# Atmospheric ion-induced nucleation of sulfuric acid and water

## Abstract

[1] Field studies show that gas phase nucleation is an important source of new particles in the Earth's atmosphere. However, the mechanism of new particle formation is not known. The predictions of current atmospheric nucleation models are highly uncertain because the models are based on estimates for the thermodynamics of cluster growth. We have measured the thermodynamics for the growth and evaporation of small cluster ions containing H_{2}SO_{4} and H_{2}O, and incorporated these data into a kinetic aerosol model to yield quantitative predictions of the rate of ion-induced nucleation for atmospheric conditions. The model predicts that the binary negative ion H_{2}SO_{4}/H_{2}O mechanism is an efficient source of new particles in the middle and upper troposphere. The ion-induced HSO_{4}^{−}/H_{2}SO_{4}/H_{2}O mechanism does explain nucleation events observed in the remote middle troposphere, but does not generally predict the nucleation events observed in the boundary layer.

## 1. Introduction

[2] Aerosol is ubiquitous in the Earth's lower atmosphere. It affects human health, visibility, atmospheric chemistry, and climate. Aerosol influences climate directly by scattering and absorbing radiation, and indirectly by acting as cloud droplet nuclei and affecting cloud properties. Gas phase nucleation is an important source of particles in the Earth's atmosphere [*Kulmala et al.*, 2004]. New particle formation is correlated with enhanced levels of gas phase sulfuric acid [*Weber et al.*, 1995, 2001a] and low aerosol surface area (the dominant sulfuric acid sink) [*Clarke*, 1992; *de Reus et al.*, 1998; *Hegg et al.*, 1993; *Perry and Hobbs*, 1994; *Hoppel et al.*, 1994; *Clarke*, 1993; *Covert et al.*, 1992; *Keil and Wendisch*, 2001]. Water is also implicated in the formation of new particles because it is abundant and it significantly suppresses the vapor pressure of sulfuric acid. Nucleation in the background troposphere is common in regions of cloud outflow [*Weber et al.*, 2001b; *Clarke et al.*, 1998a; *Hegg et al.*, 1990; *Keil and Wendisch*, 2001]. These regions are favorable for nucleation due to enhanced solar radiation, high relative humidity (RH), and reduced particle surface area. In the middle and lower troposphere, nucleation frequently occurs where classical binary H_{2}SO_{4}/H_{2}O nucleation theory does not predict new particle formation [*Weber et al.*, 2001a; *Clarke et al.*, 1998b; *Weber et al.*, 1997, 1999]. This has led to the suggestions that other species (e.g., NH_{3}) are involved [*Weber et al.*, 1998; *Kulmala et al.*, 2000; *Coffman and Hegg*, 1995; *Marti et al.*, 1997], or that an ion mechanism is important [*Yu and Turco*, 2001]. Ion-induced nucleation (IIN) may explain a possible connection between the flux of galactic cosmic rays (the main source of ions in the background atmosphere) and global cloudiness [*Carslaw et al.*, 2002; *Harrison and Carslaw*, 2003]. Field measurements of the mobility spectrum of air ions [*Hõrrak et al.*, 1998] show bursts of intermediate ions (1.6–7.4 nm) that may be associated with ion-induced nucleation. *Eichkorn et al.* [2002] have detected large positive ions in the upper troposphere that they attribute to ion-mediated nucleation.

### 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., N_{2}^{+}, O_{2}^{+}, N^{+}, O^{+}) and electrons produced in the initial ionization of air are converted rapidly to cluster ions of protonated bases (e.g., H_{3}O^{+}(H_{2}O)_{n}, (CH_{3})_{2}COH^{+}(H_{2}O)_{n}, and NH_{4}^{+}(H_{2}O)_{n}) and of the conjugate bases of strong acids (e.g., NO_{3}^{−}(HNO_{3})_{x}(H_{2}O)_{y}) [*Ferguson*, 1979]. In the presence of the strong acid H_{2}SO_{4}, NO_{3}^{−}(HNO_{3})_{x}(H_{2}O)_{y} ions are converted to cluster ions of HSO_{4}^{−}. 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 μm^{2} 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 H_{2}SO_{4}/H_{2}O 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 H_{2}SO_{4} 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, H_{2}SO_{4}). 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}, [H_{2}SO_{4}] < 5 × 10^{7} 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. Kinetic Model of Ion-Induced Nucleation

