Curvature Young-Laplace equation

There are a number of relatively simple experiments with soap films that illustrate beautifully some of the implications of the Young-Laplace equation. Two of these have already been mentioned. Neglecting gravitational effects, a film stretched across a frame as in Fig. II-1 will be planar because the pressure is the same as both sides of the film. The experiment depicted in Fig. II-2 illustrates the relation between the pressure inside a spherical soap bubble and its radius of curvature by attaching a manometer, AP could be measured directly. 

The exact treatment of capillary rise must take into account the deviation of the meniscus from sphericity, that is, the curvature must correspond to the AP = Ap gy at each point on the meniscus, where y is the elevation of that point above the flat liquid surface. The formal statement of the condition is obtained by writing the Young-Laplace equation for a general point (x, y) on the meniscus, with R and R2 replaced by the expressions from analytical geometry given in... 

As with all thermodynamic relations, the Kelvin equation may be arrived at along several paths. Since the occurrence of capillary condensation is intimately, bound up with the curvature of a liquid meniscus, it is helpful to start out from the Young-Laplace equation, the relationship between the pressures on opposite sides of a liquid-vapour interface. 

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. 

In the absence of external fields (e.g. gravity), the pressure is the same everywhere in the liquid otherwise there would be a flow of liquid to regions of low pressure. Thus, AP is constant and Young-Laplace equation tells us that in this case the surface of the liquid has the same curvature everywhere. 

In the Maximum-bubble-pressure method the surface tension is determined from the value of the pressure which is necessary to push a bubble out of a capillary against the Laplace pressure. Therefore a capillary tube, with inner radius rc, is immersed into the liquid (Fig. 2.9). A gas is pressed through the tube, so that a bubble is formed at its end. If the pressure in the bubble increases, the bubble is pushed out of the capillary more and more. In that way, the curvature of the gas-liquid interface increases according to the Young-Laplace equation. The maximum pressure is reached when the bubble forms a half-sphere with a radius r/s V(j. This maximum pressure is related to the surface tension by 7 = rcAP/2. If the volume of the bubble is further increased, the radius of the bubble would also have to become larger. A larger radius corresponds to a smaller pressure. The bubble would thus become unstable and detach from the capillary tube. 

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. 

In equilibrium and neglecting gravity, the curvature of a liquid surface is constant and given by the Young-Laplace equation ... 

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. 

In spatially evolving multiphase media (e.g., during dissolution of a porous medium, or phase separation in a polymer blend), the mean curvature of the interface between two phases is of interest. Curvature is a sensitive indicator of morphological transitions such as the transition from spherical to rod-like micelles in an emulsion, or the degree of sintering in a porous ceramic material. Furthermore, important physicochemical parameters such as capillary pressure (from the Young-Laplace equation) are curvature-dependent. The local value of the mean curvature K — (1 /R + 1 /Ri) of an interface of phase i with principal radii of curvature Rx and R2 can be calculated as the divergence of the interface normal vector ,... 

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 treatment of capillary phenomena usually requires the mathematical analysis of curved fluid-fluid interfaces. As a prelimineuy to this chapter and to chapter 5, we shall now repeat and extend parts of sec. 1.2.23a, where this matter was introduced. The description of curvature is a prerequisite for applying the Young-Laplace equation [1.1.2]. 

In the general form of the Young-Laplace equation, Ap = Pa — Pb (A above and B below), and the local mean curvature is positive for a meniscus that is concave upward. The Young-Laplace equation is more often formulated in terms of the hydrostatic pressures developed in a uniform body force field, such as that of gravity near the earth s surface ... 

This is characteristic of all solutions of the Young-Laplace equation that have a nodoidal (i.e., coil-like) character and appears also in many other representative cases. The curves in the figures have been drawn for sessile drops with the denser liquid—shaded area—below. Sessile bubbles have identical configurations except that they are turned upside down, yet have the heavier fluid still on the lower side. All of these curves are parametrized with respect to their curvature at the origin, yo, which is equivalent to parametrization by the value of their starting... 

This is the Young-Laplace equation applied to a spherical surface. A more general form of this equation is used when the curvature of the interface is not spherical [Gl]. 

This equilibrium condition is known as the Young-Laplace equation. The physical significance of (2-136) is that the pressure inside a curved interface at equilibrium is larger than that outside by an amount that depends on the curvature V n and y. Now the curvature term V n can be expressed as the sum of the two principle radii of curvature of S at any point xs. on the interface, that is,... 

Although the Young-Laplace equation can be applied in a rigorous sense only to systems in which there is no fluid motion, the concept that surface tension leads to a pressure jump across an interface with nonzero curvature can be used to qualitatively anticipate the nature of many capillary flows. This qualitative use of the Young-Laplace equation is similar in spirit to the use of the hydrostatic pressure distributions to anticipate the nature of gravity-driven flows, such as the gravity front, discussed earlier in this chapter. 

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... 

When the radius of the capillary tube is appreciable, the meniscus is no longer spherical and also 9> 0°. Then, Equation (329) requires correction in terms of curvatures and it should give better results than those from the rough corrections given in Equations (330)-(332) for almost spherical menisci. Exact treatment of the capillary rise due to the curved meniscus is possible if we can formulate the deviation of the meniscus from the spherical cap. For this purpose, the hydrostatic pressure equation, AP = Apgz (Equation (328)), must be valid at each point on the meniscus, where z is the elevation of that point above the flat liquid surface (see Figure 6.1 in Chapter 6). Now, if we combine the Young-Laplace equation (Equation (325)) with Equation (328), we have... 

The particular value L = L is obtained by the substitution rd = -c/(pp) (the standard Young-Laplace equation) using the proper value of the coefficient C that de-I ines drainage curvature for different shaped central pores (as listed in Table 1-1) lo yield for a given potential ... 

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). 

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