Laplace tension

Y/d. This picture Is that of a Laplace tension resulting from a meniscus of radius of curvature d/2. Alternatively, for a force applied parallel to the Interface the viscous resistance should be... 

Fig. 4.41 (a) Schematics of a high aspect ratio nanowire, where P3 is the polarization along the Z-direction and p is the polar radius, (b) The stress-free EUTLO3 unit ceU in a bulk material in the antiferromagnetic phase, (c) EuTiOs unit cells subjected to the surface stress (Laplace tension) [11]... 

The Laplace tension and compression of the compound layer due to curvatures of internal and outer interfaces have been taken into account in the analysis. However, vacancy supersaturation is assumed to be outside the compound layer (in the B-core) at the initial stage of reaction, so voids can be nucleated. Curvature effects on vacancy concentration exist, so the difference of vacancy concentrations at the internal and external boundaries of the compound cannot be neglected. On the basis of the Gibbs-Thomson effect, the equilibrium vacancy concentration at the internal boundary of the compound layer is higher than that at the planar free surface... 

At the internal boundary, the compoimd shell is under Laplace tension with negative additional energy per atom so that... 

This effect assumes importance only at very small radii, but it has some applications in the treatment of nucleation theory where the excess surface energy of small clusters is involved (see Section IX-2). An intrinsic difficulty with equations such as 111-20 is that the treatment, if not modelistic and hence partly empirical, assumes a continuous medium, yet the effect does not become important until curvature comparable to molecular dimensions is reached. Fisher and Israelachvili [24] measured the force due to the Laplace pressure for a pendular ring of liquid between crossed mica cylinders and concluded that for several organic liquids the effective surface tension remained unchanged... 

A solid, by definition, is a portion of matter that is rigid and resists stress. Although the surface of a solid must, in principle, be characterized by surface free energy, it is evident that the usual methods of capillarity are not very useful since they depend on measurements of equilibrium surface properties given by Laplace s equation (Eq. II-7). Since a solid deforms in an elastic manner, its shape will be determined more by its past history than by surface tension forces. 

Surfactants aid dewatering of filter cakes after the cakes have formed and have very Httle observed effect on the rate of cake formation. Equations describing the effect of a surfactant show that dewatering is enhanced by lowering the capillary pressure of water in the cake rather than by a kinetic effect. The amount of residual water in a filter cake is related to the capillary forces hoi ding the Hquids in the cake. Laplace s equation relates the capillary pressure (P ) to surface tension (cj), contact angle of air and Hquid on the soHd (9) which is a measure of wettabiHty, and capillary radius (r ), or a similar measure appHcable to filter cakes. 

The emulsification process in principle consists of the break-up of large droplets into smaller ones due to shear forces (10). The simplest form of shear is experienced in lamellar flow, and the droplet break-up may be visualized according to Figure 4. The phenomenon is governed by two forces, ie, the Laplace pressure, which preserves the droplet, and the stress from the velocity gradient, which causes the deformation. The ratio between the two is called the Weber number. We, where Tj is the viscosity of the continuous phase, G the velocity gradient, r the droplet radius, and y the interfacial tension. 

Let us start with a model situation there is a single bubble in the liquid, the gas is insoluble, and there is no flow. Internal pressure Pint in the equilibrium bubble is in this case counterbalanced by external pressure Pext and surface Laplace pressure PL = 2a/r (a is surface tension) ... 

Laplace had previously deduced from his theory that the temperature coefficient of surface tension should stand in a constant ratio to the coefficient of expansion this is in many cases verified, and shows that the effect of temperature is largely to be referred to the change of density (Cantor, 1892). 

The surface tension acting on the meniscus would pull the sphere toward the plane and give rise to an attractive pressure P over the contact region, which can be calculated in terms of the Laplace equation. 

Bubble Point Large areas of microfiltration membrane can be tested and verified by a bubble test. Pores of the membrane are filled with liquid, then a gas is forced against the face of the membrane. The Young-Laplace equation, AF = (4y cos Q)/d, relates the pressure required to force a bubble through a pore to its radius, and the interfacial surface tension between the penetrating gas and the liquid in the membrane pore, y is the surface tension (N/m), d is the pore diameter (m), and P is transmembrane pressure (Pa). 0 is the liquid-solid contact angle. For a fluid wetting the membrane perfectly, cos 0 = 1. 

G. Lippmann introduced the capillary electrometer to measure the surface tension of mercury (Fig. 4.10). A slightly conical capillary filled with mercury under pressure from a mercury column (or from a pressurized gas) is immersed in a vessel containing the test solution. The weight of the mercury column of height h is compensated by the surface tension according to the Laplace equation... 

