23 2 Physical Chemistry of 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 L829/frame/ch02 Page 23 Monday, January 31, 2000 2:03 PM © 2000 by CRC Press LLC 24 Aerosol Chemical Processes in the Environment dynamics are very closely coupled. In short, chemical reactions determine particle compositions and modify their dynamics significantly, while the number, size, and composition of aerosol particles determine conditions for heterogeneous and liquid-phase chemical reactions. The formation of aerosol particles by gas-to-particle conversion (GPC) can take place through several 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 atmosphere generally with water vapor) nucleation of those low pressure vapors, (3) vapor condensation onto surfaces 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 and liquid phases. Step 2 initiates the actual phase transition (step 3) and increases aerosol particle number 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 of GPC given above. In actuality, the formation of new particles from the gas phase is only possible through homogeneous nucleation, or through nucleation initiated by molecular ion clusters too small to be classified as aerosol particles. Heterogeneous nucleation on insoluble particles initiates changes in particle size and composition distributions, but does not increase particle number concentration. 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 in the form of condensation take place between the phases. The driving force of the transition between vapor and liquid phases is the difference in vapor pressures in gas phase and at liquid surfaces. For a species not dissociating in liquid phase, the vapor 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 , Γ i 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 vapor pressure (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 may develop 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 absorb water 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 liquid surfaces are not present, supersaturation may grow until heterogeneous nucleation wets dry particle surfaces and triggers condensation. If the preexisting soluble and insoluble particle surface area is not sufficient to deplete condensable vapors rapidly enough, supersaturation may reach a point where homogeneous nucleation creates embryos of the new phase. The phase transition between vapor and liquid phases is often made easier by the presence of more 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 vapor than 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 multi- component process in the atmosphere, involving a vapor such as sulfuric acid, which has a very low 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 p X TXKe TXp T li ii i si , , (, ) (, ) ()=Γ L829/frame/ch02 Page 24 Monday, January 31, 2000 2:03 PM © 2000 by CRC Press LLC Physical Chemistry of Aerosol Formation 25 significantly lower effective supersaturations than homogeneous nucleation. Atmospheric hetero- geneous nucleation of water is, in principle, possible; the required relative humidities (R.H.) would 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. does not reach values high enough for heterogeneous nucleation of water to take place because rapid condensation on soluble aerosol particles depletes the vapor already at relative humidities below 101%. This is the process predominantly responsible for the generation of clouds and fogs in the atmosphere. 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 vapors besides 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; and condensational growth of aerosol particles, respectively. HOMOGENEOUS NUCLEATION To date, several different theories have been proposed to explain homogeneous nucleation from vapor (for review, see Reference 1). These theories can be roughly divided into microscopic and macroscopic 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 relatively simple interaction potentials, such as the Lennard-Jones potential, and are therefore of little practical value to aerosol scientists who usually deal with molecules too complex to be described by these potentials. 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 these theories can be called into question, their predictive success is in many cases reasonable. We shall therefore 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 saturation ratio 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 clusters can 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 change and 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 the surface 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: WGn A n ==+∆∆µσ L829/frame/ch02 Page 25 Monday, January 31, 2000 2:03 PM © 2000 by CRC Press LLC 26 Aerosol Chemical Processes in the Environment (2.3) In the classical theory, the constant of proportionality q 0 is assumed to be equal to N , the total number density of molecular species in the supersaturated vapor (often approximated by the monomer 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 n 2/3 . Conse- quently, the Gibbs free energy will exhibit a maximum as a function of cluster size. The cluster corresponding 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 critical one will tend to grow further. Thus, the term “nucleation rate” refers to the number of critical clusters appearing in a unit volume of supersaturated vapor in unit time. Below, the properties of the critical cluster are denoted by an asterisk. 