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Growth of atmospheric nano-particles by heterogeneous nucleationof organic vapor

J. Wang, R. L. McGraw, and C. Kuang

Atmospheric Sciences Division, Brookhaven National Laboratory, Upton, NY 11973-5000, USA

Correspondence to:J. Wang (jian@bnl.gov)

Received: 22 June 2012 – Published in Atmos. Chem. Phys. Discuss.: 3 September 2012Revised: 10 May 2013 – Accepted: 13 May 2013 – Published: 9 July 2013

Abstract. Atmospheric aerosols play critical roles in airquality, public health, and visibility. In addition, theystrongly influence climate by scattering solar radiation andby changing the reflectivity and lifetime of clouds. One ma-jor but still poorly understood source of atmospheric aerosolsis new particle formation, which consists of the formation ofthermodynamically stable clusters from trace gas molecules(homogeneous nucleation) followed by growth of these clus-ters to a detectable size (∼ 3 nm). Because freshly nucleatedclusters are most susceptible to loss due to high rate of coagu-lation with pre-existing aerosol population, the initial growthrate strongly influences the rate of new particle formation andambient aerosol population. Whereas many field observa-tions and modeling studies indicate that organics enhance theinitial growth of the clusters and therefore new particle for-mation, thermodynamic considerations would suggest thatthe strong increase of equilibrium vapor concentration due tocluster surface curvature (Kelvin effect) may prevent ambi-ent organics from condensing on these small clusters. Here,the contribution of organics to the initial cluster growth isdescribed as heterogeneous nucleation of organic moleculesonto these clusters. We find that the strong gradient in clus-ter population with respect to its size leads to positive clusternumber flux. This positive flux drives the growth of clusterssubstantially smaller than the Kelvin diameter, convention-ally considered the minimum particle size that can be grownthrough condensation. The conventional approach neglectsthe contribution from the cluster concentration gradient, andunderestimates the cluster survival probabilities by a factorof up to 60 if early growth of clusters is due to both con-densation of sulfuric acid and heterogeneous nucleation oforganic vapors.

1 Introduction

Atmospheric aerosols have adverse effects on air quality andhuman health (Pope et al., 2002). They also contribute tourban and regional haze, leading to reduction in visibility(Hinds, 1999). On both regional and global scales, atmo-spheric aerosols strongly influence climate by scattering so-lar radiation and by serving as cloud condensation nuclei(CCN) to change the reflectivity, lifetime, and coverage ofclouds (Finlayson-Pitts and Pitts, 2000; Seinfeld and Pandis,2006; Twomey, 1977; Albrecht, 1989).

New particle formation in the atmosphere significantlyinfluences the concentration of atmospheric aerosols, andtherefore their impact on climate (Kerminen et al., 2005;Laaksonen et al., 2005; Spracklen et al., 2008). Model sim-ulations show that nearly half of the global cloud conden-sation nuclei (CCN) in the atmospheric boundary layer maybe formed through new particle formation (Merikanto et al.,2009). New particle formation is a two-stage process consist-ing of formation of thermodynamically stable clusters fromtrace gas molecules (homogeneous nucleation) followed bygrowth of these clusters to a detectable size of∼ 3 nm (Mc-Murry et al., 2005; Kuang et al., 2010). Due to the high co-agulation rate of clusters smaller than 3 nm with the pre-existing aerosol population, for new particle formation totake place, these clusters need to grow sufficiently fast to es-cape removal by coagulation (Kuang et al., 2010).

It is generally accepted that homogeneous nucleation inthe troposphere involves sulfuric acid (Weber et al., 1996; Ri-ipinen et al., 2007; Kuang et al., 2008). However, the concen-tration of gaseous sulfuric acid, while sufficient for the firststep of homogeneous nucleation, is often insufficient to growthe resulting clusters (here, particles smaller than 3 nm are

Published by Copernicus Publications on behalf of the European Geosciences Union.

6524 J. Wang et al.: Growth of atmospheric nano-particles

also referred to as clusters) fast enough to survive the coagu-lation scavenging by pre-existing aerosol population (Kuanget al., 2010). Observed growth rates of nucleated particles areoften much greater than those based only on the condensationof sulfuric acid and associated inorganic compounds (Weberet al., 1997; Kuang et al., 2010, 2012; Makela et al., 2001;O’Dowd et al., 2002; Iida et al., 2008), and modeling stud-ies have indicated that condensation of low-volatility organicvapors contribute significantly to the initial growth of theclusters (Kulmala et al., 2004c; Paasonen et al., 2010). Thisenhancement is also supported by field measurements thatshow organics are often the dominant component of newlyformed particles (Smith et al., 2008, 2010). However, ther-modynamic considerations suggest that the strong increaseof equilibrium vapor concentration due to cluster surface cur-vature (Kelvin effect) would prevent ambient organics fromcondensing on these small clusters (Kulmala et al., 2004a;Zhang and Wexler, 2002). Since freshly nucleated clustersare most susceptible to loss below 3 nm due to the high co-agulation rate, the initial growth strongly influences the rateof new particle formation and ambient aerosol population.

