J. Phys. Chem. B 2001, 105, 11615-11618
11615
Tolman’s δ, Surface Curvature, Compressibility Effects, and the Free Energy of Drops† Lawrence S. Bartell Department of Chemistry, UniVersity of Michigan, Ann Arbor, Michigan 48109 ReceiVed: March 19, 2001; In Final Form: June 25, 2001
An extension of a treatment by Laaksonen and McGraw leads to an analytical expression for the variation of Tolman’s δ with drop size. According to this approach, the change in sign of δ as a drop becomes large is a consequence of compressibility. An expression for the reversible work to produce a small drop from bulk liquid is given that corrects conventional expressions for effects of compressibility and Tolman’s δ. It is shown, however, that when this expression is applied to the formation of a drop from the supersaturated vapor, the reversible work, Wrev′, in excess of volume work no longer explicitly involves the Tolman δ or the compressibility, in agreement with treatments by Gibbs and by Debenedetti and Reiss. Nevertheless, if the expression for Wrev′ is to be related to directly measurable quantities, compressibility and δ still play a role.
Introduction
Treatment
A half-century ago Tolman extended an idea of Gibbs and showed that if the radius of the surface of tension of a drop does not coincide with the equimolar radius, the surface tension must vary with drop size.1 Tolman derived a relation that has become known as the Gibbs-Tolman-Koenig-Buff differential equation2
We take as a basis the observation23a that the Kelvin equation relating the vapor pressure Pv of a nearly incompressible spherical drop to the drop radius can be quite accurately expressed as
(
)
2[δ/Rs + (δ/Rs)2 + (δ/Rs)3/3] dRs dσ ) σ 1 + 2[δ/Rs + (δ/Rs)2 + (δ/Rs)3/3] Rs
(1)
where σ is the surface tension, Rs is the radius of the surface of tension, and the parameter δ, known as the “Tolman δ”, is the separation of the equimolar surface from the surface of tension, or δ ) Re - Rs. Moreover, Tolman argued that the two surfaces must, in general, be distinct from each other, and, in a separate paper,3 he made rough estimates of the distance between them that might be expected. He assumed, provisionally, that δ can be taken as a constant in the region of interest and, solving eq 1 on this basis, arrived at his famous equation
σ ) σ∞/(1 + 2δ/Rs)
(2)
In a companion paper, Kirkwood and Buff4 developed a general statistical mechanical theory of interfacial phenomena and confirmed the essential validity of Tolman’s approach. Since the publication of Tolman’s paper, a considerable literature has ensued,5-30 either demonstrating the consequences of this phenomenon in the kinetics of nucleation and other phenomena or attacking the problem via statistical thermodynamics. Quite a few papers have proposed methods to derive the value of δ, many assuming with Tolman that δ is independent of drop size. On the other hand, several applications of statistical thermodynamics have indicated that δ depends strongly on drop size.23-30 Since the results of these treatments are based on rather complex numerical calculations, it would be helpful to be able to express δ(Rs) analytically in terms of physical properties of the system. The present paper is a step in that direction. †
Part of the special issue “Howard Reiss Festschrift”.
∆µ ) 2σ∞V h l∆/Re + (Pv - Pv∆)V h l∆
(3)
where ∆µ is the difference in chemical potential between the liquid drop (at unstable equilibrium with the vapor at pressure PV) and the bulk liquid (at the bulk equilibrium vapor pressure Pv∆), σ∞ is the limiting surface tension for a very large drop, and V h l∆ is the molar volume of the bulk liquid at the pressure ∆ Pv . It has been shown both in experiments31,32 and in density functional (DF) calculations23a,25 that eq 3 yields an accurate estimate of the equimolar radius Re, even for small nuclei. A thermodynamically more rigorous expression for the Kelvin equation is
∆µ )
∫PP + P Vh dP v
σ
∆
(4)
v
where Pσ is the Laplace pressure, 2σ/Rs. Laaksonen and McGraw (L&M)23a have shown that by comparing eqs 3 and 4, while taking into account in each equation the compressibility of the drop, they can determine the size dependence of σ, namely,
Rs V h l∆[∆µ + ∆µl] Rs σ ) ≡ C ∆ ∆ σ∞ Re〈V h l〉[∆µ - V h l (Pv - Pv )] Re
(5)
where 〈V h l〉 is the average molar volume over the interval between the pressure Pv∆ and the final Laplace-enhanced pressure, Pσ + Pv. The quantity ∆µl is the change in chemical potential of the liquid when the pressure is decreased from Pv to Pv∆, or very nearly -V h l∆(Pv - Pv∆), so that
C≈V h l∆/〈V h l〉
(6)
In deriving an expression for 〈V h l〉, L&M assumed that the density is a linear function of pressure but we take the volume to be linear, or
10.1021/jp011028f CCC: $20.00 © 2001 American Chemical Society Published on Web 08/18/2001
11616 J. Phys. Chem. B, Vol. 105, No. 47, 2001
V hl ) V h lo[1 - κ(P - Po)]
Bartell
(7)
