1050
J. Phys. Chem. B 2001, 105, 1050-1055
On the Dependence of Surface Tension of Liquids on the Curvature of the Liquid-Vapor Interface Vladimir B. Fenelonov,*,† Gennady G. Kodenyov,‡ and Vitaly G. Kostrovsky§ BoreskoV Institute of Catalysis, Prospekt Akademika, LaVrentieVa, 5, NoVosibirsk 630090, Russia ReceiVed: August 24, 1999; In Final Form: June 1, 2000
Conventional methods for measuring pore-size distribution (PSD), which are based on the capillary condensation (CC) and mercury intrusion porosimetry (MIP), are very sensitive to the numerical values of surface tension, σ, of the corresponding liquid phase. However, it is generally accepted that σ strongly depends on the curvature radius, r. Hence, PSD measuring seems unreliable if it does not involve the σ(r) dependence. Experimental studies of the surface tension, σ, for nuclei of liquids formed in the vapor upon homogeneous nucleation support the idea of no significant dependence of σ on the curvature of the interface at the range of subnanometer radii.
1. Introduction Currently, it is generally thought that the surface tension, σ, on a liquid-vapor interface depends essentially on radius of curvature, r, for all sizes within the mesopore range, that is, from 2 nm to ≈50 nm.1-11 Therefore, the results on mesopore size measurements seem doubtful if they are obtained by capillary condensation (CC) or mercury intrusion porosimetry (MIP) based on the Kelvin and Laplace-Washburn equations1 but ignoring the σ(r) dependence. Regarding this problem, Gregg and Sing said (ref 1, p 154): “...the actual value of σ will differ appreciably from the normal value (σo). (σo is surface tension for a plane surface.) The effect of using the corrected values would be to raise the calculated value of r in proportion σ(r)/ σo. The time is perhaps not yet ripe, however, to introducing this kind of correction into calculations of pore size distribution... A fuller quantitative assessment of the situation in very fine capillaries must await the development of thermodynamics of small systems. Meanwhile, enough is known to justify the conclusion that, at the lower end of the mesopore range, the calculated value of r is almost certain to be too low by many percent.” Despite the lack of a clear understanding of the dependence of σ on r,1,12,13 CC and MIP are still the bases of the most widespread methods for measuring mesopore size. The emergence of unique mesoporous mesophase materials, such as MCM-41 with monosized pores ranging between 2 and 10 nm,14-16 stimulates the resolution of this problem. In reality, the correct determination of the mesopore size, dm, in these systems does not need CC and MIP studies, that is, the use of values for σ.17,18 At the same time, the size can be calculated from the adsorption isotherms in terms of the conventional CC methods. For example, the methods suggested by Barrett et al.19 and by Broekhoff and deBoer20 for estimating the mesopore size (dBJH and dBDB, respectively) do not involve the dependence of σ(r). The comparison of these independent estimations demonstrates21 typical situations when dBJH is ≈0.2 nm lower than dBdB and ≈0.8 nm lower than dm. These are practically †
Boreskov Institute of Catalysis, Novosibirsk, Russia. Design and Technological Institute of Instrument Engineering for Geophysics and Ecology, Novosibirsk, Russia. § Institute of Solid State Chemistry and Mechanochemistry, Novosibirsk, Russia. ‡
constant differences which are not related to any σ(r) dependence; they may be corrected by a reasonable refinement of the definition of the adsorbed film thickness, t(P/Po)17,18,21 but not σ values. We will discuss these data in another communication. Besides, calculation of the mesopore size in MCM-4122 and the other fine-porous systems23 by various methods including the latest versions of the nonlocal density functional theory (DFT) can be mentioned. In this case, the comparison of the size computed by DFT and conventional CC methods at a constant relative pressure, P/Po, of the adsorbate gives rise to the same conclusion on the constant difference between the Kelvin- and DFT-based sizes, which is again independent of any σ(r) function. Thus, there is a natural question: Is this a strong dependence σ(r) in the nanosize range? This dependence is of vital importance to not only CC and MIP measurements of mesopore size but also to development of the theory of nucleation and of many other surface and capillary phenomena.23-28 The purpose of the present work is to discuss this problem. 2. Problems In his classical work,24 Gibbs speculated rather hazily about the correlation between σ and radius of curvature, r. He introduced the notion of a surface of tension (as the surface where tension acts) but made the principal calculations for an equimolar surface (a dividing surface in a monocomponent liquid-vapor system with zero excess density of the component). In general, these surfaces do not coincide. Gibbs believed, however, that his thermodynamic equations are true until the radius of curvature turns zero. In contrast, Guggenheim3 concluded, based on statistical mechanical considerations, that σ must start depending on the radius of curvature of the liquid surface when its value falls below 50 nm. Later, Tolman4 used a quasithermodynamic approach to derive cumbersome integral equations for the σ(r) function. Koenig5 extended these equations to multicomponent systems. The theoretical studies were followed-up by Buff,6 Hill,7 Kondo,8 Ono and Kondo,9 Rusanov,10 Melrose,11 and the works discussed extensively by Rowlinson and Widom.25 The most popular and frequently cited study result3-11 is equation
10.1021/jp9929972 CCC: $20.00 © 2001 American Chemical Society Published on Web 01/13/2001
Surface Tension on Liquid-Vapor Interface
(
σ(r) ≈ σ0 1 -
2δ r + ...
