Laplace Pressure Drop

Since the capillary condensate in a particular mesopore is in thermodynamic equilibrium with the vapour, its chemical potential, p°, must be equal to that of the gas (under the given conditions of T and p). As we have seen, the difference between p° and p1 (the chemical potential of the free liquid) is normally assumed to be entirely due to the Laplace pressure drop, Ap, across the meniscus. However, in the vicinity of the pore wall a contribution from the adsorption potential, 0(z), should be taken into account. Thus, if the chemical potential is to be maintained constant throughout the adsorbed phase, the capillary condensation contribution must be reduced. 

The central idea of the approach is now that the sum of the two curvatures, proportional to the Laplace pressure drop across the meniscus, is constant while the radius in the vertical y)+ direction is determined by the local height of the channel, whereas the radius in the x — z plane can be directly related to the x-derivatives of /, 9x/ and 9 /. By integration we can thus obtain the function / as a function ofx. 

Here, the microchannel is placed horizontally. Therefore, the gravitational force cannot balance the capillary forces. Hence, the capillary advance will continue till there is a channel for the liquid to propagate in (Figure 5.37). The position L = 0 is defined as the entrance of the microchannel, that is, the input reservoir. The reservoir is very wide compared to the channel size, and therefore, no Young-Laplace pressure drop is present, that is, p(x = 0) = Pq- Th following section presents the advancement time of the capillary front for capillary pumping. 

A very important thermodynamic relationship is that giving the effect of surface curvature on the molar free energy of a substance. This is perhaps best understood in terms of the pressure drop AP across an interface, as given by Young and Laplace in Eq. II-7. From thermodynamics, the effect of a change in mechanical pressure at constant temperature on the molar h ee energy of a substance is... 

Many authors have worked on drop deformation and breakup, beginning with Taylor. In 1934, he published an experimental work [138] in which a unique drop was submitted to a quasi-static deformation. Taylor provided the first experimental evidence that a drop submitted to a quasi-static flow deforms and bursts under well-defined conditions. The drop bursts if the capillary number Ca, defined as the ratio of the shear stress a over the half Laplace pressure (excess of pressure in a drop of radius R. Pl = where yint is the interfacial tension) ... 

Figure 1.18. (a) Diameter resulting from the Rayleigh instability as a function of the shear stress, (b) Shear stress as a function of the Laplace pressure of the resulting drops. The linear fit gives a slope of 0.087. (Adapted from [149].)... 

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) ... 

This technique is based on the determination of the shape of a pendant drop that is formed at the tip of a capillary. The classical form of the Young and Laplace equation relates the pressure drop (Ap) across an interface at a given point to the two principal radii of curvature, r( and r2, and the interfacial tension (Freud and Harkins, 1929) ... 

For relatively thick films (higher than about 30 nm), the pressure drop at the film is the sum of the capillary pressures at the two film interfaces. In this case, the Young-Laplace equation for the film can be written as... 

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 ... 

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 ... 

In quasi-static conditions, it has been established both theoretically [7, 12] and experimentally [6, 12] that a drop breaks when the applied stress a overcomes the product of the critical capillary number Cacr and the Laplace pressure... 

Fig. 5. Drops diameter resulting from the first mechanism versus shear stress. Everything except the stress is kept constant during the experiment. Insert. Laplace pressure after rupture as a function of the stress. The linear fit gives a slope of 11.5...

The surface stress of some solids in a liquid might be determined by measuring solubility changes of small particles [97,98]. As small liquid drops have an increased vapor pressure in gas, small crystals show a higher solubility than larger ones. The reason is that, due to the curvature of the particles surface, the Laplace pressure increases the chemical potential of the molecules inside the particle. This is described by the Kelvin equation, which can be written (Ref. 3, p. 380)... 

The difference in curvature radii creates a pressure drop in ceil capillaries in accordance with Laplace-Young s law... 

The pressure drop (p - pj) across the interface is reflected in the curvature of that interface (r), as shown in a simplified version of the Laplace equation ... 

The Lucas-Washbum equation is the simplest equation to model the rate of capillary penetration into a porous material. It is derived from Poiseuille s iaw (4) for laminar flow of a Newtonian liquid through capillaries of circular cross-section by assuming that the pressure drop (AP) across the liquid-vapor interface is given by the Laplace-Young (6) equation. In practice, depending... 

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. 

Similarly, a fluid body with a closed surface on which no net external forces act will always adopt a spherical shape. This was already concluded in Section 10.1 on thermodynamic grounds it can yet also be shown by invoking the Laplace pressure. Figure 10.21 shows a spherical drop that is deformed to give a prolate ellipsoid. Near the ends of the long axis (near a) we have pL = 2y/Ra, whereas near b, pL = y/(l/i i,b + l/- 2,b)- The latter value is the smaller one, since both /tLh and i 2,b are larger than Ra. Consequently liquid will flow from the pointed ends to the middle of the drop, until a spherical shape is attained. Only for a sphere is the Laplace pressure everywhere the same. 

FIGURE 10.21 Illustration of the increase in Laplace pressure when a spherical drop (or bubble) is deformed into a prolate ellipsoid. Cross sections are shown in thick lines the axis of revolution of the ellipsoid is in the horizontal direction. Two tangent circles to the ellipse are also drawn. 

To make an emulsion (foam), one needs oil (a gas), water, energy, and surfactant. The energy is needed because the interfacial area between the two phases is enlarged, hence the interfacial free energy of the system increases. The surfactant provides mechanisms to prevent the coalescence of the newly formed drops or bubbles. Moreover it lowers interfacial tension, and hence Laplace pressure [Eq. (10.7)], thereby facilitating breakup of drops or bubbles into smaller ones. 

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