[8] An accurate kinetic treatment of atmospheric nucleation has not been possible due to a lack of thermodynamic data for the relevant clusters. We have implemented the following approach to significantly reduce uncertainties in the prediction of IIN rates in the H_{2}SO_{4}/H_{2}O system. (1) We have measured thermodynamics for the binding of H_{2}SO_{4} and H_{2}O to cluster ions of the form HSO_{4}^{−}(H_{2}SO_{4})_{x}(H_{2}O)_{y} and H^{+}(H_{2}SO_{4})_{n}(H_{2}O)_{m}. (2) We have connected the measured small cluster thermodynamics to the bulk liquid drop limit to yield thermodynamic predictions for all cluster sizes and compositions. (3) We have developed a nucleation model that treats the kinetics of growth and evaporation of neutral and ionic clusters explicitly.

### 2.1. Model Details

[9] Modeling H_{2}SO_{4}/H_{2}O binary nucleation is simplified by noting that the atmospheric water concentration is at least 10^{4} 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 H_{2}SO_{4}. 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 (H_{2}SO_{4})_{n} and HSO_{4}^{−}(H_{2}SO_{4})_{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.

_{2}SO

_{4}. 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 H

_{2}SO

_{4}and coagulation are similar to the ones presented by

*Raes and Janssens*[1986]. The temporal evolution of the neutral clusters starting with (H

_{2}SO

_{4})

_{2}is given by

*n*

_{i}is the number of H

_{2}SO

_{4}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 H

_{2}SO

_{4}molecule from the next larger cluster (i + 1). The second term accounts for the loss of

*i*due to evaporation of one H

_{2}SO

_{4}molecule. The third term is the production of

*i*by addition of H

_{2}SO

_{4}to the next smaller cluster, and the fourth term is the loss of

*i*by reaction with H

_{2}SO

_{4}. 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 H

_{2}SO

_{4}, so that recombination of ion cluster

*I*produces a neutral cluster

*i*with the same number of H

_{2}SO

_{4}. Neutralization converts HSO

_{4}

^{−}to H

_{2}SO

_{4}.

_{4}

^{−}H

_{2}SO

_{4}is given by

_{2}SO

_{4}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.

_{2}SO

_{4}concentration is given by

_{2}SO

_{4}(see below). The second term describes the production of H

_{2}SO

_{4}from decomposition of the H

_{2}SO

_{4}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 H

_{2}SO

_{4}due to condensation on neutral and negative clusters. The eighth term is the production of H

_{2}SO

_{4}by neutralization of HSO

_{4}

^{−}and the last term is the loss of H

_{2}SO

_{4}by reaction with the NO

_{3}

^{−}cluster ions.

_{2}SO

_{4}is the reaction of OH with SO

_{2}. The rate of production of H

_{2}SO

_{4}(k

_{OH+SO2}[OH][SO

_{2}]) is closely coupled to the solar flux through the OH production. We use the OH + SO

_{2}rate coefficient recommended by

*DeMore et al.*[1997]. The H

_{2}SO

_{4}production rate is modeled with half a sinusoidal period extending from sunrise to sunset. The production rate of H

_{2}SO

_{4}is given by

_{2}SO

_{4}is zero at night, and is defined by

*M*is the maximum H

_{2}SO

_{4}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*.

_{3}

^{−}core ion. The model assumes that ion production starts with the NO

_{3}

^{−}cluster, which reacts with H

_{2}SO

_{4}to produce the hydrated HSO

_{4}

^{−}ion. The variation of the NO

_{3}

^{−}cluster ions is given by

_{3}

^{−}cluster loss due to recombination, the third term accounts for loss via reaction with H

_{2}SO

_{4}, and the last term is the loss of NO

_{3}

^{−}by coagulation with the neutral clusters.

