Capillarity

Although defined in various ways depending on the context, capillarity for current purposes will be defined as the macroscopic motion of a fluid system under the influence of its own surface and interfacial forces. Such flow is similar to other types of hydrauhc flow in that it results from the presence of a pressure differential between two hydraulically connected regions of the liquid mass (Fig. 6.1). The direction of flow is such as to decrease the pressure difference. When the difference vanishes, or when there is no longer a mechanism to reduce the difference, flow ceases. 

Capillary effects are encountered in many areas of interface and colloid science, with its importance relative to other processes (e.g., fluid dynamics) depending on the exact situation. For example, when two spherical drops of a liquid in an emulsion make contact and coalesce to form a larger drop (Fig. 6.2a), the extent and duration of flow due to the capillary phenomenon is limited and fluid dynamics is of little practical importance. When there is an extensive amount of flow, on the other hand, such as in capillary imbibition, wicking processes, or capillary displacement (Fig. 6.2b) fluid dynamics may become important. 

When a solid capillary tube is inserted into a liquid, the liquid is raised or depressed in the tube, and the height of the liquid can be determined. A glass capillary is most commonly used for this purpose because it is transparent and is completely wettable by most liquids. 

The above treatment can be applied to the simplest ideal case, where 6= 0° and the liquid completely wets the capillary wall. However, occasionally a liquid meets the circularly cylindrical capillary wall at some contact angle, 6, between the liquid and the wall, as shown 

The curvature of the interface depends on the relative magnitudes of the adhesive forces between the liquid and the capillary wall and the internal cohesive forces in the liquid. When the adhesive forces exceed the cohesive forces, 9 lies in the range 0° 9 90° when the cohesive forces exceed the adhesive forces, 90° 9 180°. When 9 90°, the cos 9 term is negative, resulting in a convex meniscus towards the vapor phase and the liquid level in the capillary falling below the liquid level in the container (capillary depression). This occurs with liquid mercury in glass where 9 = 140° and also with water in capillary tubes coated internally with paraffin wax. Thus, liquid mercury is used in the evaluation of the porosity of solid adsorbents in the mercury injection porosimetry technique (see Section 8.5). 

Since it is a very difficult task to measure d experimentally in a capillary tube, we need a relation between d and the experimentally accessible radius of the capillary tube, r. This relation can be derived by considering gravity and surface tension effects by applying fundamental Newton mechanics the complete proof is given in Section 6.1. In the case of a figure of revolution, where Rx = R2 = d, when the elevation of a general point on the surface is denoted by z, the fundamental equation is given as 

When a capillary is immersed under the surface of a liquid, the liquid will either rise in the capillary (Fig. 10.3a) or be depressed below the surface of the external liquid (Fig. 10.3b). 

As illustrated in Fig. 10.3a, the pressure above the meniscus within a capillary of radius r at point A is the same as that at point C at the surface of the liquid level external to the capillary. The small pressure difference 

When a liquid does not wet the walls of a capillary, as shown in Fig. 10.3b, the concave side of the meniscus is within the liquid at point B, which is at a higher pressure than the gas immediately above the surface. As described by equation (10.18), the liquid is depressed a distance h below the level of the external liquid. 

Equation (10.18) was derived for capillary rise or depression assuming complete wetting, that is, 6 = 180°. In the case of contact angles greater than 0° and less than 180°, equation (10.18) must be modified. As liquid moves up the capillary during capillary rise the solid-vapor interface disappears and the solid-liquid interface appears. The work required for this process is 

FIGURE 6.1 Capillary effects due to different wettabilities of the inner wall of a capillary, reflected by the contact angle of wetting, 6 (a) capillary rise, (b) no effect, and (c) capillary depression. 

Quantitative interpretation of capillary rise and depression requires the introduction and discussion of the notion capillary pressure, that is, the pressure difference across a curved interface due to the interfacial tension in that interface. 

Curvature further plays an important role in phase transitions. The formation of a new phase starts with nuclei, very small particles, droplets, or bubbles. The strong curvature retards the growth of the nuclei so that formation of the new phase therefore occurs only after superheating, supercooling, or supersaturation. 

Consider the formation of an air bubble in a liquid medium, for instance, the blowing of a soap bubble. To blow a bubble an excess pressure is applied. This excess pressure is called the capillary pressure or the Laplace pressure, The excess 

At equilibrium, the net change in Helmholtz energy F is zero, that is, at constant temperature and composition 

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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. 

Equation 11-3 is a special case of a more general relationship that is the basic equation of capillarity and was given in 1805 by Young [1] and by Laplace [2]. In general, it is necessary to invoke two radii of curvature to describe a curved surface these are equal for a sphere, but not necessarily otherwise. A small section of an arbitrarily curved surface is shown in Fig. II-3. The two radii of curvature, R and / 2[Pg.6]

Most of the situations encountered in capillarity involve figures of revolution, and for these it is possible to write down explicit expressions for and R2 by choosing plane 1 so that it passes through the axis of revolution. As shown in Fig. II-7n, R then swings in the plane of the paper, i.e., it is the curvature of the profile at the point in question. R is therefore given simply by the expression from analytical geometry for the curvature of a line... 

Equation II-7 is the fundamental equation of capillarity and will recur many times in this chapter. 

