Augmented Young-Laplace Equation Research Articles

Stable Drop Shapes under Disjoining Pressure: I. A Hierarchical Approach and Application

The contact line region as affected by the disjoining pressure has been analyzed under the assumption that it can sustain only two types of profiles. Disjoining pressure represents the extra potential in thin films that always exists in the contact angle region where a liquid drop or a wedge thins to meet the solid substrate and in turn affects the contact angle as well as the film profile. It is shown here that the integration of the augmented Young–Laplace equation to yield the above types of drop profiles under the action of disjoining pressure leads to the usual conditions of equilibrium as well as the condition of stability in the same analysis. Other inequality constraints are obtained where the stability condition does not apply. The fact that stability condition coexists with the conditions of equilibrium is pursued to show in one case that the stability modifies half of the predicted outcomes in the drop shapes. In addition, exceptions to the rule are found, which are physically meaningful, and a scale-dependent equilibrium is reported for the first time.

Quantitative Analysis of Fluid Interface–Atomic Force Microscopy

Net repulsive interactions between n-hexadecane and a poly-styrene microsphere in aqueous solutions are measured with atomic force microscopy and interpreted using the augmented Young–Laplace equation. The true separation between probe and fluid interface is implicitly computed from the force–distance data, providing a more accurate description of drop or bubble deformation. Experiments and theoretical arguments demonstrate that a fluid interface stiffens with increasing deformation and is not accurately treated as a Hookean spring. The unexpected stability of the draining aqueous film between hydrophobic bodies in electrolyte solutions is explained primarily by the deformation of the oil drop in response to the applied normal force, as well as the increased hydrodynamic resistance due to the increased drainage area.

Menisci in a Diamond-Shaped Capillary

This paper presents a closed form analytical solution to the augmented Young-Laplace equation for the meniscus profile in a capillary formed between four equal-sized tangent cylinders centered on the vertices of a square. The solution is valid for a large class of disjoining pressure isotherms and contact angles.

Equilibrium configurations of liquid droplets on solid surfaces under the influence of thin-film forces: Part II. Shape calculations

The appropriate augmented Young-Laplace equation for cylindrical and axisymmetric geometry is solved numerically to generate equilibrium drop and bubble shapes for various interaction-potential curves, P( h), where h is the distance of the liquid-vapor interface from the solid substrate. Order-of-reduction integration of the augmented Young-Laplace equation for the cylindrical geometry serves as a useful tool in determining how the characteristics of the interaction potential and the associated parameters affect droplet shapes. In particular, a graph of P( h)+ p c h, where p c is the capillary pressure, versus h codifies the wide variety of drop and bubble shapes that arise from a given interaction-potential isotherm. Examples of shapes calculated are drops or bubbles with and without films, wiggly drops or bubbles, drops or bubbles that exhibit multiple equilibrium shapes, and drops with step structures. Specific conditions are provided for each type of drop shape.

Adsorption and capillary condensation in porous media: Liquid retention and interfacial configurations in angular pores

Conventional models of liquid distribution, flow, and solute transport in partially saturated porous media are limited by the representation of media pore space as a bundle of cylindrical capillaries (BCC). Moreover, the capillary model ignores the dominant contribution of adsorptive surface forces and liquid films at low potentials. We propose two new complementary elements for improving our understanding of liquid configuration in porous media: (1) an approach for considering the individual contributions of adsorptive and capillary forces to the matric potential and (2) a more realistic model for pore space geometry. Modern interface science formalism is applied to determine the thickness of adsorbed liquid films as a function of thermodynamic conditions and specific surface area of the medium. The augmented Young‐Laplace (AYL) equation provided the necessary framework for combining adsorptive and capillary processes. A new pore space geometry composed of an angular pore cross section (for capillary processes) connected to slit‐shaped spaces with internal surface area (for adsorption processes) offers a more realistic representation of natural porous media with explicit consideration of surface area (absent in the standard BCC model). Liquid‐vapor configuration, saturation, and liquid‐vapor interfacial area were calculated for different potentials and pore (unit cell) dimensions. Pore dimensions may be easily related to measurable soil properties such as specific surface area and porosity. Rigorous calculations based on the AYL equation were simplified and led to the development of algebraic expressions relating saturation and interfacial area of liquid in the proposed pore space geometry to chemical potential. These simple expressions are amenable to upscaling procedures similar to those presently used with the BCC model.

