Young Equation Research Articles

Microscopic Origin of Capillary Force Balance at Contact Line.

We investigate the underlying mechanism of capillary force balance at the contact line. In particular, we offer a novel approach to describe and quantify the capillary force on the liquid in coexistence with its vapor phase, which is crucial in wetting and spreading dynamics. Its relation with the interface tension is elucidated. The proposed model is verified by our molecular dynamics simulations over a wide contact angle range. Differences in capillary forces are observed in evaporating droplets on homogeneous and decorated surfaces. Our findings not only provide a theoretical insight into capillary forces at the contact line, but also validate Young's equation based on a mechanical interpretation.

Nonwetting Behavior of Al-Co Quasicrystalline Approximants Owing to Their Unique Electronic Structures.

Good wetting is generally observed for liquid metals on metallic substrates, while poor wetting usually occurs for metals on insulating oxides. In this work, we report unexpected large contact angles for lead on two metallic approximants to decagonal quasicrystals, namely, Al5Co2 and Al13Co4. Intrinsic surface wettability is predicted from first principles, using a thermodynamic model based on the Young equation, and validated by the good agreement with experimental measurements performed under ultra-high vacuum by scanning electron microscopy. The atomistic details of the atomic and electronic structures at the Pb-substrate interface, and the comparison with Pb(111)/Al(111), underline the influence of the specific electronic structures of quasicrystalline approximants on wetting. Our work suggests a possible correlation of the contact angles with the density of states at the Fermi energy and paves the way for a better fundamental understanding of wettability on intermetallic substrates, which has potential consequences in several applications such as supported catalysts, protective coatings, or crystal growth.

Droplets on an elastic membrane: Configurational energy balance and modified Young equation

We study wetting of an elastic membrane whose surface can resist stretch by surface stress. Through energy minimization, we obtain the general equilibrium equations and a configurational energy balance equation at the contact line, i.e., the Young equation, modified for an elastic membrane. We use a 3D large deformation formulation with general elastic constitutive behavior for the membrane and its surfaces. Our results allow for non-flat reference membrane shape, non-uniform and anisotropic tension and dependence of surface stress on surface strain – the generalized Shuttleworth effects. We use two examples to demonstrate the application of our theory: plane strain and axisymmetric wetting of an initially flat elastic membrane. A large deformation solution for an axisymmetric droplet on a deformable membrane is obtained numerically. We show that when surface elastic moduli for solid–vapor and solid–liquid interfaces are different, continuity of membrane strain, a key assumption used in many previous studies, no longer holds at the contact line.

On the Thermodynamics of Thin Films. The Frumkin–Derjaguin Equation

The Frumkin–Derjaguin equation, which is a fundamental relationship in the theory of thin films, relates all surface tensions associated with a wetting film to its disjoining pressure and contact angle. The published ways of proving this relationship have been reviewed and their thermodynamic drawbacks have been analyzed in this work. The relationship has been rigorously derived on the basis of two approaches. The first entails the use of a new thermodynamic potential (J potential), which is defined for fluid systems as the grand thermodynamic potential combined with the product of the system volume and some pressure $$p{\kern 1pt} '$$ (here, the pressure in the mother phase of the thin film is used as $$p{\kern 1pt} '$$). The second approach is based on the Gibbs adsorption equation and entails the use of two dividing surfaces. These approaches yield identical results and suggest that not only temperature, but also all chemical potentials of the film must be fixed when calculating the work of film thinning. The dependence of the contact angle on the disjoining pressure has been considered on the basis of the Young equation. It has been shown that, as the disjoining pressure rises, the thermodynamic surface tension of the film-containing interface increases, while the contact angle decreases. The problems encountered when calculating the contact angles via the disjoining pressure isotherms have been noted.

Experimental investigation on high temperature wettability and structural behaviour of SAW fluxes using MgO–TiO2–SiO2 and Al2O3–MgO–SiO2 flux system

To establish the relationship between wettability and structure with the change in SAW flux composition, the contact angle measurement study was performed at 1700 K. For MgO–TiO2–SiO2 and Al2O3–MgO–SiO2 flux system the wetting behaviour was studied by evaluating the contact angle as well as surface tension properties. Sessile drop method was used to determine the wetting properties of SAW fluxes. Twenty-one SAW fluxes were designed & developed by applying mixture design approach of design of experiments. Chemical, phase and structural properties of SAW fluxes were measured using modern techniques such as X-ray fluorescence (XRF), X-ray diffraction (XRD) & Fourier Transform Infra-red spectroscopy (FTIR). As per the calculated contact angle value, different surface tension values for MgO–TiO2–SiO2 and Al2O3–MgO–SiO2 flux system was calculated using Young's & Boni's equations. Using Dupre's equation the adhesion energy for twenty-one basic fluxes was also calculated. Measured contact angle value increased with increase in the TiO2/MgO & TiO2/Al2O3 flux ratio. Lower contact angle gives higher wettability between the flux and the heating substrate. With increase of TiO2/SiO2 ratio up to 1.5 to 2.0 the calculated surface tension value is decreasing while after that it is increased with increase in TiO2/SiO2 ratio.

