On deviations from Young's equation

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.

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 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.

The equilibrium shape of a small liquid drop on a smooth rigid surface is governed by the minimization of energy with respect to the change in configuration, represented by the well-known Young's equation. In contrast, the equilibrium shape near the line separating three immiscible fluid phases is determined by force balance, represented by Neumann's Triangle. These two are limiting cases of the more general situation of a drop on a deformable, elastic substrate. Specifically, we have analyzed planar equilibrium shapes of a liquid drop on a deformable membrane. We show that to determine its equilibrium shape one must simultaneously satisfy configurational energy and mechanical force balance along with a constraint on the liquid volume. The first condition generalizes Young's equation to include changes in stored elastic energy upon changing the configuration. The second condition generalizes the force balance conditions by relating tensions to membrane stretches via their constitutive elastic behavior. The transition from Young's equation to Neumann's triangle is governed by the value of the elasto-capillary number, β = TRo/μh, where TRo is twice the surface tension of the solid-vapor interface, μ is the shear modulus of the membrane, and h is its thickness.

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 Neumann and Young equations for nematic contact lines

5 - Thermodynamics of contact angles on rough, heterogeneous surfaces

Discussion notes on “Disjoining pressure of planar adsorbed films”, by J.R. Henderson

This article introduces two dimensionless positive geometric parameters that characterize the shape of a liquid droplet on a flat solid surface, which formed by the surface tension. The first parameter, “shape coefficient” K, is defined by the ratio of volume to surface and is always >3 (3 is the space dimension). The second parameter, “holding limit” κ0, is defined by the fraction of osculating surface and K and is <1. The ratio of the surface tension energy of a droplet attached to a substrate in zero gravity to the energy of the same droplet floating in zero gravity is presented through these parameters as 1-(K-3)(κ0-κ)/3(1-κ0), where the material parameter κ (which appears in the Young equation κ=cosθ) indicates the decrease in liquid surface tension by the solid The relative energy of the surface tension, K and κ0, are explicitly expressed for a droplet of an elliptical rounded segment (ERS) shape through its eccentricity e, relative height χ, and relative rounding radius η. It is shown that the Young equation is a self-consistent (i.e., leading to η=0) minimum condition of the energy only in the spherical (e=0) case. The rounding, either inner or outer, is specified by the legs of a triangle with zero angles and the median as a slope line. The main result obtained is the proof that the outer rounded ERS weighty droplets with inflection points, due to weight and hydrostatic forces, cannot exist if their radii larger than 2-4 capillary length. This proscription is absent in zero gravity.

The theory of equilibrium liquid surfaces is reviewed from the macroscopic (thermodynamic) and microscopic (statistical) points of view. Although emphasis is placed on the surface tension, other surface thermodynamic quantities are treated, especially the contact angle and work of adhesion. The Gibbs scheme is used to describe the surface thermodynamics. The equivalent hydrostatic approach, the curvature dependence of the surface tension and its relation to other surface quantities are presented. The effect of solid surface topology on the contact angle is discussed, together with the controversial Young's equation. Good's and Fowkes' theories are described and criticised from the statistical point of view. The surface tension theories for simple liquids (Kirkwood and Buff), simple metals (Evans) and solid-fluid interphases (Navascues and Berry) are deduced from a single statistical mechanical formalism. The van der Waals theory of the liquid vapour interphase is presented. Thermodynamic perturbation theory for the interphase (Toxvaerd, Abraham) and the analysis of the free energy density as a functional of the interphase density (Yang, Flemming and Gibbs) are discussed.

