Young-Laplace Research Articles

Water retention and diffusion in unsaturated clays: Connecting atomistic and pore scale simulations

Molecular diffusion is the dominant solute transport process in clays and claystones that are considered as sealing materials in the deep geological disposal of radioactive waste. These materials are typically water saturated, but during construction and later, at elevated temperatures or when gas may be produced, unsaturated conditions prevail. Investigating the clay's water retention properties as well as solute transport under unsaturated conditions is therefore mandatory. Here, functional dependencies of these properties were derived from atomistic and pore-scale simulations. In the absence of tomographic maps that resolve all pores in clays, model clay structure maps with different pore size distributions were generated using a previously developed algorithm. Upscaled water retention functions and upscaled diffusion coefficients of unsaturated samples were derived from these maps based on the shifted Young-Laplace equation that considers film adsorption and capillary condensation. Pore-scale parameters (water film thickness, diffusion coefficients) used for the upscaling were taken from Grand Canonical Monte Carlo (GCMC) and molecular dynamics (MD) simulations, thus connecting molecular and pore-scale simulations. We focused on effects of the pore size distribution and of the adsorbed water film on upscaled parameters. Sample-scale diffusion coefficients were clearly reduced in unsaturated samples compared to the saturated state, with less reduction when including adsorbed water films. The reduction was stronger in samples with a narrow size distribution of the interparticle pores as compared to those with a wide distribution (but equal mean size). The results follow the trends of the experimental data, even though the scale of the simulations is still clearly smaller than that of typical experiments.

Optimization of Patterned Surfaces for Improved Superhydrophobicity through Cost-Effective Large-Scale Computations.

The pattern design of superhydrophobic surfaces can be significantly aided by computations that predict the Cassie-Baxter (CB) to Wenzel (W) transition, which is responsible for the break-down of superhydrophobic behavior. We present a computational framework for the optimization of patterned surfaces based on the energy barriers of the CB-W transitions which comprises the following elements: (a) design of structured surface patterns, for example, arrays of pillars, with parameterized geometric features such as size, pitch, slope, and roundness. (b) Computation of the wetting states with a modified Young-Laplace equation that facilitates the introduction of solid/liquid interactions for complex surface patterns and has significantly lower computational cost than other commonly used methods, such as the volume-of-fluid, phase-field, and so forth. (c) Incorporation of the modified Young-Laplace in the simplified string method, allowing the calculation of the minimum energy paths of wetting transitions which, apart from the energy barriers, also reveal the transition mechanisms (CB failure modes). (d) Accommodation of large-scale problems with good parallel performance and scalability on multicore-distributed memory systems using fast iterative solvers and the Message Passing Interface communication protocol. We demonstrate the computational framework with a shape optimization study of inverted conical frustum pillars. The optimization objective function is the resistance to the CB-W transition, which is quantified by the energy barrier-a relatively large energy barrier suggests improved superhydrophobicity. We also report the parallel performance, in terms of parallel speedup for problems ranging from three hundred thousands to 12 million degrees of freedom, solved using up to 40 processing cores.

Investigation of modified Water chemistry for improved oil recovery: Application of DLVO theory and surface complexation model

It is widely accepted that oil recovery during waterflooding can be improved by modifying the composition of the injected brine. A typical approach is diluting the formation water to a specific lower salinity. However, recent experimental studies report the adverse effect of formation water dilution on oil recovery for specific oil/brine/rock systems. The adverse effect depends on the interactions within the system and is more pronounced in carbonates. In this study, we investigated the effect of water composition on the performance of low salinity water injection by considering the complex interaction of oil, brine, and rock system.A surface complexation model (SCM) is developed to calculate the zeta-potential at oil and rock surfaces. Considering a water film between oil and rock and using DLVO theory, attractive/repulsive forces between oil/brine and brine/rock interfaces are calculated. Contact angle is predicted employing the augmented Young-Laplace equation. Our zeta potential calculations based on the SCM reproduced the experimental data of oil/brine and brine/calcite zeta potential measurements. Our contact angle calculations using the DLVO theory and the augmented Young-Laplace equation accurately estimated the dynamic trend of contact angle during low salinity flood. Modeling wettability alteration as a function of contact angle was sufficient to predict the low salinity effect. The developed model is implemented in a comprehensive compositional reactive transport simulator to validate the proposed approach.

