5 - Thermodynamics of contact angles on rough, heterogeneous surfaces

Although Young's equation, γS=γSL+γL cos θC, is one of the oldest and most-used equations of classical physics and chemistry, it is in a rather delicate position scientifically, in that it is virtually impossible to prove experimentally. This is due, of course, to the uncertainties in the measurement of γS and γSL, the solid–air and solid–liquid surface tensions. Recently the validity of Young's equation has been questioned as the result of a theoretical analysis of the three-phase contact region where the liquid–air surface tension, γL, may be modified by interaction with the nearby substrate.The present paper argues that Young's equation is a "macroscopic" equation which does not concern itself with the microscopic shape of the liquid surface in the vicinity of the three-phase contact region. The concept of a microscopic contact angle θP is introduced. θP is the angle that the free liquid surface makes with the substrate when the liquid thickness is ≪∼ 10 Å, i.e., microscopically near the three-phase contact line. The contact angle θC given by Young's equation is shown to be the angle that the liquid surface makes with the substrate at microscopically large distances from the contact line where the modification of surface tensions by interaction is negligible (> 10 Å). The connection between θP and θC is established by making only very general statements about the nature of this interaction and does not impose the unphysical restrictions on the profile shape in the neighbourhood of the three-phase contact line which lead to the conclusions of Jameson and del Cerro.

The debate on the dependence of apparent contact angles on drop contact area or three-phase contact line: A review

This study reveals the microscopic mechanical balance at the three-phase contact line (TPCL) of an interfacial nanobubble on a substrate with a wettability pattern using molecular dynamics simulations. The apparent contact angle was compared to that evaluated using Young's equation, in which the interfacial tensions were computed using a mechanical route. The comparison was conducted by changing the wettability of the substrate from hydrophilic to neutral while maintaining a hydrophobic region in the center of the substrate. When the wettability pattern pins the TPCL at the wettability boundary, the contact angle computed by Young's equation is larger than the apparent contact angle because a pinning force exists in the inward direction of the nanobubble. Conversely, on the surfaces where the wettability pattern does not pin the TPCL, the contact angle computed by Young's equation agrees with the apparent contact angle because the pinning force disappears. The distribution of principal stresses around the TPCL, which was visualized for the first time in this study, indicates that large compressive principal stresses exist between the liquid phase and the solid substrate interface, which pin the TPCL at the surface wettability boundary, and that the maximum principal stress occurs in the inward direction of the nanobubbles at the TPCL. The normalized pinning force estimated from the maximum principal stress is equivalent to that measured experimentally.

The two-century old Young's equation has been widely used in petroleum engineering to depict wettability through contact angles and surface free energies. In order to address it's inadequacies, including the unaccounted vertical component of liquid-vapor interfacial tension and to accommodate the imbalance of intermolecular forces experienced by the three-phase confluence zone, the Young's equation has been modified in the recent literature to include a line tension term in case of solid-liquid-vapor (S-L-V) systems. In this paper, we would like to report on the results of our attempts to experimentally investigate the applicability of the line tension-based modification of the Young's equation to solid-liquid-liquid (rock-oil-brine) systems of interest to the petroleum industry. Both the ambient and reservoir condition optical cells have been used, with stock tank oil and live oil respectively, to determine the drop size dependency of dynamic contact angle subtended by the oil-brine interface against the rock surface. The experimental data have been correlated with the modified Young's equation to determine the magnitude of line tension in rock-oil-brine systems. To the best of our knowledge, this is the first attempt to apply the modified Young's equation to rock-oil-brine systems and to measure line tension for a rock-live crude oil-brine system at reservoir conditions of pressure and temperature. The measured line tension for solid-liquid-liquid (S-L-L) systems, while being positive and of the same order of magnitude as in solid-liquid-vapor (S-L-V) systems, correlates well with the water-advancing contact angle and the adhesion number, a ratio of adhesion force to capillary force. This experimental study indicates that the extent of deviation from the Young's equation exhibited by rock-oil-brine systems may be directly related to the adhesion phenomenon. This study reinforces the need to include the rock-oil adhesion forces in our consideration of rock-fluid interactions, wettability and their impact on improved oil recovery processes.

In this study, a thermodynamic analysis is conducted to investigate the chemical effect, in terms of intrinsic contact angle (CA), on the superhydrophobic behavior. It is theoretically revealed that the essential effect of intrinsic CA is to promote the composite transition. In particular, a large intrinsic CA more than 90° is necessary for such transition. Furthermore, for a pillar system with an intrinsic CA smaller than 90°, composite states are not impossible but is thermodynamically unstable. Once composite states are achieved, the advancing or maximum CA always approaches 180° whether an intrinsic CA is larger or smaller than 90°. In addition, the role of intrinsic CA in the water-repellent or self-cleaning behavior such as contact angle hysteresis (CAH) and equilibrium CA has been discussed in detail.

