Shape analysis of a sessile drop on a flat solid surface

Minimization of free energy is used to calculate the equilibrium shapes of sessile liquid drops under different wetting conditions and under the influence of a gravitational body force. The total system free energy is assumed to obey an equation recently derived by Searcy [1], and the shapes of sessile drops of different sizes are calculated and compared with experimental measurements. Stationary states in the free energy are found using standard variational principles, which lead not only to an Euler-Lagrange differential equation for the drop shape, but also to the boundary conditions at the three-phase junction where the liquid, solid, and surrounding vapor meet. The analysis shows that the Searcy equation leads to the classical Young-Dupr equation for the thermodynamic contact angle. Drop shapes calculated by this approach are in excellent agreement with experimental observations of drops of pure water resting on a clean polytetrafluoroethylene (PTFE) substrate.

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.

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.

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.

ABSTRACTAn old procedure used to carry out a graphical derivation of curves, which is based on the optical properties of plane mirrors, has been adapted for the measurement of the contact angle (θ) formed between a liquid drop and a flat solid substrate in wettability experiments carried out by the so-called “sessile drop” method. The method was tested for mercury on soda-lime glass at room temperature in air as well as for Cusil (Ag-28wt.%Cu) and Incusil-ABA (Ag-27wt.%Cu-12wt.%In-2wt.%Ti) brazing alloys on pressureless-sintered silicon carbide (PLS-SiC) at 850 °C, under a vacuum of 10-4/10-5 Torr. The proposed method is fast, simple and accurate enough from high (∼140°) to relatively low (∼10°) contact angles. Although the proposed method has been tested for metal-ceramic systems, it is of general application, so that it would be useful for any liquid-solid system. The method is applicable for any temperature, pressure and atmospheric experimental conditions employed, as well as for any chemical composition of liquid and solid. It is also useful for both low and high contact angles, as well as for reactive and non-reactive systems, as long as a photograph of a liquid drop resting on a flat solid surface is available for the studied system.

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.

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.

A new method of determining the contact angle made by a sessile drop upon a horizontal surface (sessile drop contact angle)

The distribution function of orientations of a segment, which interacts with the orienting field and is within the chain with given the end-to-end distance-vector, is calculated. The number of segments per chain is finite. The Lagrange method of conditional minimization of the chain free energy (the Helmholtz function) functional is used. Constraints for the segment orientations stem from fixed the chain end-to-end distance-vector. Hence, Lagrange multipliers, energy, free energy, and entropy, for the chain with given the end-to-end distance-vector, are calculated. Then, the distribution function of values of that vector is obtained. Furthermore, an average free energy per chain inside the polymer network with given a topological structure, the system self-deformation, and modulus of elasticity are calculated and discussed in Gaussian limit, that is, for the number of segments per chain tending to infinity. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 138–144, 2010

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.

HypothesisThe development of high-resolution in situ imaging has allowed contact angles to be measured directly inside porous materials. We evaluate the use of concepts in integral geometry to determine contact angle. Specifically, we test the hypothesis that it is possible to determine an average contact angle from measurements of the Gaussian curvature of the fluid/fluid meniscus using the Gauss-Bonnet theorem. Theory and simulationWe show that it is not possible to unambiguously determine an average contact angle from the Gauss-Bonnet theorem. We instead present an approximate relationship: 2πn(1-cosθ)=4π-∫κG12dS12, where n is the number of closed loops of the three-phase contact line where phases 1 and 2 contact the surface, θ is the average contact angle, while κG12 is the Gaussian curvature of the fluid meniscus which is integrated over its surface S12. We then use the results of pore-scale lattice Boltzmann simulations to assess the accuracy of this approach to determine a representative contact angle for two-phase flow in porous media. FindingsWe show that in simple cases with a flat solid surface, the approximate expression works well. When applied to simulations on pore space images, the equation provides a robust estimate of contact angle, accurate to within 3°, when averaged over many fluid clusters, although individual values can have significant errors because of the approximations used in the calculation.