Conceptual modeling of temperature effects on capillary pressure in dead-end pores

The effect of temperature on the liquid–gas interface and consequently on the capillary pressure in unsaturated dead-end pores is conceptually modeled in this paper. Trapping of non-wetting fluid (e.g., air) in the dead-end pores impacts the capillary pressure–saturation relationship and affects the continuous flow of wetting fluids. In the dead-end pore, which is assumed to be a simple vertical cylindrical capillary tube with one end closed and the other end open to the liquid body, the dependence of solid–liquid and solid–air interfacial tensions on temperature and its subsequent effects on the contact angles are deduced. A non-linear ordinary differential equation, using the Young–Laplace equation, in terms of a contact-angle-sensitive temperature function is derived and numerically solved using the fourth order Runge–Kutta method. This temperature function is used to obtain the capillary pressure–temperature relationship for a solid–liquid–air capillary system. Two example problems, first a glass–water–air capillary system and second a polytetrafluoroethylene–n-hexadecane–air capillary system, are solved here. A linear decrease in capillary pressure with temperature is observed, suggesting that entrapped air affects capillary pressure in dead-end pores. A similar linear decrease in capillary pressure, consistent with experimental observations, is observed for open-end pores.

Effects of nano-voids and nano-cracks on the elastic properties of a host medium: XFEM modeling with the level-set function and free surface energy

This work deals with the influences of nano-heterogeneities in the form of voids/cavities or cracks on the elastic properties of a host medium. With a relatively large ratio of apparent surface to volume and particularly strong physical interactions with the surrounding medium at nano-scale, nano-heterogeneities can potentially affect the elastic properties of the parent medium (matrix) containing them in a significant manner. This has been reported by various theoretical and experimental studies, some of which are discussed in the present paper. To describe the positive (reinforcement) or negative (degradation) effect of the nano-heterogeneities from the modeling perspective, it is necessary to take into account the energy of interfaces/surfaces between nano-heterogeneities and the matrix, which, because of the relatively large extent of their apparent surface and their strong physical interaction with their neighborhood, can no longer be neglected compared to those of the volume energy. Thus, to account for the effects of interfaces/surfaces in a nanostructured heterogeneous medium, the coherent interface model is considered in the present investigation within a periodic homogenization procedure. In this interface/surface model, the displacement vector is assumed to be continuous across the interface while the stress vector is considered to be discontinuous and satisfying the Laplace–Young equations. To solve these equations coupled to the classical mechanical equilibrium problem, a numerical simulation tool is developed in a two-dimensional (2D) context using the extended finite element method and the level-set functions. The developed numerical tool is then used to carry out a detailed analysis about the effect of nano-heterogeneities on the overall mechanical properties of a medium. The nano-heterogeneities are present in the medium initially as cylindrical cavities (circular in 2D) before being reduced to plane cracks (line in 2D) by successive flattenings.

Unraveling Interaction Mechanisms between Molybdenite and a Dodecane Oil Droplet Using Atomic Force Microscopy

