Interactive comment on “ Flux induced growth of atmospheric nanoparticles by organic vapors ”

In this manuscript the authors present an interesting mechanism affecting the very initial growth of particles in atmospheric new particle formation process. This mechanism, referred to as growth due to the diffusion in cluster size space, is included in classical nucleation theory and can be derived from the net flux of clusters in heterogenous nucleation of organic vapours. Diffusion in cluster size space is basically analogous to traditional molecular diffusion, which spreads the molecules against the concentration gradient in spatial coordinates. Diffusion in cluster size space occurs against the concentration gradient in cluster size coordinate and tends to widen the cluster mode along this coordinate. In principle, this manuscript is worth publishing in ACP, since the studied mechanism may have even a significant impact on the new particle formation


Introduction
Atmospheric aerosols have adverse effects on air quality and human health (Pope et al., 2002).They also contribute to urban and regional haze, leading to reduction in visibility (Hinds, 1999).On both regional and global scales, atmospheric aerosols Figures

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Full strongly influence climate by scattering solar radiation and by serving as Cloud Condensation Nuclei (CCN) to change the reflectivity, lifetime, and coverage of clouds (Twomey, 1977;Albrecht, 1989;Finlayson-Pitts and Pitts, 2000;Seinfeld and Pandis, 2006).New particle formation in the atmosphere significantly influences the concentration of atmospheric aerosols, and therefore their impact on climate (Kerminen et al., 2005;Laaksonen et al., 2005;Spracklen et al., 2008).Model simulations show that nearly half of the global cloud condensation nuclei (CCN) in the atmospheric boundary layer may be formed through new particle formation (Merikanto et al., 2009).New particle formation is a two-stage process consisting of formation of thermodynamically stable clusters from trace gas molecules (homogeneous nucleation) followed by growth of these clusters to a detectable size of ∼ 3 nm (McMurry et al., 2005;Kuang et al., 2010).Due to the high coagulation rate of clusters smaller than 3 nm with the pre-existing aerosol population, for new particle formation to take place, these clusters need to grow sufficiently fast to escape removal by coagulation (Kuang et al., 2010).
It is generally accepted that homogeneous nucleation in the troposphere involves sulfuric acid (Weber et al., 1996;Riipinen et al., 2007;Kuang et al., 2008).However, the concentration of gaseous sulfuric acid, while sufficient for the first step of homogeneous nucleation, is often insufficient to grow the resulting clusters (here, particles smaller than 3 nm are also referred to as clusters) fast enough to survive the coagulation scavenging by pre-existing aerosol population (Kuang et al., 2010).Observed growth rates of nucleated particles are often much greater than those based only on the condensation of sulfuric acid and associated inorganic compounds (Weber et al., 1997;Makela et al., 2001;O'Dowd et al., 2002;Iida et al., 2008;Kuang et al., 2010), and modeling studies have indicated that condensation of low-volatility organic vapors contribute significantly to the initial growth of the clusters (Kulmala et al., 2004c;Paasonen et al., 2010).This enhancement is also supported by field measurements that show organics are often the dominant component of newly formed particles (Smith et al., 2008(Smith et al., , 2010)).However, thermodynamic considerations suggest that for 1-2 nm newly formed clusters, the strong increase of equilibrium vapor concentration due to Introduction

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Full cluster surface curvature (Kelvin effect) would prevent ambient organics from condensing on these small clusters (Zhang and Wexler, 2002;Kulmala et al., 2004a).Since freshly nucleated clusters are most susceptible to loss below 3 nm due to the high coagulation rate, the initial growth strongly influence the rate of new particle formation and ambient aerosol population.Here, we extend the methodology of classical nucleation theory to describe the initial condensational growth of freshly nucleated clusters by organic vapor.The initial growth is treated as heterogeneous nucleation of organic molecules onto the clusters, and the number flux of clusters through a given size is examined using the same statistical thermodynamics treatment as in nucleation theory.We find that the strong gradient in cluster population with respect to size lead to positive cluster number flux, and therefore driving the growth of clusters substantially smaller than the Kelvin diameter, which is the critical embryo size for heterogeneous nucleation and is conventionally considered as the minimum particle size that can be grown through condensation.The conventional approach, which neglects this contribution from the cluster concentration gradient, substantially underestimates the initial growth of the clusters and the rates of new particle formation and the subsequent production of CCN.

