Young-Laplace equation from curvature

For example, for a circle having its center at the origin, the equation is x2 + y2 = R2. By applying this circle equation to Equation (296) we can easily calculate the curvature of a circle, as k= 1IR and thus R, = R for a circle. 

if we vary the direction of the plane through the NN axis, this curvature will of course change, and as K varies, it achieves a minimum and a maximum (which are perpendicular to each other) known as the principal curvatures, K, and k2, and the corresponding directions are called principal directions. The principal curvatures measure the maximum and minimum bending of a surface at each point. However, it has been shown mathematically that the sum of the curvatures (k, + K, = 1 /Rt + /R2) is independent of how the first plane is oriented when the second plane is always at right angles to it, and consequently the direction of the planes is a matter of choice. 

In order to describe the curvature of three-dimensional objects in terms of principal curvatures, there are two methods mean curvature and Gaussian curvature. The mean curvature, H, is much more used in surface science and defined as the arithmetic mean of curvatures 

It has the dimension of m 1. (In some surface textbooks, mean curvature is given as H = (Ki + k2), which is wrong in mathematical definition terms.) Let R and R2 be the radii corresponding to the curvatures the mean curvature H is then given by the multiplicative inverse of the harmonic mean, 

We can also calculate the mean radius of curvature from analytical geometry since there are x-, y- and z-axes in a three-dimensional Cartesian coordinate system, without proof, we may write the mean radius of curvature as 

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In spatially evolving multiphase media (e.g., during dissolution of a porous medium, or phase separation in a polymer blend), the mean curvature of the interface between two phases is of interest. Curvature is a sensitive indicator of morphological transitions such as the transition from spherical to rod-like micelles in an emulsion, or the degree of sintering in a porous ceramic material. Furthermore, important physicochemical parameters such as capillary pressure (from the Young-Laplace equation) are curvature-dependent. The local value of the mean curvature K — (1 /R + 1 /Ri) of an interface of phase i with principal radii of curvature Rx and R2 can be calculated as the divergence of the interface normal vector ,... 

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... 

As with all thermodynamic relations, the Kelvin equation may be arrived at along several paths. Since the occurrence of capillary condensation is intimately, bound up with the curvature of a liquid meniscus, it is helpful to start out from the Young-Laplace equation, the relationship between the pressures on opposite sides of a liquid-vapour interface. 

In the Maximum-bubble-pressure method the surface tension is determined from the value of the pressure which is necessary to push a bubble out of a capillary against the Laplace pressure. Therefore a capillary tube, with inner radius rc, is immersed into the liquid (Fig. 2.9). A gas is pressed through the tube, so that a bubble is formed at its end. If the pressure in the bubble increases, the bubble is pushed out of the capillary more and more. In that way, the curvature of the gas-liquid interface increases according to the Young-Laplace equation. The maximum pressure is reached when the bubble forms a half-sphere with a radius r/s V(j. This maximum pressure is related to the surface tension by 7 = rcAP/2. If the volume of the bubble is further increased, the radius of the bubble would also have to become larger. A larger radius corresponds to a smaller pressure. The bubble would thus become unstable and detach from the capillary tube. 

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. 

When the radius of the capillary tube is appreciable, the meniscus is no longer spherical and also 9> 0°. Then, Equation (329) requires correction in terms of curvatures and it should give better results than those from the rough corrections given in Equations (330)-(332) for almost spherical menisci. Exact treatment of the capillary rise due to the curved meniscus is possible if we can formulate the deviation of the meniscus from the spherical cap. For this purpose, the hydrostatic pressure equation, AP = Apgz (Equation (328)), must be valid at each point on the meniscus, where z is the elevation of that point above the flat liquid surface (see Figure 6.1 in Chapter 6). Now, if we combine the Young-Laplace equation (Equation (325)) with Equation (328), we have... 