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: (2.4) The critical work of formation is then given by (2.5) To derive an equation for the nucleation rate, one has to consider the kinetics of cluster formation; that is, rates at which clusters of various sizes grow because of addition of monomers from the vapor (condensation), and rates at which they shrink because of evaporation. The details of the kinetics are bypassed here (for more information, see e.g., Reference 8), noting just that the steady-state nucleation rate is given by: (2.6) Here, the condensation rate R = ( kT /2 π m ) 1/2 NA * is the number of molecules impinging on a unit surface per unit time, multiplied by the surface area of the critical cluster, m is the mass of a vapor molecule, and the so-called Zeldovich factor (2.7) accounts for the difference between the steady-state and equilibrium concentrations, and for the possibility of re-evaporation of supercritical clusters. It is assumed here that the sticking probability of molecules hitting the critical cluster is unity. In the steady-state, the cluster size distribution remains constant as a function of time, which can result either from constant monomer concentration (which is a generally used approximation), or from monomer production during nucleation. Note that uncertainties in the pre-exponential factors of Equation 2.6 have a much smaller effect on the value of the nucleation rate than uncertainties in W *. Nq GkT nn =− () 0 exp ∆ r v kT S * ln = 2σ W v kT S * (ln) = 16 3 32 2 πσ J RNZ W kT=− () exp * Z kT v A = σ 2 * L829/frame/ch02 Page 26 Monday, January 31, 2000 2:03 PM © 2000 by CRC Press LLC 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 q 0 = 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., q 0 in Equation 2.3 should equal the number concentration of molecules in saturated vapor). Relatedly, Girshick and Chiu 13 and Girshick 14 considered the limiting consistency problem caused by the fact that the classical distribution of n -clusters does not return the identity N 1 = N 1 . They proposed a self-consistency corrected (SCC) model in which the work of nucleus formation is calculated from (2.8) where A 1 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. Wilemski 12 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 butanol 15 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 µ ig 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 WWkTSA SCC CNT ** ln=− − 1 σ J AkT S J SCC CNT = () exp 1 σ ∂ ∂         =− W n ig T i jg * , µ µ L829/frame/ch02 Page 27 Monday, January 31, 2000 2:03 PM © 2000 by CRC Press LLC 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 Kashchiev 16 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 Laaksonen 25 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) = (32πσ 3 v 2 )/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 ∂ ∂         =+ (ln) , kT J nm ig T i jg µ µ ∆ ∆∆ ∆µ µµ µ dn d n d d fg * **+= () 32 nCT*()=′ − ∆µ 3 fn D T*()= − ∆µ 1 WW DT CNT * * ()=− L829/frame/ch02 Page 28 Monday, January 31, 2000 2:03 PM © 2000 by CRC Press LLC Physical Chemistry of Aerosol Formation 29 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 Laaksonen 25 showed that the scaling relation (Equation 2.15) is supported by results 26 from the density functional theory of nucleation 26 ; but they did not present a general theory for calculating the temperature-dependent function D(T). However, Talanquer 27 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, ∆µ s , from the Peng-Robinson equation of state, and showed that this scaling correction (Equation 2.16) substantially improves the classical nucleation rate predictions for several nonpolar and weakly polar substances. BINARY NUCLEATION 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, Reiss 31 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, Doyle 32 derived the so-called generalized Kelvin equations for binary critical clusters. 