)
J. Phys. Chem. B, Vol. 105, No. 5, 2001 1051
(1)
or
(
σ(r) ≈ σ0 1 +
2δ -1 r + ...
)
(2)
Values for σ(r)/σ0 produced by these versions of the equation are close to one another at 2δ/r e 0.3 and almost coincide at 2δ/r e 0.1. Parameter δ here equals the difference between the curvature radius of the equimolar surface and that of the surface of tension. Equation 2 originally deduced by Tolman4 as a quasiplane approximation of the more strict and complex equation for curved surfaces. At present, many researchers refer to parameter δ as Tolman’s length and use it as the measure of thickness of the interfacial region. Tolman’s length can be defined as δ ) ξ0Dm, where Dm is the molecular diameter in the hard-sphere approximation and ξ0 is a constant. Rowlinson and Widom,25 although extensively reviewing the subject, note that Tolman’s length may be formally positive or negative depending on the relative position of the surface of tension with respect to the equimolar surface. In particular, different theoretical approximations may lead to opposite results. It is generally assumed that σ(r) < σ0 for spherical drops and σ(r) > σ0 for bubbles in a liquid. Rowlinson and Widom emphasize misinterpretation in identifying values δ of eq 1 or eq 2 with the thickness of the interfacial region for a high curVed surface, δr, because these equations are an interpolation to the quasiplane approximation (so as δ ) δrf∞ ≡ δ∞). However, δr can depend on the surface curvature. [Dependence δr(r) for drops of small radii r is demonstrated, for example, in ref 29.] Up-to-date computer-aided techniques allow characteristics of small systems to be modeled in terms of thermodynamic and statistical mechanics using modern versions of DFT, molecular Monte Carlo simulations, and molecular dynamics23,30-33 including the cluster population dynamics.28,33-37 Results of computing σ/σ0 as a function of radius of a spherical drop (Rd ) ξDm, where Dm is the diameter of molecules approximated as hard spheres) are plotted in Figure 1. The calculation performed by Bykov and Shchekin36 in terms of DFT; the molecular interaction was modeled using the Lennard-Jones6-12 pairwise potential. The results computed at δr ) Dm ) Const (i.e., ξ0 ) 1.0) on the basis of the Tolman’s integral equation (using reference data9) and approximate eq 1 are shown in Figure 1. The different computation methods seen in Figure 1 give extremely dissimilar functions σ(r). Equation 1 yields the most understated values for σ; hence, this equation is not applicable but at 2δ/r < 1.0. Plot 1 in Figure 1 corresponds to a nonmonotonic function. Initially, as the radius grows, function σ(r)/σ0 increases rapidly to reach its maximal values at σ(r)/σ0 ≈ 1.03 then starts slowly decreasing to approach σ(r)/σ0 ) 1.0. This asymptotic region may be formally described by eq 1 or eq 2 at a very low Tolman’s length, δ. Computing36 restricts the strong σ(r)/σ0 dependence to the range of drop radius lower than (2.5-3.0)Dm. DFT used in another way for analyzing the σ(r)/σ0 function29,37 yields qualitatively similar dependencies. For example, Talanquer and Oxtoby29 demonstrated a sharp decrease in σ(r)/σ0 at the drop radius of below (3.5-4.0)Dm [σ(r)/σ0 equals ≈0.94 and ≈0.97, respectively]; σ(r)/σ0 ≈ 0.71 at ξ ) 2.5, but σ(r)/σ0 ≈ 0.985 at ξ ) 4.5. A monotonic increase with initially fast and then slow approach to σ0 skipping the range of σ > σ0 is characteristic of the computed dependence in this case. This
Figure 1. σ(r)/σ0 as a function of drop radius Rd ) ξDm. Plot 1 is computed according to DFT36; plots 2 and 3 are results of the Tolman integral eqs 2 and 1 at δr ) Const, respectively.