Meniscus and Contact Angle in an Eye-Shaped Capillary

This paper presents an approximate analytical solution to the augmented Young–Laplace equation for the meniscus profile in an eye-shaped capillary. The solution is valid for nonmonotonic forms of the disjoining pressure function and shows the relationship between the meniscus profile, contact angle, and disjoining pressure. The expression derived for the contact angle reduces to the widely used Deryaguin–Frumkin formula.

Capillary and Viscoelastic Effects on Elastohydrodynamic Lubrication in Roller Nips

A viscocapillary model of liquid movement and film-splitting in deformable nips between an elatomer-covered roll and a hard roll is presented here. This “soft” elastohydrodynamic regime is described by Reynolds’ equation of lubrication flow and a simple set of spring-and-dashpot elements for the viscoelastic behavior of elastomeric roll covers. Capillary effects around the film-split are accounted for with an augmented Young-Laplace equation from film-flow theory. Effects of the liquid’s surface tension (via capillary number) and the elastomeric cover’s relaxation time (via Deborah number) are predicted and compared with available experiments. Application to the flows through the multiple roller nips of printing press systems is discussed.

Meniscus in a Narrow Slit

This note presents a closed-form analytical solution to the augmented Young–Laplace equation for the meniscus profile in a narrow slit. The solution is valid for any monotonic form of the disjoining pressure function.

Shape of an evaporating completely wetting extended meniscus

The microscopic details of fluid flow and heat transfer in the contact line region of an evaporating curved liquid film were experimentally and theoretically evaluated. The evaporating film thickness profiles were measured optically using null ellipsometry and image analyzing interferometry. These thickness profiles were analyzed using the augmented Young-Laplace equation to obtain the pressure field. Using the liquid pressure field, the evaporative mass flux profile was obtained from a Kelvin -Clapeyron model for the local vapor pressure. A correlation for the local slope (apparent contact angle) at a film thickness of 6 = 20 nm as a function of a dimensionless contact line heat sink was thereby obtained for a group of completely wetting fluids. This change in local slope leads to a decrease in the maximum value of the possible capillary suction at the base of the meniscus. A complementary macroscopic interfacial force balance was also used to describe the effects of viscous losses and interfacial forces on the local values of the apparent contact angle and curvature that are functions of the film thickness and heat flux. These two perspectives give a complete description of an evaporating, nonpolar, completely wetting curved film in the contact line region.

Two-Dimensional Meniscus in a Wedge

This paper presents a closed-form analytical solution of the augmented Young-Laplace equation for the meniscus profile in a two-dimensional wedge-shaped capillary. The solution is valid for monotonic forms of disjoining pressure which are repulsive in nature. In the limit of negligible disjoining pressure, it is shown to reduce to the classical solution of constant curvature. The character of the solution is examined and examples of practical interest which demonstrate the application of the solution to the computation of the meniscus profile in a wedge-shaped capillary are discussed.

An augmented Young-Laplace model of an evaporating meniscus in a microchannel with high heat flux

High flux evaporation from a steady meniscus formed in a 2-μm channel is modeled using the augmented Young-Laplace equation. The heat flux is found to be a function of the long-range van der Waals dispersion force that represents interfacial conditions between heptane and various substrates. Heat fluxes of (1.3–1.6) × 10 6 W/m 2 based on the width of the channel are obtained for heptane completely wetting the substrate at 100°C. Small channels are used to obtain these large fluxes. Even though the real contact angle is 0°, the apparent contact angle is found to vary between 24.8° and 25.6°. The apparent contact angle, which represents viscous losses near the contact line, has a large effect on the heat flow rate because of its effect on capillary suction and the area of the meniscus. The interfacial heat flux is modeled using kinetic theory for the evaporation rate. The superheated state depends on the temperature and the pressure of the liquid phase. The liquid pressure differs from the pressure of the vapor phase due to capillarity and long-range van der Waals dispersion forces, which are relevant in the ultrathin film formed at the leading edge of the meniscus. Important pressure gradients in the thin film cause a substantial apparent contact angle for a completely wetting system. The temperature of the liquid is related to the evaporation rate and to the substrate temperature through the steady heat conduction equation. Conduction in the liquid phase is calculated using finite-element analysis except in the vicinity of the thin film. A lubrication theory solution for the thin film is combined with the finite-element analysis by the method of matched asymptotic expansions.