Effect of line tension on axisymmetric nanoscale capillary bridges at the liquid-vapor equilibrium.

The effect of line tension on the axisymmetric nanoscale capillary bridge between two identical substrates with convex, concave, and flat geometry at the liquid-vapor equilibrium is theoretically studied. The modified Young's equation for the contact angle, which takes into account the effect of line tension, is derived on a general axisymmetric curved surface using the variational method. Even without the effect of line tension, the parameter space where the bridge can exist is limited simply by the geometry of substrates. The modified Young's equation further restricts the space where the bridge can exist when the line tension is positive because the equilibrium contact angle always remains finite and the wetting state near the zero contact angle cannot be realized. It is shown that the interplay of the geometry and the positive line tension restricts the formation of capillary bridge.

The Influence of Hydrostatic Pressure on the Contact Angle of a Sessile Bubble

Although usual pressures have typically a weak effect on the properties of condensed phases and their surface layers, a parameter has been found in the surface physical chemistry—a contact angle at a three-phase contact line—that is rather sensitive to hydrostatic pressure. Experiments with an air bubble adhered to a solid surface immersed in water have shown that an increase in the hydrostatic pressure by less than two times causes a growth of the contact angle by more than 10°, if the angle is markedly smaller than 90°. Therewith, the three-phase contact line remains immobile, and only the liquid−gas interface changes its orientation. If the angle (no matter, acute or obtuse) is close to 90°, the three-phase contact line acquires mobility as an alternative way to reach an equilibrium . A thermodynamic theory has been developed on the basis of the generalized Young equation to explain these phenomena. It has been shown that, when the three-phase contact line is fixed, a growth of the pressure in a liquid always leads to a rise in the contact angle.

Lyapunov Spectrum of Nonautonomous Linear Young Differential Equations

We show that a linear Young differential equation generates a topological two-parameter flow, thus the notions of Lyapunov exponents and Lyapunov spectrum are well-defined. The spectrum can be computed using the discretized flow and is independent of the driving path for triangular systems which are regular in the sense of Lyapunov. In the stochastic setting, the system generates a stochastic two-parameter flow which satisfies the integrability condition, hence the Lyapunov exponents are random variables of finite moments. Finally, we prove a Millionshchikov theorem stating that almost all, in a sense of an invariant measure, linear nonautonomous Young differential equations are Lyapunov regular.

Numerical Study on an Electrohydrodynamically Driven Axially Grooved Flat Miniature Heat Pipe

This study deals with the use of the electrohydrodynamic effects in flat miniature heat pipes (FMHP). A prototype is conceived and modeled. The equations of the model are developed on the basis of the mass, momentum, and energy balance equations as well as the Laplace–Young equation, which is modified in order to take into account the action of the electric field on the liquid–vapor interface. The electric field promotes the condensate flow back to the evaporator section. Hence, the capillary driving pressure, which is ensured by the capillary grooves, is diminished because the electrohydrodynamic pumping becomes responsible of the liquid flow from the condenser to the evaporator. The electric field decreases the liquid velocity and hardly affects the vapor velocity. Moreover, it increases the vapor pressure drop, and it decreases the liquid pressure drop. The effect of the fill charge on the capillary limit is particularly highlighted. For given groove dimensions, it is demonstrated that there exists an optimum fill charge that ensures a maximum capillary limit. The evolution of the flow parameters is given and discussed for different fill charges and maximum heat input powers.

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.

Experimental methods in chemical engineering: Contact angles

AbstractThe contact angle (CA) formed at equilibrium at the three‐phase line of contact between a liquid, a solid, and a gas may be expressed as a function of both the interfacial and surface tensions. Young first derived this thermodynamic relationship in 1805. In practice, multiple CA values are observed due to kinetic phenomena induced by evaporation, vapour adsorption, or swelling, and thermodynamic ones induced by roughness and surface chemical heterogeneities, even at molecular‐scale. These non‐ideal conditions result into an hysteresis, i.e., a difference between wetting and dewetting behaviours, and Young's equation rarely applies. Three measuring methods stand out for their applicability and reliability. In the sessile drop method, a syringe deposits a liquid drop on a flat surface and the contact angle is measured through optical means based on the drop shape. In the Wilhelmy balance method, the force required to immerse a solid plate in a bath of liquid is indirectly related to the contact angle. In the Washburn capillary rise method, the contact angle is derived from the rate at which a liquid rises by capillarity through a packed bed of powder. Employing probe liquids of various polarities, the free surface energy of the solid may be estimated. Over the last two years, 8600 published articles mentioned “contact angle” in their topic. Their main focus was either to develop the fundamental understanding of wetting science, or to assess the success of surface modification methods for the production of novel surfaces, composites, and membranes with enhanced wetting, adhesive, and filtration properties.