The oil-water dynamic interfacial tensions are measured by the pulsed drop (4) technique. The experimental equipment consists of a syringe pump to pump oil, with the demulsifier dissolved in it, through a capillary tip in a thermostated glass cell containing brine or water. The interfacial tension is calculated by measuring the pressure inside a small oil drop formed at the tip of the capillary. In this technique, the syringe pump is stopped at the maximum bubble pressure and the oil-water interface is allowed to expand rapidly till the oil comes out to form a small drop at the capillary tip. Because of the sudden expansion, the interface is initially at a nonequilibrium state. As it approaches equilibrium, the pressure, AP(t), inside the drop decays. The excess pressure is continuously measured by a sensitive pressure transducer. The dynamic tension at time t, is calculated from the Young-Laplace equation... 

The second effect of surface tension is that it causes the alveolus to become as small as possible. As the water molecules pull toward each other, the alveolus forms a sphere, which is the smallest surface area for a given volume. This generates a pressure directed inward on the alveolus, or a collapsing pressure. The magnitude of this pressure is determined by the Law of LaPlace ... 

Figure 17.2 Effects of surface tension and surfactant on alveolar stability, (a) Effect of surface tension. According to the law of LaPlace (P = 1ST/r), if two alveoli have the same surface tension (ST), the alveolus with the smaller radius (r), and therefore a greater collapsing pressure (P), would tend to empty into the alveolus with the larger radius, (b) Effect of surfactant. Surfactant decreases the surface tension and thus the collapsing pressure in smaller alveoli to a greater extent than it does in larger alveoli. As a result, the collapsing pressures in all alveoli are equal. This prevents alveolar collapse and promotes alveolar stability.

Fig. 4.8). At maximum pressure, the radius of the bubble is equal to the radius of the capillary rK on further increasing the pressure the difference decreases, and the gas bubble is released from the capillary. This maximum pressure pmax can be measured and correlates to the surface tension ydyn via the modified LAPLACE-equa-tion ...

The theory of surface tension, in other words, the problem how certain known facts and certain assumptions about the liquid state can be made to account for the existence of a surface tension, has been treated exhaustively by Laplace, by Gauss, and more recently by Van der Waals. The mathematical apparatus employed is very considerable, and we must confine ourselves to a statement of the... 

Granting Laplace s fundamental assumption, we see that the molecules in the interior of a liquid are subject to attraction in all directions, but that a different condition prevails in a layer at the surface, the thickness of which is smaller than the radius of molecular action. In this layer the molecules are subject to unbalanced attraction from the adjoining molecules in the interior, in other words, to an inward pull, which keeps the surface in a state of tension. If we imagine a small prominence raised somewhere in the surface, the tendency of this inward pull would be to bring it into the general... 

The relations between the intrinsic pressure and other physical constants developed in the foregoing paragraphs have been found from theoretical considerations based on Laplace s theory, that is, on the assumption of cohesive forces acting over very small distances. They are of interest to us inasmuch as there is a necessary connection between intrinsic pressure and surface tension. While no numerical expression has so far been found for this, it is obvious that high intrinsic pressures must be accompanied by high surface tensions, since the surface tension is a manifestation of the same cohesive force as causes intrinsic pressure. (See, however, equation 3, p. 27, for an empirical relation between the two.)... 

In the preceding pages we have availed ourselves of only one of the theories of surface tension, that of Laplace. It has led us directly to recognise an important property of liquids—their cohesion or intrinsic pressure—and has enabled us to establish... 

Van der Waals, whose theory has been further developed by Hulshoff and by Bakker, went one step further than Gibbs by assuming that there exists a perfectly continuous transition from one medium to the other at the boundary. This assumption limits him to the consideration of one particular case that of a liquid in contact with its own saturated vapour, and mathematical treatment becomes possible by the further assumption that the Van der Waals equation (see Chapter II.) holds good throughout the system. The conditions of equilibrium thus become dynamical, as opposed to the statical equilibrium of Laplace s theory. Van der Waals arrives at the following principal results (i) that a surface tension exists at the boundary liquid-saturated vapour and that it is of the same order of magnitude as that found by Laplace s theory (2) that the surface tension... 

The purpose of this chapter is to introduce the effect of surfaces and interfaces on the thermodynamics of materials. While interface is a general term used for solid-solid, solid-liquid, liquid-liquid, solid-gas and liquid-gas boundaries, surface is the term normally used for the two latter types of phase boundary. The thermodynamic theory of interfaces between isotropic phases were first formulated by Gibbs [1], The treatment of such systems is based on the definition of an isotropic surface tension, cr, which is an excess surface stress per unit surface area. The Gibbs surface model for fluid surfaces is presented in Section 6.1 along with the derivation of the equilibrium conditions for curved interfaces, the Laplace equation. 

Laplace could not express any opinion on the absolute values of the attraction assumed by him. At present, our knowledge of interatomic, interionic, and analogous forces is much greater than 170 years ago, and attempts to calculate surface energies and surface tensions are possible. 

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