_{4}

^{−}concentration is given by

[16] Here the first and second terms on the right-hand side are the production of HSO_{4}^{−} by reaction of NO_{3}^{−} clusters with H_{2}SO_{4} and by decomposition of HSO_{4}^{−}H_{2}SO_{4}. The third term is loss of HSO_{4}^{−} by association with H_{2}SO_{4}, the fifth term is loss by coagulation with all the neutral clusters starting with (H_{2}SO_{4})_{2}, and the last term is loss due to recombination with positive ions.

_{2}SO

_{4}evaporation/decomposition rate constants (

*k*

^{d}) were calculated from the H

_{2}SO

_{4}clustering thermodynamics and the condensation/association rate coefficient (

*k*

^{a}). For example, for the following reaction

_{2}SO

_{4}condensation/association and evaporation/decomposition rate coefficients are related to the Gibbs free energy change by

_{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 H

_{2}SO

_{4}condensation and evaporation rate coefficients were calculated by averaging the individual rate coefficients (e.g., for reaction (9)) over the equilibrium water distributions.

*f*

_{x,y}is the fraction of the equilibrium distribution of a cluster with

*x*H

_{2}SO

_{4}that 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 HSO_{4}^{−} + H_{2}SO_{4} rate coefficient (k_{298K} = 2 × 10^{−9} cm^{3} 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} cm^{3} molecule^{−1} s^{−1} was used for the small cluster limiting recombination coefficient [*Bates*, 1982]. Accomodation coefficients were typically set to 1.0 for H_{2}SO_{4} condensation on neutral and negative clusters, and for coagulation of all clusters. Rate coefficients for association of H_{2}SO_{4} 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

*Froyd and Lovejoy*, 2003b]. The enthalpy changes for the sulfuric acid clustering reactions

*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

^{−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 H

_{2}SO

_{4}and H

_{2}O binding thermodynamics extending from molecular cluster ions to the bulk, based on experimental thermodynamics for the small clusters. Bulk liquid H

_{2}SO

_{4}/H

_{2}O 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 (H_{2}SO_{4})_{x}(H_{2}O)_{y}. These terms systematically decrease the bond energies of H_{2}SO_{4} and H_{2}O 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 H_{2}SO_{4} 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 H_{2}SO_{4} molecules for 1 × 10^{7} H_{2}SO_{4} cm^{−3}, RH = 0.5, and 240 K, which is consistent with measurements by *Eisele and Hanson* [2000], who report a critical cluster “near” 4 H_{2}SO_{4} molecules.

## 3. Model Predictions for the Atmosphere

[21] Nucleation contour plots were generated by running the model from sunrise until sunset for a range of relative humidities, sulfuric acid source strengths, and preexisting aerosol surface areas. Results from about 1000 model runs were used to generate particle production contours for a given temperature. At a specific temperature, the particle production is most sensitive to RH and H_{2}SO_{4} concentration. The H_{2}SO_{4} concentration is a function of the H_{2}SO_{4} production rate and the aerosol surface area, which is an important H_{2}SO_{4} sink. To reduce the dimensionality of the system we plot the particle production contours as a function of RH and the average maximum daytime [H_{2}SO_{4}]. The averaging facilitates the representation of the data and introduces uncertainties in the particle production of typically less than 20%. We define the particle production as the sum of new particles greater than 3 nm diameter because this coincides with the minimum detectable size of the particle counters deployed in the field.

[22] Particle production contours for a range of temperatures characteristic of the troposphere are plotted in Figure 4. The concentration of H_{2}SO_{4} is typically less than 10^{8} molecule cm^{−3} in the boundary layer, and less than 10^{7} in the middle and upper troposphere (see, e.g., Table 1). Based on these [H_{2}SO_{4}], it is apparent that the HSO_{4}^{−}/H_{2}SO_{4}/H_{2}O nucleation mechanism has the potential to produce more than 10^{4} particles >3 nm per cubic centimeter in a day in the middle and upper troposphere.

