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Cite This: J. Phys. Chem. A XXXX, XXX, XXX−XXX

Effect of Heterogeneous Chemical Reactions on the Köhler Activation of Aqueous Organic Aerosols Yuri S. Djikaev* and Eli Ruckenstein† Department of Chemical and Biological Engineering, SUNY at Buffalo, Buffalo, New York 14260, United States ABSTRACT: We study some thermodynamic aspects of the activation of aqueous organic aerosols into cloud droplets considering the aerosols to consist of liquid solution of water and hydrophilic and hydrophobic organic compounds, taking into account the presence of reactive species in the air. The hydrophobic (surfactant) organic molecules on the surface of such an aerosol can be processed by chemical reactions with some atmospheric species; this affects the hygroscopicity of the aerosol and hence its ability to become a cloud droplet either via nucleation or via Köhler activation. The most probable pathway of such processing involves atmospheric hydroxyl radicals that abstract hydrogen atoms from hydrophobic organic molecules located on the aerosol surface (first step), the resulting radicals being quickly oxidized by ubiquitous atmospheric oxygen molecules to produce surface-bound peroxyl radicals (second step). These two reactions play a crucial role in the enhancement of the Köhler activation of the aerosol and its evolution into a cloud droplet. Taking them and a third reaction (next in the multistep chain of relevant heterogeneous reactions) into account, one can derive an explicit expression for the free energy of formation of a four-component aqueous droplet on a ternary aqueous organic aerosol as a function of four independent variables of state of a droplet. The results of numerical calculations suggest that the formation of cloud droplets on such (aqueous hydrophilic/hydrophobic organic) aerosols is most likely to occur as a Köhler activation-like process rather than via nucleation. The model allows one to determine the threshold parameters of the system necessary for the Köhler activation of such aerosols, which are predicted to be very sensitive to the equilibrium constant of the chain of three heterogeneous reactions involved in the chemical aging of aerosols.

1. INTRODUCTION Aerosols play an important role in the global climate system through their interaction with solar and terrestrial radiation (direct radiative forcing) as well as through their ability to serve as cloud condensation nuclei (CCN) thus affecting the reflectivity of clouds (indirect radiative forcing).1−4 It has also been implicated in human disease and mortality.1,5−9 The distribution of aerosol and cloud particles with respect to their sizes and chemical composition constitutes a major component of both regional and global climate models. Physicochemical properties of atmospheric particles can be affected by heterogeneous chemical reactions of atmospheric species thereupon or/and therein. Since such chemical reactions and the condensation of atmospheric gases/vapors on pre-existing nucleating centers can take place concurrently, it is important to develop an adequate theoretical model for these phenomena that would be simple enough to implement into climate models. Although heterogeneous condensation is the subject of significant research interest in atmospheric sciences as well as in the fundamental theory of first-order phase transitions, it has never been studied under conditions of concomitant chemical reactions between vapor-gas medium and evolving liquid particle. From this standpoint, particularly interesting are aqueous aerosols consisting of water and both hydrophobic and hydrophilic compounds. Hereafter, such an aerosol particle © XXXX American Chemical Society

will be referred to as an aqueous hydrophilic/hydrophobic organic (AHHO) aerosol. Organic aerosol (OA) is an important component of particulate matter (PM).3,4,10,11 OA can be both primary (POA, emitted as aerosol) and secondary (SOA, formed in situ in the atmosphere as condensable vapors). Sources of POA include, among others, combustion processes.12−14 SOA is formed as a result of the oxidation of both anthropogenic and biogenic organic species.15−18 Oxidation of these species leads to the formation of products that tend to contain a high degree of functionality, including hydroxyl, carbonyl, carboxy, nitrooxy, and nitro groups.19−22 Those products with sufficiently low vapor pressures will partition between the gas and aerosol phases, forming SOA and contributing to the overall PM burden. Some of these products may also partition to the aerosol aqueous phase.23 Organic materials, composing a large fraction of the atmospheric aerosol mass (20%−90%), can be located either within the liquid particle or at its surface.4,24,25 The surface layer can be hydrophilic, hydrophobic, or of composite (mixed hydrophobic/hydrophilic) character.4,26−28 Some aerosols obtain hydrophobic organics in their coating upon formation, while others obtain them in secondary processes. Received: February 5, 2018 Revised: March 28, 2018

A

DOI: 10.1021/acs.jpca.8b01276 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A

physicochemical properties of its core. The core of an organic-coated aerosol may consist either of pure organic compounds or of a mixture of organic and inorganic species. It can be either liquid (most likely aqueous, but not necessarily) or solid; in the latter case, it can be either soluble or insoluble. Organic-coated aerosols whereof the core contains a multicomponent aqueous solution are among the most widespread in the atmosphere.1−4,27,28 In what follows, all aerosol particles and droplets are assumed to be liquid; that is, we consider phenomena only below the freezing level. 2.1. Mechanism of Free-Radical Reactions of the Oxidative Aging of Aqueous Organic Aerosols. The chemical composition of an atmospheric organic or organiccoated aerosol can be extremely complex; there may be thousands of various large organic compounds in typical atmospheric aerosol, such as, proteins, lipids, free amino acids, small oligosaccharide derivatives, carboxylic/fatty acids, etc.4,40−42,56,57 Among them, carboxylic acids (and particularly amphiphilic fatty acids) constitute one of the most significant classes of ingredients in both marine aerosols37,40,42,57 and continental organic aerosols.4,27,28,56 A question, whether the exposed surfaces of OA are partially oxygenated and if they can exhibit certain water affinity, remains an object of discussion. However, one can anticipate the hydrophilic parts of the oligomeric and polymeric fragments of such an aerosol will also be embedded into its aqueous core, leaving the hydrophobic parts pointed outward 37,42,58,59 (issues concerning the morphology of the aerosol surface are beyond the scope of our current study and hence will not be discussed hereinafter). Although the effect of chemical mechanism of aging on the formation and evolution of organic aerosols has been actively investigated lately,29−35,37−39,60 many aspects of these processes remain rather obscure. Among various atmospheric gas-phase oxidants, OH has been unquestionably determined to be the most efficient in the troposphere in processing surface-located organic constituents of aerosols34,37,38,45−47,59 (although such oxidants as O3, HO2, Cl, etc. are also able to trigger the hydrophobic-to-hydrophilic conversion of organic-coated aerosols30,39,44). The aerosol phase and morphology was observed to be affected depending on the fate of the products of conversion reactions triggered by these oxidants.45,61−63 Most available laboratory experiments provide evidence on the efficient oxidation of aerosol organics through the reactions with atmospheric oxidants, while the main differences concern the extent of their volatilization and mass loss. Some of the products become volatile organics, which would lead to the reduced particle mass thus serving as a source of oxidized volatile organic compounds (OVOC); some remain in the condensed (particle) phase not necessarily surface bound but able to diffuse into the aerosol core; there are still conflicting reports concerning this aspect of heterogeneous chemical aging of aerosols.21,31,32,35−37 2.1.1. Concurrent Character of the Oxidative Aging of an Organic Aerosol and Multicomponent Condensation thereupon. In the atmosphere, the processes of heterogeneous chemical aging of organic aerosols become much more complex because of the concomitant condensation (thereupon) of water and other vapors such as, for example, volatile oxidized organic species, etc. The atmospheric hydroxyl radicals initiate a series of free radical reactions on the aerosol surfaces thus processing their inert hydrophobic coatings. Incident OH radicals react and create a large number of reactive radical sites. Further chain reactions oxidize the hydrocarbon coatings into alcohols,

The presence of hydrophobic compounds at the surface of the aerosol hinders its ability to act as a cloud condensation nucleus.4,29−32 However, there exist three major mechanisms4,30,33 whereby the hydrophobic-to-hydrophilic conversion of the organic coating of a primary aerosol may occur: (a) chemical (oxidative) aging due to heterogeneous reactions of atmospheric traces species on the aerosol surface; (b) condensation of hydrophilic species (such as sulfuric acid, nitric acid, etc.) onto the aerosol; (c) coagulation of the primary aerosol with larger, more hydrophilic particles. While the condensation and coagulation mechanisms are important mostly in highly polluted regions,4,33 the chemical aging mechanism is much more widespread and, at the same time, the least understood.4,30−32,21−37 During oxidative evolution (“oxidative aging”), gas-phase atmospheric oxidants such as hydroxyl −OH and hydroperoxy HO2 free radicals, ozone O3, may interact with the hydrophobic compounds of the aerosol organic coating and induce transformations that render those compounds hydrophilic by creating polar (particularly oxygenated) functional groups within the organic layer of the aerosol.44,10,11,34−47 As a result, the hydrophobic sites of the aerosol coating are transformed into hydrophilic ones; such a “processed” (aged) aerosol becomes capable of serving as a nucleating center for aqueous droplets. The chemical mechanism of aging could potentially affect the hydrophobic-to-hydrophilic conversion of the aerosol coating on time scales shorter than those expected for condensation and coagulation processes4 under normal atmospheric conditions.4,33 The condensation mechanism can be of particular importance in regions with a high atmospheric concentration of condensable material, such as sulfuric acid and nitric acid vapors, volatile oxidized organic species, etc. Although important in highly polluted areas, the contribution of coagulation to the hydrophobic-to-hydrophilic conversion of aerosol coating and to the aerosol activation as a condensation center can be neglected if the aerosol concentration is not too high; however, even in areas with low aerosol concentration, this contribution may become important at later stages of evolution of aerosols due to their condensational growth.4,33 Current nucleation-condensation models are inadequate for the application to condensation on organic-coated aerosols in atmosphere; they do not take into account the concomitant oxidative aging of hydrophobic aerosol coatings. In this paper we develop the thermodynamics of condensation of ternary vapor mixture of water and hydrophobic and hydrophilic organic molecules on an aqueous ternary organic atmospheric aerosols, explicitly taking into account the effect of concomitant chemical reactions thereupon. Our model for the organic aerosol (AHHO) will be based on the coupled hydrophilic/ hydrophobic organics aerosol model developed by Seinfeld, Seigneur, Pun, Couvidat, and co-workers.48−54 Our approach involves the detailed molecular modeling of surface radical processes and concomitant ternary condensation and can be expected to improve the current computer models for the distribution of aerosol and cloud particles with respect to their size and chemical composition; such a distribution constitutes a major, necessary component of both regional and global climate models.1−4,55