Use of the Kelvin-Clapeyron Equation to Model an Evaporating Curved Microfilm

A Kelvin–Clapeyron change-of-phase heat transfer model is used to evaluate experimental data for an evaporating meniscus. The details of the evaporating process near the contact line are obtained. The heat flux and the heat transfer coefficient are a function of the film thickness profile, which is a measure of both the intermolecular stress field in the contact line region and the resistance to conduction. The results indicate that a stationary meniscus with a high evaporative flux is possible. At equilibrium, the augmented Young–Laplace equation accurately predicts the meniscus slope. The interfacial slope is a function of the heat flux.

Equilibrium contact angles and film thicknesses in the apolar and polar systems: role of intermolecular interactions in coexistence of drops with thin films

A stable coexistence (or the lack of it) of a macroscopic drop with ita thin films on a substrate depends on the apolar (Lifshitz-van der Waals) and polar interactions near the contact zone. The augmented Young-Laplace (AYL) equation of capillarity incorporating the apolar and apolar interactions gives conditions for this coexistence, and the resulting equilibrium contact angles. The macroscopic (observed) contact angle is shown to depend only on the free energy of coexisting flat thin film, notwithstanding a significant curvature in the contact zone. The free energy of the film and, consequently, equilibrium contact angle are then shown to be related to the macroscopic parameters of wetting by a modified Young-Dupre equation with film pressure. For systems that are completely apolar, completely polar, or when components of the spreading coefficient, S, are of the same sign, the film pressure vanishes for S 0. In the later case, the film thickness and ita edge morphology are determined from the AYL equation and conservation of initial volume. Conditions engendering the film pressure and ita magnitude are identified when the apolar and polar spreading coefficient are of opposite signs. An interesting possibility is that macrodrops of finite angles can coexist with thin films even when the total spreading coefficient is positive.

Equilibrium thin films on rough surfaces. 1. Capillary and disjoining effects

Equilibrium menisci of wetting films on rough solid surfaces are calculated as Galerkin/finite element solutions of the augmented Young-Laplace (AYL) equation. Solutions are obtained for films on one-dimensional and two-dimensional periodic surfaces. The numerical solutions of the AYL equation indicate that wetting fluid distributions are characterized by three regimes: a thin-film regime dominated by disjoining pressure effects, a transition regime in which both disjoining pressure and capillarity are important, and a bulk regime dominated by capillarity. The interplay of capillarity, disjoining pressure, and the length scale of surface roughness is cast in the form of two dimensionless parameters, the values of which determine the distribution of the wetting fluid on the solid. Moreover, the transition from the thin films to pendular structures is traced through sequences of solutions of the AYL equation. The contributions of thin films and pendular structures to the total inventory of the wetting phase are analyzed by comparing analytical solutions of the Young-Laplace and the disjoining pressure equations to the finite element solutions of the AYL equations. 24 refs., 6 figs.

Use of the Augmented Young-Laplace Equation to Model Equilibrium and Evaporating Extended Menisci

Numerical solutions of equilibrium and nonequilibrium models based on the augmented Young-Laplace equation were successfully used to evaluate experimental data for an extended meniscus. The data for the equilibrium and nonequilibrium meniscus profiles were obtained optically using ellipsometry and image processing interferometry. A Taylor series expansion of the fourth-order nonlinear transport model was used to obtain the extremely sensitive initial conditions at the interline. The solid-liquid-vapor Hamaker constants for the systems were obtained from the experimental data. The consistency of the data was demonstrated by using the combining rules to calculate the unknown value of the Hamaker constant for the experimental substrate. The sensitivity of the meniscus profile to small changes in the environment was demonstrated. Both temperature and intermolecular forces need to be included in modeling transport processes in the contact line region because the chemical potential is a function of both temperature and pressure.

Investigation of an Evaporating Extended Meniscus Based on the Augmented Young–Laplace Equation

The microscopic details of fluid flow and heat transfer near the contact line of an evaporating extended meniscus of heptane formed between a horizontal substrate and a “washer” were studied at low heat fluxes. The film profile in the contact line region was measured using ellipsometry and microcomputer-enhanced video microscopy, which demonstrated the details of the transition between a nonevaporating superheated flat thin film and an evaporating curved film. Using the augmented Young-Laplace equation, the interfacial properties of the system were initially evaluated in situ and then used to describe the transport processes. New analytical procedures demonstrated the importance of two dimensionless parameters. Both fluid flow and evaporation depend on the intermolecular force field, which is a function of the film profile. The thickness and curvature profiles agreed with the predictions based on interfacial transport phenomena models. The heat flux distribution and the pressure field were obtained. Since there are significant resistances to heat transfer in this small system due to interfacial forces, viscous stresses, and thermal conduction, the “ideal constant heat flux” cannot be attained. The description of the pressure field gives the details of the coupling between the disjoining and capillary pressures.

