Mean-field theory of liquid droplets on roughened solid surfaces: application to superhydrophobicity

We present calculations of the density distributions and contact angles of liquid droplets on roughened solid surfaces for a lattice gas model solved in a mean-field approximation. For the case of a smooth surface, this approach yields contact angles that are well described by Young's equation. We consider rough surfaces created by placing an ordered array of pillars on a surface, modeling so-called superhydrophobic surfaces, and we have made calculations for a range of pillar heights. The apparent contact angle follows two regimes as the pillar height increases. In the first regime, the liquid penetrates the interpillar volume, and the contact angle increases with pillar height before reaching a constant value. This behavior is similar to that described by the Wenzel equation for contact angles on rough surfaces, although the contact angles are underestimated. In the second regime, the liquid does not penetrate the interpillar volume substantially, and the contact angle is independent of the pillar height. This situation is similar to that envisaged in the Cassie-Baxter equation for contact angles on heterogeneous surfaces, but the contact angles are overestimated by this equation. For larger pillar heights, two states of the droplet can be observed, one Wenzel-like and the other Cassie-like.     

Prediction of solid-fluid interfacial tension and contact angle

Two simple equations have been developed using the lattice theory and the regular solution assumption to predict the solid-vapor and solid-liquid interfacial tension. The required parameters are the liquid critical temperature and volume, the solid melting temperature and the molar volume of liquid and solid compounds. To confirm the models, the predicted solid-fluid interfacial tension values have been used to predict the contact angle of the liquid drop on the solid surface applying Young's equation. Agreement of the predicted contact angle with the experimental data reveals the reliability of the developed models.     

Wetting of real solid surfaces: new glance on well-known problems

The wetting of solid surfaces is treated. The Young and receding contact angles are experimentally unattainable values for the majority of solid surfaces. Actually, we always observe the apparent contact angle. This makes the characterization of wetting of real surfaces problematic. It is proposed in this paper to characterize wetting of real surfaces with the advancing contact angle and the minimum work of adhesion calculated according to the Dupre equation. The advancing contact angle, which depends slightly on the experimental technique used for its measurement, corresponds to the maximal solid/liquid surface tension and correspondingly to the minimal work of adhesion, calculated according to the Dupre equation.     

Contact angle interpretation in terms of solid surface tension

Recent experimental (low-rate) dynamic contact angles for 14 solid surfaces are interpreted in terms of their solid surface tensions. Universality of these experimental contact angle patterns is illustrated; other reasons that can cause data to deviate from the patterns are discussed. It is found that surface tension component approaches do not reflect physical reality. Assuming solid surface tension is constant for one and the same solid surface, experimental contact angle patterns are employed to deduce a functional relationship to be used in conjunction with the Young equation to determine solid surface tensions. The explicit form of such a relation is obtained by modifying Berthelot’s rule together with experimental data; essentially constant solid surface tension values are obtained, independent of liquid surface tension and molecular structure. A new combining rule is also derived based on an expression similar to one used in molecular theory; such a combining rule should allow a better understanding of the molecular interactions between unlike solid–liquid pairs.     

Contact angle hysteresis in multiphase systems

Contact angle (CA) hysteresis is the difference between the maximum (advancing) and minimum (receding) water CA. Hysteresis is caused by adhesion hysteresis in the solid–water contact area (2D effect) and by pinning of the solid–water–air triple line due to the surface roughness (1D effect). In this work, we show that CA hysteresis is present also in more complex systems, such as an organic liquid (oil) in contact with a solid immersed in water. In order to decouple the 1D and 2D effects, we study CA hysteresis in solid–water–air (droplet), solid–air–water (bubble), solid–water–oil, and solid–water–air–oil systems involving rough and microstructured surfaces. The comparative analysis of these systems allows decoupling the 1D and 2D effects as well as hydrogen bonding and entropic forces (water–air tension) and dispersion forces (oil–air tension).     

