Droplets Young-Laplace equation

Recent theoretical studies indicate that thermal fluctuation of a liquid/ liquid interface plays important roles in chemical/physical properties of the surface [34-39], Thermal fluctuation of a liquid surface is characterized by the wavelength of a capillary wave (A). For a macroscopic flat liquid/liquid interface with the total length of the interface of /, capillary waves with various A < / are allowed, while in the case of a droplet, A should be smaller than 2nr (Figure 1) [40], Therefore, surface phenomena should depend on the droplet size. Besides, a pressure (AP) or chemical potential difference (An) between the droplet and surrounding solution phase increases with decreasing r as predicted by the Young-Laplace equation AP = 2y/r, where y is an interfacial tension [33], These discussions indicate clearly that characteristic behavior of chemical/physical processes in droplet/solution systems is elucidated only by direct measurements of individual droplets. 

This is the Young-Laplace equation. For spherical droplets or bubbles of radius R in a dilute emulsion or foam,... 

Young-Laplace Equation. Interfacial tension causes a pressure difference to exist across a curved surface, the pressure being greater on the concave side (i.e., on the inside of a droplet). In an interface between phase A in a droplet and phase B surrounding the droplet, the phases will have pressures and If the principal radii of curvature are Ri and R2, then... 

Equation 12 is the Young-Laplace equation (3J). It shows that p > ps the pressure inside a droplet exceeds that outside. For spherical droplets in an emulsion. 

A variant is the micro-pipette method, which is also similar to the maximum bubble pressure technique. A drop of the liquid to be studied is drawn by suction into the tip of a micropipette. The inner diameter of the pipette must be smaller than the radius of the drop the minimum suction pressure needed to force the droplet into the capillary can be related to the surface tension of the liquid, using the Young-Laplace equation [1.1.212). This technique can also be used to obtain interfacial tensions, say of individual emulsion droplets. Experimental problems include accounting for the extent of wetting of the inner lumen of the capillary, rate problems because of the time-dependence of surfactant (if any) adsorption on the capillary and, for narrow capillaries accounting for the work needed to bend the interface. Indeed, this method has also been used to measure bending moduli (sec. 1.15). 

Provided that the droplet is very small the effects of gravity can be neglected and the droplet may be assumed to be spherical with radius r. If so, the Young-Laplace equation becomes... 

Kelvin Equation An expression for the vapor pressure of a droplet of liquid, RT In(p/pQ) = 2 iVfr where R is the gas constant, T is the absolute temperature, p is the vapor pressure of the liquid in bulk, pQ is that of the droplet, 7 is the surface tension, V is the molar volume, and r is the radius of the liquid. See also Young—Laplace Equation. 

Pendant or Sessile Drop Method The surface tension can be easily measured by analyzing the shape of a drop. This is often done by optical means. Assuming that the drop is axially symmetric and in equilibrium (no viscous and inertial effects), the only effective forces are gravity and surface or interfacial forces. In this case, the Young-Laplace equation relates the shape of the droplet to the pressure jump across the interface. Surface tension is, then, measured by fitting the drop shape to the Young-Laplace equation. Either a pendant or a sessile drop can be used for surface tension measurement. The pendant drop approach is often more accurate than the sessile drop approach since it is easier to satisfy the axisymmetric assumption. Similar techniques can be used for measuring surface tension in a bubble. 

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) ... 

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. 

A curved interface is an indicator of a pressure jump across the interface with higher pressure on the concave side. This can be easily seen in the case of a spherical droplet or bubble. For example consider the free body diagram of a droplet with radius R cut in half, as depicted in Fig. 4. The uniform surface tension along the circumference of the droplet is balanced by the pressure acting on the projected area nR. The balance of forces in the horizontal direction results in (2itR)y = nR )AP, or AP = Pi — Po = y/R where Pi and Po are the equilibrium pressure inside and outside of the droplet, respectively. In the case of a bubble, one obtains AP = Pi Po = 4y/R since there are two layers of surface tensions one in contact with the outside gas and one in contact with the gas inside the bubble. This simple relation can be extended to any surface with a mean curvature K = l/Ri + 1/R2 where R and R2 are the principal radii of curvature. The resulting equation is known as the Young-Laplace equation... 

The difference between the actual contact angle and the macroscopic contact angle is due to the line contributions to the free energy mentioned in section 3.2.1, in particular the line tension Tivs, which influence the contact angle. Phenomenologically, the force balance (Young-Laplace) equation (3.1) for a mesoscopic droplet with a circular base of radius R (i.e., the radius of the circular wetted area) can be written as... 

Interfacial tension measurements by the Hanging Drop Method. A bitumen droplet of known volume is formed at the tip of a hypodermic needle immersed in an aqueous solution. The shape of a drop of fiquid hanging from a syringe tip is determined from the balance of forces which include the surface tension of that liquid. The surface or interfacial tension at the liquid interface can be related to the drop shape through the Young-Laplace equation. (Hiemmenz, 1988)... 

An alternative method for measuring surface tension is to study the shape of a drop of the fluid, either a drop placed on a surface (a sessile drop) or a drop hanging from a pipette tip (pendant drop). In either case, a motionless droplet of the fluid is imaged. The only forces acting on the drop are gravity and the surface tension force. The result of these forces will determine the drop shape according to the Young-Laplace equation ... 

The Young-Laplace equation gives the pressure difference (inside-outside) across a curved surface. For spherical droplets, the equation is written as (see Appendix 4.1 for the derivation) ... 

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) ... 

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. 

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... 

In this section, we consider the influence of a gravitational field on the shape of a liquid droplet residing on a solid substrate (see Fig. 8). This topic was already addressed some 100 years ago by Bashforth and Adams [29], who supplied numerical tables for the shape of the liquid droplet. Their analysis is based on two equations the Laplace equation to describe the shape of the droplet, and Young s equation to determine the contact angle. Young s angle. 

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