J. S. Rowlinson and B. Widom, Molecular Theory of Capillarity, Clarendon Press, Oxford, 1984. 

F. Buff, The Theory of Capillarity, in Handbuch der Physik, vol. 10, Springer-Verlag, Berlin, 1960. 

The film pressure is defined as the difference between the surface tension of the pure fluid and that of the film-covered surface. While any method of surface tension measurement can be used, most of the methods of capillarity are, for one reason or another, ill-suited for work with film-covered surfaces with the principal exceptions of the Wilhelmy slide method (Section II-6) and the pendant drop experiment (Section II-7). Both approaches work very well with fluid films and are capable of measuring low values of pressure with similar precision of 0.01 dyn/cm. In addition, the film balance, considerably updated since Langmuir s design (see Section III-7) is a popular approach to measurement of V. 

A solid, by definition, is a portion of matter that is rigid and resists stress. Although the surface of a solid must, in principle, be characterized by surface free energy, it is evident that the usual methods of capillarity are not very useful since they depend on measurements of equilibrium surface properties given by Laplace s equation (Eq. II-7). Since a solid deforms in an elastic manner, its shape will be determined more by its past history than by surface tension forces. 

A homogeneous metastable phase is always stable with respect to the fonnation of infinitesimal droplets, provided the surface tension a is positive. Between this extreme and the other thennodynamic equilibrium state, which is inhomogeneous and consists of two coexisting phases, a critical size droplet state exists, which is in unstable equilibrium. In the classical theory, one makes the capillarity approxunation the critical droplet is assumed homogeneous up to the boundary separating it from the metastable background and is assumed to be the same as the new phase in the bulk. Then the work of fonnation W R) of such a droplet of arbitrary radius R is the sum of the... 

Rowlinson J S and Widom B 19H2 Molecular Theory of Capillarity (Oxford Clarendon)... 

Fleer G J, Cohen Stuart M A, Scheutjens J M H M, Cosgrove T and Vincent B 1993 Polymers at Interfaces (London Chapman and Hall) Rowlinson J S and Widoni B 1982 Molecular Theory of Capillarity (Oxford Clarendon)... 

Wolynes P G 1997 Folding nucleus and energy landscapes of larger proteins within the capillarity approximation Proc. Natl Acad. Sci. (USA) 94 6170-5... 

Transport. Wood is composed of a complex capillary network through which transport occurs by capillarity, pressure permeability, and diffusion. A detailed study of the effect of capillary stmcture on the three transport mechanisms is given in Stamm (13). 

In the United States, a number of physical tests are performed on siUcon carbide using standard AGA-approved methods, including particle size (sieve) analysis, bulk density, capillarity (wettabiUty), friabiUty, and sedimentation. Specifications for particle size depend on the use for example, coated abrasive requirements (134) are different from the requirements for general industrial abrasives. In Europe and Japan, requirements are again set by ISO and JSA, respectively. Standards for industrial grain are approximately the same as in the United States, but sizing standards are different for both coated abrasives and powders. 

Reactive Hquid infiltration (45,68,90,93,94) is similar to the CVI process used to make RBSN. Driven by capillarity, a reactive Hquid infiltrates a porous preform and reacts on free surfaces. Reactive Hquid infiltration is used to make reaction bonded siHcon carbide (RBSC), which is used in advanced heat engines and as diffusion furnace components for semiconductor wafer processing. 

In porous and granular materials, Hquid movement occurs by capillarity and gravity, provided passages are continuous. Capillary flow depends on the hquid material s wetting property and surface tension. Capillarity appHes to Hquids that are not adsorbed on capillary walls, moisture content greater than fiber saturation in cellular materials, saturated Hquids in soluble materials, and all moisture in nonhygroscopic materials. 

Capillarity. The outer surface of porous material has pore entrances of various sizes. As surface Hquid is evaporated during constant rate drying, a meniscus forms across each pore entrance and interfacial forces are set up between the Hquid and material. These forces may draw Hquid from the interior to the surface. The tendency of Hquid to rise in porous material is caused pardy by Hquid surface tension. Surface tension is defined as the work needed to increase a Hquid s surface area by one square meter and has the units J/m. The pressure increase caused by surface tension is related to pore size ... 

Several micromanometers, based on the liquid-column principle and possessing extreme precision and sensitivity, have been developed for measuring minute gas-pressure differences and for cahbrating low-range gauges. Some of these micromanometers are available commercially. These micromanometers are free from errors due to capillarity and, aside from checking the micrometer scale, require no cahbration. See Doolittle, op. cit., p. 21. 

Surface evaporation can be a limiting factor in the manufacture of many types of products. In the drying of paper, chrome leather, certain types of synthetic rubbers and similar materials, the sheets possess a finely fibrous structure which distributes the moisture through them by capillary action, thus securing very rapid diffusion of moisture from one point of the sheet to another. This means that it is almost impossible to remove moisture from the surface of the sheet without having it immediately replaced by capillary diffusion from the interior. The drying of sheetlike materials is essentially a process of surface evaporation. Note that with porous materials, evaporation may occur within the solid. In a porous material that is characterized by pores of diverse sizes, the movement of water may be controlled by capillarity, and not by concentration gradients. 

Capillar) cells Fliininatur plale.s rioiini humidiried air... 

Hollow carbon nanotubes (CNTs) can be used to generate nearly onedimensional nanostrutures by filling the inner cavity with selected materials. Capillarity forces can be used to introduce liquids into the nanometric systems. Here, we describe experimental studies of capillarity filling in CNTs using metal salts and oxides. The filling process involves, first a CNT-opening steps by oxidation secondly the tubes are immersed into different molten substance. The capillarity-introduced materials are subsequently transformed into metals or oxides by a thermal treatment. In particular, we have observed a size dependence of capillarity forces in CNTs. The described experiments show the present capacities and potentialities of filled CNTs for fabrication of novel nanostructured materials. 

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