Secondary organic aerosol (SOA) formed from partitioning of oxidation products of anthropogenic and biogenic volatile organic compounds (VOCs) accounts for a substantial portion of atmospheric particulate matter. In describing SOA formation, it is generally assumed that VOC oxidation products rapidly adopt gas-aerosol equilibrium. Here we estimate the equilibration timescale,τeq, of SOA gas-particle partitioning using a state-of-the-art kinetic flux model.τeqis found to be of order seconds to minutes for partitioning of relatively high volatility organic compounds into liquid particles, thereby adhering to equilibrium gas-particle partitioning. However,τeqincreases to hours or days for organic aerosol associated with semi-solid particles, low volatility, large particle size, and low mass loadings. Instantaneous equilibrium partitioning may lead to substantial overestimation of particle mass concentration and underestimation of gas-phase concentration.
 Organic aerosol is ubiquitous in the atmosphere [Goldstein and Galbally, 2007; Zhang et al., 2007]. The major component of OA is secondary organic aerosol (SOA), the formation of which involves the multi-generation gas-phase oxidation of volatile organic compounds (VOCs), leading to an array of lower volatility oxidation products that partition between the gas and condensed phases [Jimenez et al., 2009; Kroll and Seinfeld, 2008]. The aerosol condensed phase is generally a mixture of organic and inorganic compounds, such as sulfate, nitrate, ammonium, and water. It is generally assumed that the semi-volatile organic oxidation products rapidly establish a gas-particle equilibrium partitioning [Odum et al., 1996; Donahue et al., 2006]. The theory upon which the equilibrium partitioning is based relies on the implicit assumption that the condensed phase is homogeneously mixed.
 Recent evidence has emerged of the existence of semi-solid SOA [Virtanen et al., 2010; Saukko et al., 2012]. Glass transition temperatures of α-pinene related SOA compounds range over 260–310 K [Koop et al., 2011]. It has been found that α-pinene secondary organic aerosol does not evaporate in a thermodenuder as predicted by equilibrium partitioning theory [Cappa and Wilson, 2011], and unexpectedly slow evaporation of ambient and laboratory-generatedα-pinene SOA has also been observed [Vaden et al., 2011]. Perraud et al. observed non-equilibrium formation and growth ofα-pinene SOA, consistent with semi-solid behavior. Conceptually, it has been shown that the aerosol phase state, as characterized by its viscosity or bulk diffusivity, can assume liquid, semi-solid, or glassy solid state depending on ambient relative humidity (RH) and temperature [Koop et al., 2011; Mikhailov et al., 2009; Shiraiwa et al., 2011].
 Given these observations of aerosol phase states, an overriding question is the effect of this phase state on the common assumption of gas-particle partitioning equilibrium, specifically the effect of mass transport in the bulk of amorphous semi-solid particles. The present work provides a theoretical analysis of the equilibration timescale of SOA partitioning in liquid, semi-solid, and amorphous solid particles using the kinetic flux model KM-GAP [Shiraiwa et al., 2012a] (see auxiliary material), which resolves mass transport in both gas and particle phases. The model allows a systematic evaluation of the equilibration timescale as a function of SOA volatility, bulk-phase diffusivity, surface accommodation coefficient, and particle size.
 In the present study we evaluate numerically the time to establish gas-particle equilibrium for a large range of SOA parameters. While the results for any numerical simulation strictly reflect only the conditions of that simulation, by varying volatility and aerosol phase state over a wide range it is possible to infer general conclusions concerning the establishment of gas-particle equilibrium and how the rate depends on the key properties of the SOA system. While data sets indicating the existence of semi-solid aerosol phases are emerging, it is not yet possible to link the type of detailed simulations that we present here to specific experiments due to the shortage of measurements of SOA viscosity. Nonetheless, we are able to anticipate from the numerical simulations the effect of SOA volatility and aerosol phase state on the nature of the SOA growth that will occur.
