Curved liquid surfaces

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. 

The Young1-Laplace2 equation relates the pressure difference between the two phases AP and the curvature of the surface  

R and Ro are the two principal radii of curvature. AP is also called Laplace pressure. Equation (2.5) is also referred to as the Laplace equation. 

1 Thomas Young, 1773-1829. English physician and physicist, professor in Cambridge. 

2 Kerre-Simon Laplace, Marquis de Laplace, 1749-1827. French natural scientist. 

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In a small-diameter capillary tube, wetting forces produce a distortion of the free liquid surface, which takes a curvature. Across a curved liquid surface, a difference of pressure exists and the variation of pressure across the surface is the Laplace or capillary pressure, given by ... 

If one dips a tubing into water (or any fluid) and applies a suitable pressure, then a bubble is formed (Figure 2.4). This means that the pressure inside the bubble is greater than the atmospheric pressure. Thus, curved liquid surfaces induce effects that need special physicochemical analyses in comparison to flat liquid surfaces. It must be noted that, in this system, a mechanical force has induced a change on the... 

The consequence of Laplace pressure is very important in many different processes. One example is that, when a small drop comes into contact with a large drop, the former will merge into the latter. Another aspect is that vapor pressure over a curved liquid surface, pcur, will be larger than on a flat surface, pf,at. A relation between pressure over curved and flat liquid surfaces was derived (Kelvin equation) ... 

The measurement of the dimensions of curved liquid surfaces or bubbles. 

The first objection, that of uonequilibriuni, has received a partial rebuttal from Rodebush (R5, R6, R7). The motions of translation and rotation tend to stabilize a cluster, as can be shown by considerations of the entropy of these two effects. It is also pointed out that water is an unusual material because the liquid molecules in the bulk material have a tetrahedral arrangement. Thus a tiny bubble will be surrounded by unsatisfied hydrogen bonds in the curved liquid surface. The effect on entropy of the interfacial organization probably means that the use of a constant molecular heat of vaporization X as used by Bernath and others is in error. 

A more accurate value, using the same method, involves a volumetric flask as the vessel and an analytical balance for the weighings. The volumetric flask has a narrow neck that makes accurate measurement easy liquid is added until the bottom of the curved liquid surface (the meniscus) appears to just touch the mark that is etched on the neck (Figure 7-1). 

The cause for this change in vapor pressure is the Laplace pressure. The raised Laplace pressure in a drop causes the molecules to evaporate more easily. In the liquid, which surrounds a bubble, the pressure with respect to the inner part of the bubble is reduced. This makes it more difficult for molecules to evaporate. Quantitatively the change of vapor pressure for curved liquid surfaces is described by the Kelvin equation ... 

When applying the Kelvin equation, it is instructive to distinguish two cases A drop in its vapor (or more generally a positively curved liquid surface) and a bubble in liquid (a... 

In prepared catalysts the pore sizes may be quite uniform. However, in most naturally occurring materials there is a wide range of pore sizes. The actual pore size distribution can be obtained from methods such as porosimetry, in which a nonwetting liquid (usually mercury) is pumped into a solid sample [12,13,15,26,30,55]. The solid is considered to be composed of a bundle of capillaries. For each capillary, the Laplace equation (see Section 3.2.2) gives the pressure drop across a curved liquid surface ... 

FIGURE 3.29 A schematic view from above the disk of a passive capillary burst valve. A liquid flows in a channel or capillary and is pinned at the discontinuity where the channel meets a chamber or a wider channel. Sufficient fluidic pressure must be exerted by the centrifugal pump to overcome the pressure of curved liquid surfaces and to wet the walls of the chamber with liquid. This pressure is achieved at a characteristic rate of rotation or burst frequency, C0c, above which the liquid exits the channel and enters the chamber. CO, depends on the hydraulic diameter (dH) of the capillary and the amount of liquid in the channel and therefore provides a means of gating the flow of liquid [1042]. Reprinted with permission from the American Chemical Society. 

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. 

Figure 1.8. The principal radii of curvature R and R2 at a point Q on a curved liquid surface.

Let a small portion of a curved liquid surface ABCD (Fig. 13. VIIIG), with AB and CD equal, parallel, and at right angles to AD and BC, be in equilibrium under the surface tension a and a difference of pressure p between the two sides of the surface. If the surface ABCD is displaced so ... 

