Effect of Meniscus Geometry on Equilibrium Pressures of the Lennard

Langmuir 2008, 24, 9375-9380

9375

Effect of Meniscus Geometry on Equilibrium Pressures of the Lennard-Jones Liquids Vladimir Kh. Dobruskin† 11 Pashosh Street, Beer-YacoV 30700, Israel ReceiVed February 20, 2008. ReVised Manuscript ReceiVed June 6, 2008 The theory of phase transitions on interfaces relates equilibrium vapor pressures and enthalpies of condensation. Since the latter are size- and shape-dependent properties and their variations are determined by variations of energies of autoadsorption on liquid menisci, a knowledge of these energies enable one to calculate the equilibrium pressures in capillaries. The novel mathematical model for a computation of the adsorption energies is developed on the basis of the Lennard-Jones potential for molecular interactions. The model is applied to hemispherical, conical, cylindrical, semi-elliptical, and paraboloidal menisci in nanocapillaries. Predicted equilibrium vapor pressures are favorably compared with the independent experimental data.

1. Introduction The unusual properties of nanometer-sized materials have generated tremendous interest in both the scientific and technological communities. The materials offer unique and entirely different electrical, optical, mechanical, magnetic, thermodynamic, and catalytic properties compared with conventional micro- or millimeter-size materials.1–3 With decreasing particle size, the heat of fusion4,5 and cohesive energy6,7 decrease and the melting response moves toward lower temperatures.8–10 A cluster-dependent change of the chemical potential of metal particles has been demonstrated.11 The unusual phases are observed when “ordinary” materials are fabricated in nanocrystalline form.12,13 Recently, it has been shown14–16 that the enthalpy of condensation, Hcon, is also a size-dependent property and the difference between Hcon for two liquid bodies R and β of the same chemical nature, ∆HRβ, is equal to the variation of / , on their surfaces: the autoadsorption energies, ∆εRβ / ∆HRβ ≡ ∆Hcon(β) - ∆Hcon(R) = -∆εRβ

(1)

/ Here, ∆εRβ ) εβ/ - εR/ , where εβ/ and εR/ are the energies of autoadsorption on β and R, respectively. The term “autoadsorption” refers to adsorption of vapor on the surface of its own condensed phase. As accepted in the adsorption theory, the asterisk (*) indicates values of parameters corresponding to the well † Telephone: (972)-506-864 642. Fax: (972)-8-930 2675. E-mail: [email protected].

(1) Handbook of nanostructured materials and nanotechnology; Nalwa, H. S., Ed.; Academic Press: London, 2000; Vol. 1-5. (2) Springer handbook of nanotechnology; Bhushan, B., Ed.; Spinger-Verlag: Berlin, 2004. (3) Moriatry, P. Rep. Prog. Phys. 2001, 64, 297–381. (4) Luo, W.; Hu, W.; Xiao, S. J. Phys. Chem. C 2008, 112, 2359–2369. (5) Ercolessi, F.; Andreoni, W.; Tosatti, E. Phys. ReV. Lett. 1991, 66, 911–914. (6) Jiang, Q.; Li, J. C.; Chi, B. Q. Chem. Phys. Lett. 2002, 366, 551–554. (7) Qi, W. H.; Wang, M. P.; Hu, W. Y. Matter. Lett. 2004, 58, 1745–1749. (8) Takagi, M. J. Phys. Soc. Jpn. 1954, 9, 359–364. (9) Sun, J.; Simon, S. L. Termochim. Acta 2007, 463, 32–40. (10) Lai, S. L.; Guo, J. Y.; Petrova, V.; Ramanath, G.; Allen, L. H. Phys. ReV. Lett. 1996, 77, 99–102. (11) Schafer, R. Z. Phys. Chem. 2003, 217, 989–999. (12) Mayo, M. J.; Suresh, A.; Porter, W. D. ReV. AdV. Mater. Sci. 2003, 5, 100–109. (13) Sun, C. Q. Prog. Solid State Chem. 2007, 35, 1–159. (14) Dobruskin, V. Kh. Langmuir 2003, 19, 4004–4013. (15) Dobruskin, V. Kh. Langmuir 2005, 21, 2887–2894. (16) Dobruskin, V. Kh. J. Phys. Chem. B 2006, 110, 19582–19585.