2. THEORETICAL MODEL AND METHODS The chemical aging of organic aerosols depends not only on the composition of its outer, surface layer but also on the B


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For simplicity, an initial AHHO aerosol will be considered to be a spherical particle of a liquid ternary solution, consisting of water and hydrophilic and hydrophobic organic molecules. Clearly, the latter will be mostly (if not exclusively) located on the aerosol surface, thus forming hydrophobic patches on the aerosol. These patches are assumed not to cover the surface completely so that there is a dynamic exchange of water and hydrophilic organic molecules between aerosol and air. This assumption imposes a restriction on the mole fraction of hydrophobic component in the solution (the number of hydrophobic molecules in the aerosol must be smaller than the number necessary to form a complete hydrophobic monolayer on the aerosol surface), and hence, a restriction on the density of hydrophobic molecules in the surrounding air. The latter is composed of molecules of condensable species (water and hydrophilic and hydrophobic organics) as well as of many noncondensable species (oxygen, nitrogen, etc.), including such trace gases as ozone, chlorine, hydroxyl radicals, etc. Previously, a chemical mechanism was suggested37,59 whereby the hydrophobic organic molecules on the aerosol surface can be “atmospherically” processed. The mechanism involves a variety of sequential chemical reactions between surface-located organic molecules and atmospheric gaseous species, with atmospheric hydroxyl radicals (and, to a much lesser extent, Cl atoms) playing a crucial role of a “trigger”. Incident OH radicals react and create a large number of “free” radical sites on the aerosol organic coating. These radicals are very reactive. The ensuing chain reactions oxidize the hydrophobic molecules so that they become alcohols, aldehydes, alkenes, ketones, carboxylic acid functional groups, etc. so that the hydrophobic component of the aerosol is gradually transformed into hydrophilic one. This significantly increases the hygroscopicity of the aerosol and hence its ability to serve as a CCN. Certainly, there may occur some polymerization92−94 of the intermediate products/radicals on the aerosol surface during this processing. Explicitly taking account of such polymerization would significantly complicate our model and would make it less transparent, thus preventing us from achieving the goal (i.e., studying the effect of heterogeneous chemical reactions on the Köhler activation of aqueous organic aerosols). Nevertheless, its effect will be implicitly incorporated in our model via the aggregate equilibrium constant of the chain of three chemical reactions, first three steps of the mechanism of chemical aging considered below; this aggregate equilibrium constant will be one of the input parameters of our model. The first step of the chemical aging of an AHHO aerosol may consist37,59 of hydrogen atom abstraction from the hydrophobic moiety of a surface-located molecule by an OH radical. One can write this reaction as

aldehydes, alkenes, and ketones and transform the hydrophobic coating into a hydrophilic layer. The “processed” AHHO aerosol can serve as a cloud condensation nucleus. The processing of the organic aerosol surface by atmospheric radicals creates hydrophilic sites on the surface that can bind water. As a result, water clusters can form owing to the formation of complexes “O-(H2O)n” (n = 1, 2, ...) between water molecules, impinging onto the aerosol and an oxygen atom present within such groups as −CHO, −CH2OH, −COOH, etc., in the processed alkyl tails of molecules coating the aerosols. In such “complexes” oxygen atoms serve as microscopic active centers for the condensation of water molecules on the aerosol. Thus, water condensation on organic aerosols becomes possible. If the water vapor supersaturation is not too low (about or higher than 1%), the heterogeneous formation of a water cluster comprising hundreds of molecules would take no longer than several seconds.67 In turn, the intensified water condensation on the processed aerosol enhances the condensation of SOA compounds (gas-phase OVOCs and SVOCs) onto the aerosol; there is a dynamic exchange of organic molecules (components of the aerosol) between aerosol and surrounding air. Therefore, it is reasonable to consider multicomponent condensation on and chemical aging of aqueous organic aerosols as occurring concurrently.37,60 2.1.2. An Aqueous Hydrophilic−Hydrophobic Organic (AHHO) Aerosol Model. For an organic aerosol we will hereafter adopt an AHHO aerosol model reminiscent to that first proposed by Seinfeld and co-workers48−50 and then further developed and modified by Seigneur and co-workers;51−54 it was the first fully coupled module for the gas-particle partitioning of organic compounds, the Model to Predict the Multiphase Partitioning of Organics (MPMPO).49 According to MPMPO, secondary oxidation organic compound present in the air, because of low vapor pressures and high polarities, may exhibit ability to simultaneously partition into both the organic and aqueous condensed phases. Accordingly, in the framework of the AHHO aerosol model, an organic aerosol particle consists of water and both hydrophobic and hydrophilic organic compounds. The distinction between hydrophobic and hydrophilic is based on structural and physical characteristics of the compound. Both hydrophilic and hydrophobic aerosol components may be of various oxidation degrees, semivolatile, or of low volatility, but still present in the surrounding air. In the present work we will consider the condensation of a ternary vapor mixture of water and two low-volatility organic compounds on such a liquid organic aerosol with allowance for concomitant heterogeneous chemical reactions occurring between hydrophobic organic molecules, located on the aerosol surface, and atmospheric gaseous species. The latter are not directly involved in the nucleation-condensation process but stimulate it by processing the hydrophobic patches on the aerosol surface and rendering it more hygroscopic. 2.2. Initial and “Processed” AHHO Aerosols. Clearly, the exact chemical composition of atmospheric organic aerosols depends on where and when (i.e., in which area of the marine boundary layer and in what season of the year) they are formed; even for aerosols formed in the same area at the same time it can vary from one aerosol particle to another. This composition can be extremely complex; there may be thousands of various organic compounds in a typical atmospheric aerosol.27

OH(g) + HR/aerosol ⇌ H 2O(g) + R·/aerosol

(1)

where HR denotes the hydrophobic molecules, with the radical “R·” being essentially the whole molecule less one of the hydrogen atoms, denoted by “H”, in its hydrophobic moiety. This is an exothermic reaction; in the case where, for example, the hydrophobic molecule HR is that of hexanoic acid, it is accompanied by the release of ∼20 kcal/mol. The C−H bond strength depends on whether the hydrogen is primary, secondary, or tertiary; the dissociation energies for these bonds are ∼98, 94, and 92 kcal/mol, respectively.64 Besides, a secondary C−H bond reacts ∼40% faster65,66 if it is bonded to two −CH2− groups rather than to one −CH2− and one −CH3. However, one should keep in mind that the C


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2.3. Free Energy of Formation of a Droplet on an AHHO Aerosol. Let us consider an initial AHHO aerosol (a spherical particle of ternary liquid solution) within a vapor-gas medium (air) composed of a ternary mixture of condensable vaporswater and hydrophilic and hydrophobic organics (components 1, 2, and 3, respectively), as well as noncondensable speciesnitrogen oxide, hydroxyl radicals, oxygen, nitrogen dioxide (components 5, 6, 7, and 8, respectively). All other gases/species in the air will not be considered hereafter. As substantiated above, in the framework of the above model, it is reasonable to treat the ternary condensation of components 1, 2, and 3 on and surface processing of the aerosols as occurring simultaneously. The number of component i in the initial AHHO aerosol of radius R0 will be denoted by qi (i = 1, 2, 3). Since component 2 is hydrophilic, it is most likely to be distributed more or less uniformly within the aqueous droplet, while molecules of component 3 will be located mostly on the aerosol surface (although their dissolution within the aerosol core can not be excluded). The number of molecule components 1, 2, and 3 in the aerosol may change due to their condensation from and their evaporation into the atmosphere. Besides, the hydrophobic molecules HR on the aerosol surface are simultaneously processed by reactions with atmospheric OH radicals and O2 and NO molecules. As clear to eqs 1−(3), these reactions produce additional components in the evolving droplet, namely, radicals R·, RO2·, and RO·. Since all three reactions are relatively fast, one can assume that the number of intermediate radicals R· (product of reaction 1) and RO2· (product of reaction 2) in the droplet will be negligible compared to the number of “final” radical RO· (product of reaction 3), so that the entire chain of reactions 1−(3) gives rise to the presence of only one additional component in the droplet, namely, radical RO·; hereafter, it will be referred to as component 4. The number of “molecules” of component i (i = 1, 2, 3, 4) in the evolving droplet will be denoted by νi; any physical quantities for component i in the system will be marked by subscript i (i = 1,···, 8) In the framework of the classical nucleation theory (based on the capillarity approximation),68 the reversible work W of formation of a droplet on a ternary AHHO aerosol can be determined as69−72 the difference Xfin − Xin, where Xin is the appropriate thermodynamic potential of the initial system (an initial AHHO aerosol within the vapor-gas mixture (air)), and Xfin is the value of the same thermodynamic potential of the system after the aerosol evolved into a droplet with variables ν1, ν2, ν3, ν4; the choice of the potential X depends (strictly speaking) on the thermodynamic conditions under which the droplet formation occurs (see the Appendix for more details). One can thus obtain the following expression for W, which is a function of four independent variables of state of the droplet, W ≡ W(ν1, ν2, ν3, ν4), and also depends on the initial parameters of the AHHO aerosol and vapor-gas medium (air) (see the Appendix for details):

collisions of OH radicals, impinging onto the aerosols, with the −CH3 group and the adjacent −CH2− group are much more probable than the collisions with −CH2− groups situated “deeper” in the alkyl tail; this is due to the particular orientation of carboxylic acid molecules in the aerosol coating. The general formalism developed hereafter is independent of whether it is the abstraction of a primary atom or a secondary one or a tertiary one. On the second step of the chain reactions processing the hydrophobic coating of the aerosol, the surface-bound radical R· produced by reaction 1 is rapidly oxidized by O2 molecules thus producing a new surface-bound radical RO2·: O2 (g) + R· /aerosol ⇌ RO2 ·/aerosol