2. Gas-Particle Partitioning
 The volatility of a compound i can be expressed by the effective saturation mass concentration Ci* = 106Mipio/760 RT, where Mi (g mol−1) is the molecular weight of compound i, pio (Torr) is the saturation vapor pressure of pure compound i, R (m3 atm mol−1 K−1) is the gas constant, and T(K) is temperature. Note that ideal mixing condition is considered for simplicity. At gas-particle equilibrium,pi = ps,i, namely Cig = Cis, where pi and Cig are the partial pressure and mass concentration in the vapor phase, respectively, and ps,i and Cis are the partial pressure and mass concentration of i just above the particle surface, respectively. Note that ps,iobeys Raoult's law in equilibrium with the near-surface bulk, which is resolved by KM-GAP. Atps,i < pi (Cis < Cig), species iwill diffuse from the gas to the particulate phase (diffusion-limited growth). Ifpi changes slowly and ps,i follows pi instantaneously (ps,i ≈ pi), the particle still grows (quasi-equilibrium growth). The equilibration timescaleτeq can be defined as the e-folding time for relaxation of the partial pressure gradient.
 We illustrate the evaluation of τeqfor condensation of a semi-volatile compound generated by oxidation of a parent VOC. We assume that the parent VOC, with an initial concentration of 1011 cm−3, is converted to the semi-volatile product with a given first-order rate coefficient,kg. Conversion of this first-generation product to higher generation compounds need not be considered. The physical and kinetic parameters assumed for the system are given inTable 1. To investigate the effect of volatility of the oxidation product on τeq, C* is varied over the range of 10−5–105μg m−3. The initial number and mass concentrations of non-volatile pre-existing particles are taken as 104 cm−3 and 1 μg m−3, respectively. The initial particle size distribution is assumed log-normal with mean diameterDmean = 50 nm and standard deviation σ= 1.5. The nominal aerosol-phase bulk diffusion coefficient (Db) is assumed to be 10−8 cm2 s−1, corresponding to that of a viscous-liquid droplet [Koop et al., 2011], and the surface accommodation coefficient αs,0 = 1 (the fraction of impinging molecules that are adsorbed at the particle surface). Note that surface tension is set to be 0.04 N m−1 and the use of different values within the range of 0.02–0.07 N m−1 gives practically the same results.
Table 1. Properties and Kinetic Parameters of the Oxidation Product VOC Used in the Simulations for SOA Growth
αs,0 is varied for determining the effect of surface accommodation.
Db is varied for representing the effect of particle phase state.
surface accommodation coefficient on free-substratea
first-order rate coefficient of conversion for VOC → SVOC
M (g mol−1)
 In the nominal simulation of a very low volatility product (C* = 10−3μg m−3), gas-particle equilibrium is reached at ∼104 s (Figure 1a). Particle growth is controlled by gas-phase diffusion becauseCg > Csin the course of the particle growth. The evolution of the particle size distribution exhibits the narrowing characteristic of diffusion-limited particle growth [Seinfeld and Pandis, 2006; Zhang et al., 2012] (Figure 1b). In the case of an intermediate volatility oxidation product (C* = 103μg m−3), the gas-phase partial pressure gradient vanishes within ∼1 s, and as the mass concentration of the product in the gas phase,Cg, continues to increase due to the conversion of the parent VOC, the mass concentration of the product just above the particle surface, Cs, tracks the change in Cg essentially instantaneously (Figure 1c). In this case, the gas-phase rate of formation of the oxidation product controls particle growth (so-called quasi-equilibrium growth). Due to the relatively high volatility assumed for the condensing species, the particle grows only slightly (Figure 1d).
Figure 1eshows the comparable results for a semi-volatile oxidation product (C* = 10 μg m−3). Particle growth is initially governed by gas-phase diffusion until ∼102 s. After this initial period, the peak of the size distribution increases with increasing width (Figure 1f), which is characteristic of quasi-equilibrium growth. The larger particles absorb more vapor thus grow preferentially, as the equilibrium vapor pressure over their surface is smaller than that over smaller particles due to the Kelvin effect [Zhang et al., 2012].
