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Correlation between Saturation Pressures and Dimensions of Nanoparticles. Are the Fundamental Equations Really Fair? Vladimir Kh. Dobruskin* St. Pashosh 11, Beer-Yacov, 70300 Israel Received October 24, 2002. In Final Form: February 11, 2003 The relationships between the geometric form and saturation pressure are derived on the basis of thermodynamics and molecular models. A comparison of the relationships with the classical Laplace and Kelvin equations shows that there is a profound conceptual diversity between approaches: the classical equations contain the surface tension, γ, as a parameter, whereas developed equations include the difference of the potential energies of a nanoparticle and a semi-infinite planar body with respect to the gas phase, which is equal to the difference of energies of autoadsorption on surfaces of bodies. The latter circumstance affords ground for calculating the change of interest proceeding from well-developed methods of adsorption theory. Molecular models are discussed for simple spherical bodies and applied to the processes of nucleation and growth of drops. In the cases of (i) water and mercury drops or capillaries with diameters of ≈1 µm and (ii) cyclohexane meniscuses on mica cylinders, the classical and new relations predict close results, which are in agreement with experiments; nevertheless, for organic substances, especially at elevated temperatures, noticeable deviations must be expected. The divergence of the approaches makes its appearance in the case of condensation in narrow capillaries with radii ≈1-10 nm when the new equation provides a correct description of experimental facts whereas the Kelvin model predicts unrealistic effects. Considering the growth of surface areas due to the extension or adsorption shows that γ does not take into account all changes associated with alterations of surface areas, and it is this fact that underlies the probable defect in the classical equations.
1. Introduction The fundamental equation of Young-Laplace
∆p ) γ
(
)
1 1 + R1 R2
(1)
the Kelvin equation, and the Gibbs theory of capillarity underlie the physical chemistry of surface phenomena.1-3 Here γ is the surface tension, ∆p is the difference of pressures due to the surface curvature, and R1 and R2 are two radii of curvature. In the present paper, we shall consider the behavior of simple spherical bodies for which both radii of curvature are equal to the sphere radius, r, and the Laplace and Kelvin equation take the forms
∆p )
2γ r
p 2γV RT ln ) ps r
(2) (3)
where ps is the saturation pressure. The surface tension of a one-component liquid is usually determined as
γ)
(dG dA )
T,p
(4)
where dG is the change of the Gibbs potential due to the * Tel: 972-056-864642. E-mail:
[email protected]. (1) Adamson, A. W.; Gast, A. P. Physical Chemistry of Surfaces; John Wiley: New York, 1997. (2) Heimenz, P. C. Principles of Colloid and Surface Chemistry; Marcel Dekker: New York, 1986. (3) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, 1989.
change of the surface area, dA. There is a vast literature on physical chemistry of surfaces, which cannot be quoted here, and we shall refer mostly to the fundamental treatise of Adamson.1 Equation 2 follows from the assumption that at equilibrium the work of gas compression, if the surface is virtually displaced, is equal to the decrease of free surface energy of the bubble surface:1
∆p4πr2 dr ) 8πrγ dr
(5)
Our intension to revise the classical approach arises from the conflict between the classical equations and relationships between the geometric form and saturation pressure, derived in the present paper on the basis of thermodynamic and molecular approaches. The quantitative divergence between the two groups of equations is not great for characteristic sizes of about 1 µm and could be probably attributed to the simplifications adopted in the molecular model, but the conceptual diversity is profound: the Laplace and Kelvin equations contain the surface tension as a parameter, whereas developed equations include the difference of the potential energies of a nanoparticle and a semi-infinite planar body with respect to the gas phase, which is equal to the difference of energies of autoadsorption on surfaces of bodies. It is quite natural that the first suspicion falls on the new equations; but in our opinion, this is not the case and the defect exists in the fundamental equations. The analysis of condensation in capillaries with radii less than 10 nm confirms the validity of the new approach. The remainder of our paper is organized as follows. In the next section, we present the thermodynamic derivation of the relationship between the saturation pressure, on one hand, and the curvature and dimension of a body, on the other hand. Parameters of the relationship are
10.1021/la0267490 CCC: $25.00 © 2003 American Chemical Society Published on Web 03/20/2003
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discussed in section 3. The following three sections are devoted to the molecular model of interactions with the simplest bodies: a sphere, a spherical shell, and two converging spheres. In the final three sections, the classical and new approaches are compared against experiments and the discrepancies are discussed. 2. General Thermodynamic Approach Derivation of the Fundamental Equation. Let us compare the semi-infinite slab and a small nanoparticle of the same chemical nature. The terms “semi-infinite slab” and “nanoparticle” are used here to denote the macroscopic body with a flat surface and the body of small dimensions with curved borders, respectively. For either of the bodies, one may use the Clausius-Clapeyron equation,4 which relates the saturation pressure, p, the temperature, T, and the enthalpy of the phase transition, ∆H:
d ln pslab ∆Hslab ) dT RT2
(6)
and
d ln pnano ∆Hnano ) dT RT2
(7)
Here ∆Hslab and ∆Hnano are positive. Subtracting eq 6 from eq7 gives
pnano d ln pslab ∆Hn-s ) dT RT2
(8)
where ∆Hn-s is the difference of the enthalpies of the phase transition in the cases of the nanoparticle and slab. ∆Hn-s takes its origin only from the geometry of bodies, all other possible variables such as temperature and chemical nature being identical. Due to the operation of subtraction, kinetic energies of bodies and all possible contributions to ∆Hn-s, except those that depend on the geometry, annihilate and ∆Hn-s is equal to the difference of the changes of potential energies, U, of two bodies on the phase transition:
∆Hn-s ) ∆Un-s
(9)
In general, the forces between molecules may be loosely classified into three categories:5 pure electrostatic in origin arising from the Coulomb forces between charges, polarization forces that arise from induced dipole moments, and quantum mechanical covalent forces. The latter, being local by nature, do not depend on the form and size of the body, and for covalent solids ∆Un-s ) 0, whereas for the first two categories, a strong geometric effect for small bodies is to be expected. We shall restrict ourselves only to consideration of bodies formed by the van der Waals forces, and calculations will be carried out for the situation when additive dispersion forces are responsible for the body properties. Nevertheless, all conclusions will remain valid also for the Coulomb forces. If attraction is determined mainly by the London interactions, ∆Hn-s is (4) Moelwyn-Hughes, E. A. Physical Chemistry, 2nd ed.; Pergamon Press: Oxford, 1961. (5) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: London, 1995.
