Heterogeneous nucleation on aerosol particles

JOURNAL OF CHEMICAL PHYSICS

VOLUME 114, NUMBER 3

15 JANUARY 2001

Heterogeneous nucleation on aerosol particles Kira Padilla Facultad de Quı´mica, UNAM. Me´xico, D. F, 04510, Me´xico

V. Talanquer Department of Chemistry, University of Arizona, Tucson, Arizona 85721

共Received 24 August 2000; accepted 24 October 2000兲 We have applied density functional theory in statistical mechanisms to study the heterogeneous nucleation of supersaturated vapors on spherical aerosol-like substrates. Our calculations reveal the inadequacy of the classical nucleation theory in describing the condensation of droplets on very small particles. The latter approach is particularly inaccurate both at high supersaturations and in the vicinity of the wetting transition. Comparisons are also made with recent experimental results for the condensation of supersaturated vapors on submicrometer particles. © 2001 American Institute of Physics. 关DOI: 10.1063/1.1332995兴

I. INTRODUCTION

sembling aerosol particles. We show that the macroscopic capillary approximation fails to predict the nucleation behavior under several conditions, particularly for the very small particles and close to the wetting transition. We discuss the importance of further theoretical studies to fully understand the experimental results for those kinds of systems. The outline of the papers is as follows. In Sec. II we describe the density functional employed to study the system and we present some useful results for the interfacial properties of the bulk fluid. The classical nucleation theory is introduced in Sec. III and applied to the particular case of a van der Waals fluid. Section IV then turns to the predictions of the density functional approach for the heterogeneous nucleation on aerosol particles. Comparisons with the classical theory and available experimental data are also outlined in this section. Finally, Sec. V presents our conclusions.

Interest in the study of the heterogeneous nucleation of supersaturated vapors and solutions has increased in the last few years. This burst of attention has been driven by the recognition of the central role that nucleation events play in atmospheric and environmental processes of concern world wide. In particular, cloud formation processes in the atmosphere are strongly influenced by the heterogeneous nucleation induced by fine aerosols.1 Our theoretical understanding of heterogeneous nucleation has been deeply rooted in the classical nucleation theory as developed by Volmer,2 Turnbull,3 and Gretz.4 These authors considered the formation of a critical nucleus on a flat wall as the rate limiting step in the process of nucleation, with an energetic cost that is estimated in a macroscopic capillary approximation. This classical approach was later extended by Fletcher to the study of nucleation on aerosol particles, which included the effect of particle size.5 Recently, Lazaridis presented a modified version of Fletcher’s theory, incorporating the effect of the line tension on the work of formation of the critical nucleus.6 In spite of the development of new experimental techniques for determining nucleation rates, experiments on heterogeneous nucleation on aerosol particles are rare.7–9 Nevertheless, recent experiments have revealed that the classical predictions for critical supersaturations are systematically flawed in several cases, predicting too high a value for the barrier to nucleation for all particle sizes.10–14 In addition, the classical theory is fundamentally unsatisfactory because it fails to include molecular level effects. Density functional theory has been used in the last several years to develop a nonclassical approach to nucleation.15 In this theory, the properties of the critical nucleus are obtained from the structure of the free energy of a nonuniform fluid. We have successfully applied this approach to the study of homogeneous nucleation of simple and complex fluids,1 and to the heterogeneous nucleation of supersaturated vapors on planar walls.16 Here, we extend our work to the study of gas-to-liquid nucleation on spherical substrates re0021-9606/2001/114(3)/1319/7/$18.00

II. MODEL SYSTEM

Let us consider a van der Waals model for a simple fluid whose bulk Helmholtz free energy density is given by17 f 关 ␳ 兴 ⫽kT ␳ 关 ln ␳ ⫺1⫺ln共 1⫺b ␳ 兲兴 ⫺a ␳ 2 ,

共1兲

where ␳ is the number density of particles in the system, k is Boltzmann’s constant, T is the absolute temperature, and a and b are phenomenological parameters. The latter parameters, a and b, are chosen in order to include, respectively, the effect of long range attractive forces between particles and the molecular size. At fixed temperature T, chemical potential ␮, and pressure P, the properties of the coexisting vapor and liquid phases are uniquely determined by the conditions16

␮ v 共 ␳ v ,T 兲 ⫽ ␮ l 共 ␳ l ,T 兲 ,

共2a兲

P v 共 ␳ v ,T 兲 ⫽ P l 共 ␳ l ,T 兲 .

