1.1. Atmospheric Ions

[3] Wilson‘s [1911] early work showed that ions are effective nucleating agents. Ions are likely aerosol precursors because they form very stable clusters, due to the strong electrostatic interaction with polar ligands. Ions are formed by cosmic rays at the rate of 1–30 ion pairs cm⁻³ s⁻¹ in the background lower atmosphere with maximum rates at about 10 km [Rosen et al., 1985]. Other ion sources, such as radioactive decay, lightning, transmission lines, and combustion, can produce locally elevated ion concentrations. The reactive ions (e.g., N₂⁺, O₂⁺, N⁺, O⁺) and electrons produced in the initial ionization of air are converted rapidly to cluster ions of protonated bases (e.g., H₃O⁺(H₂O)_n, (CH₃)₂COH⁺(H₂O)_n, and NH₄⁺(H₂O)_n) and of the conjugate bases of strong acids (e.g., NO₃⁻(HNO₃)_x(H₂O)_y) [Ferguson, 1979]. In the presence of the strong acid H₂SO₄, NO₃⁻(HNO₃)_x(H₂O)_y ions are converted to cluster ions of HSO₄⁻. Ions are lost by recombination with ions of the opposite polarity, and by reaction with aerosol. Ion‐ion recombination is the dominant ion loss process for aerosol surface areas below about 10 μm² cm⁻³. For an ion production rate of 5 pair cm⁻³ s⁻¹ and no background aerosol, the steady state ion concentration is about 1800 ions cm⁻³, and the ion lifetime with respect to recombination is about 350 s. The ion production rate ultimately limits the maximum nucleation rate for IIN.

1.2. Gas Phase Nucleation: Neutrals and Ions

[4] At the molecular level, gas phase homogeneous nucleation involves a series of reversible elementary reactions leading from the gas phase molecules through molecular clusters to stable drops or crystals. The spontaneity of gas phase homogeneous nucleation is determined by the Gibbs free energy for formation of the relevant clusters from their gas phase constituents. Small clusters are generally less stable than the bulk because of limited solvation. Therefore, for moderate supersaturations, there is often a barrier on the Gibbs free energy surface for cluster growth. Gibbs free energy surfaces for the H₂SO₄/H₂O neutral and negative ion systems calculated with our kinetic model (vide infra) are shown for a specific set of conditions in Figure 1. For these conditions there is a significant barrier on the neutral coordinate, and no barrier on the negative ion coordinate. In this case, growth of the cluster ions is mostly limited by the rate of H₂SO₄ addition, whereas the initial growth of the neutral clusters is strongly inhibited due to efficient evaporation of the small clusters. The critical cluster, which sits on top of the barrier, has an effective vapor pressure equal to the partial pressure of the ligand (in this case, H₂SO₄). Clusters larger than the critical cluster grow spontaneously because the partial pressure of the ligand exceeds the effective vapor pressures of the clusters. The clusters that are smaller than the critical cluster have effective vapor pressures greater than the partial pressure of the ligand, and evaporation is spontaneous. Ion cluster growth will generally have smaller Gibbs free energy barriers than the corresponding neutral system due to the strong electrostatic interactions between the ion and the ligands. The height and position of the barrier are functions of the supersaturation. The barrier generally moves toward smaller clusters and decreases in height as the supersaturation increases. The rate of nucleation increases exponentially as the height of the barrier decreases.

[5] For atmospheric conditions (ion production rate = 1–30 ion pairs cm⁻³ s⁻¹, [H₂SO₄] < 5 × 10⁷ molecule cm⁻³), cluster ions accumulate less than 100 sulfuric acid molecules before they are lost to recombination. Therefore the pathway for IIN involves ion cluster growth followed by recombination that produces a stable neutral cluster, larger than the critical cluster, that continues to grow [Arnold, 1980; Turco et al., 1998]. This is an effective mechanism to bypass a nucleation barrier on the neutral coordinate (see pathway b in Figure 1).