independent of temperature and one can integrate eq 8:
pnano ∆Hn-s ln )+ const pslab RT
(10)
Since pressures are equal when ∆Hn-s ) 0, const ) 0 and we obtain the final simple relationship:
∆Hn-s pnano )ln pslab RT
(11)
which relates the equilibrium pressures to the difference of energies of phase transition of two bodies. Concept of Autoadsorption. Equation 11 is the generalization of the condensation equation6-8 suggested for calculating monolayer adsorption by means of comparison of adsorption on nanoparticles or in porous media with the adsorption properties of the reference nonporous body:
( ) p ln ref p
a
)
∆* RT
(12)
where p and pref are the equilibrium pressures of adsorbate on surfaces of the body of interest and the reference body at a given amount adsorbed, a, and ∆* is the difference of the potential energies of adsorption on the surfaces in the Henry limit. For any body, we may consider the saturation pressure as the equilibrium pressure corresponding to autoadsorption of a molecule on the surface of its own condensed phase. The concept of autoadsorption was widely exploited in the Russian scientific literature in the 1960s and 1970s.9-17 Shimulis14-17 applied the Hill equation for delocalized adsorption to autoadsorption of vapors of noble gases on the surfaces of their own condensed phases. Rusanov and his group10-13 investigated surface properties of water and organic liquids and autoadsorption of vapors on the surface layers. For autoadsorption, pref in eq 12 is equal to the equilibrium pressure for the macroscopic semiinfinite slab pslab; the subscript a in this equation may be omitted, since the chemical nature of the surface remains invariable. Equations 9, 11, and 12 may be simultaneously valid only if
∆Hn-s ≡ ∆Un-s ≡ -∆*
(13)
Due to the particular significance of eq 13, we shall examine it in the framework of the molecular approach. For definiteness, look at the condensation of a gas. The energy of an atom in a gas phase at rest an infinite distance from the body is taken to be zero. As the thermodynamic results do not depend on the way the change occurs but only upon the initial and final states, we can imagine that the gas condenses in two stages: on the first stage, (6) Dobruskin, V. Kh. Langmuir 2003, 19, 2134-2146. (7) Dobruskin, V. Kh. Langmuir 1998, 14, 3847-3857. (8) Dobruskin, V. Kh. Carbon 2001, 39, 583-591. (9) Vitol, E. N.; Orlova, K. B. Kolloidn. Zh. 1993, 55, 11-15. (10) Rusanov, A. I.; Kochurova, N. N.; Anis’kova, M. A. Vestn. Leningr. Univ., Fiz., Khim. 1976, 2, 75-79. (11) Khabarov, V. N.; Rusanov, A. I.; Kochurova, N. N. Kolloidn. Zh. 1976, 38, 120. (12) Khabarov, V. N.; Rusanov, A. I.; Kochurova, N. N. Kolloidn. Zh. 1975, 37, 407. (13) Khabarov, V. N.; Rusanov, A. I.; Kochurova, N. N. Kolloidn. Zh. 1975, 37, 92-98. (14) Shimulis, V. I.; Riofrio, V. Zh. Fiz. Khim. 1971, 45, 783-786. (15) Riofrio, V.; Shimulis, V. I. Zh. Fiz. Khim. 1970, 44, 1779-1782. (16) Shimulis, V. I. Zh. Fiz. Khim. 1968, 42, 2881-2886. (17) Shimulis, V. I. Zh. Fiz. Khim. 1967, 41, 376-383.
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In the theory of adsorption, calculating ∆* is based mainly on the summation of the Lennard-Jones (LJ) 12-6 potential function for the interaction energy, , between single atomic or molecular species:18-21
) 4/aa
Figure 1. Effect of the surface curvature on the energies of autoadsorption. When atom is placed in a point A out of the sphere, its separation AD from all points of the convex shell surface is greater than its separation AC from the tangent surface P, except the tangent point T; correspondently, the energy of adsorption of atom A on the planar surface exceeds that on the convex surface. When atom is placed in a point B inside the shell and interacts with a concave surface, the result is antipodal.