Here, we extend the methodology of classical nucleationtheory to describe the initial condensational growth of freshlynucleated clusters by organic vapor. The contribution of or-ganics to the initial growth is treated as heterogeneous nucle-ation of organic molecules onto the clusters, and the numberflux of clusters through a given size is examined using thesame statistical thermodynamics treatment as in nucleationtheory. We find that the strong gradient in cluster populationwith respect to size leads to positive cluster number flux, andtherefore drives the growth of clusters substantially smallerthan the Kelvin diameter, which is the critical embryo size forheterogeneous nucleation and is conventionally consideredas the minimum particle size that can be grown through con-densation. This conventional approach, which neglects thecontribution from the cluster concentration gradient, substan-tially underestimates the initial growth of the clusters and thesurvival probabilities of these clusters to 3 nm or CCN sizes.

2 Methods

Conventionally, the growth rate (GR) due to condensation iscalculated from the difference between the concentrations ofcondensing species far from and at the particle surface cor-rected for the effect of particle curvature (Seinfeld and Pan-dis, 2006):

dDp

dt=

1

2D2p

(Dp + Dm

)2cµv

[C∞ − CSexp

(4σv

kT Dp

)]≡ GRcond. (1)

Here,Dp is the particle diameter,Dm is the diameter of thecondensing molecule,cµ is the mean relative speed of theparticle and condensing molecule,v is the volume of thecondensing molecule,C∞ is the concentration of condens-

ing species far from the particle surface,CS is the saturationconcentration over a flat surface, and GRcond is the conven-tional condensational growth rate. The enhancement of thesurface vapor concentration due to particle curvature (Kelvin

effect) is described by the factor exp(

4σvkT Dp

), whereσ is sur-

face tension,k is the Boltzmann constant, andT is absolutetemperature. In Eq. (1), the term1

D2p

(Dp + Dm

)2 is included

to account for the increase in collision diameter, which be-comes non-negligible when the sizes of the particle and con-densing molecule are comparable (Nieminen et al., 2010).Here we assume condensed organics do not form a solutionwith existing species in the particle (i.e., the Raoult’s effecton equilibrium organic vapor pressure is neglected). Previ-ous studies indicate that the early growth is often dominatedby organics during NPF events, suggesting particles con-sisting mostly of organics. Therefore, the Raoult’s effect onequilibrium organic vapor pressure may be modest over theparticle size range of interest (Riccobono et al., 2012). TheKelvin factor increases exponentially with decreasingDp andis huge for newly formed 1–2 nm clusters, reaching as high as105 for typical ambient organic species (Zhang and Wexler,2002; Kulmala et al., 2004a). As a result, the concentration

differenceC∞ − CSexp(

4σvkT Dp

)in Eq. (1) becomes nega-

tive, preventing organics from condensing onto the clusters(Zhang and Wexler, 2002). The particle diameter at which

the quantityC∞−CSexp(

4σvkT Dp

)equals zero is referred to as

the Kelvin diameter. The Kelvin diameter, which is typicallygreater than 2 nm for ambient organic vapors, represents theminimum size at which particles would grow through con-densation according to Eq. (1). In this traditional view, con-densation of organic vapors cannot contribute to the growthof clusters smaller than the Kelvin diameter, and the result-ing particle number flux, derived asJcond=

dNdDp

GRcond, is

zero or even negative, wheredNdDp

is the particle number sizedistribution, andN is the number concentration of particleswith diameters less thanDp. It has been suggested that othermechanisms, such as the lowering of vapor pressure throughRaoult’s law (Kulmala et al., 2004a, b) (nano-Kohler the-ory), heterogeneous chemical reactions (Zhang and Wexler,2002), and adsorption of organics on cluster surface (Wangand Wexler, 2013) may help to overcome this large Kelvineffect by facilitating growth to above the Kelvin diameter, atwhich point the organic vapor can condense conventionallyaccording to Eq. (1).

