Pressure Young-Laplace equation

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. 

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. 

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. 

The oil-water dynamic interfacial tensions are measured by the pulsed drop (4) technique. The experimental equipment consists of a syringe pump to pump oil, with the demulsifier dissolved in it, through a capillary tip in a thermostated glass cell containing brine or water. The interfacial tension is calculated by measuring the pressure inside a small oil drop formed at the tip of the capillary. In this technique, the syringe pump is stopped at the maximum bubble pressure and the oil-water interface is allowed to expand rapidly till the oil comes out to form a small drop at the capillary tip. Because of the sudden expansion, the interface is initially at a nonequilibrium state. As it approaches equilibrium, the pressure, AP(t), inside the drop decays. The excess pressure is continuously measured by a sensitive pressure transducer. The dynamic tension at time t, is calculated from the Young-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. 

Relation 9.77 is usually called the Washburn equation [55,237], One should consider it as a special case of the fundamental Young-Laplace equation [3,9-11], Washburn was the first to propose the use of mercury for measurements of porosity. Now, it is a common method [3,8,53-55] of psd measurements for a range of sizes from several hundreds of microns to 3 to 6 nm. The lower limit is determined by the maximum pressure, which is applied in a mercury porosimeter the limiting size of rWl = 3 nm is achieved under PHg = 4000 bar. The measurements are carried out after vacuum treatment of a sample and filling the gaps between pieces of solid with mercury. Further, the hydraulic system of a device performs the gradual increase of PHg, and the appropriate intmsion of mercury in pores of the decreasing size occurs. 

The Young-Laplace equation gives the equilibrium pressure difference (mechanical equilibrium) at the menisci between liquid water in membrane pores and vapor in the adjacent phase ... 

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. 

The controlled drop tensiometer is a simple and very flexible method for measuring interfacial tension (IFI) in equilibrium as well as in various dynamic conditions. In this technique (Fig. 1), the capillary pressure, p of a drop, which is formed at the tip of a capillary and immersed into another immiscible phase (liquid or gas), is measured by a sensitive pressure transducer. The capillary pressure is related to the IFT and drop radius, R, through the Young-Laplace equation [2,3] ... 

For relatively thick films (higher than about 30 nm), the pressure drop at the film is the sum of the capillary pressures at the two film interfaces. In this case, the Young-Laplace equation for the film can be written as... 

In the absence of external fields (e.g. gravity), the pressure is the same everywhere in the liquid otherwise there would be a flow of liquid to regions of low pressure. Thus, AP is constant and Young-Laplace equation tells us that in this case the surface of the liquid has the same curvature everywhere. 

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... 

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. 

In this chapter we get to know the second essential equation of surface science — the Kelvin5 equation. Like the Young-Laplace equation it is based on thermodynamic principles and does not refer to a special material or special conditions. The subject of the Kelvin equation is the vapor pressure of a liquid. Tables of vapor pressures for various liquids and different temperatures can be found in common textbooks or handbooks of physical chemistry. These vapor pressures are reported for vapors which are in thermodynamic equilibrium with liquids having planar surfaces. When the liquid surface is curved, the vapor pressure changes. The vapor pressure of a drop is higher than that of a flat, planar surface. In a bubble the vapor pressure is reduced. The Kelvin equation tells us how the vapor pressure depends on the curvature of the liquid. 

The capillary pressure of interest in water-air-GDM systems is the difference between the pressures of the liquid and gas phases across static air-water interfaces within a GDM. This pressure difference is fundamentally related to the mean curvature H of the air-water interfaces through the well-known Young-Laplace equation 22... 

As a consequence of surface tension, there is a balancing pressure difference across any curved surface, the pressure being greater on the concave side. For a curved surface with principal radii of curvature rj and r2 this pressure difference is given by the Young-Laplace equation, Ap = y(llrx + l/r2), which reduces to Ap = 2y/r for a spherical surface. 

The Young-Laplace equation forms the basis for some important methods for measuring surface and interfacial tensions, such as the pendant and sessile drop methods, the spinning drop method, and the maximum bubble pressure method (see Section 3.2.3). Liquid flow in response to the pressure difference expressed by Eqs. (3.6) or (3.7) is known as Laplace flow, or capillary flow. 

The Young-Laplace equation (3.4/3.5) shows that, pA>pB, the pressure inside a bubble or drop exceeds that outside. For a sphere, Ap=pA - Pb = 2y/R, so that Ap varies with the radius, R. Thus the vapour pressure of a drop should be higher, the smaller the drop. This is shown by a related equation, the Kelvin equation [13,26], which is described here. 

In primary oil recovery from underground reservoirs, the capillary forces described by the Young and Young-Laplace equations are responsible for retaining much of the oil (residual oil) in parts of the pore structure in the rock or sand. It is these same forces that any secondary or enhanced (tertiary) oil-recovery-process strategies are intended to overcome [2,133,421,690,691]. In an oil-bearing reservoir the relative oil and water saturations depend upon the distribution of pore sizes in the rock. The capillary pressure in a pore is... 

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 ,... 

According to the Young-Laplace equation, a pressure difference Ap is required to support a stable, curved interface with curvature radii r, and r2 (by convention, positive for convex interfaces and negative for concave ones),... 

Another important factor is capillary forces. The pressure increase Ap across the liquid-vapor meniscus in a cylindrical pore of radius r is described by the Young-Laplace equation. 

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. 

The thermod niamical and mechanical interpretations of interfacial tensions point to two ways of measurement either the reversible work or the force required to extend cin interface should be determined. In practice it is often easier to balance the interfacial force against others, say against gravity. In general, the results can be interpreted in terms of the capillary pressure Ap created by the interfacial tension across curved interfaces. From sec. 1.2.23 we know that the required relation is given by the Young-Laplace equation (1.2.23.191... 

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). 

The above equation is intuitively understandable because it is the basis for the Kelvin equation If the pressure difference is equated with the Young-Laplace equation it yields the Kelvin one. It should be noted here that Eq.(5) does not suffer from the incorrectness of the Kelvin equation for nanopores because it does not include any pore-size related factor. 

In the general form of the Young-Laplace equation, Ap = Pa — Pb (A above and B below), and the local mean curvature is positive for a meniscus that is concave upward. The Young-Laplace equation is more often formulated in terms of the hydrostatic pressures developed in a uniform body force field, such as that of gravity near the earth s surface ... 

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