The minimum free energy (MFE) of ribonucleic acids (RNAs) increases at an apparent linear rate with sequence length. Simple indices, obtained by dividing the MFE by the number of nucleotides, have been used for a direct comparison of the folding stability of RNAs of various sizes. Although this normalization procedure has been used in several studies, the relationship between normalized MFE and length has not yet been investigated in detail. Here, we demonstrate that the variation of MFE with sequence length is not linear and is significantly biased by the mathematical formula used for the normalization procedure. For this reason, the normalized MFEs strongly decrease as hyperbolic functions of length and produce unreliable results when applied for the comparison of sequences with different sizes. We also propose a simple modification of the normalization formula that corrects the bias enabling the use of the normalized MFE for RNAs longer than 40 nt. Using the new corrected normalized index, we analyzed the folding free energies of different human RNA families showing that most of them present an average MFE density more negative than expected for a typical genomic sequence. Furthermore, we found that a well-defined and restricted range of MFE density characterizes each RNA family, suggesting the use of our corrected normalized index to improve RNA prediction algorithms. Finally, in coding and functional human RNAs the MFE density appears scarcely correlated with sequence length, consistent with a negligible role of thermodynamic stability demands in determining RNA size.

The asymmetry between the left and right sides seems to be a general principle of organization of the nervous systems in Bilateria, providing the foundations for a plethora of leftward and rightward biases in behaviour as documented in species ranging from Caenorhabditis elegans nematodes to humans. Several theories have been put forward to account for the existence and maintenance in the evolution of the asymmetric organization of the brain at both individual and population levels. However, what is missing in theorizing about the evolution of brain asymmetry is an overarching general hypothesis that may subsume all different aspects of current models. Here, we tried to provide an overarching general framework based on the energy and free-energy minimization principle, which proved so valuable in other areas of neuroscience. We found that at the individual level the antisymmetric singlet configuration realizes the lowest energy state of the system, whereas at the group level, the spontaneous emergence of directional asymmetry arises as a consequence of the minimization of the free energy of the system, which guarantees its stability and equilibrium. We thus argue that the various phenomenological aspects of brain asymmetry that have been captured in biology-e.g. sparing of neural tissue, control of unitary motor responses and, at the population level, evolutionarily stable strategies described by mathematical games theory-may be thought of as the manifestation of a more general principle of energy minimization generating, among others, asymmetry of the brains.

Aqueous solutions of different polymers can separate and form aqueous two-phase systems (ATPS). ATPS provide an aqueous, biocompatible, and mild environment for separation and fractionation of biomolecules. The interfacial tension between the two aqueous phases plays a major role in ATPS-mediated partition of biomolecules. Because of the structure of the two aqueous phases, the interfacial tensions between the phases can be 3-4 orders of magnitude smaller than conventional fluid-liquid systems: ∼1-100 μJ/m(2) for ATPS compared to ∼72 mJ/m(2) for the water-vapor interface. This poses a major challenge for the experimental measurements of reproducible interfacial tension data for these systems. We address the need for precise determination of ultralow interfacial tensions by systematically studying a series of polymeric ATPS comprising of polyethylene glycol (PEG) and dextran (DEX) as the phase-forming polymers. Sessile and pendant drops of the denser DEX phase are formed within the immersion PEG phase. An axisymmetric drop shape analysis (ADSA) is used to determine interfacial tensions of eight different ATPS. Specific criteria are used to reproducibly determine ultralow interfacial tensions of the ATPS from pendant and sessile drops. Importantly, for a given ATPS, pendant drop and sessile drop experiments return values within 0.001 mJ/m(2) indicating reliability of our measurements. Then, the pendant drop technique is used to measure interfacial tensions of all eight ATPS. Our measured values range from 0.012 ± 0.001 mJ/m(2) to 0.381 ± 0.006 mJ/m(2) and vary with the concentration of polymers in equilibrated phases of ATPS. Measurements of ultralow interfacial tensions with such reproducibility will broadly benefit studies involving partition of different biomolecules in ATPS and elucidate the critical effect of interfacial tension.

Detailed statistical contact angle analyses; “slow moving” drops on inclining silicon-oxide surfaces