Location | Ref.^{a} |
Altitude, km | [H_{2}SO_{4}], 10^{6} cm^{−3} |
RH, % | Temp., °C | Surface Area, μm^{2} cm^{−3} |
Local Time | Max. Observed Particle Concentration,^{b} cm^{−3} |
Model Predictions,^{c} cm^{−3} |
---|---|---|---|---|---|---|---|---|---|

Mauna Loa Observatory | A | ground | 7.8–9.5 | 49–53 | 11–12 | 4–18 | 1200 | 60 UF1 | <1 |

Idaho Hill | B | ground | 8.4–9.8 | 28–29 | 10 | 40–60 | 1200 | 200 UF1 | <1 |

ACE 1 | C | 0.043 | 4.4 | 58 | 4.4 | 1316 | 400 UF1 | <10 | |

Macquarie Island | 0.55 | 7.3 | 17 | 2.8 | 1325 | 2200 UF1 | <10 | ||

PEM Tropics A Flight 19 | D | 0.16–0.18 | 3.7–55 | 88–98 | 23–25 | 7–15 | 1400 | 35 000 UCN | <1600 |

ACE 1 Flight 17 | E | 3.2–4.1 | 2.0–6.0 | 36–62 | −11–−17 | <10 | 1600–1800 | 4000 UF2, 10 000 UCN | <30 000 |

ACE 1 Flight 27 | E | 4.0–4.2 | 12–19 | 45–65 | −19–−20 | <12 | 1200–1400 | 2000 UF2, 6000 UCN | <35 000 |

ACE 1 Flight 16 | F | 5.8–6.0 | 1.5–2.5 | 30–80 | −22–−24 | 2–8 | 1500–1600 | 230 UF1, 4000 UCN | <40 000 |

PEM Tropics B Flight 13 | G | 0.3 5.5 | 20–40 4–8 | 70–80 40–90 | 26 4–−8 | 20–60 5–20 | 1230–1300 1130–1230 | 0 UF1, 400 UCN 6 UF1, 1200 UCN | <2000 <38 000 |

- a
A [
*Weber et al.*, 1995]; B [*Weber et al.*, 1997]; C [*Weber et al.*, 1998]; D [*Clarke et al.*, 1998b]; E [*Clarke et al.*, 1998a]; F [*Weber et al.*, 2001a]; G [*Weber et al.*, 2001b]. - b UF1 = 3–4 nm diameter; UF2 = 3–10 nm; UCN = >3 nm.
- c UCN particle production predicted by the model. See text for details.

[23] The rate of nucleation depends strongly on the thermodynamics for cluster growth. The position and height of the nucleation barrier is sensitive to the temperature, RH and [H_{2}SO_{4}]. Low temperatures, high RH, and high [H_{2}SO_{4}] decrease the height of the barrier and facilitate nucleation. Figure 5 shows the conditions where the nucleation barrier for the ion coordinate is eliminated and the growth of all cluster ions is spontaneous. For example, for temperatures below 240 K and RH greater than 0.3 there is no barrier to ion growth for [H_{2}SO_{4}] greater than about 10^{6} molecule cm^{−3}.

[24] The minimum [H_{2}SO_{4}] required to produce 100 particles in 12 hours of daylight is around 2 × 10^{6} molecule cm^{−3} under the most favorable nucleation conditions (e.g., 220 K, see Figure 4). There is no barrier to IIN for these conditions (see Figure 5) and the particle production is determined by the rates of ion production and H_{2}SO_{4} condensation. In the warm humid boundary layer, greater than 10^{8} H_{2}SO_{4} molecule cm^{−3} are required to make just 100 particles cm^{−3} in a day. There are barriers to nucleation for these conditions, and the particle production is a stronger function of temperature, RH, and [H_{2}SO_{4}].

[25] The particle production by IIN is self-limiting. As the production rate increases, the surface area increases and eventually reaches levels where small particles are effectively scavenged and 3 nm particle production is suppressed. The maximum particle production rates are about 40,000 particle cm^{−3} day^{−1} with diameters greater than 3 nm for the conditions of Figure 4.

