Charging state of the atmospheric nucleation mode: Implications for separating neutral and ion-induced nucleation

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112,, doi: /2007jd008649, 2007 Charging state of the atmospheric nucleation mode: Implications for separating neutral and ion-induced nucleation Veli-Matti Kerminen, 1 Tatu Anttila, 1 Tuukka Petäjä, 2 Lauri Laakso, 2 Stéphanie Gagné, 2 Kari E. J. Lehtinen, 3,4 and Markku Kulmala 2 Received 12 March 2007; revised 29 May 2007; accepted 8 August 2007; published 8 November [1] One way of getting information about the relative roles of neutral and ion-induced nucleation in the atmosphere is to measure the charging state of aerosol particles, i.e., the extent by which the charged fraction of aerosol particles differs from the corresponding equilibrium charged fraction. By using a theoretical approach, we investigated how the charging state of a growing nucleation mode behaves under different atmospheric conditions. We found that the time evolution of the nuclei charging state is governed by two parameters: the initial nuclei charging state determined by the nucleation mechanism and the parameter K that is directly proportional to the cluster ion concentration and inversely proportional to the nuclei growth rate. We demonstrated that if the value K is larger than a certain threshold value (2 4 nm 1 ), any information about the initial charging state of the nuclei, and thereby about the nucleation mechanism, will be lost by the time the nuclei grow into the measurement size range (>3 nm). By making a few simplifying assumptions, we derived an analytical expression for the functional dependence of the nuclei charging state as a function of the nuclei size. We demonstrated that the derived expression can usually be fitted into experimental data on nuclei charging states. When the value of K is small enough, the obtained fitting can be extrapolated successfully down to sizes where the nucleation has taken place to obtain information about the relative importance of neutral and ion-induced nucleation. Citation: Kerminen, V.-M., T. Anttila, T. Petäjä, L. Laakso, S. Gagné, K. E. J. Lehtinen, and M. Kulmala (2007), Charging state of the atmospheric nucleation mode: Implications for separating neutral and ion-induced nucleation, J. Geophys. Res., 112,, doi: /2007jd Introduction [2] Observations made during the last decade or so demonstrate clearly that the production of new aerosol particles by nucleation and subsequent nuclei growth is a frequent phenomenon that takes place in most atmospheric environments [Kulmala et al., 2004a]. Model simulations have shown that this phenomenon is capable of affecting particle number concentrations all the way up to global scales [Adams and Seinfeld, 2002; Lauer and Hendricks, 2006; Spracklen et al., 2006]. More regionally, atmospheric aerosol formation is likely to affect the population of particles affecting cloud properties and thereby climate [Kerminen et al., 2005; Laaksonen et al., 2005]. [3] Because of the high variability in the availability of different low-volatile trace gases between different environments, atmospheric aerosol formation is likely to involve 1 Research and Development, Finnish Meteorological Institute, Helsinki, Finland. 2 Department of Physics, University of Helsinki, Helsinki, Finland. 3 Kuopio Unit, Finnish Meteorological Institute, Kuopio, Finland. 4 Also at Department of Physics, University of Kuopio, Kuopio, Finland. Copyright 2007 by the American Geophysical Union /07/2007JD multiple different pathways. One such pathway is ioninduced, or ion-mediated, nucleation. The role of ions in atmospheric aerosol formation has remained a controversial topic [Tammet et al., 1988; Hõrrak et al., 1998; Turco et al., 1998; Yu and Turco, 2001; Laakso et al., 2002; Lee et al., 2003; Kazil and Lovejoy, 2004; Laakso et al., 2004; Lovejoy et al., 2004; Eisele et al., 2006; Iida et al., 2006; Kanawade and Tripathi, 2006; Yu, 2006]. Some of the studies made so far suggest that the contribution of ion-induced nucleation to the total nucleation rate is important or even dominant, whereas some of the studies indicate that the role of ions in aerosol formation is negligible. [4] Until very recently, ion-induced nucleation rates have been estimated on the basis of modeling studies only. The modeling approach is, however, subject to large uncertainties because the lack of reliable kinetic and thermodynamic data from the sub-3 nm diameter size range. An alternative way of approaching the problem is to use the available measurements from the 3 20 nm size range and, by using these data, to try to interpret what has been taking place at sizes below 3 nm. [5] One and perhaps the most straightforward way to estimate the role of ion-induced nucleation is to use the concept of a charging state. The charging state is defined as the ratio between two quantities: the charged fraction of 1of12

2 particles at a certain diameter d p (f + (d p )orf (d p )) and the corresponding equilibrium charged fraction (f +,eq (d p ) or f,eq (d p )). If a population of particles is formed by ioninduced nucleation, its initial charging state is well above unity; that is, the particles are overcharged. Likewise, a particle population formed by neutral nucleation has a charging state smaller than unity; that is, the particles are undercharged. At present, the particle charging state can be determined down to about 3 nm diameter by simultaneous measurements of air ion and particle number size distributions [Mäkelä etal., 2003; Iida et al., 2006; Vana et al., 2006; Laakso et al., 2007]. [6] The main difficulty in applying the charging state to making interpretations about the role of ion-induced nucleation is its dynamic nature: when a population of nuclei formed in the atmosphere grows in size, its charging state immediately starts to evolve toward unity and eventually all the information about the initial nucleation mechanism will be lost [e.g., Iida et al., 2006; Laakso et al., 2007]. The purpose of this paper is twofold. First, we (1) aim to find out which quantities or parameters determine the charging state of a growing nucleation mode under different atmospheric conditions and (2) illustrate under which conditions the measured charging state could tell us something about the original nucleation mechanism. Second, by relying on a few simplifying assumptions, we will derive two analytical expressions for the functional dependence of the charging state on the particle size. Such expressions are useful when trying to extrapolate measured charging states down to sizes at which nucleation actually takes place. 