Equilibrium contact angles of liquid droplets on ideal rough solids

This work proposes a theoretical model for predicting the apparent equilibrium contact angle of a liquid on an ideal rough surface that is homogeneous and has a negligible body force, line tension, or contact angle hysteresis between solid and liquid. The model is derived from the conservation equations and the free-energy minimization theory for the changes of state of liquid droplets. The work of adhesion is expressed as the contact angles in the wetting process of the liquid droplets. Equilibrium contact angles of liquid droplets for rough surfaces are expressed as functions of the area ratios for the solid, liquid, and surrounding gas and the roughness ratio and wetting ratio of the liquid on the solid for the partially and fully wet states. It is found that the ideal critical angle for accentuating the contact angles by the surface roughness is 48°. The present model is compared with existing experimental data and the classical Wenzel and Cassie-Baxter models and agrees with most of the experimental data for various surfaces and liquids better than does the Wenzel model and accounts for trends that the Wenzel model cannot explain.     

微结构疏水表面上液滴的表观接触角

虽然微结构疏水表面上Cassie-Baxter状态液滴的表观接触角已有理论预测公式,但实验研究发现,由于微结构疏水表面上的瑕疵,以及液滴受到的各种轻微扰动等原因,很容易造成Cassie-Baxter状态液滴局部区域出现Wenzel状态的情况(即混合状态),而有关混合状态液滴表观接触角的研究还比较少.本文通过对比相同微结构疏水表面上Cassie-Baxter状态和混合状态液滴表观接触角的大小,发现将Cassie-Baxter预测公式中的固液接触面积分数F换成最外缘三相接触线处的局部固液接触面积分数F′,则能同时较好地预测上述两种情况下液滴的表观接触角;通过进一步的研究发现,表观接触角的大小仅与最外缘三相接触线处的固液接触状态有关,而与其他处的固液接触状态无关.该结果对于进一步认识微结构疏水表面上液滴的表观接触角以及润湿性质具有重要意义.     

A stable intermediate wetting state after a water drop contacts the bottom of a microchannel or is placed on a single corner

It is considered that, after a water drop contacts the base of a roughness groove, water should immediately fill this roughness groove. Subsequently, Cassie-Baxter wetting state is transited to that of Wenzel. Accordingly, one of the criteria used to judge the transition from Cassie-Baxter to Wenzel states is whether a water drop has contact with the base of a roughness groove. In this work, through theoretical and experimental investigations, we show that this transition criterion does not always hold true in the case of microchannels. We first theoretically prove that, when an angle criterion is satisfied, there may exist an intermediate wetting state inside a microchannel after a water drop contacts the bottom of the microchannel. In this wetting state, water does not completely fill the microchannel, and air pockets still exist in its bottom corners. Also, the wetting state is stable in the sense that its energy state is lower than that of the Wenzel model. According to the angle criterion, such intermediate states may exist, for example, in microchannels with vertical sidewalls, when contact angles on the inner surfaces of these microchannels are larger than 135°. In addition to microchannels, the aforementioned intermediate state may also exist on a single corner (which is formed by a horizontal plate and an inclined plate), when the angle criterion is met. After theoretical modeling, we then conduct four types of tests on single corners and microchannels to validate the angle criterion. In these tests, once the angle criterion is met, stable intermediate states are observed on the corresponding samples. In addition, it is found from the two types of tests conducted on microchannels that, once Laplace pressure inside a water drop is gradually reduced, such an intermediate wetting state may be transited back to the original Cassie-Baxter state. On the other hand, the Wenzel state may not have such a reversal transition unless an additional force is applied to overcome energy barrier between Wenzel and Cassie-Baxter states.     

Effects of geometrical characteristics of surface roughness on droplet wetting

Surface roughness is known to alter the wettability on a solid substrate. In general, either Wenzel or Cassie-Baxter theory is adopted to describe the apparent contact angle. Following the minimum free energy pathway associated with the imbibition process, we have derived a generalized expression for the apparent contact angle on a textured surface and the liquid-gas contact area within the groove that plays a key role. Depending on the geometrical characteristics of the grooves, the surface wetting falls into three regimes: (i) single stable state which is either Wenzel (completely wetted roughness) or Cassie-Baxter (completely nonwetted roughness) state, (ii) two stable states (Wenzel and Cassie-Baxter) separated by an energy barrier, and (iii) single stable state with partially wetted roughness. The sufficient condition for each regime is derived and several groove geometries are given to show the free energy path. Alteration in the geometric parameters may lead to the wetting crossover. We also show that the Cassie-Baxter can occur at a hydrophilic surface for particular pore shapes.     

