AEROSOL CHEMICAL PROCESSES IN THE ENVIRONMENT - CHAPTER 2 docx

Aerosol Formation Markku Kulmala, Timo Vesala, and Ari Laaksonen CONTENTS Introduction ...23 Homogeneous Nucleation ...25 One-Component Nucleation...25 Classical Theory ...25 Self-Consis

Aerosol Formation

Markku Kulmala, Timo Vesala, and Ari Laaksonen

CONTENTS

Introduction 23

Homogeneous Nucleation 25

One-Component Nucleation 25

Classical Theory 25

Self-Consistency 27

Nucleation Theorem 27

Scaling Correction to Classical Theory 28

Binary Nucleation 29

Classical Theory 29

Explicit Cluster Model 31

Hydration 32

Nucleation Rate 32

Heterogeneous Nucleation 33

Binary Heterogeneous Nucleation on Curved Surfaces 33

Free Energy of Embryo Formation 33

Nucleation Rate 35

Nucleation Probability 36

The Effect of Active Sites, Surface Diffusion, and Line Tension on Heterogeneous Nucleation 36

Activation 38

Condensation 40

Vapor Pressures and Liquid Phase Activities 40

Mass Flux Expressions 42

Uncoupled Solution 42

Semi-Analytical Solution 43

Acknowledgments 44

References 45

INTRODUCTION

The formation and growth of aerosol particles in the presence of condensable vapors represent processes of major importance in aerosol dynamics The emergence of new particles from the vapor changes both the aerosol size and composition distributions The size distribution of the aerosol at

a given time is affected by particle growth rates, which in turn are governed partially by particle compositions Aerosol deposition, which transfers chemical species from the atmosphere, is influ-enced by particle size It is easy to see, therefore, that the physical and chemical aspects of aerosol

24 Aerosol Chemical Processes in the Environment

dynamics are very closely coupled In short, chemical reactions determine particle compositionsand modify their dynamics significantly, while the number, size, and composition of aerosol particlesdetermine conditions for heterogeneous and liquid-phase chemical reactions

The formation of aerosol particles by gas-to-particle conversion (GPC) can take place throughseveral different mechanisms, including (1) reaction of gases to form low vapor pressure products(e.g., the oxidation of sulfur dioxide to sulfuric acid), (2) one- or multicomponent (in the atmospheregenerally with water vapor) nucleation of those low pressure vapors, (3) vapor condensation ontosurfaces of preexisting particles, (4) reaction of gases at the surfaces of existing particles, and (5)chemical reactions within the particles Steps 1, 4, and 5 affect the compositions of both vapor andliquid phases Step 2 initiates the actual phase transition (step 3) and increases aerosol particlenumber concentration, while step 3 increases aerosol mass

The purpose of this chapter is to focus on Steps 2 and 3 of the rather generalized picture ofGPC given above In actuality, the formation of new particles from the gas phase is only possiblethrough homogeneous nucleation, or through nucleation initiated by molecular ion clusters toosmall to be classified as aerosol particles Heterogeneous nucleation on insoluble particles initiateschanges in particle size and composition distributions, but does not increase particle numberconcentration Soluble aerosol particles may grow as a result of equilibrium uptake of vapors(mostly water), but only when the vapor becomes supersaturated can significant mass transfer inthe form of condensation take place between the phases

The driving force of the transition between vapor and liquid phases is the difference in vaporpressures in gas phase and at liquid surfaces For a species not dissociating in liquid phase, thevapor pressure (p l,i) at the surface of aerosol particle is given by:

(2.1)

Here, X i is the mole fraction of component i, Γi is the activity coefficient, Ke i is the Kelvin effect(increase of saturation vapor pressure because of droplet curvature), and p s,i is the saturation vaporpressure (relative to planar surface) For more detailed discussion of liquid phase activities, refer

to the chapter subsection “Vapor Pressure and Liquid Phase Activities.”