2. Analytical Expression for the Particle Charging State 2.1. Underlying Assumptions [7] In order to investigate how the charging state of the nucleation mode particles depends on their size under different conditions, the following idealized system depicted in Figure 1 will be considered. To start with, we assume there is a very intense, short-term nucleation event that produces a narrow (close to monodisperse) distribution of growing nucleation mode particles around diameter d 0. A more continuous production of new nuclei can be thought of as a sequence of such short-term events, so in that sense our idealization does not seriously limit the applicability of the results obtained later. In line with atmospheric observations [e.g., Israël, 1970, 1973; Hõrrak et al., 2003; Hirsikko et al., 2005; Komppula et al., 2007], we further assume that there is a persistent mode of both negative and positive cluster ions at slightly smaller sizes, typically at around 1 nm. [8] Depending on the nucleation mechanism, the recently formed nucleation mode may contain different proportions of neutral, negatively charged and positively charged particles. A measure of this proportion is the charging state of this mode. As the nucleation mode grows in size by vapor condensation and possibly by some other processes, its total particle number concentration decreases because of coagulation. At the same time, the relative proportions of neutral and charged particles in this mode change because of their coagulation with cluster ions and with each other. As a Figure 1. A schematic picture of the size distributions of cluster ions and growing nucleation mode particles in the idealized system used in our investigation. The number concentrations of neutral, negatively and positively charged particles in the growing nucleation mode at the time t are denoted by N 0 (t), N (t) and N + (t), respectively, whereas the concentrations of negative and positive cluster ions are denoted by N C and N C +, respectively. result of all these processes, the charging state of the nucleation mode changes and, after a sufficient growth of the mode to larger sizes, approaches the value of unity being indicative of charge equilibrium. [9] In order to derive an analytical expression for the particle charging state in the above-mentioned scenario, a number of simplifying assumptions needs to be made: (1) The number concentration of positive and negative cluster ions is the same and remains constant during the considered time period, i.e., N + C = N C = N ± C ; (2) the number concentration of positively and negatively charged particles in the growing nucleation mode is the same, i.e., N + (d p )= N (d p )=N ± (d p ); (3) the ion-ion recombination coefficient a between two particles/ions of opposite sign is constant, i.e., a(d p,1, d p,2 )=a; (4) the ion-aerosol attachment coefficient a 0 between the cluster ions and neutral particles of diameter d p scales as a 0 (d p )=a 0 (d 0 ) (d p /d 0 ) l, where l is constant; (5) all particles in the growing nucleation mode have either 0 or 1 charges; (6) the equilibrium charged fraction of particles in the growing nucleation mode is equal to the ratio a 0 (d p )/a; (7) the charged fraction of particles in the growing nucleation mode is substantially below unity; (8) the number concentration of charged particles in the growing nucleation mode is substantially smaller than the number concentration of cluster ions; (9) coagulation of nucleation mode particles with larger preexisting particles (intermodal coagulation) does not significantly perturb the distribution of the charged fraction; (10) nuclei selfcoagulation is negligible, except for nuclei of opposite charge; (11) neutral and charged particles grow at the same rate; and (12) particle growth rate is independent of the particle size. [10] The first two of these assumptions have been made just to simplify the problem. The direct consequence of these assumptions is that charged fractions and equilibrium charged fractions are the same for negatively and positively charged particles, i.e., f + (d p )=f (d p )=f ± (d p ) and f +,eq (d p )= f,eq (d p )=f ±,eq (d p ). In the real atmosphere the concentra- 2of12

3 tions of positive and negative cluster ions, as well as the concentrations of positively and negatively charged particles in any size region, differ from each other slightly due mainly to the different mean size of positive and negative cluster ions [e.g., Reischl et al., 1996]. However, as shown in section 2.2, the generality of the conclusions made in this paper does not suffer from these two assumptions. [11] Assumptions 3 and 4 are not strictly true but, to our best knowledge based on theoretical calculations [e.g., Hoppel and Frick, 1986; Tammet and Kulmala, 2005], can be considered as good as any other simplifying assumption concerning the functional form of a(d p,1, d p,2 ) or a 0 (d p ). In all the analysis presented later, we will use the value of cm 3 s 1 for a, the value of cm 3 s 1 for a 0 at d p = 1 nm and the value of 1.2 for l. Similar to assumptions 1 and 2, these selections should not affect the generality of our later conclusions. [12] Assumption 5 is clearly valid for nucleation mode particles [Hoppel and Frick, 1986; Reischl et al., 1996]. When assumption 1 is valid, assumption 6 follows directly from the set of aerosol and ion balance equations, as demonstrated by Hoppel and Frick [1986]. [13] In view of the above, assumptions 1 6 appear justified when investigating the nuclei charging state. The validity of assumptions 7 12, however, depends on atmospheric conditions. This issue will be returned to in section 4.1 when investigating the accuracy of the analytical expressions derived in the next section Derivation of Analytical Expressions [14] The set of balance equations describing the time evolution of the charge distribution in a monodisperse aerosol is relatively simple [see, e.g., Hoppel and Frick, 1986]. In our system, we