Pressure induced transition between superhydrophobic states: configuration diagrams and effect of surface feature size

The stability of wetting states, namely the Cassie state (partial wetting) and the Wenzel state (complete wetting) of surfaces with protrusions, is determined by comparing the total free energy of a liquid drop in terms of their apparent contact angles for different protrusion features. It is found that when the area fraction of the topographical features and the intrinsic contact angle for a flat surface are large, the Cassie state is favored, but it can be either the metastable or stable state. It is shown that the transition from the Cassie state to the Wenzel state requires the application of a pressure to the meniscus between the surface protrusions. The critical transition pressure increases not only with increasing area fraction and intrinsic contact angle, but also with decreasing protrusion size. During the transition, a high-pressure gas can be trapped around the protrusions that can cause the Cassie state to be recovered after the release of the applied pressure. The analysis shows that a droplet can 'hang' upside-down when the protrusion size is very small; namely, the protrusions can pin the meniscus. These results are discussed relative to the advancing and receding contact angle.     

Mechanical and superhydrophobic stabilities of two-scale surfacial structure of lotus leaves

To understand why lotus leaf surfaces have a two-scale structure, we explore in this paper two stability mechanisms. One is the stability of the Cassie-Baxter wetting mode that generates the superhydrophobicity. A recent quantitative study (Zheng et al., Langmuir 2005, 21, 12207) showed that the larger the slenderness ratio of the surface structures was, the more stable the Cassie-Baxter wetting mode would be. On the other hand, it is well-known that more slender surface structures can only sustain lower critical water pressures for structure buckling, or Euler instability, while in the natural environments, the water pressure impacting on the lotus surface can reach a fairly high value (105 Pa in a heavy rain). Our analysis reveals that the two-scale structure of the lotus leaf surfaces is necessary for keeping both the structure and the superhydrophobicity stable. Furthermore, we find that the water-air interfacial tension makes the slender surface structure more instable and the two-scale structure a necessity.     

The apparent contact angle of water droplet on the micro-structured hydrophobic surface

The apparent contact angle of Cassie-Baxter state water droplets can be calculated by the existing theoretical formula, but due to the defects of the micro-structured hydrophobic surface and some inevitable tiny disturbances in the experiment, Cassie-Baxter state water droplets will appear partly in Wenzel state, that is, the mixed state water droplets. In this paper, apparent contact angles of Cassie-Baxter state and mixed state water droplets on micro-structured hydrophobic surfaces are compared. The rese...     

Wetting and wetting transitions on copper-based super-hydrophobic surfaces

Rough and patterned copper surfaces were produced using etching and, separately, using electrodeposition. In both of these approaches the roughness can be varied in a controlled manner and, when hydrophobized, these surfaces show contact angles that increase with increasing roughness to above 160 degrees . We show transitions from a Wenzel mode, whereby the liquid follows the contours of the copper surface, to a Cassie-Baxter mode, whereby the liquid bridges between features on the surface. Measured contact angles on etched samples could be modeled quantitatively to within a few degrees by the Wenzel and Cassie-Baxter equations. The contact angle hysteresis on these surfaces initially increased and then decreased as the contact angle increased. The maximum occurred at a surface area where the equilibrium contact angle would suggest that a substantial proportion of the surface area was bridged.     