dependence used for calculating the rates of nonane nucleation gave values that were in good agreement with the experimental data.29 However, very contradictory data on experimental checking of the σ(r) dependence have been reported. For example, Nielsen and Sarig38 discovered 17% decrease in the surface tension for water drops of 3.2 nm radius. In their experiments on impregnation of mesoporous Vycor spheres, Wingrave et al.39 observed the surface tension increase by 5 to 50% more than σ0 for water and five n-alkanes. (Their computations were unreliable because of numerous very disputable assumptions.) Dubinin2 used eq 2 to improve CC methods for computing the pore size distribution. Parameter δ was calculated by equation
δ ) 0.9165
() VS NA
1/3
(3)
where VS is the molar solidlike volume and NA is Avogadro’s number. This equation was deduced by Ahn et al.40 based on the transient-state theory of the significant liquid structures they developed assuming that δ > 0 for a concave interface and δ < 0 for a convex one. The application of these assumptions to Dubinin’s work resulted in the almost parallel shift of the PSD plots toward the range of coarser pores. There is no proof for the validity of such a shift. Recently, Galarneau et al.41 reproduced Dubinin’s method. They also used eqs 2 and 3 for correcting BdB computations of the mesopore diameter, dBdB, for several MCM-41 silicates.20 The authors believe that an excellent agreement between dBdB (σ was calculated at δ ) 0.33 nm) and the mesopore size calculated as dV ) 4VMe/ABET is evidence of the valid consideration. In the latter relationship, ABET is the total of Brunauer-Emmett-Teller (adsorption isotherm) (BET)-specific surface area, and VMe is the mesopore volume estimated from the adsorption isotherm at the top of the mesopore-filling step. Unfortunately, this agreement can not be accepted as a reliable confirmation of the validity of eqs 2 and 3 because the authors also based their speculations on many disputable assumptions. First, the specific surface area, even though calculated accurately from the starting region of the adsorption isotherm, is indeed a sum of the mesopore surface area, AMe, and the external surface area, Aext, left after the mesopores were filled.17,18 The nitrogen adsorption isotherms41 show that Aext contributes considerably to this total surface area. Besides, the values of VMe used by the authors are greater than the real mesopore volume, which can be estimated more correctly by the method suggested in refs 17 and 18. For these reasons, the values computed by the
1052 J. Phys. Chem. B, Vol. 105, No. 5, 2001
Fenelonov et al.
authors for dV seem unconvincing. Again, they calculated ABET based on a nontypical area, ω ) 0.135 nm2, of the surface fragment covered by one molecule in the monolayer. This problem was discussed in detail previously.42 Finally, more correct equations for the statistical thickness of a polymolecular film have been suggested recently.17,18 Fisher and Israelashvili26,43,44 reviewed many publications devoted to verification of the σ(r) dependence and reported confirmatory measurements for liquid bridges between crossed mica cylinders. In this case concave menisci were formed with the radius of curvature, r, ranging between 19 and 4.2 nm. (The situation is similar to that of the meniscus in a capillary because r is negative.) Very precise measurements of the force induced by the Laplace pressure [using a surface forces apparatus (SFA)] for a pendular ring of liquids allowed the authors to conclude for several organic liquids that the effective surface tension remained unchanged down to radii of curvature as low as 4.2 nm. Later, in his SFA experiments, Christenson45 found that the Laplace equation held for water down to the radii of 2 nm. Most of the above-discussed experiments were conducted with a three-phase (solid-liquid-vapor) system. The results obtained for these systems depend on specific solid-liquid and solid-vapor interaction, on contact angles, surface roughness, etc. It would be desirable to minimize the number of parameters for more correct verification. That is the case of measuring σ in the processes of homogeneous nucleation. 