Here we do not attempt to dispute or ascertain the relativeimportance of various mechanisms for the early growth ofnucleated clusters described in previous studies. Instead, wefocus on an alternative mechanism that has not been consid-ered. We show that the particle number flux calculated usingthe conventional condensational growth rate represents thetotal number flux only when the particle size is substantiallygreater than the Kelvin diameter. When the cluster size is

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J. Wang et al.: Growth of atmospheric nano-particles 6525

smaller than or close to the Kelvin diameter, heterogeneousnucleation of organic vapors onto the clusters makes a secondcontribution that drives growth even for clusters smaller thanthe Kelvin diameter. This second contribution is due to thegradient in cluster concentration with respect to cluster size.It is not accounted for in conventional growth rate calcula-tions, although it is included in classical nucleation theory.During the heterogeneous nucleation of organic vapors, thenet flux from clusters withg condensed organic molecules tothose withg + 1, Jg is (Seinfeld and Pandis, 2006):

Jg = βgfg − γg+1fg+1, (2)

wherefg is the corresponding number concentration of theinitial clusters (i.e., seed) plusg molecules of condensate,βg

is the size-dependent per-particle condensation rate, andγg isthe size-dependent per-particle evaporation rate. Equation (2)can be rewritten to separate contributions to the flux fromdrift and diffusion in cluster size-space (detailed derivationgiven in Appendix A1):

Jg =

DRIFT︷ ︸︸ ︷fg

(βg − γg

)DIFFUSIONINCLUSTERSIZESPACE︷ ︸︸ ︷−∇g

[(βg + γg

)2

fg

]. (3)

As dNdDp

=fg

∇gDp, rewriting the particle flux asJg =

dNdDp

GReff =fg

∇gDpGReff defines an effective growth rate

GReff:

GReff =Jg(

fg

/∇gDp

) =

DRIFT︷ ︸︸ ︷(∇gDp

)(βg − γg

)DIFFUSIONINCLUSTERSIZESPACE︷ ︸︸ ︷−

∇gDp

fg

∇g

[(βg + γg

)2

fg

]. (4)

The lead term on the right side of Eq. (4) describes the drift inthe force field given by the gradient of the cluster free energywith respect to cluster size, and is essentially the conven-tional condensational growth rate described by Eq. (1) (de-tails given in Appendix A2). The second term on the right isrelated to the gradient in cluster concentration with respectto cluster size. Note that traditional molecular diffusion the-ory describes the tendency of molecules to spread in spaceand is due to the gradient of concentration with respect tothe space coordinate. As an analogy to traditional moleculardiffusion, the second term is referred to as the contributiondue to diffusion in cluster size space, with the size-dependentdiffusion coefficientβg+γg

2 (McGraw, 2001; Goodrich, 1964;Friedlander, 2000; Lifshitz and Pitaevskii, 1981). Similarly,this diffusion due to the concentration gradient spreads thecluster population along the size coordinate. The second termis significant when the cluster size distribution is steep (i.e.,strong gradient infg with respect to cluster size), which oc-curs for clusters below or near the Kelvin diameter (conven-tional critical embryo size of the heterogeneous nucleation).

As shown later, below the Kelvin diameter, the drift term re-sults in evaporation to a smaller size since

(βg − γg

)< 0;

in this size regime, only the diffusion term makes a posi-tive contribution to the number flux. Conventional growthrate calculations consider only the drift term, which under-estimates the full growth rate because the diffusion contribu-tion to growth is positive during new particle formation. Thiscontribution becomes especially significant near or below theKelvin diameter.

3 Results and discussions

The particle growth rate and its contributions from both driftand diffusion with respect to cluster size are calculated usingparameters of gas-phase organic species (Table 1) that arewithin the typical ranges observed in earlier studies. Previousanalyses of field measurements indicate ambient organic va-por concentrations (C∞) between 1× 107 and 3× 108 cm−3

(Paasonen et al., 2010; Kulmala et al., 2001), and saturationvapor concentrations(CS) less than∼ 105

