Young-Laplace method

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 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] ... 

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. 

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. 

Mercury porosimetry is the most suitable method for the characterization of the pore size distribution of porous materials in the macropore range that can as well be applied in the mesopore range [147-155], To obtain the theoretical foundation of mercury porosimetry, Washburn [147] applied the Young-Laplace equation... 

Henderson-Hasselbalch equation lahn-Teller effect Lee-Yang-Parr method Lineweaver-Burk method Mark-Houwink plot Meerwein-Ponndorf theory Michaelis-Menten kinetics Stem-Volmer plot van t Hoff-Le Bel theory Wolff-Kishner theory Young-Laplace equation Ziegler-Natta-type catalyst... 

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). 

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. 

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... 

Wenzel s relation has been confirmed in terms of the first two laws of thermodynamics. Huh and Mason, in 1976, used a perturbation method for solving the Young-Laplace equation while applying Wenzel s equation to the surface texture. Their results can be reduced to Wenzel s equation for random roughness of small amplitude. They assume that hysteresis was caused by nonisotropic equilibrium positions of the three-phase contact line, and its movement was predicted to occur in jumps. On the other hand, in 1966, Timmons and Zisman attributed hysteresis to microporosity of solids, because they found that hysteresis was dependent on the size of the liquid molecules or associated cluster of molecules (like water behaves as an associated cluster of six molecules). 

Henderson—Hasselbalch equation Jahn—Teller effect Lineweaver—Burk method Mark-Houwink plot Meerwein—Ponndorf theory Michaelis—Menten kinetics Stern—Volmer plot van t Hoff—Le Bel theory Wolff—Kishner theory Young—Laplace equation Ziegler—Natta-type catalyst... 

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. 

Many methods of measuring surface and interfacial tensions are based on the use of the Young-Laplace equation. This relates to the surface tension the difference in pressure Ap that is observed across any arbitrarily curved interface. If the curvature is described by two radii R and Ri (see figure 2.2) and the interfacial tension is y, then the work of maintaining this interface is balanced by the work needed to displace the interfacial plane along the radius. Thus we find... 

This relationship among the two bulk phase pressures, the mean curvature, and the interfacial tension is fimdamental to the study of fluid interfaces. It is often referred to as the Laplace or the Young-Laplace equation and is the basis for several methods of measuring interfacial tension. Note that with the interface cnrved in the manner shown in Figure 1.2, H is negative and Equation 1.22 implies that Pa > Pb- Thus the pressure inside a drop or bubble always exceeds the pressure outside. 

Unlike the Young-Laplace analysis, which was based on mechanical equilibrium, this method describes the dynamic sorption of a liquid into a porous medium. The analysis is based on balancing viscous and capillary forces, and it neglects inertia or gravitational effects. The mass of absorbed liquid m in time t is related to the fluid density p, dynamic viscoscity 0, and surface tension G v at the liquid-vapor interface ... 

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