Langmuir 2005, 21, 2887-2894
2887
Effect of Surface Area on Equilibrium Pressure. Adsorption Approach Vladimir Kh. Dobruskin* Pashosh st. 13, Beer-Yacov 70300, Israel Received September 17, 2004. In Final Form: November 26, 2004 Changing droplet radii in a liquid-vapor system is due to the phase transition on the droplet surface. As a variation of the internal energy does not depend on the way the change occurs, we can imagine that a gas condenses on a droplet surface in two stages: in the first stage, autoadsorption occurs on the liquid surface, and in the second stage, adsorbed molecules transfer into the volume by diffusion. Assuming that the energetic effects of the diffusion are independent of the surface curvature, one may conclude that if two liquid bodies differ only with respect to their geometry, the difference of enthalpies of condensation on their surfaces, ∆Hbd, is equal to the variation of energies of autoadsorption. An estimation of ∆Hbd for the simple bodies is presented, and the relationship between the saturation pressure and droplet radii is derived. In the range of micrometer dimensions, the new equation and the Kelvin model lead to close results; for nanocapillaries, the Kelvin equation predicts a divergence of hysteresis loops, whereas the new equation adequately describes the observations. The classical model presumes that a surface area, A, affects the free energy, while the new approach is based on the assumption that A is the repository for the internal energy.
1. Introduction The fundamental equation of Young-Laplace, the Kelvin equation, and the Gibbs theory of capillarity underlie the physical chemistry of surface phenomena,1-5 which is currently one of the most active fields of research in physics, chemistry, and biology that attracts the attention of those engaged in both basic science and applications. There is a vast amount of literature on the problem, which cannot be quoted here, and we shall refer mostly to the fundamental treatise of Adamson.2 In the present paper, we are concerned with the simple spherical bodies for which radii of curvature are equal to the sphere radius, r, and the Laplace and Kelvin equations take the forms
2γ r
(1)
p 2γVm ) ps r
(2)
∆p ) RT ln
Here γ is the surface tension, ∆p is the difference of pressures due to the surface curvature, Vm is the molar volume, and ps is the saturation pressure. Although the equations play the dominant position in the theory, there are some questions related to their applications. According to Finn,6 the author of a comprehensive monograph on capillary surfaces, the Laplace equation has never undergone direct experimental testing. Nev* E-mail:
[email protected]. (1) Atkins, P. W.; de Julio, P. Atkins’ Physical Chemistry; Oxford University Press: Oxford, 2002. (2) Adamson, A. W.; Gast, A. P. Physical Chemistry of Surfaces; John Wiley: New York, 1997. (3) Heimenz, P. C. Principles of Colloid and Surface Chemistry; Marcel Dekker: New York, 1986. (4) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, 1989. (5) Moelwyn-Hughes, E. A. Physical Chemistry, 2nd ed.; Pergamon Press: Oxford, 1961. (6) Finn, R. Equilibrium Capillary Surfaces; Mir: Moscow, 1989; p 31 (in Russian)
ertheless, there were numerous attempts to verify its logical corollary, the Kelvin equation. Along with vast evidence of the validity of the equation, there are experimental data that show a deviation between the theory and practice.7 Although the latter data were criticized,8 the critic did not present any alternative measurements. Summarizing experimental results, Everett with co-workers 9 and Adamson,2 the authors of the fundamental studies on surface phenomena, draw attention to the fact that the situation of its verification is still conflicting! One may come to the same conclusion considering a capillary condensation in porous media. Even the nature of condensation in the idealized case of a single, infinitely long, cylindrical pore is poorly understood in the framework of the Kelvin model.10 In an empty cylindrical pore, the capillary condensation starts with a cylindrical meniscus; however, since the filled capillary has a hemispherical meniscus, evaporation occurs with a hemispherical meniscus. From here, the equilibrium pressures of condensation, pc, and evaporation (desorption), pe, do not coincide with one another giving rise to the capillary hysteresis.2,11,12 Nevertheless, the Kelvin equation, being applied to the cylindrical and hemispherical menisci, predicts a noncoincidence of pc and pe for any value of radii, that is, a divergence of hysteresis branches, while thousands of experiments carried out with different adsorbents and adsorbates show that the capillary branches, on contrary, converge at pressures corresponding to condensation in the narrow capillaries!11,12 Due to this striking discrepancy, it was even declared11 that “the complexities of the pore structure in real systems are often so great as to defeats a rational analysis”. (7) Coleburn, N. L.; Shereshefsky, J. L. J. Colloid Interface Sci. 1972, 38, 84-90. (8) Melrose, J. C. Langmuir 1989, 5, 293-295. (9) Everett, D. H.; Haynes, J. M.; McElroy, P. J. Sci. Prog. (Oxford) 1971, 59, 279. (10) Evans, R.; Marconi, U. M. B.; Tarasona, P. J. Chem. Soc., Faraday Trans. 2 1986, 82, 1763-1787. (11) Steele, W. A. The Interaction of Gases with Solid Surface; Pergamon Press: Oxford, 1974. (12) Gregg, S. J.; Sing, K. S. W. In Surface and Colloid Science; Matijevic, E., Ed.; John Wiley: New York, 1976; Vol. 6, pp 231-360.