Model and experimental studies for contact angles of surfactant solutions on rough and smooth hydrophobic surfaces

Despite the practical need, no models exist to predict contact angles or wetting mode of surfactant solutions on rough hydrophobic or superhydrophobic surfaces. Using Gibbs' adsorption equation and a literature isotherm, a new model is constructed based on the Wenzel and Cassie equations. Experimental data for aqueous solutions of sodium dodecyl sulfate (SDS) contact angles on smooth Teflon surfaces are fit to estimate values for the adsorption coefficients in the model. Using these coefficients, model predictions for contact angles as a function of topological f (Cassie) and r (Wenzel) factors and SDS concentration are made for different intrinsic contact angles. The model is also used to design/tune surface responses. It is found that: (1) predictions compare favorably to data for SDS solutions on five superhydrophobic surfaces. Further, the model predictions can determine which wetting mode (Wenzel or Cassie) occurred in each experiment. The unpenetrated or partially penetrated Cassie mode was the most common, suggesting that surfactants inhibit the penetration of liquids into rough hydrophobic surfaces. (2) The Wenzel roughness factor, r, amplifies the effect of surfactant adsorption, leading to larger changes in contact angles and promoting total wetting. (3) The Cassie solid area fraction, f, attenuates the lowering of contact angles on rough surfaces. (4) The amplification/attenuation is understood to be due to increased/decreased solid-liquid contact-area.     

Analytical modeling and thermodynamic analysis of robust superhydrophobic surfaces with inverse-trapezoidal microstructures

A polydimethylsiloxane (PDMS) elastomer surface with perfectly ordered microstructures having an inverse-trapezoidal cross-sectional profile (simply PDMS trapezoids) showed superhydrophobic and transparent characteristics under visible light as reported in our previous work. The addition of a fluoropolymer (Teflon) coating enhances both features and provides oleophobicity. This paper focuses on the analytical modeling of the fabricated PDMS trapezoids structure and thermodynamic analysis based on the Gibbs free energy analysis. Additionally, the wetting characteristics of the fabricated PDMS trapezoids surface before and after the application of the Teflon coating are analytically explained. The Gibbs free energy analysis reveals that, due to the Teflon coating, the Cassie-Baxter state becomes energetically more favorable than the Wenzel state and the contact angle difference between the Cassie-Baxter state and the Wenzel state decreases. These two findings support the robustness of the superhydrophobicity of the fabricated Teflon-coated PDMS trapezoids. This is then verified via the impinging test of a water droplet at a high speed. The dependencies of the design parameters in the PDMS trapezoids on the hydrophobicity are also comprehensively studied through a thermodynamic analysis. Geometrical dependency on the hydrophobicity shows that overhang microstructures do not have a significant influence on the hydrophobicity. In contrast, the intrinsic contact angle of the structural material is most important in determining the apparent contact angle. On the other hand, the experimental results showed that the side angles of the overhangs are critical not for the hydrophobic but for the oleophobic property with liquids of a low surface tension. Understanding of design parameters in the PDMS trapezoids surface gives more information for implementation of superhydrophobic surfaces.     

Impact of surface forces on wetting of hierarchical surfaces and contact angle hysteresis

The influence of the long-range surface forces on the wetting of multi-scale partially wetted surfaces is discussed. The possibility of partial wetting is stipulated by a specific form of the Derjaguin isotherm. Equilibrium of a liquid meniscus inside a cylindrical capillary is used as a model. The interplay of capillary and disjoining pressures governs the equilibrium of the liquid in the nano- and micrometrically scaled pores constituting the relief of the surface. It is shown that capillaries with a radius smaller than a critical one will be completely filled by water, whereas the larger capillaries will be filled only partially. Thus, small capillaries will show the Wenzel type of wetting behavior, while the same liquid inside the large capillaries will promote the Cassie-Baxter type of wetting. Consideration of disjoining/conjoining pressure allows explaining of the “rose petal effect”, when a high apparent contact angle is accompanied with a high contact angle hysteresis.     