depth. It was found14–18 that (1) the energy of autoadsorption on the flat surface of a bulk liquid, ε/b, is equal in magnitude to the total surface energy, Es, and (2) the energy of autoadsorption on / / / /εslab), where εsp and the droplet surface, ε/d, is equal to ε/d ) Es(εsp / εslab are the Lennard-Jones dispersion components of the energies / ) and on of autoadsorption on the spherical droplet surface (εsp / the flat surface (εslab) of the bulk liquid simulated by a semiinfinite slab, respectively. From here

( )

∆ε/db ) Es - Es

ε/sp

/ εslab

≡ Esλ

(2)

where the coefficient λ

λ)1-

ε/sp ε/slab

(3)

is a geometrical factor that depends on geometrical parameters of liquid samples. The ratio of the dispersion components (a / / /εslab, dispersion ratio) for a sphere and a semi-infinite slab, εsp 14–16 and hence λ are known from the adsorption theory. The thermodynamic relationship between an equilibrium pressure and an enthalpy of condensation is given by the Clausius-Clapeyron equation.19–21 Applying it (1) to a bulk liquid and (2) to the droplet of the same chemical nature and (3) subtracting one expression from another, one obtains the following relation14–16 between an equilibrium pressure over droplets, p, the variation of autoadsorption energies ∆ε/db, and the parameters of Es and λ:

RgT ln

p ) ∆ε/db ≡ Esλ ps

(4)

where ps is the saturation pressure of a bulk liquid, Rg is the gas constant, and T is the absolute temperature. It is a general thermodynamic relation that is applicable to any equilibrium liquid objects and not only to the spherical droplets. (17) Dobruskin, V. Kh. Langmuir 2003, 19, 2134–2146. (18) Dobruskin, V. Kh. Carbon 2001, 39, 583–591. (19) Atkins, P. W.; de Julio, P. Atkins’ Physical Chemistry; Oxford University Press: Oxford, 2002. (20) Callen, H. B. Thermodynamics and Introduction to Thermostatics; John Wiley: New York, 1985. (21) Berry, R. S.; Rice, S. A.; Ross, J. Physical Chemistry; Oxford University Press: New York, 2000.

10.1021/la800546u CCC: $40.75  2008 American Chemical Society Published on Web 07/30/2008

9376 Langmuir, Vol. 24, No. 17, 2008

Dobruskin

To take advantage of eq 4, one needs to know the variation of the autoadsorption energies between a liquid body of interest and the bulk liquid. To the present day, the calculations have been performed for simple bodies such as spheres,14 cylinders,18 and slits.22–25 It has been shown14–16 that the theory is in good agreement with the experimental facts; it describes the equilibrium pressures of nanosize objects with characteristic dimensions about 1 nm where the classical Kelvin equation fails. The purposes of the present paper are (1) to extend the theory to more complicated objects such as hemispherical, conical, semi-elliptical, and paraboloidal menisci in nanocapillaries and (2) to compare the theory with experiments. The specific novel model for calculation of the autoadsorption energies based on the Lennard-Jones potential for molecular interactions will be developed.

2. Mathematical Model 2.1. Formulation of the Problem. Our objective is to calculate the energy of dispersion interactions of an adsorbate with bodies of interest. In most cases, the solution is based on the summation of the Lennard-Jones (LJ) potential, ε, between single atomic or molecular species:22–24