where κ is the isothermal compressibility of the liquid, and Po, the standard pressure. Because the compressibility correction is small, the two treatments lead to essentially the same expression for the average volume
h lo{1 - (κ/2)[Pσ + Pv + Pv∆ - 2Po]} 〈V h l〉 ) V
(8)
so that V h l∆/〈V h l〉 is very nearly [1 - (k/2)(Pσ + Pv - Pv∆)] ≈ (1 - (k/2)Pσ). Knowing all quantities entering eq 5 from density functional (DF) calculations on a Lennard-Jones liquid, L&M showed that the equation gave an excellent representation of surface tension over the whole range of drop radii at the reduced temperatures of 0.6 and 0.8. We extend their approach one small step further to derive an analytical expression for δ(Rs) applicable to drops obeying eq 7. As noted above, the constant C in eq 5 reduces very nearly h l〉 so that to V h l∆/〈V
σ ) σ∞
( )(
)
Rs 1 Rs + δ 1 - κσ/Rs
(9)
It is expedient to remove the small correction term κσ/Rs from the right hand side of eq 9 by solving the equation for σ and, for the sake of compactness in the following text, to express the result in terms of the roughly comparable dimensionless variables u ≡ δ/Rs and x ≡ κσ∞/Rs, to obtain
σ ) σ∞{1 - [1 - 4x(1 + u)-1]1/2}/2x
(10)
By differentiating the expression for σ with respect to Rs and comparing the result with eq 1, we secure a differential equation for the Tolman δ. The result is tractable if we expand it in terms of the small variables u and x to obtain
δ′ + f1 + f2 + f3 + f4 + ... ) 0
(11)
where the terms fn are
f1 ) u + x f2 ) -4u2 - 2ux + 6x2 f3 ) (38/3)u3 - 4u2x - 30ux2 + 33x3 f4 ≈ -(122/3)u4 + 54u3x + 78u2x2 - 248ux3 + 122x4, etc. (12) Results If we consider only the terms linear in 1/Rs in eq 11 (e.g., through f1) we get a zeroth-order solution which should be a good approximation when Rs is large, namely,
δ(Rs) ≈ k/Rs - κσ∞
(13)
where k is an undetermined integration constant depending upon physical properties of the system. The behavior of this solution already resembles those of DF solutions. If we include the second degree terms as well, the solution becomes
δ ) [(0.2Rs + κσ∞)[M-0.3,1(z) + k2W-0.3,1(z)] + 0.3RsM0.7,1(z) - 0.25k2RsW0.7,1(z)]/[M-0.3,1(z) + k2W-0.3,1(z)] (14)
Figure 1. Solutions for the size-dependent Tolman δ corresponding to various stages of approximation of eq 11. The higher and lower light solid curves represent solutions through fn for f1 and f2, respectively. The higher and lower dashed curves plot solutions through f3 and f4. The heavy solid curve is a numerical solution derived from the functional relationship between eqs 1 and 10. The horizontal dashed line is the asymptotic solution -κσ∞. Conditions were similar to those of ref 23 for a liquid of Lennard-Jones spheres at a reduced temperature of 0.8.
where Mλ,µ(z) and Wλ,µ(z) are Whittaker confluent hypergeometric functions with arguments of z ) 10x, or 10κσ∞/Rs, and k2 is an undetermined dimensionless integration constant. Terms fn in eq 12 of higher degree than quadratic can be taken approximately into account by substituting the zerothorder solution into the variable u and carrying out the resultant simple integrations for the correction terms for δ (Rs). Density functional solutions imply positive values of k in eq 13, making δ(Rs) at very small radii substantially larger than κσ∞.23a-29 The solution of eq 14 differs little from that of eq 13 as long as Rs is very large compared with κσ∞. Since κσ∞ is typically of the order of magnitude of 0.025 nm for small molecules such as argon, benzene, and water, this restriction is not very severe. The task of relating constant k to physical properties of the system remains to be done. Figure 1 illustrates the behavior of δ(Rs) when the undetermined integration constants k and k2 are adjusted to make results comparable with those of the Lennard-Jones system of L&M,23a taking the value of κσ∞ to be 0.025 nm. It appears that the zeroth-order solution, eq 13, is satisfactory for Rs down to about 1.5 nm and the quadratic solution, down to about 0.9 nm. At small R, higher degree solutions oscillate markedly with n as n increases. The average of the cubic and quartic solutions is good down to about 0.5 nm, or about 40 atoms. It gives a good account of the σ/σ∞ data of L&M, which go down to about 0.6 nm, as it should because it is based on the same approximation. However, the δ(R) implied by eq 11 falls off more slowly with drop size than the δ(R) calculated for nonane drops by Gra´na´sy via CahnHilliard theory.27 The object of Tolman’s original treatment, of course, was to investigate how the surface tension of drops depended on the separation between the surface of tension and the equimolar surface. Figure 2 illustrates how the surface tension depends on drop size if δ is nonvanishing and assumed to be constant as Tolman assumed it was for the sake of argument or, instead, if δ is allowed to vary with drop size as required by statistical thermodynamics.