Methods
Conventionally, the growth rate (GR) due to condensation is calculated from the difference between the concentrations of condensing species far from and at the particle surface corrected for the effect of particle curvature (Seinfeld and Pandis, 2006): Here D p is the particle diameter, D m is the diameter of the condensing molecule, cµ is the mean relative speed of the particle and condensing molecule, v is the volume of Figures

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Full the condensing molecule, C ∞ is the concentration of condensing species far from the particle surface, C S is the saturation concentration over a flat surface, and GR cond is the conventional condensational growth rate.The enhancement of the surface vapor concentration due to particle curvature (Kelvin effect) is described by the factor exp 4σv kT D p , where σ is surface tension, k is the Boltzmann constant, and T is absolute temperature.
In Eq. ( 1), the term 1 2 is included to account for the increase in collision diameter, which becomes non-negligible when the sizes of the particle and condensing molecule are comparable (Nieminen et al., 2010).The Kelvin factor increases exponentially with decreasing D p and is huge for newly formed 1-2 nm clusters, reaching as high as 10 5 for typical ambient organic species (Zhang and Wexler, 2002;Kulmala et al., 2004a).As a result, the concentration difference becomes negative, preventing organics from condensing onto the clusters (Zhang and Wexler, 2002).The particle diameter at which the quantity C ∞ − C S exp 4σv kT D p equals zero is referred to as the Kelvin diameter.The Kelvin diameter, which is typically greater than 2 nm for ambient organic vapors, represents the minimum size at which particles would grow through condensation according to Eq. (1).In this traditional view, condensation of organic vapors cannot contribute to the growth of clusters smaller than the Kelvin diameter, and the resulting particle number flux, derived as is zero or even negative, where dN dD p is the particle number size distribution, and N is the number concentration of particles with diameters less than D p .It has been suggested that other mechanisms, such as the lowering of vapor pressure through Raoult's law (Kulmala et al., 2004a,b) (nano-K öhler theory) and heterogeneous chemical reactions (Zhang and Wexler, 2002) may help to overcome this large Kelvin effect by facilitating growth to above the Kelvin diameter, at which point the organic vapor can condense conventionally according to Eq. ( 1).
Here we don't attempt to dispute or ascertain the relative importance of various mechanisms for the early growth of nucleated clusters described in previous studies.Introduction

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Full Instead, we focus on an alternative mechanism that has not been considered.We show that the particle number flux calculated using the conventional condensational growth rate represents the total number flux only when the particle size is substantially greater than the Kelvin diameter.When the cluster size is smaller than or close to the Kelvin diameter, heterogeneous nucleation of organic vapors onto the clusters makes a second contribution that drives growth even for clusters smaller than the Kelvin diameter.This second contribution is due to the gradient in cluster concentration with respect to cluster size.It is not accounted for in conventional growth rate calculations, although it is included in classical nucleation theory.During the heterogeneous nucleation of organic vapors, the net flux from clusters with g condensed organic molecules to those with g + 1, J g , is (Seinfeld and Pandis, 2006): where f g is the corresponding number concentration of the initial clusters (i.e.seed) plus g molecules of condensate, β g is the size-dependent per-particle condensation rate, and γ g is the size-dependent per-particle evaporation rate.Equation ( 2) can be rewritten to separate contributions to the flux from drift and diffusion in cluster sizespace (detailed derivation given in the Supplement): Rewriting the particle flux as

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Full The lead term on the right side of Eq. ( 4) describes the drift in the force field given by the gradient of the cluster free energy with respect to cluster size, and is essentially the conventional condensational growth rate described by Eq. ( 1) (details given in the Supplement).The second term on the right is related to the gradient in cluster concentration with respect to cluster size.Note that traditional molecular diffusion theory describes the tendency of molecules to spread in space and is due to the gradient of concentration with respect to the space coordinate.As an analogy to traditional molecular diffusion, the second term is referred to as the contribution due to diffusion in cluster size space, with the size-dependent diffusion coefficient (Goodrich, 1964;Lifshitz and Pitaevskii, 1981;Friedlander, 2000;McGraw, 2001).This term is significant when the cluster size distribution is steep (i.e.strong gradient in f g with respect to cluster size), which occurs for clusters below or near the Kelvin diameter, the critical embryo size of the heterogeneous nucleation.As shown later, below the Kelvin diameter, the drift term results in evaporation to a smaller size since β g − γ g < 0; in this size regime only the diffusion term makes a positive contribution to the number flux.Conventional growth rate calculations consider only the drift term, which underestimates the full growth rate because the diffusion contribution to growth is positive during new particle formation.This contribution becomes especially significant near or below the Kelvin diameter.