10.1021/la047675q CCC: $30.25 © 2005 American Chemical Society Published on Web 02/22/2005
2888
Langmuir, Vol. 21, No. 7, 2005
Dobruskin
Recently, a new equation13 for calculating equilibrium pressures of droplets, p, has been derived
RT ln
p ) ∆bd* ps
(3)
where ∆db* is a function of radii, which is equal to a difference between the energy of autoadsorption on the surface of the semi-infinite planar body and that on the droplet surface ∆db* > 0. Equations 3 and 2 predict similar values of equilibrium pressures over micrometer droplets; in the case of capillary condensation, the latter equation, in contrast to the Kelvin model, correctly describes the capillary hysteresis eliminating discrepancies between the theory and experiments. 13 A similar equation was earlier applied for adsorption in slitlike micropores;14,15 it was used for the discussion of some problems of nucleation,13 adsorption on nanoparticles, and adsorption contributions to the particle agglomeration.16 For those specific applications, it was derived from (i) isotherms of FowlerGuggenheim and quasi-chemical approximation,14 in the framework of (ii) thermodynamics15 and (iii) statistical mechanics.15 The applicability of the equation to the description of the variety of phenomena gives grounds to assume that, being laid on the solid thermodynamic foundation, it may be used as a basis for a new approach to surface effects. In the present paper, we intend (i) to develop the consistent thermodynamic basis for a new theoretical approach to the problem and (ii) to compare it with the classical model. The statistical mechanical derivation of eq 3 for droplets will be presented, and the fields of validity of the classical and new models will be outlined. We shall consider the behavior of molecular liquid objects with dimensions of nanoparticles, namely, droplets and liquids in cylindrical capillaries, and our interest will be centered on the effect of size on the equilibrium pressure. 2. Basic Postulates for the New Approach In formulation of the first law,1,17-19 the system is characterized by its internal energy alone: the work, dw, and quantity of heat, dq, gained by the system in an infinitesimal change of its state results in the variation of its internal energy, dU
dU ) dq + dw
(4)
It therefore seems plausible to adopt the following postulate (1): A surface area is the repository for the internal energy. It is the most universal hypothesis concerning a contribution of a surface area to energetic processes in the system. In the third section, an explicit expression for the change of U with radii will be given. The equilibrium thermodynamics concerns the idealized quasi-static processes;18 therefore, calculating a quasiequilibrium pressure associated with uniform droplets necessitates assuming that droplets and their vapor are in equilibrium, that is, the radii and surface area are invariable. Because this assumption should be valid for any droplet radii, it means that between the state of a (13) Dobruskin, V. Kh. Langmuir 2003, 19, 4004-4013. (14) Dobruskin, V. Kh. Langmuir 1998, 14, 3847-3857. (15) Dobruskin, V. Kh. Langmuir 2003, 19 (6), 2134-2146. (16) Dobruskin, V. Kh. Langmuir 2002, 18 (11), 4356-4361. (17) Berry, R. S.; Rice, S. A.; Ross, J. Physical Chemistry, Oxford University Press: New York, 2000. (18) Callen, H. B. Thermodynamics and Introduction to Thermostatics; John Wiley: New York, 1985. (19) Kaufmann, M. Thermodynamics; Marcel Dekker: New York, 2002.