(2)

Although the rate constants of reactions 1 and (2) are approximately the same,3 the number density of oxygen molecules in the atmosphere is much higher (by more than 12 orders of magnitude) than that of hydroxyl radicals. Therefore, every radical R· produced by reaction 1 is almost immediately oxidized by reaction 2. The further evolution of the surface-bound radical RO2·, produced by reaction 2, may vary but always results in the formation of water-soluble and/or volatile species and/or hydrophilic radicals.37 Those transformations may involve trace gases and radicals such as NO, NO2, O3, and HO2, which have much smaller number densities than that of atmospheric oxygen (by 6, in moderately polluted zones, to 10, in remote areas), orders of magnitude.64 However, the rate constants of their reactions with radicals RO2· are of the same order of magnitude that the rate constant of reaction 2 itself.37,64 As a possible pathway, for concreteness, one can notice that RO2· radicals readily oxidize37 NO to NO2 with an exothermicity of −12 kcal/mol (−50 kJ/mol): RO2 · /aerosol + NO(g) ⇌ RO·/aerosol + NO2 (g)

(3)

The rate constant for peroxyl radical oxidation of NO is ∼8× 10−12 cm3/s (see ref 64a). Alternatively, the surface-bound peroxyl radicals (RO2·) can also readily oxidize ozone (O3) to generate alkoxy radicals or oxidize sulfur dioxide (SO2) to produce SO3 (and/or eventually organic sulfates, see refs 37 and 59 for a more detailed discussion of various reactive channels of radicals RO2· and RO·). As a result of the chain of heterogeneous reactions 1−(3), the surface-located hydrophobic molecule HR converted to a radical RO·; although the latter may still contain hydrophobic parts, there now appears a highly hydrophilic site on its formerly hydrophobic moiety. Thus, one can expect that the radicals RO· will be more soluble in aqueous media so that they can now relocate into the core of the aerosol. Clearly, the chemical aging of atmospheric organic aerosols is an extremely complex, multifaceted phenomenon. It can occur via a multitude of pathways involving myriads of initial reagents, intermediate compounds, radicals, and final stable products. Photochemical processes, triggered on the aerosol surface as well as in its interior, may play an important (if not crucial) role90 (especially in micro- and nanodroplets) in various pathways whereby an organic aerosol undergoes chemical aging under appropriate atmospheric conditions. However, in the model that is being presented in this paper, we consider a specific pathway37 involving reactions 1−(3), whereby a relatively narrow class of aqueous organic aerosols can undergo initial steps of chemical aging. Photochemical processes are not explicitly involved in this pathway.

4

W = −kBT ∑ νi ln i=1

ζi + σ(χ1 , χ2 , χ3 ) χi fi (χ1 , χ2 , χ3 )

× A(ν1 , ν2 , ν3 , ν4) + W0(q1 , q2 , q3)

(4)

Here kB is Boltzmann’s constant, T is the temperature, and ζi = Pi/Pi∞ (i = 1, 2, 3) is the saturation ratio of the condensable D


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of formation of the nucleus Wc = W(ν1c, ν2c, ν3c, ν4c) roughly determines the height ΔWc ≡ Wc − W0 of the activation barrier that the droplet must overcome to become a cloud droplet. When the activation barrier is very low, ΔWc ≲ (2 ÷ 3)kBT, the process of transformation of an aerosol into a cloud droplet is called “Köhler activation”81 (this is the analogue of spinodal decomposition in the fundamental theory of first-order phase transitions). When the activation barrier is too high, ΔWc ≳ 100kBT, initial aerosols overcome the barrier with a negligible (virtually zero) rate. Since the activity coefficients f i ≡ f i(χ1, χ2, χ3) and surface tension σ(χ1, χ2, χ3) satisfy the Gibbs adsorption equation,82 set (6) equations for determining the coordinates of the saddle point ν1c, ν2c, ν3c, ν4c can be reduced to the set

component i in the vapor mixture, with Pi being its partial pressure and Pi∞ its equilibrium vapor pressure; χi = νi/(ν1 + ν2 + ν3 + ν4) (i = 1,...,4) is the mole fraction of component i in the droplet (clearly, χ1 + χ2 + χ3 + χ4 = 1), whereas f i(χ1, χ2, χ3) is the activity coefficient of component i in the four-component solution of composition determined by χ1, χ2, and χ3; σ(χ1, χ2, χ3) and A(ν1, ν2, ν3, ν4) = 4πR2d(ν1, ν2, ν3, ν4) are the surface tension and surface area of the evolving droplet of radius Rd ≡ Rd(ν1, ν2, ν3, ν4). Although there is no condensable component 4 in the air, the quantity ζ4 can be loosely (for the sake of convenience) called “the saturation ratio” of component 4; it is defined as ζ3ζ5ζ6ζ7Keq

ζ4 =

ζ1ζ8

(5)

kBT ln[χi fi (χ1 , χ2 , χ3 )] =

where Keq is the equilibrium constant of the chain of reactions 1−(3), and ζj = Pj/Pj0 (j = 5, 6, 7, 8), with Pj being the partial pressure of component j in the air and Pj0 its standard partial pressure (at given temperature T) for which the standard Gibbs free energy change of reactions 1−(3) is assumed to be known. Finally, the term W0(q1, q2, q3) on the right-hand side (RHS) of eq 4 depends only on the initial parameters of the aerosol (q1, q2, q3); it can be interpreted as the free energy of the initial aerosol and can serve as a reference level for measuring the free energy of formation of a droplet thereon. As clear from eq 4, W is a function of four independent variables of state of a droplet: ν1, ν2, ν2, and ν4. Although W is the free energy of formation of a droplet on a pre-existing AHHO aerosol, the latter is composed of the same kind of molecules as the condensable vapors. Hence, if the activation of the initial AHHO aerosol (which results in its becoming a CCN, i.e., an irreversibly growing cloud droplet that eventually precipitates as a raindrop) occurs via nucleation (i.e., in a fluctuational way), one can expect73−75 the function W = W(ν1, ν2, ν3, ν4) to determine a free-energy surface (in a fivedimensional space) which has only a saddle point, unlike the case of multicomponent heterogeneous condensation, where the free energy surface has both a well point and a saddle point.76−78 Hereafter, the quantities for the saddle point will be marked with subscripts “c”. Note that eq 4 for W is obtained without approximations that would affect the accuracy of the model (see Appendix); only approximations intrinsic to the capillarity approximation (i.e., CNT) are involved in deriving eq 4. The coordinates of the saddle point ν1c, ν2c, ν3c, ν4c of the free-energy surface (determined by the function W = W(ν1, ν2, ν3, ν4)) are obtained (as usual in CNT68−80) as the solution of four simultaneous equations ∂W ∂νi

=0 c

2viσ(χ1 , χ2 , χ3 ) R d(ν1 , ν2 , ν3 , ν4)

(i = 1, 2, 3, 4)

(7)

where νi is the volume per molecule of component i in the droplet (recall that χi = νi/(ν1 + ν2 + ν3 + ν4)). Alternatively, considering χ1, χ2, χ3, and R to be independent variables of state of the droplet (instead of ν1, ν2, ν3, ν4), one can rewrite set (7) as ⎫ 1 1 ln[χ1 f1 (χ1 , χ2 , χ3 )] = ln[χ2 f2 (χ1 , χ2 , χ3 )] ⎪ v1 v2 ⎪ ⎪ 1 1 ln[χ2 f2 (χ1 , χ2 , χ3 )] = ln[χ3 f3 (χ1 , χ2 , χ3 )]⎪ ⎪ v2 v3 ⎬ ⎪ 1 ln[χ3 f3 (χ1 , χ2 , χ3 )] ⎪ v3 ⎪ 1 = ln[(1 − χ1 − χ2 − χ3 )f4 (χ1 , χ2 , χ3 )] ⎪ ⎪ v4 ⎭ Rd =

(8)

2σ(χ1 , χ2 , χ3 )v(χ1 , χ2 , χ3 ) 4

kBT ∑i = 1 χi fi (χ1 , χ2 , χ3 )

(9)

where v(χ1, χ2, χ3) ≡ χ1v1 + χ2v2 + χ3v3 + χ4v4 (with χ4 = 1 − χ1 − χ2 − χ3) is the volume per molecule in the solution of composition χ1, χ2, χ3. According to eq 8, the mole fractions χ1c, χ2c, χ3c of the nucleus can be found without knowing the surface tension of the droplet, in agreement with the multicomponent CNT.77,83 This makes eqs 8 and (9) significantly more convenient for determining the coordinates of the saddle point (i.e., the size of and composition of the solution in the nucleus). However, note that set (7) or eqs 8 and (9) can be accurately used only if the activity coefficients f i (i = 1, 2, 3, 4) and surface tension σ(χ1, χ2, χ3) exactly satisfy the Gibbs adsorption equation. Otherwise, if approximations for f i (i = 1, 2, 3, 4) and σ(χ1, χ2, χ3) are such that the perturbation to the Gibbs adsorption equation is not sufficiently small, using set (7) or eqs 8 and (9) may result in large inaccuracies in determining the location of the saddle point and hence in the height Wc of the activation barrier. In this case the parameters of the nucleus (i.e., the location of the saddle point of W) must be determined by solving the set of simultaneous eqs 6.