Tolman’s δ and the Compressibility of Drops
J. Phys. Chem. B, Vol. 105, No. 47, 2001 11617
Figure 2. Dependence of surface tension on drop size. The heavy curve corresponds to the numerical solution for the δ(Rs) of Figure 1. The lower light curve was calculated from Tolman’s well-known eq 2, taking δ to be constant at 0.3 nm, a solution corresponding to eq 1 when δ is assumed to be constant. The upper light curve was calculated from eq 9 assuming a constant value of 0.3 nm for δ. Only the heavy curve closely approximates the density functional solution.
What is interesting to note is that as a droplet becomes larger, the Tolman δ implied by eq 9 changes sign and approaches the asymptotic limit -κσ∞. A reversal of sign in δ at large R is also characteristic of DF solutions.23a-30 If eq 9 can be trusted, this reversal is a consequence of compressibility. A similar connection has been suggested in prior Cahn-Hilliard treatments.14,33 Other studies34-38 have noted that κσ∞ is a fundamental characteristic length in liquids though the connection with the Tolman δ seems not to have been explicitly made. [Note: After submission of this manuscript, a reviewer pointed out a paper26b which had been missed in the original literature search. In this paper it was shown that if certain scaling relations23b are adopted, the limiting value of the Tolman length derived is δ∞ ) - κσ∞/3. On the other hand, if higher-order terms are retained in the expansion of W*(∆µ), the limiting value turns out to be δ∞ ≈-κσ∞, the same result found here from the relation of Laaksonen and McGraw, eq 5.] It is worth considering what happens as the compressibility goes to zero. The above solution for an incompressible liquid indicates only that δ vanishes in the limit of large Rs. Since the integration constant is still undetermined, nothing more can be inferred from this approach. A paper has appeared, however, which purports to show that the surface tension must be independent of drop size, i.e., that the Tolman δ vanishes, for an incompressible liquid.39 This was done by expressing the Gibbs adsorption isotherm for a drop of density F as
(∂µ∂F) dF + A dσ ) 0
(15)
and pointing out that, for an incompressible liquid, dF is zero and, hence, dσ must also be zero. Unfortunately, the resultant equation is indeterminate and therefore is unable to support the argument because it involves, essentially, simultaneously multiplying and dividing by zero. That is, the quantity (∂µ/∂F) can be expressed as
∂µ/∂F ) (∂µ/∂P)(∂P/∂F) ) V h (∂P/∂F)
WrI )
∫PP +P σ
o
o
-P dV
) nV h oκ[(Pσ + Po)2 - (Po)2]/2
Considerations of Compressibility
no
the surface of the drop being exposed to a constant external pressure Po. It is convenient to express the number of moles in terms of the size of the drop, so that n ) (4/3)πRe3/V h , where V h is given by eq 7. The process can be broken down into two steps, I and II. In step I, n moles of liquid are compressed from the initial standard pressure of Po to the final pressure in the interior of the drop, namely, Pσ + Po. Next, in step II the liquid is forced through a variable diaphragm originally suggested by Gibbs.40 The aperture of this diaphragm is adjusted during the expulsion of liquid so as to maintain the radius of curvature of the expelled portion of the liquid to be constant at Rs, the final drop radius. Therefore, the surface tension is constant as is also the pressure exerted to expel the liquid, at 2σ(Rs)/Rs + Po, during the entire process of formation of the drop. Accordingly, the total reversible work involved is the Helmholtz free energy change ∆AI+II ) WrI + WrII, where the work to compress the drop is found from the isothermal equation of state, eq 7, to be
(16)