− 106 cm−3 (Kul-mala et al., 1998; Kerminen et al., 2000; Anttila and Ker-minen, 2003). The conventional growth rate from Eq. (1),(GRcond), the effective growth rate (GReff) based on the totalflux J , and its resolution into the distinct contributions fromdrift and diffusion (GRdrift and GRdiff ) are shown in Fig. 1 asa function of cluster diameter. The total fluxJ is derived fromequilibrium cluster distribution andβg (Seinfeld and Pandis,2006). GRdrift agrees well with the commonly used GRcondbased on Eq. (1). Both GRcond and GRdrift decrease with de-creasingDp and reach zero at a Kelvin diameter of∼ 1.9 nmdetermined by the parameter values in Table 1. Therefore,according to the conventional formula, the smallest parti-cle size that can be grown through the condensation of or-ganic vapors is 1.9 nm in this case. The contribution fromthe diffusion term GRdiff is determined by cluster size dis-tribution fg and the gradient offg with respect tog (i.e.,the 2nd term on the RHS of Eq. 4). The cluster size distribu-tion (Fig. A1, Appendix A3) is derived fromβg andγg understeady state (McGraw, 2001), a reasonable assumption forsizes near or below the critical embryo size of the heteroge-neous nucleation (i.e., Kelvin diameter) (Seinfeld and Pandis,2006), where GRdiff is significant. Based on the method fromShi et al. (1990) and parameters from Table 1, the character-istic time to reach steady state is calculated as 87 s, whichis substantially shorter than the typical time scale for varia-tion of organic vapor concentration or the initial cluster (i.e.seed) population. Becausefg appears in both the numera-tor and denominator, only the shape offg (i.e., steady statecluster size distribution) is required to derive GRdiff . As Dpdecreases, the negative gradient infg with respect tog be-comes stronger, and therefore GRdiff increases drastically.As a result, the overall growth rate GReff remains 5 % and1 % of the maximum growth rate (i.e., 11 nm h−1, occurs at3.5 nm in this case) at sub-KelvinDp of 1.68 and 1.57 nm,

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6526 J. Wang et al.: Growth of atmospheric nano-particles

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

Parameter Value

σ 0.04 N m−1

v 135 cm3 mole−1

T 293.15 KCS 106 cm−3

C∞ 108 cm−3

24

Figures 1

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30

2

4

6

8

10

Dp (nm)

GR

(nm

/hr)

GRcond

GRdrift

GRdiff

GReff

2

Figure 1. Comparison of growth rates. The conventional growth rate (GRcond) based on equation 3

(1), the effective growth rate (GReff), and the contributions to GReff due to drift (GRdrift) and 4

diffusion in cluster size space (GRdiff) as functions of cluster size. GRcond and GRdrift vanish at 5

the Kelvin diameter, and the difference between GReff (red) and GRcond (black) is essentially the 6

contribution from diffusion GRdiff (green). All growth rates are derived using the parameters 7

listed in Table 1. 8

Fig. 1. Comparison of growth rates. The conventional growth rate(GRcond) based on Eq. (1), the effective growth rate (GReff), andthe contributions to GReff due to drift (GRdrift) and diffusion incluster size space (GRdiff ) as functions of cluster size. GRcond andGRdrift vanish at the Kelvin diameter, and the difference betweenGReff (red) and GRcond(black) is essentially the contribution fromdiffusion GRdiff (green). All growth rates are derived using the pa-rameters listed in Table 1.

respectively. This indicates that the conventional approachsubstantially overestimates the minimum size of the particlesthat grow through condensation of organics. Furthermore,even at particle sizes just above the Kelvin diameter, GRcondremains much smaller than the overall GReff. As shown later,this translates into a significant underestimation of the clus-ter survival probabilities calculated using GRcond alone. Itis worth noting that while GRdiff depends only on the spec-tral shape offg, the corresponding particle number flux andthe total fluxJ are linearly proportional tofg. Assuming thecluster growth is dominated by the heterogeneous nucleationof organics, we calculate the concentrations of sub-Kelvinclusters between 1.5 and 1.7 nm, and between 1.7 and 1.9 nmas 5000 cm−3 and 250 cm−3, respectively, to reproduce theaverage new particle formation rate of∼ 0.5 cm−3 s−1 ob-served from 14 March to 16 May 2011 in Hyytiala, south-ern Finland (Kulmala et al., 2013). The magnitude of thesecluster concentrations is broadly consistent with the averagemeasurements in these size ranges (Kulmala et al., 2013).

25

(a) Traditional view:

Cluster size

Fre

e E

nerg

y

Kelvin Diameter

(b) Heterogeneous nucleation:

Cluster size

Fre

e E

nerg

yKelvin Diameter

1

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. (a) In traditional view, particles below the Kelvin diameterevaporate and only particles with diameter greater than the Kelvindiameter can grow through condensation;(b) based on heteroge-neous nucleation theory, even particles below the Kelvin diametercan overcome the energy barrier and grow to larger sizes due to thenet forward number flux.