ε ) 4ε/aa

[( ) ( ) ] σaa s

12

-

σaa s

6

(5)

where s is the separation between the nuclei of the species, ε/aa is the depth of the energy minimum for the atom-atom interaction, and σaa is the distance at which εaa ) 0. Because coordinates of liquid species and their distribution are unknown, the application of the LJ equation to liquids is a challenging problem. Nevertheless, if we restrict ourselves only to a calculation of λ (and not to calculation of absolute values of adsorption energies), the significant simplifications are made possible. / / / Note that eq 3 is equivalent to λ ) (εslab - εsp )/εslab . Owing to the operation of subtraction, approximations, which bring in / / / the equal inaccuracies into εslab and εsp , cancel each other in (εslab / - εsp). This opens up possibilities for the following simplifications: (1) for the integration of eq 5 over the liquid volume instead of the summation, (2) for application of a lattice, one-centered model of a liquid with no regard to the radial distribution functions, and (3) for neglecting many-body effects as well as retardation effects. In fact, it means that estimating λ for liquids may be performed by the same methods as a calculation of energies of adsorption on solids. For adsorption on the semi-infinite slab, which simulates the reference body, this way gives24 ε/slab ) (2
10/9)πnε/aa, where n is the number of interacting centers per unit volume. The letter / parameter appears also in the expression for εsp after integration / / of eq 5, but the division of the energy difference (εslab - εsp ) by / / εslab eliminates the vague parameters of n and εaa from the dispersion ratio. 2.2. Notation System. Generally, the total adsorption energies and their dispersion components do not coincide: for example, a significant part of the total autoadsorption energy for water is associated with the hydrogen bonds and dipole-dipole interactions. Nevertheless, the same conventional symbol ε is used for both the total adsorption energy and for its dispersion component. To discriminate these values, the dispersion energies are labeled by subscripts sp, slab, con, cyl, hsp, sel, and par, which are abbreviations for sphere, slab, cone, cylinder, hemisphere, semi(22) Steele, W. A. The Interaction of Gases with Solid Surface; Pergamon Press: Oxford, 1974. (23) Avgul, N. N.; Kiselev, A. V.; Poshkus, D. P. Adsorption of Gases and Vapors on Homogenous Surfaces, Khimia: Moscow, 1975. ( Russian) (24) Crowell, A. D. J. Chem. Phys. 1954, 22, 1397–1399. (25) Everett, D. H.; Powl, J. C. J. Chem. Soc., Faraday Trans. 1 1976, 72, 619–636.

Figure 1. Schematic diagram of liquid rotation about the Z-axis.

ellipsoid, and paraboloid, respectively; otherwise, the symbol ε refers to the total adsorption energy. 2.3. Interactions with Paraboloidal Meniscus. Consider a liquid column with a radius r rotating with the angular velocity, δ, about the vertical axis OZ passing through the cylinder center and introduce the cylindrical coordinates shown in Figure 1. Let F and φ be the polar radius and polar angle, respectively, Z be the axis of revolution, and -H be the Z-coordinate of the cylinder bottom. We assume that a liquid does not interact with the wall and its surface is flat when δ ) 0. For such an ideal liquid, the meniscus is a paraboloid of revolution; the meniscus height, z, in an arbitrary point depends only on the polar radius F and is given by the equation of a parabola, z ) (δ2/2g)F2 ≡ βF2, where g is the acceleration of gravitation and β ≡ δ2/2g is the parameter of the parabola.26 At equilibrium, the chemical potential of the liquid and energies of gas-liquid interactions are constant throughout the meniscus. For this reason, it is expedient to calculate the adsorption energy for the molecule located in the point C(0, z0) on the axis of revolution where δ ) 0. For the sake of convenience, all lengths will be expressed in the reduced forms with σaa as a scale parameter. If F, φ, and z are the variable coordinates of points inside the liquid, then the elementary volume of the liquid, dV ) F dF dφ dz, interacts with the molecule in point C with the energy

[( 1s ) - ( 1s ) ]F dF dφ dz 12

dε ) 4ε/aan

6

(6)

where s ) (F2 + (z0 - z)2)1/2 is the reduced separation between the point C and elementary volume (AC in Figure 1). The total energy is found by the integration over a liquid volume:

εpar )

∫0r ∫-HβF ∫02π 4ε/aa[( 1s )

12

2

-

( 1s ) ]nF dφ dz dF 6

(7)

Since the integration of the inverse 12 and 6 powers is a difficult problem, one can apply the known mathematical trick,27 which reduces the evaluation of eq 7 to the differentiation of the simpler integral, SI, determined as follows:

SI )