Results and discussions
The particle growth rate and its contributions from both drift and diffusion with respect to cluster size are calculated using parameters of gas-phase organic species (Table 1) that are within the typical ranges observed in earlier studies.Previous analyses of field measurements indicate ambient organic vapor concentrations (C ∞ ) between 1 × 10 7 and 3 × 10 8 cm −3 (Kulmala et al., 2001;Paasonen et al., 2010), and saturation vapor concentrations (C s ) less than ∼ 10 5 -10 6 cm −3 (Kulmala et al., 1998;Kerminen et al., 2000;Anttila and Kerminen, 2003).The conventional growth rate from Eq. ( 1) (GR cond ), 22819 Introduction

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Full the effective growth rate (GR eff ) based on the total flux J, and its resolution into the distinct contributions from drift and diffusion (GR drift and GR diff ) are shown in Fig. 1 as a function of cluster diameter.GR drift agrees well with the commonly used GR cond based on Eq. ( 1).Both GR cond and GR drift decrease with decreasing D p and reach zero at a Kelvin diameter of ∼ 1.9 nm determined by the parameter values in Table 1.
Therefore, according to the conventional formula, the smallest particle size that can be grown through the condensation of organic vapors is 1.9 nm in this case.The contribution from the diffusion term GR diff is determined by cluster size distribution f g and the gradient of f g with respect to g.The cluster size distribution is derived from β g and γ g under steady state (McGraw, 2001), a reasonable assumption for sizes near or below the critical embryo size of the heterogeneous nucleation (i.e.Kelvin diameter) (Seinfeld and Pandis, 2006), where GR diff is significant.Because f g appears in both the numerator and denominator, only the shape of the cluster size distribution is required to derive GR diff .As D p decreases, the negative gradient in f g with respect to g becomes stronger, and therefore GR diff increases drastically.As a result, the overall growth rate GR eff remains substantial even at D p as small as ∼ 1.5 nm.This indicates that the conventional approach substantially overestimates (by 20 % in this case) the minimum size of the particles that grow through condensation of organics.Furthermore, even at particle sizes just above the Kelvin diameter, GR cond remains much smaller than the overall GR eff .As shown later, this translates into a significant underestimation of the new particle formation rate calculated using GR cond alone.
The main mechanism for the growth of the particles through heterogeneous nucleation is also illustrated in Fig. 2. The conventional approach (i.e.GR cond takes into consideration only the drift term, which is negative below the Kelvin diameter, since β g − γ g < 0. Therefore, in the conventional view, particles below the Kelvin diameter will evaporate because it is energetically favorable, and only particles above the Kelvin diameter (i.e.larger than the critical embryo size of the heterogeneous nucleation) can grow through condensation.In heterogeneous nucleation theory, for an individual particle that is smaller than the Kelvin diameter, the per particle evaporation rate is greater Introduction

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Full than the per particle condensation rate, i.e. γ g > β g .Therefore it is more likely for the individual particle to evaporate than to grow.However, because of the strong (negative) gradient in cluster population with respect to size (f g > f g+1 ), the total condensation flux (i.e.β g f g ) is greater than the total evaporation flux (i.e.γ g+1 f g+1 ) and the resulting net flux is positive (i.e.J g = β g f g − γ g+1 f g+1 > 0).This net forward flux induced by heterogeneous nucleation can effectively move sub-Kelvin particles over the energy barrier and allow them to grow to larger sizes.The impact of the gradient in the cluster size distribution on the number flux is previously established in both heterogeneous and homogeneous nucleation theories (Frenkel, 1946;Goodrich, 1964;Shizgal and Barrett, 1989;Ruckenstein and Nowakowski, 1990), and it is the very same principle in which gas phase molecules can overcome the energy barrier and form larger thermodynamically stable clusters during homogeneous nucleation (i.e. the first step of new particle formation).
It is worth noting that given the positive heterogeneous nucleation flux, GR eff is greater than zero at all cluster sizes.Here we define D p,lower as the particle size at which GR eff is 1 % of its maximum value (the maximum GR eff occurs at 3.5 nm in the above case), and D p,upper as the particle size at which the relative difference between GR eff and GR cond decreases to 10 % (2.24 nm for the above case).The size range from D p,lower to D p,upper represents where the overall growth rate may be substantially underestimated if the contribution from GR diff is not taken into consideration.Figure 3a shows D p,lower , D p,upper , and the Kelvin diameter as the ambient organic vapor concentration C ∞ increases from 1 × 10 7 to 3 × 10 8 cm −3 , a representative range suggested from earlier studies (all other parameters remain the same, as listed in Table 1).Similarly, the dependencies of the three diameters on surface tension σ are presented in Fig. 3b for σ ranging from 0.03 to 0.06 N m −1 .As expected, the Kelvin diameter, or the critical embryo size for heterogeneous nucleation, decreases with increasing C ∞ or decreasing σ.The fractional differences between D p,lower , D p,upper , and the Kelvin diameter increase as the Kelvin diameter decreases, suggesting a stronger impact from the diffusion term GR diff for systems with smaller Kelvin diameter.Calculations are also Introduction