have larger preexisting particles in addition to the narrow nucleation mode with a mean diameter of d p, and the balance equations may be written as: dn 0 dn ¼ 2a d c ; d p N N C þ a d p; d p N 2 2a 0 d p N0 N C ; 0:5K 0 d p ; d p N 2 0 K 0; d p ; d p N0 N CoagS 0 d p N0 ð1þ ¼ a 0 d p N0 N C þ K 0; d p ; d p N0 N a d p ; d p N 2 a d c ; d p N N C 0:5K : ; d p ; d p N 2 CoagS d p N [15] Here K 0, K 0,± and K ±,± are the self-coagulation coefficients between neutral nuclei, between neutral and charged nuclei, and between nuclei having the same charge, respectively. The term CoagS is the so-called coagulation sink of nuclei into the larger preexisting particles [Kulmala et al., 2001]. The value of CoagS depends on the nuclei size and preexisting particle number size distribution, and it is expected to be somewhat larger for the charged nuclei compared with the neutral nuclei (CoagS ± > CoagS 0 ). The factor 2 in front of some of terms comes from the fact that there are two different combinations of cluster ions and nucleation mode particles that influence this term. ð2þ [16] By neglecting the self-coagulation terms according to assumption 10 in section 2.1, equations (1) and (2) reduce into dn 0 dn ¼ 2a d c ; d p N N C þ a d p; d p N 2 2a 0 d p N 0 N C CoagS 0 d p N0 ; ð3þ ¼ a 0 d p N0 N C a d p; d p N 2 a d c ; d p N N C CoagS d p N : ð4þ [17] By assuming then that f ± 1 (assumption 7 in section 2.1), the time evolution of the charged fraction in the growing nucleation mode can be written as df ¼ 1 N 0 dn f dn 0 : ð5þ N 0 [18] By combining equations (3) (5), the right-hand side of equation (5) becomes equal to: a 0 f a þ 2f a 0 f a N N C 2f 2 a f 2 a N N C N C þ ½CoagS 0 CoagS Šf ð6þ [19] On the basis of the assumptions that both f ± and N ± /N ± C are substantially below unity (assumptions 7 and 8 in section 2.1), terms 3 6 in (6) can be considered small compared with terms 1 and 2, so they can be neglected. When the preexisting aerosol loading is low enough (assumption 9 in section 2.1), also the coagulation sink terms are small and equation (6) reduces into: df ¼ ½a 0 f ašn C : ð7þ [20] It should be noted that equation (7) does not contain any interaction term between negatively and positively charged particles in the growing nuclei mode. As a result, provided that assumptions 7 9 are valid, equation (7) is applicable also when the concentrations of negatively and positively charged particles in the nucleation mode differ from each other (i.e., when assumption 2 is not valid). In that case equation (7) can be written separately for f and f +. [21] The time evolution of the charging state, S, in the growing nucleation mode can be written as ds ¼ d f f ;eq ¼ 1 df f ;eq S df ;eq : ð8þ [22] By using equation (7) and noting that f ±,eq = a 0 /a (assumption 6 in section 2.1) we get ds ¼ ð1 SÞaN C S da 0 : ð9þ a 0 3of12

4 [23] Next, following the approach by Kerminen and Kulmala [2002], we change the coordinate system by writing d ¼ dd p d d ¼ GR d p : dd p dd p ð10þ [24] Here GR is the particle growth rate which, according to assumption 11 made in section 2.1, is the same for both charged and neutral particles. With help of equation (10) and using assumption 4 made in section 2.1, we finally get where ds d p ¼ K K þ l=d p S; ð11þ dd p an C K ¼ : ð12þ GR d p [25] Equations (11) and (12) determine the size dependency of the charging state in a growing nucleation mode. The shape of the curves S(d p ) is affected essentially by two processes: the recombination of cluster ions with nucleation mode particles of opposite sign (factor a) and the attachment of cluster ions with neutral nucleation mode particles (factor a 0 ). The overall influence of these two processes on S(d p ) is tied to the parameters K (unit: nm 1 ) and l, the latter of which tells us how rapidly a 0 and equilibrium charged fraction (ratio a 0 /a) increase with an increasing particle size. [26] Equation (11) can be solved analytically only if the particle growth rate is independent of particles size, i.e., GR(d p ) = GR = constant (assumption 12 in section 2.1). When this is the case, the solution of equation (11) is S d p l Gd 0 ; d p ¼ 1 K l dp l exp Kd p þ ð S0 1Þ dl 0 exp K d p d 0 ; ð13þ where S 0 is the value of S at d p = d 0 and G is a sum term given by X 1 Gd 0 ; d p ¼ n¼0 Kd p nþl ð Kd0 ðn þ lþn! Þ nþl d l p : ð14þ [27] The value of l depends slightly on the particle size range considered. On the basis of theoretical calculations by Hoppel and Frick [1986] and Tammet and Kulmala [2005], we may estimate that l is in the range for nuclei smaller than about 10 nm in diameter. In a special case when we have l = 1, equation (13) reduces to a form 1 S d p ¼ 1 þ 1 þ ð S 0 1ÞKd 0 exp K d p d 0 : ð15þ Kd p Kd p [28] In summary, we have now three ways of calculating the charging state of growing nucleation mode particles as a function of their size in our system: explicit treatment of the balance equations (1) and (2), or by using the analytical expressions (13) or (15). Compared with explicit treatment of the balance equations, the accuracy of expression (13) depends essentially on the validity of assumptions 7 12 made in section 2.1. The expression (15) differs from expression (13) only in terms of the parameter l and, if accurate enough, would be very valuable for practical purposes. The accuracies of both analytical expressions will be investigated in more detail in section General Behavior of the Particle Charging State [29] The analysis made in the previous section suggests that the charging state of nucleation mode particles depends essentially on two parameters: the initial charging state of the mode (S 0 ) and the parameter K given by equation (12). The value of S 0 is expected to depend on the nucleation mechanisms, being larger for cases in which the contribution of ion-induced nucleation to the overall nucleation rate is higher [Iida et al., 2006; Vana et al., 2006; Yu, 2006; Laakso et al., 2007]. The parameter K tells essentially how rapidly the growing nucleation mode approaches charge equilibrium. Since K is directly proportional to the number concentration of ion clusters and inversely proportional to the nuclei growth rate, charge equilibrium should be achieved faster for slowly growing nuclei populations and when high concentrations of cluster ions are