Gas Diffusion. The second mechanism for foam coalescence in porous media, gas diffusion, pertains primarily to the stagnant, trapped bubbles. According to the Young—Laplace equation, gas on the concave side of a curved foam film is at a higher pressure and, hence, higher chemical potential than that on the convex side. Driven by this difference in chemical potential, gas dissolves in the liquid film and escapes by diffusion from the concave to the convex side of the film. The rate of escape is proportional to film curvature squared and, therefore, is rapid for small bubbles (16, 26). 

We have thus far restricted our discussion to plane interfaces. However, because of the existence of surface tension, there will be a tendency to curve the interface, as a consequence of which there must be a pressure difference across the surface with the highest pressure on the concave side. The expression relating this pressure difference to the curvature of the surface is usually referred to as the Young-Laplace equation. It was published by Young in 1805 and, independently, by Laplace in 1806. From a calculation of the p-V work required to expand the curved surface and so change its surface area, it is relatively straightforward to show that this equation may be written... 

As noted, there is also an excess pressure from surface tension that is associated with the interface curvature. From the Young-Laplace equation for the plane case considered, this excess pressure is (p ) = cr/R, where R is the local radius of curvature of the wave. For small displacements R = —d lldx (Eq. 10.3.1), from which... 

Equation (3.6) is called Young-Laplace equation, in which R is the harmonic mean of the principal radii of curvature. The capillary pressure promotes the release of atoms or molecules from the particle surface. This leads to a decrease of the equilibrium vapour pressure with increasing droplet size Kelvin equation) ... 

Normal stress condition. The stress at any point on the interface in a direction normal to the interface as we cross from one phase to the other, jumps by an amount equal to the normal stress due to interfacial curvature (this is in turn given by the Young-Laplace equation and depends on the geometry) the jump in pressure at a point on the interface occurs as we cross into the phase which contains the center of curvature. 

For a liquid surface, if no force acts normal to a tensioned surface, the surface must remain flat. However, if the pressure on one side of the surface differs from pressure on the other side, the pressure difference times surface area results in a normal force. The surface tension forces must cancel the force due to pressure and the surface must be curved. Figure 2.18 shows how the curvature of a tiny patch of surface leads to a net component of surface tension forces acting normal to the center of the patch. When all the forces are balanced, the result is known as the Young-Laplace equation ... 

For quasi-static process, if one ignores the gravity, the pressure on each side of an interface is a constant at any time. This implies that the curvature of the interface is constant (from the Young-Laplace equation). Thus, the interface is a circular arc. From (4.16), it is easy to show that the contact point X is related to 6 as. 

Let us assume the bubble to have quasistatic motion, that is, the velocity of the bubbles is nearly zero, and the bubble remains close to equilibrium for all bubble positions. The channel is also assumed to be smooth, that is, free from sharp corners. The pressure at the left and right of the bubble is Pi and P, respectively. The pressure difference across the bubble is AP ,. The surface tension for the liquid-gas, solid-liquid, and solid-gas interface is ajg, pressure difference AP = P, - P across the interface according to the Young-Laplace equation is... 

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 Laplace-Young equation refers to a spherical phase boundary known as the surface of tension which is located a distance from the center of the drop. Here the surface tension is a minimum and additional, curvature dependent, terms vanish (j ). The molecular origin of the difficulties, discussed in the introduction, associated with R can be seen in the definition of the local pressure. The pressure tensor of a spherically symmetric inhomogeneous fluid may be computed through an integration of the one and two particle density distributions. 

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... 

Up till now we considered only flat interfaces. The surface tension aspires to bend the surface. As a result, there is a negative pressure jump as we go through the interface from the side where the centre of curvature is located. The expression for the pressure jump is called the Laplace-Young equation ... 

This result is in agreement with the result obtained by the application of the Laplace-Young equation, Eq. (1.1). For a spherical bubble the principal radii of curvature, Ri and R2, are equal to its radius, that is / i = / 2 = r. So the excess pressure across each surface of the bubbles is, from Eq. (1.1),... 

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