Figure 1. Effect of the surface curvature on the energies of autoadsorption. The separation AC between the atom A and points of the convex surface is greater than its separation AD from the tangent surface P, except the tangent point T; correspondently, the energy of adsorption on the planar surface exceeds that on the convex droplet surface.
vapor and the state of a bulk liquid there is an infinite number of intermediate states differing with respect to particle radii. From here, the transition from nuclei of condensation to the bulk liquid may be thought of as a quasi-static locus in the thermodynamic configuration space that consists of an ordered succession of equilibrium states with growing radii, each of which is considered as a distinct quasi-equilibrium state of the same substance. Although the quasi-static process of the transition from the nuclei to the bulk liquid is an idealized concept, quite distinct from a real physical process, it is possible to contrive a real process that has a close relationship to the idealized one. Thus, radii of capillaries of an ordinary active carbon (meso- and macropores) fall in the range of 1-30000 nm, and the set of liquid islands arising at the successive growth of equilibrium pressure gives a model of equilibrium states. Another example is the growing droplets of low-volatile substances at low temperatures that can attain equilibrium with the vapor; for volatile substances, especially at elevated temperatures, the method, which follows, will be valid only within a quasistatic approximation. A formulation of the second postulate employs a notion of autoadsorption.20,21 This term denotes adsorption of vapors on the surface of its own condensed phase (for example, adsorption of water vapor on the surfaces of either ice or liquid water). In the case of autoadsorption, molecules only touch the surface without entering the surface layer. Let us show that the energy of autoadsorption varies with droplet radii. Consider a segment of a sphere, a tangent surface P to the sphere, and the molecule A outside the sphere and compare the energies of autoadsorption, *, on the curved shell and planar tangent surface. The energy of adsorption is expressed as the sum of the interactions of the adsorbate with all molecules of the body,11,22,23 but only contributions of the molecule placed in the tangential point are identical for planar and curved surfaces. For other points, the energy of interaction with the curved surface differs from that with the plane due to alterations of separations between interacting species: compared to the planar surface, for droplets separations increase and interactions decrease. In the case of a concave capillary surface, the effect is antipodal. Consider a vapor condensation on a droplet surface. As variation of the internal energy does not depend on the way the change occurs but only upon the initial and final states, we can imagine that a gas condenses in two stages: in the first stage, autoadsorption occurs on the (20) Shimulis, V. I. Zh. Fiz. Khim. 1967, 41, 376-383. (21) Khabarov, V. N.; Rusanov, A. I.; Kochurova, N. N. Kolloidn. Zh. 1975, 37, 92-98. (22) Avgul, N. N.; Kiselev, A. V.; Poshkus, D. P. Adsorption of Gases and Vapors on Homogeneous Surfaces; Khimia: Moscow, 1975 (Russian). (23) Everett, D. H.; Powl, J. C. J. Chem. Soc., Faraday Trans. 1976, 72, 619.
Effect of Size on Equilibrium Pressure
Langmuir, Vol. 21, No. 7, 2005 2889
liquid surface, and in the second stage, the molecules adsorbed are transferred into the volume by diffusion. We introduce postulate 2, which will be verified by the success of the derived theory: The energetic effect of the diffusion (the second stage) is independent of the droplet radii. From here, if two liquid bodies differ only with respect to their geometry, the difference of enthalpies of condensation on their surfaces is equal to the variation of energies of autoadsorption. Let the subscripts b and d denote a bulk liquid and droplets, respectively, and the subscript db refers to the difference of parameters of the bulk liquid and droplets. Then
∆Hdb ≡ ∆Hb - ∆Hd ) -∆db* ≡ ∆db*
(5)