∫0r ∫-HβF ∫02π s2 +1 R F dφ dz dF 2

(8)

where R is the subsidiary parameter. The trick is based (1) on the relationship27

d dR

R) dV ∫ab f(V, R) dV ) ∫ab ∂f(V, ∂R

(9)

which is valid for the convergent integrals, and (2) on the following identity: (26) The Feynman Lectures on Physics. Exercises; Levanuk, A. P., Ed.; Mir: Moscow, 1969; p 506. (Russian) (27) Smirnov, V. I. Course of the Higher Mathematics; Izdatelstvo TekhnikoTeoreticheskoi Literaturi: Moscow, 1957; Vol. 1-5. (Russian)

Effect of Meniscus Geometry

lim

Rf0

Langmuir, Vol. 24, No. 17, 2008 9377

(-1)n-1 ∂n-1(s2 + R)-1 1 1 ) lim 2 ) 2n Rf0 (s + R)n (n - 1)! ∂Rn-1 s

(10)

εsel )

∫0r ∫-Hb-b√1-F ⁄r ∫02π 4ε/aa[( 1s )

12

2 2

-

( 1s ) ]nF dφ dz dF 6

(15)

The attractive and repulsive terms of the LJ potential then reduce to the 2- and 5-fold differentiations of the SI with respect to R, and the desired integral (eq 7) takes the form

3. Parameters of Model for Nonspherical Molecules 3.1. Total Surface Energy. The total surface energy Es relates to the surface tension as follows28–30

/

εpar ) 2nε/aa lim

Rf0

(

∂2SI 4nεaa ∂5SI lim 5! Rf0 ∂R5 ∂R2

(11)

The integration of eq 8 over variables φ and z leads to

SI ) 2π

∫0r

[

F arctan -

-H - z0

√R + F2

√R + F2

]

+

[

F arctan

-H + βF2

√R + F2

√R + F2

]

)

dF (12)

Further integration of eq 12 over F and calculation of the second and fifth derivatives of SI in respect to R are carried out by a computer program. The final explicit expression is extremely cumbersome; it cannot be presented here and is used only for the computer calculations. Since the intermolecular forces decay as the seven degree of separations, only a surface layer with a thickness of 5÷10 molecules contributes to the adsorption energy; hence, one can assign any value of |H| > 10 to the cylinder bottom. For the numerical calculations, we shall take H ) -100. 2.4. Interactions with Hemispherical, Semi-Ellipsoid, and Conical Menisci. The stated above method is applicable to a calculation of energies of adsorption on all figures of revolution. Similarly, geometrical figures with liquidlike walls often arise after the pore walls have been covered by the preadsorption layers. For instance, the residual free space of the cylindrical capillary covered by the liquid film may be thought of as a hollow cylinder with a liquidlike wall; near-spherical or ellipsoidal pores of zeolites and hollows between globules of silica gels are the other cases in point. Since the intermolecular forces sharply decay with a separation, the energy of adsorption on the solid body covered by the adsorption film is practically indistinguishable from the energy of adsorption on the identical, hypothetical body made up entirely of the liquid. Apart from a paraboloid, we shall consider conical, hemispherical, and semi-ellipsoidal menisci. The conical surface may be viewed as that formed by the rotation of a straight line z ) kF, where k is the coefficient; the hemispherical one with a radius r may be thought of as the surface formed by the rotation of a circular arc z ) r - (r2 - F2)1/2. The semi-ellipsoid of revolution is generated by the elliptic arc z ) b - b[1 - (F2/r2)]1/2, where r and b are the horizontal and vertical semiaxes, respectively. In these cases, eq 7 takes the following forms

εcon ) εhsp )

∫0r ∫-HkF ∫02π 4ε/aa[( 1s )

( 1s ) ]nF dφ dz dF (13) 1 1 4ε [( ) - ( ) ]nF dφ dz dF s s 12

∫0r ∫-Hr-√r -F ∫02π 2

2

/ aa

-

12

6

6

(14) (28) Adamson, A. W.; Gast, A. P. Physical Chemistry of Surfaces; John Wiley: New York, 1997.

