Derivation of the Young-Laplace 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 

This expression is independent of the absolute orientation of AB and CD. Integration over the borderline (only 90° rotation of the four segments) gives the total vertical force, caused by the surface tension  

In equilibrium, this downward force must be compensated by an equal force in the opposite direction. This upward force is caused by an increased pressure AP on the concave side of trcPAP. Equating both forces leads to 

These considerations are valid for any small part of the liquid surface. Since the part is arbitrary the Young-Laplace equation must be valid everywhere. 

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This is the derivation of the Young-Laplace equation for a spherical interface from surface thermodynamics. Now, if we consider the general case, where /q A jc2 and d( /q - K-f) A 0, for any curved figure, Equation (314) applies for this condition and it has been proved that... 

We can derive the simplest form of the Young-Laplace equation for a spherical vapor bubble in equilibrium with liquid in a one-component system (or a liquid drop in air) from Newton mechanics. In the absence of any external field such as gravitational, magnetic or electrical fields, the bubble will assume a spherical shape, and the force acting towards the boundary of the bubble (or liquid drop) from the interior of the bubble is given as... 

Later, Neumann developed the static Wilhelmy plate method which depends on capillary rise on a vertical wall, to measure 6 precisely. A Wilhelmy plate whose surface is coated with the solid substrate is partially immersed in the testing liquid, and the height of the meniscus due to the capillary rise at the wall of the vertical plate is measured precisely by means of a traveling microscope or cathetometer. If the surface tension or the capillary constant of the testing liquid is known, then the contact angle is calculated from the equation, which is derived from the Young-Laplace equation... 

The most important application of the Young-Laplace equation is possibly the derivation of the Kelvin equation. The Kelvin equation gives the vapour pressure of a curved surface (droplet, bubble), P, compared to that of a flat surface, P °. The vapour pressure (P) is higher than that of a flat surface for droplets but lower above a liquid in a capillary. The Kelvin equation is discussed next. 

The Kelvin equation is derived from the Young-Laplace equation and the principles of phase equilibria. It gives the vapour pressure, P, of a droplet (curved surface) over the ordinary vapour pressure (P ) for a flat surface (see Appendix 4.2 for the derivation) ... 

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. 

A theoretical model for the heterogeneous nucleation was proposed by Hsu [10] for the growth of pre-existing nuclei in a cavity on a heated surface. The model included the effect of nmi-uniform superheated liquid. The equation for the activation curve of bubble nucleation was derived by combining the Clausius-Qapeyron and the Young—Laplace equations. Then, by substituting the linear temperature profile into the equation, the range of active cavity sizes on the heated surface was obtained. 

In most cases of interest, the surface excess mass E is small, so that the acceleration and body force terms may be neglected. Then Equation 1.40 simplifies to two conditions. One of them, V y = 0, requires that interfacial tension be uniform. The other is the Young-Laplace equation (Equation 1.22), which was obtained previously from thermodynamics for situations where body force and acceleration terms were unimportant. That the same equation (Equation 1.22) results from independent thermodynamic and mechanical derivations implies that interfacial tension must have the same value whether it is defined as in Equation 1.9 from energy considerations or as in Equation 1.39 from force considraations. Simply put, the force and energy definitions of interfacial tension are eqnivalrait, a conclusion emphasized in the work of Buff (1956). 

We will derive the Young-Laplace equation, which in general terms gives the pressure difference across a curved surface, for the specific case of a liquid spherical drop, having a radius R and a surface tension y. The pressure inside the droplet is designated as Pi and the pressure outside as P2. 

We provide here a derivation of the Kelvin equation based on the Young-Laplace equation and phase equilibrium principles. 

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 values generally used are y = 485 dyn cm (1 dyn cm" = 1 m-N m ) and 0 = 140 . As a result of its non wetting properties with regard to numerous solids, mercury was chosen for this operation. It can be noted that Washburn s equation is derived from the more general equation proposed by Young-Laplace relating the pressure difference across a meniscus to its radius via the following expression ... 

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. 

An alternative derivation of Young s equation follows the same route as the derivation of the Laplace equation using a notional change of the location of the dividing surface. Consider the surface free energy of the system depicted in Fig. 4. Around the line of three-phase contact a cylinder is drawn with length L and radius R, and implicitly we assume that R and L approach... 

Young-Laplace equation (pressure Applied in the derivation of the Kelvin... 

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-Laplace equation) and is considered as the basic equation of capillarity. Equation (8) is equivalent to Eq. (6b) with the influence of air accounted for in Eq. (8). 

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... 

Continuous models including intermolecular forces, in particular, the diffuse interface model provide a sound theoretical basis for studying equilibrium capillary phenomena in fluids. We have shown that these models can be extended in a natural way to study a thoroughly dynamical spreading process. The lubrication limit, where the contact angle is small, allows us to derive consistently an equation of motion for the liquid-vapor interface interacting with the solid surface. In the static limit, this equation yields back the equilibrium Young-Laplace theory. 

In the last part of this section, we show that the profile determined in the previous subsection from the Laplace equation and Young s law indeed corresponds to a minimum in the free energy. A convenient method of proving this fact is that of functional differentiation. This derivation [11] of Young s law and Laplace s law in the presence of gravity is closely related to the derivation first outlined by Gibbs [4] and later given by Johnson [8] (see also Ref. 9). 

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