present. In the following we illustrate briefly how the charging state S of a growing nucleation mode depends on the nuclei size d p and how the behavior of S(d p ) is connected with the parameters S 0 and K. [30] The calculations presented below have been made by solving numerically the balance equations (3) and (4). In order to do this in practice, we rely on the validity of assumptions 1 6 made in section 2.1 and keep further the aerosol concentrations low enough such that role of coagulation in affecting the charging state remains negligible. These restrictions can be considered justified for our illustrative purposes. [31] To start with, let us investigate the overall shape of the curves representing S(d p ). A few examples of such curves using different values of S 0 and different nuclei growth rates are presented in Figure 2. We may see that nuclei populations formed initially by neutral nucleation (S 0 < 1) approach smoothly charge equilibrium (S = 1) and, as one might expect, reach it at smaller sizes when the nuclei grow slower (larger values of K). Nuclei populations formed partially by ion-induced nucleation (S 0 > 1) behave differently: their charging state decreases to values below unity, reaches a minimum that depends on both K and S 0, and approaches finally the value of unity similar to nuclei populations being formed by neutral nucleation alone. The drop of the charging state below unity means that the two main processes (ion-ion recombination and ion-aerosol attachment) that alter the charged fraction of the growing nuclei population are in a way slower than the rate at which the equilibrium charged fraction of the same nuclei population is changed. This feature is common to all solutions of equation (11), provided that the parameter l is positive. In this context, it should be noted that the dominance of ion- 4of12

5 Figure 2. Charging state of the growing nucleation mode as function of nuclei size, S(d p ). The initial charging state of the nuclei at 1 nm has been set equal to 0.1, 3 or 10, and the concentrations of both positive and negative cluster ions have been set equal to 500 cm 3. Solid and dashed lines represent nuclei growing at the rate of 10 and 1 nm h 1, respectively. induced nucleation, i.e., more than 50% of the nuclei have been formed by ion-induced nucleation, corresponds to the value of S of about in the diameter range nm (in mobility or Millikan diameter). In mass or Tammet s diameter the corresponding size range is nm. However, used instruments (ion-dmps [see Laakso et al., 2007]) utilized Millikan diameter, and therefore nm is the proper size range. Also atmospheric nucleation seems to start at that size range [e.g., Kulmala and Tammet, 2007]. [32] For a given value of K, the family of S(d p ) curves starting from different initial charging states S 0 approach each other and eventually merge together at some point in the S d p plane. This apparent merging point depends only on K and is located always below the line S = 1. When the nuclei grow in size beyond the merging point, the memory of the initial nuclei charging state has been completely lost and no information about the contribution of ion-induced nucleation can be obtained any more. [33] The nuclei charging state can currently be measured down to sizes of about 3 nm [Iida et al., 2006; Vana et al., 2006; Laakso et al., 2007], whereas growing nuclei are expected to be formed at sizes below 2 nm. In order to find out how this influences our ability to make interpretations about the role of ion-induced nucleation based on available experimental data, we need to estimate the magnitude of K under different atmospheric conditions. The concentrations of cluster ions in the lower troposphere are typically in the range cm 3 [e.g., Hõrrak et al., 2003; Hirsikko et al., 2005; Wilding and Harrison, 2005], whereas typical nuclei growth rates are in the range 1 20 nm h 1 [Kulmala et al., 2004a]. By using these values, we may estimate that the values of K are expected to be in the range nm 1 in the lower troposphere. [34] Figures 3 and 4 illustrate how well the memory of the initial charging state of the nuclei is conserved into the 3 6 nm size range. We may see that if the growing nuclei are formed at 1 nm (Figure 3), different initial charging states can still be separated at the 3 nm size if the value of K is smaller than about 1 2 nm 1. By the time nuclei reach 6 nm size, this information is completely lost if K is larger than 1 nm 1. If the nuclei form initially at larger sizes, the memory about their initial charging state is conserved better (Figure 4). In all the cases, however, information about initial charging state of the nuclei cannot be obtained from the size range 3 6 nm any more if the value of K is larger than about 2 4 nm 1. [35] The memory effect associated with the nuclei charging state has been investigated also by Iida et al. [2006]. By using a very detailed theoretical framework for nuclei dynamics, they found that the nuclei should grow faster than about nm h 1 to conserve some information about their initial charging state up to the diameter range 3 5 nm. This translates into the value of K smaller than about 1nm 1, when taking into account the cluster ion concentration of 800 cm 3 assumed by Iida et al. [2006]. We may conclude that our results concerning the memory effect of the nuclei charging state are fully consistent with the results by Iida et al. [2006] obtained using a somewhat different approach. [36] Finally, a large set of simulations were made in which the parameter K was kept constant but the relative Figure 3. Charging state of nuclei at (top) 3 nm and (bottom) 6 nm as a function of the parameter K. Different lines represent different initial charging states S 0 at 1 nm. 5of12