[6] In classical nucleation theory, the thermodynamics of clusters are approximated with the liquid drop model [Seinfeld and Pandis, 1998] that assumes that the molecular clusters are spherical drops with surface tension and density equal to the bulk liquid. These assumptions are inappropriate for small molecular clusters, and lead to large uncertainties in nucleation rates derived from classical theory. For ionic clusters, the thermodynamics are calculated with the Thomson equation [Holland and Castleman, 1982], which approximates the ionic clusters as charged dielectric spheres with surface tension and density of the bulk liquid. Experimental studies have confirmed that the Thomson equation is generally not accurate for small ion clusters [Holland and Castleman, 1982].

[7] Yu and Turco [2001] and Laakso et al. [2002] have recently modeled the role of ions in the formation of particles in the troposphere. They conclude that the ionic mechanism is capable of explaining observations of tropospheric nucleation. However, their predictions have large uncertainties because the calculations are not based on accurate clustering thermodynamics.

2.1. Model Details

[9] Modeling H₂SO₄/H₂O binary nucleation is simplified by noting that the atmospheric water concentration is at least 10⁴ times greater than the sulfuric acid concentration, so that the clusters equilibrate with water, and the growth and evaporation of the clusters is limited by addition and loss of H₂SO₄. Our thermodynamic measurements indicate that the positive ions are less likely to nucleate than the negative ions [Froyd and Lovejoy, 2003a], so the positive ions are treated as a single species, and the neutral and negative ion clusters are treated explicitly. The structure of the model is depicted schematically in Figure 2. For the neutral and negative clusters, the model uses 20–40 bins that increment by one sulfuric acid molecule, representing hydrated (H₂SO₄)_n and HSO₄⁻(H₂SO₄)_n‐1, respectively. The molecular resolution of these bins gives a full kinetic treatment of nucleation. In the next 40–60 bins the number of sulfuric molecules increases geometrically, typically by a factor of 1.5 in order to account for particles up to about 1 μm diameter.

[10] All clusters equilibrate with water and grow and evaporate by addition and loss of H₂SO₄. Negative clusters coagulate with neutral clusters, recombine with positive ions, and are formed by the coagulation of smaller negative and neutral clusters. Neutral clusters coagulate with both neutral and negative clusters, and are formed by the recombination of cluster ions and by the coagulation of smaller neutral clusters. The differential equations describing condensation and evaporation of H₂SO₄ and coagulation are similar to the ones presented by Raes and Janssens [1986]. The temporal evolution of the neutral clusters starting with (H₂SO₄)₂ is given by

The delta function is defined by

where n_i is the number of H₂SO₄ molecules in cluster i. Lowercase indices refer to neutral clusters, and uppercase indices refer to negative cluster ions. The first term on the right‐hand side of equation (1) describes the production of neutral cluster i by the loss of an H₂SO₄ molecule from the next larger cluster (i + 1). The second term accounts for the loss of i due to evaporation of one H₂SO₄ molecule. The third term is the production of i by addition of H₂SO₄ to the next smaller cluster, and the fourth term is the loss of i by reaction with H₂SO₄. The fifth and sixth terms describe the production of i by coagulation of smaller neutral clusters. The seventh term is the loss of cluster i by coagulation with all other clusters, and the eighth term accounts for the loss of cluster i by coagulation with all of the negative clusters. The last term describes the gain of cluster i due to recombination of negative ion cluster I with a positive ion. It is assumed that the positive ions do not contain H₂SO₄, so that recombination of ion cluster Iproduces a neutral cluster i with the same number of H₂SO₄. Neutralization converts HSO₄⁻ to H₂SO₄.

[11] The evolution of the negative clusters starting with HSO₄⁻H₂SO₄ is given by

where the H₂SO₄ association and decomposition terms are similar to those of the neutrals. In contrast to the neutrals, the negative ions are produced by coagulation of negative with neutral clusters and lost by coagulation with only the neutrals. The negative clusters are also lost by recombination with positive ions.