Here ∆Hd and ∆Hb are enthalpies of condensation on the surface of the bulk liquid and droplets; ∆Hdb ≡ ∆Hb - ∆Hd is the difference of the enthalpies and ∆bd* ≡ d* - b* is the difference of the energies of autoadsorption on the surfaces. Note that the adsorption energy is taken upward from the potential minimum and, by convention, it is positive, whereas the changes of the potential energy on adsorption are negative. Since for condensed phases U and H are practically equivalent, eq 5 states that the variation of the internal energy with radii is also equal to the difference of the autoadsorption energies. 3. Effect of Radii on the Enthalpy of Condensation In general, the intermolecular forces may be loosely classified into three categories:24 pure electrostatic in origin arising from the Coulomb forces between charges, polarization forces that arise from induced dipole moments, and quantum mechanical covalent forces. Accordingly, the total energy of autoadsorption is the sum of the different constituents determined by the chemical nature. Due to the lack of a reliable theory for energy calculations we shall resort to the bypass method for estimating ∆bd*: b* will be expressed through experimental parameters and then this value, along with dispersion contributions, will be used for evaluating d* and ∆bd*. Energy of Autoadsorption on the Surface of a Bulk Liquid. The thickness of a liquid surface region, for which thermodynamic properties differ from those in the volume, is determined only by the intermolecular forces, and it is independent of the quantity of a liquid. Hence, to keep this thickness constant with the addition of an adsorption layer, an equal quantity of a liquid from the surface region transfers into the volume. One may see that there is a one-to-one correspondence between the number of autoadsorbed molecules and the number of molecules migrating from the surface layer toward the interior. Hence, the energetic effect of autoadsorption is equal to the difference between the energy of molecules on the surface and the energy in the volume. This quantity is called the excess surface energy or the total surface energy and usually labeled as Es.2-5,25 Finally, we have
b* ) Es
(6)
According to thermodynamics, Es is a function of γ:2-5
dγ dT
Pay attention that this argumentation is valid only for a bulk liquid. In the case of droplets, the number of molecules on the surface exceeds the number of molecules in any layer located in the depth (for instance, only one molecule may be placed in the center of a droplet). Hence, the number of “sinking” molecules for droplets is less than the number of adsorbed molecules, a one-to-one correspondence between autoadsorption and migration in the liquid is disturbed, and d* < Es. Dispersion Constituents of the Autoadsorption Energy. In the case of dispersion forces, calculating is based on the summation of the Lennard-Jones 12-6 potential function for the interaction energy, , between single atomic or molecular species:11,22,23
) 4aa*
[( ) ( ) ] σaa s
12
-
σaa s
6
(8)
where s is the distance between the nuclei of the atoms, aa* is the depth of the energy minimum for the atomatom interaction, and σaa is the distance at which aa ) 0. The asterisk (*) refers to the parameters of the well depth. In further calculations, we shall take σaa as a scale parameter, put σaa ) 1 in eq 8, and express lengths in the reduced form. Because the final purpose is an estimation of the energy difference ∆bd* ) d* - b*, some simplifications, which lead to the similar errors in each of the terms and cancel each other in the required value of ∆bd*, may be introduced in calculations. In particular, the interactions of adsorbate with a condensed phase should be calculated by the summation of terms given by eq 8 over all atoms of the phase. But here the simpler operation of integration may be used for calculating ∆bd*. For the planar semi-infinite slab, which is a model of a bulk liquid, this way leads to the 9-3 potential function22,23
slab )
3 2 19 1 * 1/2 slab 15 z z 10
3
[ ( ) ( )]
(9)
where z is the reduced separation of the molecule from the surface layer and slab* is the energy minimum, which determines the adsorption energy on the slab surface23
slab* )
2(101/2) πnaa* 9
(10)
Here n is the number of interacting centers per unit volume. For the interaction between a molecule and a sphere of reduced radius R, the dispersion energy, sp, is given as follows13
{
16πnaaR3 1 + sp(R,z) ) 3 ((R + z)2 - R2)3 15(R + z)6 + 63(R + z)4R2 + 45(R + z)2R4 + 5R6 15[(R + z)2 - R2]9
}
(7)
(11)
(24) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: London, 1995. (25) Dobruskin, V. Kh. Carbon 2001, 39, 583-591.
It is convenient to take slab* as a scale energy parameter and express energy values also in the reduced form. Then, eq 11 becomes
Es ) γ - T
2890
Langmuir, Vol. 21, No. 7, 2005
Dobruskin
{
24R3 1 + 1/2 10 ((R + z)2 - R2)3 15(R + z)6 + 63(R + z)4R2 + 45(R + z)2R4 + 5R6
sp(R,z) )
}
15[(R + z)2 - R2]9
(12) Because reduced lengths are used, the geometrical radius is linked with R as r ) σaaR. Figure 3 demonstrates that the maximum values of reduced energies approach unit with an increase of sphere radii. For a cylinder, the reduced dispersion energy is described by the hypergeometric function 2F1[R,β;γ;z]25
cyl(R,z) )
{
[ [
] ]}
Figure 2. The reduced energies of interactions sp between an atom and spheres with R of 1.5, 2, 3, 5, 10, and 100 as a function of reduced separation z.