6 cannot currently be measured at sizes below 3 nm, the accuracy of expressions (13) and (15) will be investigated in the size range of 3 10 nm. [39] Figure 5 illustrates the accuracy of expressions (13) and (15) under conditions that are typical for nucleation events observed in Hyytiälä, Southern Finland [Dal Maso et al., 2005; Hirsikko et al., 2005]. We may see that the charging state predicted by expression (13) is very accurate, even though slight deviations from the calculated values of S are seen at high initial charging states. Expression (15) consistently overpredicts the nuclei charging state but captures the overall shape of the curve S(d p ) very well. The error in the value of S predicted by expression (15) depends mainly on S 0 and to a lesser extent on K and d p. For S 0 < 1 this error is relatively small but it reaches a range of 10 20% when S 0 approaches the value of 10. [40] The overall accuracy of expression (13) demonstrates that assumptions 7 12 made in section 2.1 were justified, at least for conditions typically encountered in a boreal forest environment. In order to get a more complete picture on the Figure 4. Same as Figure 3 except that the initial charging states have been taken at 2 nm. magnitudes of the terms GR, N ± C and a in it were varied. The simulations showed that in the size range 3 10 nm, the resulting changes in the values of S(d p ) were usually very minor (<1%) and remained below 5% over the whole reasonable range of values of GR, N ± C and a. This confirms our theoretical argument that the parameters K and S 0 together unambiguously determine the charging state of a near-monodisperse population of growing nucleation mode particles. 4. Applicability of the Analytical Expressions 4.1. Accuracies of the Analytical Expressions [37] In the previous section we illustrated numerically that the charging state of a growing nucleation mode, i.e., the shape of the curve S(d p ), is determined by the parameters K and S 0, as suggested by our theoretical arguments. Here we will investigate how accurately the analytical expressions derived in section 2.2 imitate the real behavior of S(d p ) under different atmospheric conditions. [38] The exact size at which nuclei are formed and start to grow is not known at the moment, even though there are indications that it lies somewhere between 1 and 2 nm. In the following we will start our simulations from 1.5 nm, which means that the initial charging state S 0 refers to the value of S(d p )atd p = 1.5 nm. Since the nuclei charging state Figure 5. Charging state of the growing nucleation mode in the size range 3 10 nm when setting the parameter K equal to (top) 0.3 or (bottom) 1. The initial charging state at 1.5 nm has been set equal to 25 (the set of curves marked by A), 5 (B) or 0.1 (C). The different curves have been determined by explicit solution of equations (1) and (2) (solid line), by using expression (13) (dashed line), or by using expression (15) (dotted line). 6of12

7 Figure 6. Error in the value of S(d p )atd p = 3 nm (solid lines) and at d p = 6 nm (dashed lines) as a function of S 0 when using expression (13) (two bottom curves) or expression (15) (two top curves). applicability of expressions (13) and (15), we next will study each of the assumptions 7 12 individually Charged Fraction (Assumption 7) [41] It is clear that in case of pure ion-induced nucleation, the initial charged fraction is (close to) unity and the validity of assumption 7 breaks down for the smallest particle sizes. The influence of this phenomenon on the accuracy of expressions (13) and (15) is depicted in Figure 6. [42] Regardless of d p and K, expression (13) is very accurate when S 0 remains smaller than about 10. The error in the value of S(d p ) predicted by expression (13) exceeds 10% in the range S 0 = and finally goes up close to or even above 100% when approaching the limit of pure ion-induced nucleation. Such a limit corresponds to S 0 of about 160 at 1.5 nm with the selection of the functions a and a 0 made here. [43] The error in S(d p ) resulting from using expression (15) increases somewhat faster than that resulting from using expression (13). When S 0 = 100, this error varies from <20% up to about 50% depending on the values of d p and K. [44] We may conclude that the errors in S(d p ) associated with assumption 7 do not seriously deteriorate the applicability of expressions (13) or (15), except when the contribution of ion-induced nucleation to the growing nuclei population is very high. In that case expressions (13) and (15) overestimate rather than underestimate the contribution of ion-induced nucleation Charged Nuclei Versus Cluster Ions (Assumption 8) [45] Measurements of ambient air ion number size distributions have demonstrated that during nucleation events, the concentrations of ions in the diameter range nm are substantially enhanced and may become comparable to the concentrations of cluster ions [Hõrrak et al., 1998, 2003; Hirsikko et al., 2005]. This challenges the validity of assumption 8. [46] Figure 7 shows the error in the value of S(d p ) predicted by expressions (13) and (15) as a function of the ratio N nuc /N C ±, where N nuc refers to the initial number concentration of particles in the growing nucleation mode. As expected, the accuracy of expressions (13) and (15) decreases with increasing N nuc /N C ±, the pattern being qualitatively similar for different values of d p and K. At moderate initial charging states (S 0 < 10 20), the error in S(d p ) remains reasonable for N nuc /N C ± of about This means that expressions (13) and (15) can safely be used as far as the number concentration of nucleation mode particles do not exceed a few thousand nuclei cm 3. [47] Large errors in the predicted values of S(d p ) emerge if large N nuc /N C ± ratios occur simultaneously with large values of S 0. Such conditions are, however, very unlikely, since ion-induced nucleation is not capable of producing very large nuclei populations under conditions typical for the lower troposphere Intermodal Coagulation (Assumption 9) [48] Coagulation of growing nucleation mode particles with larger preexisting particles affects the charging state of the nucleation mode, since the rate at which charged nucleation mode particles coagulate with the preexisting particle population differs from the corresponding rate of neutral nucleation mode particles. In order to estimate the overall effect of this phenomenon, we simply write: F ¼ CoagSðN Þ=CoagSðN 0 Þ: ð16þ [49] The factor F depends in a complicated way on the preexisting particle number size distribution, the distribution of charges over the preexisting particle population, and the coagulation kernels between all pairs of particles having different number of elementary charges. On the basis of literature data on such coagulation kernels and charging probabilities [e.g., Hoppel and Frick, 1986; Tammet and Kulmala, 2005], we may estimate that the factor F is likely to be greater than unity but unlikely to exceed the value of two. Figure 7. Error in the value of S(d p )atd p = 3 nm as a function of the ratio N nuc /N C ± when assuming K =0.3nm 1 (solid lines) or K = 2 nm 1 (dashed lines). The curves marked by A represent expression (15), and the curves marked by B represent expression (13). The value of S 0 was set equal to 10. 7of12