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. Wilemski 34 showed how the derivatives are removed by the correct use of the Gibbs adsorption equation, and Mirabel and Reiss, 35 and Nishioka and Kusaka 36 later argued that there are even more fundamental thermodynamical reasons for the derivative terms not to appear (see also Reference 28). The discussion below, however, follows Wilemski’s derivation. The change of Gibbs free energy of formation of a spherical binary liquid cluster from the vapor 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, ∆µ i 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 of the cluster are assumed to be the same as for macroscopic systems with plane surfaces; and possible effects on density and surface tension caused by the curvature of the cluster are neglected (the capillarity approximation). Following Wilemski 34,37 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 . DT CT s () ()= − ∆µ 2 2 ∆∆ ∆Gn n r=+ + 11 2 2 2 4µµπσ nnn ii s i b =+ L829/frame/ch02 Page 29 Monday, January 31, 2000 2:03 PM © 2000 by CRC Press LLC 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 the Gibbs adsorption isotherm (see, for example, References 34 and 37), (2.20) (2.21) one obtains the binary Kelvin equations (2.22) The partial molecular volumes v i are related to the cluster radius as follows: (2.23) Note that the v i values depends on composition, and they are determined at X b . From the Kelvin equations, one obtains (2.24) which can be solved numerically to find the bulk mole fraction X* b of the critical cluster at given gas-phase chemical potentials. For the radius and the free energy of formation of the critical cluster, one has (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 ∂ ∂       = ∆G n i n j 0 nd nd bl bl 11 22 0µµ+= nd nd Ad sl sl 11 22 0µµσ++= ∆µ σ i i v r += 2 0 * 4 3 3 11 2 2 πrnvnv=+ ∆∆µµ 12 21 vv= r v i i * =− 2σ µ∆ ∆Gr**= 4 3 2 πσ nd nd Ad ll 11 22 0µµσ++= L829/frame/ch02 Page 30 Monday, January 31, 2000 2:03 PM © 2000 by CRC Press LLC Physical Chemistry of Aerosol Formation 31 the mole fraction range), the predictions of the theory become unphysical. For example, with constant 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 Kelvin equations, to construct the free energy surface (∆G – n 1 – n 2 ) with Equation 2.17, and locate the saddle 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 = n 2 /(n 1 + n 2 ). If this is not the case, a method is needed to calculate X b for a general (n 1 , n 2 )-cluster. Flageollet-Daniel et al. 39 proposed to treat water/alcohol clusters in terms of a microscopic model, allowing for enrichment of the alcohol at the surface of the cluster, and at the same time depleting the interior of alcohol. Laaksonen and Kulmala 40 have proposed an alternative explicit cluster model and demonstrated 41 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 unimolecular surface layer and an interior bulk core with (2.28) The volume of a cluster is calculated assuming a spherical shape. The numbers of molecules in the 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 σ i 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 the distribution between surface and interior molecules as the partial molecular areas are taken as curvature dependent (for details, see Reference 41). The cluster model predicts, for a given set of total 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 the interior composition. These quantities and the total number of molecules are then used to determine the binary nucleus by creating a saddle surface in three-dimensional (∆G – n 1 – n 2 ) space and searching 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 the interior, and thereby also the mole fraction of the critical cluster. Surprisingly enough, it seems that this is sufficient for correcting the unphysical predictions of the classical theory, at least qualitatively (although one should bear in mind that the approximate nature of Equation 2.30 and the equations describing the molecular areas might contribute). Laaksonen 41 found that the theory produced 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 nnn ii s i b =+ ArnAnA ss ==+4 2 11 22 π σ σσ X Xv Xv Xv Xv b ss ss () = − () + − () + 1 1 11 2 2 12 L829/frame/ch02 Page 31 Monday, January 31, 2000 2:03 PM © 2000 by CRC Press LLC 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 one should therefore treat both the theoretical predictions and experimental results of the methanol nucleation with caution. Another system with a tendency to associate is the binary sulfuric acid/water mixture, which is important in ambient aerosol formation. The very high enthalpy of mixing 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. In other words, it is energetically more difficult to form a critical nucleus out of hydrates than out of monomers (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 (2.31) where the activities are given by A il = p i,sol /p i,s and A ig = p i /p i,s ; and p i , p i,s , and p i,sol denote the partial pressure, saturation vapor pressure, and vapor pressure over the solution, respectively. The correction for the acid activities has the following form: (2.32) The correction factor C h due to hydration is given by: (2.33) 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 the derivative 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 the fraction 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 hydrate distributions 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 ∆µ i ig il kT A A =− ln −       =− −kT A A kT A A kT C ag al cor ag al h ln ln ln C Kp KK Kp Kp KK Kp h w sol h w sol h w h w h n a = ++…+×…× ++…+×…×         1 1 112 112 ,, L829/frame/ch02 Page 32 Monday, January 31, 2000 2:03 PM © 2000 by CRC Press LLC [...]