Experimental Study on Contact Angle Patterns: Liquid Surface Tensions Less Than Solid Surface Tensions

The interpretation of contact angles in terms of solid surface tensions is not trivial. In the past, we and others have postulated that contact angles should be measured with liquid of surface tension larger than the anticipated solid surface tension, i.e., gamma(lv)>gamma(sv). This has recently been disputed. It is also not entirely obvious how to proceed experimentally since gamma(sv) is not known initially. Typically, one starts with a liquid of high gamma(lv) (such as water) and goes lower. We have stopped in the past when the contact angles became small. A question arises as to what would happen if we would go on. Contact angles of liquids with gamma(lv) less than or near gamma(sv) were measured on eight polymer-coated solid surfaces. The experimental contact angle patterns for gamma(lv)gamma(sv) were compared. Results suggest that contact angle interpretation in terms of solid surface tensions requires contact angles to be measured for gamma(lv)>gamma(sv) because the Young equation is not applicable for gamma(lv)相似文献   

The Neumann and Young equations for nematic contact lines

The Neumann and Young equations for three-phase nematic contact lines have been derived using the momentum balance equation and classical liquid crystal physics theories. The novel finding is the presence of bending forces, originating from the anchoring energy of nematic interfaces, and acting on the contact line. The classical Neumann triangle or tensile force balance becomes in the presence of a nematic phase the Neumann pentagon, involving the usual three tensile forces and two additional bending forces. The Young equation that describes the static contact angle of a fluid in contact with a rigid solid is again a tensile force balance along the solid, but for nematics it also involves an additional bending force. The effects of the bending forces on contact angles and wetting properties of nematic liquid crystals are thoroughly characterized. It is found that in terms of the spreading coefficient, bending forces enlarge the partial wetting window that exists between dewetting and spontaneous spreading. Bending forces also affect the behaviour of the contact angle, such that spreading occurs at contact angles greater than zero and dewetting at values greater than pi. Finally, the contact angle range in the partial wetting regime is always less than pi.     

The Neumann and Young equations for nematic contact lines

The Neumann and Young equations for three-phase nematic contact lines have been derived using the momentum balance equation and classical liquid crystal physics theories. The novel finding is the presence of bending forces, originating from the anchoring energy of nematic interfaces, and acting on the contact line. The classical Neumann triangle or tensile force balance becomes in the presence of a nematic phase the Neumann pentagon, involving the usual three tensile forces and two additional bending forces. The Young equation that describes the static contact angle of a fluid in contact with a rigid solid is again a tensile force balance along the solid, but for nematics it also involves an additional bending force. The effects of the bending forces on contact angles and wetting properties of nematic liquid crystals are thoroughly characterized. It is found that in terms of the spreading coefficient, bending forces enlarge the partial wetting window that exists between dewetting and spontaneous spreading. Bending forces also affect the behaviour of the contact angle, such that spreading occurs at contact angles greater than zero and dewetting at values greater than pi. Finally, the contact angle range in the partial wetting regime is always less than pi.     

Contact angle hysteresis

In thermodynamic equilibrium, the contact angle is related by Young's equation to the interfacial energies. Unfortunately, it is practically impossible to measure the equilibrium contact angle. When for example placing a drop on a surface its contact angle can assume any value between the advancing Θ_a and receding Θ_r contact angles, depending on how the drop is placed. Θ_a − Θ_r is called contact angle hysteresis. Contact angle hysteresis is essential for our daily life because it provides friction to drops. Many applications, such as coating, painting, flotation, would not be possible without contact angle hysteresis. Contact angle hysteresis is caused by the nanoscopic structure of the surfaces. Here, we review our current understanding of contact angle hysteresis with a focus on water as the liquid. We describe appropriate methods to measure it, discuss the causes of contact angle hysteresis, and describe the preparation of surfaces with low contact angle hysteresis.     