[12] The temporal variation of the H₂SO₄ concentration is given by

The first term on the right‐hand side is the photochemical production rate of H₂SO₄ (see below). The second term describes the production of H₂SO₄ from decomposition of the H₂SO₄ dimer. The third term describes the production from the evaporation of neutral clusters larger than the dimer, and the fourth term is the production from decomposition of the negative cluster ions. The fifth term accounts for the loss due to dimerization. The sixth and seventh terms describe the loss of H₂SO₄ due to condensation on neutral and negative clusters. The eighth term is the production of H₂SO₄ by neutralization of HSO₄⁻ and the last term is the loss of H₂SO₄ by reaction with the NO₃⁻ cluster ions.

[13] In the troposphere, the rate‐limiting step for the production of gas phase H₂SO₄ is the reaction of OH with SO₂. The rate of production of H₂SO₄(k_OH+SO2[OH][SO₂]) is closely coupled to the solar flux through the OH production. We use the OH + SO₂ rate coefficient recommended by DeMore et al. [1997]. The H₂SO₄ production rate is modeled with half a sinusoidal period extending from sunrise to sunset. The production rate of H₂SO₄ is given by

The delta function ensures that the photochemical production of H₂SO₄ is zero at night, and is defined by

where M is the maximum H₂SO₄ production rate, L is the time between sunrise and sunset, D is the length of the day (i.e., 24 hours), and i is the day index. Sunrise is at t = iD.

[14] As described in the introduction, atmospheric ionization produces unstable ions that rapidly convert to clusters based on the NO₃⁻ core ion. The model assumes that ion production starts with the NO₃⁻ cluster, which reacts with H₂SO₄to produce the hydrated HSO₄⁻ ion. The variation of the NO₃⁻ cluster ions is given by

where S is the ion pair production rate, the second term is the NO₃⁻ cluster loss due to recombination, the third term accounts for loss via reaction with H₂SO₄, and the last term is the loss of NO₃⁻ by coagulation with the neutral clusters.

[15] The temporal variation of the HSO₄⁻ concentration is given by

[16] Here the first and second terms on the right‐hand side are the production of HSO₄⁻ by reaction of NO₃⁻ clusters with H₂SO₄ and by decomposition of HSO₄⁻H₂SO₄. The third term is loss of HSO₄⁻ by association with H₂SO₄, the fifth term is loss by coagulation with all the neutral clusters starting with (H₂SO₄)₂, and the last term is loss due to recombination with positive ions.

[17] H₂SO₄ evaporation/decomposition rate constants (k^d) were calculated from the H₂SO₄ clustering thermodynamics and the condensation/association rate coefficient (k^a). For example, for the following reaction

the H₂SO₄ condensation/association and evaporation/decomposition rate coefficients are related to the Gibbs free energy change by

where K_c is the equilibrium constant in concentration units and ΔG° is the Gibbs free energy change for the reaction. For each cluster, the equilibrium water distribution was calculated based on the ambient temperature, relative humidity, and measured water clustering thermodynamics. The effective H₂SO₄condensation and evaporation rate coefficients were calculated by averaging the individual rate coefficients (e.g., for reaction (9)) over the equilibrium water distributions.

where f_x,y is the fraction of the equilibrium distribution of a cluster with x H₂SO₄that has y waters, and k_x,y^a is the corresponding condensation rate coefficient. Similar averaging was performed to derive evaporation rate constants.

[18] Brownian coagulation coefficients (k^c) for neutral and charged particles were calculated by using Fuch’s approach [Fuchs, 1964; Seinfeld and Pandis, 1998; Yu and Turco, 1998]. The ion‐neutral coagulation coefficients were matched to the Su and Chesnavich [1982] prediction for the HSO₄⁻ + H₂SO₄ rate coefficient (k_298K = 2 × 10⁻⁹ cm³ molecule⁻¹ s⁻¹) by scaling the ion/dipole term in the interaction potential. Recombination coefficients were calculated with Fuch’s Brownian coagulation formula for oppositely charged particles [Fuchs, 1964]. A value of 1.6 × 10⁻⁶ cm³ molecule⁻¹ s⁻¹ was used for the small cluster limiting recombination coefficient [Bates, 1982]. Accomodation coefficients were typically set to 1.0 for H₂SO₄ condensation on neutral and negative clusters, and for coagulation of all clusters. Rate coefficients for association of H₂SO₄ to neutral and charged clusters, coagulation of neutral and charged clusters, ion‐ion recombination, and decomposition are shown as a function of cluster size in Figure 3. Ion production rates as a function of altitude were taken from Rosen et al. [1985]. The set of differential equations describing cluster growth, evaporation, and coagulation was integrated by using a semi‐implicit extrapolation method suitable for stiff sets of equations [Press et al., 1992].