非离子表面活性剂Triton X-100溶液在不同生长期小麦叶片表面的润湿行为

选择不同生长期小麦叶片,利用座滴法研究了非离子表面活性剂Triton X-100在小麦叶片表面接触角,考察浓度对接触角、粘附张力、固-液界面张力及润湿状态的影响。研究表明,在低浓度下,表面活性剂分子在气-液界面吸附量(ΓLV)和固-液界面吸附量(Γ'SL)相似,但吸附量较少形成了不饱和吸附层,接触角保持不变,其润湿状态为Cassie-Baxter状态;当浓度进一步增加,液滴突破叶片表面三维立体结构中存在的钉扎效应,取代空气层而处于Wenzel状态,接触角陡降,同时Γ'SL/ΓLV远大于1;当浓度超过临界胶束浓度(CMC)时,表面活性剂分子在气-液界面和固-液界面形成饱和吸附层,并产生毛细管效应,使溶液在小麦叶片三维立体结构中产生半渗透过程,此时接触角保持不变。     

Control over wettability of polyethylene glycol surfaces using capillary lithography

We demonstrate that wettability of poly(ethylene glycol) (PEG) surfaces can be controlled using nanostructures with various geometrical features. Capillary lithography was used to fabricate PEG nanostructures using a new ultraviolet (UV) curable mold consisting of functionalized polyurethane with acrylate group (MINS101m, Minuta Tech.). Two distinct wetting states were observed depending of the height of nanostructures. At relatively lower heights (< 300 nm for 150 nm pillars with 500 nm spacing), the initial contact angle of water was less than 80 degrees and the water droplet easily invaded into the surface grooves, leading to a reduced contact angle at equilibrium (Wenzel state). At relatively higher heights (> 400 nm for 150 nm pillars with 500 nm spacing), on the other hand, the nanostructured PEG surface showed hydrophobic nature and no significant change in contact angle was observed with time (Cassie state). The presence of two wetting states was also confirmed by dynamic wetting properties and contact-angle hysteresis. The wetting transition from hydrophilic (bare PEG surface) to hydrophobic (PEG nanostructures) was described by the Cassie-Baxter equation assuming that enhanced hydrophobicity is due to the heterogeneous wetting mediated by an air pocket on the surface. The measured contact angles in the Cassie state were increased with increasing air fraction, in agreement with the theoretical prediction.     

From hygrophilic to superhygrophobic: theoretical conditions for making high-contact-angle surfaces from low-contact-angle materials

The possibility of making high-contact-angle, rough surfaces from low-contact-angle materials has recently been suggested and demonstrated. A thermodynamic analysis of this possibility in terms of feasibility and stability is presented. It turns out that only roughness topographies that conform to a feasibility condition which is developed in the present paper can support this phenomenon. Even under conditions that support the phenomenon, the high-contact-angle state may not be stable, and transition from the heterogeneous (Cassie-Baxter) wetting regime to the homogeneous (Wenzel) regime with a lower contact angle may occur. In addition, it is suggested to use the general terms hygrophilic and hygrophobic (based on the Greek prefix hygro- that means liquid) to describe low- and high-contact-angle surfaces, respectively.     

The Cassie equation: how it is meant to be used

A review of literature shows that the majority of papers cite a potentially incorrect form of the Cassie and Cassie-Baxter equations to interpret or predict contact angle data. We show that for surfaces wet with a composite interface, the commonly used form of the Cassie-Baxter equation, cosθ(c)=f(1)cosθ-(1-f), is only correct for the case of flat topped pillar geometry without any penetration of the liquid. In general, the original form of the Cassie-Baxter equation, cosθ(c)=f(1)cosθ(1)-f(2), with f(1)+f(2)≥1, should be used. The differences between the two equations are discussed and the errors involved in using the incorrect equation are estimated to be between ~3° and 13° for superhydrophobic surfaces. The discrepancies between the two equations are also discussed for the case of a liquid undergoing partial, but increasing, levels of penetration. Finally, a general equation is presented for the transition/stability criterion between the Cassie-Baxter and Wenzel modes of wetting.     