共2b兲

The corresponding phase diagram for the van der Waals fluid is characterized by a region of liquid–gas phase coexistence that ends at the critical temperature kT c ⫽8a/27b. 1319

© 2001 American Institute of Physics

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J. Chem. Phys., Vol. 114, No. 3, 15 January 2001

K. Padilla and V. Talanquer

The properties of inhomogeneous states in this system can be obtained by assuming that the Helmholtz free energy F is a function of the local density ␳ (r). In a gradient square approximation, the grand potential of the system can be written as18,19 ⍀ 关 ␳ 共 r兲兴 ⫽F 关 ␳ 共 r兲兴 ⫺ ␮ ␳ 共 r兲 ⫽

冕

dr关 f 关 ␳ 共 r兲兴 ⫺ ␮ ␳ 共 r兲兴 ⫹

⫹

冕

As

K 2

冕

⌽ 关 ␳ 共 r兲兴 da,

dr关 ⵜ ␳ 共 r兲兴 2 共3兲

where f 关 ␳ (r) 兴 is the local Helmholtz free energy density and where the square-gradient term accounts for the free energy cost associated with spatial variations of the density. If we assume that the interactions between the fluid and the aerosol particle are sufficiently short-ranged that their contribution to the free energy ⌽ 关 ␳ (r) 兴 depends only on the density at contact ␳ 0 , we can take19 ⌽ 关 ␳ 0 兴 ⫽⫺h ␳ 0 ⫺ 21 g 0 ␳ 20 ,

共4兲

where h represents the surface field, and g is a measure of the surface enhancement of molecular interactions at the substrate. A consistent comparison of our results with those of the classical theory for gas to liquid nucleation on aerosol particles requires the calculation of several macroscopic parameters for the van der Waals model. In particular, the interfacial tension of the liquid–gas interface ␥ lg , the interfacial tensions of the liquid–substrate interfaces ( ␥ sg and ␥ sl ), and the line tension ␶. These quantities are needed to estimate the classical barrier to nucleation and the contact angle of a droplet resting on a solid surface. In our model, the interfacial tension ␥ i j of the i j-interface of area A i j is given by the grand potential difference per unit area, A i j ␥ i j ⫽⍀ 关 ␳ 共 r兲兴 ⫺⍀ u 关 ␳ 兴 ,

共5兲

where ⍀ u 关 ␳ 兴 is the grand potential of the uniform system, and ⍀ 关 ␳ (r) 兴 is the grand potential of the equilibrium density profile for that particular interface. The corresponding density profiles can be obtained by minimizing the grand potential in Eq. 共3兲 and solving the associated Euler–Lagrange equation,18,20

␦ ⍀ 关 ␳ 共 r兲兴 d f 关 ␳ 共 r兲兴 ⫽ ⫺ ␮ ⫺Kⵜ 2 ␳ 共 r兲 ⫽0, ␦ ␳ 共 r兲 d ␳ 共 r兲

共6兲

under appropriate boundary conditions. The mathematical and numerical methods commonly used to solve Eq. 共6兲 have been described extensively in previous works.20,21 The resulting interfacial tensions allow us to evaluate the bulk contact angle ␪ 0 as a function of the surface field h using the Young equation

␥ sl ⫺ ␥ sg ⫹ ␥ lg cos ␪ 0 ⫽0.

