical solutions of the force balance agree well with SE simulation outcomes.

Capillary rise in a microchannel of arbitrary shape and wettability: hysteresis loop

Capillary rise in an asymmetric microchannel, in which both contact angle (wettability) and open angle (geometry) can vary with position, is investigated based on free-energy minimization. The integration of the Young-Laplace equation yields the general force balance between surface tension and gravity. The former is surface tension times the integration of cos θ(u) along the contact line, where θ(u) depicts the local difference between contact angle and open angle. The latter comes from the total volume right underneath the meniscus. For the same channel height, multiple solutions can be obtained from the force balance. However, the stable height of capillary rise must satisfy stability analysis. Several interesting cases have been studied, including short capillary, truncated cone, hyperboloid, and two different plates. As the tube length is smaller than Jurin's height, the angle of contact will be tuned to fulfill the force balance. While only one stable state is seen for divergent channels, two stable states can be observed for convergent channels. Three regimes can be identified for the plot of the stable height of capillary rise against the channel height. The higher height dominates in the short channel regime, while the lower height prevails in the tall channel regime. However, both solutions are stable in the intermediate regime. Surface Evolver simulations and experiments are performed to validate our theoretical predictions. Our results offer some implications for water transport to the tops of tall trees. A small bore at the uppermost leaf connected to a larger xylem conduit corresponds to a convergent channel, and two stable heights are possible. The slow growth of the tree can be regarded as a gradual rise of the convergent channel. Consequently, the stable height of capillary rise to the top of a tall tree can always be achieved.

Droplet compression and relaxation by a superhydrophobic surface: contact angle hysteresis

In this article, the contact angle hysteresis (CAH) of acrylic glass is experimentally and theoretically studied through the compression-relaxation process of droplets by using a superhydrophobic surface with negligible CAH effect. In contrast to the existing technique in which the volume of the droplet changes during the measurement of CAH, this procedure is carried out at a constant volume of the droplet. By observing the base diameter (BD) and the contact angle (CA) of the droplet during the compression-relaxation process, the wetting behavior of the droplet can be divided into two regimes, the contact line withdrawal and the contact line pinning regimes, depending on the gap thickness (H) at the end of the compression process. During the compression process, both regimes possess similar droplet behavior; the contact line will move outward and the BD will expand while the CA remains at the advancing angle. During the relaxation process, the two regimes are significantly different. In the contact line withdrawal regime, the contact line will withdraw with the CA remaining at the receding angle. In the contact line pinning regime, however, the contact line will be pinned at the final position and the CA will decline to a certain value higher than the receding angle. Furthermore, the advancing pinning behavior can also be realized through a successive compression-relaxation process. On the basis of the liquid-induced defects model, Surface Evolver simulations are performed to reproduce the behavior of the droplet during the compression-relaxation process; both contact line withdrawal and pinning regimes can also be identified. The results of the experiment and simulation agree with each other very well.

Drops sitting on a tilted plate: receding and advancing pinning

The wetting behavior of a liquid drop sitting on an inclined plane is investigated experimentally and theoretically. Using Surface Evolver, the numerical simulations are performed based on the liquid-induced defect model, in which only two thermodynamic parameters (solid-liquid interfacial tensions before and after wetting) are required. A drop with contact angle (CA) equal to θ is first placed on a horizontal plate, and then the plate is tilted. Two cases are studied: (i) θ is adjusted to the advancing CA (θ(a)) before tilting, and (ii) θ is adjusted to the receding CA (θ(r)) before tilting. In the first case, the uphill CA declines and the downhill CA remains unchanged upon inclination. When the tilted drop stays at rest, the pinning of the receding part of the contact line (receding pinning) and the depinning of the advancing part of the contact line (advancing depinning) are observed. The free energy analysis reveals that upon inclination, the reduction of the solid-liquid free energy dominates over the increment of the liquid-gas free energy associated with shape deformation. In the second case, the downhill CA grows and the uphill CA remains the same upon inclination. Advancing pinning and receding depinning are noted for the tilted drop at rest. The free energy analysis indicates that upon inclination, the decrease of the liquid-gas free energy compensates the increment of the solid-liquid free energy. The experimental results are in good agreement with those of simulations.

Comparison of sessile drop and captive bubble methods on rough homogeneous surfaces: a numerical study

Quasi-static experiments using sessile drops and captive bubbles are the most employed methods for measuring advancing and receding contact angles on real surfaces. These observable contact angles are the most easily accessible and reproducible. However, some properties of practical surfaces induce certain phenomena that cause a built-in uncertainty in the estimation of advancing and receding contact angles. These phenomena are well known in surface thermodynamics as stick-slip phenomena. Following the work of Marmur (Marmur, A. Colloids Surf., A 1998, 136, 209-215), where the stick-slip effects were studied with regard to sessile drops and captive bubbles on heterogeneous surfaces, we developed a novel extension of this study by adding the effects of roughness to both methods for contact angle measurement. We found that the symmetry between the surface roughness problem and the chemical heterogeneity problem breaks down for drops and bubbles subjected to stick-slip effects.