8 [50] The influence of the preexisting particle population on the accuracy of expression (13) is illustrated in Figure 8. As a measure of the magnitude of the preexisting particle population, the so-called condensation sink (CS) has been used [e.g., Kulmala et al., 2001]. Typical values of CS are <0.001 s 1 in remote marine and continental environments, <0.005 s 1 in rural continental sites, somewhat larger in urban areas, and in the range s 1 in megacities and other heavily polluted environments [e.g., Kulmala et al., 2005]. From Figure 8 we may see that intermodal coagulation is unlikely to affect significantly the nuclei charging state in remote or rural or environments, but probably do so in urban and other polluted areas especially if the factor F is substantially larger than unity Nuclei Self-Coagulation (Assumption 10) [51] It has been shown that in case all the nucleation mode particles are uncharged, their self-coagulation starts to influence the dynamics of this mode when the total nuclei number concentrations reaches the range of cm 3 [Kerminen and Kulmala, 2002; Kerminen et al., 2004]. In section we showed that the charging state of the nucleation mode is affected by much lower nuclei number concentrations of about cm 3. Taken together, these two things demonstrate that nuclei self-coagulation needs not to be taken into account when considering the charging state of a growing nucleation mode Growth Rates of Neutral and Charged Nuclei (Assumption 11) [52] Theoretical calculations predict that condensation of vapor molecules, especially of sulfuric acid and its hydrates, is faster for charged particles compared with neutral particles [Yu and Turco, 2000; Laakso et al., 2003; Nadykto and Yu, 2003; Nadykto et al., 2004]. This means that charged nuclei are expected to grow faster in size than what neutral nuclei do. [53] The factor EF, termed the condensation enhancement factor, by which condensation into charged particles is enhanced compared with condensation into neutral particles depends on a number of poorly known quantities. Despite these uncertainties, the value of EF is expected to be close to unity for nuclei larger than about 3 5 nm in diameter, which means that the charging state is not significantly affected by this phenomenon in the size range that can be measured using current instrumentation. Below 3 nm, the value of EF increases rapidly with decreasing particle size and may reach values larger than 2 when the nuclei diameter approaches 1 nm [Laakso et al., 2003; Nadykto and Yu, 2003; Nadykto et al., 2004]. [54] Some idea about the influence of charged-enhanced condensation on the particle charging state, as observed in the size range >3 nm, is obtained from the work by Iida et al. [2006]. According to their simulations, the maximum effect of this phenomenon is to enhance the value of S(d p ) by a factor EF calculated roughly at the diameter d 0. This limit is reached at extremely rapid nuclei growth rates corresponding to very low values of our parameter K. At growth rates relevant to most atmospheric systems, the value of S(d p ) is enhanced to a substantially lesser degree. We conclude that relying on assumption 11 may cause a slight overestimation of the nuclei charging state when extrapolating the measurement data to below 3 nm, and that this effect depends on the magnitudes of both K and EF. Figure 8. Error in the value of S(d p )atd p = 3 nm as a function of condensation sink, CS, when using expression (13). Different combinations of the parameters K and F have been used. The value of S 0 was set equal to (top) 10 or (bottom) Constant Growth Rate (Assumption 12) [55] As mentioned earlier, expressions (13) and (15) cannot be derived unless the nuclei growth rate is constant with particle size. In a real atmosphere this is probably not the case below 10 nm diameter, as suggested by recent observations [e.g., Kulmala et al., 2004b; Hirsikko et al., 2005]. [56] In order to estimate the potential errors resulting from assumption 12, we made a number of simulations in which the particle growth rate was allowed to vary with particle size. These simulations showed very clearly that the most serious problem in this regard is the general lack of information on the particle growth rates for sizes below 3 nm in diameter. For example, if the real particle growth rate in the size range <3 nm were only half of that in the size range >3 nm, assuming a constant GR (equal to that in the size range >3 nm) would lead to a significant underprediction (overprediction) of the extrapolated value of S 0 at 1.5 nm for overcharged (undercharged) cases. Typical errors in these values were found to be 20 50% for the real S 0 (1.5 nm) of 1 10 and 30 70% for the real S 0 (1.5 nm) of The errors were bigger for larger values of K as one might expect. [57] In case we have some information on the functional form of GR(d p ), the errors resulting from assumption 12 can 8of12