2
(R - z) 21 9 11 27 ; 1; π 2F1 , 2 2 2(101/2) 288R9 R2 (R - z)2 3 5 1 , ; 1; F (13) 1 2 2 2 3R3 R2
For the given R, the maximum of * for spheres and cylinders is searched by the numerical method with respect to z. It usually occurs at the reduced equilibrium separation z* ≈ 0.858. Notation System. It is generally accepted to denote the adsorption energy and the dispersion energy by the Greek symbol . Such a practice usually does not lead to confusion when these values coincide. In our case, the adsorption and dispersion energies can be different: for example, a significant part of the total surface energy of water is associated with the hydrogen bonds. Nevertheless, we retain the conventional definitions. Note that energies are discriminated only by subscripts: b*, d*, and ∆bd* refer to the adsorption energies, whereas slab, sp, and cyl relate to the dispersion energies. There is no special indication for reduced and absolute values discriminated from the text. Effect of Radii on Enthalpy of Condensation. The ratio of the autoadsorption energy to its dispersion constituent depends only on the chemical nature, and it is independent of the curvature. For the bulk liquid, b* ≡ Es (eq 6) and the ratio is equal to Es/slab*, whereas for the droplet it is d*/sp*. Then
d* = slab* sp* Es
(14)
From here, we obtain the following expression for the autoadsorption energy on the droplet surface
d* ) Es
sp* slab*
(15)
and for the requested value of ∆db* ) b* - d*
∆db* ) Es - Es
sp* ≡ Esλ slab*
(16)
where
sp* >0 λ≡1slab*
Figure 3. The effect of radii on the relative enthalpy change, (Hd - Hb)/Es.
From eqs 5, and 16 one has
∆Hbd ∆Hd - ∆Hb ≡ )λ Es Es
(18)
Here Es is responsible for the individual property of the liquid. For instance, for a droplet with R ) 5, sp*/slab* ≈ 0.76 (Figure 2) and λ ) 0.24; in particular, for water2 Es ) 7451 J‚mol-1 and ∆Hbd ) 0.24Es ) 1788 J‚mol-1. The ratio ∆Hbd/Es (Figure 3) shows the effect of radii on the enthalpy of condensation. For R < 50-100, the curve sharply drops with radii; beyond R ≈ 100 it flattens out and, within the error less than 1%, (∆Hd - ∆Hb)/Es for R > 50 may be approximated by the simplest function
∆Hbd 1.431 )λ≈ Es R
(19)
Since intermolecular forces make their appearance within separations not more than 3σaa-5σaa, estimation of λ is valid even for objects with dimensions about a nanometer. If 1 is the energy of interactions between an external molecule and a dense sphere and 2 is that between a molecule inside a hollow sphere and the sphere shell, then 1 ) -2, provided that radii are equivalent and the shell thickness exceeds 3-5σaa. For interactions between molecules in the gas phase and a hemispherical meniscus, adsorption energies, 3, are invariable above the surface of the equilibrium meniscus and 3 can be easy calculated for the case of the molecule placed above the center of the meniscus. For this molecule, 3 ) 2 because of a negligible contribution of the missing hemisphere. Hence, the same formulas describe interactions with a hemispherical meniscus, a droplet, and a spherical cavity. 4. Effect of Radii on Entropy of Droplets
(17)
The change of entropy of droplets with respect to the bulk liquid, ∆Sbd, is calculated just like for any phase
Effect of Size on Equilibrium Pressure
Langmuir, Vol. 21, No. 7, 2005 2891
Table 1. Effect of Radii on Entropy of Droplets r, nm
A, m2‚mol-1
∆Hdb, J‚mol-1
∆Sdb, J‚mol-1‚K-1
0.5 1 5 10 50
1.08 × 105 5.4 × 104 1.08 × 104 5.4 × 103 1.08 × 103
3815 2308 547 276 59.6
12.8 7.74 1.84 0.926 0.200
(
Es ) γ - T
transition
(20)
where ∆Hdb is given by eq 18. For water, the effect of radii on entropy of droplets at T ) 298 is shown in Table 1; the value of σaa ) 0.2649 nm is taken from ref 13. When water clusters with r ) 0.5 nm coalesce into the bulk liquid (r f ∞ and A f 0), entropy decreases on ∆S ≈ -12.8 J‚mol-1‚K-1. For a bulk water at 298 K, S ) 70 J‚mol-1‚K-1 and, hence, the absolute entropy of clusters comes to ≈82.8 J‚mol-1‚K-1. The entropy changes are significant only for a coalescence of nanodroplets, especially at low temperatures; even for r > 50 nm (R > 190) ∆Sdb < 0.2 J‚mol-1‚K-1, and for microdroplets this value is close to zero. 5. Derivation of the Main Equation Equation 3 is derived13 proceeding from the ClapeyronClausius equation. Substitution of eqs 5 and 18 into eq 3 establishes the correlation between the equilibrium pressure pd and autoadsorption energies
ln
pd Esλ ) ps RT
6. Comparison of the Approaches
Es ) χL
(
dγ γ-T dT
(
)
(22)
where