Molybdenite (MoS2) is a mineral that has drawn great interest because of its potential application in various fields. To facilitate the flotation of molybdenite, the mineral pulp is commonly treated with nonpolar oil additives to promote hydrophobicity and to form an oil bridge between ultrafine molybdenite particles for agglomeration. In this study, dodecane was chosen as a model oil to investigate the flotation mechanisms of molybdenite with nonpolar oil. The interaction forces between a micrometer-sized dodecane droplet and the molybdenite basal plane in various electrolyte solutions were directly measured by the atomic force microscope droplet probe technique. The effects of added salts, ionic strength, and solution pH on interaction forces were evaluated by considering van der Waals, electrical double-layer (EDL), and hydrophobic forces. The experimentally measured force curves were found to agree well with the Reynolds lubrication model and the augmented Young-Laplace equation. The results show that the competition between repulsive EDL forces and attractive hydrophobic forces was directly responsible for oil-molybdenite attachment behavior. High pH and low salinity (<24 mM NaCl) led to strong repulsive EDL forces, which stabilized the interaction and prevented the attachment of oil to molybdenite. Both low pH and high salinity facilitated the attachment of oil to molybdenite through the depression of EDL force, allowing attractive hydrophobic force to dominate. The hydrophobic attraction was quantified with an exponential decay length of 1.0 ± 0.1 nm. Furthermore, calcium ions decreased the magnitude of the surface potentials of both oil and molybdenite more than that seen with the same ionic strength of sodium ions, suggesting the suppressed EDL repulsion. This study provides quantitative information about the surface forces between oil and the molybdenite basal plane and an improved understanding of the fundamental interaction mechanisms governing molybdenite recovery by mineral flotation.

Sessile droplets shape response to complex body forces

The majority of studies on forced wetting of sessile droplets refers to application of a steadily increasing normal or tangential force or to a specific combination of these forces arising from tilting the substrate. The above constitute well-defined test conditions but are not representative of what is encountered in most industrial applications. To approach realistic industrial conditions and so also expand existing wetting theories, an evaluation of droplet shape deformation under the influence of a complicated evolution of body forces, is performed herein. To this aim, two sets of experiments are carried out in Kerberos device creating force fields by combination of gravitational and centrifugal forces: i) by oscillations of the tilting angle at constant rotation speed and ii) by alternating small step-increases of the rotation speed and the tilting angle, one after the other, so as to trail the symmetric side profile (SSP) curve of a droplet. The latter is done in three or six steps following two different paths: increasing first either the rotation speed or the tilting angle. The aim of the experiments is, on one hand, to explore droplet deformation under cycles of increasing/decreasing tangential forces and, on the other hand, to analyze the effect of residual tangential forces on droplet shape. Several features related to the droplet shape evolution are investigated such as contact angles, side and top contour profiles and droplet 3-dimensional shape. The resulting droplet profiles are analyzed using numerical solutions of the Young-Laplace equation. It is found that the 3-dimensional Young Laplace equation can describe very accurately the experimental profiles.

Bio-inspired lightweight polypropylene foams with tunable hierarchical tubular porous structure and its application for oil-water separation

Inspired by nature, the biomimetic polypropylene (PP) foams with tunable hollow tubular structure, hierarchical porous structure and rough inner surface characteristics were designed and massively fabricated (with a speed exceed 0.5 m/s) via environment-friendly supercritical carbon dioxide (scCO2) continuous extrusion foaming. Additionally, the inner diameter of the as-prepared biomimetic tubular PP foams can be easily adjusted from extreme low (∼0.4 mm) to very high (∼20.0 mm) size through a fixed 2.0 mm circle extruder die by carefully controlling the radially distributed temperature and melt viscoelasticity. Furthermore, we also theoretically analyzed the reasons of the formation of such biomimetic hierarchical tubular structure during foaming and drew up the corresponding schematic diagram. More interestingly, thanks to the above biomimetic structure, the as-prepared PP foam strips exhibited both superior absorption and filtration ability during the oil-water separation, in which the deionized water could be filtered quickly through the foam strips while the oil was absorbed in seconds. Besides, the relationship between pore size and hydrophobicity of polymer foams was theoretically analyzed by the Young-Laplace equation. More importantly, we believe that the design principles and processing strategies reported here can also be applied to other materials and it has great potential application in the environmental sewage treatment.

Atomistic Study of Dynamic Contact Angles in CO2-Water-Silica System.