The main mechanism for the growth of the particlesthrough heterogeneous nucleation is also illustrated in Fig. 2.The conventional approach (i.e., GRcond) takes into consider-ation only the drift term, which is negative below the Kelvindiameter, since

(βg − γg

)< 0. Therefore, in the conventional

view, particles below the Kelvin diameter will evaporate be-cause it is energetically favorable, and only particles abovethe Kelvin diameter (i.e., larger than the critical embryo sizeof the heterogeneous nucleation) can grow through conden-sation. In heterogeneous nucleation theory, for an individ-ual particle that is smaller than the Kelvin diameter, the perparticle evaporation rate is greater than the per particle con-densation rate, i.e.,γg > βg. Therefore it is more likely forthe individual particle to evaporate than to grow. However,because of the strong (negative) gradient in cluster popu-lation with respect to size (fg > fg+1), the total condensa-tion flux (i.e.,βgfg) is greater than the total evaporation flux(i.e., γg+1fg+1) and the resulting net flux is positive (i.e.,Jg = βgfg − γg+1fg+1 > 0). This net forward flux inducedby heterogeneous nucleation can move sub-Kelvin particles

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J. Wang et al.: Growth of atmospheric nano-particles 6527

over the energy barrier and allow them to grow to largersizes. The impact of the gradient in the cluster size distribu-tion on the number flux is previously established in both het-erogeneous and homogeneous nucleation theories (Frenkel,1946; Goodrich, 1964; Shizgal and Barrett, 1989; Rucken-stein and Nowakowski, 1990), and it is the very same prin-ciple in which gas phase molecules can overcome the en-ergy barrier and form larger thermodynamically stable clus-ters during homogeneous nucleation (i.e., the first step of newparticle formation).

It is worth noting that given the positive heterogeneous nu-cleation flux, GReff is greater than zero at all cluster sizes.Here we defineDp,lower as the particle size at which GReffis 1 % of its maximum value (the maximum GReff occurs at3.5 nm in the above case), andDp,upperas the particle size atwhich the relative difference between GReff and GRcond de-creases to 10 % (2.24 nm for the above case). The size rangefrom Dp,lower to Dp,upperrepresents where the overall growthrate may be substantially underestimated if the contributionfrom GRdiff is not taken into consideration. Figure 3a showsDp,lower, Dp,upper, and the Kelvin diameter as the ambientorganic vapor concentrationC∞ increases from 1× 107 to3× 108 cm−3, a representative range suggested from earlierstudies (all other parameters remain the same, as listed in Ta-ble 1). Similarly, the dependencies of the three diameters onsurface tensionσ are presented in Fig. 3b forσ ranging from0.03 to 0.06 N m−1. As expected, the Kelvin diameter, or thecritical embryo size for heterogeneous nucleation, decreaseswith increasingC∞ or decreasingσ . The relative differencesbetweenDp,lower, Dp,upper, and the Kelvin diameter increaseas the Kelvin diameter decreases, suggesting a stronger im-pact from the diffusion term GRdiff (i.e., steeper gradient incluster concentration respective to cluster sizes) for systemswith smaller Kelvin diameter. Calculations are also carriedout by varying the organics saturation vapor concentrationand molar volume, and show similar results.

The contribution from GRdiff can have a significant impacton the overall GReff and therefore on the time required forclusters to grow to detectable sizes. The growth time directlycontrols the survival probability of freshly formed clusters,which is defined as the probability that a cluster will grow toa detectable size (nominally 3 nm) before being scavenged bythe pre-existing aerosol (McMurry et al., 2005; Weber et al.,1997; Kerminen and Kulmala, 2002). Because freshly nucle-ated clusters are most susceptible to loss below 3 nm, the sur-vival probability strongly influences the direct impact of at-mospheric nucleation on the ambient aerosol population. Asshown in Fig. 1, GRdiff contributes substantially to the overallgrowth rate GReff for clusters near or smaller than the Kelvindiameter. To examine its impact on the survival probability,we numerically solved the aerosol general dynamic equationfor an aerosol population growing through simultaneous con-densation and coagulation (Friedlander, 2000; Gelbard andSeinfeld, 1978; Kuang et al., 2008, 2009), explicitly account-ing for the size-dependent growth rates in Fig. 1. Aerosol loss

26

107

108

1

2

3

4

C

(cm-3)

Dp (

nm

)

Dp,lower

Kelvin DiameterD

p,upper

0.03 0.04 0.05 0.06

1

2

3

(N/m)

Dp (

nm

)