2.2. Cluster Thermodynamics

[19] The growth and evaporation kinetics for the clusters were calculated from the cluster thermodynamics as described above. The enthalpy and entropy changes for the clustering reactions

for x ≤ 6 and y ≤ 10 were derived from the temperature dependence of the equilibrium constants measured in an ion‐molecule flow reactor coupled to a quadrupole mass spectrometer [Froyd and Lovejoy, 2003b]. The enthalpy changes for the sulfuric acid clustering reactions

for x ≤ 5 were derived from a master equation analysis of the temperature and pressure dependence of the thermal decomposition reactions (13) measured in a quadrupole ion trap mass spectrometer [Curtius et al., 2001; Lovejoy and Curtius, 2001]. Reaction entropies were calculated with standard formulae by using ab initio (HF/6 − 31 + G(d)) geometries and scaled vibrational frequencies [Curtius et al., 2001]. The reaction enthalpies and entropies for sulfuric acid clustering to the hydrated clusters

were derived from the experimental results for reactions (12) and (13) by using thermodynamic cycles. The experimental thermodynamics for the small cluster ions deviated significantly from the Thomson predictions, but converged towards the Thomson predictions for the larger clusters. The Thomson equation over predicted the water binding for the small clusters by up to 7 kcal mol⁻¹, and under predicted the sulfuric acid binding by up to 11 kcal mol⁻¹. The set of experimental cluster ion thermodynamics was connected to the Thomson predictions for larger clusters by adding small terms to the Thomson equation that decayed exponentially with cluster size. The details of the interpolation scheme are described by Froyd [2002]. This analysis yielded a complete set of H₂SO₄ and H₂O binding thermodynamics extending from molecular cluster ions to the bulk, based on experimental thermodynamics for the small clusters. Bulk liquid H₂SO₄/H₂O thermodynamics were derived from vapor pressures calculated with the On‐line Aerosol Inorganics Model [Carslaw et al., 1995]. The thermodynamics, density [Perry and Chilton, 1973], and surface tension [Sabinina and Terpugow, 1935] of bulk sulfuric acid‐water solutions were parameterized as a function of composition and temperature.

[20] The thermodynamics of the neutral clusters are evaluated with the liquid drop model modified to give nucleation rates consistent with the experimental results of Ball et al. [1999]. The terms, 4exp(−x/5) and 5exp(−y/5) (units of kcal mol⁻¹), are added to the liquid drop Gibbs free energies for the addition of sulfuric acid and water, respectively, to the clusters (H₂SO₄)_x(H₂O)_y. These terms systematically decrease the bond energies of H₂SO₄ and H₂O and inhibit the neutral nucleation relative to predictions based on the liquid drop model. With these modifications, nucleation of the neutrals is negligible for the conditions examined in this study. The Ball et al. [1999] experiments were performed with H₂SO₄ concentrations significantly higher than found in the atmosphere. Therefore the predictions of the direct neutral nucleation with our model are quite uncertain due to the exponential dependence of the nucleation rate on the cluster thermodynamics. Conversely, the IIN rate is significantly less sensitive to the neutral thermodynamics. The IIN rate is sensitive to the position of the neutral nucleation barrier, which is a relatively weak function of the cluster thermodynamics (vide infra). The present modified liquid drop thermodynamics predict a neutral critical cluster containing 5 H₂SO₄ molecules for 1 × 10⁷ H₂SO₄cm⁻³, RH = 0.5, and 240 K, which is consistent with measurements by Eisele and Hanson [2000], who report a critical cluster “near” 4 H₂SO₄ molecules.