3. Surface Tension of Critical Nuclei at Homogeneous Vapor-Liquid Phase Transitions The idea of experimental determination of surface tension, σ, by studying homogeneous nucleation is as follows. It is general knowledge that spontaneous formation of a liquid drop in the supersaturated vapor involves the stage of formation of nuclei of critical size, the nucleation rate being determined by the probability of emergence of such nuclei. In its turn, the rate of isothermal homogeneous nucleation obeys the equation28,46,47 (which is called the first fundamental nucleation theorem in ref 28):
(dd lnln SI ) ) g* + 1 T
(4)
where I is the number of aerosol particles generated in a unit of time through a unit volume or the nucleation rate; S ) P/P∞ is the supersaturation ratio; P and P∞ are the partial pressure of vapor in the system and the pressure of the saturated vapor, respectively; g* is the number of molecules in a nucleus of the critical size. Equation 4 was first deduced in terms of thermodynamics46 and then verified in terms of statistical mechaniscs.28 It allows the number of molecules, g*, in a critical nucleus, which are in a nonstable equilibrium with the supersaturated vapor, to be determined based on experimental values of ∆ ln I and ∆ ln S. The equimolecular radius of the nucleus, r*, is found from the relationship
g*V0 ) 4πr*3/3
(5)
where V0 is the average volume occupied by one molecule of the liquid. Because of importance of eq 4, its deduction is presented in the application after the article of Bedanov et al.47 If the size of the critical nucleus, and the temperature and pressure of the supersaturated vapor P are known, one can check directly the applicability of the Kelvin formula1,24:
ln S )
2V0σ kTr*
or
ln S )
( )
2 4πσ 3V0 3 kT 4π
2/3
g*-1/3 (6)
to ultimately small bodies of subnanometer size. In this equation, k is the Boltzmann constant. Such checking was made first at individual points46,48-50 and then in a certain range of supersaturation ratios.47 Nucleation processes were studied using a specially designed flow-diffusion chamber. The chamber was a metal tube of 155mm length and 4-mm diameter. The tube walls were thermostated at a preset temperature TS at the accuracy of (0.1 K. Gas carrier (nitrogen, argon) was fed into the tube as two coaxial flows at different temperatures. A flow of pure gas at temperature TS was supplied into the peripheral and another flow saturated by the vapor of the compound under study at temperature TH > TS entered the central part of the tube. Required supersaturation ratio was achieved by varying the central flow temperature TH. Supersaturation ratio grew up as TH increased at TS ) Const, and the aerosol particles were formed at the outlet of the diffusion chamber. The second diffusion chamber of the same design was used for enlarging the aerosol droplets to the standard size of 1 µm. To achieve the enlargement, a flow of hot argon saturated with ethyleneglycol vapor to prevent the homogeneous nucleation of ethyleneglycol was fed into the second chamber. As a result, ethyleneglycol vapor condensed only on the aerosol drops generated in the first diffusion chamber. The primary experimental result was intensity of light scattered by particles at the outlet of the second chamber as a function of temperature TH of the first chamber hot flow. The simple geometry and the laminary nature of the resulting gas flow provided reliable calculations of profiles of concentration, temperature, and supersaturation ratio. They were found by solving equations of vapor diffusion and heat conductivity in the