Dp,lower

Kelvin DiameterD

p,upper

a b

1

Figure 3. Variation of Dp,lower, Dp,upper, and the Kelvin diameter over representative ranges of C 2

(a) and (b). All other parameters remain the same as listed in Table 1. 3

4

Fig. 3.Variation ofDp,lower, Dp,upper, and the Kelvin diameter overrepresentative ranges ofC∞ (a) and σ (b). All other parametersremain the same as listed in Table 1.

rates were determined exclusively from coagulation with thepre-existing aerosol, modeled with a size distribution typi-cal of an urban aerosol (Jaenicke, 1993), and scaled to givea Fuchs surface area (AFuchs) of 330 µm2 cm−3, the aver-age value for NPF events observed in Mexico City duringthe MILAGRO measurement campaign (Kuang et al., 2010).The effects of reduced cluster concentrations (due to co-agulation) on the diffusion contribution to growth are ne-glected in this calculation. Early work on the multistate ki-netics of nucleation in the presence of background aerosol(McGraw and Marlow, 1983) provides a model-tested crite-rion for when these effects are important. Empirically, it wasfound that whenAFuchs/f1a1 < 1, heref1 is the concentra-tion of monomer anda1 the surface area per monomer, clus-ter scavenging can be neglected, and classical nucleation the-ory, based on the same condensation and evaporation fluxes,applies even with scavenging by background aerosol present.An equivalent dimensionless parameter,L, was introducedby McMurry et al. (2005) for a different purpose, namelyas a criterion for new particle formation in the sulfur-richAtlanta atmosphere: new particle formation was typicallyobserved whenL was less than unity but not whenL wasgreater. The combination of these independent findings sug-gests that when new particle formation is observed to occur,i.e., L ≈ AFuchs/f1a1 < 1, so that the time scale for nucle-ation is less than the time scale for cluster scavenging, thelatter process can be neglected. Conversely, when the scav-enging rate is high and needs to be included, it is unlikelythat new particle formation will occur anyway.

Here we assume that early growth of clusters is due to bothcondensation of sulfuric acid and heterogeneous nucleationof organic vapors during new particle formation events, andthe survival probabilities are derived using GRSA+ GRcondand GRSA+ GReff, where GRSA represents the growth rateattributed to sulfuric acid condensation, and GRcond andGReff are the conventional growth rate and the effectivegrowth rate due to organics presented in Fig. 1. Earlier fieldstudies show that the growth enhancement factor0, definedas (GRSA+ GRcond)/GRSA, measured at 3 nm mostly rangesfrom 5 to 20 (Kuang et al. (2010); also note that GRcond and

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6528 J. Wang et al.: Growth of atmospheric nano-particles

Table 2.The ratio of the cluster survival probability derived from GReff to that based on GRcondunder sulfuric acid and organics concentra-tions representative of NPF events. The survival probability is calculated for 1.5 nm thermodynamically stable clusters growing to form 3 nmparticles.

Case # C∞ CS [H2SO4], Growth rate AFuchs, Ratio in cluster(organics, cm−3) (organics, cm−3) (cm−3) enhancement (µm2 cm−3) survival probability

1 (urban) 1.0× 108 1.0× 106 5.4× 107 5 330 1.62 (urban) 1.0× 108 1.0× 106 2.4× 107 10 330 4.63 (urban) 1.0× 108 1.0× 106 1.1× 107 20 330 624 (remote) 1.0× 107 1.0× 105 5.4× 106 5 33.5 1.65 (remote) 1.0× 107 1.0× 105 2.4× 106 10 33.5 4.76 (remote) 1.0× 107 1.0× 105 1.1× 106 20 33.5 66

GReff are essentially the same at 3 nm). The calculations arecarried out for three sulfuric acid concentrations (5.4× 107,2.4× 107, and 1.1× 107 cm−3), which correspond to0 val-ues of 5, 10, and 20 for the value of GRcond shown in Fig. 1.For thermodynamically stable clusters of 1.5 nm (Kulmalaet al., 2007), the ratio of the cluster survival probability de-rived from GReff to that based on GRcond is given in Table 2.The survival probability and therefore the rate of new particleformation are significantly underestimated when the conven-tional growth rate GRcond is applied to organics. This under-estimation of the survival probability becomes more severewith increasing0, as the relative contribution of organics tototal growth increases, reaching factors of 1.6, 4.8, and 62 for0 values of 5, 10, and 20, respectively.