which obviously blows up if the liquid is incompressible. Another aspect of compressibility enters the expression for the free energy of a small drop. We assume a process in which a drop containing n moles is formed from the bulk liquid with
≈ (2/3)πRe3κPσ2
(17)
for very small drops. It has been shown elsewhere41 that the reversible work, WrII, to produce such a drop from the compressed liquid is
WrII )
(
)
3 4πσ 2Re + Rs2 3 Rs
(18)
This work is associated not only with the motion of the piston expelling the drop but also with the opening and closing of the aperture.41 Besides yielding a very simple expression for the dominant term in the free energy of formation of a drop, eq 18 leads to an elementary derivation of eq 1.41 From the above relations, the Helmholtz free energy to produce a drop from the bulk liquid in its standard state can be written as
(
)
3 2 4πσ 2Re ∆AI+II ) πRe3κPσ2 + + Rs2 3 3 Rs
(19)
through linear terms in the compressibility, where σ is the sizedependent surface tension. For drops the size of critical nuclei in condensation, WrI is perhaps several percent of WrII. Although the work to form a drop from the bulk liquid is a quantity of some significance in capillary theory, a quantity more often applied in practice, especially in nucleation theory, is the reversible work to form a critical nucleus from the supersaturated vapor. This quantity, particularly the reversible work Wrev′ in excess of volume work Wv, has been treated by Gibbs40 and, recently, with special attention to rigor, by Debenedetti and Reiss.42 Of particular interest is the fact that the GibbsDebenedetti-Reiss expression for Wrev′ shows no explicit dependence on either the compressibility or the Tolman δ, quantities that enter the work of formation of such a nucleus from the bulk liquid, according to the present treatment. Therefore it is worth calculating the former quantity from the latter, both to check the consistency of the present treatment and to find whether, by comparing the two approaches, anything can be learned about the relationship between the Tolman δ and compressibility.
11618 J. Phys. Chem. B, Vol. 105, No. 47, 2001
Bartell
To calculate Wrev′ from the reversible work of producing a drop from bulk liquid in its standard state, one simply subtracts from ∆AI+II the reversible work ∆Avap of producing the supersaturated vapor from the bulk liquid, as well as the volume work Wv, involved in the formation of a critical nucleus from the vapor. If, for simplicity, fugacities are expressed as vapor pressures, the expression for ∆Avap is
∆Avap ) nPoV h lo - nRT + nRT ln(Pv/Pv∆)
(20)
where the Kelvin equation can be applied to simplify the last term on the right hand side of eq 20 to yield
nRT ln(Pv/Pv∆) )
∫P
Po
+ Pσ
o
V dP
) nV h loPσ(1 - (κ/2)Pσ) ) nV h lPσ(1 - (κ/2)Pσ)/(1 - κPσ) ) (4/3)πRe3Pσ(1 + (κ/2)Pσ)
(21)
through terms linear in compressibility, and Wv is simply
h lo + nRT Wv ) -nPoV
(22)
The assembled terms, then, are
Wrev′ ) ∆AI+II - ∆Avap - Wv
[
(
)]
2Re3 ) (2/3)πRe κPσ + (4/3)πσ + Rs2 Rs 3
2
-
h lo - nRT + (4/3)πRe3Pσ(1 + (κ/2)Pσ)] [nPoV h lo + nRT] [-nPoV ) (4/3)πRs2σ
(23)
or exactly the Gibbs-Debenedetti-Reiss expression for Wrev′. Since only Rs is left, Re having been eliminated, no explicit involvement of the Tolman δ remains, and the dependence on compressibility cancels. Although nothing new is found to relate δ to the compressibility, at least the present treatment is seen to be consistent with the prior, alternative treatments of Wrev′. Even though neither compressibility nor Tolman’s δ explicitly enters the expression for Wrev′, as a practical matter neither of these quantities can be ignored. For one thing, the compressibility plays a role in determining δ(Rs). If δ(Rs) is known, then both Rs and σ(Rs) can be inferred from more readily available quantities. That is, the radius which is most accessible experimentally is Re, not Rs,43,44 and the surface tension σ which is directly accessible to measurement is that for very large drops, not that for submicroscopic drops. Therefore it is worthwhile