The dynamic wetting for the CO2-water-silica system occurring in deep reservoirs is complex because of the interactions among multiple phases. This work aims to quantify the contact angle of CO2-water flow in the silica channel at six different flow velocities using molecular dynamics. The dynamic contact angle values at different contact line velocities are obtained for the CO2-water-silica system. By calculating the rates of the adsorption-desorption process of CO2 and water molecules on the silica surface using molecular dynamics simulations, it has been found that the results of the dynamic contact angle can be explained by the molecular kinetic theory and predicted from the equilibrium molecular simulations. Moreover, the capillary pressure at different contact line velocities is predicted according to the Young-Laplace equation. The change in contact angles at different velocities is compared with empirical equations in terms of capillary number. The results of this study can help us better understand the dynamic process of the multiphase flow at the nanoscale under realistic reservoir conditions.

Predictability of sub-bandage pressure in compression therapy based on material properties

The pressure developed beneath any compression system is governed by the tension in the compression fabric. Depending on the type of material – cotton or elastomer – compression systems may react differently under the same stress. The previous work of our group and experience related to compression systems showed that classification of bandage materials is required for manufacturers who want to create products with specific compression levels. One of the mathematical descriptions of the expected pressure due to a tensile force in the fabric is the modified Young–Laplace equation. However, the pressure achieved by real compression systems hardly matches with the estimated values. Two groups of elastic bandages – short stretch and long stretch – were selected and investigated within this work. The tensile forces were measured using a dynamometer, and the pressures deduced at a given maximal load (3 N/cm) were compared with the pressures measured beneath the bandage on living subjects. The results show that the interface pressure can be predicted from the tensile force at the inflexion point of the hysteresis curves. Calculated pressures were in a therapeutic range but were significantly different from practically applied pressures, indicating large differences between calculated and real pressures on the leg.

Evaluating the dual porosity of landfilled municipal solid waste.

The objective of this paper was to study the pore-size distribution of municipal solid waste (MSW) and provide a basis for understanding the mechanism of preferential flow in MSW. Two methods were used to investigate the pore-size distribution in MSW. The first was an indirect method based on the soil-water characteristic curve (SWCC) and the second was a direct method using nuclear magnetic resonance (NMR). Samples taken from different depths of a landfill were used. In the SWCC method, SWCCs of the matrix region were obtained by a pressure plate extractor and SWCCs of the macropore region were determined by water breakthrough tests. A SWCC equation is proposed by modifying the Van Genuchten equation to consider the dual-porosity feature of MSW and the pore-size distribution was obtained based on the Young-Laplace equation. In the NMR method, the pore-size distribution of MSW was obtained by analyzing the T2 curves. The results of the two methods were qualitatively similar and both showed a dual-porosity characteristic of MSW. The average macropore radii of shallow, middle, and deep MSWs obtained from the SWCC method are 0.193, 0.184, and 0.173 mm, and those obtained from the NMR method are 0.213, 0.138, and 0.145 mm, respectively. The proportion of macropores decreases with depth. The average micropore radii given by the SWCC method are 0.022, 0.011, and 0.008 mm, and those given by the NMR method are 0.013, 0.011, and 0.008 mm, respectively. As the depth and fill age increase, the average micropore size becomes smaller and the proportion of micropores increases. The volume ratios of macropores obtained by the two methods are quantitatively quite different. The discrepancy is mainly caused by the different test principles adopted by the two methods.