adsorption occurs onto the liquid surface, and on the second stage, molecules adsorbed are transferred into the liquid volume by diffusion, the total effect being
H ) auto + Henter
(14)
For two bodies that differ only by the geometry, the energetic effects of the second stages, Henter, are identical, since they do not depend on the curvature, whereas the effects of the first stages, /auto, do depend on the curvature (see below). When calculating ∆Hn-s, the Henter values of the two bodies cancel each other and we may see that ∆Hn-s does equal the difference of the potential energies of autoadsorption on the surfaces of the bodies. Finally, we obtain
ln
/ pnano ∆n-s ) pslab RT
(15)
This relationship allows taking advantage of well-known methods of the adsorption theory for calculating ∆Hn-s ≡ -∆*. Let us show that /auto depends on the body geometry. Consider a segment of a sphere, a tangent surface to the sphere, and molecules outside and inside of the sphere (Figure 1) and compare /auto for the curved shells and planar tangent surface. The autoadsorption energy is the sum of the interactions of the adsorbate with all molecules of the body, but only contributions of the molecule of the body placed in the tangency point are identical for planar and curved surfaces. For other points, the energy of interaction with the curved surfaces differs from that with a slab due to alterations of separations between interacting species: compared to the planar surface, separations decrease and energies increase in the case of the concave surface, whereas for a convex surface, the effect is antipodal. 3. Parameters of the Fundamental Equation To make use of eq 15, one should know pslab and estimate ∆* for two bodies, which differ only with respect to their geometry. For liquids,
pslab ≡ pref ≡ ps
(16)
[( ) ( ) ] σaa s
12
-
σaa s
6
(17)
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. In the further calculations, the value of σaa will be taken as a scale parameter, and we shall put σaa ) 1 in eq 17. The interactions of the adsorbate with a condensed phase should be calculated by the summation of terms given by eq 17 over all atoms of the phase, but we shall use the simpler operation of integration. Transferring the energy sum to an integral artificially “smears out” the discrete and bumpy structure into a continuum and may lead to significant errors in the absolute energy values. For example, the successful application of the integration in the case of the graphite layer is only due to the extremely dense packing of carbons caused by strong covalent bonds.18-21 Since our final purpose is estimation of not the absolute value but the difference of adsorption energies of two bodies, for example, a slab and a sphere, we assume that integration, being applied to calculating adsorption energies of both bodies, introduces similar errors, which cancel each other in the final results. Besides, other possible inaccuracies (the retardation effect, for instance) also vanish or diminish in the required value of ∆*. In the case of liquids, there is a serious additional problem. Just as for calculating the internal and surface energies of liquids,1,22 accurate calculated potential energies of autoadsorption should be based on the radial distribution function taking into account the peculiarity of the liquid state. Nevertheless, again the simplified approach, being applied, for example, to a liquid in a macroscopic container and to a liquid droplet, introduces the same inaccuracy into both values of ∆*, which afresh cancel each other in the desired value of ∆*, which finally will look more respectable than each of the terms deserves to. The value of ∆* is calculated as
∆* ) * - /slab
(18)
where * and /slab are the energies of gas-body interactions. It is convenient to transform eq 18,
(
∆* ) /slab
)
* - 1 ≡ /slab(f - 1) /slab
(19)
and take /slab as a scale energy parameter. Then, the geometry factor, (f - 1), which takes into account deviations due to forms and sizes, is just equal to the reduced difference of energies of interactions with two bodies:
∆* ) ∆/red ) f - 1 / slab
(20)
Since we shall calculate only reduced energy values, the subscript “red” will be further omitted. In the case of (18) Crowell, A. D. J. Chem. Phys. 1954, 22, 1397-1399. (19) Everett, D. H.; Powl, J. C. J. Chem. Soc., Faraday Trans. 1976, 72, 619. (20) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon Press: Oxford, 1974.
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interactions with the planar semi-infinite slab, the LJ potential leads to the well-known 9-3 potential function19
slab )
( ) [ ( ) ( )] 3 2 19 1 /slab 15 z z x10
3
(21)
be given below (section 9). In fact, Es is the excess energy of the surface region with respect to the bulk volume, which numerically coincides with the excess energy gained by a molecule on falling from an infinite distance to the surface. Combining eqs 15, 16, 19, 20, and 23, we obtain the relationship that is suitable for practical calculations:
where z is the reduced separation of the molecule from the surface layer, and /slab is the energy minimum, that is, the well depth, which determines the adsorption energy on the slab surface:19
/slab )
2x10 πn/aa 9
(22)
Let us now come back to eq 19. Although we believe that approximations and simplifications involved annul each other when we do subtraction, the same approximations do not allow accurate estimating of the absolute value of /slab. For this reason, we shall use the molecular approach only for calculating the geometry factor (f -1) by eq 20 and /slab given by eq 22 will be really used only as the energy scale parameter, whereas the absolute values of /slab in eq 19 will be substituted by experimental values taken from the literature. We believe that such an approach is a way to bypass obstacles that are insuperable at present. The Potential Energy of Adsorption on the Planar Surface /slab. One may consider a molecule in the surface region as being in a state intermediate between that in the vapor phase and in the interior. As long ago as 1886, Stefan23 assumed that the energy gained by a molecule on being transferred from the bulk liquid to its surface must be equal to the energy gained during a transfer from the position on the surface to the vapor. The same conclusion follows from the approximate approach proposed by Skapski.24 He has shown that some properties of liquids can be calculated directly from the heat vaporization at absolute zero and the configuration of nearest neighbors. In the bulk of a close-packed liquid and in the surface layer of this liquid, each of the molecules is surrounded by 12 and 9 nearest neighbors, respectively. A molecule adsorbed on the surface of close-packed spheres at the position of the minimum potential energy (without entering the surface layer!) has three nearest neighbors. Note that the emphasis is placed on the fact that a molecule of adsorbate does not enter the surface layer. For liquids, the condensation starts on the surface and then the adsorbate transfers into the volume by diffusion; molecules entering into the surface layer and diffusion into the volume do not relate to adsorption. One may see that the number of nearest neighbors lost by a molecule on moving from the volume to the surface is equal to that on desorption into the gas phase (3 ) 3). Hence, the energy changes for these processes are identical. A difference between the energy of molecules in the bulk liquid and that on the surface of the liquid is usually interpreted as the total surface energy, Es,1 and we shall accept Es as the change of the potential energy on adsorption on the slab surface,
Es ) /slab
(23)
Another confirmation of the validity of this relation will (21) Avgul, N. N.; Kiselev, A. V.; Poshkus, D. P. Adsorption of Gases and Vapors on Homogeneous Surfaces; Khimia: Moscow, 1975 (in Russian). (22) Atkins, P. W. Physical Chemistry; Mir: Moscow, 1980 (Russian edition).