resulting flow. An experimental curve was recounted as a dependence of the nucleation rate on supersaturation ratio, I(S). The slopes of plots of lg I vs lg S were used for calculating the number of molecules in the critical nucleus, g*, by eq 4. More procedural details are given elsewhere.46-49 Results of experimental examinations, carried out for 15 years, from 1973 to 1988, initially have shown agreement, within the experimental accuracy, between the nucleus size, defined from the experiments with the help of eq 4, and the calculated one according to Kelvin eq 6. Later with the improvement of the experimental technique and equipment it became clear that Kelvin’s equation, which treated the nuclei as spherical droplets with the same density and surface tension as for the bulk liquid, was not accurate for such small particles. It lead to the systematic decrease of the equilibrium vapor pressure over the surface of a critical nucleus or, in other words, it decreased the radius of a critical nucleus existing in equilibrium with the supersaturated vapor up to 10-15% (30-45% decreasing number of molecules in a critical nucleus). Results processing using the least-squares method showed that supersaturation data as the functions of g*-1/3 for the given nucleation temperature fell on straight lines within 5% accuracy:
ln S ) RR0
(g*1 )
1/3
+δ
(7)
where R0 is the coefficient for g*-1/3 in Kelvin eq 6; R and δ are some constants for the given substance and nucleation temperature. Usually R slightly exceeds 1 and ranges from 1 to 1.15 and δ does not exceed several percent of the main term. The presence of such constants testifies that genuine dependence
Surface Tension on Liquid-Vapor Interface of vapor pressure logarithm slightly differs from the direct proportion to inverse critical radius, ln S ∼ g*-1/3. It is evident that there is the main term proportional to g*-1/3 and corresponding to the right part of Kelvin eq 6 plus small corrections containing higher powers of inverse radius g*-h, where h > 1/3. As a result of the examination performed, it was found that the dependence of pressure of the vapor in equilibrium with a nucleus versus its size (eq 7) differs from the Kelvin formula but the difference is not high. It does not assume some radical changes in the nucleation theory. An improvement of the theory consists of introduction of the mentioned microscopic corrections to the Kelvin equation and the corresponding relatively small corrections to the free energy of the critical nucleus formation, that is, to the exponent of nucleation rate too. But this examination did not reveal if the Kelvin equation would retain its appearance eq 6 and corrections are presented only in σ and V0; or this equation looks like eq 7 and other corrections not concerning σ and V0 exist, or corrections to the mentioned and other parameters exist too. The ratios of the classical theoretical nucleation rate to the experimental one for phthalates47,50 Ith(S,T)/Iexp(S,T) increased rapidly with increasing temperature, as in ref 51. An intermediate step in calculating this ratio is the calculation of g*(S,T). On the other hand, if we could determine g* independently, we could determine the function Ith(g*,T)/Iexp(g*,T). It appeared that, in contrast with the former relationship, the latter ratio became constant for each investigated substance over a wide measured range of temperature T and supersaturation ratios S:Ith(g*,T)/ Iexp(g*,T) ) Const. Here in Ith we used surface tension and the volume of the embryo molecule of the bulk liquid, despite the number of molecules in the nuclei varying more than 2-fold. A consequence of this result is evidently the correctness of the equation for the nucleation rate expressed with T and g* variables as: I ) A(g*,T) exp{-4π(3V0/4π)2/3σg*2/3/3kT}, where A(g*,T) is a slowly changing preexponential factor. So, we again convinced ourselves