( dTdγ )

Es ) γ - T

(16)

Es in eqs 2–4 is measured in units of energy per mole, whereas γ is usually given in units of energy per m2. To convert Es to the energy per mole,one needs to know the area, am, occupied by one mole in the surface layer. If molecules are treated as spheres, then28–31 am ) χL01/3Vm2/3, where L0 is the Avogadro number and χ is the steric factor that is close to unity. In such a case

(

Es (J mol-1) ) am γ -

dγ dT

)

(17)

where γ is given in usual units of energy per m2. It is clear that this approach is valid for simple spherical molecules, but it may introduce a significant error in cases of complex, nonspherical molecules with specific orientations on liquid surfaces. In this case, one needs to know the area occupied by a molecule in the surface layer, ω, from a reliable, independent source. The requested value of Es (J mol-1) is then found as

(

Es ) ωL0 γ -

dγ dT

)

(18)

Consider the parameters of isopropyl alcohol and toluene which will be further used for a verification of the theory. According to the approach developed by Harkins,32 the hydroxyl group of isopropyl alcohol must be turned toward the bulk alcohol and the hydrocarbon groups of the surface layer away from the liquid (Figure 2). The area occupied by one molecule should then be close to 2aCH3 ) 0.42 nm2, where aCH3 ) 0.21 nm2 is the area of methyl groups on the surface.28 The study of surface tensions of alcohols also leads to ω ) 0.43 nm2.28,33 The detailed study of vapor-liquid interfaces of aromatic compounds34 shows that the orientation of the aromatic ring of these compounds at the vapor-liquid interfaces is tilted to the surface plane, the plane of the aromatic ring does not lie in the interfacial plane, and the substituent CH3 groups favor the liquid side of the interface. In the former paper,16 it has been suggested that we can take ω ) 0.25 nm2 as a first approximation for toluene. In a further treatment, eq 17 is used for simple nearspherical molecules and eq 18 is used for molecules with a specific orientation (Table 1). 3.2. The Lennard-Jones Diameters. One should take into account that for nonspherical molecules σaa are just the empirical constants, which are useful for purposes of calculation.35,36 For (29) Heimenz, P. C. Principles of Colloid and Surface Chemistry; Marcel Dekker: New York, 1986. (30) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, 1989. (31) Moelwyn-Hughes, E. A. Physical Chemistry, 2nd ed.; Pergamon Press: Oxford, 1961. (32) Harkins, W. D. The Physical Chemistry of Surface Films; Reinhold Publishing: New York, 1952, p 18. (33) Posner, A. M.; Anderson, J. R.; Alexander, A. E. J. Colloid Sci. 1952, 7, 623–628. (34) Hommel, E. L.; Allen, H. C. Analyst 2003, 128, 750–755. (35) Reid, R. C.; Sherwood, T. K. The Properties of Gases and Liquids; McGraw-Hill: New York, 1958; p 270. (36) Hirschfelder, J. O.; Curtiss, C. F.; Bird, K. B. Molecular theory of Gases and Liquids; John Wiley: New York, 1954; pp 1110-1112.

9378 Langmuir, Vol. 24, No. 17, 2008

Dobruskin

Figure 2. Hypothetical surface of isopropyl alcohol. Table 1. Some Parameters of Liquids substances

T (K)

γ (mJ m-2)

dγ/dT (mJ K-1 m-2)

Es (J mol-1)

σaa (nm)

isopropyl alcohol toluene water

293

21.32

-0.078

11 200a

0.5114b

293 298

28.52 72.14

-0.12 -0.16

9580a 7450c

0.5790b 0.2649d

a

Equation 18.

b

Equation 19. c Reference 18.

d

/ / Figure 4. Reduced autoadsorption energy εsel /εslab (eq 15) on surfaces of ellipsoids as a function of the ratio of its semiaxes r/b. The curve refers to the ellipsoid with r ) 5. Inset: Semi-ellipsoid. Here, ac ) r and oc ) b. When r ) b, the semi-ellipsoid (sel) is turned into a hemisphere (hsp).

Reference 35.