共7兲

Figure 1 illustrates typical results for a van der Waals fluid in contact with a flat substrate with marginal surface enhancement (g⫽0), at a constant reduced temperature T r ⫽T/T c

FIG. 1. The bulk contact angle ␪ 0 as a function of the surface field h for a system with marginal surface enhancement (g⫽0). The reduced temperature T r ⫽T/T c ⫽0.5 in this case. The locations of the wetting (h/kT c ⫽0.8677) and drying (h/kT c ⫽⫺0.9711) transitions are indicated.

⫽0.5. We have set the dimensionless constant K/(kT c b 5/3) ⫽4 in all our calculations, which produces reasonable estimates of the surface tension for real liquids at several temperatures. The system exhibits a wetting 共drying兲 first-order transition at the value of the surface field h where the contact angle ␪ 0 ⫽0° ( ␪ 0 ⫽180°). In the following sections, this figure can serve as a useful reference to characterize the wetting properties of the substrate given a particular value of the surface field. The line tension of a thermodynamic state in which a pair of coexisting bulk phases are in contact with a planar substrate is given by17

␶ ⫽min lim

␳ R i j →⬁

冕

A

da⍀ 关 ␳ 共 r兲兴 ⫺

␥i jRi j , 兺 i, j

共8兲

where the distances R i j are the lengths of the two-phase i j-interfaces within the area of integration (sl, sg, and lg). This area is in a plane perpendicular to the three-phase contact line, and its sides are perpendicular to any intersecting two-phase interface 共see Ref. 22 for more details兲. The evaluation of ␶ requires that Eq. 共6兲 be solved allowing for density variations in two directions perpendicular to the three-phase contact line. The resulting nonlinear partial differential equations can be solved using a nonlinear multigrid method: the full approximation storage algorithm.23 For systems with g⫽0, the line tension shows a rapid increase close to the drying and wetting boundaries, and reaches a minimum at a small negative value of h/kT c 共see Fig. 3 in Ref. 16兲. In systems with a negative surface enhancement, g⬍0, ␶ exhibits qualitatively similar behavior but becomes more negative for small and intermediate absolute values of the surface field. In general, the line tension increases with temperature for constant values of h. We will



Heterogeneous nucleation on aerosol particles

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Thus, for positive values of the line tension, the contact angle ␪ * is always greater than its macroscopic limit ␪ 0 as given by the Young equation 关Eq. 共7兲兴. The height of the barrier to nucleation corresponds to the maximum value of ⌬⍀ in Eq. 共9兲 under the equilibrium condition defined by Eq. 共11兲. The corresponding critical nucleus has a radius r * d that satisfies the Laplace relationship for mechanical stability r d* ⫽2 ␥ l v /⌬ P,

共13兲

and the work of formation results in

*⫽ ⌬⍀ CL FIG. 2. Schematic representation of a liquid cluster of radius r d and contact angle ␪ on a spherical aerosol particle of radius R p . The continuous external lines in this figure outline a cross section of the type of conical container in which the system is embedded in our density functional calculations.

8 ␲ ␥ 3l v 3⌬ P

use these results in the next section to estimate the barrier to nucleation as predicted by the classical theory for nucleation of a van der Waals fluid on spherical substrates.

In classical nucleation theory 共CNT兲 the work of formation of an embryo of the liquid phase on the surface of a spherical aerosol particle is given by5,6 共9兲

where V is the volume of the nucleating droplet, ⌬ P⫽ P l ⫺ P is the pressure difference between the liquid in the droplet and the surrounding metastable vapor, and L slg corresponds to the length of the three-phase contact line. The original classical theory of heterogeneous nucleation on aerosol particles as developed by Fletcher neglected the effect of the line tension.5 Recently, Lazaridis considered the influence of this additional term in the free energy, introducing the following revised approach.6 Referring to Fig. 2, we see that V⫽ 13 ␲ r 3d 共 2⫺3 cos ⌿⫹cos3 ⌿ 兲共10a兲

A lg ⫽2 ␲ r 2d 共 1⫺cos ⌿ 兲 ,

共10b兲

A sl ⫽2 ␲ R 2p 共 1⫺cos ␾ 兲 ,

共10c兲

L slg ⫽2 ␲ R p sin ␾ ,

共10d兲

where r d represents the radius of the spherical droplet and R p is the radius of the aerosol particle. For a given volume V, the work of formation in Eq. 共9兲 is minimum for the spherical sector with a contact angle ␪ * given by

␥ sl ⫺ ␥ sg ⫹ ␥ lg cos ␪ * ⫹

␶ ⫽0, R p tan ␾

共11兲

where tan ␾ ⫽

r d sin ␪ * R p ⫺r cos ␪ *

.