that the Kelvin equation is incorrect and conclude that the surface tension is independent of the nucleus size. So we used Kelvin eq 6 for the surface-tension evaluation for different substances on the experimentally defined supersaturation ratio S and number of molecules in critical nucleus g* from expression 4. Nucleus density was assumed to be the same as for the bulk liquid. The resulting surface tensions were slightly coarse because of insertion of the sum of the corrections mentioned above in σ, but it would be justified because genuine correction to the surface tension σ would be the same order or less than that named above. Principal experimental results are summarized in Table 1. Table 1 includes the data on temperature T and supersaturation S; g*, r*, and σ calculated by equations 4, 5, and 6, respectively. Reference data on σ0 measured for macroscopic liquids are given for comparison. One of the columns of Table 1 shows the estimated relative difference between σ and σ0 (in percent):
∆σ σ - σ0 ) × 100% σ0 σ0 The relative difference ∆σ/σ0 reaches 14%, even though it is 3-5% for some experiments. Undoubtedly the scatter in ∆σ/σ0 is due to the accuracy of the individual experiments. The differences of obtained σ and σ0 may result wholly or in part from the contribution to σ of the above-mentioned small corrections to the Kelvin equation. Accordingly, we are
J. Phys. Chem. B, Vol. 105, No. 5, 2001 1053 TABLE 1: Principal Experimental Results σ (mJ/m2) T, (°K) 250 260 269 280 293 303 313 323 333 343 353 293 303 313 323 333 343 353 303 308 318 323 283 288 298
283 288 283 293 303 313 333 288
S 46600 16318 8103 3072 351.66 199.39 120.35 75.89 50.59 32.11 21.38
g*
r* σ, present (nm) study
σ0, ref data
dibutylphthalate, C6H4(COOC4H9)2 9.51 0.992 42.8 37.8 10.62 1.031 41.5 36.9 11.74 1.068 41.0 36.1 13.50 1.122 39.8 35.1 18.5 1.25 33.8 20.5 1.29 32.6 22.7 1.338 31.5 29.4 25.00 1.382 30.5 27.8 1.431 29.4 33.0 1.515 28.3 38.5 1.594 27.2
refs 47 47 47 47 27
∆σ/σ0 (%) 13.2 12.5 13.6 13.4 7.1 -
glycerin, C3H8O3 40.28 29.25 0.944 59.4 59.4 64.0 29.20 33.31 0.986 58.4 59.0 22.86 36.00 1.015 57.3 58.5 17.08 39.60 1.045 56.2 58.0 13.30 45.61 1.152 54.9 57.4 10.03 54.80 1.160 53.6 56.7 8.48 58.70 1.191 52.2 55.9
60 62 60 60 60 60 60 60
0.0 -7.2 -1.0 -2.0 -3.1 -4.3 -5.5 -6.6
nanodecane, C19H30 1.319 27.3 27.8 1.332 26.6 27.3 1.450 25.3 26.4 1.485 24.6 26.0
60 60 60 60
-1.8 -2.5 -4.2 -5.4
ethylene glycol, C2H6O2 8.06 58.04 1.086 49.0 7.51 58.44 1.088 48.4 48.0 (293) 6.55 61.36 1.106 47.3 47.99 48.8 48.7
61 62 63 64
(0.8) -1.4 -3.1 -2.9
diethylene glycol, C4H10O3 9.59 56.31 1.316 24.9 4.20 72.72 1.398 26.0 26.28
65
-1.0
27
12.6 -
345.92 258.34 113.68 82.33
141.29 118.60 96.78 81.20 56.06
16.65 17.20 22.13 27.78
triethylene glycol, C6H14O4 22.67 1.062 46.9 24.86 1.095 48.3 25.53 1.105 48.2 42.8 27.39 1.131 49.1 31.12 1.180 49.9
diphenylmethane, C13H12 88.00 27.44 1.224 39.4 37.56 (299)
256.6 262.7
19.0 12.9
19.6 21.8
254.0 261.6 267.1
6.9 6.0 5.1
57.6 57.1 68.7
260.9
5.0
65.6
226.0
14.8
43.1
65
(+4.9)
2-pentanol, C5H12O 0.95 27.0 0.98 25.0 26.3 (287.3) 65
(-4.9)
n-butanol, C4H10O 1.20 26.5 1.28 27.1 1.36 26.6 24.4 (290) 26.2 (273)
65 66
(+9.0) (+1.5)
n-propanol, C3H8O 1.25 29.2 26.4 23.32 (298) 23.8 (298)
39 (+10.6) 62 (+25.2) 64 (+22.7)
0.676
water, H2O 94.67 (273) 67 69.2 (ice) 75.8 (water)
convinced that the true surface tension σ is not distinguished from σ0 or the difference between σ and σ0 is less than 14%. The state of uninterrupted fluctuations is characteristic of the nucleation clusters because of their small size. In this state, their temperature, density, shape, etc., differ remarkably from the average equilibrium parameters. Bedanov et al.59 believed that it is precisely these fluctuations rather than changes in the surface tension that are responsible for the increase of the
1054 J. Phys. Chem. B, Vol. 105, No. 5, 2001
Fenelonov et al.