3.1. Comparison With Observations

[27] Weber et al. [1999] recently summarized observations of middle and lower troposphere nucleation events that include measurements of ultrafine particles, H₂SO₄ concentration, RH and temperature. These data are tabulated in Table 1with data for several other events published more recently. A direct comparison between the model predictions and observations is problematic because the history of the air parcel containing the fresh ultrafine particles is uncertain, and the range of observed conditions may not accurately represent the conditions during nucleation. We calculated particle production for the observations listed in Table 1 by running the model from sunrise until the observation time for RH’s, surface areas, and H₂SO₄ production rates characteristic of the observation conditions (see Table 1). A contour plot of the particle production as a function of RH and [H₂SO₄] for the ACE 1 flight 27 nucleation event is presented in Figure 6. The model predicts up to 35 000 particle cm⁻³ greater than 3 nm diameter for these conditions, in agreement with the observed concentration of up to 6000 particle cm⁻³. The temporal evolution of the H₂SO₄ and particle concentrations for a set of conditions characteristic of the flight 27 event is presented in Figure 7. For these conditions, the H₂SO₄ concentration peaks at 1.5 × 10⁷ molecule cm⁻³at about 1100 local time. The concentration of particles greater than 3 nm diameter reaches a maximum around 1400 hours at about 30 000 cm⁻³. These particles appear over about 4 hours, giving apparent production rates of about 2 cm⁻³ s⁻¹. This is close to the limit dictated by the ion production rate of 5 ion pairs cm⁻³ s⁻¹. For these conditions there is no barrier to IIN for [H₂SO₄] above 6 × 10⁶molecule cm⁻³.

[28] Similar analyses were performed for the other nucleation observations, and the predicted ranges of particle production are listed in Table 1. The wide range in the predictions is mostly a result of the sensitivity of the model to the [H₂SO₄] combined with relatively large uncertainties in [H₂SO₄]. Quoted uncertainties (2σ) in the H₂SO₄ measurements are ±42% for p > 940 mbar and ±62% for p < 940 mbar [Mauldin et al., 1998].

[29] In general the model reproduces the observed particle production for events with atmospheric temperatures below about 270 K, but underpredicts the production for most of the lower‐altitude, higher‐temperature events. The low‐altitude nucleation event observed during PEM Tropics B flight 13 is predicted by the model. This event had high [H₂SO₄] and surface areas, representative of very strong H₂SO₄ sources. The observed nucleation rates (∼1 cm⁻³ s⁻¹) [Weber et al., 1999] derived for the other low‐altitude events and not predicted by the model, are comparable to ion production rates. These events may be due to an efficient ionic mechanism involving additional species (e.g., NH₃) or multicomponent neutral homogeneous nucleation [see, e.g., Kulmala et al., 2000].

[30] Observations of new particle formation in the upper troposphere are not accompanied by measurements of H₂SO₄. Hermann et al. [2003] observed frequent enhanced concentrations of 4–9 nm particles in large regions of the upper troposphere, suggestive of widespread recent nucleation. Schröder and Ström [1997] observed bursts of >7 nm diameter aerosol in the midlatitude upper troposphere. In the absence of H₂SO₄ measurements, the propensity for IIN may be evaluated with Figure 8, which correlates particle production with the H₂SO₄source strength and the preexisting aerosol surface area. For a typical midlatitude upper tropospheric noontime H₂SO₄ production rate of 700 molecule cm⁻³ s⁻¹, Figure 8 shows that IIN is important (>100 particle cm⁻³ day⁻¹) for surface areas below about 4 μm² cm⁻³, consistent with the Schröder and Ström [1997]observations. The midday H₂SO₄ production rate was calculated for the following conditions: 220 K, 200 mbar, noontime [OH] = 0.4 pptv [Brune et al., 1998] and [SO₂] = 30 pptv [Thornton et al., 1999]. Lee et al. [2003] report good agreement between the predictions of our model and extensive observations of ultrafine particles in the upper troposphere and lower stratosphere, accompanied by measurements of OH and SO₂.

