Surface force capillary rise method

Both the Wilhelmy and capillary rise methods for determining 7 have been based on the concept of surface tension as a force. While this point of view is useful for describing the experimental methods we have discussed, it is only one way of interpreting 7. An energetic interpretation is also possible that makes surface tension amenable to the powerful methods of thermodynamics. 

The capillary rise method is a classical method for determining surface tension that has important applications. For capillary rise in a narrow capillary (Figure 3.7) a force balance yields ... 

While determining the liquid height, it is better to measure with a falling (or receding) meniscus, so that the liquid level is initially raised above its equilibrium value by a slight suction above the capillary tube, and then left to equilibrate. On the other hand, two-armed capillary tubes, connected with a cross tube above the liquid level, are also used to ensure that the pressure in both arms of the glass apparatus is the same. An interesting modification of the capillary rise method is to measure the pressure, AP, that is required to force the meniscus down until it is on the same level as the plane surface of liquid outside the capillary tube. This method is useful to compare the surface tension of water and its dilute solutions. 

The detachment of a ring or a plate (a Wilhelmy plate) from the surface of a liquid or solution is a static surface tension measurement method, which gives the detachment force of a film of the liquid and its extension from the liquid surface. These methods are less accurate than the capillary rise method, but they are normally employed in most surface laboratories because of their ease and rapidity. 

Equation 14 therefore represents an equilibrium dominant balance between the gravitational and surface tension forces. Thus, given that p and cos 0 are known for a given fluid, the capillary rise can be determined. It is also evident from Eq. 14 that the capillary rise is dependent on the capillary radius. For small capillaries, H can therefore become relatively large. This capillary rise method is therefore very useful and accurate for measuring the surface tension of liquids. 

Classically, the approach used to calculate capillary flow has been to determine the curvature of liquid interfaces in the system and calculate Pcap from Equation (6.1). Those values could then be used to calculate the direction and magnitude of the driving forces. In systems of simple geometry such as liquids which form spherical interfaces and smooth cylindrical solid surfaces, the technique works out very well. Perhaps the best known example of such a system is the capillary rise method for determining the surface tension of a hquid, illustrated in Figure 6.10. In this system, capillary forces cause the hquid to rise in the tube due to differences in curvature of the liquid-air interface within the tube (a small radius of curvature) and that in the reservoir... 

FIGURE 6.10. In the capillary rise method of surface tension measurement, surface tension effects canse the wetting hquid to rise in the small capillary to a height that just balances the hydrodynamic force dne to gravity (a). For non-wetting liquids such as mercury, a depressing effect is observed (Z>). 

The capillary rise method was the earliest technique by which surface tension was measured and, indeed, was the technique by which the force itself was recognized. If a narrow tube of radius r is partially inserted into a liquid, the liquid rises up inside the tube to some equilibrium position as shown in Fig. 22. This occurs because the attractive interaction of the wetting liquid (aqueous solution) with the solid surface is stronger than that of the gas phase. Gravity opposes the rise, and the equilibrium height H corresponds to the minimum free energy of the system. The treatment is based on the Laplace equation that gives the pressure difference across a curved interface due to the surface or interfacial tension of the liquid [62]. Let us assume that we have a spherical bubble Of gas in a liquid... 

A thin film of water spreads up the inside walls of the capillary because of strong adhesive forces between water and glass (water wets glass). The pressure below the meniscus falls slightly. Atmospheric pressure then pushes a column of water up the tube to eliminate the pressure difference. The smaller the diameter of the capillary, the higherthe liquid rises. Because its magnitude is also directly proportional to surface tension, capillary rise provides a simple experimental method of determining surface tension, described in Exercise 122. 

For this reason, the surface tension has units of either energy per unit area or force per unit length. The surface tension of a liquid is measured in one of several ways capillary rise, ring detachment, or drop weight. Each method of surface tension measurement is outlined in Figure 9.2. With capillary rise, the fluid is suspended by the surface... 

The methods so far discussed have required more or less tabular solutions, or else correction factors to the respective ideal equations. Further, if one needs to make continuous measurements, then it is not easy to use some of these methods (such as capillary rise or bubble method). The most useful method of measuring the surface tension is by the well-known Wilhelmy plate method. If a smooth and flat plate-shaped metal is dipped in a liquid, the surface tension forces will be found to give rise to a tangential force. Figure 1.20. This is because a new contact phase is created between the plate and the liquid. 

The simplest and most common method of determining the surface tension of a liquid is the capillary tube rise method. If a glass capillary tube is immersed in a liquid such as water, the liquid in the capillary tube will rise above the outside level of the liquid. This is due to the greater liquid-solid force than the liquid-liquid intermolecular forces, and the liquid tends to wet as much of the solid... 

The surface tension of solvents, o, represents the work that has to be applied to the solvent in order to increase its surface area by one unit, but the units of are generally given as the force acting normal to a unit length, that is, mNm. The phase on the other side of the liquid surface is implied to be the vapor, but air at atmospheric pressure is usually used with no appreciable difference. There are several methods used to measure the surface tension, such as the force applied to a ring or plate touching the surface or the capillary rise of the solvent. The values of at 25 C for the solvents dealt with here are listed in Table 3.4, and their temperature dependence is negative but rather small. 

The surface tension measurement techniques can be divided into the following three categories (i) Force Methods, which include the truly static methods of the capillary rise and Wilhelmy plate methods, as well as the dynamic detachment methods of the Du Nouy ring and drop weight, (ii) Shape Methods, which include the pendant or sessile drop or bubble, as well as the spinning drop methods, and (iii) Pressure Methods, which are represented by the maximum bubble pressure method. These techniques are summarized in the following sections of this chapter. 

Another numerical study of free-surface flow patterns in narrow channels was conducted by Yang et al. [185]. They considered the flow of bubbles of different size driven by body forces, for example the rising of bubbles in a narrow capillary due to buoyancy. The lattice Boltzmann method [186] was used as a numerical scheme... 

These equations may be compared with those for cylinders, see for instance [1.3.21. For flat plates one does not have to worry about complications of the details of the profile, but this advantage is offset by the much lower rise. Typically, h is of order i.e. h = O (mm) and y is proportional to whereas it scales with ah in capillaries. Over the last few decades laser-optical techniques for scanning the meniscus and establishing h down to about 10" mm have become available In a modem variant of the Wilhelmy plate technique, to be described in sec. 1.8a, the force needed to pull the plate out of the liquid is measured as a function of the height above the zero level. In this way the surface tension and contact angle can be determined simultaneously. Alternatively, the method can be used to obtain contact angles, i.e. from [1.3.161 after y has been measured by some other technique. 

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