(i = 1, 2, 3, 4) (6)

The droplet, whereof the variables (ν1, ν2, ν3, ν4) coincide with the coordinates of the saddle point, is referred to as a “nucleus”. The evolution of the droplet during its fluctuational growth (when it becomes an irreversibly growing droplet) occurs along the path that passes in the vicinity of the saddle point. This path may not coincide with the path of the steepest descent of the five-dimensional free energy surface W(ν1, ν2, ν3, ν4); neither does it have to pass exactly through the saddle point (ν1c, ν2c, ν3c, ν4c) itself.73,78,81 Nevertheless, the droplet evolution path passes close enough to the saddle point so that the free energy

3. NUMERICAL EVALUATIONS AND DISCUSSION To illustrate the above presented theory with numerical results, we performed calculations for an initial AHHO aerosol E


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The Journal of Physical Chemistry A consisting of q1 = 1.286 × 108 molecules of water, q2 = 6.3 × 10 5 molecules of 2-methylglyceric acid [C 4 H 8 O 4 , or CH2(OH)−C(CH3)(OH)−COOH], and q3 = 1.4 × 106 molecules of 3-methyl-4-hydroxy-benzoic acid [C8H8O3, or CH3−C6H3−OH−COOH]; according to Couvidat et al.,54 2methylglyceric acid can be classified as a hydrophilic SOA compound, whereas 3-methyl-4-hydroxy-benzoic acid can be characterized as a hydrophobic compound. These q1, q2, and q3 values correspond to the initial aerosol size R0 ≈ 0.1 μm with the mass fraction of organic compounds ∼0.1. As a thought experiment, this model AHHO aerosol was then placed in a vapor-gas medium (air) containing a ternary vapor mixture of water, 2-methylglyceric acid, and 3-methyl-4-hydroxy-benzoic acid, as well as nitrogen oxide, hydroxyl radicals, oxygen, and nitrogen dioxide (components 1, 2, 3, 5, 6, 7, and 8, respectively) at temperature T = 293.15 K. The densities/ partial pressures of noncondensable gaseous atmospheric species were assumed to be fixed and taken to be such that the parameters ζ5 = 1.01, ζ6 = 1.01, ζ7 = 1.01, and ζ8 = 1.01. Since molecules of 3-methyl-4-hydroxy-benzoic acid are largely hydrophobic,54 they will be mostly (although probably not exclusively) located at the aerosol surface, with the methyl groups −CH3 exposed to the air. It is hence reasonable to consider the abstraction of the H atom from the methyl group of a 3-methyl-4-hydroxy-benzoic acid molecule as the first step (see eq 1) in the hydrophobic-to-hydrophilic conversion of the latter. The radical R· in eqs 1−(3) can then be identified as

The effect of the droplet surface tension on condensation/ nucleation phenomena has been well investigated.75 Aiming mainly at the qualitative sensitivity studies of the Köhler activation with respect to the aggregate equilibrium constant Keq, one can conjecture that the effect of radicals R4 (resulting from the hydrophobic-to-hydrophilic conversion of 3-methyl-4hydroxy-benzoic acid) on the surface tension will be roughly similar to the effect of a hydrophilic component on the surface tension of its aqueous solutions. Taking this into consideration, we modeled the surface tension σαβ(χ1, χ2, χ3) of the fourcomponent solution “water/2-methylglyceric acid/3-methyl-4hydroxy-benzoic acid/radical R4” with the surface tension of a model ternary solution “water/hydrophilic solute (which would represent 2-methylglyceric acid and radicals R4 combined) /hydrophobic solute (which would represent 3-methyl-4hydroxy-benzoic acid)”. As such, we chose the solution of water, n-pentyl acetate (surrogate hydrophobic solute), and methanol (surrogate hydrophilic solute). An analytical expression for its surface tension σ̃ as a function of its composition was obtained by Santos et al.86 e3 σ ̃ ≡ σ (̃ x w , xp) = x wσw* + xpσp* + xmσm* + σwpm

where σw*, σp*, and σm * are the surface tensions of pure liquid of water, n-pentyl acetate, and methanol, respectively, and xw, xp, and xm are their mole fractions in the solution (with xm = 1 − e3 xw − xp); σe3 wpm ≡ σwpm (xw, xp) is the excess surface tension of a ternary solution. The latter is approximated86 as σe3 wpm (xw,xp) = σewp + σewm + σepm + σe3, with σewp = xwxp(Awp + Bwp(1 + (xw − xp)Cwp)), σewm = xwxm(Awm + Bwm(1 + (xw − xm)Cwm)), σepm = xpxm(Apm + Bpm(1 + (xp − xm)Cpm)), σe3 = xwxpxm[D1 + D2(xw − xp) + D3(xp − xm)]/[1 + D4(xw − xp)], and Awp = −48.458, Awm = 108.530, Apm = 38.907, Bwp = 0, Bwm = −178.258, Bwp = −40.357, Cwp = 0, Cwm = −0.335, Cpm = −0.019, D1 = −23.998, D2 = −156.136, D3 = −66.983, D4 = −1.567 (for σ̃ measured in units of dyn/cm). In the framework of this approach, in eqs 4, (7), and (9) we adopted an approximation σ(χ1, χ2, χ3) ≈ σ̃(χ1, χ3) (neglecting the curvature effect on σ as well). It is worth emphasizing that, according to eq 8, the composition coordinates of the saddle point (namely, χ1c, χ2c, χ3c) do not depend on σ. However, the linear size (radius) of the nucleus and the height of the nucleation barrier are both very sensitive to σ. Therefore, this is a very rough approximation suitable only for qualitative (sensitivity) analysis of the free energy W. The numerical evaluations were performed for two distinct cases. First, the four-component solution in the evolving droplet was assumed to be ideal; in this case all activity coefficients in eqs 4−(9) are set to be equal to 1, that is, f i(χ1, χ2, χ3) = 1 (i = 1, 2, 3, 4). Then, we recurred to the UNIFAC method84,85 to build the activity coefficients of components 1 to 4 in the liquid solution. The results of our calculations are illustrated in Figures 1−4 (in the approximation of ideal solution in the droplet) and in Figures 5 and 6, which present the results for both the ideality approximation and UNIFACapproximated activity coefficients. For the sake of better visualization, in Figures 1−4 the four independent variables of the free energy W were chosen to be ν ≡ ν1 + ν2 + ν3 + ν4, χ1, χ2, and χ3; in Figures 1 and 2, the ordinate axis represents the dimensionless ratio ΔW/(kBTq), with ΔW ≡ W − W0 (see eq 4) and q = q1 + q2 + q3.

R· ≡ −CH 2−C6H3−OH−COOH

whereas component 4 of the aqueous solution in the evolved droplet can be identified as the radical −OCH 2−C6H3−OH−COOH

Thus, one can consider the four-component solution in the evolved droplet as a liquid mixture of functional groups for all of which all the relevant parameters are given in the updated tables of the UNIFAC method for activity coefficients.84,85 Since the main goal of our investigation was to better understand how heterogeneous chemical reactions on organic aerosols affect the hygroscopicity of the latter and their evolution into CCN, we calculated the dependence of the height of the activation barrier ΔWc on the equilibrium constant of the chain of chemical reactions 1−(3), Keq, for a fixed set of “saturation” ratios ζi (i = 1, 8, i ≠ 4). Furthermore, at given temperature T for any aerosol of given size and composition q ≡ {q1, q2, q3} and parameters ∼ ζ1 ≡ {ζ2 , ζ3 , ζ5 , ζ6 , ζ7 , ζ8}, the height of the nucleation barrier ∼ disappears at some ζ1 ≳ ζ1K, where ζ1K ≡ ζ1K(q , T , ζ1) is the threshold value of the saturation ratio ζ1 necessary for the ∼ Köhler activation of that aerosol q at given T and ζ1 . It is defined via the equation ΔWc(ζ1)|ζ1= ζ1K = 0

L i k e w is e , o n e c a n d e fi n e t h e t h r e s h o ld v a lu e s ∼ ζ2K ≡ ζ2K(q , T , ζ͠ 2) and ζ3K ≡ ζ3K(q , T , ζ3) of saturation ratios ζ2 and ζ3 necessary for the Köhler activation of the ∼ aerosol q at given T and at given ζ͠ 2 and ζ3, respectively, where ∼ ζ͠ 2 ≡ {ζ1 , ζ3 , ζ5 , ζ6 , ζ7 , ζ8} and ζ3 ≡ {ζ1 , ζ2 , ζ5 , ζ6 , ζ7 , ζ8}. We considered fixed ζ2 and ζ3 but calculated the dependence of ζ1K on Keq (see below). F


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Figure 1. Typical profiles of the free energy surface, determined by the function W = W(ν, χ1, χ2, χ3) (given by eq 4), in the (cross-section) planes χ1 = χ1c, χ2 = χ2c, χ3 = χ3c (a) showing the function of a single variable W(ν, χ1c, χ2c, χ3c) and ν = νc, χ2 = χ2c, χ3 = χ3c (b) showing the function of a single variable W(νc, χ1, χ2c, χ3c). The ordinate axis represents the dimensionless ratio ΔW/(kBTq), with ΔW ≡ W − W0 and q = q1 + q2 + q3.