A generalized variational approach for predicting contact angles of sessile nano-droplets on both flat and curved surfaces

Derivations of Young's equation have traditionally neglected the Laplace pressure and its influence on the curvature of sessile droplets under thermodynamic equilibrium. Omission of the Laplace pressure results in overestimating the line tension by several orders of magnitude and contributes to significant errors in predicting the contact angle for droplet volumes ranging from micro-liters to atto-liters. This paper addresses this issue and offers a correct inclusion of VdP as a virtual work term in the free energy variation at the liquid-vapor boundary for both flat and curved surfaces. While the Laplace pressure is constant at the liquid-vapor interface as a condition of equilibrium, the variation of the Laplace pressure is not zero, as it influences both the shape (spherical cap) and contact angle of sessile droplets. Inclusion of this term leads naturally to a definition of the line tension as a volume dependent term, and more importantly to a correct prediction in both sign and magnitude of the line tension value. The inclusion of VdP work predicts a cubic relationship between the cosine of the contact angle and droplet line radius. This new model extends existing theories on the behavior of nanosized droplets and its predictions exhibit quantitative agreement with experimental results for nonane, dodecane, fullerene, and glycerol trioleate over a range of 15 orders of magnitude in droplet volume. Finally, we also theorize the existence of two different equilibrium contact angles at lower droplet volumes. These two values diverge with the inverse line radius.

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.

Robust Oil-Fouling Resistance of Amorphous Cellulose Surface Underwater: A Wetting Study and Application.

The antioil-fouling characteristic of an amorphous cellulose (a-cellulose) surface was elucidated using the sessile droplet method (static) and a modified Wilhelmy plate technique (dynamic). As compared to other hydrophilic surfaces (cellulose acetate, poly(vinyl alcohol), and glass), the oil (poly(dimethylsiloxane)) contact angle on the a-cellulose surface underwater shows the largest value (170.5 ± 5.0°), having the smallest deviation from its theoretical value (180.0°) as estimated by Young's equation. Also, the a-cellulose surface demonstrates the strongest affinity with water in an oil medium (stable hydrophilicity). Moreover, the work of adhesion between the receding oil phase and a-cellulose underwater is quantified to be 10.3 mN/m, approximately one-fourth of that in air (42.0 mN/m). The overall wetting study suggests a rather low oil/solid/water three-phase contact line (TPL) friction in the direction that water displaces oil. A proposed mechanism attributes these phenomena to the water-accessible rigid cellulose chains and supramolecular structure of a-cellulose. The former hinders molecular rearrangement during processing or upon exposure to oil, such that its polar hydroxyl groups are readily accessible to water, thereby retaining its hydrophilicity. The latter allows water to diffuse across the TPL, forming the hydration shells that weaken the van der Waals interactions between oil and cellulose chains. Such findings of the a-cellulose surface can be exploited to fabricate mesh membranes with high water permeation flux (375.4 ± 13.5 L m-2 h-1 Pa-1), high oil/water separation efficiency (93-98%), and long-lasting stability, which is suitable for offshore oil spill remediation.

Mechanics of Good Trade Execution in the Framework of Linear Temporary Market Impact

We define the concept of good trade execution and we construct explicit adapted good trade execution strategies in the framework of linear temporary market impact. Good trade execution strategies are dynamic, in the sense that they react to the actual realisation of the traded asset price path over the trading period; this is paramount in volatile regimes, where price trajectories can considerably deviate from their expected value. Remarkably however, the implementation of our strategies does not require the full specification of an SDE evolution for the traded asset price, making them robust across different models. Moreover, rather than minimising the expected trading cost, good trade execution strategies minimise trading costs in a pathwise sense, a point of view not yet considered in the literature. The mathematical apparatus for such a pathwise minimisation hinges on certain random Young differential equations that correspond to the Euler-Lagrange equations of the classical Calculus of Variations. These Young differential equations characterise our good trade execution strategies in terms of an initial value problem that allows for easy implementations.

A heuristic approach for nanodrops on a smooth solid surface.