pnano Es(f - 1) Es∆* ln ≡ ) ps RT RT
(24)
The total surface energy is nearly temperatureindependent in the wide range not too close to the critical temperature, Tc, and eventually drops to zero at Tc. The total surface enthalpy, Hs ) Gs + TSs, and Es are practically indistinguishable due to the trifling pv-term. The value of Es is calculated as follows:1,4,23
dγ Es ) γ - T dT
(25)
and is equal to the surface tension extrapolated to 0 K. It is worth drawing attention to the remarks of Benson and Yun25 and Johnson.26 They point out that there is a confusing lack of uniformity in the use of the terms total surface energy, surface free energy, and surface tension by various authors and “a general lack of rigor in discussing surface energetics has contributed immeasurable to the confusion.”26 4. Energy of Adsorption on the Sphere Surface There are numerous publications on calculation of the Hamaker constant1,5,27-29 based on the London or the Lifshitz theories,30 but these papers consider only the attractive term and the final results contain, to some extent, a vague parameter of equilibrium separation between bodies. The expression taking into account both attractive and repulsive contributions was obtain by Henderson and co-workers31 for the interaction, sp, between a molecule and a sphere of reduced radius R, the molecule being a distance z from the sphere:
{
16πnaaR3 1 + 3 ((R + z)2 - R2)3 15(R + z)6 + 63(R + z)4R2 + 45(R + z)2R4 + 5R6
sp )
15[(R + z)2 - R2]9
}
(26) Here n is the number of interacting centers per unit volume of solid. The trivial alterations are introduced in the original equation of Henderson and co-workers31 to adjust it to the definitions of z and n accepted in the adsorption theory. We recall that in all our calculations the energy and distance scales are taken so that the energy of adsorption on a planar body /slab and the LJ parameter (23) Moelwyn-Hughes, E. A. Physical Chemistry; Inostrannaya Literatura: Moscow, 1962; Vol. 2, p 815 (Russian edition). (24) Skapski, A. S. J. Chem. Phys. 1948, 16, 389-393. (25) Benson, G. C.; Yun K. S. In The Solid-Gas Interface; Flood, E. A., Ed.; Marcel Dekker: New York, 1967; Vol. 1, pp 203-264. (26) Johnson, R. E. J. Am. Chem. Soc. 1959, 63, 1655-1658. (27) Witte, N. S. J. Chem. Phys. 1993, 99, 8168-8182. (28) Arunachalam, V.; Marlow, W. H.; Lu, J. X. Phys. Rev. E 1998, 58, 3451-3457. (29) Chen, J.; Anandarajah, A. J. Colloid Interface Sci. 1996, 180, 519-523. (30) Lifshitz, E. M. Soviet Phys. JETP (English Transl.) 1956, 2, 73-83. (31) Henderson, D.; Duh, D.; Chu, H.; Wasan, D. J. Colloid Interface Sci. 1997, 185, 265-268.
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Figure 2. The energies of interaction between an atom and spheres sp as a function of reduced separation z for spheres with radii of 1.5, 2, 3, 5, 10, and 100. Positive adsorption energies -/sp are equal to the absolute values of well depths, which occur at z ≈ 0.86. Dashed lines correspond to /sp ) -0.5 and /sp ) -1. For clusters with R > 2, |/sp| > 0.5.
Figure 3. The change of energy of interactions between the molecule and the sphere with respect to the planar body ∆* as a function of sphere radii.
σaa are equal to unities, that is, all energies and distances are dimensionless. With /slab (eq 22) taken as a scale energy parameter, eq 26 becomes
sp )
24 I(R, z) x10
(27)
where
{
1 + ((R + z)2 - R2)3 15(R + z)6 + 63(R + z)4R2 + 45(R + z)2R4 + 5R6
I(R, z) ) R3 -
15[(R + z)2 - R2]9
Figure 4. Equilibrium relative pressures of clusters of water (stars) and benzene (diamonds) as a function of their radii at T ) 298 K.
}
(28) The well depth for interactions with the sphere, /sp, is found from eq 27 by numerical calculation. The asterisk (*) refers to the parameters of the well depth. The energies of interactions between a molecule and spheres as a function of reduced separation z are depicted in Figure 2. Energies sp reach minima /sp at z* ≈ 0.86. When radii of spheres increase, the well depths are amplified, approaching the well depth of the slab. The sphere with R ) 1.5 corresponds to the cluster with 3 atoms along its diameter, which models the dense packing of 13 spherical atoms in the space. In this model, there are 12 atoms forming the surface layer gathered round the central atom. Generally, the number of atoms around the central one for the dense packing, N, is given as N ) 10n2 + 2,32 where n is the number of surrounding layers. Hence, the sphere with R ) 2.5 accommodates 5 atoms along its diameters (two layers around the central atom) and contains 42 + 1 ) 43 atoms. The physical sizes of clusters may be estimated proceeding from σaa ≈ 0.4 nm: diameters of clusters with R ) 1.5, 2.5, and 10 are equal to 1.2, 2, and 4 nm, respectively. One can discuss the energetic processes in terms of either the negative potential energy or the positive adsorption energies, although such a duality may sometimes lead to misunderstanding. By convention, the adsorption energy is positive and taken upward from the bottom of the well depth. Of greater interest than /sp is the change of reduced potential energies with respect to (32) Wells, A. F. Structural Inorganic Chemistry; Mir: Moscow, 1987; Vol. 1, p 181 (Russian edition).