Three-dimensional menisci in polygonal capillaries

The shapes of gravity-free, three-dimensional menisci are computed from the augmented Young-Laplace equation. Incorporation of disjoining thin-film forces in the Young-Laplace relation eliminates the contact line, thereby eliminating the free boundary from the problem. To calculate a meniscus with finite contact angles, the conjoining/disjoining pressure isotherm must also contain an attractive, sharply varying, spike function. The width of this function, w, reflects the range of the thin-film forces. In the limit of w approaching zero, a solution of the Young-Laplace equation is recovered. The proposed calculation method is demonstrated for menisci in two different types of capillaries. In the first case, the capillary is regular-polygonal in cross section with either 3, 4, or 6 sides and with contact angles Φ ranging from 0 to 45°. In the second case, the capillary is rectangular in section with aspect ratios ranging from 1.2 to 5 and with Φ = 0°, 15°, or 30°. Gas-liquid menisci inside a square glass capillary of 0.5 mm inscribed radius are measured optically for air bubbles immersed in a solution of di- n-butyl phthalate and mineral oil. This liquid mixture exhibits a zero contact angle with the wall and matches the refractive index of the glass capillary, permitting precise visual location of the interface. Excellent agreement is found with the numerical results which further demonstrates that the limiting process of the proposed method is valid. Because it avoids the issue of locating the contact line, solution of the augmented Young-Laplace equation is a simple and powerful method for the calculation of three-dimensional menisci.

Two-dimensional menisci in nonaxisymmetric capillaries

Calculation of meniscus shapes is often difficult because the contact line is usually a free boundary. We describe a method which avoids this difficulty by using a disjoining force to eliminate the free contact line. Thus, a free-boundary problem is converted to a problem with a known domain boundary. We solve numerically the augmented Young—Laplace equation and recover the solution of the Young—Laplace problem with a contact line by a limiting procedure. The method is demonstrated by calculating the shapes of two-dimensional gravity-free menisci in elliptic, eye-shaped, and star-shaped capillaries. The numerical results are confirmed by comparison with previously known analytic solutions.

Axisymmetric shapes and stability of pendant and sessile drops in an electric field

The equilibrium shapes and stability of an axisymmetric conducting drop of fixed volume that is surrounded by a fluid insulator and is pendant or sessile on a face of a parallel-plate capacitor are governed by electrical and gravitational Bond numbers, N e and G. These measure, respectively, the relative importances of electrical and gravitational forces compared to surface tension forces. When both N e and G vanish, equilibrium drop shapes are sections of a sphere and are stable with respect to all perturbations having an infinitesimal amplitude. When N e and G are nonzero, drop shape and stability can be found by solving simultaneously, by Galerkin's method with finite element basis functions, the free boundary problem composed of the augmented Young—Laplace equation for surface shape and the Laplace equation for electric field. Asymptotic analysis for small values of N e , when G = 0, agrees well with numerical calculations and predicts that drops that are hemispherical in the absence of electric field evolve into a family of spheroidal shapes when the contact angle is prescribed and a family of conical shapes when the contact line is fixed. The locus of singular points marking the loss of existence of static drops is determined. The theoretical results are found to be in excellent agreement with available experiments. As expected, the results show that drops with their contact lines fixed are more stable than ones whose contact angles are prescribed.

Axisymmetric shapes and stability of charged drops in an external electric field

A highly conducting charged drop that is surrounded by a fluid insulator of another density can be levitated by suitably applying a uniform electric field. Axisymmetric equilibrium shapes and stability of the levitated drop are found by solving simultaneously the augmented Young–Laplace equation for surface shape and the Laplace equation for the electric field, together with constraints of fixed drop volume, charge, and center of mass. The means are a method of subdomains, finite element basis functions, and Galerkin’s method of weighted residuals, all facilitated by a large-scale computer. Shape families of fixed charge are treated systematically by first-order continuation. Previous analyses by Abbas et al. in 1967 and Abbas and Latham in 1969, in which the shapes of levitated drops are approximated as spheroids, are corrected. The new analysis shows that drops charged to less than the Rayleigh limit lose shape stability at turning points, with respect to external field strength, and that the instability seen in experiments of Doyle et al. in 1964 and others is not a bifurcation to a family of two-lobed shapes, but rather is a related imperfect bifurcation.