共 1⫺cos ␾ 兲 . sin ␾

共14兲

冉

冊冋冉冊冉冊册冉冊

1⫺mx g

⫹3mx 2

3

⫹x 3 2⫺3

x⫺m ⫺1 , g

x⫺m x⫺m ⫹ g g

3

共15兲

where

III. CLASSICAL NUCLEATION THEORY

⫺ 13 ␲ R 3p 共 2⫺3 cos ␾ ⫹cos3 ␾ 兲 ,

f 共 m,x 兲 ⫹2 ␲ R p ␶

In this relationship, the geometrical factor f (m,x) is a function of m⫽cos ␪* and x⫽R p /r d* , and is given by f 共 m,x 兲 ⫽1⫹

⌬⍀⫽⫺V⌬ P⫹ ␥ lg A lg ⫹ 共 ␥ sl ⫺ ␥ sg 兲 A sl ⫹ ␶ L slg ,

2

共12兲

g⫽ 共 1⫹x 2 ⫺2mx 兲 1/2.

共16兲

If we further assume that the liquid is incompressible and the surrounding metastable vapor is ideal, we can take ⌬ P⫽ ␳ l kT ln S,

共17兲

where S⫽ P/ P sat is the supersaturation, the ratio of the actual vapor pressure P to the equilibrium pressure at coexistence P sat . In classical nucleation theory the rate of formation of critical droplets per aerosol particle per unit time is written as

* /kT 兲 , J⫽J 0 exp共 ⫺⌬⍀ CL

共18兲

where the particular structure of the pre-exponential factor depends on the assumed mechanism for particle deposition 共like direct condensation of molecules from the vapor phase or diffusion of adsorbed molecules on the surface of the aerosol particle兲. We have explored the predictions of CNT for the properties of critical nuclei at different values of the surface fields h,g, and the radius R p of aerosol particles surrounded by a van der Waals fluid. In general, results for CNT show a strong dependence on the particular value of the line tension. For positive values of ␶, the theory predicts that the microscopic contact angle ␪ * decreases with the radius of the critical nucleus r * for a given particle size. The inverse trend is observed in systems with highly negative values of ␶. The * is always a monotonic predicted barrier to nucleation ⌬⍀ CL decreasing function of the saturation S and, for most systems, it becomes smaller as the surface field h increases 共given aerosol particles of a certain size兲. However, if the line tension of the system is positive, CNT predicts an inversion in * as a function of h for values of the the behavior of ⌬⍀ CL surface field close to that associated with the wetting transition. As shown in Fig. 3 for aerosol particles with three different radii R p at T r ⫽0.5 and S⫽4.5, the classical work of


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* at constant FIG. 3. Classical work of formation of critical nuclei ⌬⍀ CL supersaturation, S⫽4.5, as a function of the surface field h on aerosol particles with different radii R p /b 1/3⫽R * p (g⫽0 and T r ⫽0.5).

formation of the critical nucleus increases in the vicinity of the wetting transition (h/kT c ⫽0.8677), where the height of the barrier to nucleation seems to be determined by the energy cost of creating the three-phase line. The location of the minimum shifts to higher values of the surface field at lower * becomes a monotonic decreassupersaturations, and ⌬⍀ CL ing function of h for states close to the binodal. In all cases, the work of formation of the critical nucleus increases when the radius of the aerosol particle decreases in a given supersaturated vapor.