In a second set of calculations we examine the impactof the growth due to diffusion in size space (GRdiff ) onthe survival probability under more pristine conditions. Inthese calculations, the H2SO4 concentrations in the abovecalculations are reduced by a factor of 10 to 5.4× 106,2.4× 106, and 1.1× 106 cm−3, respectively. The concentra-tion 1.1× 106 cm−3 is near the lower end of sulfuric acidconcentration ranges observed during new particle formationevents in remote locations (Kuang et al., 2010). The con-centration and saturation concentrations of organic vapor arealso reduced by a factor of 10 to maintain the same growthrate enhancement0. If the original organics concentrationsare maintained, the growth rate enhancement will be toolarge and inconsistent with field observations, and the impactdue to GRdiff will be even greater, as the growth due to organ-ics will have an even larger contribution to the total particlegrowth. For the pristine conditions, the survival probabilityis derived using a pre-existing aerosol size distribution witha typical shape of a rural aerosol (Jaenicke, 1993), but scaledto give an average Fuchs surface area of 33.5 µm2 cm−3 forthe new particle formation events observed at Hyytiala dur-ing QUEST IV campaign (Kuang et al., 2010). For thermo-dynamic stable clusters of 1.5 nm, the ratio of the survivalprobability derived from GReff to that based on GRcond issimilar to that under a more polluted environment, and rangesfrom 1.6 to 66 as0 increases from 5 to 20. Importantly, for a

given new particle formation event, the ratio of survival prob-abilities to 3 nm also represents the ratio in production ratesof particles at larger sizes, such as cloud condensation nuclei(CCN,∼ 100 nm), because GRcond and GReff are essentiallythe same for particles larger than 3 nm.

4 Conclusions

We have shown that for typical organic vapors in the atmo-sphere, the diffusion of clusters in size space (GRdiff ) canhave a significant positive contribution to the overall parti-cle growth during new particle formation events. This dif-fusion contribution can lead to substantial growth for sub-Kelvin clusters, which may be one of the mechanisms thatallows ambient organics to contribute to the growth of freshlyformed clusters despite the strong Kelvin effect. The conven-tional approach, which neglects this diffusion contribution,underestimates the early growth rate of clusters. Assumingearly growth of clusters is due to both condensation of sul-furic acid and heterogeneous nucleation of organic vapors,we show the growth rate calculated using the conventionalapproach may lead to an underestimation of cluster survivalprobability to 3 nm or CCN size by a factor of up to 60. Givenits importance, this contribution of cluster diffusion in sizespace to initial particle growth needs to be included whenmodeling the rate of new particle formation and the subse-quent production of CCN – a critical determinant of aerosolindirect effects on climate.

Appendix A

A1 Contributions to heterogeneous nucleation flux

The separation of the heterogeneous nucleation flux into“drift” and “diffusion in cluster size space” contributions isdescribed in a number of earlier works in classical nucleationtheory (Frenkel, 1946; Goodrich, 1964; Shizgal and Barrett,1989; Ruckenstein and Nowakowski, 1990) and textbooks

Atmos. Chem. Phys., 13, 6523–6531, 2013 www.atmos-chem-phys.net/13/6523/2013/

J. Wang et al.: Growth of atmospheric nano-particles 6529

(Friedlander, 2000; Seinfeld and Pandis, 2006). A similarseparation is also given in McGraw (2001). The separationof the flux following earlier works is briefly described below(Goodrich, 1964; Ruckenstein and Nowakowski, 1990). Thenet flux from clusters withg condensed molecules to thosecontainingg + 1 condensed molecules is (Seinfeld and Pan-dis, 2006):

Jg = βgfg − γg+1fg+1. (A1)

Based on the continuum approximation,J is approximatedas a continuous function ofg, and the gradient inJg withrespect tog is given by

dJg

dg= Jg − Jg−1 = βgfg − γg+1fg+1 − βg−1fg−1 + γgfg. (A2)

Performing a second order Taylor expansion of the 2nd andthird term of the right hand side of Eq. (A2) gives

γg+1fg+1 = γgfg +d

dg

(γgfg

)+

1

2

d2

dg2

(γgfg

)βg−1fg−1 = βgfg −

d

dg

(βgfg

)+

1

2

d2

dg2

(βgfg

). (A3)

Combining Eqs. (A2) and (A3) gives

dJg

dg=

d

dg

(βgfg

)−

1

2

d2

dg2

(βgfg

)−

d

dg

(γgfg

)(A4)

−1

2

d2

dg2

(γgfg

)=

d

dg

[(βg − γg

)fg

]−

1

2

d2

dg2

[(βg + γg

)fg

]Integrating the above Eq. (A4) gives the desired separation

of the flux into contributions from the drift in the force fieldand diffusion in cluster size space:

Jg=fg

(βg−γg

)−

1

2

d

dg

[(βg+γg

)fg

](A5)

=

DRIFT︷ ︸︸ ︷fg

(βg − γg

)DIFFUSIONINCLUSTERSIZESPACE︷ ︸︸ ︷−

1

2∇g

[(βg + γg

)fg

].