The wetting properties of a liquid in contact with a solid are commonly described by Young's equation, which defines the relationship between the angle made by a fluid droplet onto the solid surface and the interfacial properties of the different interfaces involved. When modeling such interfacial systems, several assumptions are usually made to determine this angle of contact, such as a completely rigid solid or the use of the tension at the interface instead of the surface free energy. In this work, we perform molecular dynamics simulations of a Lennard-Jones liquid in contact with a Lennard-Jones crystal and compare the contact angles measured from a droplet simulation with those calculated using Young's equation based on surface free energy or surface stress. We analyze cases where the solid atoms are kept frozen in their positions and where they are allowed to relax and simulate surfaces with different wettability and degrees of softness. Our results show that using either surface free energy or surface stress in Young's equation leads to similar contact angles but different interfacial properties. We find that the approximation of keeping the solid atoms frozen must be done carefully, especially if the liquid can efficiently pack at the interface. Finally, we show that to correctly reproduce the measured contact angles when the solid becomes soft, the quantity to be used in Young's equation is the surface free energy only and that the error committed in using the surface stress becomes larger as the softness of the solid increases.

Understanding fluid behavior at the liquid-solid interface is essential for advancing micro/nanoscale technologies, yet current models struggle to capture the complexity introduced by surface heterogeneity. In this study, we investigate the interplay between nanoscale surface roughness, wettability, and hydrodynamic slip by using molecular dynamics simulations on nanopatterned silica surfaces. By systematically varying the droplet size and surface geometry, we characterize the influence of nanoscale pinning on contact angle hysteresis and determine the macroscopic equilibrium contact angles for six distinct rough surfaces. Parallel simulations of force-driven water flow through nanochannels reveal the local and average slip lengths associated with each surface. For smooth surfaces, slip length correlates well with contact angle, aligning with predictions from well-known theoretical models by Huang et al. and Sendner et al. However, on nanopatterned surfaces, significant deviations arise due to contact line pinning, which disrupts the expected correlation between wettability and slip. We observe a clear decoupling between hydrophobicity and interfacial mobility, which can be exemplified by rose petal-like behavior where high contact angles are accompanied by strong pinning and minimal slip. These findings underscore the limitations of existing theoretical models when applied to heterogeneous interfaces and emphasize the need for refined frameworks that account for pinning effects. This study offers such a framework, providing new insight into interfacial transport and guiding the design of functionally structured surfaces.

The Effect of Drop (Bubble) Size on Advancing and Receding Contact Angles for Heterogeneous and Rough Solid Surfaces as Observed with Sessile-Drop and Captive-Bubble Techniques

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.

The Neumann and Young equations for nematic contact lines

Dynamic wetting characteristics during droplet formation in a microfluidic T-junction

By extending the theoretical framework derived in our previous study [Imaizumi et al., J. Chem. Phys. 153, 034701 (2020)], we successfully calculated the solid-liquid (SL) and solid-vapor (SV) interfacial tensions of a simple Lennard-Jones fluid around solid cylinders with nanometer-scale diameters from single equilibrium molecular dynamics systems in which a solid cylinder was vertically immersed into a liquid pool. The SL and SV interfacial tensions γSL - γS0 and γSV - γS0 relative to that for bare solid surface γS0, respectively, were obtained by simple force balance relations on fluid-containing control volumes set around the bottom and top ends of the solid cylinder, which are subject to the fluid stress and the force from the solid. The theoretical contact angle calculated by Young's equation using these interfacial tensions agreed well with the apparent contact angle estimated by the analytical solution to fit the meniscus shape, showing that Young's equation holds even for the menisci around solids with nanoscale curvature. We have also found that the curvature effect on the contact angle was surprisingly small while it was indeed large on the local forces exerted on the solid cylinder near the contact line. In addition, the present results showed that the curvature dependence of the SL and SV interfacial free energies, which are the interfacial tensions, is different from that of the corresponding interfacial potential energies.

The present paper aims to study the influence of pore-scale fiber structure on the Cassie-Baxter contact angle of droplets on PTFE fibrous hydrophobic membranes. The fiber structure is obtained by the Confocal Laser Scanning Microscope (CLSM), and an optical test bench is built to measure the Cassie-Baxter contact angles of a droplet on a PTFE membrane. The free energy of the solid-liquid interface is measured by the surface tension integral on the droplet profile. An iterative algorithm for liquid-vapor interface shrinking is brought forward to analyze the liquid-vapor interface area between the droplet and the membrane, and the interface parameters of the Cassie-Baxter model are recognized. The experiment shows that the Cassie-Baxter contact angles and the contact line radii along the droplet circumference are in a normal distribution due to random fiber structures, and the variance of the normal distribution increases with the membrane pore diameter. The normal distribution of Cassie-Baxter’s contact angle indicates the constraint of the solid-liquid interface to the liquid-vapor interface. On the same normal contact pressure condition, the liquid-vapor area ratio of the droplet and membrane interface changes little, and Cassie-Baxter’s contact angle increases with the solid-liquid interface area ratio.

This study has been achieved on industrial products (glassfibres coated with different sizings). Contact angle measurements and contact angle hysteresis were obtained with three liquids (glycerol, tricresylphosphate and mineral oil). Fibres are perfectly wetted with tricresylphosphate and mineral oil contrary to glycerol. For this last liquid a large distribution of contact angle and contact angle hysteresis were observed. Three parameters were used to analyse the results: places where filaments were extracted in the roving (inside or at the periphery), roughness and heterogeneity of the filaments surfaces (chemical heterogeneity). Statistical measurements of contact angles have shown heterogeneity in the roving. Atomic force microscopy (AFM) measurements have shown that roughness has only a weak contribution to the large distribution observed on the wetting results with glycerol. This large distribution and the contact angle hysteresis were due to the chemical heterogeneity of the fibres' surfaces.

Evaluation of unusual change in contact angle between MgO and molten magnesium