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Full The contribution from GR diff can have a significant impact on the overall GR eff and therefore on the time required for clusters to grow to detectable sizes.The growth time directly controls the survival probability of freshly formed clusters, which is defined as the probability that a cluster will grow to a detectable size (nominally 3 nm) before being scavenged by the pre-existing aerosol (Weber et al., 1997;Kerminen and Kulmala, 2002;McMurry et al., 2005).Because freshly nucleated clusters are most susceptible to loss below 3 nm, the survival probability quantifies the direct impact of atmospheric nucleation on the ambient aerosol population.As shown in Fig. 1, GR diff contributes substantially to the overall growth rate GR eff for clusters near or smaller than the Kelvin diameter.To examine its impact on the survival probability, we numerically solved the aerosol general dynamic equation for an aerosol population growing through simultaneous condensation and coagulation (Gelbard and Seinfeld, 1978;Friedlander, 2000;Kuang et al., 2008Kuang et al., , 2009)), explicitly accounting for the size-dependent growth rates in Fig. 1.Aerosol loss rates were determined exclusively from coagulation with the preexisting aerosol, modeled with a size distribution typical of an urban aerosol (Jaenicke, 1993), and scaled to give a Fuchs surface area (A Fuchs ) of 330 µm 2 cm −3 , the average value for NPF events observed in Mexico City during the MILAGRO measurement campaign (Kuang et al., 2010).The effect of reduced cluster concentrations (due to coagulation) on the diffusion contribution to growth has been neglected in this calculation.
Since both sulfuric acid and organic vapors contribute to particle growth during new particle formation events, the survival probabilities are derived using GR SA + GR cond and GR SA + GR eff , where GR SA represents the growth rate attributed to sulfuric acid condensation, and GR cond and GR eff are the conventional growth rate and the effective growth rate due to organics presented in Fig. 1.Earlier field studies show that the growth enhancement factor Γ, defined as (GR SA + GR cond )/GR SA , measured at 3 nm mostly ranges from 5 to 20 (Kuang et al., 2010, also note that GR cond and GR eff are essentially the same at 3 nm).The calculations are carried out for three sulfuric acid Figures

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Full concentrations (5.4 × 10 7 , 2.4 × 10 7 , and 1.1 × 10 7 cm −3 ), which correspond to Γ values of 5, 10, and 20 for the value of GR cond shown in Fig. 1.For thermodynamically stable clusters of 1.5 nm (Kulmala et al., 2007), the ratio of the cluster survival probability derived from GR eff to that based on GR cond is given in Table 2.The survival probability and therefore the rate of new particle formation are significantly underestimated when applying the conventional growth rate GR cond to organics.This underestimation of the survival probability becomes more severe with increasing Γ, as the relative contribution of organics to total growth increases, reaching factors of 1.6, 4.8, and 62 for Γ values of 5, 10, and 20, respectively.In a second set of calculations we examine the impact of the growth due to diffusion in size space (GR diff ) on the survival probability under more pristine conditions.In these calculations, the H 2 SO 4 concentrations in above calculations are reduced by a factor of 10 to 5.4 × 10 6 , 2.4 × 10 6 , and 1.1 × 10 6 cm −3 , respectively.The concentration 1.1×10 6 cm −3 is near the lower end of sulfuric acid concentration ranges observed during new particle formation events in remote locations (Kuang et al., 2010).The concentration and saturation concentrations of organic vapor are also reduced by a factor of 10 to maintain the same growth rate enhancement Γ.If the original organics concentrations are maintained, the growth rate enhancement will be too large and inconsistent with field observations, and the impact due to GR diff will be even greater, as the growth due to organics will have an even larger contribution to the total particle growth.For the pristine conditions, the survival probability is derived using a pre-existing aerosol size distribution with a typical shape of a rural aerosol (Jaenicke, 1993), but scaled to give an average Fuchs surface area of 33.5 µm 2 cm −3 for the new particle formation events observed at Hyyti äl ä during QUEST IV campaign (Kuang et al., 2010).For thermodynamic stable clusters of 1.5 nm, the ratio of the survival probability derived from GR eff to that based on GR cond is similar to that under a more polluted environment, and ranges from 1.6 to 66 as Γ increases from 5 to 20.Importantly, for a given new particle formation event, the ratio of survival probabilities to 3 nm also represents the ratio in production rates of particles at larger sizes, such as cloud condensation nuclei, (CCN, Figures