A2 Comparison of GRdrift and GRcond

The size-dependent per-particle condensation rateβg is givenby (Seinfeld and Pandis, 2006):

βg =π

4

(Dp + Dm

)2cµC∞. (A6)

The size-dependent per-particle evaporation rate is related tothe condensation rate by (Seinfeld and Pandis, 2006):

γg =βg−1CS

C∞

exp

[σ

(ag − ag−1

)kT

], (A7)

whereag is the surface area of the growing cluster consistingof the initial cluster (seed) andg monomers of condensate.

15

Inserting equation (A8) into equation (A7), gg can be written as: 1

1 exp exp

4exp

g S g g S gg

g S

p

C a C a

C kT C kT

C v

C kTD

b s b sg

b s

-

¥ ¥

¥

æ ö æ ö ÷ ÷ç ç÷ ÷= »ç ç÷ ÷ç ç÷ ÷ç çè ø è øé ùê ú= ê úê úë û

(A10) 2

Combining equations (A6), (A9), and (A10), we can show that the traditional condensation 3

growth rate is essentially the contribution of drift term to the total growth rate: 4

( )( ) ( )2

2

41exp

2drift g p g g p m S condp p

vGR D D D c v C C GR

D kTDm

sb g ¥

é ùæ ö÷çê ú÷ç= - = + - =÷ê úç ÷÷çè øê úë û (A11) 5

A3. Steady state cluster size distribution fg 6

1.5 1.6 1.7 1.8 1.9 210

1

102

103

Cluster size (nm)

f g (

cm-3

)

7

Figure A1. An example of steady state cluster size distribution at Sub-Kelvin size range derived 8

using parameters from Table 1. 9 Fig. A1. An example of steady state cluster size distribution at sub-Kelvin size range derived using parameters from Table 1.

Letting Ds denote the size of the initial cluster, the size ofthe growing cluster is:

Dp =

(D3

s + gD3m

)1/3.

Given its small volume, addition of a monomer leads to asmall increase in cluster diameter. Employing the continuumapproximation at largeg, we have

ag − ag−1 = ∇ga = ∇g

[π

(D3

s + gD3m

)2/3]

(A8)

=2

3π

(D3

s + gD3m

)−1/3D3

m = 41

6πD3

m

(D3

s + gD3m

)−1/3

=4v

Dp.

Similarly, the gradient of the size of the growing cluster isgiven by

∇gDp = ∇g

[(D3

s + gD3m

)1/3]

=1

3

(D3

s + gD3m

)−2/3D3

m

=

π6 D3

mπ2 D2

p=

vπ2 D2

p. (A9)

Inserting Eq. (A8) into Eq. (A7),γgcan be written as

γg =βg−1CS

C∞

exp

(σ∇ga

kT

)≈

βgCS

C∞

exp

(σ∇ga

kT

)=

βgCS

C∞

exp

[4σv

kT Dp

]. (A10)

Combining Eqs. (A6), (A9), and (A10), we can show that thetraditional condensation growth rate is essentially the contri-bution of drift term to the total growth rate:

GRdrift =(∇gDp

)(βg − γg

)(A11)

=1

2D2p

(Dp + Dm

)2cµv

[C∞−CSexp

(4σv

kT Dp

)]=GRcond.

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6530 J. Wang et al.: Growth of atmospheric nano-particles

A3 Steady state cluster size distributionf g

The steady state cluster size distributionfg is derived fromβg andγg using parameters from Table 1. It is worth not-ing that GRdiff depends only on the spectral shape offg,whereas the corresponding flux and the total flux is linearlyproportional to the value offg. The steady state cluster dis-tribution shown in Fig. A1 corresponds to a total flux of0.5 cm−3 s−1, the average value observed during NPF eventsfrom 14 March to 16 May 2011 in Hyytiala, southern Finland(Kulmala et al., 2013).

Acknowledgements.This work was supported by the U.S. Depart-ment of Energy’s Atmospheric Science Program (Office of Science,OBER) under contract DE-AC02-98CH10886.

Edited by: T. Karl

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