Pressure difference Young-Laplace equation

Equation (6.27) is the Laplace equation, or Young-Laplace equation, which defines the equilibrium condition for the pressure difference over a curved surface. In Section 6.2 we will examine the consequences of surface or interface curvature for some important heterogeneous phase equilibria. 

The Young-Laplace equation gives the equilibrium pressure difference (mechanical equilibrium) at the menisci between liquid water in membrane pores and vapor in the adjacent phase ... 

Recent theoretical studies indicate that thermal fluctuation of a liquid/ liquid interface plays important roles in chemical/physical properties of the surface [34-39], Thermal fluctuation of a liquid surface is characterized by the wavelength of a capillary wave (A). For a macroscopic flat liquid/liquid interface with the total length of the interface of /, capillary waves with various A < / are allowed, while in the case of a droplet, A should be smaller than 2nr (Figure 1) [40], Therefore, surface phenomena should depend on the droplet size. Besides, a pressure (AP) or chemical potential difference (An) between the droplet and surrounding solution phase increases with decreasing r as predicted by the Young-Laplace equation AP = 2y/r, where y is an interfacial tension [33], These discussions indicate clearly that characteristic behavior of chemical/physical processes in droplet/solution systems is elucidated only by direct measurements of individual droplets. 

In this chapter we get to know the second essential equation of surface science — the Kelvin5 equation. Like the Young-Laplace equation it is based on thermodynamic principles and does not refer to a special material or special conditions. The subject of the Kelvin equation is the vapor pressure of a liquid. Tables of vapor pressures for various liquids and different temperatures can be found in common textbooks or handbooks of physical chemistry. These vapor pressures are reported for vapors which are in thermodynamic equilibrium with liquids having planar surfaces. When the liquid surface is curved, the vapor pressure changes. The vapor pressure of a drop is higher than that of a flat, planar surface. In a bubble the vapor pressure is reduced. The Kelvin equation tells us how the vapor pressure depends on the curvature of the liquid. 

The capillary pressure of interest in water-air-GDM systems is the difference between the pressures of the liquid and gas phases across static air-water interfaces within a GDM. This pressure difference is fundamentally related to the mean curvature H of the air-water interfaces through the well-known Young-Laplace equation 22... 

As a consequence of surface tension, there is a balancing pressure difference across any curved surface, the pressure being greater on the concave side. For a curved surface with principal radii of curvature rj and r2 this pressure difference is given by the Young-Laplace equation, Ap = y(llrx + l/r2), which reduces to Ap = 2y/r for a spherical surface. 

The Young-Laplace equation forms the basis for some important methods for measuring surface and interfacial tensions, such as the pendant and sessile drop methods, the spinning drop method, and the maximum bubble pressure method (see Section 3.2.3). Liquid flow in response to the pressure difference expressed by Eqs. (3.6) or (3.7) is known as Laplace flow, or capillary flow. 

According to the Young-Laplace equation, a pressure difference Ap is required to support a stable, curved interface with curvature radii r, and r2 (by convention, positive for convex interfaces and negative for concave ones),... 

Young-Laplace Equation. Interfacial tension causes a pressure difference to exist across a curved surface, the pressure being greater on the concave side (i.e., on the inside of a droplet). In an interface between phase A in a droplet and phase B surrounding the droplet, the phases will have pressures and If the principal radii of curvature are Ri and R2, then... 

The above equation is intuitively understandable because it is the basis for the Kelvin equation If the pressure difference is equated with the Young-Laplace equation it yields the Kelvin one. It should be noted here that Eq.(5) does not suffer from the incorrectness of the Kelvin equation for nanopores because it does not include any pore-size related factor. 

Finally, we note that the surface body force term constitutes the sum of the surface-excess body force and the bulk-phase body force vector densities. The surface-excess body force is the 2D analog of continuum body forces in 3D fluids (e.g., gravitational force, electromagnetic force, etc). This force is often neglected. The bulk-phase body force has no counterpart for 3D fluids, as it denotes the stresses applied intimately at the interface by the surrounding 3D bulk phases. The normal component of this force equals the pressure difference between the two bulk phases, a relationship often referred to as the Young-Laplace equation. 

