Curved Liquid Surfaces Young-Laplace Equation

3 Curved Liquid Surfaces Young-Laplace Equation 

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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. 

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. 

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... 

A theoretical model for the heterogeneous nucleation was proposed by Hsu [10] for the growth of pre-existing nuclei in a cavity on a heated surface. The model included the effect of nmi-uniform superheated liquid. The equation for the activation curve of bubble nucleation was derived by combining the Clausius-Qapeyron and the Young—Laplace equations. Then, by substituting the linear temperature profile into the equation, the range of active cavity sizes on the heated surface was obtained. 

For a liquid surface, if no force acts normal to a tensioned surface, the surface must remain flat. However, if the pressure on one side of the surface differs from pressure on the other side, the pressure difference times surface area results in a normal force. The surface tension forces must cancel the force due to pressure and the surface must be curved. Figure 2.18 shows how the curvature of a tiny patch of surface leads to a net component of surface tension forces acting normal to the center of the patch. When all the forces are balanced, the result is known as the Young-Laplace equation ... 

The Young-Laplace equation gives the pressure difference across a curved surface and has many applications, e.g. in the understanding/control of liquid rise in capillaries, capillary forces in soil pores and nucleation and growth of aerosols in atmosphere. The importance of curvature is evident if we think, for example, that a bubble with radius equal to 0.1 mm in champagne results in a pressure difference of 1.5 kPa, enough to sustain a column of water of 15 cm height. 

The most important application of the Young-Laplace equation is possibly the derivation of the Kelvin equation. The Kelvin equation gives the vapour pressure of a curved surface (droplet, bubble), P, compared to that of a flat surface, P °. The vapour pressure (P) is higher than that of a flat surface for droplets but lower above a liquid in a capillary. The Kelvin equation is discussed next. 

We will derive the Young-Laplace equation, which in general terms gives the pressure difference across a curved surface, for the specific case of a liquid spherical drop, having a radius R and a surface tension y. The pressure inside the droplet is designated as Pi and the pressure outside as P2. 

Because so many applications of surfactants involve surfaces and interfaces with high degrees of curvature, it is often important to understand the effect of curvature on interfacial properties. What is usually considered to be the most accurate procedure for the determination of the surface tension of liquids, the capillary rise method, depends on a knowledge of the relationship between surface curvature and the pressure drop across curved interfaces. Because of the existence of surface tension effects, there will develop a pressure differential across any curved surface, with the pressure greater on the concave side of the interface that is, the pressure inside a bubble will always be greater than that in the continuous phase. The Young-Laplace equation... 

Because the liquid wets and spreads over the solid surfaces, pores will be formed in the liquid. The reduction of the liquid-vapor interfacial area provides the driving force for shrinkage or densification of the compact. If the pore in the liquid is assumed to be spherical with radius of r, the pressure difference across the curved surface is given by the Young and Laplace equation ... 

A liquid bridge has a curved surface and therefore the pressure on both sides of the surface is not the same with the pressure difference AP being given by the Laplace-Young equation ... 

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