Table 1. Effect of Radii on Equilibrium Pressures supersaturation, p/ps radius of cluster
∆*
water at 275 K
ethanol at 273 K
benzene at 273 K
6.81 4.99 3.97 3.34 2.92 2.63 4.2
4.29 3.38 2.85 2.49 2.25 2.08 2.3
18.36 11.45 8.10 6.23 5.08 4.3
R ) 1.5 0.5889 R ) 2.0 0.4934 R ) 2.5 0.4234 R ) 3.0 0.3704 R ) 3.5 0.3289 R ) 4.0 0.2957 experimental supersaturation,a p/ps a
Reference 1.
the slab, ∆/sp, which determines the geometry factor (eq 20). This value is given as follows:
∆/sp ≡ f - 1 )
24 I(R, z*) - 1 x10
(29)
The plot of ∆/sp as a function of radii is shown in Figure 3. Sometimes we shall drop the subscript sp when it does not influence the understanding. In terms of adsorption energies, the values of ∆/sp for a convex sphere are negative. Figure 4 shows equilibrium pressures of water and benzene at T ) 298 K as functions of nanoparticle radii, which are calculated by eq 24 proceeding from known values of ∆/sp and Es. The calculated supersaturation for the finest clusters of water, benzene, and ethyl alcohol with R in the range 1.5-4.0 are given in Table 1. A comparison of these values with the experimental supersaturations of critical nuclei1 shows that radii of critical nuclei fall in the range R ≈ 2.5-3.5. To compare derived relationships with the classical equations, we shall approximate eq 29 by the simplest
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Langmuir, Vol. 19, No. 9, 2003 4009
Figure 5. Effect of shell thickness h on the energy of adsorption on the internal surface of the spherical shell. The radius of the hollow is taken to be 10. For h > 5, the energy of adsorption is practically independent of thickness.
functions,
∆* )
a Rn
(30)
where a is the empirical coefficient. For example, a ) -1.4477 for R ≈ 1000. 5. Energy of Adsorption on the Surface of the Spherical Shell We shall consider a spherical shell, with the external radius R and the shell thickness h, and two positions of the atom with respect to the shell: (i) outside the spherical shell and (ii) inside the internal hollow. The internal radius of the shell is equal to R - h. When the atom is outside the shell, the energy of interactions with the shell, out, is equal to the energy of interactions with the dense sphere of radius R with the deduction of interactions with missing masses in the hollow, both terms being given by eq 27,
out )
24 [I(R, z) - I(R - h, z)] x10
(31)
If we consider only energies of adsorption at equilibrium separation z*, then the energy of interactions of a molecule inside the hollow with the shell, /in, for large radii (R . h) due to the symmetry is found as
/in ) -/out
(32)
The effect of shell thickness on the energy of interactions is plotted in Figure 5, which shows that if h exceeds five molecular diameters, the adsorption energies become practically independent of wall thickness. This value of h may be considered as an approximate range of intermolecular forces. Hence, in the case of thick films (h > 5) with large R, energies of adsorption on the dense sphere and spherical shell are equivalent. According to Adamson,1 the wall thickness of a thin soap bubble is about 10 nm, that is, about 25 molecular diameters (h ) 25); hence, in this case, approximation leads to the nearly exact result. Analogously, the energies of adsorption on the surfaces of the sphere and hemisphere for large R are identical because of the negligible contribution of the second hemisphere; if one compares interactions with the droplet and with the hemispherical meniscus in the capillary of the same radius, the absolute values of energies of interactions will be equal to each other. 6. Adsorption in the Space between Two Spheres It has long been recognized that the adsorption energy in pores is enhanced due to the overlapping of the
Figure 6. Effect of the separation between the atom and the sphere with radius R on the energy of autoadsorption in the space between two approaching spheres. The plot corresponds to R ) 10, and the distance between sphere centers is S ) 22. When the atom is in the range of separations from 0.76 to 1.23, the total energy of interactions with both spheres exceeds the energy of interaction with the planar infinite body.
adsorption potential of pore walls. Polanyi33 was the first to put forward this idea, which underlies the theories of adsorption in porous media. This assumption was extended to adsorption in the space between flat nanoparticles, which may come together during a random walk, forming “vagabond micropores”.34 It is obvious that this phenomenon may take place also in the case of a random walk of spherical nanoparticles, and our objective now is to examine the processes in the space between such particles. Since the comparative probability of triple collisions is negligible, we shall consider only a double convergence. Let the distance between centers of two converging spheres be S, and the molecule is placed on the straight line passing through the centers. In this case, the reduced energy of adsorption in the space between converging spheres, sp-sp, is calculated as follows:
sp-sp(R, S, z) )
24 [I(R, z) + I(R, S - 2R - z)] (33) x10
where (S - 2R - z) is the distance between the atom and the surface of the second sphere. The plot of sp-sp(R, S, z) as a function of z is shown in Figure 6. The enhancement of adsorption energy in the space between spheres is similar to the same effect in pores or between colloid particles,19,35 and the space between spheres may be viewed as an open pore with curved walls. The potential well between two spheres has a maximum value when the distance between the atom and each of the spheres is equal to z* ≈ 0.858; for the smallest spheres, z* slightly exceeds 0.858. As a whole, the space around the converging spheres is energetically nonuniform with respect to the adsorption potential, which gains the maximum between particles. The maximum interaction energy in the space between / , is twice as large then for two converging spheres, sp-sp / a separate particle sp. The effect of the sphere radius on the maximum well depth in this space is shown in Figure 7. Since for the cluster with R ) 1.5, |/sp| ) 0.4111 < 0.5, there are no positions in the space between such clusters / | > 1 and the interaction energy is everywhere |sp-sp (33) Polanyi, M. Trans. Faraday Soc. 1932, 28, 316. (34) Dobruskin, V. Kh. Langmuir 2002, 18, 4356-4361. (35) Wertheim, M. S.; Blum, L.; Bratko, D. In Micellar Solution and Microemulsions. Structure, Dynamics, and Statistical Thermodynamics; Chen, S., Rajagopalan, R., Eds; Springer-Verlag: New York, 1990; pp 99-110.