If the partial pressure of species i in the gas phase (p g,i) is higher than p l,i, a net mass flux maydevelop from gas phase to liquid phase A prerequisite is the existence of (enough of) liquid surfaces;this is the case if the aerosol contains a sufficient amount of soluble particles that are able to absorbwater and other vapors at subsaturated conditions (i.e., grow along their Köhler curves) Conden-sation will then start as soon as the vapor becomes effectively supersaturated However, if liquidsurfaces are not present, supersaturation may grow until heterogeneous nucleation wets dry particlesurfaces and triggers condensation If the preexisting soluble and insoluble particle surface area isnot sufficient to deplete condensable vapors rapidly enough, supersaturation may reach a pointwhere homogeneous nucleation creates embryos of the new phase

The phase transition between vapor and liquid phases is often made easier by the presence ofmore than one condensing species The reason for this can be understood from Equation 2.1: mixing

in the liquid phase tends to lower the equilibrium vapor pressure p l,i of species i compared with

p s,i, and therefore effective saturation takes place at lower vapor densities in multicomponent vaporthan in vapor containing a single species

Homogeneous nucleation may create new particles in air with low aerosol concentration, but

a high effective supersaturation is needed Therefore, homogeneous nucleation is always a component process in the atmosphere, involving a vapor such as sulfuric acid, which has a verylow saturation vapor pressure and can form droplets with water even at low relative humidities.Homogeneous nucleation of pure water requires relative humidities of several hundred percent,and is thus out of the question in the atmosphere Heterogeneous nucleation can take place at

multi-p l i, = ΓX i i( , )T X Ke T X p i( , ) s i,( )T

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Physical Chemistry of Aerosol Formation 25

significantly lower effective supersaturations than homogeneous nucleation Atmospheric geneous nucleation of water is, in principle, possible; the required relative humidities (R.H.) would

hetero-be just a few percent over one hundred (depending on the surface characteristics of the particles)and lower still if some other vapor were to participate However, usually the atmospheric R.H doesnot reach values high enough for heterogeneous nucleation of water to take place because rapidcondensation on soluble aerosol particles depletes the vapor already at relative humidities below101% This is the process predominantly responsible for the generation of clouds and fogs in theatmosphere Here, the starting point of condensation is not a genuine nucleation process, and can

be called activation of soluble aerosol particles Note also that in the case of activation, other vaporsbesides water may have an effect: by depressing the equilibrium vapor pressure of the particles(p l,i), they may lower the threshold R.H at which activation takes place

This chapter focuses on the various aspects of aerosol formation by gas-to-particle conversion.Subsequent chapter sections are devoted to a review of theoretical investigations on one- and two-component homogeneous nucleation; heterogeneous nucleation; activation of soluble particles; andcondensational growth of aerosol particles, respectively

HOMOGENEOUS NUCLEATION

To date, several different theories have been proposed to explain homogeneous nucleation fromvapor (for review, see Reference 1) These theories can be roughly divided into microscopic andmacroscopic ones From a theoretical point of view, the microscopic approach is more fundamental,

as the phenomenon is described starting from the interactions between individual molecules.However, microscopic nucleation calculations have thus far been limited to molecules with relativelysimple interaction potentials, such as the Lennard-Jones potential, and are therefore of little practicalvalue to aerosol scientists who usually deal with molecules too complex to be described by thesepotentials The macroscopic theories, on the other hand, rely on measurable thermodynamic quan-tities such as liquid densities, vapor pressures, and surface tensions This enables them to be used

in connection with real molecular species; and although certain assumptions underlying thesetheories can be called into question, their predictive success is in many cases reasonable We shalltherefore focus on the macroscopic nucleation theories below

O NE -C OMPONENT N UCLEATION

Classical Theory

The first quantitative treatment that enabled the calculation of nucleation rate at given saturationratio and temperature was developed by Volmer and Weber,2 Farkas,3 Volmer,4,5 Becker and Döring,6

and Zeldovich,7 and is called the classical nucleation theory The classical theory relies on the

capillary approximation: it is assumed that the density and surface energy of nucleating clusterscan be represented by those of bulk liquid According to the classical theory, the reversible work

of forming a spherical cluster from n vapor molecules is equal to the Gibbs free energy changeand can be written as:

(2.2)

where the chemical potential change between the liquid and vapor phases is given by ∆µ = –kT ln

S, S is the saturation ratio of the vapor, k is the Boltzmann constant, T is temperature, σ is thesurface tension of bulk liquid, A denotes the surface area of the cluster with a volume of V = nv,and v is the liquid-phase molecular volume The equilibrium number concentration of n-clusters

is given by:

W =∆G n=n∆µ+Aσ

26 Aerosol Chemical Processes in the Environment

(2.3)

In the classical theory, the constant of proportionality q0 is assumed to be equal to N, the totalnumber density of molecular species in the supersaturated vapor (often approximated by themonomer density)