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Full ∼ 100 nm), because GR cond and GR eff are essentially the same for particles larger than 3 nm.

Conclusions
We have shown that for typical organic vapors in the atmosphere, the diffusion of clusters in size space (GR diff ) can have a significant positive contribution to the overall particle growth during new particle formation events.The conventional approach, which neglects this diffusion contribution, underestimates the rate of new particle formation and the subsequent production of CCN by a factor of up to 60.In addition, this diffusion contribution can lead to substantial growth for sub-Kelvin clusters, which may be one of the mechanisms that allow ambient organics to contribute to the growth of freshly formed clusters despite the strong Kelvin effect.Given its importance, this contribution of cluster diffusion in size space to initial particle growth need to be included when modeling the rate of new particle formation and the subsequent production of CCN, a critical determinant of aerosol indirect effects on climate.Full   (1), the effective growth rate (GR eff ), and the contributions to GR eff due to drift (GR drift ) and diffusion in cluster size space (GR diff ) as functions of cluster size.GR cond and GR drift vanish at the Kelvin diameter, and the difference between GR eff (red) and GR cond (black) is essentially the contribution from diffusion GR diff (green).All growth rates are derived using the parameters listed in Table 1.1), the effective growth rate (GR eff ), and the contributions to GR eff due to drift (GR drift ) and diffusion in cluster size space (GR diff ) as functions of cluster size.GR cond and GR drift vanish at the Kelvin diameter, and the difference between GR eff (red) and GR cond (black) is essentially the contribution from diffusion GR diff (green).All growth rates are derived using the parameters listed in Table 1.Introduction

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Full   1.
Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | carried out by varying the organics saturation vapor concentration and molar volume, and show similar results.
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Figure 1 .
Figure 1.Comparison of growth rates.The conventional growth rate (GR cond ) based on equation

Fig. 1 .
Fig. 1.Comparison of growth rates.The conventional growth rate (GR cond ) based on Eq. (1), the effective growth rate (GR eff ), and the contributions to GR eff due to drift (GR drift ) and diffusion in cluster size space (GR diff ) as functions of cluster size.GR cond and GR drift vanish at the Kelvin diameter, and the difference between GR eff (red) and GR cond (black) is essentially the contribution from diffusion GR diff (green).All growth rates are derived using the parameters listed in Table1.
Figure 2. (a) In traditional view, particles below the Kelvin diameter evaporate and only 2 particles with diameter greater than the Kelvin diameter can grow through condensation, (b) 3 Based on heterogeneous nucleation theory, even particles below the Kelvin diameter can 4 overcome the energy barrier and grow to larger sizes due to the net forward number flux. 5

Fig. 2 .Figure 3 . 4 Fig. 3 .
Fig. 2. (a) In traditional view, particles below the Kelvin diameter evaporate and only particles with diameter greater than the Kelvin diameter can grow through condensation, (b) based on heterogeneous nucleation theory, even particles below the Kelvin diameter can overcome the energy barrier and grow to larger sizes due to the net forward number flux.

Table 1 .
Parameter values for model organic species used in calculation of particle fluxes and growth rates.

Table 2 .
The ratio of the cluster survival probability derived from GR eff to that based on GR cond under sulfuric acid and organics concentrations representative of NPF events.The survival probability is calculated for 1.5 nm thermodynamically stable clusters growing to form 3 nm particles.