Regulating capillary pressure to achieve ultralow areal mass loading metallic lithium anodes

Placing metallic Li inside 3D porous scaffold is effective to improve its reversibility. However, in most cases, excessive Li jeopardizes energy density of cells, and causes deterioration of electrochemical performances, due to uncontrollable thermal infusion process. Herein, by analyzing parameters that influence capillary pressure acting on molten Li using Young-Laplace equation, a novel strategy to control Li content in 3D scaffolds by regulating pore size and surface lithiophilicity of scaffolds is proposed. Especially, choosing lightweight 3D graphene with tunable surface lithiophilicity as the scaffold can generate appropriate capillary pressure to achieve ultralow areal mass loading (1.8 mg cm−2) of Li. The as-prepared Li anode possesses comparable weight with thinnest commercial Li foil, but exhibits improved reversibility to sustain 700 h of stable stripping/plating at 3.0 mAh cm−2. Moreover, the relationship between areal Li loading and lithiophilicity of graphene scaffold is established. Thus, the areal loadings of Li can be tuned in a wide range from 1.8 mg cm−2 to over 9.0 mg cm−2, making this novel anode applicable to pair with a variety of high areal loading (∼10 mg cm−2) cathodes, including lithium iron phosphate, Li-rich layered oxide and sulfur, showing great application potentials in practical Li metal batteries.

A Physics-Based Estimation of Mean Curvature Normal Vector for Triangulated Surfaces

In this note, we derive an approximation for the mean curvature normal vector on vertices of triangulated surface meshes from the Young-Laplace equation and the force balance principle. We then demonstrate that the approximation expression from our physics-based derivation is equivalent to the discrete Laplace-Beltrami operator approach in the literature. This work, in addition to providing an alternative expression to calculate the mean curvature normal vector, can be further extended to other mesh structures, including non-triangular and heterogeneous meshes.

Measuring porosities of chromatographic columns utilizing a mass-based total pore-blocking method: Superficially porous particles and pore-blocking critical pressure mechanism

Experimentally determined total, interstitial and intraparticle porosity values are necessary to equate theory, simulation and experimental column performance. This paper reports a study of a mass-based technique for determining total, interstitial and intraparticle porosity measurements based on the total pore-blocking (TPB) method. Commercially available superficially porous particle (SPP) columns, in a variety of small-pore and wide-pore materials, with both hydrophobic and hydrophilic surfaces, are utilized as samples. The results are compared with previously determined literature values for a number of columns and contrasted with HPLC-based elution methods. This method uses only a high-precision balance and an HPLC pump.A simple theoretical analysis of the TPB method using the Young-Laplace equation shows the pressure bounds and flow rate constraints of the method which ensure pore blocking stability. The results suggest that particles with small-pore diameters can be analyzed over a range of solvent clearing pressures and flow rates. However, wide-pore materials, typically with pore diameters in excess of 400 Å, have very low critical pressures and are difficult to determine without losing the pore blocking component. Small mass differences between clearing solvents are shown to present a challenge for measuring the interstitial volume.

Precisely Tailoring Bubble Morphology in Microchannel by Nanoparticles Self-assembly

Precisely tailoring bubble morphology is always a long-standing great challenge. In this work, a facile and scalable method to generate nonspherical bubbles with long-term stability is proposed. Taking advantage of the electrostatic interaction between silica nanoparticles (SNPs) and cationic surfactants, the SNPs are decorated with surfactants and endowed with interfacial activity. Due to the rearrangement of surfactants, the decorated SNPs transform to a kind of Janus particles at the gas–liquid interface. By precisely manipulating the surface activity, packing density, and jamming of Janus SNPs at the bubble surface, four different shapes such as oblaten-like, bullet-like, tadpole-like, and worm-like bubble were obtained continuously in the microchannel. Herein, our method to generate bubbles with a prescribed shape poses opportunities for gas microreactor, cavity material, gas storage, and provide a platform to study the applicable scope of the Young–Laplace equation.