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(eq 20) is described by hypergeometric functions 2F1[R, β; γ; z]:
∆/cyl )
{ () [ () [
]
2 21 σ0 9 9 11 (R - 0.858) 27 π F1 , ;1; R2 2x10 288 r 2 2 2 2 1 σ0 3 3 5 (R - 0.858) F1 , ;1; - 1 (34) 3 r 2 22 R2
]}
which also may be approximated by a simple function like that given by eq 30:
Figure 7. Difference between the maximal autoadsorption energy in the space between converging spheres and the autoadsorption energy on the planar surface as a function of cluster radii. For clusters with R ) 1.5 and R ) 2, these differences are negative and close to zero, respectively; for clusters with R g 2.5, there are points in the space between converging spheres where the adsorption energy exceeds that for a planar surface.
where less than that for the planar surface. For R ) 2, / | ) 2|/sp| ) 1.0132 practically |/sp| ) 0.5066 and |sp-sp / coincides with slab. Only if a cluster consists of at least 5 molecules along its diameter (R g 2.5) is there a part of the space where the energy of adsorption exceeds /slab. For such clusters, molecules evaporated from the external periphery of either of the nuclei with R g 2.5 will tend to condense in the space between converging droplets even at p < ps. This process prevents droplet evaporation and, on the contrary, may result in droplet merging. Consequently, there is a critical diameter Rcrit ≈ 2.5 so that the clusters with R < Rcrit are unstable and incapable of growing at the saturation pressure, whereas clusters with R g Rcrit may grow due to adsorption in the interparticle space. This phenomenon probably determines the critical dimensions of nuclei and the conditions of their formation as was shown in section 4. Unlike the semirigorous kinetic treatment of Becker and Doring,1 the present approach does not involve rather arbitrary assumptions for estimating critical supersaturations. Of course, our consideration relates only to the simplest situation when T ) constant; it does not take into account adsorption (condensation) heat and the change of the numerical concentrations of particles. We also do not touch here the role of electrical charges on condensation, although this factor may be easily accounted for by introducing the additional term in ∆*. The adsorption mechanism of droplet growth was probably first suggested by Hage who considered droplet formation as a secondary adsorption process, independent of condensation.36 7. Comparison with Experiments Our aim is to compare pressures predicted by classical and new equations with known experimental data. Usually, experiments are carried out with drops or capillaries with r ≈ 1 µm and we shall take a ) 1.4477 for calculating ∆* by eq 30 in the case of interactions with a sphere. Apart from a sphere, we intend to discuss two examples of interactions between a molecule and a cylinder. This problem has been examined in our earlier publication.8 Just as in the case of spheres, interactions of molecules placed inside and outside the cylinder are opposite by sign. The reduced difference of energies of interactions (36) Van der Hage, J. C. H. J. Colloid Interface Sci. 1983, 91, 384.
∆/cyl )
0.983 R
(35)
For large R, a ) 0.983. Note that R is reduced radius; hence, the physical radius r is equal to r ) σaaR and eq 24 takes the form
ps aσaaEs RT ln ) p r
(36)
The Kelvin eq 3 is compared with eq 36 in Table 2 for a number of liquids and temperatures. In fact, it is enough to compare 2γV from eq 3 with aσaaEs from eq 36. For nitrogen, water, benzene, and ethyl alcohol, values of γ and Es are reported by Adamson,1 whereas for mercury and carbon tetrachloride, values are taken from MoelwynHughes.4 For toluene37 and cyclohexane,38 Es is calculated proceeding from known γ at different temperatures (eq 25). The procedure of calculation is given by Adamson and Moelwyn-Hughes.1,4 σaa for all substances, except toluene, is taken from the literature;39,40 for toluene, this value is obtained from atomic volumes.39 Table 2 shows that for water and mercury, which have been used in experiments designated for verification of classical equations, both approaches give close results. The same conclusion may be drawn from the analysis of experiments of Fisher and Israelachvili who have measured the mean radius of curvature of cyclohexane condensed between crossed mica cylinders.41 The measured radii were within (6% of those predicted by the Kelvin equation. If we make use of eq 36 for interactions with the cylinder (eq 35), the radii calculated fall into the same range of (6%. For other organic liquids, one can see deviations between equations: the new equation predicts larger lowering of pressures. Benzene, ethyl alcohol, and carbon tetrachloride are used here as examples for theoretical and not for experimental comparisons, since the corresponding data were not found in the literature. The most famous attempts to verify the Kelvin equation have been undertaken over 44 years (1928-1972) by Shereshefsky and co-workers, who reported that, in the case of toluene, the equilibrium radii were 2.9-5.442 and even 10-30 times43 greater than the values calculated from the Kelvin equation. Shereshefsky’s publications were subject to the rigorous criticism of Melrose.44 In our (37) Korosi, G.; Kovats, E. J. Chem. Eng. Data 1981, 26, 323. (38) Chemical Handbook; Nikolski, B. P., Ed.; Khimia: Leningrad, 1962 (in Russian). (39) Reid, R. C.; Sherwood, T. K. The properties of Gases and Liquids; McGraw-Hill: New York, 1958; pp 270-271. (40) Hirschfelder, J. O.; Curtiss, C. F.; Bird, K. B. Molecular theory of Gases and Liquids; John Wiley: New York, 1954; pp 1110-1112. (41) Fisher, L. R.; Israelachvili, J. N. J. Colloid Inteface Sci. 1981, 80, 528. (42) Folman, M.; Shereshefsky, J. L. J. J. Phys. Chem. 1955, 59, 607-610. (43) Coleburn, N. L.; Shereshefsky, J. L. J. Colloid Interface Sci. 1972, 38, 84-90. (44) Melrose, J. C. Langmuir 1989, 5, 293-295.