Figure 2 shows τeq as a function of C*. τeq increases as C*decreases, as the partial pressure gradient between the gas phase and the particle surface is larger for smaller C* [Meng and Seinfeld, 1996; Marcolli et al., 2004; Zhang et al., 2012]. The effects of reaction rate constant kg are investigated by varying kg in the range of 0.1–10−7 min−1. τeq is independent of kg as long as τeq < kg−1. The slope of τeq decreases when τeq > kg−1, because the rate of formation of the oxidation product drops due to consumption of the parent VOC that would otherwise increase linearly. τeqof ELVOC and LVOC is of order hours, and their growth may be governed by gas-phase diffusion, whileτeqof SVOC and IVOC is the order of seconds, thus controlled by quasi-equilibrium growth.τeqcan be regarded as the timescale for transition from kinetically-limited growth to quasi-equilibrium growth.
 To examine the influence on τeq of surface accommodation coefficient αs,0 and particle phase state, as reflected by the bulk diffusivity Db, we consider, for convenience, the growth of mono-disperse particles of initial diameter 200 nm from a condensing oxidation product ofC* = 10 μg m−3. The initial gas-phase VOC concentration is 1011 cm−3 and initial particle number concentration is 104 cm−3. In calculation of τeq, αs,0 and Db are varied over the ranges of 10−3 −1 and 10−21–10−5 cm2 s−1, respectively. Note that typical values of Db of organics are 10−10–10−5 cm2 s−1 for liquid, 10−20–10−10 cm2 s−1for semi-solid, and <10−20 cm2 s−1 for solid [Shiraiwa et al., 2011].
 The value of αs,0 has a major impact on τeqfor liquid and semi-solid particles with relatively highDb (Figure 3). Decrease of αs,0 by an order of magnitude leads to roughly an order of magnitude increase in τeqSOA growth is limited by gas-phase diffusion atαs,0 ≈ 1, but becomes limited by surface accommodation at smaller αs,0. In this αs,0- limited regime,τeq is insensitive to Dbbecause surface-bulk exchange and bulk diffusion are more rapid than surface accommodation. WhenDbdecreases below a certain threshold, the timescales for surface-bulk exchange and bulk diffusion become longer than that of gas-phase diffusion and accommodation. In this case,τeq is insensitive to αs,0 but sensitive to Db. In the Db- limited regime, decrease ofDb by an order of magnitude leads to roughly an order of magnitude increase in τeq; for the conditions of the simulation, τeqis the order of minutes for semi-solid particles withDb ≈ 10−15 cm2 s−1, increasing to days and longer for particles with Db < 10−20 cm2 s−1.
 We investigate the impact on τeqof size and concentration of pre-existing particles, over the range of 30–1000 nm and 0.1–100μg m−3, respectively (Figure 4). Initial conditions are the same as those of the prior calculation with αs,0 = 1 and C* = 10 μg m−3. The growth of mono-disperse particles is considered.Figure 4 shows τeq for viscous liquid (Db = 10−8 cm−3) and semi-solid (Db = 10−15 cm−3) particles. A larger diameter leads to longer τeq: for example, at a typical ambient concentration of 1 μg m−3, τeq is on the order of minutes for 100 nm liquid particles, increasing to the order of hours for 1 μm particles (Figure 4a). In this comparison ambient particle mass concentration is held constant, so increasing particle size translates to a decrease of the number and surface area concentration of particles, and a decrease of total accommodation of molecules to the surface. An increase of particle concentration also leads to an increase of surface area concentration, thereby leading to a decrease of τeqNote that all particles are assumed to be spherical in our analysis; if SOA particles are non-spherical with a larger surface area,τeq would be correspondingly smaller.
 Typical SOA mass concentrations in laboratory chamber experiments lie in the range of 10–100 μg m−3. Over this range, τeqfor accumulation mode particles is on the order of minutes for both liquid and semi-solid particles. Therefore, SOA particles formed at these mass loadings will be in gas-particle equilibrium. Typical ambient organic mass concentrations in Beijing [Sun et al., 2010], Mexico City [Jimenez et al., 2009], Los Angeles Basin [Hersey et al., 2011], Hyytiälä, Finland [Raatikainen et al., 2010], and Amazon Basin [Chen et al., 2009] are indicated in Figure 4. With the exception of highly polluted urban areas such as Beijing and Mexico City, ambient organic mass concentrations are typically <10 μg m−3, and τeqis of order minutes for liquid particles and hours or more for semi-solid particles. The time required to reach equilibrium is sufficiently long such that atmospheric SOA constituents may exist in a non-equilibrium state, particularly for semi-solid aerosol particles.