IV. NUCLEATION: DENSITY FUNCTIONAL APPROACH

We will now compare the prediction of CNT with those generated by the density functional theory 共DFT兲. In this approach, critical droplets formed on the surface of an aerosol particle immersed in a metastable vapor correspond to saddle point solutions of the grand potential ⍀ 关 ␳ (r) 兴 in an open system.15 The corresponding density profiles satisfy the Euler–Lagrange equation 关Eq. 共6兲兴 and, given the symmetry of the system, one can expect them to be a function of both the radial distance to the center of the aerosol particle, r, and the polar angle ␾ 共see Fig. 2兲. Away from the binodal, the solution of Eq. 共6兲 is more complicated because any relaxation procedure is now unstable. However, saddle point solutions to the grand potential become minima of the Helmholtz free energy F⫽⍀⫹ ␮ ␳ when the total number of particles in the system N is fixed.16,24 Our system can then be thought of as being contained inside a conical container with perfectly nonwetting– nondrying walls as represented in Fig. 2. The size of the container, V c , is chosen to assure that its walls do not alter the distribution of matter inside or outside the system. The restriction


FIG. 4. Density profile ␳ (r, ␾ ) for a critical nucleus condensed on a spherical particle with radius R p /b 1/3⫽10.0. The surface fields h/kT c ⫽0.6 and g⫽0 in this particular case.

N⫽

冕␳ Vc

共 r兲 dr,

共19兲

can then be introduced into Eq. 共6兲 to eliminate the dependence on the chemical potential ␮.16 The resulting Euler– Lagrange can be solved by a nonlinear multirigid method23 for given values of N and T to obtain the density profile ␳ (r, ␾ ) for the critical nucleus. Given the symmetry of our system, the solution of the corresponding nonlinear partial differential equation is obtained using polar coordinates (r, ␾ ), by working on a curved lattice consisting of concentric spherical layers with a constant grid spacing ⌬r⫽1/40 and ⌬ ␾ ⫽1/60, for the finest grid. A typical result for ␳ (r, ␾ ) for a nucleating cluster on a spherical particle of radius R p ⫽10.0 is depicted in Fig. 4. The work of formation of the critical droplet is then given by the grand potential difference

* ⫽⍀ 关 ␳ 共 r, ␾ 兲兴 ⫺⍀ 关 ␳ 共 r 兲兴 , ⌬⍀ DFT

共20兲

where ⍀ 关 ␳ (r) 兴 is the grand potential of the surrounding metastable vapor, with density ␳ (r), in contact with the aerosol particle. The thermodynamic properties of this phase can be obtained by calculating the chemical potential of the system 共see Ref. 16 for details兲. The rate of nucleation at a given supersaturation is estimated by means of Eq. 共18兲, where we employ the same pre-exponential factor J 0 as in the classical theory. Figure 5共a兲 illustrates the behavior of the work of forma* as a function of the supersatution of critical nuclei ⌬⍀ DFT ration S for aerosol particles of radius R p /b 1/3⫽3.0 and with a marginal surface enhancement g⫽0. Results for several values of the surface field h have been included in this figure, along with the barrier to homogeneous nucleation in the corresponding metastable vapor. In contrast with the predictions




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Surface adsorption on a substrate reduces the extension of the metastable region available for nucleation. This effect becomes less pronounced as the radius of the spherical particles decreases, as depicted in Fig. 5共b兲. In this figure we * as a function of S for aerosol show the behavior of ⌬⍀ DFT particles of different sizes and with surface fields h/kT c ⫽0.8 and g⫽0. The value of S at the surface spinodal, where * ⫽0, increases monotonously from its limit for a flat ⌬⍀ DFT surface 共for R p →⬁) to its value at the bulk spinodal 共for R p ⫽0). Hence, the barrier to nucleation at a given supersaturation is always higher for the critical nuclei formed on the smaller aerosol particles. The corresponding cluster size, as measured by the clusters’ excess number of particles i e⫽