Saturation Pressures and Dimensions
Langmuir, Vol. 19, No. 9, 2003 4011
Table 2. Comparison between Classical and New Equations M
σaa, cm × 108
T, K
F, g cm-3
V, cm3 mol-1
γ, mJ m-2
Es, J mol-1
H2O Hg
18 200.6
2.649 2.898
C6H12a C6H6 C6H5 CH3b C2H5OH CCl4
84.16 78.11 92.14 46.07 153.82
6.093 5.27 5.79 4.455 5.881
293 273 393 293 293 293 293 273 393 75
1.0 13.595 13.352 0.779 0.879 0.867 0.789 1.6326 1.3902 0.808
18 14.755 15.024 108.04 88.86 106.4 58.39 94.22 110.65 34.67
72.25 463 448 25.16 28.88 28.52 22.75 27.99 14.33 9.71
7451 30081 31897 9802 11248 11968 5609 12298 12298 2449
substance
N2 a
28.01 b
3.681
2γV × 104 J cm mol-1
aEsσaa × 104 J cm mol-1
2.619 13.663 13.462 5.391 5.133 6.069 2.657 5.274 3.171
2.859 12.620 13.382 5.871c 8.559 10.03 3.617 10.471 10.471
c
Cyclohexane. Toluene. Interactions with the cylinder (eq 35).
opinion, interpretations of results given by both Shereshefsky and Melrose are incorrect. The main reason for “some startling discrepancies”1 in Shereshefsky’s experiments originates from the erroneous comparison between the radius of curvature calculated by the Kelvin equation and the measured “radius” of capillary r. It is only for the spherical meniscus in the cylindrical capillary that these values are equivalent, whereas Shereshefsky applied coneshaped capillaries. As a result, no firm conclusion can be drawn from his data. Adamson points out that the situation with the experimental verification of the Kelvin equation, which is the logical corollary of Laplace’s equation, is still conflicting, whereas Everett, Haynes, and McElroy45 in their conclusion draw attention to the fact that the Kelvin equation still lacks experimental verification! Sometimes the practice of application of the Kelvin equation might be rather interpreted as an illustration that there are some problems associated with this equation. For example, it is reported that the Kelvin equation correctly describes condensation in a slit with a width of 0.8 nm.5 Note that the pore size is taken as the distance between centers of atoms forming a pore wall and, since a usual atom size is ≈0.4 nm, a pore with a diameter of 0.8 nm may accommodate only one molecule. It is self-evident that there are no physical reasons to attribute the notion of surface curvature to one or two molecules! Considering data related to drops, one should bear in mind that (i) the system of small drops and vapors cannot be strictly in equilibrium with liquid vapors at constant temperature and cannot be in equilibrium with respect to appreciable interchange of mass, either between any of the drops or between any one drop and the vapor, and (ii) the inevitable polydispersity of drops has a strong effect on the pressure and prevents the interpretation of small quantitative distinctions in favor of either of the equations due to the proximity of their numerical coefficients. As a whole, experimental data taken from the literature and related to drops and capillaries with radii about 1 µm do not allow drawing firm unambiguous conclusions in favor of either classical or new equations.
models for (i) the cylindrical meniscus
ps γV RT ln ) pa r
(37)
and (ii) the spherical meniscus
ps 2γV RT ln ) pd r
(38)
lead to a specific prediction, which can be tested against experimental data:20
() pa ps
2
)
pd ps
(39)
Equation 39 is independent of the size of the capillary and predicts the divergence of hysteresis branches. No porous system conforms to this model. Although many artificial assumptions were introduced to save the Kelvin model (“ink-bottle” pores, impurities, bundle model, irreversible change in the pore structure, and so forth),1,20,46 a quantitative explanation of peculiarities of hysteresis loops has yet not been given. It was generally accepted to use only the desorption branch for calculating mesopore radii by eq 38 and, in fact, ignore the adsorption branch. Besides, it has long been recognized that eq 38, being applied to the desorption branch, underestimates the radii of mesopores, and the corrected equation was suggested, which adds the film thickness to the Kelvin results. Our objective now is to examine the condensation in cylindrical capillaries in the framework of the developed model. With a practically zero error, we can take z* ) 0.858 in both eqs 29 and 34. From eq 24, we obtain the following expressions for calculating equilibrium pressures for adsorption (pa) and desorption (pd) branches of isotherms:
pa Es∆/cyl ln ) ps RT
(40)
pd (24/x10)Es[I(R, 0.858) - 1] ln ) ps RT
(41)
8. Adsorption in Mesopores The general characteristic of adsorption in mesopores is the hysteresis loop. Consider a model of adsorption in individual infinite cylindrical mesopores with both ends open.1,20,46 In this case, capillary condensation (adsorption) starts with a cylindrical meniscus at the pressure pa, but capillary evaporation (desorption) occurs with a hemispherical meniscus at the pressure pd.20,46 The Kelvin (45) Everett, D. H.; Haynes, J. M.; McElroy, P. J. Sci. Prog. (Northwood, U.K.) 1971, 59, 279. (46) 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.
and
Equilibrium adsorption (dashed lines) and desorption (solid lines) pressures of nitrogen as functions of R are depicted in Figure 8. Similar curves are obtained also for other adsorbates. The left ends of the adsorption and desorption branches converge at p/ps ) 0.3-0.35, 0.18, and 0.36-0.40 for nitrogen, benzene, and water, respec-
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Langmuir, Vol. 19, No. 9, 2003
Figure 8. Effect of capillary radii on the equilibrium nitrogen pressures at 75 K for the adsorption (dashed line) and desorption (solid line) branches of isotherms. Filling of the capillary with R ) 10 takes place at p/ps ) 0.737, whereas emptying (desorption) occurs at p/ps ) 0.593. Both branches of hysteresis loops converge at p/ps ≈ 0.3-0.35.