A heuristic approach is developed to obtain a simple equation for the contact angle of a nanodrop on a smooth planar solid surface. First, nanodrops of various fluids in contact with various solid surfaces are considered on the basis of nonlocal density functional theory (DFT). Along with the traditional (apparent) contact angle, θa, which the drop profile makes with the solid surface, another one, θd, formed by the smooth part of the drop profile and the horizontal plane separating that part from the oscillatory part of the profile was examined. For each of the contact angles, a separate simple equation resembling the Young equation for the macroscopic drops but containing, instead of surface tensions, the microscopic parameters of intermolecular interactions, temperature, and average density of the fluid was hypothesized and the parameters of this equation were determined using the results of DFT calculations. It was shown that predictions of these equations coincide with the results provided by DFT.

Analytical derivation of a pH dependent contact angle equation and the relationship to industrial processes of energy concerns

Regarding Young's equation, three fundamental interfacial energies of interfacial tensions can be identified. One of these interfacial energies relevant to our purpose is that related to the solid-liquid interface and it can be pH dependent. Intermolecular theory of surface tension links the solid-liquid interfacial tension or energy to surface charge species of solids. These species are electrostatic and depend on the pH of the aqueous medium in contact with the solid surface. Therefore, a theoretical relationship between contact angle and pH of some kind is worth exploring based on the fundamental theories of pH dependent solid-liquid interfacial tension, given the pH dependent electric double layer.A review of the literature dealing with the pH-dependent contact angle shows that polynomial curves have been fitted to contact angle versus pH data. Based on established theoretical works dealing with surface charge theories related to the electric double layer found in the literature, we have derived a pH dependent contact angle equation that describes trends in published experimental data with a high degree of regression coefficient. We have also theoretically and experimentally validated our model. The industrial significance of our model lies in the fact that the solution for pH, when the derivative is equated to zero, gives the point of zero charge pH of solid surface, which is an approach that has already been used in published works without any theoretical justification. The relationship of our model to industrial operations of energy and environmental significance has been discussed.

Achievement of nearly super hydrophobicity in plasma enhanced chemical vapour deposited one and two dimensional carbon nanostructures

Plasma enhanced chemical vapour deposition (PECVD) has been employed in order to synthesis different one, two and mixed dimensional carbon nanostructures on catalyst modified silicon substrate. The microstructures of all the as prepared samples were characterized by field emission scanning electron microscopes. The bonding and hybridization states of all the samples were analysed with the help of Fourier transformed infrared spectroscopy and Raman spectroscopy in depth. It has been shown that the roughness of the silicon substrates as well as the thickness of the catalyst layer casts a marked effect on the sample morphology of the as synthesized samples.The response of all the samples towards water droplets has been studied in detail and it has been shown that it is possible to achieve nearly super hydrophobicity in the samples simply by varying the deposition condition, surface roughness as well as catalyst layer.The surface energies of all the samples were calculated following modified Young's equation by measuring the contact angles of two different liquids. Also efforts has been given in order to relate the hydrophobic response of the samples with the constituent components of the surface energy.This systematic effort can play an important role in determining the structure energy relationship in PECVD synthesized carbon nanomaterials.

Assessment of localization and degradation of ZnO nano-particles in the PLA/PCL biocompatible blend through a comprehensive rheological characterization

The aim of this work was to investigate the effect of addition of various amounts of zinc oxide (ZnO) nanoparticles on the polylactide/polycaprolactone (PLA/PCL) blend prepared by melt mixing method. The morphological, rheological and degradation of prepared nanocomposites were studied. The morphology was characterized as dispersed PCL droplets into the PLA matrix, and according to Young's equation, zinc oxide nanoparticles (ZnO-NPs) was anticipated to preferentially locate in the matrix phase. The results of ATR-FTIR and the hydrolytic degradation tests demonstrated that, addition of ZnO-NPs introduced an accelerated degradation on the PLA/PCL blends. Subsequently, comprehensive linear viscoelastic evaluations were carried out to support these results and also give a deeper insight on the possible influences associated with the ZnO-NPs incorporation. Rheological results showed that with increasing nanoparticles, the elastic modulus and the complex viscosity decreased, which it was correlated to the degradation and chain scission of polymeric chains induced by ZnO-NPs.

Wettability of rock, oil and brine system based on density functional theory

The composition of an ionic aqueous solution can modify the wettability behavior of a liquid/liquid/solid system. This behavior is of great importance for oil extraction and some separation processes. Here we model the contact angle and wettability on the basis of a tridimensional classical density functional theory (DFT) and Young equation for a brine/oil/solid surface system. We propose the inclusion of dispersion interaction between surface-ion and oil-ion. The equilibrium contact angle is found through the minimization of the grand potential. The approach presented here is promising and due to its versatility can be expanded to systems with some peculiarities such as roughness and thin film.