9 Figure 9. Overcharging detected with an Ion-DMPS during 18 July 2006 in Hyytiälä, Finland, for positive particles. (a) Experimental points and associated errors marked with dots and boxes around them. The two lines display the medium of the 1000 fittings made using either expression (13) or (15). (b and c) Histograms of the fitted values of S 0 at 2 nm and K based on 1000 fittings with expression (15). (d and e) Histograms of the fitted values of S 0 at 2 nm and K based on 1000 fittings with expression (13). be reduced considerably. One way to do this is solve the coupled set of equations (11) and (12) together, which is considerably easier than solving the original balance equations (1) and (2). Another way is to divide the size range (d 0, d p ) into a small number (n) of sub size ranges i (d p,i1, d p,i ), for which GR i and thereby K i can be determined separately. After obtaining the values of K i, one may apply expression (13) or (15) in a series by calculating S n1 (d p,n1 ) from S(d p ) using K n, then calculating S n2 (d p,n2 ) from S n1 (d p,n1 ) using K n1, and continuing like that until one ends up to the value of S 0 (d 0 ) Application to Field Measurements [58] Here we illustrate briefly how our theoretical curves (equations (13) and (15)) can be fitted into real measurement data, and how extrapolation of the fitted curves to smaller particle sizes is affected by errors in the measurement data. For experimental data, we employ measurements made with a new ion-dmps instrument which is able to measure the particle charging state down to 3 nm in diameter [Laakso et al., 2007]. The measurements were conducted at the SMEARII station in Hyytiälä, a boreal forest site in southern Finland [see Hari and Kulmala, 2005]. [59] There are two types of errors in the ion-dmps data: an error in particle size and error in the measured charging state. We assumed that the sizing error of the ion-dmps consists of half wihs of the Differential Mobility Analyzer transfer function and errors due to flow fluctuations. The error in the charging state results from Poisson counting statistics of the CPC, being dependent on measured concentrations of ions and neutralized particles as well as the CPC integration time. In the two examples discussed below, the range of errors associated with the particle size and charging state have been marked with squares around the actual measurement points (Figures 9a and 10a). [60] The two unknowns to be determined when fitting the curves given by equation (13) or (15) into experimental data are the parameter K and the charging state S 0 at some reference diameter d 0. In order to estimate how the values of K and S 0 (d 0 ) are affected by measurement errors, we fitted a single experiment for 1000 times. Between fittings, the experimental data points were relocated randomly within their assumed error range. [61] In our first example (Figure 9), a clear overcharging in particle sizes slightly above 3 nm was observed. The value of the parameter K, obtained from fittings, was equal to 0.56 ± 0.07 nm 1 (mean ± standard deviation) when using the fitting equation (13) and 0.51 ± 0.07 nm 1 when using the less accurate fitting equation (15). The corresponding 9of12

10 Figure 10. Undercharging detected with an Ion-DMPS during 2 May 2005 in Hyytiälä, Finland, for negative particles. (a e) Same as in Figure 9. values of S 0 at d 0 = 2 nm were equal to ± 2.56 and ± These values are suggestive of a significant, but probably not dominant, contribution by ion-induced nucleation. The relatively small variability of S 0 in this case suggests further that measured charging states can be extrapolated down to smaller particle sizes with a reasonable accuracy, at least in overcharged cases having a sufficiently low value of the parameter K. [62] In our second example particles were undercharged over the whole measurement size range (Figure 10). In this case the fitted values of K were 0.84 ± 0.06 nm 1 when using expression (13) and slightly smaller (0.74 ± 0.06 nm 1 ) when using expression (15). The fitted values of S 0 at d 0 = 2 nm were very similar (0.36 ± 0.29 and 0.35 ± 0.28) between the two fitting equations. We may see the majority of the fittings predicted negative vales of S 0 at 2 nm. This, seemingly an unphysical result can be explained as follows. Consider a subset of curves determined by expression (15) that decrease monotonically with decreasing particle size. All of these curves predict negative values of S when d p becomes small enough. Some of these curves (the unphysical ones) become negative very rapidly after d p decreases below 3 nm, whereas others (the physical ones) represent different cases in which the contribution of ion-induced nucleation is either very small or nonexistent. The problem is that in the size range >3 nm, the curves representing unphysical solutions to S(d p ) are not very far from the curves representing real solutions to S(d p ). This can be explained by noting that S(d p ) is proportional to 1/d p, and exhibits thus large sensitivity to the values of K and S(d 0 ) at small particle sizes (see equations (13) and (15)). We emphasize that this feature reflects the nature of the processes determining the charging state rather than approximations underlying the applied equations. As a result, small errors in experimentally determined values of S(d p )inthe size range >3 nm and/or in the fitting procedure may give us a seemingly unphysical solution (negative value of S 0 ). However, more important than the negative values of S 0 is the fact that practically none of the fittings resulted in a value of S 0 that was substantially larger than unity. This means that regardless of measurement errors, the fittings consistently predicted that the role of ion-induced nucleation had been negligible in this case Discussion [63] There are number of issues that should be kept in mind when extrapolating measured charging states to lower particle sizes in order to estimate the relative contributions of ion-induced and neutral nucleation. [64] First of all, if nucleated particles grow very slowly (the value of K is larger than about 2 4 nm 1 ), no extrapolation should be attempted at all, since in such cases all information about the relative importance of neutral and ion-induced nucleation have been lost before the time nucleated particles reach measurable sizes (>3 nm). Similarly, if particles grow slower in the size range <3 nm compared with the size range >3 nm, assuming a constant particle growth rate over the whole particle size range might 10 of 12

