Cylindrical meniscus

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. 

Fig. 3.14 (a) A cone-shaped pore with hemispherical meniscus, (b) A wedge-shaped pore with cylindrical meniscus. 

According to the classical treatment of Cohan [8], which is the basis of the conventional BJH method [14], capillary condensation in an infinite cylindrical pore is described by the Kelvin equation using cylindrical meniscus, while desorption is associated with spherical meniscus. In large pores the following asymptotic equation is expected to be valid [8] Pd/Po = (PA/P0)2, where Pd/Po and Pa/Po are the relative pressures of the desorption and adsorption, respectively. An improved treatment [9-11, 13], originated from Deijaguin [9], takes into... 

IUPAC defines the lower limit of mesopores as 2 nm [1] which was considered as the limit below which the adsorption will occur by volume filling. However, in our recent article, based on the tensile stress hypothesis, we have shown that this limit is different than IUPAC limit. Using the mechanical stability criterion for the cylindrical meniscus (during adsorption), the critical size is obtained from... 

Fig. 3.3 shows a symmetrical plane-parallel film in contact with a cylindrical meniscus L. The plane of symmetry of the film coincides with the basic film surface of tension. Let us assume that at the other surfaces of tension the surface tension everywhere is constant and equal to that of the meniscus [Pg.96]

The adsorption and desorption isotherms have been calculated for the Nj sorption at 77K in cylindrical pores of MCM-41 materials in the range 1-12 nm. The points of spinodal and equilibrium transitions are plotted in Fig. 2. There are several features worth noticing. As the pore size increases, the line of spinodal desorption saturates at the value corresponding to the spinodal decomposition of the bulk liquid. The line of equilibrium capillary condensation asymptotically approaches the Kelvin equation for the spherical meniscus and the line of spontaneous capillary condensation asymptotically approaches the Kelvin equation for the cylindrical meniscus. This asymptotic behavior is in agreement with the classical scenario of capillary hysteresis [12] capillary condensation occurs spontaneously after the formation of the cylindrical adsorption film on the pore walls while evaporation occurs after the formation of the equilibrium meniscus at the pore end. Most interestingly, the NLDFT predictions of equilibrium and spontaneous capillary condensation transitions for pores wider than 6 nm are approximated by the semi-empirical equations of the Deijaguin-Broekhoff-de Boer theory [13]. 

Thus Eq. (4) and (5) provide two criticahties based on a purely thermodjmamic analysis of the adsorption in the cylindrical pore. During adsorption or desorption, although the thermodynamic stabihty criteria are satisfied, the condition for mechanical stability of the meniscus has to be satisfied. The mechanical stabihty criteria for the cylindrical meniscus (during adsorption) and hemispherical meniscus (during desorption) are given by [4,6,7,12]... 

Another pore filling model based upon capillary equilibrium in cylindrical pores has recently been proposed in which the condition of thermodynamic equilibrium is modified to include the effects of surface layering and adsorbate-adsorbent interactions [135-137]. Assuming that the vapor-liquid interface is represented by a cylindrical meniscus during adsorption and by a hemispherical meniscus during desorption, and invoking the Defay-Prigogene expression for a curvature-dependent surface tension [21], the equilibrium condition for capillary coexistence in a cylindrical pore is obtained as... 

Foster, A.G. (1952). Sorption hysteresis. Part 11. The role of the cylindrical meniscus effect. J. Chem. Soc. Lond. Part 11, 1806-12. 

For adsorption, liquid is formed via surface layering and at the inception of condensation from a vapour filled pore (radial filling rather than vertical filling), the meniscus takes the cylindrical shape as shown in Figure 3.9-5. For this cylindrical meniscus, two planes are drawn passing through the normal vector from any point on the liquid surface. One plane cuts the pore, and hence the principal radius is r, = r. The other plane perpendicular to the former will cut the liquid interface along the pore axial direction, and hence its principal pore radius is r2 = oo. Therefore, the radius of curvature for the case of adsorption is... 

An alternative view was suggested by Cohan who suggested that capillary condensation occurs along both adsorption and desorption branches of the isotherm, the difference being due to a difference in the shape of the meniscus. During adsorption the pore fills radially and a cylindrical meniscus is formed as sketched in Figure 2.12a. Under these conditions dv/ds = r-(rather than jl as assumed in the Kelvin equation) and with 0 = 0 ... 

For a cylindrical meniscus, r, = r, V2 = 00, and the contractive force F = nrQO (and not 2nrod). When the particles are completely immersed in the liquid, rj = rj = r, and F = 0. The opposite limiting case of a nearly dry meniscus, that is, when rj 0 and rj 0 (r, > 0 and rj < 0), yields rf = 2ro I r21 (from the Pythagorean theorem) and F 2 nr o. Under incomplete wetting conditions, F= cos 9(1 - rfr. These types of meniscus attractive forces between particles are frequently encountered in nature and in many practical applications. These forces play a critical role in the structural and rheological properties of soils and the ground. 

For parallel plates or open slit-shaped capillaries, a meniscus cannot be formed during adsorption, but during desorption a cylindrical meniscus is already present, hence adsoiption is delayed to produce hysteresis. During desorption (Figure 3.5) rj = r and r2 = hence ... 

During adsorption a meniscus caimot be formed but during desorption a cylindrical meniscus is present. During desorption the Kelvin equation takes the form [31,41] ... 

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... 