* as a function of the FIG. 5. Work of formation of critical clusters ⌬⍀ DFT supersaturation S for aerosol particles with marginal surface enhancement g⫽0 at T r ⫽0.5. Results for: 共a兲 particles of a set radius R p /b 1/3⫽R * p ⫽3.0 and different surface fields h; 共b兲 particles of different sizes, R * p , but similar surface field h/kT c ⫽0.8. The solid line in 共a兲 depicts the height of the barrier to homogeneous nucleation.

of CNT for this kind of system 共see Fig. 3兲, results from DFT for the height of the barrier to nucleation do not show any non monotonic behavior close to the bulk wetting transition (h/kT c ⫽0.8677 at T r ⫽0.5). For all supersaturations, the work of formation of the critical clusters decreases as h increases, and eventually becomes zero at a surface spinodal. The surface spinodals correspond to inflection points of the Helmholtz free energy F 关 ␳ (r, ␾ ) 兴 for any value of the surface potential ⌽( ␳ 0 ). 25,26 For repulsive substrates (h⬍0), heterogeneous nucleation on the aerosol particle is the most likely outcome at low supersaturations. As was the case for nucleation on a flat wall, however, homogeneous nucleation of free liquid clusters is favored close to the bulk spinodal.

冕

Vc

dr共 ␳ 共 r, ␾ 兲 ⫺ ␳ 共 r 兲兲 ,

共21兲

is also larger for the smaller particles. The effect of the substrate’s size and curvature on the properties of the critical nucleus become practically negligible for aerosol particles with R⬎100b 1/3. When compared with the results of DFT, CNT fails by predicting too high a value for the barrier to nucleation at high supersaturation and too low a value at low supersaturations. This result is illustrated in Fig. 6共a兲, where we compare the rates of nucleation predicted by both theories at several supersaturations and different values of the surface field for particles with R p /b 1/3⫽10.0. For systems in which the line tension is largely positive 共negative兲, CNT tends to overestimate 共underestimate兲 the height of the barrier to nucleation at all supersaturations. Deviations in the predictions of both theories become more strongly dependent on the value of the supersaturation S in the smaller aerosol particles, as illustrated in Fig. 6共b兲 for particles with h/kT c ⫽0.4 and g⫽0 at T r ⫽0.5. Recently, Chen and co-workers have reported interesting results for the heterogeneous nucleation of water and n-butanol on monodisperse submicrometer particles of different materials (SiO2, TiO2, SiC兲.10–14 Their experimental results for the critical supersaturation S cr at several temperatures are consistently lower than those predicted by the classical theory even for perfectly wet particles. These authors point out that the discrepancy between theory and experiment cannot be accounted for by the effect of the line tension on the barrier to nucleation, or by the effect of surface diffusion on the nucleation rate pre-exponential factor 关J 0 in Eq. 共18兲兴. Intrigued by these results, we also explored the behavior of the critical supersaturation S cr for nucleation of the van der Waals fluid on aerosol particles of different sizes and for different surface fields. Assuming that the actual rate of nucleation is determined mainly by the height of the barrier to nucleation, we define S c as the value of the supersaturation at which the work of formation of the critical nuclei ⌬⍀ * is equal to an arbitrary reference value. 共In particular, we choose ⌬⍀ * /kT⫽40.0.) Results for aerosol particles with a surface enhancement g⫽0 at T r ⫽0.5 are depicted in Fig. 7; the solid line in this figure follow the predictions of CNT in the limit of perfectly wet particles. For this type of system, a surface field h/kT c ⫽0.8677 leads to total wetting when R p


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FIG. 7. The critical supersaturation S cr as a function of the radius R p for aerosol particles with a marginal surface enhancement g⫽0 at T r ⫽0.5. The solid line depicts the predictions of the classical theory for perfectly wet particles.

completely wet particles, however, are always below those predicted by our theory in systems that exhibit bulk partial wetting. From this perspective, our model does not seem to take into account all of the relevant factors that determine the low critical supersaturations characteristic of some real fluids. V. CONCLUSIONS

FIG. 6. The ratio J CL /J DFT of the classical J CL and the density functional theory J DFT rates of nucleation as a function of the supersaturation S for aerosol particles with marginal surface enhancement g⫽0 at T r ⫽0.5. Results for 共a兲 particles of as set radius R p /b 1/3⫽R * p ⫽10.0 and different surface fields h; 共b兲 particles of different sizes, R * p , but similar surface field h/kT c ⫽0.4.