Figure 2. Typical profiles of the free energy surface, determined by the function W = W(ν, χ1, χ2, χ3) (given by eq 4), in the (cross-section) planes ν = νc, χ1 = χ1c, χ3 = χ3c (a) showing the function of a single variable W(νc, χ1c, χ2, χ3c) and ν = νc, χ2 = χ2c, χ2 = χ2c (b) showing the function of a single variable W(νc, χ1c, χ2c, χ3). The ordinate axis represents the dimensionless ratio ΔW/(kBTq), with ΔW ≡ W − W0 and q = q1 + q2 + q3.

Figures 1 and 2 show the typical profiles of the free energy surface, determined by the function W = W(ν, χ1, χ2, χ3) (given by eq 4), in the planes (cross sections): χ1 = χ1c, χ2 = χ2c, χ3 = χ3c (Figure 1a), ν = νc, χ2 = χ2c, χ3 = χ3c (Figure 1b), ν = νc, χ1 = χ1c, χ3 = χ3c (Figure 2a), and ν = νc, χ1 = χ1c, χ2 = χ2c (Figure 2b); these profiles represent functions of a single variable: W(ν, χ1c, χ2c, χ3c) (Figure 1a), W(νc, χ1, χ2c, χ3c) (Figure 1b), W(νc, χ1c, χ2, χ3c) (Figure 2a), and W(νc, χ1c, χ2c, χ3) (Figure 2b). The saturation ratios of condensable components in these figures were ζ1 = 0.65, ζ2 = 0.1, ζ3 = 0.1, and the parameters ζ5 = 1.01, ζ6 = 1.01, ζ7 = 1.01, ζ8 = 1.01; the coordinates of the saddle point droplet are νc/q ≈ 3.48, χ1c ≈ 0.645 55, χ2c ≈ 0.099 7637, χ3c ≈ 0.098 8412. As expected, the free energy as a function of ν has a maximum (Figure 1a). Such a dependence is observed for any χ1, χ2, and χ3 and for any saturation ratios of condensable components (vapors), provided that they are lower than their threshold values (necessary for the Köhler activation). This is a well-known form of the size dependence of free energy of droplet formation in a theory of homogeneous nucleation, both unary and multicomponent67,68,74,75 (in the case where the first-order phase transition occurs in a fluctuational way). The variable ν1 can thus play the role of an unstable variable of state of a droplet (which is always single in any first-order phase transition).

Figures 1b and 2a,b present the dependence of W on one of the mole fractions (either χ1 or χ2 or χ3) at three other variables fixed and equal to their values at the saddle point. As seen, all these functions are concave downward, so that the variables χ1, χ2, and χ3 can be considered as stable variables of state of a droplet (analogous to the composition variables in a theory of multicomponent nucleation, both homogeneous and heterogeneous.73,74,76,77 Figures 3 and 4 further illustrate the unstable character of the variable ν and the stable character of the variables χ1, χ2, and χ3 with the contour plots of the free energy surface. The unstable character of the variable ν is clearly visible in Figure 3; on the one hand, Figure 3a shows W versus variables ν and χ1 at fixed χ2 = χ2c and χ3 = χ3c, Figure 3b shows W versus variables ν and χ2 at fixed χ1 = χ1c and χ3 = χ3c, and Figure 3c shows W versus variables ν and χ3 at fixed χ1 = χ1c and χ2 = χ2c. On the other hand, the stable character of the variables χ1, χ1, and χ3 is clear from Figure 4; Figure 4a shows W versus variables χ1 and χ2 at fixed ν = νc and χ3 = χ3c, Figure 4b shows W versus variables χ1 and χ3 at fixed ν = νc and χ2 = χ2c, and Figure 4c shows W versus variables χ2 and χ3 at fixed ν = νc and χ1= χ1c. The contours are plotted with a uniform step. The corresponding plots in the case of the UNIFAC approximation for the activity coefficients would have a similar shape; for example, at ζ1 = 0.98, ζ2 = 0.0065, ζ3 = 0.0065, ζ5 = 1.008, ζ6 = 1.01, ζ7 = 1.01, ζ8 = 1.01, and Keq = 1.34, the coorodinates of the saddle-point G


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Figure 3. Illustration of the unstable character of the variable ν with the contour plots of the free energy surface: (a) W vs variables ν and χ1 at fixed χ2 and χ3; (b) W vs variables ν and χ2 at fixed χ1 and χ3; (c) W vs variables ν and χ3 at fixed χ1 and χ2. The contours are plotted with a uniform step.

Figure 4. Illustration of the stable character of the variables χ1, χ2, and χ3 with the contour plots of the free energy surface: (a) W vs variables χ1 and χ2 at fixed ν and χ3; (b) W vs variables χ1 and χ3 at fixed ν and χ2; (c) W vs variables χ2 and χ3 at fixed ν and χ1. The contours are plotted with a uniform step.

droplet are νc/q ≈ 129.8, χ1c ≈ 0.977 962, χ2c ≈ 0.006 539 74, χ3c ≈ 0.006 425 88.

As some clarification to Figures 1−4, we note that eq 4 for W, the free energy of formation of a droplet, determines W as a H


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Figure 6. Dependence of the threshold saturation ratio ζ1K of water vapor (necessary for the Köhler activation of the aqueous hydrophilic/ hydrophobic organic aerosol) on the overall equilibrium constant Keq of the chain of chemical reactions 1−(3) in the case of an ideal solution approximation (Figure 5a) and in the case of UNIFAC approximation for activity coefficients (Figure 5b) (All the parameters of the system (except for Keq) were fixed as detailed in the text).

Figure 5. Dependence of the height of the activation barrier on the aggreagate equilibrium constant, shown as ΔWc/(kBTq) vs Keq: (a) the case of an ideal solution approximation for activity coefficients; (b) the case of the UNIFAC approximation for activity coefficients.

function of four independent variables of state of a droplet ν1, ν2, ν3, ν4. This function of four variables, W = W(ν1, ν2, ν3, ν4), determines a free energy surface in an abstract five-dimensional (5D) space, with four of the dimensions being ν1, ν2, ν3, ν4, and the fifth being W. This 5D surface has one single saddle point, with the coordinates ν1c, ν2c, ν3c, ν4c, Wc. Since we live in a three-dimensional (3D) space, the closest way to visualize that abstract 5D free energy surface is to fix any two of the variables of the droplet and consider W as a function of other two variables; in other words, we must fix any couple of variables ν1, ν2, ν3, ν4 and consider W as a function of two other variables. Such a function of two variables would determine a surface in our real-life 3D space, which can be visualized with the help of either 3D plots or contour plots. With the latter method, Figures 3a−c and 4a−c show such visualizations of W, obtained by picking various pairs from four variables ν1, ν2, ν3, ν4 and setting them equal to their values at the saddle point, and considering W as a function of remaining two variables. Thus, all Figures 3a−c and 4a−c represent all possible real-life 3D visualizations of an abstract 5D surface in the vicinity of the single saddle point of the latter. Likewise, Figures 1 and 2 represent all possible two-dimensional (2D) visualizations of the same abstract 5D surface in the vicinity of its single saddle point; these visualizations are obtained by setting any three (of four) variables of state of the droplet equal to their values at the saddle point and considering W as a function of a single (remaining) variable.

To clarify the sensitivity of the aerosol activation to the chemical composition of the aerosol hydrophobic coating, we considered an initial AHHO aerosol (q1, q2, q3) in the atmosphere with given ζ1, ζ2, ζ3, ζ5, ζ6, ζ7, and ζ8, and calculated the height of the activation barrier Wc for various Keq values from in the range from 1.2 to 1.3. The dependence of Wc on Keq is shown in Figure 5 as ΔWc/(kBTq) versus Keq; Figure 5a represents the case of an ideal solution approximation for activity coefficients (for atmospheric conditions ζ1 = 0.65, ζ2 = 0.1, ζ3 = 0.1, ζ5 = 1.01, ζ6 = 1.01, ζ7 = 1.01, and ζ8 = 1.01), whereas Figure 5a represents the case of an ideal solution approximation for activity coefficients (for atmospheric conditions ζ1 = 0.98, ζ2 = 0.065, ζ3 = 0.065, ζ5 = 1.008, ζ6 = 1.01, ζ7 = 1.01, and ζ8 = 1.01). One can note that the dependence of Wc on Keq is extremely nonlinear. The activation barrier is extremely high at smaller value of Keq, making it virtually impossible for aerosols to overcome it and become rain droplets. However, for given atmosepheric conditions, the height Wc quickly decreases with increasing Keq; after reaching the values below ∼100kBT, the height of the barrier continues to decrease with increasing Keq, but very slowly, with ΔWc attaining 0 at some threshold value K0eq, which would correspond to the Köhler activation of aerosols. Thus, one can suggest that the transformation of very large variety of aqueous hydrophilic/hydrophobic organic aerosols, whereof the chemical composition is such that the I