Effect of Surface Depressions on Wetting and Interactions between Hydrophobic Pore Array Surfaces

The surface structure is known to significantly affect the long-range capillary forces between hydrophobic surfaces in aqueous solutions. It is, however, not clear how small depressions in the surface will affect the interaction. To clarify this, we have used the AFM colloidal probe technique to measure interactions between hydrophobic microstructured pore array surfaces and a hydrophobic colloidal probe. The pore array surfaces were designed to display two different pore spacings, 1.4 and 4.0 μm, each with four different pore depths ranging from 0.2 to 12.0 μm. Water contact angles measured on the pore array surfaces are lower than expected from the Cassie-Baxter and Wenzel models and not affected by the pore depth. This suggests that the position of the three-phase contact line, and not the interactions underneath the droplet, determines the contact angle. Confocal Raman microscopy was used to investigate whether water penetrates into the pores. This is of importance for capillary forces where both the movement of the three-phase contact line and the situation at the solid/liquid interface influence the stability of bridging cavities. By analyzing the shape of the force curves, we distinguish whether the cavity between the probe and the surfaces was formed on a flat part of the surface or in close proximity to a pore. The pore depth and pore spacing were both found to statistically influence the distance at which cavities form as surfaces approach each other and the distance at which cavities rupture during retraction.     

Measurement of the surface tension of liquid marbles

The capillary rise and Wilhelmy plate methods have been used to study the "surface tension" of water marbles encapsulated with polytetrafluoroethylene (PTFE) powders of 1-, 35-, and 100-μm particle size. With the capillary rise technique, a glass capillary tube was inserted into a water marble to measure the capillary rise of the water. The Laplace pressure exerted by the water marble was directly measured by comparing the heights of the capillary rise from the marble and from a flat water surface in a beaker. An equation based on Marmur's model was proposed to calculate the water marble surface tension. This method does not require the water contact angle with the supporting solid surface to be considered; it is therefore a simple but efficient method for determining liquid marble surface tension. The Wilhelmy method was used to measure the surface tension of a flat water surface covered by PTFE powder. This method offers a new angle for investigating liquid marble shell properties. A discussion on the nature and the realistic magnitude of liquid marble surface tension is offered.     

Analysis of the relationship between liquid droplet size and contact angle

The purpose of this paper is to present a consistent theoretical concept that can explain the various physical phenomena associated with the effect of droplet size on contact angle for droplets on solid surfaces, and with the geometry of the liquid/gas/solid contact line in general. Two droplet geometries have been considered: uniformly elongated droplets and axisymmetric droplets. It has been shown that the contact angle for elongated droplets is size-independent and, thus, satisfies the Young equation for constant material and interfacial properties. On the other hand, whereas the contact angle for axisymmetric droplets is size-dependent and does not satisfy the original Young equation, it is shown that this contact angle can still be predicted for any combination of droplet and substrate materials, and a given mass of the droplet. The theoretical work has been combined with the development of numerical schemes of solving the Laplace-Young equation for various droplet geometries. The proposed approach has been applied to different material/substrate combinations and validated against several sets of experimental data. As a result, a method has been developed for predicting the contact angle of both long and axisymmetric sessile droplets of arbitrary sizes for given liquid/solid/gas properties.     

Design and fabrication of micro-textures for inducing a superhydrophobic behavior on hydrophilic materials

Artificial superhydrophobic surfaces are typically fabricated by tuning the surface roughness of intrinsically hydrophobic surfaces. We report here the design and fabrication of micro-textures for inducing a superhydrophobic behavior on hydrogen-terminated Si surfaces with an intrinsic water contact angle of approximately 74 degrees . The micro-textures consist of overhang structures with well-defined geometries fabricated by microfabrication technologies, which provide positions to support the liquid and prevent the liquid from entering into the indents between the micro-textures. As a result, water is in contact with a composite surface of solid and air, which induces the observed macroscopic superhydrophobic behavior.     

Application of Young’s equation to two immiscible liquids resting on a solid substrate

The geometry of two immiscible liquid drops (water and mercury) resting on a solid substrate is reported. The problem under investigation is a very simple situation of a non-miscible two-phase liquid system which is properly documented. The simple experimental observation of water micro drop on top of a mercury drop on polypropylene substrate is discussed. The static situation of the drops is explored, and a new equation is derived for such situation by applying Young’s equation to the contact angles. The obtained equation simply represents the additive summation of Young equations for two independent drops, and may be applied for the case when the interfaces of two drops are close to each other and hence interact. The forces at the contact lines of each drop are treated separately as well as in correlation. This paper shows a schematic representation of two-liquid phase contact angle system. The value of this work may be appreciated from a pedagogical point of view.     