evaporation rate of small clusters compared with the Kelvin equation predictions that they observed in the numerical experiments. As a whole, the experimental studies demonstrate the validity of the Kelvin formula with the error not higher than 12-14% to drops of radii below 0.7-1.0 nm, the surface tension being constant or changed by less than 14%. Admissible deviations are comparable with those accounted for by the effect of temperature, microimpurities, etc. 4. Conclusions We believe that this brief review and the experimental data obtained support the idea of no significant dependence of the surface tension of liquids on the curvature of their surface at the range of r g 1 nm, which would have a remarkable impact upon the CC and MIP results of pore size and PSD measurements. The main drawbacks of these methods are most likely to result from the assumptions based on the shape of menisci, the geometrical models for the porous space, the insufficient account of cooperative processes of adsorption and desorption in the system of interlaced pores,68,69 the incorrect choice of equations for the statistical thickness of the polymolecular films, and the reference standards used for comparative analysis of the adsorption isotherms (such as Sing’s dS method1). Studies of mesoporous materials, such as MCM-41, pillared clays, opals and the like, which are characterized by a simple and regular geometry, make it possible to overcome these drawbacks of modern methods for PSD analysis based on CC and MIP. Application. The system of kinetic equations of the steadystate isothermal nucleation55 was used as a basis. The solution of this system looks like
( ) ( ) G-1
I)
1
∑ g)1 b n
-1
)
g g
-1
1G-1 1
∑ b g)1 s n
Z(g*) )
x( ) 2 1 ∂ ln ng 2π ∂g2
Consequently we obtain the standard notion of the steady-state isothermal nucleation rate55-58:
I ) bsg*Z(g*)ng*
∂ ln ng* d ln(sg*bZ(g*)) d ln I ) + d ln n1 ∂ ln n1 d ln n1 If we recall that b ∼ n1 and sg* ∼ g*2/3 we will find with the help of eq 9 that
d ln[g*2/3 Z(g*)] d ln I ) g* + 1 + d ln n1 d ln n1
∏ i)2
n g ) n1
( ) bi-1 aisi
)
n1s1bg-1 sg
g
ln ng ) ln n1 + (g - 1) ln b + ln
∏ i)2 a
(9)
i
where ng is the equilibrium nuclei size distribution; bg ) b(n1) is the quantity of precipitated molecules per a surface area unit in a unit of time; ag is the quantity of evaporated molecules per a surface area unit in a unit of time for the clusters with g molecules in it; sg is the surface area of such a cluster. The condensation rate per surface area unit b is directly proportional to the supersaturated vapor density and is independent of the cluster size, b ∼ n1 ) Sn∞(T), where n∞(T) is the saturated vapor density over the flat surface. The evaporation rate from a cluster surface area unit ag grows as the cluster size decreases, after the decrease of the evaporation potential energy U(g) caused by a reduction of the number of neighboring molecules, ag ∼ exp[-u(g)/kT]; big droplets tend to the evaporation rate for flat surfaces. Such behavior of b and ag leads to the fact that the function 1/ng has the sharp maximum in the point g ) g*, determined by condition ag*sg* ) bsg*-1. To the left from this point all ag/b presented in product are higher than 1 and the function 1/ng increases quickly; to the right from this point ag/b are less than 1 and the function decreases rapidly. The summation of this type of functions is reduced to the multiplication of its value in the maximum point by the effective width ∆ ) Z(g*)-1, where Z(g*) is the Zeldovich factor
[ ] s1
g
1
∏ s i)2 a g
i
Because b ) βn1(kT/2πm)1/2, where β is an accommodation coefficient and m is molecule mass, then
[x ] [ ] kT
ln ng ) g ln n1 + (g - 1) ln β
1
(12)
Since ln S ) lnn1 - lnn∞(T) and lnn∞(T) ) Const at T ) const then d ln n1 ) d ln S. We in fact obtained the result if we can show that the last term in eq 12 is negligibly small. So lets take the logarithm of expression 9
and g
(11)
After differentiation ln I with respect to ln n1 and using the condition for the minimum of the nuclei size distribution: (∂ ln ng/∂g)g)g* ) 0, we have
(8)
g g
(10)
g)g*
2πm
s1
g
1
∏ s i)2 a
+ ln
g
g
After differentiating previous equation with respect to g we have
∂ ln ng ) ln n1 + f1(g) ) ln S + f(g) ∂g
(
where f(g) ) (∂/∂g){ln βx(kT/2πm)
g-1
)
(13)
g
(s1/sg) ∏ (1/ag)]} + i)2