In supersaturated vapor, the first term of Equation 2.2 is negative and proportional to the number

of molecules in the cluster, whereas the second term is positive and proportional to n2/3 quently, the Gibbs free energy will exhibit a maximum as a function of cluster size The clustercorresponding to the maximum is called critical, as it is in unstable equilibrium with the vapor;clusters smaller than the critical one will tend to decay, whereas clusters larger than the criticalone will tend to grow further Thus, the term “nucleation rate” refers to the number of criticalclusters appearing in a unit volume of supersaturated vapor in unit time Below, the properties ofthe critical cluster are denoted by an asterisk

Conse-The radius r* of the critical cluster can be located by setting the derivative of ∆G with respect

to n zero, resulting in the so-called Kelvin equation:

of the kinetics are bypassed here (for more information, see e.g., Reference 8), noting just that thesteady-state nucleation rate is given by:

(2.6)

Here, the condensation rate R = (kT/2πm)1/2NA* is the number of molecules impinging on a unitsurface per unit time, multiplied by the surface area of the critical cluster, m is the mass of a vapormolecule, and the so-called Zeldovich factor

N n=q0exp(−∆G kT n )

*ln

= 2σ

*( ln )

= 163

3 2 2

πσ

J=RNZexp(−W*kT)

Z kT

v A

= σ 2

*

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Physical Chemistry of Aerosol Formation 27

Self-Consistency

Later investigators (Courtney,9 Blander and Katz,10 and Reiss et al.,11) have pointed out that with

q0 = N, Equation 2.3 does not obey the law of mass action (see also Reference 12), and have argued

that a correct treatment of nucleation kinetics results in multiplication of I in Equation 2.6 by a

factor of 1/S (i.e., q0 in Equation 2.3 should equal the number concentration of molecules in saturated

vapor) Relatedly, Girshick and Chiu13 and Girshick14 considered the limiting consistency problem

caused by the fact that the classical distribution of n-clusters does not return the identity N1 = N1

They proposed a self-consistency corrected (SCC) model in which the work of nucleus formation

is calculated from

(2.8)

where A1 is the surface area of a (spherical) monomer in liquid phase and CNT denotes classical

theory The nucleation rate is then

(2.9)

Note that, although the SCC approach offers a correction for both the mass action consistency and

limiting consistency problems, the choice of Equation 2.8 must be regarded as somewhat arbitrary

Wilemski12 argued that the mass action consistency problem is more serious than the limiting

consistency problem because mass action consistency is fundamentally necessary, while limiting

consistency is not a fundamental property that must be satisfied by a distribution

The predictive powers of the classical theory and the SCC model appear quite similar, although

the classical theory predicts lower nucleation rates than the SCC model Both theories predict the

critical supersaturations S cr (supersaturation at which the nucleation rate reaches a certain level) of

some substances rather well and others not so well; the classical theory seems to succeed especially

with butanol15 and the SCC model with toluene.13 The prediction of correct nucleation rates is

usually more difficult than that of critical supersaturations because J is generally a very steep

function of S, and thus both of the above theories predict in some cases nucleation rates differing

from the experimental ones by several orders of magnitude A common problem with both theories

is the incorrect temperature dependence of the predicted S cr found with many substances In any

case, the fact that almost-correct critical supersaturations are predicted by theories relying on the

capillarity approximation is quite remarkable in itself

N UCLEATION T HEOREM

An important new development in nucleation studies is the rigorous proof of the so-called

Nucle-ation Theorem, given by Oxtoby and Kashchiev.16 The Nucleation Theorem relates the variation

of work of formation of the critical cluster with its molecular content:

(2.10)

Here n i denotes the number of molecules belonging to species i in a multicomponent vapor, and

µig is the gas-phase chemical potential of species i in a multicomponent vapor This result was first

proposed for one-component systems by Kashchiev,17 who assumed that the surface energy of the

jg

*

,

µ µ

28 Aerosol Chemical Processes in the Environment

nucleus is only weakly dependent on supersaturation Viisanen et al.18 derived the Nucleation

Theorem using statistical mechanical arguments that assume that the critical cluster and the

sur-rounding vapor can be treated as if decoupled This approach was generalized to binary systems

by Strey and Viisanen,19 and Viisanen et al.20 extended Kashchiev’s original derivation to the

two-component case However, only with the work of Oxtoby and Kashchiev16 was it was realized that

the Nucleation Theorem is a completely general thermodynamic statement free of any specific

model-related assumptions, and holds down to the smallest nucleus sizes The Nucleation Theorem

is particularly useful because it allows the measurement of numbers of molecules in critical clusters