Am ) χL1/3Vm2/3
(23)
is the surface area occupied by 1 mol in the surface layer, L is the Avogadro number, and χ is the steric factor that is close to unity.2 If one puts χ ) 1, Am may be transformed to
L1/3Vm2/3 ) (L/Vm)1/3Vm ≈ Vm/σaa
(24)
since Vm/L is a volume occupied by one molecule, and eq 22 takes the form
)
(26)
It is a transformed form of eq 21, which is valid for R g 50-100 (r g 15-30 nm). A comparison of this equation with the Kelvin model (eq 2) shows that both equations predict a decay of RT ln p/ps as r-1 but have different coefficients, 1.431(γ - T∂γ/∂T) and 2γ. Though the agreement between the two sets of coefficients (Table 2) is not very close, it may nevertheless be reported as satisfactory, especially in view of (i) the uncertainty in χ for complex molecules and (ii) the accepted one-centered model of dispersion interactions. The typical values of dγ/dT are so that at usual temperatures the coefficients prove to be rather close and only at low temperatures (nitrogen) the divergence growths. Condensation in Cylindrical Capillaries. As mentioned in the Introduction section, for spherical and cylindrical menisci the equilibrium pressures of condensation pc and evaporation (desorption) pe do not coincide with one another giving rise to the capillary hysteresis.2,11,12 For the spherical meniscus, the Kelvin pressure is given by eq 2; while for the cylinder, the model takes the form11
RT ln
Equilibrium Pressure of Droplets. The effect of radii on the equilibrium pressure is shown in Figure 4. Points correspond to the Kelvin equation and a solid line refers to the new equation. For water and mercury, the approaches predict very close results; for organics there are divergences, especially for the finest droplets. Note that namely water and mercury were used for verification of the Kelvin equation;2 for benzene reliable data are absent. The underlying reasons of such a behavior become evident after transformation of eq 21. One must take into account that Es in eq 21 is expressed in J‚mol-1, whereas γ in eq 2 is given in J‚m2. If one uses the experimental values of γ [J‚m2], then2,5
Vm2/3
(25)
dγ Vm p ≈ 1.431 γ - T ps dT r
(21)
The statistical mechanical derivation of the equation is given in the Appendix.
1/3
)
Substituting eqs 19 and 25 into eq 21 and keeping in mind that r ) Rsaa, one obtains for R g 50
RT ln ∆Sbd ) ∆Hbd/T
dγ Vm dT σaa
p γVm ) ps r
(27)
Applying eqs 2 and 27 to the hysteresis branches, one obtains the specific, independent of radii and adsorbates relation between pc and pe11
() pc ps
2
)
pe ps
(28)
which, in contrast to observations, predicts the divergence of hysteresis branches (see Figure 5). Note that eqs 2 and 27 are the special cases of the general relation that follows from the Young-Laplace equation2
RT ln
(
)
1 p 1 ) γVm + ps rcur1 rcur2
(29)
where rcur1 and rcur2 are the radii of the meniscus curvature. For a sphere, rcur1 ) rcur2 ) r, while for a cylinder of infinite length, rcur1 ) ∞ and rcur2 ) r. The equations are derived by means of substitution of these values into eq 29. Note that eqs 2 and 27 are the equations of “equal weight”: it is an impossible situation when one of them is valid while the second is wrong. Hence, the underlying reason of the divergence with experiments may be rooted only in drawbacks of the model. In the framework of the new model, RT ln pc/ps ) Esλcyl and RT ln pe/ps ) Esλsp, where λsp is given by eq 17, whereas λcyl is obtained by substituting cyl for sp in eq 17. Equilibrium pressures calculated by these equations are depicted in Figure 6. The similar curve for nitrogen is presented in ref 13. In contrast to the Kelvin model, the left ends of adsorption and desorption branches converge at R ) 2.5 corresponding to p/ps 0.35, 0.18, and 0.35 for nitrogen, benzene, and water, respectively. These results
2892
Langmuir, Vol. 21, No. 7, 2005
Dobruskin
Figure 4. Effect of radii on equilibrium pressures of droplets: solid line, eq 21; points, the Kelvin equation (eq 2). Table 2. Comparison of the Equations for R > 50
substance water2 mercury5 mercury5 nitrogen2 ethyl alcohol2 benzene2
dγ/dT, γ, T, K mJ‚m-2 mJ‚K-1‚m-2 293 273 393 75 293 293
-0.16 -0.087 -0.162 -0.23 -0.086 -0.13
77.75 463 448 9.71 22.75 29.88
1.431(γ T dγ/dT), mJ‚m-2
2γ, mJ‚m-2
171 697 732 38.1 68.6 97.3
145 925 896 19.4 45.5 57.8
Figure 5. The Kelvin model for adsorption (eq 27, solid line) and desorption (eq 2, dashed line) in cylindrical capillaries. In contrast to experiments, branches diverge at low pressures.