Figure 3. Geometrical factors λ and pressures of water vapor log p/ps for paraboloidal and conical menisci as functions of decimal logarithms of parameters (log β and log k). Panels (a) and (c) refer to a paraboloid; panels (b) and (d) relate to a cone.

complex molecules, σaa may be approximately calculated from the empirical equation

σaa ) 1.18Vb1⁄3

(19) 35

where Vb is the Le Bas value obtained from atomic volumes.

4. Results and Discussion 4.1. Conical and Paraboloidal Surfaces. Figure 3 demonstrates the geometrical factors for paraboloidal (Figure 3a) and conical (Figure 3b) menisci and the equilibrium water pressures over the paraboloid (Figure 3c) and the cone (Figure 3d); the corresponding values are shown as functions of decimal logarithms of parameters, log β and log k. When a funnel-shaped opening increases (k and β tend to zero), the paraboloid and the cone become akin to a planar surface and λ approaches 0; when the funnel-shaped opening tends to zero (k f ∞ or β f ∞), the dispersion ratio approaches the limit value of 3.67906. In practice, the limit values are already reached when k ≈ 5 for a cone and β ≈ 1.5 for a paraboloid. Equilibrium water pressures decrease with narrowing funnel opening and reach p/ps ≈ 3 × 10-4. 4.2. Hemispherical and Semi-Ellipsoidal Surfaces. Consider a semi-ellipsoid of revolution with semiaxes of r and b (inset of Figure 4). Figure 4 shows the effect of the ratio r/b on the * * autoadsorption energy εsel /εslab . Here, the ellipsoid with r ) 5 is taken as an example.

Figure 5. Geometrical factors λ for the hemispherical (points, eq 14) and cylindrical (line, eq 20) menisci as functions of true cylinder radii r. A dotted line shows values of λ for the hemispherical meniscus used in the previous publications (see section 4.4) Inset: Cylindrical and hemispherical menisci in the empty and filled cylinders. Table 2. Approximate Calculations of Geometrical Factors geometrical figure

geometrical factor

hemispherical meniscus

/ / λhsp ≡ 1- (εhsp /εslab ) ≈ (-1.4540/r) - (1.1666/r2) 3 - (1.0605/r )

paraboloidal meniscus

/ / λpar ≡ 1 - (εpar /εslab ) ≈ -2.90β (for β < 0.01)

conical meniscus

/ / λcon ≡ 1 - (εcon /εslab ) ≈ -1.60k (for k < 0.01)

spherical droplet

/ / λsp ≡ 1 - (εsp /εslab) ≈ (1.4428/r) - (1.1514/r2) + (0.4687/r3)

cylindrical meniscus

/ / λcyl ≡ 1 - (εcyl /εslab ) ≈ (-0.7362/r) - (0.2863/r2) - (1.1341/r3)

The reduced energy falls from 3.679 for the meniscus extended / / along the vertical axis (b < 0.01) to εsel /εslab ≈ 1 for large values of b when the ellipsoid degenerates to a nearly flat surface. When r ) b, the semi-ellipsoid is turned into a hemisphere; in this case, / / εsel /εslab ) 1.339. The hemispherical menisci are of primary concern to the theory of capillary condensation. The condensation in a cylindrical capillary occurs once its surface has been covered by the preadsorption film; in this case, a meniscus is cylindrical in shape. However, when condensation is completed, the adsorbed liquid forms a hemispherical meniscus and desorption (evaporation) proceeds from the hemisphere (inset of Figure 5). It is believed that due to the different shapes of menisci the equilibrium

Effect of Meniscus Geometry

Langmuir, Vol. 24, No. 17, 2008 9379 Table 3. Capillary Radii at Level of Menisci