→⬁ 共flat wall, ␪ 0 ⫽0), but smaller particles are partially wet due to the effect of a positive line tension. In this case, DFT predicts a higher critical supersaturation than CNT for all particle sizes. For values of h closer to the surface spinodal (h/kT c ⫽1.1 in Fig. 7兲, the latter pattern can be reversed but only in systems in which the bulk contact angle has already vanished. For systems with negative surface enhancement g ⬍0, which is probably the most natural case, our results exhibit the same qualitative behavior. In this case, the free * , which has energy contribution of the line tension to ⌬⍀ DFT a smaller positive value or even a negative value, reduces the nonclassical critical supersaturation in systems with a small but definite bulk contact angle. Classical values of S cr for

We have used DFT to study the heterogeneous nucleation of a van der Waals fluid on spherical substrates resembling small aerosol particles. Comparison of our results with those of a CNT show that the capillary approximation tends to overestimate the effect of the line tension on the work of formation of critical nuclei. Deviations between DFT and the classical predictions for the height of the barrier to nucleation become more pronounced for smaller aerosol particles, particularly at high supersaturations and close to the bulk wetting transition. In spite of their shortcomings, CNT predictions that incorporate the effect of the line tension represent a considerable improvement over those generated when the latter free energy contribution is not taken into account. Although density functional theory naturally incorporates the effect of size and curvature on the microscopic interfacial and line tension of the critical droplets, our model seems unable to fully explain the available experimental results for the heterogeneous nucleation of fluids on submicrometer aerosol particles. It is clear that a more realistic modeling of the intermolecular interactions between particles in the system 共fluid–solid and fluid–fluid兲 is needed to adequately represent the behavior of real fluids. The introduction of long-range potentials should enhance the nonclassical effects, in particular when their range becomes comparable to the size of the critical nucleus.



Because of its relevance in environmental studies, most of the experimental work on condensation of supersaturated fluids on aerosol particles has been focused on the nucleation of polar fluids on diverse substrates. Unfortunately, the nucleation behavior of polar systems tends to deviate from that of simpler fluids.27,28 In fact, we still lack a satisfactory theoretical model that can explain the remarkably high critical supersaturations in the homogeneous nucleation of polar fluids.29,30 From this point of view, it would be desirable to have access to accurate measurements of gas–liquid nucleation rates for simple fluids 共nonpolar, nonassociating兲 on well-characterized, clean, solid substrates. This information would help us understand and discriminate the most influencing factors in the heterogeneous nucleation on aerosol particles. ACKNOWLEDGMENTS

Support to K.P. from the DGPA at UNAM and from CONACYT is gratefully acknowledged. 1

A. Laaksonen, V. Talanquer, and D. W. Oxtoby, Annu. Rev. Phys. Chem. 46, 489 共1995兲. 2 M. Volmer, Z. Elektrochem. 35, 555 共1929兲. 3 D. Turnbull, J. Chem. Phys. 18, 198 共1950兲. 4 R. D. Gretz, J. Chem. Phys. 45, 3160 共1966兲. 5 N. H. Fletcher, J. Chem. Phys. 29, 572 共1958兲. 6 M. Lazaridis, J. Colloid Interface Sci. 155, 386 共1993兲. 7 P. C. Mahata and D. J. Alofs, J. Atmos. Sci. 32, 116 共1975兲. 8 B. Y. H. Liu, D. Y. H. Pui, R. L. McKenzie, J. K. Agarwal, F. G. Pohl, O. Preining, G. Reischl, W. Szymansky, and P. E. Wagner, Aerosol. Sci. Technol. 3, 107 共1984兲.


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