Meniscus, 5.22

In the case of the fine fibre (Fig. 1.11a), gravity can be neglected and the surface generated has zero curvature. Its profile z x) can be determined from the statement that the tension is conserved  

In the case of the plane surface (Fig. 1.11), we equate the hydrostatic and capillary pressures at A. This leads to 7/i = —pgz. The curvature is given by 

This equation for 6 — 6 gives the height h of the meniscus. For total wetting, 0E = 0 and h 0-E = 0) = /2/ . Once again, it can be checked that qso — 7sl carries the weight of the liquid lifted per unit length of the contact line. 

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Since the blocked gas inside of the capillary is dissolving in the liquid and then diffusing towards the exit of the channel, the meniscus of the liquid crosses the position l and goes deeper. This second stage of capillary filling with liquid is called diffusive imbibition and plays an important role in PT processes. The effect of diffusive imbibition upon PT sensitivity has been studied in [7]. 

There are two approaches to explain physical mechanism of the phenomenon. The first model is based on the existence of the difference between the saturated vapor pressures above two menisci in dead-end capillary. It results in the evaporation of a liquid from the meniscus of smaller curvature ( classical capillary imbibition) and the condensation of its vapor upon the meniscus of larger curvature originally existed due to capillary condensation. 

The second mechanism can be explained by the wall liquid film flow from one meniscus to another. Thin adsorptive liquid layer exists on the surface of capillary channel. The larger is a curvature of a film, the smaller is a pressure in a liquid under the corresponding part of its film. A curvature is increasing in top s direction. Therefore a pressure drop and flow s velocity are directed to the top. 

At first we tried to explain the phenomenon on the base of the existence of the difference between the saturated vapor pressures above two menisci in dead-end capillary [12]. It results in the evaporation of a liquid from the meniscus of smaller curvature ( classical capillary imbibition) and the condensation of its vapor upon the meniscus of larger curvature originally existed due to capillary condensation. We worked out the mathematical description of both gas-vapor diffusion and evaporation-condensation processes in cone s channel. Solving the system of differential equations for evaporation-condensation processes, we ve derived the formula for the dependence of top s (or inner) liquid column growth on time. But the calculated curves for the kinetics of inner column s length are 1-2 orders of magnitude smaller than the experimental ones [12]. 

This report presents the results of investigations aimed at the creation of the surface wave transducer for the automated control. The basic attention is drawn to the analysis of the position of the front meniscus of the contact liquid when the surface waves excite through the slot gap and to the development of system for acoustic contact creation. 

The use of the surface ultrasonic waves seems to be convenient for these purposes. However, this method has not found wide practical application. Peculiarities of excitation, propagation and registration of surface waves created before these time great difficulties for their application in automatic systems of duality testing. It is connected with the fact that the surface waves are weakened by soil on the surface itself In addition, the methods of testing by the surface waves do not yield to automation due to the difficulties of creation of the acoustic contact. In particular, a flow of contact liquid out of the zone of an acoustic line, presence of immersion liquid, availability of chink interval leads to the adsorption and reflection of waves on tlie front meniscus of a contact layer. The liquid for the acoustic contact must be located only in the places of contact, otherwise the influence on the amplitude will be uncontrolled. This phenomenon distorts the results of testing procedure. 

For exciting the surface waves the traditional method of transforming of the longitudinal wave by the plastic wedge is used. The scheme of surface waves excitation is shown in fig. 1. In particular, it is ascertained that the intensity of the excitation of the surface wave is determined by the position of the extreme point of the exit of the acoustic beam relatively to the front meniscus of the contact liquid. The investigations have shown, that under the... 

Firstly, this is the supplying of energy needed for retaining the necessary volume of the liquid. Secondly, this is the choice of magnetic field topography in which the variation of meniscus position is small enough for different liquid quantities. 

The topic of capillarity concerns interfaces that are sufficiently mobile to assume an equilibrium shape. The most common examples are meniscuses, thin films, and drops formed by liquids in air or in another liquid. Since it deals with equilibrium configurations, capillarity occupies a place in the general framework of thermodynamics in the context of the macroscopic and statistical behavior of interfaces rather than the details of their molectdar structure. In this chapter we describe the measurement of surface tension and present some fundamental results. In Chapter III we discuss the thermodynamics of liquid surfaces. 

Similarly, the identical expression holds for a liquid that completely fails to wet the capillary walls, where there will be an angle of contact between the liquid and the wall of 180°, a convex meniscus and a capillary depression of depth h. 

A slightly more general case is that in which the liquid meets the circularly cylindrical capillary wall at some angle 6, as illustrated in Fig. II-7. If the meniscus is still taken to be spherical in shape, it follows from simple geometric consideration that / 2 = r/cos 6 and, since R = / 2, Eq. II-9 then becomes... 

The exact treatment of capillary rise must take into account the deviation of the meniscus from sphericity, that is, the curvature must correspond to the AP = Ap gy at each point on the meniscus, where y is the elevation of that point above the flat liquid surface. The formal statement of the condition is obtained by writing the Young-Laplace equation for a general point (x, y) on the meniscus, with R and R2 replaced by the expressions from analytical geometry given in... 

This is exact—see Problem 11-8. Notice that Eq. 11-14 is exactly what one would write, assuming the meniscus to be hanging from the wall of the capillary and its weight to be supported by the vertical component of the surface tension, 7 cos 6, multiplied by the circumference of the capillary cross section, 2ar. Thus, once again, the mathematical identity of the concepts of surface tension and surface free energy is observed. 