It should be noted that the pressure is always greater on the concave side of the interface irrespective of whether or not this is a condensed phase.) The phenomena due to the presence of curved liquid surfaces are called capillary phenomena, even if no capillaries (tiny cylindrical tubes) are involved. The Young-Laplace equation is the expression that relates the pressure difference, AP, to the curvature of the surface and the surface tension of the liquid. It was derived independently by T. Young and P. S. Laplace around 1805 and relates the surface tension to the curvature of any shape in capillary phenomena. In practice, the pressure drop across curved liquid surfaces should be known from the experimental determination of the surface tension of liquids by the capillary rise method, detailed in Section 6.1. 

This simple form of the Young-Laplace equation shows that if the radius of the sphere increases, AP decreases, and when sph—> °°, AP —> 0, so that when the curvature vanishes and transforms into a flat Euclidean plane, there will be no pressure difference, and the two phases will be in hydrostatic equilibrium as stated above. 

On the other hand, the liquid surface in the capillary tube mostly takes the form of a concave spherical cap, as seen in Figure 4.7. In other terms, we can attribute the rise of a liquid in a capillary tube as simply the automatic recording of the pressure difference, AP, across the meniscus of the liquid in the tube, the curvature of the meniscus being determined by the radius of the tube and the angle of contact, 0, between the liquid and the capillary wall. If the capillary tube is circular in cross section and not too large in radius, then the meniscus will be completely hemispherical, that is 0=0° and r = Ri = R2 in the Young-Laplace equation (Equation (325)) giving... 

Gas Diffusion. The second mechanism for foam coalescence in porous media, gas diffusion, pertains primarily to the stagnant, trapped bubbles. According to the Young—Laplace equation, gas on the concave side of a curved foam film is at a higher pressure and, hence, higher chemical potential than that on the convex side. Driven by this difference in chemical potential, gas dissolves in the liquid film and escapes by diffusion from the concave to the convex side of the film. The rate of escape is proportional to film curvature squared and, therefore, is rapid for small bubbles (16, 26). 

Young-Laplace Equation The fundamental relationship giving the pressure difference across a curved interface in terms of the surface or interfacial tension and the principal radii of curvature. In the special case of a spherical interface, the pressure difference is equal to twice the surface (or interfacial) tension divided by the radius of curvature. Also referred to as the equation of capillarity. 

We have thus far restricted our discussion to plane interfaces. However, because of the existence of surface tension, there will be a tendency to curve the interface, as a consequence of which there must be a pressure difference across the surface with the highest pressure on the concave side. The expression relating this pressure difference to the curvature of the surface is usually referred to as the Young-Laplace equation. It was published by Young in 1805 and, independently, by Laplace in 1806. From a calculation of the p-V work required to expand the curved surface and so change its surface area, it is relatively straightforward to show that this equation may be written... 

The forces that give rise to the phenomena spoken of appear because of the alteration in stresses at the interface between two immiscible fluid phases. For a curved interface there is a difference in pressure between the two fluids given by the Young-Laplace equation. This pressure difference is termed the capillary pressure, and since the normal stress component at the interface must be continuous, then that pressure added to the hydrostatic pressure must balance. A balance can always be achieved under static conditions. In addition, the tangential stress must also be continuous at the interface. However, if there... 

Now the total pressure drop driving the bubble is also given by the difference in pressure drops at the front and rear menisci. In particular, Pi Px [p2 Po) Pi Po)> whence from the Young-Laplace equation... 

Many methods of measuring surface and interfacial tensions are based on the use of the Young-Laplace equation. This relates to the surface tension the difference in pressure Ap that is observed across any arbitrarily curved interface. If the curvature is described by two radii R and Ri (see figure 2.2) and the interfacial tension is y, then the work of maintaining this interface is balanced by the work needed to displace the interfacial plane along the radius. Thus we find... 

As indicated above, the Young-Laplace equation (Equation 1.22) is one fundamental result obtained from the theory of interfaces. Because this equation relates interfadal tension to the pressure difference between fluids at each point along an interface, it can be used with the equations of hydrostatics to calculate the shape of a static interface. Or, if interfadal shape can be determined... 

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