Capillary bridges between unequal-sized spherical particles: Rupture distances and capillary forces

This study focuses on capillary bridges between unequal-sized spherical particles in the pendular regime where the capillary bridge surface is axisymmetric. An analytical theory as well as closed-form expressions have been developed for the rupture distance and the capillary force. These expressions have been validated with a very large dataset of numerical solutions of the governing Young-Laplace equation, generated through a high-resolution integration method.For small capillary bridge volumes and contact angles smaller than 20∘, it has been shown that the meridional profile of the capillary bridge between unequal-sized particles can be accurately described by part of ellipses and that the contact radii for the large and the small particle are approximately equal. Contrary to the widely used toroidal approximation, the developed analytical theory takes into account the governing Young-Laplace equation, according to which the capillary force is constant along the capillary bridge. The analytical theory rigorously shows that expressions developed for cases with equal-sized particles can be directly employed in cases with unequal-sized particles by the use of the Derjaguin radius when the capillary bridge volume is small. Expressions for the rupture distance and the capillary force have also been derived analytically.For large capillary bridge volumes the use of the Derjaguin radius is not sufficient to accurately describe the properties of capillary bridges between particles with unequal sizes. By curve-fitting to the large dataset of numerical solutions of the Young-Laplace equation, a closed-form expression for the rupture distance has been developed, that accounts for the influence of the particle size ratio and that is accurate over a wide range of capillary bridge volumes. Expressions in the literature for the capillary force between unequal-sized spherical particles have been rigorously evaluated. With the new expression for the rupture distance, an improved closed-form expression for the capillary force has been formulated that is accurate for a wide range of small and large capillary bridge volumes (more specifically, when the ratio of the capillary bridge volume to the cubic root of the Derjaguin radius is smaller than 0.5) and separation distances, for contact angles smaller than 40∘.

Micro-Surface and -Interfacial Tensions Measured Using the Micropipette Technique: Applications in Ultrasound-Microbubbles, Oil-Recovery, Lung-Surfactants, Nanoprecipitation, and Microfluidics.

This review presents a series of measurements of the surface and interfacial tensions we have been able to make using the micropipette technique. These include: equilibrium tensions at the air-water surface and oil-water interface, as well as equilibrium and dynamic adsorption of water-soluble surfactants and water-insoluble and lipids. At its essence, the micropipette technique is one of capillary-action, glass-wetting, and applied pressure. A micropipette, as a parallel or tapered shaft, is mounted horizontally in a microchamber and viewed in an inverted microscope. When filled with air or oil, and inserted into an aqueous-filled chamber, the position of the surface or interface meniscus is controlled by applied micropipette pressure. The position and hence radius of curvature of the meniscus can be moved in a controlled fashion from dimensions associated with the capillary tip (~5–10 μm), to back down the micropipette that can taper out to 450 μm. All measurements are therefore actually made at the microscale. Following the Young–Laplace equation and geometry of the capillary, the surface or interfacial tension value is simply obtained from the radius of the meniscus in the tapered pipette and the applied pressure to keep it there. Motivated by Franklin’s early experiments that demonstrated molecularity and monolayer formation, we also give a brief potted-historical perspective that includes fundamental surfactancy driven by margarine, the first use of a micropipette to circuitously measure bilayer membrane tensions and free energies of formation, and its basis for revolutionising the study and applications of membrane ion-channels in Droplet Interface Bilayers. Finally, we give five examples of where our measurements have had an impact on applications in micro-surfaces and microfluidics, including gas microbubbles for ultrasound contrast; interfacial tensions for micro-oil droplets in oil recovery; surface tensions and tensions-in-the surface for natural and synthetic lung surfactants; interfacial tension in nanoprecipitation; and micro-surface tensions in microfluidics.

Gibbsian Thermodynamic Study of Capillary Meniscus Depth

In the presence of gravity or other external fields, liquid surface curvature deviates from a spherical shape and the surface configuration can be found by numerical integration of the Young–Laplace equation and the typical initial point for integration is the apex of the interface. The meniscus shape in large Bond number systems, which have the central portion of the interface flattened, cannot be determined with the apex as the initial point for integration. Here we find the depth of capillary menisci by considering an initial point for integration to be at the three-phase-contact-line (TPCL) and evaluate the curvature at the TPCL by free energy analysis and inspect the effect of different parameters on the interface shape. A new parameter—which is the deviation of equilibrium curvature at the TPCL from the spherical shape (SR)—is introduced and inspected and it was found that at a Bond number of 13 the maximum deviation, approximately 0.8 of spherical curvature, takes place while for large enough Bond numbers the curvature at the three-phase contact line is near the spherical shape (0.95 < SR < 1). A potential application of this approach is to measure the capillary rise at the TPCL to find the surface tension in high Bond number systems such as those with low surface/interfacial tensions.