Capillary forces offer a coherent explanation for the drying periods of many materials. If a tapered capillary is filled with water and exposed to a current of air, the meniscus at the smaller end remains stationary while the tube empties from the wider end. A similar situation exists in a wet particulate bed and the phenomenon is explained by the concept of suction potential. A negative pressure exists below the meniscus of a curved liquid surface which is proportional to the surface tension, X, and inversely proportional to the radius of curvature, r. (The meniscus is assumed to be a part of a hemisphere.) This negative pressure or suction potential may be expressed as the height of liquid, expressed by Eq. (25),... 

Curved Liquid Surfaces 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. 

Houllevigue9 and Satterly10 calculated the change of latent heat of evaporation 7e over a curved liquid surface as compared with the normal value over a plane surface. 

Vapor Pressure at Strongly Curved Liquid Surfaces... 

The vapor pressure (p°)r at strongly curved liquid surfaces with radius r can be calculated from the Gibbs-Thomson equation ... 

Fig. 7.14 Illustration of (a) the sunk-curved liquid surface whai stirring the Newtonian fluids and (b) the convex-curved liquid surface when stirring polymer melt or concentrated solutions...

As the proportion of liquid to particles is increased the liquid is free to move and the attractive force between particles decreases (funicular). When there is sufficient liquid to completely fill the interstitial pores between the particles (capillary) the granule strength falls further as there are fewer curved liquid surfaces and fewer boundaries for surface tension forces to act on. Clearly when the particles are completely dispersed in the liquid (droplet) the strength of the structure is very low. 

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. 

Another aspect is that vapor pressure over a curved liquid surface, pc rve will be larger than on a flat surface, Pdaf A relation between pressure over curved mdftat liquid surface was derived (Kelvin equation) ... 

Capillary cohesion phenomenon — Kelvin equation. The theory of capillary cohesion and Kelvin equation are the theoretical basis of physical vapor adsorption. When the steam of adsorbate contacts with porous solid surface, it will form liquid film of the adsorbate on the surface adsorption field. The films in the pore bend variously with the pore diameter, while the films in the outer surface of particles are relatively flat. The film thickness of liquid of adsorption increases with increase in vapor pressure. When it reaches a certain moment, the gravity between the curved liquid surfaces sufficiently liquidity the vapor from gaseous automatically, and completely fill the pores. This phenomenon is known as capillary cohesion. 

It can see from the above-mentioned discussion that capillary cohesion is closely related to the curved liquid surface. The pressme boimdary causes capillary cohesion — the critical vapor pressure relates to the cmvatme radius of liquid surface. Kelvin equation has been derived from thermodynamics, where the curvature radius (rjs) of the meniscus of hemispherical (concave) liquid and the equilibrium vapor pressure (p) has the following relationships ... 

Most other methods of measuring surface tmision also depend on the production of a curved liquid surface, by die combined effect of gravity and a solid boundary, a the calculation of pressure difference given by Laplace s equation. One recent method is radically different this is the study of the speed of capillary waves on the surface by using them to scatter li t from a laser. A wave of length A on the surface of an incompressible invisdd liquid moves with a phase velocity c given by ... 

The BET (Brunauer-Emmett-Teller)-type isotherm in Figure 3.3.8B reflects multilayer adsorption of the adsorbate. After a monolayer adsorbate coverage is achieved in the adsorbent pores, additional molecular layers are formed on top of the adsorbed monolayer by condensation of vapors. In adsorbents having small-diameter pores, multilayer condensation of the adsorbate vapor can fill the pore completely with the liquid adsorbate. This phenomenon is called capillary condensation (Figure 3.3.8C). Consider the curved interface between the vapor phase and the condensed phase of species i in the micropore in this figure. The vapor pressure of the condensed liquid above the concave curved liquid surface in the capillary F utved less than that over a plane condensed liquid surface F pi ... 

As a result of the action of surface tension, -y, there exists across a curved liquid surface a pressure difference, Ap, that depends on the curvature of the surface. The pressure is greater on the inside of the surface. If the principal radii of curvature of the surface are r, and / 2, then the Laplace equation can be written Ap = y[(l/r,) + (l/rj)]. 

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