While Eq. 11-14 is exact, its use to determine surface tension from capillary rise experiments is not convenient. More commonly, one measures the height, h, to the bottom of the meniscus. 

Approximate solutions to Eq. 11-12 have been obtained in two forms. The first, given by Lord Rayleigh [13], is that of a series approximation. The derivation is not repeated here, but for the case of a nearly spherical meniscus, that is, r h, expansion around a deviation function led to the equation... 

The first term gives the elementary equation (Eq. 11-10). The second term takes into account the weight of the meniscus, assuming it to be spherical (see Problem II-3). The succeeding terms provide corrections for deviation firom sphericity. 

The general case has been solved by Bashforth and Adams [14], using an iterative method, and extended by Sugden [15], Lane [16], and Paddy [17]. See also Refs. 11 and 12. In the case of a figure of revolution, the two radii of curvature must be equal at the apex (i.e., at the bottom of the meniscus in the case of capillary rise). If this radius of curvature is denoted by b, and the elevation of a general point on the surface is denoted by z, where z = y - h, then Eq. II-7 can be written... 

Perhaps the best discussions of the experimental aspects of the capillary rise method are still those given by Richards and Carver [20] and Harkins and Brown [21]. For the most accurate work, it is necessary that the liquid wet the wall of the capillary so that there be no uncertainty as to the contact angle. Because of its transparency and because it is wet by most liquids, a glass capillary is most commonly used. The glass must be very clean, and even so it is wise to use a receding meniscus. The capillary must be accurately vertical, of accurately known and uniform radius, and should not deviate from circularity in cross section by more than a few percent. 

The basic observation is that a thin plate, such as a microscope cover glass or piece of platinum foil, will support a meniscus whose weight both as measured statically or by detachment is given very accurately by the ideal equation (assuming zero contact angle) ... 

It should be noted that here, as with capillary rise, there is an adsorbed film of vapor (see Section X-6D) with which the meniscus merges smoothly. The meniscus is not hanging from the plate but rather fiom a liquidlike film [53]. The correction for the weight of such film should be negligible, however. 

A modification of the foregoing procedure is to suspend the plate so that it is partly immersed and to determine from the dry and immersed weights the meniscus weight. The procedure is especially useful in the study of surface adsorption or of monolayers, where a change in surface tension is to be measured. This application is discussed in some detail by Gaines [57]. Equation 11-28 also applies to a wire or fiber [58]. 

The cases of the sessile drop and bubble are symmetrical, as illustrated in Fig. n-16. The profile is also that of a meniscus 0 is now positive and, as an... 

Show that the second term in Eq. 11-15 does indeed correct for the weight of the meniscus. (Assume the meniscus to be hemispherical.)... 

Derive Eq. 11-14 from an exact analysis of the meniscus profile. Hint Start with Eq. II-12 and let p = y", where y" = pdpjdy. The total weight W is then given by W = lApgirj xydx. 

Derive the equation for the capillary rise between parallel plates, including the correction term for meniscus weight. Assume zero contact angle, a cylindrical meniscus, and neglect end effects. 

Derive, from simple considerations, the capillary rise between two parallel plates of infinite length inclined at an angle of d to each other, and meeting at the liquid surface, as illustrated in Fig. 11-23. Assume zero contact angle and a circular cross section for the meniscus. Remember that the area of the liquid surface changes with its position. 

A liquid of density 2.0 g/cm forms a meniscus of shape corresponding to /3 = 80 in a metal capillary tube with which the contact angle is 30°. The capillary rise is 0.063 cm. Calculate the surface tension of the liquid and the radius of the capillary, using Table II-l. 

Equation 11-30 may be integrated to obtain the profile of a meniscus against a vertical plate the integrated form is given in Ref. 53. Calculate the meniscus profile for water at 20°C for (a) the case where water wets the plate and (b) the case where the contact angle is 40°. For (b) obtain from your plot the value of h, and compare with that calculated from Eq. 11-28. [Hint Obtain from 11-15.]... 

While Eq. III-18 has been verified for small droplets, attempts to do so for liquids in capillaries (where Rm is negative and there should be a pressure reduction) have led to startling discrepancies. Potential problems include the presence of impurities leached from the capillary walls and allowance for the film of adsorbed vapor that should be present (see Chapter X). There is room for another real effect arising from structural peiturbations in the liquid induced by the vicinity of the solid capillary wall (see Chapter VI). Fisher and Israelachvili [19] review much of the literature on the verification of the Kelvin equation and report confirmatory measurements for liquid bridges between crossed mica cylinders. The situation is similar to that of the meniscus in a capillary since Rm is negative some of their results are shown in Fig. III-3. Studies in capillaries have been reviewed by Melrose [20] who concludes that the Kelvin equation is obeyed for radii at least down to 1 fim. 

A familiar (and biblical [SO]) example is the formation of tears of wine in a glass. Here, the evaporation of the alcohol from the meniscus leads to a local raising of the surface tension, which, in turn, induces a surface and accompanying bulk flow upward. 

The material of interest is dissolved in a volatile solvent, spread on the surface and allowed to evaporate. As the sweep moves across, compressing the surface, the pressure is measured providing t versus the area per molecule, a. Care must be taken to ensure complete evaporation [1] and the film structure may depend on the nature of the spreading solvent [78]. When the trough area is used to calculate a, one must account for the area due to the meniscus [79]. Barnes and Sharp [80] have introduced a remotely operated barrier drive mechanism for cleaning the water surface while maintaining a closed environment. 

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