The Contact Angle Variation of Floating Particles Makes It Difficult to Use the Neumann Condition To Quantify the Air-Water Interface Deformation in Three-Dimensional Space.

Capillary force is critical to the floatability of particles at the air-water interface. Quantification of the capillary force requires solving the Young-Laplace equation using suitable boundary conditions (BCs) at the triple contact line. For axisymmetric (two-dimensional, 2D) systems, such as single spheres floating at an initially flat air-water surface, both the Dirichlet (constant contact depth) and Neumann (constant contact angle) BCs can be applied. For three-dimensional (3D) systems, Neumann BCs (NBCs) have been successfully used. In this paper, we have challenged the use of NBCs for the 3D deformation of the air-water surface induced by floating particles, which always exhibit intrinsic contact angle (CA) hysteresis that is significantly amplified in 3D systems. Specifically, we designed and conducted the experiments using single prismatic particles, which allowed for the determination of two characteristic CAs at the two diagonal axes with a high degree of certainty. We calibrated the numerical solution to the 3D Young-Laplace equation using the deformed air-water interface profiles at the two diagonal axes and then validated the numerical solution for the capillary force on the floating particles with the measured force. We obtained reliable data for the CA along the three-phase contact line (TPCL), which displayed a significant distribution. We also discussed the findings that were significant to floating spheres in asymmetric systems, such as pairs of floating spheres. This paper provides experimental and theoretical evidence that the CA is not constant along the contact line in a 3D geometry, which invalidates the use of NBCs for 3D systems of floating particles. This study highlights the significance of the CA variation known as CA hysteresis, which should be considered when predicting the floatability of particles at the air-water interface.

Interpretation of Young's equation for a liquid droplet on a flat and smooth solid surface: Mechanical and thermodynamic routes with a simple Lennard-Jones liquid.

In this study, we carried out molecular dynamics simulations of a cylindrical Lennard-Jones droplet on a flat and smooth solid surface and showed that Young's equation as the relation among solid-liquid, solid-vapor, and liquid-vapor interfacial tensions γSL, γSV, and γLV, respectively, was applicable only under a very restricted condition. Using the fluid stress-tensor distribution, we examined the force balance in the surface-lateral direction exerted on a rectangular control volume set around the contact line. As the mechanical route, the fluid stress integrals along the two control surfaces normal to the solid-fluid interface were theoretically connected with γSL and γSV relative to the solid-vacuum interfacial tension γS0 by Bakker's equation extended to solid-related interfaces via a thought experiment, for which the position of the solid-fluid interface plane was defined at the limit that the fluid molecules could reach. On the other hand, the fluid stress integral along the control surface lateral to the solid-fluid interface was connected with γLV by the Young-Laplace equation. Through this connection, we showed that Young's equation was valid for a system in which the net lateral force exerted on the fluid molecules from the solid surface was zero around the contact line. Furthermore, we compared γSL - γS0 and γSV - γS0 obtained by the mechanical route with the solid-liquid and solid-vapor works of adhesion obtained by the dry-surface method as one of the thermodynamic routes and showed that both routes resulted in a good agreement. In addition, the contact angle predicted by Young's equation with these interfacial tensions corresponded well to the apparent droplet contact angle determined by using the previously defined position of the solid-fluid interface plane; however, our theoretical derivation indicated that this correspondence was achieved because the zero-lateral force condition was satisfied in the present system with a flat and smooth solid surface. These results indicated that the contact angle should be predicted not only by the interfacial tensions but also by the pinning force exerted around the contact line.