observations

calculations

observations

calculations

p/ps alcohol

cap. 1

cap. 2

Kelvin’s model

new model

p/ps toluene

cap. 1

cap. 2

Kelvin’s model

new model

0.9966 0.9959 0.9954 0.9947 0.9933 0.9911 0.9879 0.9845

2.30 1.50 1.30 1.35 1.20 1.25 1.00 1.00

2.60 1.80 1.60 1.30 1.30 1.30 1.10 1.00

0.39 0.33 0.29 0.25 0.20 0.15 0.11 0.086

2.00 1.70 1.48 1.28 1.02 0.76 0.56 0.45

0.9951 0.9939 0.9927 0.9911 0.9892 0.9882 0.9863

1.50 1.20 0.95 1.10 1.00 0.90 0.9

1.50 1.30 1.25 1.25 1.00 0.95 0.85

0.51 0.41 0.34 0.28 0.23 0.21 0.18

1.36 1.10 0.90 0.74 0.62 0.56 0.48

pressures of adsorption and desorption do not coincide with one another, generating a hysteresis loop of capillary condensation. The values of geometrical factors for hemispherical and cylindrical surfaces are presented in Figure 5. The expression for the autoadsorption energy inside the infinite cylindrical capillary of a radius r was derived earlier18

ε/cyl / εslab

)

{

[

]

21 9 11 (r - 0.858)2 27 F π , ;1; 1 2 9 2 2 r2 2√10 288r

[

1 3 5 (r - 0.858)2 F , ;1; 1 2 2 2 3r3 r2

]}

(20)

where 2F1 is the hypergeometric function. Since condensation starts after the cylinder is covered by the monolayer preadsorption film, the true reduced radius of cylinder, rt, is equal to r + 1. One may see that the factors coincide at r < 3 and r > 100, and hence, the hysteresis loop falls in the range 3 < rt < 100. Relative pressures of vapors in capillaries with r ) 3 are in good agreement with experimental p/ps values of the beginning hysteresis loops.14–16 This adsorption pattern is consistent with observations. 4.3. Approximations. One can avoid the cumbersome calculations of geometrical factors and take advantage of the simple approximate expressions, which produce least-squares fits to the exact numerical values (Table 2). 4.4. Notice. According to the classical theory, pressures p over (1) a hemispherical meniscus in capillaries and over (2) a spherical droplet are given by the Kelvin equation28–30 RT ln p/ps ) (2γVm(1/r1 + 1/r2), where r1 and r2 are the radii of curvature and Vm is the liquid molar volume. Here, the plus sign relates to the convex sphere, whereas the minus sign refers to the interactions of a molecule with the concave hemisphere. In the wake of the classical theory, we have believed14–16 that if / / for a sphere λsp ) 1 - εsp /εslab, then for a concave hemisphere / / / / λhsp ) -(1 - εsp/εslab). Since the ratio εsp /εslab was known from 14–16 the literature

εsp(r, z) / εslab

)

{

1 24r3 ( √10 (r + z)2 - r2)3

15(r + z)6 + 63(r + z)4r2 + 45(r + z)2r4 + 5r6 15[(r + z)2 - r2]9

}

(21)

we found the value of λhsp and used it for the calculations.14–16 The exact values of the factor (points in Figure 5) and previously applied values of λhsp (a dotted line in Figure 5) are practically identical for r > 20; nevertheless, the divergence between them becomes visible for r < 20. Although these corrections have little effect on the capillary phenomena, it is expedient to note that there is no simple symmetry between interactions with the concave and convex liquid surfaces in nanocapillaries. 4.5. Experimental Verification. The most famous attempts to measure the equilibrium pressure in conical capillaries have

Figure 6. Schematic illustration of menisci in a cone-shape capillary. Solid lines for the capillary (anVqc) and paraboloid menisci (npq and abc); dotted lines for a hemispherical surface and two tangents to the surface; and a dot-dashed line for the axes of revolution. Since tangents to a sphere in points a and c must be parallel to one another, the hemispherical meniscus cannot be inscribed in the cone-shape capillary. In contrast to the hemisphere, the paraboloidal meniscus, which is tangent to the cone wall, is allowable for the cone-shape capillary. ad and ns are the capillary radii at levels of paraboloidal menisci; db and sp are the meniscus heights.