[26] In this work, background particles are typically modeled with a lognormal distribution centered around 60 nm diameter. For typical atmospheric conditions, H_{2}SO_{4} uptake to this aerosol is in the free molecular regime and the first order loss of H_{2}SO_{4} is given approximately by the surface area times 5.5 × 10^{−5} s^{−1} um^{−2} cm^{3}.

### 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, H_{2}SO_{4} concentration, RH and temperature. These data are tabulated in Table 1 with 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 H_{2}SO_{4} production rates characteristic of the observation conditions (see Table 1). A contour plot of the particle production as a function of RH and [H_{2}SO_{4}] 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 H_{2}SO_{4} and particle concentrations for a set of conditions characteristic of the flight 27 event is presented in Figure 7. For these conditions, the H_{2}SO_{4} concentration peaks at 1.5 × 10^{7} molecule cm^{−3} at 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 [H_{2}SO_{4}] above 6 × 10^{6} molecule 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 [H_{2}SO_{4}] combined with relatively large uncertainties in [H_{2}SO_{4}]. Quoted uncertainties (2σ) in the H_{2}SO_{4} 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 [H_{2}SO_{4}] and surface areas, representative of very strong H_{2}SO_{4} 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., NH_{3}) 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 H_{2}SO_{4}. *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 H_{2}SO_{4} measurements, the propensity for IIN may be evaluated with Figure 8, which correlates particle production with the H_{2}SO_{4} source strength and the preexisting aerosol surface area. For a typical midlatitude upper tropospheric noontime H_{2}SO_{4} 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 μm^{2} cm^{−3}, consistent with the *Schröder and Ström* [1997] observations. The midday H_{2}SO_{4} production rate was calculated for the following conditions: 220 K, 200 mbar, noontime [OH] = 0.4 pptv [*Brune et al.*, 1998] and [SO_{2}] = 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 SO_{2}.

### 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 × 10^{4} particles cm^{−3} > 3 nm dia. in 4 hours for the following PEM A conditions: 298 K, RH = 0.95, 7 um^{2} cm^{−3}, 10^{4} H_{2}SO_{4} cm^{−3} s^{−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 × 10^{4} H_{2}SO_{4} cm^{−3} s^{−1} and clean air (about 1 um^{2} cm^{−3}), *Laakso et al.* [2002] predict the production of >10^{4} 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. [H_{2}SO_{4}] = 1.1 × 10^{7} molecule cm^{−3}). In this case, there is no barrier to nucleation and the nucleation rate is limited by the ion production and H_{2}SO_{4} 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. [H_{2}SO_{4}] ≈ 5 × 10^{6} 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 [H_{2}SO_{4}] 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 (>10^{4} 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 H_{2}SO_{4} decreased the particle production by 20% for the conditions of ACE 1 flight 27 (T = 253K, RH = 0.55, max. [H_{2}SO_{4}] = 1.1 × 10^{7} molecule cm^{−3}). At lower particle production rates (e.g., with T = 253 K, RH = 0.55, max. [H_{2}SO_{4}] = 5 × 10^{6} 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 ([H_{2}SO_{4}] = 5 × 10^{6} molecule 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.

## 4. Summary

[35] We have developed a kinetic model of ion induced nucleation that is based on experimental thermodynamics of small ion clusters of H_{2}SO_{4} and H_{2}O. This model should be a significant improvement over the current modeling strategies that are based on estimates of cluster thermodynamics. The model predicts that IIN of H_{2}SO_{4} and H_{2}O is an efficient source of new particles in the middle and upper troposphere, thereby establishing links between particle formation and ions and H_{2}SO_{4}. Fossil fuel combustion is the major source of atmospheric SO_{2}, which directly impacts the H_{2}SO_{4} production rate. In the free troposphere, ions are formed predominately by cosmic rays, which are modulated by solar activity [*Carslaw et al.*, 2002]. Therefore this work establishes a quantitative basis for understanding the influence of solar and anthropogenic activity on particle formation in the Earth's atmosphere.

## Acknowledgments

[36] The authors are grateful to D. Hanson, R. Weber, and F. Eisele for useful discussions. This work was supported in part by NOAA's Climate and Global Change Program.