11 lead to a significant underestimation of the potential contribution of ion-induced nucleation to the total nucleation rate. We recommend that when available, information about the size dependency of particle growth rates should be made use of (see the discussion in section 4.1.6). [65] Second, unlike in case of traditional thermodynamic nucleation, the nucleation diameter d 0 is ill defined for dynamic nucleation mechanisms such as ion-induced nucleation. In principle this is not a problem because the equilibrium charged fraction, and thereby the value of S corresponding to any specific contribution from ion-induced nucleation, can be calculated for all diameters d 0 below 3 nm. In practice, however, a problem exists because the charging states of freshly formed nuclei will be changed by both ion-ion recombination and ion-aerosol attachment. This means that using a wrong value of the nucleation diameter d 0 may lead to a wrong value of the initial nuclei charging state, even when all other criteria needed for a successful extrapolation are met. Our remedy for this problem is that the value of S 0, and thereby the contribution of ion-induced nucleation, will be calculated for at least two different diameters d 0 (for example 1.5 and 2 nm). This procedure would reveal the sensitivity of the calculated contribution of ion-induced nucleation to the selection of d 0 for each case under consideration. [66] Finally, it should be noted that many of the assumptions made in deriving expressions (13) and (15) are not valid in the limit of pure ion-induced nucleation. The simulations made in section 4.1 demonstrated that regardless of this problem, expressions (13) and (15) are able to reveal quite reliably whether ion-induced nucleation have dominated over neutral nucleation or not. A more quantitative assessment (with a typical accuracy of 10 20%) of the contribution of ion-induced nucleation to the total nucleation rate is possible in many cases when this contribution is of the order of 50% or below. 5. Summary and Conclusions [67] Neutral atmospheric aerosol particles cannot currently be measured down to sizes where the actual nucleation process takes place. As a result, indirect measurement methods are needed in order to get information about the nucleation rate and relevant nucleation mechanisms. In assessing the role ion-induced nucleation, a useful measurable quantity in this context is the particles charging state: a larger charging state of a growing nuclei population indicates that a bigger fraction of these nuclei have been formed via ion-induced nucleation [Mäkelä etal., 2003; Iida et al., 2006; Vana et al., 2006; Yu, 2006; Laakso et al., 2007]. [68] Irrespective of its initial value, the charging state of a nuclei population evolves toward unity as the nuclei reside in the atmosphere and grow in size. In this paper we showed that the speed of this process depends on a single parameter K that is directly proportional to the concentration of ion clusters and inversely proportional to the nuclei growth rate. Another important result of our analysis is that when the value of K is larger than a certain threshold (2 4 nm 1 ), practically all information about the initial charging state of the growing nuclei population will be lost by the time the nuclei reach measurable sizes (>3 nm diameter). In such cases there is no way of making interpretation about the potential importance of ion-induced nucleation using current instrumental techniques. On the other hand, if K is smaller than this threshold, measured charging states in the size range 3 10 nm can in principle be extrapolated down to the sizes at which nuclei were initially born in the atmosphere. [69] We showed that by making a few simplifying assumptions, the charging state of a growing nuclei mode can be described with an analytical expression. Two such expressions were derived here: a more accurate one and a simpler but less accurate one. Both the expressions give the nuclei charging state as a function of the nuclei size, provided that the parameter K and the value of the charging state at some reference nuclei size d 0, S 0 (d 0 ), are known. Because of our simplifying assumptions, the derived analytical expressions are not valid when the measured air mass is polluted or when either the nucleation rate or nuclei growth rate is highly variable. Under most conditions, however, the derived expressions are expected to be applicable to so-called regional nucleation events [see, e.g., Kulmala et al., 2004a] taking place in clean or moderately polluted environments. [70] The nuclei charging state can currently be measured with reasonable accuracy at a few selected sizes above 3 nm diameter [e.g., Laakso et al., 2007]. We demonstrated that the analytical expression derived here can be fitted to measurements to obtain the values of K and S 0 (d 0 ). When these two parameters are known, the analytical expression can be extrapolated below 3 nm in order to get an estimate of the nuclei charging state at sizes where the actual nucleation has been taking place. [71] In principle, the tools introduced here can be used to estimate the relative contributions of neutral and ion-induced nucleation to atmospheric new particle formation when suitable measurement data are available. However, a few uncertainties in applying these tools still exist. First, the nuclei growth rate is likely to depend on their size, making the value of K different in the size range <3 nm as compared with the size range >3 nm. Some help for this problem could be obtained from existing ion spectrometer data, on the basis of which the nuclei growth rate below 3 nm can be estimated [e.g., Hirsikko et al., 2005]. More emphasis should clearly be put on investigating how representative, or accurate, the nuclei growth rates deduced from ion spectrometer data really are. Second, the exact sizes at which neutral and ion-induced nucleation take place in the atmosphere are not very well known. However, some recent experimental [see, e.g., Kulmala et al., 2007; Kulmala and Tammet, 2007] and theoretical studies [Kulmala et al., 2006] show that activation of existing charged and neutral clusters might be key to understand atmospheric nucleation. In this case the nucleation (activation) will occur at size range nm in Millikan diameter. Third, it is also possible that ions contribute to nucleation without changing the nuclei charging state compared with neutral nucleation. This would be case, for example, if the recombination of ion clusters were a significant source of growing nuclei. It is clear that the existing uncertainties could be reduced considerably if particle charging state measurements could be extended to smaller sizes, preferably close to 2 nm. Simultaneously, it is important to develop further both the theoretical frameworks and models simulating the interactions between ion clusters, neutral and charged nuclei as well as vapor molecules. 11 of 12