been undertaken by Shereshefsky and co-workers.37–41 We shall take advantage of one of his papers40 where the detailed information on the apparatus and procedure are presented as well as the tabulated experimental data related to toluene and isopropyl alcohol. The measurements were based on the equilibrium established between the condensed liquid in a coneshaped capillary and its vapor over a bulk solution of the liquid and nonvolatile dibutyl phthalate. The contact angle on glass, θ, was found to be zero. Experimental values of vapor pressures corresponding to the observable capillary widths for two capillaries denoted as cap. 1 and cap. 2 are presented in Table 3. Shereshefsky and Folman40 found that the Kelvin equation fails in description of concave capillaries, and the experimental radii were 2.9-11.6 times greater than those predicted by the Kelvin equation from the experimental values of p/ps. Now, consider these data in the light of the new model. First of all, let us show that menisci in cone-shaped capillaries cannot be spherical in shape. It follows from the fact that the walls of the meniscus and the walls of the capillary must be tangent to one another. If a meniscus has been spherical, then tangents to the sphere in the points a and c (Figure 6) would have been (37) Shereshefsky, J. L. J. Am. Chem. Soc. 1928, 50, 2966–2980. (38) Shereshefsky, J. L. J. Am. Chem. Soc. 1928, 50, 2980–2985. (39) Shereshefsky, J. L.; Carter, C. P. J. Am. Chem. Soc. 1950, 72, 3682–3686. (40) Folman, M.; Shereshefsky, J. L. J. Phys. Chem. 1955, 59, 607–610. (41) Coleburn, N. L.; Shereshefsky, J. L. J. Colloid Interface Sci. 1972, 38, 84–90.

9380 Langmuir, Vol. 24, No. 17, 2008

Dobruskin

parallel (dotted lines); however, cone walls, aV and cV, intersect and the sphere cannot be inscribed in the cone. The most credible speculation is that the meniscus is similar to a paraboloid, which may be inscribed into the cone. The parameter of a parabola β may be found from coordinates of points a, b, and c as follows. According to Finn,42 the author of Monograph on Equilibrium Capillary Surfaces, the dimensionless height of the meniscus, h/r, must obey the inequality: (1 - sin θ)/cos θ - fθB < h/r < (1 - sin θ)/cos θ. Here, f(θ) ) [(1 - sin θ)/cos2 θ][(1 - sin3 θ)/3cos3 θ - sin θ/2cos θ], h is the height of the meniscus with a radius r, and B ≡ (dg/γ)r2, where d is the density of the liquid. As θ ) 0 and r is about 1 µm, one has, as an example, for toluene f(θ) )1/3 and B ≈ 3 × 10-7. Hence, h/r = 1, and the meniscus height h in the Shereshefsky observations was close to the capillary radius r. Substitution h ) r in the parabola equation leads to r ) βr2. From here, the requested parameter β is given as

β)

1 r

(22)

For example, in experiments with toluene, the radius changes from 0.85 to 1.5 µm; hence, β ranges between β ) 0.579/(0.85 × 103) ) 0.000681 and β ) 0.000386 (we recall that r is taken in the reduced form with σaa ) 0.579 nm). The capillary radii, which correspond to the equilibrium pressures in conical capillaries, are presented in the column headed as “new model”. (42) Finn, R. Equilibrium Capillary Surfaces; Mir: Moscow, 1989; p 45. (Russian edition)

One may see that the model is in good agreement with the observations when the menisci are situated in the upper parts of the capillaries, but the accord is impaired in the vicinity of the capillary vertex. It is quite possible that it is a consequence of a shape distortion of the capillaries and deviations from regular cones. Shereshefsky and Folman40 report experimental profiles of glass capillaries (radii via length) that demonstrate such distortions in the vicinity of the capillary vertex. Nevertheless, a comparison of experimental radii with those predicted (1) by the Kelvin equation and (2) by the new model unambiguously demonstrates the advantage of the latter.

5. Conclusions The developed model provides a calculation of the effect of shape and size of menisci on the energy of autoadsorption on the liquid surface. Since variations of these energies determine changing enthalpies of condensation, the theory of phase transitions on interfaces, which relates an equilibrium pressure to the enthalpy of condensation, enables one to calculate the equilibrium pressures in capillaries. Because the concept of the adsorption energy remains valid even for extremely curved objects, the model is also applicable for nanocapillaries when the classical Kelvin equation fails. Predicted equilibrium vapor pressures are favorably compared with the independent experimental data. LA800546U