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The Journal of Physical Chemistry A aggregate equilibrium constant Keq spans a relatively wide range, will occur via nucleation, that is, by overcoming the activation barrier. On the one hand, if the chemical composition of aerosols is such that the aggregate equilibrium constant Keq is below that range, they will not be able to serve as CCN. On the other hand, if the composition of aerosols is such that Keq > K0eq, they will become irreversibly growing (rain) droplets via Köhler activation (i.e., in a barrierless way). As discussed above, for an aerosol of given size and composition, the height ΔWc of the activation barrier becomes equal to zero at some threshold saturation ratios ζiK (i = 1, 2, 3) (see the definitions above), such that, if any ζi ≳ ζiK, the aerosol becomes a cloud condensation nucleus (i.e., an irreversibly growing droplet) via the Köhler activation rather than via nucleation. For fixed ζ2 = 0.065 and ζ3 = 0.065, Figure 6 illustrates the dependence of ζ1K on the overall equilibrium constant Keq of the chain of chemical reactions 1−(3) in the case of an ideal solution approximation (Figure 6a) and in the case of UNIFAC approximation for activity coefficients (Figure 6b) (the other parameters of the system were chosen as described in the beginning of this Section). One can notice that ζ1K is quite sensitive to Keq. This sensitivity suggests that heterogeneous chemical reaction may have a significant impact on the activation of aqueous hydrophilic/hydrophobic organic aerosols as CCN. Furthermore, it transpires from this Figure that different AHHO aerosols, whereof the hydrophobic compounds undergo hydrophobic-to-hydrophilic conversion via heterogeneous chemical reactions with slightly different rates (hence with slightly different equilibrium constants of the chain (1)−(3)), would become irreversibly growing cloud droplets at signif icantly different atmospheric conditions. Therefore, it seems to be essential that various heterogeneous chemical reactions on the surface of atmospheric organic aerosols be included into any model for the distribution of aerosol and cloud particles with respect to their size and chemical composition. Some relatively recent investigations report the acceleration of some chemical reactions in and on microdroplets (see, e.g., ref 90 and references therein), which suggests that rate constants of those reactions may depend on the radii of droplets (of micrometer size). Although in our model the rates and rates constants of reactions 1−(3) are not the object investigation, they do enter the model implicitly through the equilibrium constants of the corresponding reactions and hence through the aggregate equilibrium constant Keq of the entire chain of reactions 1−(3). However, the latter (Keq) is an input parameter of our model and thus is assumed to be known. The main goal of our model is to investigate the sensitivity of the Köhler activation of aqueous hydrophilic/hydrophobic organic aerosols to that equilibrium constant; the more adequate/ accurate the input parameter Keq is, the more accurate the predictions of our model are.

The most probable pathway of such processing involves atmospheric hydroxyl radicals that abstract hydrogen atoms from organic molecules coating the aerosol (first step), the resulting radicals being quickly oxidized by ubiquitous atmospheric oxygen molecules to produce surface-bound peroxyl radicals (second step). These two reactions play a crucial role in the Köhler activation81 of the aerosol. The further evolution of surface-bound radicals may vary, but it always results in the formation of water-soluble and/or volatile species and/or hydrophilic radicals. For example, the surfacebound radicals (formed via the second reaction) can readily oxidize atmospheric NO to NO2. Taking these three reactions into account, one can derive an explicit expression for the free energy of formation of a fourcomponent aqueous organic droplet on a ternary aerosol as a function of four independent variables of state of a droplet. This function determines a surface (in a 5D space) that has a topology analogous to the free energy surface in a theory of homogeneous multicomponent condensation; it has a saddle point if condensation is not barrierless. Our model is developed in the framework of classical nucleation theory (CNT) based on the classical equilibrium (Gibbsian) thermodynmamics; virtually all (if not all) atmospheric models use CNT to describe the nucleationcondensation phenomena in the atmosphere (i.e., the formation and evolution of liquid atmospheric particles via nucleation, condensation, and/or Kö hler activation).2,3,64,75,87,88 The application of the CNT-based approach to the thermodynamics of nucleation-condensation phenomena in the atmosphere (where thermodynamic conditions are, strictly speaking, nonequilibrium89) can be justified taking into account the hierarchy of time scales; the characteristic time scales of the nucleation-condensation phenomena (when they are of any importance at all) are usually by many orders of magnitude smaller (ranging from microseconds to seconds) than the characteristic time scales of changes of thermodynamic conditions in the surrounding air. This allows one to assume that the predictions of such CNT-based thermodynamic models for nucleation/condensation/Köhler activation in the atmosphere are valid for any instantaneous set of thermodynamic parameters characterizing not too large air parcels. To illustrate the proposed thermodynamic model, we have considered a model aqueous hydrophilic/hydrophobic organic (AHHO) aerosol consisting of water, 2-methylglyceric acid (as a hydrophilic compound), and 3-methyl-4-hydroxy-benzoic acid (as a hydrophobic organic compound), in the vapor-gas medium containing vapors of these compounds and typical atmospheric gaseous species. On the one hand, the numerical calculations show that the total number of “molecules” (three original components and radicals produced by the chain of heterogeneous chemical reactions) in a droplet plays the role of an unstable variable of its state; the dependence of the free energy of droplet formation on that variable is similar to the size dependence of the formation free energy in a theory of homogeneous nucleation. On the other hand, as expected from the multicomponent CNT, the mole fractions of three components in the four-component droplet can be considered as its stable variables of state. The proposed model allows one to determine the threshold value of Köhler activation for the saturation ratio of each of the condensable components in the air (vapors of water and hydrophilic and hydrophobic organic compounds) as a function of all other parameters of the system. The numerical results

4. CONCLUSIONS We have studied the thermodynamics of the condensation of an atmospheric ternary vapor mixture of water with two organic compounds, one hydrophobic and one hydrophilic, on an initial aerosol consisting of ternary liquid solution of these components. The hydrophobic organic coating of such an aerosol can be processed by chemical reactions with atmospheric species; this affects the hygroscopicity of aerosols and hence their ability to become CCN via Köhler activation. J


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of these two phenomena and its impact on the transformation of aqueous hydrophilic/hydrophobic organic aerosols into irreversibly growing droplets (eventually raindrops). We believe that such an approach more adequately describes multicomponent nucleation/condensation/Köhler activation processes in the atmosphere. Although the water vapor is the absolutely dominant condensable component in the atmosphere, there may be some other condensable species affecting the hydrophobic-tohydrophilic conversion of the aerosol coating. For example, in the free troposphere, low temperatures favor the condensation of OVOCs or semivolatile organic compounds that may be chemically produced locally or elsewhere (either emitted directly by a source or resulted from chemical processing in the atmosphere).3,4 Furthermore, nitrogen oxides and sulfur oxides are typically products of combustion, volcano eruptions, etc; they are widely present in the atmosphere, and in some areas and sometimes in relatively high concentrations. Sulfur dioxide and nitrogen oxides, NOx, are toxic acid gases; there are well-identified mechanisms of chemical reactions (involving such reagents as H2O, HO, HO2, O2, etc.) whereby they transform into sulfuric acid and nitric acid (and possibly nitrous acid, to a lesser extent) vapors.3,64 As a result, a vapor/gas mixture of sulfuric acid, nitric acid, oxidized volatile and semivolatile organic compounds, and water can constitute an important part of the atmospheric medium wherein the chemical aging of the organic-coated aerosol takes place. The H2SO4/HNO3 molecules (as well as OVOCs, SVOCs) impinging onto the aerosol can attach even to its hydrophobic coating and thus create additional hydrophilic sites for water condensation. This effect is often referred to as the condensational mechanism of the aerosol aging (hydrophobic-tohydrophilic conversion of the aerosol coating), which enables it to serve as nucleus for a water condensation thereupon. Thus, one should consider these processes (i.e., impingement of water, H2SO4, and HNO3, molecules, SVOCs, OVOCs, etc. onto the aerosol and their attachment thereto) as a part of multicomponent condensation on an organic aerosol whereof the surface layer is undergoing the hydrophobic-to-hydrophilic conversion via chemical reactions triggered by atmospheric trace radicals; regardless of the terminology chosen, it must be kept in mind that all the processes involved have a concurrent character and must be treated as such in any model. Note that the model presented in this paper can be further generalized to include the possibility of more channels of chemical aging of aqueous organic aerosols involving more different kinds of hydrophobic or surfactant species. This would certainly lead to the increase of the number of independent variables of state of the droplet, and new aggregate equilibrium constants of every channel of chemical reactions would enter the model. Such generalizations would result in a significantly more complicated and less transparent model, but it would also become even more adequate for describing atmospheric nucleation/condensation/Köhler activation phenomena involving aqueous organic aerosols. Importantly, it would remain analytical within the framework of CNT and thus amenable to be implemented into the current computer models and to improve their predictions for the distribution of aerosol and cloud particles with respect to their size and chemical composition; such a distribution constitutes a major, necessary component of both regional and global climate models. Clearly, the coagulation of liquid droplets, as well as their fission,91 enabled by their surfactant coatings, would certainly