3.2. Comparison With Other Models

[31] Yu and Turco [2001] have simulated the Clarke et al. [1998b] PEM A observations of ultrafine particles in the marine boundary layer (see Table 1). Yu and Turco predict the formation of 2 × 10⁴ particles cm⁻³ > 3 nm dia. in 4 hours for the following PEM A conditions: 298 K, RH = 0.95, 7 um² cm⁻³, 10⁴ H₂SO₄ cm⁻³s⁻¹, and 2 ions cm⁻³ s⁻¹. Our model predicts <100 particle cm⁻³ in 12 hours for the same conditions. Similarly, comparisons with predictions by Laakso et al. [2002]for warm conditions show very different results. For example, for 293 K, RH = 0.5, 10 ion cm⁻³ s⁻¹, 2.3 × 10⁴ H₂SO₄ cm⁻³ s⁻¹ and clean air (about 1 um² cm⁻³), Laakso et al. [2002] predict the production of >10⁴ particles cm⁻³ in 3 hours. In contrast our model predicts <100 particle cm⁻³ in 12 hours. The large differences in the model predictions are not surprising considering the sensitivity of the rate of nucleation to the thermodynamics of the small cluster ions. Yu and Turco [2001]do not treat evaporation explicitly in their kinetic model, but scale the condensation rate coefficients to impose some resistance to cluster ion growth. Laakso et al. [2002] apply classical nucleation theory based on thermodynamics derived with the Thomson equation. The present work uses a full kinetic treatment of the condensation and evaporation of the cluster ions based on experimentally measured cluster ion thermodynamics.

3.3. Model Output Uncertainties

[32] Random, normally distributed errors of 0.5 kcal mol⁻¹ in the Gibbs free energies of negative ion clustering lead to a standard deviation of 2% in the predicted number of particles greater than 3 nm diameter for the conditions of ACE 1 flight 27 (Table 1: T = 253 K, RH = 0.55, max. [H₂SO₄] = 1.1 × 10⁷ molecule cm⁻³). In this case, there is no barrier to nucleation and the nucleation rate is limited by the ion production and H₂SO₄ condensation, and is insensitive to the clustering thermodynamics. In contrast, when the particle production is less efficient, the model predictions are more sensitive to the clustering thermodynamics. For example, for conditions with net particle production of about 1000 particle cm⁻³ (253 K, RH = 0.55, max. [H₂SO₄] ≈ 5 × 10⁶ molecule cm⁻³), a 0.5 kcal mol⁻¹ random uncertainty in the cluster ion thermodynamics leads to a 50% uncertainty in the particle production. The model predictions are relatively insensitive to the coagulation rate constants. Doubling the coagulation rate constants reduces the particle production (>3 nm diameter) by 15% for the conditions of the ACE 1 flight 27. Changing the sulfuric acid accommodation coefficient from 1.0 to 0.5 increases the [H₂SO₄] required for significant particle production (e.g., 1000 particles cm⁻³ > 3 nm dia.) by about two times, and decreases the net particle production at the highest productions (>10⁴ cm⁻³) by about a factor of two.

[33] The pathway for IIN involves ion cluster growth followed by recombination to produce a neutral cluster. If the resultant neutral cluster is smaller than the critical cluster it will evaporate, and if it is larger than the critical cluster it will continue to grow. Therefore the rate of IIN is a function of the size of the neutral critical cluster and is dependent on the neutral thermodynamics. Systematically making the neutral clusters all 1 kcal mol⁻¹ less stable with respect to adding H₂SO₄ decreased the particle production by 20% for the conditions of ACE 1 flight 27 (T = 253K, RH = 0.55, max. [H₂SO₄] = 1.1 × 10⁷ molecule cm⁻³). At lower particle production rates (e.g., with T = 253 K, RH = 0.55, max. [H₂SO₄] = 5 × 10⁶ molecule cm⁻³) the particle production decreased by 50% for the same modification of the neutral thermodynamics as described above. These modifications of the neutral thermodynamics increased the size of the critical cluster ([H₂SO₄] = 5 × 10⁶molecule cm⁻³) from 5 to 8 sulfuric acid molecules.

[34] It is recognized that sectional aerosol models are subject to errors associated with numerical diffusion [Tsang and Rao, 1988]. In this work, where the formation rate of 3 nm diameter particles is the main result, the errors are not significant because most of the particle growth and evaporation occur in the linear bins.