This is possible because the rate of nucleation depends on the work of nucleus formation and on

a pre-exponential kinetic factor, which in turn is only weakly dependent on supersaturation It can

be shown that

(2.11)

where m is between 0 and 1 Measurements of molecular content of critical clusters have been

performed by Viisanen and Strey,15 Viisanen et al.,18,20,21 Strey and Viisanen,19 Strey et al.,22,23 and

Hruby et al.24 These studies have shown that with one-component nuclei, the Kelvin equation

predicts the critical nucleus size surprisingly well, down to about 40 to 50 molecules

Scaling Correction to Classical Theory

Applying the Nucleation theorem to a general form of reversible work of critical nucleus formation

W* = W CNT – f(n*, ∆µ), where the function f gives the departure from classical theory, McGraw

and Laaksonen25 derived the following differential equation:

(2.12)

In the classical theory, f = 0, and the equation can be solved to give n* CNT = C(T)∆µ–3 The

temperature-dependent function C(T) is identified with the help of the Kelvin relation to be C(T)

= (32πσ3v2)/3

In general, f is non-zero; and without additional information, Equation 2.12 cannot be solved.

However, in the special case that each side of Equation 2.12 vanishes separately, a class of

homogeneous solutions is obtained for n* and the product fn*:

(2.13)(2.14)

As ∆µ → 1, one must have n* → n* CNT and, hence, C ′(T) = C(T) Thus, in the generalized theory,

the number of molecules in the critical nucleus is the same as in the classical theory, in agreement

with experiments The work of nucleus formation, on the other hand, becomes

(2.15)

that is, the difference from the classical theory being given by a function that depends on temperature

only and not only supersaturation This is also in accord with experiments, as the classical theory

predicts with many substances the supersaturation dependence of nucleation rate reasonably well.(The predictions of the supersaturation dependence and the molecular content of the nucleus are

of course linked by the Nucleation Theorem.)

McGraw and Laaksonen25 showed that the scaling relation (Equation 2.15) is supported byresults26 from the density functional theory of nucleation26; but they did not present a general theory

for calculating the temperature-dependent function D(T) However, Talanquer27 pointed out that

D(T) can be obtained by requiring that the work of nucleus formation, W*, goes to zero at the

spinodal line (see Reference 28) The function D(T) then becomes

(2.16)

Talanquer obtained the chemical potential at the spinodal, ∆µs, from the Peng-Robinson equation

of state, and showed that this scaling correction (Equation 2.16) substantially improves the classicalnucleation rate predictions for several nonpolar and weakly polar substances

B INARY N UCLEATION

Classical Theory

Classical binary nucleation theory (extension of the Kelvin equation to two-component systems)was first used by Flood,29 Volmer,5 and Neumann and Döring.30 Unaware of the earlier work, Reiss31

considered binary nucleation, and noted that the growing binary clusters can be thought of as moving

on a saddle-shaped free energy surface, the saddle point corresponding to the critical cluster Building

on Reiss’ work, Doyle32 derived the so-called generalized Kelvin equations for binary criticalclusters These equations contained derivatives of surface tension with respect to particle composi-tion In 1981, Renninger et al.33 noted that the equations of Doyle were thermodynamically incon-sistent, and that the correct binary Kelvin equations do not contain any compositional derivatives

of surface tension This is, incidentally, in accord with the early German investigators Wilemski34

showed how the derivatives are removed by the correct use of the Gibbs adsorption equation, andMirabel and Reiss,35 and Nishioka and Kusaka36 later argued that there are even more fundamentalthermodynamical reasons for the derivative terms not to appear (see also Reference 28) Thediscussion below, however, follows Wilemski’s derivation

The change of Gibbs free energy of formation of a spherical binary liquid cluster from thevapor phase is expressed as (e.g., see Reference 31)

(2.17)

where n i denotes the number of molecules of the ith species in the cluster, ∆µi is the change of the

chemical potential of species i between the vapor phase and the liquid phase taken at the pressure outside of the cluster, r is the radius of the cluster, and σ is the surface tension The properties ofthe cluster are assumed to be the same as for macroscopic systems with plane surfaces; and possibleeffects on density and surface tension caused by the curvature of the cluster are neglected (thecapillarity approximation) Following Wilemski34,37 and Zeng and Oxtoby,26 one can write the total

number of molecules of species i in the cluster as:

(2.18)