Interfacial Free Energies in Protein Solution Droplets on FEP-Teflon by Axisymmetric Drop Shape Analysis by Profile—IgG versus BSA

Determination of the Surface Tension of Microporous Membranes Using Contact Angle Measurements

This paper was prepared for the Oilfield Chemistry Symposium of the Society of Petroleum Engineers of AIME, to be held in Denver, Colo., May 24–25, 1973. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of where and by whom the paper is presented. Publication elsewhere after publication in the JOURNAL paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon requested to the Editor of the appropriate journal, provided agreement to give proper credit is made. provided agreement to give proper credit is made. Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers office. Such discussions may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines. Abstract This distinction between low energy surfaces on which liquids exhibit finite contact angles and high energy surfaces on which liquid spreading occurs is extended to the vapor-solid interface, where it is assumed that multilayer adsorption at saturation pressure of the vapor occurs in the liquid spreading case, while submonolayer adsorption is characteristic of the nonspreading case. An empirical relationship is established between contact angle and adsorption at saturation pressure. Thermodynamic quantities of interest in wetting phenomena are calculated on the basis of energetic parameters in the adsorption expression. Introduction Wetting refers to the interaction between liquid and solid phases in the presence of saturated vapor. Traditionally, the subject has been approached from the observational standpoint (does a liquid drop spread on the subject solid or does it retain a semblance of its drop character?). Interpretation has centered on the mechanical properties of the interfacial systems in analogy to treatments applied to the more accessible liquid-vapor interface. This approach is reflected by the Young Equation, , (1) where terms refer to interfacial tensions, the particular interface being identified by subscripts indicating the phases present and being the angle (measured through the liquid) characterizing the three phase line of contact. Interfacial tensions are recognized as numerically equivalent to interfacial free energies but Eq. 1 remains an expression which reflects but does not quantize the wetting phenomenon since SV and SL are not amenable to measurement. Melrose has cataloged the thermodynamic relationships relevant to the energetics of wetting in a common notational structure and, in so doing, called attention to the energetic parameters which should be accessible to parameters which should be accessible to experiment. These are embodied in the following two equations: (2) (3)

Poly(3,3,3‐trifluoropropylsilsesquioxane) (Prf‐T) and poly(3,3,4,4,5,5,6,6,6‐nonafluorohexylsilsesquioxane) (Hxf‐T) were synthesized via the hydrolysis of trichloro‐3,3,3‐trifluoropropylsilane and trichloro‐3,3,4,4,5,5,6,6,6‐nonafluorohexylsilane, respectively. The related resin‐like fluorosilicone polymers containing dimethylsiloxy units were also prepared. Contact angle measurements were carried out after the polymers were crosslinked by condensation. The solid surface tensions were determined from the combined Owens‐Wendt/Young equation using water and methylene diiodide contact angles. The value obtained for Prf‐T resin (20 mN/m) is similar to the value for poly(methylsilsesquioxane) as the unfluorinated silsesquioxane analogue (21,5 mN/m). On the other hand, the Hxf‐T polymer shows a very low surface tension, 7 mN/m, lower than that of the corresponding polysiloxane with linear structure, poly[methyl‐(3,3,4,4,5,5,6,6,6‐nonafluorohexyl)siloxane]. The surface tensions based on other approaches, such as the Girifalco‐Good‐Fowkes‐Young equation and the Zisman critical surface tension of wetting, were also determined.

While static wettability is well treated with Young's equation via its static contact angle, theoretical analyses for wetting dynamics are not yet reaching consensus due to a singularity of the spreading forces worked at the vapor/liquid/solid contact line. One plausible explanation to overcome the singularity problem is that there is a so-called precursor film spreading outside the apparent contact line. After its first finding in 1919, many researchers have attempted to visualize its shape. However, because its length and thickness are as small as micrometer and nanometer-order, respectively, its visualization stillremains a challenging issue especially for low-viscosity liquids. In the present study, we developed a differential laser interference microscope, which has a thickness resolution of approximately 2nm at the best, and applied it to the wetting front of 10 cSt of silicone oil spreading on a silicon wafer with an almost constant spreading velocity. As a result, the precursor film of 14µm long and 108nm thick was clearly visualized. While the macro contact line has a finite advancing contact angle of 4.0°, the gradient of the precursor film surface gradually decreased and converged to ~ 0.1° at the micro-contact angle. The shape of the precursor film was independent of the time after the dropping for the range of 600s ± 10%, which is consistent to theoretical estimation. The present study demonstrated that our interferometer simultaneously achieved nanometer thickness resolutions, micrometer in-plane spatial resolution, and at least a millisecond temporal resolution with a simple optical setup.