Laplace Equation Of Capillarity Research Articles

Microscopic Expression of the Surface Tension of Nano-Scale Cylindrical Liquid and Applicability of the Laplace Equation

There is no consensus on whether the macroscopic Laplace Equation of capillarity is applicable for nanoscale systems. The microscopic expression for the radius and surface tension of the surface of tension for cylindrical liquid were deduced on Gibbs theory of capillarity. The radii and tensions of the surfaces of tension and the differences between internal and outside pressure for several argon liquid cylinders consisting of different numbers of atoms with Lennard-Jones (LJ) potential under the temperature of 90 K were obtained by combination of molecular dynamics simulation and calculation. The results suggested that Laplace equation could be applicable in nanoscale with fairly good approximation.

Axisymmetric Shape Analysis of Sessile Droplets: A Metrology for High-Temperature Wetting and Nonwetting Systems

An optimization process for the solution of the Laplace equation of capillarity by numerical analysis is described. The approach is based on the image analysis of droplets that provides the “best” initial graphical estimate from parameters of the system, which are then optimized. This optimization process yields the contact angle, the surface tension, and the work of adhesion for any experimental droplet that has an equilibrium axisymmetric shape (i.e., a sessile or pendant drop). For the optimization, the Levenberg-Marquardt (LM) and the Gauss-Newton (GN) methods are compared through an objective function. The LM has the lowest number of iterations and produces the smallest errors for the optimization. It can converge to a solution even when different combinations of initial parameters are chosen for optimization. The best results are obtained when all the parameters are optimized to generate a solution. To illustrate the process, sessile drops of molten aluminum resting on sapphire single crystal (0001) are described and analyzed.

Adsorption of alkanes from the vapour phase on water drops measured by drop profile analysis tensiometry

In this paper it is demonstrated that short chain alkanes (pentane, hexane, heptane, octane) can be adsorbed from a saturated vapour phase at the water–air interface. Measurements with the drop profile analysis tensiometry demonstrate that this molecular adsorption transfers into a condensation leading to a thin alkane film at the drop surface. At sufficient amounts, i.e. after such a film has reached a respective thickness, the condensed liquid starts to drain and an oil lens of alkane is formed at the water drop apex. The formation of such oil lenses is visually observed and leads to the situation that the obtained surface tensions are only effective values because the Laplace equation of capillarity does no longer describe the profile of the drop/oil lens. This is clearly demonstrated by the standard deviation of the profile fitting and the distribution of the deviation between experimental and calculated profile.

A New Method for Measuring Solvent Diffusivity in Heavy Oil by Dynamic Pendant Drop Shape Analysis (DPDSA)

Summary This paper presents a new experimental method and its computational scheme for measuring solvent diffusivity in heavy oil under practical reservoir conditions by DPDSA. In the experiment, a see-through windowed high-pressure cell is filled with a test solvent at desired pressure and temperature. Then, a heavy-oil sample is introduced through a syringe delivery system to form a pendant oil drop inside the pressure cell. The subsequent diffusion of the solvent into the pendant oil drop causes its shape and volume to change until an equilibrium state is reached. The sequential digital images of the dynamic pendant oil drop are acquired and digitized by applying computer-aided digital image-acquisition and-processing techniques. Physically, variations of the shape and volume of the dynamic pendant oil drop are attributed to the interfacial tension reduction and the well-known oil-swelling effect as the solvent gradually dissolves into heavy oil. Theoretically, the interfacial profile of the dynamic pendant oil drop is governed by the Laplace equation of capillarity, and the molecular diffusion process of the solvent into the pendant oil drop is described by the diffusion equation. An objective function is constructed to express the discrepancy between the numerically predicted and experimentally observed interfacial profiles of the dynamic pendant oil drop. The solvent diffusivity in heavy oil and the mass-transfer Biot number are used as adjustable parameters and thus are determined once the minimum objective function is achieved. This novel experimental technique is tested to measure diffusivities of carbon dioxide in a brine sample and a heavy-oil sample, respectively. It should be noted that, with the present technique, a single diffusivity measurement can be completed within an hour and only a small amount of oil sample is required. The interface mass-transfer coefficient at the solvent/heavy-oil interface can also be determined. In particular, this new technique allows the measurement of solvent diffusivity in an oil sample at constant prespecified high pressure and temperature. Therefore, it is especially suitable for studying the mass-transfer process of injected solvent into heavy oil during solvent-based post-cold heavy-oil production (post-CHOP).