Contact angle kinetics of human albumin solutions at solid surfaces

Contact angle kinetics of sessile drops of albumin solution on hydrophilic acetal and hydrophobic FC 721 surfaces were measured using axisymmetric drop shape analysis. Young's equation is used to calculate the solid/liquid interfacial tension from measured contact angles and surface tensions as a function of time. The change in solid/liquid interfacial tension is a result of protein adsorption. It indicates that at the hydrophilic acetal surface the albumin molecules, interact only weakly, whereas the interaction with the hydrophobic FC 721 surface is quite strong.     

Nanodrop on a nanorough solid surface: density functional theory considerations

The density distributions and contact angles of liquid nanodrops on nanorough solid surfaces are determined on the basis of a nonlocal density functional theory. Two kinds of roughness, chemical and physical, are examined. The former considers the substrate as a sequence of two kinds of semi-infinite vertical plates of equal thicknesses but of different natures with different strengths for the liquid-solid interactions. The physical roughness involves an ordered set of pillars on a flat homogeneous surface. Both hydrophobic and hydrophilic surfaces were considered. For the chemical roughness, the contact angle which the drop makes with the flat surface increases when the strength of the liquid-solid interaction for one kind of plates decreases with respect to the fixed value of the other kind of plates. Such a behavior is in agreement with the Cassie-Baxter expression derived from macroscopic considerations. For the physical roughness on a hydrophobic surface, the contact angle which a drop makes with the plane containing the tops of the pillars increases with increasing roughness. Such a behavior is consistent with the Wenzel formula developed for macroscopic drops. For hydrophilic surfaces, as the roughness increases the contact angle first increases, in contradiction with the Wenzel formula, which predicts for hydrophilic surfaces a decrease of the contact angle with increasing roughness. However, a further increase in roughness changes nonmonotonously the contact angle, and at some roughness, the drop disappears and only a liquid film is present on the surface. It was also found that the contact angle has a periodic dependence on the volume of the drop.     

Comparison of contact angle hysteresis of different probe liquids on the same solid surface

Advancing and receding contact angles of water, formamide and diiodomethane were measured on 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC) layers deposited on three different solid supports—glass, mica and poly(methyl methacrylate). Up to five statistical monolayers were deposited on the surfaces by spreading DPPC solution. It was found that even on five statistical DPPC monolayers, the hysteresis of a given liquid depends on the kind of solid support. Also on the same solid support the contact angle hysteresis is different for each probe liquid used. The AFM images show that the heights of roughness of the DPPC films cannot be the primary cause of the observed hysteresis because the heights are too small to cause the observed hystereses. It is believed that the hysteresis is due to the liquid film present right behind the three-phase solid surface/liquid drop/gas (vapour) contact line and the presence of Derjaguin pressure. The value of contact angle hysteresis depends on both the solid surface and liquid properties as well as on intermolecular interactions between them.     

Effects of the addition of electrolyte on liquid infiltration in a hydrophobic nanoporous silica gel

In this letter, we report the experimental results of pressure induced infiltration in the hydrophobic nanopores of a silica gel. The infiltration pressure increases with the prolonged surface treatment time, whereas the infiltration volume is not dependent on the surface coverage. When temperature increases, if the liquid phase is pure water, the infiltration pressure would decrease, which is in agreement with the classic contact angle measurement results at large solid surfaces. As an electrolyte is added, however, the variation in infiltration pressure is negligible over a broad temperature range.     

Drop size dependence of contact angles on two fluoropolymers

Young’s equation predicts that the contact angle of a liquid drop is independent of its size. Nevertheless, large drop size dependences of contact angles have been observed, especially for millimetre-sized drops, on a variety of solid surfaces. We report new measurements of drop size dependence of contact angles for several liquids on two fluoropolymer surfaces, Teflon AF 1600 and EGC-1700. We demonstrate a new strategy for contact angle measurement that allows detection of approximately 0.1° changes in the contact angle during the growth of a drop. We find that on the surfaces examined, drop size dependence of contact angles is ten times smaller than on all previously studied fluoropolymers at the millimetre scale. The data are insensitive to various attempted surface modifications. We discuss the interpretation of the data and possible physical sources.