Surfaces Laplace equation

An approximate treatment of the phenomenon of capillary rise is easily made in terms of the Young-Laplace equation. If the liquid completely wets the wall of the capillary, the liquids surface is thereby constrained to lie parallel to the wall at the region of contact and the surface must be concave in shape. The... 

The exact treatment of capillary rise must take into account the deviation of the meniscus from sphericity, that is, the curvature must correspond to the AP = Ap gy at each point on the meniscus, where y is the elevation of that point above the flat liquid surface. The formal statement of the condition is obtained by writing the Young-Laplace equation for a general point (x, y) on the meniscus, with R and R2 replaced by the expressions from analytical geometry given in... 

Equations II-12 and 11-13 illustrate that the shape of a liquid surface obeying the Young-Laplace equation with a body force is governed by differential equations requiring boundary conditions. It is through these boundary conditions describing the interaction between the liquid and solid wall that the contact angle enters. 

Surface properties enter tlirough the Yoimg-Laplace equation of state for the surface pressure ... 

The Laplace equation, which defines tire pressure difference, AP, across a curved surface of radius, r. 

When the void space in an agglomerate is completely filled with a Hquid (Fig. Ic), the capillary state of wetting is reached, and the tensile strength of the wet particle matrix arises from the pressure deficiency in the Hquid network owing to the concave Hquid interfaces at the agglomerate surface. This pressure deficiency can be calculated from the Laplace equation for chcular capillaries to yield, for Hquids which completely wet the particles ... 

In the case of the free jet, the solution for the Aaberg exhaust system can be found by solving the Laplace equation by the method of separation of variables and assuming that there is no fluid flow through the surface of the workbench. At the edge of the jet, which is assumed to be at 0—0, the stream function is given by Eq. (10.113). This gives rise to... 

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 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. 

Equation (6.27) is the Laplace equation, or Young-Laplace equation, which defines the equilibrium condition for the pressure difference over a curved surface. In Section 6.2 we will examine the consequences of surface or interface curvature for some important heterogeneous phase equilibria. 

The fact that the curvature of the surface affects a heterogeneous phase equilibrium can be seen by analyzing the number of degrees of freedom of a system. If two phases a and are separated by a planar interface, the conditions for equilibrium do not involve the interface and the Gibbs phase rule as described in Chapter 4 applies. On the other hand, if the two coexisting phases a and / are separated by a curved interface, the pressures of the two phases are no longer equal and the Laplace equation (6.27) (eq. 6.35 for solids), expressed in terms of the two principal curvatures of the interface, defines the equilibrium conditions for pressure ... 

The fundamental property of liquid surfaces is that they tend to contract to the smallest possible area. This property is observed in the spherical form of small drops of liquid, in the tension exerted by soap films as they tend to become less extended, and in many other properties of liquid surfaces. In the absence of gravity effects, these curved surfaces are described by the Laplace equation, which relates the mechanical forces as (Adamson and Gast, 1997 Chattoraj and Birdi, 1984 Birdi, 1997) ... 

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 ... 

Consider a one dimensional oil droplet pressed against a solid surface and surrounded by an aqueous solution of surfactant (Fig. 8). The shape of the oil-water interface far away from the sohd surface is governed by the Laplace equation [18] and close to the sohd surface is augmented by an additional term relating to the film energy given by the structural disjoining pressure [12] ... 

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 equilibrium curvature of a liquid surface or meniscus depends not just on its surface tension but also on its density and the effect of gravity. The variation in curvature of a meniscus surface must be due to hydrostatic pressure differences at different vertical points on the meniscus. If the curvature at a given starting point on a surface is known, the adjacent curvature can be obtained from the Laplace equation and its change in hydrostatic pressure Ahpg. In practice the liquid... 

Use the Laplace equation to calculate the spherical radius of the soap film which is formed by the contact of two bubbles with radii of 1 and 3 cm. Assume that the soap bubbles have a surface tension of 30 mj m . Draw a sketch of the contacting bubbles to help you. 

Equation (10.4) is a special case of a more general concept represented by the Young and Laplace equation. A sphere possesses a constant radius of curvature. For an area element belonging to a nonspherical curved surface there can exist two radii of curvature (rj and 2)- If the two radii of curvature are maintained constant while an element of the surface is stretched along the x-axis from x to x + dx and along the y-axis from y to y + dy the work performed will be... 

The Young and Laplace equation, equation (10.14), reduces to equation (10.4) for the special case of a sphere with r 1 equal to r2. For a bubble, the right-hand side of equations (10.14) and (10.4) should be multiplied by 2 to allow for the fact that there are two surfaces being stretched, the interior and the exterior. 

Apply the Laplace equation to determine the pressure across a curved surface. 

The LaPlace Equation. The concept of surface energy allows us to describe a number of naturally occurring phenomena involving liquids and solids. One such situation that plays an important role in the processing and application of both liquids and solids is the pressure difference that arises due to a curved surface, such as a bubble or spherical particle. For the most part, we have ignored pressure effects, but for the isolated surfaces under consideration here, we must take pressure into account. 

Adding Eqs. (2.66) and (2.67) together to arrive at the work against surface tension, equating them with the pressure-volume work in Eq. (2.65) and simplifying leads to the Laplace equation. ... 

The Laplace equation in this form is general and applies equally well to geometrical bodies whose radii of curvature are constant over the entire surface to more intricate shapes for which the Rs, are a function of surface position. In the instance of constant radii of curvature across the surface, Eq. (2.68) reduces for several common cases. For spherical surfaces, R = R2 = R, where R is the radius of the sphere, and Eq. (2.68) becomes ... 

The Young Equation. The principle of balancing forces used in the derivation of the Laplace equation can also be used to derive another important equation in surface thermodynamics, the Young equation. Consider a liquid droplet in equilibrium... 

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. 

The Laplace equation applied specifically to spherical surfaces can be derived in a variety of ways. Example 6.1 considers an alternative derivation that points out the thermodynamic character of the result quite clearly. 

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