Lattice boltzmann study on the contact angle and contact line dynamics of liquid-vapor interfaces.

The moving contact line problem of liquid-vapor interfaces was studied using a mean-field free-energy lattice Boltzmann method recently proposed [Phys. Rev. E 2004, 69, 032602]. We have examined the static and dynamic interfacial behaviors by means of the bubble and capillary wave tests and found that both the Laplace equation of capillarity and the dispersion relation were satisfied. Dynamic contact angles followed the general trend of contact line velocity observed experimentally and can be described by Blake's theory. The velocity fields near the interface were also obtained and are in good agreement with fluid mechanics and molecular dynamics studies. Our simulations demonstrated that incorporating interfacial effects into the lattice Boltzmann model can be a valuable and powerful alternative in interfacial studies.

The effects of capillary force and gravity on the interfacial profile in a reservoir fracture or pore

This paper presents a mathematical model for determining the interfacial profile between two immiscible fluids in a reservoir fracture or pore with a random orientation. This model is derived from the Laplace equation of capillarity by considering the effects of the capillary force, the gravity, and the wettability of fluids on reservoir rocks. The fourth-order Runge–Kutta method is employed to obtain the numerical solution. The mathematical model together with its numerical scheme is verified by the measured interfacial profiles between a heavy oil sample and a water phase in a circular tube. The numerical results show that the equilibrium shape of the interfacial profile depends on the dimensionless Bond number, the orientation of the reservoir fracture or pore and the contact angle. In particular, it is found that the Bond number, which is defined as the ratio of the gravity to the capillary force, has a strong effect on the shape of the interfacial profile. For a sufficiently small Bond number, the interfacial profile is governed by the capillary force and the contact angle formed with reservoir rocks. In this case, the orientation of the reservoir fracture or pore has no influence on the interfacial profile. Based on the numerical results, a universal critical Bond number is obtained and further used in determining the so-called critical permeability of hydrocarbon reservoirs. The interfacial profile in a reservoir fracture or pore is dominated by the capillary force as long as the reservoir permeability is smaller than this critical permeability.

Contact angle calculations from the contact/maximum diameter of sessile drops

This paper presents two algorithms for computing the contact angle of sessile liquid drops given data about the drops. The first yields the contact angle given the volume, surface tension and maximum diameter (or contact diameter) of a single drop. This algorithm is an extension of existing algorithms based on knowledge of the maximum diameter or of the contact diameter of a drop. A sensitivity analysis is included for this algorithm, allowing estimates to be made of the error in computed contact angle caused by errors in the measurement of the volume and/or diameter. The second algorithm requires only the volume and maximum or contact diameter of two different drops as input, and it produces both the contact angle and surface tension as output. Both algorithms are based on Newton's method applied to a function whose value is computed by solving a system of ordinary differential equations obtained from the Laplace equation of capillarity. The techniques are applicable to both hydrophobic and hydrophilic surfaces. Copyright © 2000 John Wiley & Sons, Ltd.

An analytical solution of the laplace equation for the shape of liquid surfaces near a stripwise heterogeneous wall

There is a special case of liquid in contact with a heterogeneous solid wall for which the Laplace equation of capillarity in three dimensions can be integrated analytically to obtain the exact shape of the liquid surface. It is the case when the liquid surface is minimal and periodic, the wall being plane and consisting of alternating parallel strips of two different materials. The solution presented is based on the method of conformal mapping. By examination of the results the classical theory of capillarity is shown to suffer a high curvature breakdown which occurs at the ends of the four-phase (solid-solid-liquid-vapor) lines. The apparent need to consider the role of line tension in the study of contact angle hysteresis is discussed.