Dobruskin
Figure 10. Schematic diagram of autoadsorption on the flat surface of a macroscopic liquid. ABDC and CDFE are the initial volumes of the bulk liquid and the surface region, respectively. When an adsorption layer, EKLF, is formed on the liquid surface, an equal quantity of liquid, CGHD, transfers into the bulk liquid to keep the thickness of the surface region, CE ) GK, invariable. There is a one-to-one correspondence between adsorption on the surface and the transport of molecules from the surface region into the bulk volume.
Figure 9. Widths of the hysteresis loop, pa/ps - pd/ps, for nitrogen at 75 K as a function of reduced pore radii. Both branches converge at R ) 2.4 when pa/ps - pd/ps ) 0; the maximum of the width equals 0.151 at R ) 7.39.
tively, in excellent agreement with experimental pressures, 0.35,46 0.17,47 and 0.40;48 whereas for the right ends of the curves there is a small difference ∆ ) (pa/ps - pd/ps) at p f ps. Due to the peculiarity of the adsorption experiment, the terminal adsorption point is the first point of desorption and both branches coincide in this point, but ∆ makes its appearance in the occurrence of the plateau because desorption starts only after lowering the pressure below pd. The plateau length depends on the terminal adsorption point, adsorbate, and pore size distribution. The width of the hysteresis loop, pa/ps - pd/ ps, reaches the maximum at the intermediate pore radius (Figure 9). For nitrogen, water, and benzene, the maximums of the widths fall on reduced radii 7.39, 6.68, and 7.98, respectively. As a whole, one may see that the new model, unlike the Kelvin equation, predicts real effects, which correspond to the real observations: the convergence of branches at low pressures, the maximum of loop width at the intermediate radius, and the presence of a plateau at p f p s. 9. Are the Fundamental Equations Really Fair? The surface tension γ determined by eq 4 may be measured in the hypothetical experiment as the work done per unit area when a wall of the macroscopic containerparallelepiped with a one-component liquid moves, for example, to the right. On the other hand, a new surface does not spring up from nowhere and its growth in extent (47) Dubinin, M. M. Adsorption Properties and Pore Structure; Izdatel‘stvo VAHZ: Moscow, 1965 (in Russian). (48) Dubinin, M. M. Carbon 1980, 18, 355-364.
Figure 11. Schematic diagram of autoadsorption on the sphere surface. OD is the initial radius of the sphere, and AD is the thickness of the surface region. When the additional adsorption layer with the thickness DC is formed on the liquid surface, the liquid inside the shell ring, AB, transfers into the bulk to keep the thickness of surface region, AD ) BC, invariable. Since the volume of the shell ring DC exceeds that of AB, part of the adsorbed molecules, depending on the radius, stay in the surface region and the energetic effect of the transport in the liquid is equal to Esf, where f is the geometry factor.
takes place from the bulk liquid: if the surface area grows by dA, the number moles in the bulk liquid decreases by dn. The extension of the container leads to the lowering liquid level and decreasing hydrostatic pressure of the liquid.49 If the wall of the container continues moving to the right, the base of the container inevitably will begin exerting a strong influence on the liquid properties when the thickness of the bulk liquid arrives at several molecular diameters. In this case, the properties of the bulk liquid and the surface region change, and if one wants to prevent further changing properties and γ, one has to insert a liquid on extension. If one tried to measure γ of the spherical drop by means of the same hypothetical experiment, one would immediately come to the conflict: it is impossible to retain (49) Flood, E. A. In The Solid-Gas Interface; Flood, E. A., Ed.; Marcel Dekker: New York, 1967; Vol. 1, p 43.
Saturation Pressures and Dimensions
the form and extent of the drop with a constant mass. The work of drop extension, if extension takes place, certainly involves not only the change of free energy due to the growth of surface area but also that due to the increase of mass, and it is not equal to γ dA as accepted in stipulation 5. The radius of droplets may be really changed only due to condensation or evaporation. Let us consider the formation of additional layers on the surface of the liquid due to adsorption (condensation). We start from the flat surface (Figure 10). Since the surface area of the planar surface and the thickness of the surface region hold invariable on adsorption, the adsorbed mass is equal to the mass transferred from the surface region in the bulk volume. In the course of transferring, potential energies of molecules diminish from those on the surface to those in the bulk volume. If we add the kinetic contributions, which are the same for both regions, to the change of potential energy, we shall see that the energetic change in a liquid on adsorption is equal to the change of total surface energy. The latter just follows from the definition of Es as the excess energy of the surface region1 (see section 3). Hence, there is a one-to-one correspondence between Es and /slab (eq 23).
Langmuir, Vol. 19, No. 9, 2003 4013
In the case of adsorption on the curved convex surface (Figure 11), the surface area increases and, depending on the radii, part of the molecules stay in the surface region with the elevated energy, diminishing the net energetic effect to Esf (eq 20), where f < 1 is the true geometry factor; the unity in eq 20 appears due to our intention to calculate ∆*. In the case of adsorption on the concave surface, the situation is antipodal: the surface area contracts and more molecules transfer into the volume than adsorb on the surface; hence, the net energetic effect exceeds that for the planar and convex surfaces. It follows from the foregoing that the surface tension does not take into account all changes associated with alterations of surface areas, and it is this fact that underlies the probable defect in the classical equations. Acknowledgment. The author is grateful to Dr. Yu. Rozenfeld from Micro Tag Temed Ltd. for very fruitful and helpful discussions. LA0267490