Here, n s

i and n b

i are the numbers of surface and interior (“bulk”) molecules of the ith species

in the cluster, respectively The above-mentioned thermodynamic quantities are determined using

the bulk mole fraction X b

30 Aerosol Chemical Processes in the Environment

The saddle point on the free energy surface can be found by setting

(2.19)

Applying these conditions to Equation 2.17 and making use of the Gibbs-Duhem equation and theGibbs adsorption isotherm (see, for example, References 34 and 37),

(2.20)(2.21)one obtains the binary Kelvin equations

(2.25)

(2.26)

Finally, as noticed by Laaksonen et al.,38 the total number of molecules in the critical cluster can

be calculated using Equations 2.23 and 2.27

(2.27)

(which follows from the addition of Equations 2.20 and 2.21)

The predictions of binary classical nucleation theory have been found to be qualitatively correct

in the case of nearly ideal mixtures.19 However, for systems in which surface enrichment of one

of the components takes place (marked by considerable nonlinear variation of surface tension over

+2 =0

*

43

∆G*= 4 r*3

2

1 µ1+ 2 µ2+ σ=0

the mole fraction range), the predictions of the theory become unphysical For example, withconstant alcohol vapor concentration in a water/alcohol system, addition of water vapor will

suddenly result in lowering the predicted nucleation rate, associated with a prediction of negative

total number of water molecules in the critical cluster

Explicit Cluster Model

An alternative (classical) way to find the critical cluster would be, instead of using the Kelvinequations, to construct the free energy surface (∆G – n1 – n2) with Equation 2.17, and locate thesaddle point, for example, with the help of a computer This can be readily done with systems

exhibiting no surface enrichment, that is, if X b = X = n2/(n1 + n2) If this is not the case, a method

is needed to calculate X b for a general (n1, n2)-cluster Flageollet-Daniel et al.39 proposed to treatwater/alcohol clusters in terms of a microscopic model, allowing for enrichment of the alcohol atthe surface of the cluster, and at the same time depleting the interior of alcohol Laaksonen andKulmala40 have proposed an alternative explicit cluster model and demonstrated41 that for a number

of water/alcohol systems, the agreement with the cluster model and experiments is rather good.The cluster model describes a two-component liquid cluster as composed of a unimolecularsurface layer and an interior bulk core with

(2.28)

The volume of a cluster is calculated assuming a spherical shape The numbers of molecules inthe surface layer are determined from

(2.29)

where A i is the partial molecular area of species i The surface composition is assumed to be

connected to the surface tension of a bulk binary solution via a phenomenological relationship:

(2.30)

Here, X s = n s

2/(n s

1 + n s

2), and σi denotes the surface tension of pure i This description is, in effect,

an approximation to the Gibbs adsorption isotherm The cluster size is allowed to affect thedistribution between surface and interior molecules as the partial molecular areas are taken ascurvature dependent (for details, see Reference 41) The cluster model predicts, for a given set oftotal numbers of molecules at fixed gas temperature, the numbers of interior and surface molecules

in the cluster The surface tension, liquid phase activities, and density are calculated using theinterior composition These quantities and the total number of molecules are then used to determinethe binary nucleus by creating a saddle surface in three-dimensional (∆G – n1 – n2) space andsearching the saddle point

The principal difference between the cluster model and the classical theory is that in the former,the cluster size is allowed to affect the relative fractions of the molecules at the surface and in theinterior, and thereby also the mole fraction of the critical cluster Surprisingly enough, it seemsthat this is sufficient for correcting the unphysical predictions of the classical theory, at leastqualitatively (although one should bear in mind that the approximate nature of Equation 2.30 andthe equations describing the molecular areas might contribute) Laaksonen41 found that the theoryproduced well-behaved activity plots (plots of vapor phase activities at which the nucleation rate

is constant at given temperature), and that the predicted nucleation rates were within 6 orders of

n i n i s n

i b

32 Aerosol Chemical Processes in the Environment

magnitude of the measured rates in several water/alcohol systems Furthermore, Viisanen et al.20

showed that the explicit cluster model predicts almost quantitatively correct numbers of molecules

in critical water/ethanol clusters over the whole composition range

Hydration

The association of molecules in the vapor phase can significantly affect their nucleation behavior