In a cylindrical pore the meniscus will be spherical in form, so that the two radii of curvature are equal to one another and therefore to r (Equation (3.8)). From simple geometry (Fig. 3.8) the radius r of the core is related to r by the equation... 

Fig. 3.8 Relation between r of the Kelvin equation (Equation (3.20)) and the core radius r for a cylindrical pore with a hemispherical meniscus 6 is the angle of contact.

Fig. 3.11 Capillary condensation in cylindrical pores, (a) Cylinder closed at one end, B. The meniscus is hemispherical during both capillary condensation and capillary evaporation, (h) and (c) Cylinder open at both ends. The meniscus is cylindrical during capillary condensation and hemispherical during capillary evaporation. Dotted lines denote the...

The variant of the cylindrical model which has played a prominent part in the development of the subject is the ink-bottle , composed of a cylindrical pore closed one end and with a narrow neck at the other (Fig. 3.12(a)). The course of events is different according as the core radius r of the body is greater or less than twice the core radius r of the neck. Nucleation to give a hemispherical meniscus, can occur at the base B at the relative pressure p/p°)i = exp( —2K/r ) but a meniscus originating in the neck is necessarily cylindrical so that its formation would need the pressure (P/P°)n = exp(-K/r ). If now r /r, < 2, (p/p ), is lower than p/p°)n, so that condensation will commence at the base B and will All the whole pore, neck as well as body, at the relative pressure exp( —2K/r ). Evaporation from the full pore will commence from the hemispherical meniscus in the neck at the relative pressure p/p°) = cxp(-2K/r ) and will continue till the core of the body is also empty, since the pressure is already lower than the equilibrium value (p/p°)i) for evaporation from the body. Thus the adsorption branch of the loop leads to values of the core radius of the body, and the desorption branch to values of the core radius of the neck. 

Both the cone-shaped and the wedge-like pore give rise to simple, hysteresis-free behaviour. The meniscus is nucleated at the apex of the cone (Fig. 3.14(a)) or at the intersection of the two planes of the wedge (Fig. 3.14(b)), giving a spherical meniscus in the first case and a cylindrical one in the second. In both systems the process of evaporation is the exact reverse of that of condensation, and hysteresis is therefore absent. 

The radius of curvature r of the meniscus and its depth H are expressed for a cylindrical micro-channel as... 

To explain the role of the medium capillary pressure upon foam coalescence, consider a flat, cylindrical, stationary foam lamella of thickness, 2h, circa 1000 A, and radius, R (i.e., 50 to 100 /xm), subject to a capillary pressure, P, at the film meniscus or Plateau border, as shown in Figure 3. The liquid pressure at the film meniscus is (P - P ), where P is the gas pressure. g c g... 

Figure 4. Schematic representation of a meniscus of mercury in a cylindrical pore and at the rim of an enlargment of the pore. Modified from Kloubek [11],...

The penetration of mercury in MCM-41, a material with smooth cylindrical pores, takes place at the pressure indicated by the Washbum-Laplace model, indicating that this model is still valid at the scale of a few nanometers. When the pore surface is pitted with micropores or when the pores are interconnected, like in the case of SBA-15, the Washbum-Laplace model underevaluates the size of the pores, due to the excess energy needed for advancement of the meniscus beyond the surface defects. 

Figure 2. Capillary hysteresis of nitrogen in cylindrical pores at 77 K. Equilibrium desorption (black squares) and spinodal condensation (open squares) pressures predicted by the NLDFT in comparison with the results of Cohan s equation (the BJH method) for spherical (crosses and line) and cylindrical (line) meniscus.

The pore shape is generally assumed to be either cylindrical or slit-shaped in the former case, the meniscus is hemispherical, and hence r, is equal to r2 in the latter case, the meniscus is hemicylindrical, and thus r, is equal to the width of the slit and r2 is... 

Many technologically important solids (including silica) have a micropore structure that is well below the lower limit of pore size at which the Kelvin equation (equation 5) is applicable. The basis for the Kelvin equation is a thermomechanical equilibrium across the hemispherical meniscus of a capillary condensate within a cylindrical pore. Below a pore size of 2 nm diameter, the liquid cannot be considered as a fluid with bulk properties because of the forces exerted by the wall. Theoretical calculations suggest that the properties of fluids in microporous structures are highly dependent on the size of the pore. 

When the radius of the tube is appreciable, equation (2) requires correction, because the meniscus is no longer spherical. In Fig. 57, let b be the radius of curvature of the lowest point O of the meniscus (the two radii will be equal at this point, since the tube is cylindrical and the point is on the axis of revolution). The pressure immediately under the centre of the... 

The measurement of the height of the bottom of the meniscus has theoretically to be made relative to a plane surface of liquid in communication with the capillary tube. It is not always recognized that a very large surface is required, if it is to be sufficiently plane the cylindrical tubes of 20 mm. diameter often used have a decided capillary rise. Tubes of 40 mm. diameter or more must be used if the surface is to be considered as plane. Richards and Carver have, however, tested Rayleigh s formula (10) for wide tubes, and found it to agree with experiment within a few thousandths of a millimetre for a tube 38 mm. diameter this formula can be used with fair accuracy down to tubes of 25 mm. diameter, as a correction to the level in the wide tube. The actual measurement of the difference in height between the two levels requires an accurate cathetometer and suitable illumination of the menisci (see Richards and Coombs). 

---