Pressure* difference across curved surfaces

Figure 3. Pressure differences across curved surfaces in a foam lamella.

Ultimately, the surface energy is used to produce a cohesive body during sintering. As such, surface energy, which is also referred to as surface tension, y, is obviously very important in ceramic powder processing. Surface tension causes liquids to fonn spherical drops, and allows solids to preferentially adsorb atoms to lower tire free energy of tire system. Also, surface tension creates pressure differences and chemical potential differences across curved surfaces tlrat cause matter to move. 

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... 

We have restricted our discussion to plane interfaces till now. However, because of the existence of surface tension, there will be a tendency to curve the interface. As a consequence of this, there must be a pressure difference across the surface. This pressure difference is given by Young-Laplace equation as... 

For some types of wetting more than just the contact angle is involved in the basic mechanism of the action. This is true in the laying of dust and the wetting of a fabric since in these situations the liquid is required to penetrate between dust particles or between the fibers of the fabric. TTie phenomenon is related to that of capillary rise, where the driving force is the pressure difference across the curved surface of the meniscus. The relevant equation is then Eq. X-36,... 

Jets discharging dose to the plane of the ceiling or wall are common in ventilation practice. The presence of an adjacent surface restricts air entrainment from the side of this surface. This results in a pressure difference across the jet, which therefore curves toward the surface. The curvature of the jet increases until it attaches to the surface. This phenomenon is usually referred to as a Coanda effect. The attached jet or, as it is commonly called, wall jet, can result from air supply through an outlet with one edge coincident with the plane of the wall or ceiling fFig. 7.27). Jets supplied at some distance from the surface or at some angle to the surface can also become attached (Fig. 7.28)... 

The pressure applied produces work on the system, and the creation of the bubble leads to the creation of a surface area increase in the fluid. The Laplace equation relates the pressure difference across any curved fluid surface to the curvature, 1/radius and its surface tension y. In those cases where nonspherical curvatures are present, the more universal equation is obtained ... 

Since it is relatively easy to transfer molecules from bulk liquid to the surface (e.g. shake or break up a droplet of water), the work done in this process can be measured and hence we can obtain the value of the surface energy of the liquid. This is, however, obviously not the case for solids (see later section). The diverse methods for measuring surface and interfacial energies of liquids generally depend on measuring either the pressure difference across a curved interface or the equilibrium (reversible) force required to extend the area of a surface, as above. The former method uses a fundamental equation for the pressure generated across any curved interface, namely the Laplace equation, which is derived in the following section. 

The presence of surface tension has an important implication for the pressures across a curved interface and, as a consequence, for phase equilibria involving curved interphase boundaries. The equation that relates the pressure difference across an interface to the radii of curvature, known as the Laplace equation, is derived in Section 6.4, and the implications for phase equilibria are considered for some specific cases in Section 6.5. 

Adsorption hysteresis is often associated with porous solids, so we must examine porosity for an understanding of the origin of this effect. As a first approximation, we may imagine a pore to be a cylindrical capillary of radius r. As just noted, r will be very small. The surface of any liquid condensed in this capillary will be described by a radius of curvature related to r. According to the Laplace equation (Equation (6.29)), the pressure difference across a curved interface increases as the radius of curvature decreases. This means that vapor will condense... 

We start by describing an important phenomenon If in equilibrium a liquid surface is curved, there is a pressure difference across it. To illustrate this let us consider a circular part of the surface. The surface tension tends to minimize the area. This results in a planar geometry of the surface. In order to curve the surface, the pressure on one side must be larger than on the other side. The situation is much like that of a rubber membrane. If we, for instance, take a tube and close one end with a rubber membrane, the membrane will be planar (provided the membrane is under some tension) (Fig. 2.4). It will remain planar as long as the tube is open at the other end and the pressure inside the tube is equal to the outside pressure. If we now blow carefully into the tube, the membrane bulges out and becomes curved due to the increased pressure inside the tube. If we suck on the tube, the membrane bulges inside the tube because now the outside pressure is higher than the pressure inside the tube. 

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 correction factor < > is required because on detachment (a) the drop does not completely leave the tip, (b) the surface tension forces are seldom exactly vertical and (c) there is a pressure difference across the curved liquid surface147. (f> depends on the ratio r/Vm. Values of have been determined empirically by Harkins and Brown148,149. It can be seen that values of r/Vm between about 0.6 and 1.2 are preferable (Figure 4.8). 

Figure 3.26 Pressure differences across the curved surfaces in a foam lamella leading to Pa> Pb and liquid flow towards Plateau borders at the expense of film thinning.