suggest that the considered process of activation of such an organic aerosol is quite sensitive to the equilibrium constant of the three-step chain of heterogeneous chemical reactions; the latter have thus quite important impact on the hygroscopicity of organic aerosols partially covered with patches of hydrophobic organic compounds. Consequently, different AHHO aerosols, whereof the hydrophobic compounds undergo hydrophobic-tohydrophilic conversion via heterogeneous chemical reactions with slightly different rates, would become irreversibly growing cloud droplets at signif icantly different atmospheric conditions. Therefore, taking such reactions into account in atmospheric models would markedly affect their predictions for the Köhler activation of organic aerosols and their transformation into CCN. Thus, the proposed thermodynamic model can be useful to improve the current computer models for the distribution of aerosol and cloud particles with respect to their size and chemical composition. An adequate model for such a distribution constitutes a major, necessary component of both regional and global climate models. The important role of OH radicals in altering the hygroscopicity of submicron organic aerosols and hence their activity as CNN also transpired from experimental studies.29,47a It was reported that the aging of sub-micrometer organic aerosols (triggered by heterogeneous reactions with OH radicals) may significantly alter their activity as CCN via the production of both water-soluble and surface-active species. The former, by dissolving into the droplet core, lower the water activity therein (according to Raoult’s law). The latter (which are present on the SOA surface from the very beginning of their existence) reduce the aerosol surface tension which, in turn, reduces the equilibrium vapor pressure over the aerosol thus increasing its hygroscopicity and hence ability to act as a CCN. The experimental procedure29,47a seems to be appropriate to study the Köhler activation81 of model aerosols; hence, one could experimentally verify the predictions of our theoretical model concerning the threshold values of the system parameters necessary for their Köhler activation. Clearly, extreme care must be taken in comparing theoretical predictions with experimental results to make sure that both data sets are for similar (if not identical) aerosols under similar external conditions. Note that the above results (for the activation of an aqueous hydrophilic/hydrophobic organic aerosol) markedly differ from the predictions for the role of heterogeneous chemical reactions on the activation of marine aerosols (with participation of only water vapor as a condensable component).60 As we previously reported,60 the process of activation of a marine aerosol is suggested to depend on the chemical composition of its surface layer only via the macroscopic surface tension of the evolving aerosol and to weakly depend on the microscopic chemical characteristics of surfactant molecules and radicals. This corroborated the previous findings36 suggesting that one could omit some chemical species of aerosols (and other details of their chemical composition) in models that investigated aerosol effects on climate. Previously, on the one hand, various models have been developed for heterogeneous chemical aging of organic aerosols. On the other hand, theory of heterogeneous multicomponent condensation on various kinds of foreign particles has been also well-investigated. However, in the atmosphere these two multifaceted phenomena oftentimes occur concurrently. In this paper, we have presented the first analytical model taking into account the concomitant character K


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The Journal of Physical Chemistry A W = Ffin − Fin

affect the distribution of particles with respect to their independent variables of state, and, hence, these effects should be taken into account if the kinetics (i.e., the temporal evolution) of an ensemble of droplets is the object of investigation. However, the model developed in this paper is purely thermodynamic, aiming at studying a specific effect on the Köhler activation of a single, mechanically stable aerosol particle; the history of how the particle has come to exist (whether it was formed via coagulation, or fission, or as a primary or secondary organic aerosol) is not important for our model.

(A1)

where Fin is the Helmholtz free energy of the initial system (an initial AHHO aerosol within the vapor-gas mixture (air)), and Ffin is the Helmholtz free energy of the system after the aerosol evolved into a droplet with variables ν1, ν2, ν3, ν4. Let us introduce the Greek superscripts α and β to mark quantities in the of the aerosol/droplet (α) and vapor-gas mixture (β). Physical quantities for component i in the system will be marked by subscript i (i = 1,···, 8). Denote the number of molecules of component i (i = 1, 2, 3) in the initial AHHO aerosol (of ternary solution) by qi. The initial free energy of the system (an initial AHHO aerosol in the air, i.e., in the vapor-gas mixture of water, hydrophilic organic, hydrophobic organic, nitrogen oxide, hydroxyl radicals, oxygen, and nitrogen dioxide) is

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APPENDIX. DERIVATION OF EQUATION 4 FOR THE FREE ENERGY OF DROPLET FORMATION W In the framework of CNT,68,75 the reversible work W of formation of a droplet on a ternary AHHO aerosol can be determined as69−72 the difference Xfin − Xin, where Xin is the appropriate thermodynamic potential of the initial system (an initial AHHO aerosol within the vapor-gas mixture (air)), and Xfin is the value of the same thermodynamic potential of the system after the aerosol evolved into a droplet with variables ν1, ν2, ν3, ν4 (the choice of the potential X may, in principle, depend on the thermodynamic conditions under which the droplet formation occurs; see the Appendix for more details). Strictly speaking, the choice of the thermodynamic potential X whereof the change determines the reversible work W of formation of a droplet on a pre-existing aerosol depends on a thermodynamic ensemble corresponding to the actual physical system of interest.95,96 In the atmosphere, the total volume V of such a system [“aerosol/droplet + air (multicomponent vapor/ gas)”] can be chosen at will (big enough to ensure constant chemical potentials or pressure at the boundaries of that volume); this would correspond to an open system (grand thermodynamic ensemble, or constant μVT ensemble). In the laboratory experiments, the volume V of the system is fixed by the experimental equipment, and the experimental setting determines whether nucleation occurs in the canonical, or grand thermodynamic, or Gibbs ensemble. However, as shown by Lee et al.,95 in the thermodynamic limit (i.e., in a large enough, macroscopic system, such as imaginary air parcels in the atmosphere) the use of either the Gibbs or Helmholtz free energy or grand thermodynamic potential is acceptable for the calculation of the reversible work of formation W of a particle of a new phase. They proved that the differences between W, calculated in these ensembles, become negligible in the thermodynamic limit. Thus, for convenience and transparency of the derivation, the work of formation can be calculated in the constant N, V, T ensemble, where N is the total number of molecules in the system (air parcel), V is its volume, and T is the temperature (assumed to be uniform in the system, including the droplet). Strictly speaking, in this system (air parcel) it is not possible to fix the total number of molecules. However, the variations of the total number of molecules (during the formation of a nanoor even microscopic droplet on a pre-existing aerosol) that might arise at the nucleation stage will be rather negligible, compared to the initial number of molecules in the system, hence the equivalence of the resulting W (since the condition of the thermodynamic limit are satisfied).95 Therefore, the process of droplet formation is assumed to occur in a canonical ensemble. For the reversible work of formation of a droplet on an initial ternary organic aerosol one can thus write

3

Fin =

8

∑ Ni0μi β (P0β , T ) + ∑ Ni0μi β (P0β , T ) i=1

i=5 3

∑ qiμiα (P α , T , χi0 ) − P0βV0β − P αV α + σ0A 0

+

i=1

(A2)

where the subscript or superscript 0 identifies quantities in the initial state of the system; N is the number of molecules in the vapor mixture (air); μ is the chemical potential; P and V are the total pressure of the vapor-gas medium and its volume, respectively; T is the temperature of the system; χ0 ≡ (χ01, χ02) is a couple of mole fractions of components 1 and 2 in the initial aerosols (the mole fraction of component three is χ03 = 1 − χ01 − χ02); σ and A are the surface tension and surface area of the aerosol. While the initial AHHO aerosol is being processed by reactions 1−(3), there is a concomitant process of condensation of molecules of water and hydrophilic and hydrophobic organic on the aerosol; these concurrent processes result in the formation of a larger droplet of a four-component solution; denote the number of molecules of component i in the droplet by νi(i = 1, 4). The free energy of the system after the formation of such a droplet becomes 3

Ffin =

8

∑ Niμ1β (P β , T ) + ∑ Niμi β (P β , T ) i=1

i=5 4

+

α α β β α α ∑ νμ i i (P , T , χ ) − P V − P V + σ A i=1

(A3)

where χ denotes a set of mole fractions (χ1, χ2, χ3) of components 1, 2, and 3 in the initial aerosols (with χ4 = 1 − χ1 − χ2 − χ3). Since the mother phase (vapor-gas medium) is large compared to the droplet, the removal of the small amount of material needed for the droplet formation does not change the pressure or composition of the vapor mixture. Therefore Pi ≡ Piβ = Piβ0 , P ≡ P β = P0β , μi β (P0β , T ) = μi β (P β , T ) (i = 1, 8, i ≠ 4)

(A4)

In virtue of stoichiometric eqs 1−(3) one can also write L


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where Pi is the partial pressure of component i in the air, whereas, on the one hand, the T-dependent Pi∞ is either the equilibrium vapor pressure of a condensable component (i = 1, 2, 3) or the standard partial pressure of a non-condensable component (i = 5, 8) at which the standard Gibbs free energy change of reactions 1−(3) is assumed to be known. On the other hand, the dependence of the chemical potential μiα (i = 1, 4) of component i on the solution composition χ has the form2,82

N1 = N10 − (ν1 − q1) + ν4 N2 = N20 − (ν2 − q2) N3 = N30 − (ν3 − q3) − ν4 N5 = N50 − ν4 N6 = N60 − ν4 N7 = N70 − ν4 N8 = N80 + ν4

μiα = μiα (χ ) = μ͠ iα + kBT ln χi fi (χ ) (i = 1, 4)

(A5)

Denote the volume of per molecule of component j or i in a liquid phase by vj0 (j = 1, 2, 3) or vi(i = 1, 4), so that 3

V 0α =

where μ͠ iα is the chemical potential in some standard (reference) state, and f i(χ) is the activity coefficient of this component; similar expression can be written for μαi0 = μαi (χ0) (i = 1, 2, 3). Let us denote the change of the Gibbs free energy in reactions 1, (2), and (3) by Δg1, Δg2, and Δg3, respectively:

4

∑ qjvj0 , V α = ∑ νivi j=1

(A6)

i=1

As usual, assuming the droplet incompressible, and taking into account that vj0 = ∂μαj0/∂P and vi = ∂μαi /∂P, we have

Δg(1) = ν4(μ1β + μRα· − μ3α − μ6β ) ⎫ ⎪ ⎪ (2) α α β Δg = ν4(μRO· − μR· − μ7 ) ⎬ 2 ⎪ α β ⎪ Δg(3) = ν4(μ4α + μ8β − μRO − μ ) · 5 ⎭

μi0α (P0α , T , χ0 ) = μi0α (P , T , χ0 ) + vi0(P0α − P) (i = 1, 2, 3) μiα (P α , T , χ ) = μiα (P , T , χ ) + vi(P α − P)

(i = 1, 4)

2

(A7)

According to eq A4 and the equality Vβ0 + Vα0 = Vβ + Vα, it is clear that Pβ0Vβ0 = PVβ + PVα − PVα0 . Thus, one can show that

(A8)