ln n∞(T). In deriving eq 13 we used no additional assumptions except T ) Const and (∂I/∂t) ) 0. The function f(g) in the vicinity of g* with width ∆ ) Z-1 can be approximated by power function
f(g) ) Agh here A and h are empirical fitting constants. Then
Z(g*) )
Ah (2π )
1/2
g*(h-1)/2
(14)
On setting expression 11 equal to 0 we obtain the supersaturated vapor pressure over a critical nucleus: ln S ) -Ag*h (at h ) -1/3, the last relationship becomes the Kelvin equation known from thermodynamics.) Notice that supersaturation ratio increases with a decrease of the critical nucleus size and hence h < 0. It is easy to verify that
Surface Tension on Liquid-Vapor Interface
J. Phys. Chem. B, Vol. 105, No. 5, 2001 1055
1 1 + d ln[Z(g*)g*2/3] 6h 2 ) d ln n1 ln S
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(15)
Based on expression 15 for the typical nucleation experiments with g* = 10-100, and ln S ranges from 1 to 10, the following conclusions can be reached: (1) If the Kelvin formula is correct then h ) -1/3 and expression 15 strictly equals 0; (2) if h < -1/3 then expression 15 is less than 1 and far less than g*; (3) the absolute value of expression 15 will be of order of 1 and far less than g* even in the strong dependence of the critical nucleus size versus supersaturation ratio (for h > -1/3 up to h ) -1/12): g* ∼ 1/(ln S)12 or r* ∼ 1/(ln S)4. The known experimental data and existing theoretical concepts do not provide any reasons to expect significant changes in the nucleus size dependence on supesaturation ratio: the Kelvin formula approximates this dependence for the nucleus clusters correctly; at least h cannot be widely different (several times) from -1/3. Thus, it is shown that the second term in eq 12 is negligibly small and, consequently, expression 4 is valid. This deduction was based on kinetic not thermodynamic approach and did not require any assumption about the spherical shape of clusters; liquid incompressibility inside clusters and so on. It is important that eq 4 always holds including also the small clusters where the Kelvin formula can fail. Acknowledgment. The authors are grateful to INTAS and RFBR (Projects 97-0676 and 98-03-32390) for the support of this work. References and Notes (1) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity, 2nd ed.; Academic Press: London, 1982. (2) Dubinin, M. M. In The Modern Theory of Capillarity; Rusanov, A. I., Goodrich, F. G., Eds.; Khimia Publ.: Leningrad, 1980; p 100. (3) Guggengeim, E. A. Trans. Faraday Soc. 1940, 36, 407. (4) Tolman, R. C. J. Chem. Phys. 1949, 17, 118, 333. (5) Koenig, F. O. J. Chem. Phys. 1950, 18, 449. (6) Buff, F. P. J. Chem. Phys. 1951, 19, 1591. (7) Hill, T. L. J. Phys. Chem. 1952, 56, 526. (8) Kondo, S. J. Chem. Phys. 1956, 25, 662. (9) Ono, S.; Kondo, S. Molecular Theory of Surface Tension in Liquids; Springer-Verlag: Berlin, 1960. (10) Rusanov, A. I. Phase Equilibriums and Surface Phenomena; Khimiya: Leningrad, 1967. (11) Melrose, J. C. Ind. Eng. Chem. 1968, 60, 53. (12) Webb, P. A.; Orr, C. Analytical Methods in Fine Particle Size Measurement; Micromeritics: Norcross, 1997. (13) Allen, T. Particle Size Measurement, 3rd ed.; Chapman and Hall: London, 1981. (14) Kresge, C. T.; Leonowicz, M. E.; Roth, W. J.; Vartuli, J. C.; Beck, J. S. Nature 1992, 359, 710. (15) Beck, J. C.; Vartuli, J. C.; Roth, W. J.; Leonowicz, M. E.; Kresge, C. T.; Schmitt, K. D.; Chu, C. T.-W.; Olson, D. H.; Sheppard, E. W.; McCullen, S. B.; Higgins, J. B.; Schlenker, J. L. J. Am.Chem. Soc. 1992, 114, 10834. (16) Inagaki, S.; Fukushima, Y.; Kuroda, K. J. Chem. Soc., Chem. Commun. 1993, 680. (17) Fenelonov, V. B.; Romannikov, V. N.; Derevyankin, A. Yu. MMM 1999, 28, 57. (18) Kruk, M.; Jaroniec, M.; Sayari, A. Langmuir 1997, 13, 1373. (19) Barrett, E. P.; Joyner, L. G.; Halenda, P. H. J. Am. Chem. Soc. 1951, 73, 373. (20) Broekhoff, J. C. P.; de Boer, J. H. J. Catal. 1967, 9, 8; 1968, 10, 373. (21) Derevyankin, A. Yu. Ph.D. Thesis. Boreskov Institute of Catalysis, Novosibirsk, 1999. (22) Ravikovitch, P. J.; Domhnaill, O.; Neimark, A. V.; Schu¨th, F.; Unger, K. Langmuir 1995, 11, 4765. (23) Gubbins, K. E. In Physical Adsorption: Experiment, Theory and Application; Fraissard, J., Conner, C. W., Eds.; Kiuver: Dordrecht, 1997; p 65. (24) The Collected Works of J. Gibbs; Longmans, Green: New York, 1928.