3. Kinetic Versus Instantaneous Partitioning
 To evaluate the common assumption of instantaneous gas-particle partitioning and investigate the impact of kinetic effects on predicted mass concentrations in gas and particulate phases, here we compare the comprehensive results of the kinetic flux model to those of an equilibrium gas-particle partitioning model. The evolution of mass concentration is represented by condensation of semi-volatile VOC generated by oxidation of a parent VOC with the same conditions as inFigure 1 and Table 1. In the instantaneous gas-particle partitioning model, the VOC oxidation product instantaneously partitions into the particle phase, and gas-phase and bulk diffusion kinetics are ignored.
 For IVOC (C* = 103μg m−3), both models give almost identical results, because SOA growth is governed by the quasi-equilibrium growth mechanism (Figure S1a). For LVOC, however, the instantaneous partitioning model overestimates the particle phase concentration by an order of magnitude and underestimates the gas-phase concentration up to several orders of magnitude before equilibration is established (Figure S1b). For partitioning of SVOC (C* = 10 μg m−3) into semi-solid particles (Db = 10−15 cm−3), the assumption of instantaneous equilibrium overestimates the particle phase concentration by one order of magnitude (Figure S1c). These results clearly establish the validity of instantaneous gas-particle partitioning for relatively high volatility compounds partitioning into liquid particles, but this assumption breaks downs for partitioning of low and semi-volatile compounds into liquid and semi-solid particles, leading to overestimation of the particle phase concentration and underestimation of the gas-phase concentration.
 It has been reported that the growth of freshly-nucleated particles is governed by kinetically-limited growth rather than quasi-equilibrium growth [Riipinen et al., 2011]. Indeed, the evolution of the size distribution in nucleation events often shows the narrowing characteristic of diffusion-limited growth. An apparentC* of ultrafine particles is estimated to be 10−3–10−2μg m−3 [Pierce et al., 2011; Brock et al., 2011], resulting in an equilibration timescale of hours. Such low-volatility compounds may be generated by chemical reactions in the condensed phase in addition to gas-phase formation with subsequent condensation [Kalberer et al., 2004; Ervens et al., 2011; Riipinen et al., 2012]. The formation of oligomers and other multifunctional organic substances with high molecular mass and low vapor pressure is one mechanism that could lead to solidification, resulting in an increase of particle viscosity and decrease of bulk diffusivity [Pfrang et al., 2011; Koop et al., 2011], which could significantly affect the condensation and evaporation kinetics. The evolution of the particle phase due to reactive uptake and condensed phase chemistry and phase change in the course of particle growth are not included in this study.
 Slow equilibration conditions are more likely to be prevalent in remote areas with low aerosol mass concentrations, for example, in boreal forests, where SOA has been found to exhibit amorphous solid behavior [Virtanen et al., 2010], and in the mid and upper troposphere, where SOA most likely undergoes a glass transition [Zobrist et al., 2008; Koop et al., 2011]. Moreover, slow equilibration may be more relevant under flow tube conditions with reaction times of minutes than for chamber studies with considerably longer time scales. In addition to SOA growth, the results obtained in this study are directly applicable to SOA evaporation. Several studies have observed unusually slow evaporation of ambient and laboratory-generated SOA [Grieshop et al., 2007; Vaden et al., 2011]. These observations are consistent with the evaporation timescale of semi-solid SOA, as shown inFigures 3 and 4b. Note that slow evaporation may be also due to highly complex multi-component mixtures with high molecular mass and low vapor pressure [Widmann et al., 1998]. Such slow evaporation timescales can potentially affect volatility measurements using a thermodenuder, in which organics may not be in a glassy state but remain highly viscous [Riipinen et al., 2010; Cappa and Wilson, 2011; Saleh et al., 2011].
 This work was supported by National Science Foundation grant AGS-1057183. MS is supported by the Japan Society for the Promotion of Science (JSPS) Postdoctoral Fellowship for Research Abroad. The authors thank Andreas Zuend, Xi Zhang and Ulrich Pöschl for helpful discussions.
 The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.