Young and Laplace equation

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 

If the area element is stretched due to an increase in internal pressure relative to the external pressure there will also be displacement along the z-axis as the surface expands. The work performed will be 

neglecting the product of differentials, equation (10.7) reduces to 

Because the two radii of curvature r and r2 remain unchanged, one can write 

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. 

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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) ... 

Two methods are used to measure the pore size distribution in a powder mercury porosimetry and adsorption-desorption hysteresis. Both methods utilize the same principle capillary rise. A nonwetting liquid requires an excess pressure to rise in a narrow capillary. The pressure difference across the interface is given by the Young and Laplace equation [15]. 

Because ceramic powders usually have macropores, mercury porosimetry is more suitable than gas adsorption. The principle of the technique is the phenomenon of capillary rise, as shown schematically in Fig. 4.4 [19]. When a liquid wets the walls of a narrow capillary, with contact angle, 9 < 90°, it will climb up the walls of the capillary. If the liquid does not wet the walls of a capillary, with contact angle, 9 > 90°, it will be depressed. When a nonwetting liquid is used, it is necessary to force the liquid to flow up the capillary to the level of the reservoir by applying a pressure. For a capillary with principal radii of curvature rj and r2 in two orthogonal directions, the pressure can be obtained by using the Young and Laplace equation ... 

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 ... 

Equations 6.4 and 6.5 are special forms (i.e., for the geometries of a sphere and a cylinder, respectively) of the Young and Laplace equation for an arbitrary curved... 

Young and Laplace (1805) derived meniscus curvature equation. 

To derive the equation of Young and Laplace we consider a small part of a liquid surface. First, we pick a point X and draw a line around it which is characterized by the fact that all points on that line are the same distance d away from X (Fig. 2.6). If the liquid surface is planar, this would be a flat circle. On this line we take two cuts that are perpendicular to each other (AXB and CXD). Consider in B a small segment on the line of length dl. The surface tension pulls with a force 7 dl. The vertical force on that segment is 7 dl sin a. For small surface areas (and small a) we have sin a d/R where R is the radius of curvature along AXB. The vertical force component is... 

When applying the equation of Young and Laplace to simple geometries it is usually obvious at which side the pressure is higher. For example, both inside a bubble and inside a drop, the pressure is higher than outside (Fig. 2.7). In other cases this is not so obvious because the curvature can have an opposite sign. One example is a drop hanging between the planar ends of two cylinders (Fig. 2.7). Then the two principal curvatures, defined by... 

The shape of a liquid surface is determined by the Young-Laplace equation. In large structures we have to consider also the hydrostatic pressure. Then the equation of Young and Laplace becomes... 

To describe wettability in a porous reservoir rock requires inclusion of both the fluid surface interaction and curvature of pore walls. Both are responsible for the capillary rise seen in porous media. The fundamental equation of capillarity is given by the equation of Young and Laplace [2]... 

Binder removal during thermal degradation has features that are similar to those encountered in the drying of a moist granular material. Let us consider a model in which interconnected pores of two different radii are present (Fig. 6.57a). Even though the pores have different radii (r and r ), initially liquid evaporates from them at the same rate so that the radii of the menisci (r ) are equal. The capillary tension in the liquid is given by the equation of Young and Laplace ... 

Any review on the shape of a liquid droplet on top of a solid surface has to start with the pioneering work by P. S. Laplace and Sir Thomas Young almost two centuries ago [1,2], Young and Laplace set out to describe the phenomenon of capillary action in which the liquid inside a small capillary tube may rise several centimeters above the liquid outside the tube [3], To understand this elfect, two fundamental equations were derived by Young and Laplace. The first equation, known as the Laplace or Young Laplace equation [1], relates the curvature at a certain point of the liquid surface to the pressure difference between both sides of the surface, and we consider it next in more detail. The second equation is Young s equation [2], which relates the contact angle to the surface tensions involved. 

Equation (8) can also be derived differently, e.g., as presented in Ref. 19, where gravitational effect is directly included in terms of the Bond number (defined later on in this section). Equation (8) was first introduced in 1805 by Young and Laplace (hence Young Laplaee equation) and is eonsidered as the basic equation of capillarity. Equation (8) is equivalent to Eq. (6b) with the influence of air accounted for in Eq. (8). 

The classical analysis of Young and Laplace of static wetting problems rests on the characterization of each interface by a macroscopic surface tension. At the intersection of three bulk phases, the three phase contact line is at rest only if the capillary forces represented by these surface tensions balance. When the three phases are a solid substrate S, a wetting liquid L and a vapor V, the mechanical equilibrium condition parallel to the solid gives the Young-Dupr6 equation for the contact angle Oq... 

Mercury porosimetry is based on the capillary rise phenomenon whereby an excess pressure is required to cause a non-wetting liquid to climb up a narrow capillary. The pressure difference across the interface is given by the equation of Young and Laplace [3 sic] and its sign is such that the pressure is less in the liquid than in the gas (or... 

At the interface of water and air, there are unbalanced forces owing to the surface tension of the fluid. In the unsaturated zone, this transition zone is known as the capillary fringe or tension-saturated zone and these unbalanced forces result in capillary rise of groundwater and dissolved constituents. Assuming a constant hemispherical fluid meniscus and pore radius, the equation of Young and LaPlace can be combined with an equation for hydrostatic pressure to produce a relation for computing the capillary rise of fluid ... 

The equation of Young and Laplace describes one of the fundamental laws in interface science If an interface between two fluids is curved, there is a pressure difference across it provided the system is in equilibrium. The Young-Laplace equation relates the pressure difference between the two phases AP and the curvature of the interface. In the absence of gravitation, or if the objects are so small that gravitation is negligible, the Young-Laplace equation is... 

There are a number of relatively simple experiments with soap films that illustrate beautifully some of the implications of the Young-Laplace equation. Two of these have already been mentioned. Neglecting gravitational effects, a film stretched across a frame as in Fig. II-1 will be planar because the pressure is the same as both sides of the film. The experiment depicted in Fig. II-2 illustrates the relation between the pressure inside a spherical soap bubble and its radius of curvature by attaching a manometer, AP could be measured directly. 

Returning to equilibrium shapes, these have been determined both experimentally and by solution of the Young-Laplace equation for a variety of situations. Examples... 

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