Relation between surface tension and the pressure differences across a curved liquid surface. We must now return to a most important consequence of the existence of free surface energy, which was known to Young and Laplace, and is the foundation of the classical theory of Capillarity, and of most of the methods of measuring surface tension. If a liquid surface be curved the pressure is greater on the concave side than on the convex, by an amount which depends on the surface tension and on the curvature. This is because the displacement of a curved surface, parallel to itself, results in an increase in area as the surface moves towards the convex side, and work has to be done to increase the area. This work is supplied by the pressure difference moving the surface. 

The meniscus rise z can be related to the instantaneous contact angle 6 by means of the Laplace equation (1.20) relating the pressure difference across a curved surface at point Q to the curvature. The pressure inside the liquid is higher than that in the vapour phase for the configuration shown in Figure 1.8, but the opposite occurs in the case of a wetting liquid on a vertical plate (Figure 1.9.a) i.e., Pl(z) < Pv(z). Moreover, because of the cylindrical symmetry, 1 /R2 - 0, so... 

In this model also the decrease of the pore radius due to the formation of an adsorbed layer is incorporated. Flow 1 in Fig. 9.9 is the case of combined Knudsen molecular diffusion in the gas phase and multilayer (surface) flow in the adsorbed phase. In case 2, capillary condensation takes place at the upstream end of the pore (high pressure Pi) but not at the downstream end (P2), and in case 3 the entire capillary is filled with condensate. The crucial point in cases 3 and 4 is that the liquid meniscus with a curved surface not only reduces the vapour pressure (Kelvin equation) but also causes a hydrostatic pressure difference across the meniscus and so causes a capillary suction pressure Pc equal to... 

The pressure difference across a curved interface due to the surface or interfacial tension of the solution is given by the 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. 

As a consequence of surface tension, there Is a balancing pressure difference across any curved Interface. Thus, the vapor pressure over a concave liquid surface will be smaller than that over a corresponding flat surface. This vapor pressure difference can be calculated from the Kelvin s equation ... 

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. 

Here we shall only consider some of the basic principles. We have already seen (Chapter 5, page 73) that the pressure difference across a curved surface separating two phases a and /i (Figure 12.2) is given by the Laplace equation. In the ease ol a soap film this is... 

To describe a liquid meniscus and hence to obtain the interfacial tension from the drop shape the Laplace equation is used as the mechanical equilibrium condition for two homogeneous fluids separated by an interface [188]. It relates the pressure difference across a curved interface to the surface tension and the curvature of the interface... 

There will actually also be a torque term in the surface tension, just as there is for curved dislocations, but this term is usually ignored. To make this force more physical to us, we will regard this surface energy as a surface tension. The pressure difference across a curved interface is real and occurs in all materials. In small gas-hlled voids created by implanting Xe into MgO (or Si or Al), the Xe will be crystalline if the void is small enough because the internal pressure is so high the Xe can interact with other defects just like any small crystal inside a crystalline matrix, unless the defect is a crack A simple example is calculated in Table 13.2. 

There is a pressure difference across a curved GB just as there is across a surface. 

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 ... 

The equilibrium drop shape is related to the surface tension of the drop through its curvature. In other words, the tendency for the drop to assume a curved interfacial shape is due to its surface tension. A consequence of this is the existence of a pressure difference across the interface... 

The fact that a tension exists at a liquid-fluid interface imphes that, if it is curved, there will be a difference in hydrostatic pressure across the interface. Laplace derived an expression for the pressure difference across a curved interface in terms of surface tension and curvature. The equation, referred to as the Laplace equation, is... 

Surface tension causes a pressure difference to exist across a curved surface, with the greatest pressure being on the inside of a bubble. The pressure difference across an interface between one phase (A), having pressnre pj, and another phase (B), having pressure p, for spherical bubbles of radius R, is given by ... 

It is found that there exists a pressure difference across the curved interfaces of liquids (such as drops or bubbles). For example, if one dips a tube into water (or any fluid) and applies a suitable pressure, then a bubble is formed (Figure 1.13). This means that the pressure inside the bubble is greater than the atmosphere pressure. It thus becomes apparent that curved liquid surfaces induce effects, which need special physicochemical analyses in comparison to flat liquid surfaces. It must be noticed that in this system a mechanical force has induced a change on the surface of a liquid. This phenomenon is also called capillary forces. Then one may ask, does this also require similar consideration in the case of solids The answer is yes, and will be discussed later in detail. For example, in order to remove liquid, which is inside a porous media such as a sponge, one would need force equivalent to these capillary forces. Man has been fascinated with bubbles for many centuries. As seen in Figure 1.13, the bubble is produced by applying a suitable pressure, AP, to obtain a bubble of radius R, where the surface tension of the liquid is y. 

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