to have a means of estimating the magnitude and behavior of the Tolman δ. Acknowledgment. This work was supported by a grant from the National Science Foundation. I gratefully acknowledge that some perceptive remarks of Professor Howard Reiss stimulated the above search for a relation between compressibility and the Tolman δ. I am obligated to Professor David Oxtoby for some crucial criticisms of an earlier draft. I thank Dr. Laszlo Gra´na´sy for helpful discussions and for bringing several relevant references to my attention, and Professor Dimo Kashchiev for suggestions about the presentation of results. References and Notes (1) Tolman, R. C. J. Chem. Phys. 1949, 17, 333. (2) Mandell, M. J.; Reiss, H. J. Stat. Phys. 1975, 13, 107. (3) Tolman, R. C. J. Chem. Phys. 1949, 17, 118. (4) Kirkwood, J. C.; Buff, F. P. J. Chem. Phys. 1949, 17, 338. (5) Choi, D. S.; Jhon, M. S.; Eyring, H. J. Chem. Phys. 1970, 53, 2608. (6) Vogelsberger, W. Z. Phys. Chem. 1980, 261, 1217. (7) Rowlinson, J. S. J. Phys. A 1984, 17, L357. (8) Blokhuis, E. M.; Bedeaux, D. J. Chem. Phys. 1992, 97, 3576. (9) Blokhuis, E. M.; Bedeaux, D. Mol. Phys. 1993, 80, 705. (10) Romero-Rochin, V.; Varea, C.; Robelo, A. Mol. Phys. 1993, 80, 821. (11) Varea, C.; Robelo, A. Mol. Phys. 1993, 85, 477. (12) Ginoza, M. J. Phys. Condens. Mattter 1994, 6, 1439. (13) Rowlinson, J. S. J. Phys. Condens. Mattter 1994, 6, A1. (14) Iwamatsu, M. J. Phys. Cond. Matter 1994, 6, L173. (15) McClurg, R. B.; Flagan, R. C.; Goddard, W. A., III. J. Chem. Phys. 1996, 105, 7648. (16) Kalikmanov, V. I. J. Chem. Phys. 1995, 103, 4250. (17) Kalikmanov, V. I. Phys. ReV. E 1997, 55, 3068. (18) Groenewold, J. Physica A 1995, 214, 356. (19) Baldoni, F. Acta Mech. 1996, 117, 145. (20) Van Gliessen, A. E.; Blokhuis, E. M.; Bukman, D. J. J. Chem. Phys. 1998, 108, 1148. (21) Bykov, T. V.; Zeng, X. C. J. Chem. Phys. 1999, 111, 3705. (22) Baidakov, V. G.; Boltachev, G. S. Phys. ReV. E 1999, 59, 469. (23) (a) Laaksonen, A.; McGraw, R. Europhys. Lett. 1996, 35, 367. (b) McGraw, R.; Laaksonen, A. J. Chem. Phys. 1997, 106, 5284. (24) Hadjiagapiou, I. J. Phys. Condens. Matter 1994, 6, 5303. (25) Talanquer, V.; Oxtoby, D. W. J. Chem. Phys. 1995, 99, 2865. (26) (a) Koga, K.; Zeng, X. C.; Shchekin, A. X. J. Chem. Phys. 1998, 109, 4063. (b) Koga, K.; Zeng, X. C. J. Chem. Phys. 1999, 110, 3466. (27) Gra´na´sy, L. J. Chem. Phys. 1998, 109, 9660. (28) Harrowell, P.; Oxtoby, D. W. J. Chem. Phys. 1984, 80, 1639. (29) McGraw, R.; Laaksonen, A. J. Chem. Phys. 1997, 106, 5284. (30) ten Wolde, P. R.; Frenkel, D. J. Chem. Phys. 1998, 109, 9901 (31) Viisanen, Y.; Strey, R.; Reiss, H. J. Chem. Phys. 1993, 99, 4680. (32) Strey, R.; Wagner, P. E.; Viisanen, Y. J. Phys. Chem. 1994, 98, 7748. (33) Gra´na´sy, L. Private communication, 1999. (34) Mayer, S. W. J. Phys. Chem. 1963, 67, 2160. (35) Fisk, S.; Widom, B. J. Chem. Phys. 1969, 50, 3219. (36) Egelstaff, P. A.; Widom, B. J. Chem. Phys. 1970, 53, 2667. (37) Mon, K. K.; Stroud, D. J. Chem. Phys. 1981, 57, 2078. (38) Zeng, X. C.; Stroud, D. J. Chem. Phys. 1989, 90, 5208. (39) Schmelzer, J. W. P.; Gutzow, I.; Schmelzer, J., Jr. J. Colloid Interface Sci. 1996, 178, 657. (40) Gibbs, J. W. Collected Works; Longmans Green: New York, 1928; Vol. 1, p 257. (41) Bartell, L. S. J. Phys. Chem. 1987, 91, 5985. (42) Debendetti, P. G.; Reiss, H. J. Chem. Phys. 1998, 108, 5498. (43) Kashchiev, D. J. Chem. Phys. 1982, 76, 5098. (44) Oxtoby, D. W.; Kashchiev, D. J. Chem. Phys. 1994, 100, 7665.