... (2. 75) where Aig is the gas phase activity The solution for the preceding set of equations is63 I1 = (1 − B 22 )b1 ( A1g − A1 ) + B 12 b2 ( A2 g − A2 ) 1 − B 22 − B11 + B11 B 22 − B 12 B21 (2. 76) I2 = (1 − B11 )b2 ( A2 g − A2 ) + B21b1 ( A1g − A1 ) 1 − B 22 − B11 + B11 B 22 − B 12 B21 where bi = Bij = −4 πrMi β M ,i Di pis, f RT∞ Ai Li L j Mi 2 4 πrβ T k∞ RT∞ (2. 77) bi Note that although the mass fluxes now... and S23 by S 12 = 2 πr 2 (1 − cos ψ ) (2. 37) 2 S23 = 2 πRp (1 − cos φ) (2. 38) Here, cos ψ = − cos φ = ( (r − R m) (R p (2. 39) d p − rm ) (2. 40) d 2 d = Rp + r 2 − 2rRp m ) 1/ 2 (2. 41) where r is the radius of the embryo and Rp the radius of the solid surface (see Figure 2. 1) One should notice that the heterogeneous nucleation theory gives the same value for the critical radius as the homogeneous theory... pressure In calculating the mass fluxes, the disturbance of the saturation equilibrium can be formally taken into account by a correction factor (see Equation 2. 72) © 20 00 by CRC Press LLC L 829 /frame/ch 02 Page 42 Monday, January 31, 20 00 2: 03 PM 42 Aerosol Chemical Processes in the Environment MASS FLUX EXPRESSIONS Several mass flux expressions and growth equations can be found in the literature The well-known... between the embryo and the surrounding substrate The embryo formation occurs via three successive stages In the beginning, the surface area of the interface between the liquid embryo and the substrate (S23) is smaller than the surface area of the active site (S) When the embryo has grown enough, so that S23 = S, the contact angle, and thereby the form of the embryo, start to change Finally, after the transformation... 20 00 2: 03 PM 34 FIGURE 2. 1 Aerosol Chemical Processes in the Environment A cluster (2) on aerosol particle (3) in gas phase (1) σij and Sij 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– 23 )/σ 12, and the values of S 12. ..L 829 /frame/ch 02 Page 33 Monday, January 31, 20 00 2: 03 PM Physical Chemistry of Aerosol Formation 33 I = RAV FZ exp( − ∆G * kT ) (2. 34) Here, F is the total number of molecular species in the vapor, and RAV is the average condensation rate For nonassociating vapors, one obtains RAV = R1 R2 R1 sin 2 Φ + R2 cos 2 Φ (2. 35) where Φ is the angle between the n2-axis and the direction of cluster growth at the. .. (2. 48) L 829 /frame/ch 02 Page 36 Monday, January 31, 20 00 2: 03 PM 36 Aerosol Chemical Processes in the Environment where mµ is the reduced mass of the two molecules For V, Lazaridis et al. 52 used the modified Lennard-Jones potential of polar molecules, resulting in τo = 2. 55 × 10–13 s, which corresponds to water–water interaction For E, they used the latent heat of condensation (see Reference 53) The minimum... of Aerosol Formation 45 REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 Laaksonen, A., Talanquer, V., and Oxtoby, D.W., Annu Rev Phys Chem., 46, 489, 1995 Volmer, M and Weber, A., Z Phys Chem., 119, 27 7, 1 926 Farkas, L., Z Phys Chem., 125 , 23 6, 1 927 Volmer, M., Z Elektrochem., 35, 555, 1 929 ... at 27 3K when surface diffusion was allowed for The discontinuity of two or more volume phases is connected to surface tension Line tension, on the other hand, arises from the discontinuity between two or more surface phases The Gibbs free energy, which takes into account the line tension (σt), is ∆G* = σt 2 πr *2 σ 12 f ( m, x ) − S + 2 πRp σ t sin φ Rp tan φ 23 3 (2. 58) The inclusion of a positive line... expressions without the dependence on the droplet temperature is desirable Expressions in this form can be derived by replacing the © 20 00 by CRC Press LLC L 829 /frame/ch 02 Page 44 Monday, January 31, 20 00 2: 03 PM 44 Aerosol Chemical Processes in the Environment explicit droplet temperature dependence by the dependence on the products of latent heats and mass fluxes Following this approach, linearizing with respect . ()= − ∆µ 2 2 ∆∆ ∆Gn n r=+ + 11 2 2 2 4µµπσ nnn ii s i b =+ L 829 /frame/ch 02 Page 29 Monday, January 31, 20 00 2: 03 PM © 20 00 by CRC Press LLC 30 Aerosol Chemical Processes in the Environment The saddle. Xv b ss ss () = − () + − () + 1 1 11 2 2 12 L 829 /frame/ch 02 Page 31 Monday, January 31, 20 00 2: 03 PM © 20 00 by CRC Press LLC 32 Aerosol Chemical Processes in the Environment magnitude of the measured rates in several. References 34 and 37), (2. 20) (2. 21) one obtains the binary Kelvin equations (2. 22) The partial molecular volumes v i are related to the cluster radius as follows: (2. 23) Note that the v i values depends