It is known that methanol, for example, has a considerable enthalpy of self-association, and oneshould therefore treat both the theoretical predictions and experimental results of the methanolnucleation with caution Another system with a tendency to associate is the binary sulfuricacid/water mixture, which is important in ambient aerosol formation The very high enthalpy ofmixing of these species causes them to form hydrates in the gas phase The hydrates consist of one

or more sulfuric acid and several water molecules, and have a stabilizing effect on the vapor Inother words, it is energetically more difficult to form a critical nucleus out of hydrates than out ofmonomers (although from a kinetic viewpoint, hydration does make nucleation a little bit easier).Jaecker-Voirol et al.42 deduced a correction for the classical free energy of cluster formation,taking into account the effect of hydration The hydrates were assumed to contain one sulfuric acid

and one or more water molecules Expressing the chemical potential difference of species i with

the help of liquid and gas phase activities, one obtains

where the subscripts w and a refer to water and acid, respectively, K i is the equilibrium constant

for hydrate formation, and h is the number of water molecules per hydrate Jaecker-Voirol et al.42

noted that an approximate expression is obtained for the equilibrium constants by taking thederivative of ∆G of a hydrate with respect to the number of water molecules Kulmala et al.43

extended the classical hydration model into systems where the gas phase number concentrations

of acid and water molecules may be of the same order of magnitude They also showed that thefraction of free molecules to the total number of molecules in the vapor can be solved numerically,rendering the equilibrium constants unnecessary However, the resulting sulfuric acid hydratedistributions were shown to be similar to those calculated by Jaecker-Voirol et al.42

Nucleation Rate

The nucleation rate in a binary system is:44

∆µi

ig il

Here, F is the total number of molecular species in the vapor, and R AV is the average condensationrate For nonassociating vapors, one obtains

(2.35)

where Φ is the angle between the n2-axis and the direction of cluster growth at the saddle point of

the free energy surface The Zeldovich nonequilibrium factor Z can be obtained from second

derivatives of ∆G at the saddle point (see Reference 44 for details) An approximate expression for

Z was given by Kulmala and Viisanen,45 who derived the classical binary equations starting fromthe “average monomer” concept In this case the Zeldovich factor is reduced to that of the one-

component case (Equation 2.7 with v = (1 – X)v1 + Xv2) and S = (A 1g /A 1l)1–X (A 2g /A 2l)X Thecorresponding approximate growth angle is the steepest descent angle with tan Φ = X/(1 – X).

Although it is true that the nucleation rate is governed primarily by the exponent of ∆G, one

should be careful when using the kinetic expressions Kulmala and Laaksonen46 showed that variouskinetic expressions presented in the literature produced H2O/H2SO4 nucleation rates differing fromeach other by several orders of magnitude In a recent study, Vehkamäki et al.47 used numericalmethods to solve the binary kinetic equations exactly They found that in the water/sulfuric acidsystem the exact rates were within one order of magnitude of those produced by the analyticalexpressions of Stauffer.44

HETEROGENEOUS NUCLEATION

The quantification of heterogeneous nucleation is even more difficult than that of homogeneousnucleation This is due to the complexity of interactions between the nucleating molecules and theunderlying surface The heterogeneous nucleation rate is strongly dependent on the characteristics

of the surface, and it is extremely difficult to produce well-defined surfaces for experimentalinvestigations The lack of experimental data, on the other hand, makes it difficult to verify anytheoretical ideas It seems probable that in the future more information of the details of heteroge-neous nucleation phenomena will be acquired through molecular dynamics or Monte Carlo simu-lations, rather than through laboratory experiments (see Reference 48)

A further complication, compared to laboratory conditions, emerges when one tries to carryout calculations of heterogeneous nucleation at ambient conditions: in the lab, the surface materials

at least are known, but this is usually not the case in the atmosphere The uncertainties associatedwith atmospheric heterogeneous nucleation calculations can therefore by very large However, someguidance can be acquired about the conditions at which heterogeneous nucleation can take placeusing the classical nucleation theory, which is reviewed below Also considered are some factorsthat may be important, but are left out of the classical description

B INARY H ETEROGENEOUS N UCLEATION ON C URVED S URFACES

Free Energy of Embryo Formation

In classical nucleation theory, the Gibbs free energy of formation of a liquid embryo from a binarymixture of vapors onto a curved surface is given by the expression49:

= − ln − ln + 12σ12+(σ23−σ13) 23

34 Aerosol Chemical Processes in the Environment

σij and S ij are the surface free energy and surface area of the interface, respectively, between phases

i and j The gas phase is indexed by 1, the liquid phase embryo by 2, and the substrate by 3 The

contact angle θ is given by cos θ = m = (σ13–σ23)/σ12, and the values of S12 and S23 by

(2.37)(2.38)Here,