are in the excellent agreement with experimental values of p/ps 0.35,12 0.17,27 and 0.40.28 Contribution of Heat and Work. Since the work of reversible changing the surface area by the infinitesimal quantity dA
dw ) γdA
(30)
at constant pressure and temperature increases the free energy of the system
dw ) (dG)p,T
(31)
it was assumed28 that the change of the total molar free energy of a one-component liquid with a changeable surface area is made up of dG for the bulk phase
dG )
(∂G ∂p )
T
dp +
(∂G ∂T )
p
dT
(32)
plus γ times dA28
dG )
(∂G ∂p )
T,A
dp +
(∂G ∂T )
p,A
dT +
(∂G ∂A )
p,T
dA
(33)
(26) Dubinin, M. M. Adsorption Properties and Pore Structure; Izdatel’stvo VAHZ.; Moscow, 1965 (in Russian). (27) Dubinin, M. M. Carbon 1980, 18, 355-364. (28) Landau, L. D.; Lifshitz, E. M. Statistical Physics; Pergamon Press: London, 1958.
This equation implies that γ dA accounts for the surface contribution to the free energy. Although G and U are linked by definition
G ) U + pV - TS ≡ H - TS
(34)
the approaches based on the first postulate and eq 33 are not equivalent. Compare a reversible increase of the molar surface area of a bulk liquid carried out in two ways: (1) by the container expansion and (2) by the evaporation of a liquid inside the container with fixed diathermal walls and the following condensation of the vapor into droplets. For example, (1) a bulk liquid, for which A = 0 m2/mol, is stretched until its surface area reaches 1 m2/mol and (2) the same bulk sample evaporates and the vapor condenses into the droplets with A ) 1 m2/mol. The excess surface energies of both samples with A ) 1 m2/mol are equal to one another. What are the other energetic features of the processes? In thermodynamics work and heat are forms of the energy that are perceived by an observer located in the surroundings when the energy enters or leaves a system.17 The observer does not care what happens to this particular energy before or after it crosses the boundary, only the crossing process itself.17 On the container expansion, mechanical work is observed in the surroundings (probably, depending on constrains, along with a heat flux); in the case of evaporation and condensation, the observer perceives the heat flux alone, as the walls are fixed and no work is done in the surroundings. Hence, in the latter case the droplet/vapor interface arises due to heat transfer associated with the phase transition; it is by no means the result of work done. This effect is compatible with the first postulate, since ∆U is initiated by both heat and work transferred through the system walls (eq 4), but it is in disagreement with eq 33, which relates an increase of A only to a work done. The equilibrium thermodynamics concerns with quasi-static processes for which a heat flux is associated with an entropy change18
dq ) T dS
(35)
Therefore, the fundamental difference between two ways of changing a surface area is the following: If mechanical work is responsible for a surface area variation, the entropy of a liquid stays constant; if heat flux is coupled with the variation, the entropy of droplets changes. Table 1 shows that changes of entropy and latent heats of droplet formations are significant only for r < 10 nm (less than 35-40 water diameters). As we saw, the difference between the models of capillary condensation makes its appearance namely in this range and only the new model taking into account heat fluxes correctly describes the hysteresis loop in the finest capillary. It is only for constant p and T, ∆G determines the maximal work of a system, but a droplet-vapor system at these conditions is in equilibrium and changes of droplet
Effect of Size on Equilibrium Pressure
Langmuir, Vol. 21, No. 7, 2005 2893
Figure 6. Correlation between capillary radii and equilibrium pressures of condensation (solid lines) and evaporation (dashed lines).