Condensation-induced wetting state and contact angle hysteresis on superhydrophobic lotus leaves

In this paper, we demonstrate how condensed moisture droplets wet classical superhydrophobic lotus leaf surfaces and analyze the mechanism that causes the increase of contact angle hysteresis. Superhydrophobic lotus leaves in nature show amazing self-cleaning property with high water contact angle (>150°) and low contact angle hysteresis (usually <10°), causing droplets to roll off at low inclination angles, in accordance with classical Cassie–Baxter wetting state. However, when superhydrophobic lotus leaves are wetted with condensation, the condensed water droplets are sticky and exhibit higher contact angle hysteresis (40–50°). Compared with a fully wetted sessile droplet (classical Wenzel state) on the lotus leaves, the condensed water droplet still has relatively large contact angle (>145°), suggesting that the wetting state deviates from a fully wetted Wenzel state. When the condensed water droplets are subjected to evaporation at room conditions, a thin water film is observed bridging over the micropillar structures of the lotus leaves. This causes the dew to stick to the surface. This result suggests that the condensed moisture does not uniformly wet the superhydrophobic lotus leaf surfaces. Instead, there occurs a mixed wetting state, between classical Cassie–Baxter and Wenzel states that causes a distinct increase of contact angle hysteresis. It is also observed that the mixed Cassie–Baxter/Wenzel state can be restored to the original Cassie–Baxter state by applying ultrasonic vibration which supplies energy to overcome the energy barrier for the wetting transition. In contrast, when the surface is fully wetted (classical Wenzel state), such restoration is not observed with ultrasonic vibration. The results reveal that although the superhydrophobic lotus leaves are susceptible to being wetted by condensing moisture, the configured wetting state is intermediate between the classical Cassie–Baxter and Wenzel states.     

Influence of nanoscale particle roughness on the stability of pickering emulsions

The wetting behavior of solid surfaces can be altered dramatically by introducing surface roughness on the nanometer scale. Some of nature's most fascinating wetting phenomena are associated with surface roughness; they have inspired both fundamental research and the adoption of surface roughness as a design parameter for man-made functional coatings. So far the attention has focused primarily on macroscopic surfaces, but one should expect the wetting properties of colloidal particles to be strongly affected by roughness, too. Particle wettability, in turn, is a key parameter for the adsorption of particles at liquid interfaces and for the industrially important use of particles as emulsion stabilizers; yet, the consequence of particle roughness for emulsion stability remains poorly understood. In order to investigate the matter systematically, we have developed a surface treatment, applicable to micrometer-sized particles and macroscopic surfaces alike, that produces surface coatings with finely tunable nanoscale roughness and identical surface chemistry. Coatings with different degrees of roughness were characterized with regard to their morphology, charging, and wetting properties, and the results were correlated with the stability of emulsions prepared with coated particles of different roughness. We find that the maximum capillary pressure, a metric of the emulsions' resistance to droplet coalescence, varies significantly and in a nonmonotonic fashion with particle roughness. Surface topography and contact angle hysteresis suggest that particle roughness benefits the stability of our emulsions as long as wetting occurs homogeneously (Wenzel regime), whereas the transition toward heterogeneous wetting (Cassie-Baxter regime) is associated with a loss of stability.     

Wetting transitions in two-, three-, and four-phase systems

We discuss wetting of rough surfaces with two-phase (solid-liquid), three-phase (solid-water-air and solid-oil-water), and four-phase (solid-oil-water-air) interfaces mimicking fish scales. We extend the traditional Wenzel and Cassie-Baxter models to these cases. We further present experimental observations of two-, three-, and four-phase systems in the case of metal-matrix composite solid surfaces immersed in water and in contact with oil. Experimental observations show that wetting transitions can occur in underwater oleophobic systems. We also discuss wetting transitions as phase transitions using the phase-field approach and show that a phenomenological gradient coefficient is responsible for wetting transition, energy barriers, and wetting/dewetting asymmetry (hysteresis).     

Patterned nonadhesive surfaces: superhydrophobicity and wetting regime transitions

Nonadhesive and water-repellent surfaces are required for many tribological applications. We study mechanisms of wetting of patterned superhydrophobic Si surfaces, including the transition between various wetting regimes during microdroplet evaporation in environmental scanning electron microscopy (ESEM) and for contact angle and contact angle hysteresis measurements. Wetting involves interactions at different scale levels: macroscale (water droplet size), microscale (surface texture size), and nanoscale (molecular size). We propose a generalized formulation of the Wenzel and Cassie equations that is consistent with the broad range of experimental data. We show that the contact angle hysteresis involves two different mechanisms and how the transition from the metastable partially wetted (Cassie) state to the homogeneously wetted (Wenzel) state depends upon droplet size and surface pattern parameters.