Substituting eqs A2 and (A3) into eq A1 and taking account of eqs A4−(A8), one can obtain W = −[(ν1 − q1) −

ν4]μ1β

− [(ν3 − q3) +

ν4]μ3β

4

− ν4μ8β +

where

≡

−

ν4μ5β

(A14)

q2)μ2β −

ν4μ6β

−

Thus, the third term on the RHS of eq A10 represents the pure contribution ΔG ≡ ν4Δg from the sequence of chemical reactions 1−(3) to W, the free energy of formation of the droplet (ν1, ν2, ν3, ν4). Taking eqs A11, (A12) into account, one can show that

ν4μ7β

α α ∑ νμ i i − ∑ qiμi 0 + σA − σ0A 0

μjβ (P ,

Δg = ν4(μ1β + μ4α + μ8β − μ3α − μ5β − μ6β − μ7β )

3

i=1

μjβ

− (ν2 −

(A9)

i=1

T ) (j = 1, 8, j ≠ 4), μi0α ≡

⎡ ζ ζ χ f (χ ) ⎤ 4 4 ⎥ Δg = Δ͠ g + kBT ln⎢ 1 8 ⎢⎣ ζ5ζ6ζ7 χ3 f3 (χ ) ⎥⎦

μi0α (χ0 )

=μi0α (P , T , χ0 ) (i = 1, 2, 3), and μiα ≡ μiα (χ )= μiα (P , T , χ ) (i = 1, 4). Note as well that the surface tensions σ0 and σ and surface areas A0 and A are functions of the composition and size of the corresponding initial aerosol and evolved droplet: σ0 = σ0(χ0), A0 = A0(q1, q2, q3), and σ = σ(χ), A = A(ν1, ν2, ν3, ν4). Alternatively, eq A9 can be rewritten in the form 3

W=

3

+

ν4(μ1β

⎡ ζ ζ χ f (χ ) 1 ⎤ 4 4 ⎥ Δg = kBT ln⎢ 1 8 ⎢⎣ ζ5ζ6ζ7 χ3 f3 (χ ) Keq ⎥⎦

i=1

+

μ4α

+

μ8β

−

μ3α

− μ5β − μ6β − μ7β )

+ ν4(μ3α − μ3β ) + σA − σ0A 0

P μi = μi (P) = μi (Pi ∞) + kBT ln i Pi ∞ β

β

(A16)

Taking account of eqs A11, (A12), (A14), and (A16), denoting ζi = Pi /Pi ∞(i = 1, 8, i ≠ 4), and introducing the parameter ζ4 as

(A10)

For the pressure dependence of the gas/vapor phase chemical potentials one can use the representation2,82 β

(A15)

where Δ͠g is the aggregate change of the Gibbs free energy of reactions 1−(3) in the standard state. Defining the aggregate equilibrium constant of reactions 1−(3) via Keq = exp[−Δ͠g /kBT ], the last equation can be rewritten as

∑ (νi − qi)(μiα − μi β ) + ∑ qi(μiα − μi0α ) i=1

(A13)

where μαR· and μαRO2· are the chemical potentials of intermediate, “short-lived” products of chemical reactions 1 and (2), respectively. These do not enter the expression for the aggregate change Δg of the Gibbs free energy in the sequence of chemical reactions 1−(3):

−P αV α − P βV β + P0αV 0α + P0βV0β = (P − P α)V α − (P − P0α)V 0α

(A12)

ζ4 =

(i = 1, 8, i ≠ 4)

ζ3ζ5ζ6ζ7Keq ζ1ζ8

(A17)

one can thus rewrite eq A10 as

(A11) M


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The Journal of Physical Chemistry A 4

W = −kBT ∑ νi ln i=1

(13) Schauer, J. J.; Kleeman, M. J.; Cass, G. R.; Simoneit, B. R. T. Measurement of emissions from air pollution sources, 1, C1 through C29 organic compounds from meat charbroiling. Environ. Sci. Technol. 1999, 33, 1566−1577. (14) Schauer, J. J.; Kleeman, M. J.; Cass, G. R.; Simoneit, B. R. T. Measurement of emissions from air pollution sources, 2, C1 through C30 organic compounds from medium duty diesel trucks. Environ. Sci. Technol. 1999, 33, 1578−1587. (15) Hoffmann, T.; Odum, J. R.; Bowman, F.; Collins, D.; Klockow, D.; Flagan, R. C.; Seinfeld, J. H. Formation of organic aerosols from the oxidation of biogenic hydrocarbons. J. Atmos. Chem. 1997, 26, 189−222. (16) Odum, J. R.; Hoffmann, T.; Bowman, F.; Collins, D.; Flagan, R. C.; Seinfeld, J. H. Gas/particle partitioning and secondary organic aerosol yields. Environ. Sci. Technol. 1996, 30, 2580−2585. (17) Odum, J. R.; Jungkamp, T. P. W.; Griffin, R. J.; Flagan, R. C.; Seinfeld, J. H. The atmospheric aerosol-forming potential of whole gasoline vapor. Science 1997, 276, 96−99. (18) Griffin, R. J.; Cocker, D. R., III; Flagan, R. C.; Seinfeld, J. H. Organic aerosol formation from the oxidation of biogenic hydrocarbons. J. Geophys. Res. 1999, 104, 3555−3567. (19) Atkinson, R. Gas-phase tropospheric chemistry of organic compounds. J. Phys. Chem. Ref. Data, Monogr. 1994, 2. (20) Atkinson, R. Gas-phase tropospheric chemistry of volatile organic compounds, 1, Alkanes and alkenes. J. Phys. Chem. Ref. Data 1997, 26, 215−290. (21) Smith, D. F.; Kleindienst, T. E.; McIver, C. D. Primary product distributions from the reaction of OH with m-, p-xylene, 1,2,4-and 1,3,5-trimethylbenzene. J. Atmos. Chem. 1999, 34, 339−364. (22) Yu, J.; Cocker, D. R., III; Griffin, R. J.; Flagan, R. C.; Seinfeld, J. H. Gas phase ozone oxidation of monoterpenes: Gaseous and particulate products. J. Atmos. Chem. 1999, 34, 207−258. (23) Saxena, P.; Hildemann, L. M. Water-soluble organics in atmospheric particles: A critical review of the literature and application of thermodynamics to identify candidate compounds. J. Atmos. Chem. 1996, 24, 57−109. (24) Murphy, D. M.; Cziczo, D. J.; Froyd, K. D.; Hudson, P. K.; Matthew, B. M.; Middlebrook, A. M.; Peltier, R. E.; Sullivan, A.; Thomson, D. S.; Weber, R. J. Single-particle mass spectroscopy of tropospheric aerosol particles. J.Geophys.Res. 2006, 111, D23532. (25) Zhang, Q.; Jimenez, J. L.; Canagaratna, M. R.; Allan, J. D.; Coe, H.; Ulbrich, I.; Alfarra, M. R.; Takami, A.; Middlebrook, A. M.; Sunet, Y. L.; et al. Ubiquity and dominance of oxygenated species in organic aerosols in anthropogenically influences Northern Hemisphere midlatitudes. Geophys. Res. Lett. 2007, 34 (6), L13801. (26) Chacon-Madrid, H. J.; Donahue, N. M. Fragmentation vs functionalization: chemical aging and organic aerosol formation. Atmos. Chem. Phys. 2011, 11, 10553−10563. (27) Pandis, S. N.; Donahue, N. M.; Murphy, B. N.; Riipinen, I.; Fountoukis, C.; Karnezi, E.; Patoulias, D.; Skyllakou, K. Introductory lecture: Atmospheric organic aerosols: insights from the combination of measurements and chemical transport models. Faraday Discuss. 2013, 165, 9−24. (28) Seinfeld, J. H.; Pankow, J. F. Organic atmospheric particulate matter. Annu. Rev. Phys. Chem. 2003, 54, 121−140. (29) Ruehl, C. R.; Wilson, K. R. Surface organic monolayers control the hygroscopic growth of submicrometer particles at high relative humidity. J. Phys. Chem. A 2014, 118, 3952−3966. (30) Petters, M. D.; Prenni, A. J.; Kreidenweis, S. M.; DeMott, P. J.; Matsunaga, A.; Lim, Y. B.; Ziemann, P. J. Chemical aging and the hydrophobic-to-hydrophilic conversion of carbonaceous aerosol. Geophys. Res. Lett. 2006, 33, L24806. (31) Rudich, Y. Laboratory perspectives on the chemical transformations of organic matter in atmospheric particles. Chem. Rev. 2003, 103, 5097−5124. (32) Rudich, Y.; Donahue, N. M.; Mentel, T. F. Aging of organic aerosol: bridging the gap between laboratory and field studies. Annu. Rev. Phys. Chem. 2007, 58, 321−352.

ζi + σ(χ1 , χ2 , χ3 ) χi fi (χ1 , χ2 , χ3 )

× A(ν1 , ν2 , ν3 , ν4) + W0(q1 , q2 , q3)

(A18)

On the RHS of this equation, the last term, W0(q1, q2, q3), contains only quantities independent of the variables ν1, ν2, ν3, ν4 of state of the droplet; it can be interpreted as the free energy of formation of an initial aerosol, characterized by the parameters q1, q2, q3. This is eq 4 in the main text. Note that although component 4 (radical R4) is not present in the vapor mixture, the quantity ζ4 can be loosely (for the sake of uniformity) called “the saturation ratio” of component 4.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: idjikaev@buffalo.edu. (Y.S.D.) ORCID

Yuri S. Djikaev: 0000-0002-0645-4105 Notes

The authors declare no competing financial interest. † E-mail: feaeliru@buffalo.edu. (E. R.)

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