radii are impossible. Hence, alterations of droplet radii occur only with variation either p or T (or both of them), but ∆G in these cases does not bear useful information and it is not equal to work done. The same conclusion may be drawn from the basic concepts of thermodynamics: being a path function, the amount of work associated with different processes may be different, even though each of the processes initiates in the same initial state and each terminates in the same final state.17 Changing Extensive Parameters. A description of a thermodynamic system requires specifications of the “walls” that separate it from the surroundings and provide its boundary conditions. It is by means of manipulations of walls that extensive parameters of the system are altered and processes associated with a redistribution of some quantity among the system and surroundings are initiated.18 For instance, when the new surface is formed by the mechanical expansion, the process may be thought of as a redistribution of a volume (surface) between the container and the surroundings. But changing the surface area of droplets in the gas-vapor system in no sense relates to a redistribution of a surface between a system and the surroundings. It is the phase transition on the droplet surface that causes the droplet radius to be changed. Independent Variables. The thermodynamic state of a system is uniquely determined when a complete set of macroscopic coordinates, independently controllable by the experimenter, is given.17,18 For droplets, r and A are in a one-to-one correspondence
A ) 3Vm/r
(36)
Substitution of eq 36 into eq 2 leads to
φ(p,T,A) ) RT ln p/ps - 2/3γA ) 0
(37)
The very existence of this relationship between A, p, and T (we stress that it is the experimental fact) attests that only any two of these variables (and not three!) constitute a complete set of independent variables. This situation is very closely analogous to that pertaining to the perfect gas case when only any two of three parameters linked by the equation of state f(p,V, T) ) 0 form such a set.1,17,18 Hence, the free energy of the droplets should be a function of two independent variables. If we chose p and T as independent variables, eq 33 for the case of a changeable surface area should be transformed to
dG )
(∂G ∂p )
T,A
dp +
(∂G ∂T )
p,A
dT
(38)
In comparison with eq 32, the additional constraints are imposed on the entropy, S ) -(∂G/∂T)p, and on the volume, V ) (∂G/∂p)T, which become functions of A. When A tends to infinity, eq 38 refers to the gas, whereas for A f 0 it
relates to the bulk liquid; for intermediate values, the equation addresses droplets. 7. Conclusions There may be two ways of changing the surface area: the way associated only with a mechanical work and the way coupled with a heat flux. The experiments with the expansion of soap films and the Langmuir microbalances relate to the first group, whereas adsorption in porous media, nucleation, and capillary condensation are the examples of processes for which appearance/disappearance of the surface area is closely associated with heat fluxes of the phase transition. The first postulate accounts both for work and for heat fluxes across the system walls and provides a consistent description of the surface phenomena, while eq 33 is valid only for mechanical deformations. The approaches become equivalent when thermal effects can be neglected. Appendix. Statistical Mechanical Approach In the case of autoadsorption, one should take into account the following: (1) As the surface layer remains invariable in the course of autoadsorption, the liquid can be regarded as an inert adsorbent. (2) “Adsorbent” and “adsorbate” are chemically identical. On the solid surface a similar situation is realized when the fractional adsorption θ ) 1; e.g., the surface is covered by the monolayer. (3) Due to the molecular transfer from the surface into the volume, only the monolayer autoadsorption takes place on the liquid surface. (4) The second postulate is equivalent to the statement that the planar surface and the surface of droplets have the same arrangement of adsorption sites. Hence, autoadsorption is visualized as a monolayer adsorption of a vapor in the potential field near the surface of the inert adsorbent. The method developed by Steele for solid adsorbents11 will be used further for calculating the equilibrium pressure. It is convenient to consider the adsorbed film as a distinct phase with a known volume V(s), containing a known number of molecules, N(s), at fixed temperature T. A superscript s denotes that the thermodynamic properties relate to the adsorbed phase. The thermodynamic properties of the system are calculated from the canonical partition function, Q(N,V,T)11
Q(N,V,T) ) ZN(s)/N!Λ3N
(A1)
where Λ is the thermal deBroglie wavelength and ZN(s) is the configurational integral N
ZN(s) )
∫V exp[-(∑us(ri) + i)1
∑
u(rij))] dr1 dr2 ... drN (A2)
1ei
