KL-parameterization of atmospheric aerosol size distribution University of Tartu, Institute of Physics Growth of nanometer particles during weak stationary formation of atmospheric aerosol ACKNOWLEDGMENTS: This research was in part supported by the Estonian Science Foundation through grant 8342 and the Estonian Research Council Project SF s08. Special thanks to Kaupo Komsaare, Urmas Hõrrak, Marko Vana, and Markku Kulmala for help with data. (The presentation is compiled from fragments of the poster)

1. INTRODUCTION 1.1. Motivation Intermediate atmospheric ions (charged fine nanometer particles between 1.5–7.5 nm) are thoroughly studied during burst events of new particle formation when high concentrations ensure strong signal in mobility analyzers. The new instrument SIGMA (Tammet, 2011) offers a standard deviation of noise about five times less than the BSMA and makes measurements during quiet periods possible. We have a dataset of measurements for about one year (Hõrrak et al., 2011; Tammet et al., 2012) and wish to understand what is possible to conclude about new particle formation during quiet periods between burst events. An additional aim is to explain intermediate ion balance with a simple and intelligible model. The mathematical approach is an alternative for recent studies (Leppä et al., 2011; Gagné et al., 2012) and the equations will be derived from scratch while including only unavoidable components.

1.2. Simplifications The probability of having two elementary charges on an intermediate atmospheric ion is negligibly low. The attachment coefficient of an opposite charged small ion to a 7.5 nm intermediate ion is about 1.2×10 –6 cm 3 s –1 and the typical concentration of small air ions is about 500 cm –3. It follows characteristic time of recombination less than half an hour. Quiet periods of aerosol formation typically last many hours and the steady state model seems to be an acceptable tool in the present study. Key simplifications are:  chemical composition and internal structure of nanoparticles are not discussed,  nanoparticles are considered as neutral or singly charged spheres,  the nanoparticle-nanoparticle coagulation is neglected,  background aerosol particles are assumed to be in equilibrium charging state,  parameters of positive and negative ions are expected to be equal,  all processes are assumed to be in the steady state.

1.3. Symbols d is the diameter of a nanometer particle and d bkg is the diameter of a particle of background aerosol. n(d) = dN(d) / dd, where N(d) is the number concentration of particles, which diameter does not exceed d. Neutral particles are marked with index 0 and charged particles of one polarity with index 1. Correspondingly, the distributions of neutral, charged and total particles are n 0 (d), n 1 (d), and n total (d) = n 0 (d) + 2 n 1 (d). GR(d) = dd / dt is the growth rate of an individual particle. Sometimes the growth rate is measured by the growth of the population mean diameter. This would lead to a different quantity. The growth rate of singly charged particles (an average of two polarities) GR 1 may considerably exceed GR 0 due to their ability to entrap different growth units depending on the electric charge of the growing particle. GF(d) = GR(d) × n(d) is the growth flux of particles through the diameter d. β(d) is the attachment coefficient of small ions to a nanometer particle. c is the concentration of small ions of one polarity. The small ions are not in the focus in the following discussion and their concentrations appear only in combination with an attachment coefficient. Ionization and recombination of small ions are symmetric. Thus and the effect of small ions in the steady state aerosol balance appears to be nearly polarity-symmetric. S bkg0 (d) and S bkg1 (d) are coagulation sinks of neutral and charged nanometer particles on the pre-existing background aerosol.

2. EQUATIONS Let us consider a size section ( d a … d b ) and fraction concentrations The components of particle flux into the section are:

Polarity-asymmetric equations

2. EQUATIONS Let us consider a size section ( d a … d b ) and fraction concentrations The components of particle flux into the section are:

In the steady state, the sum of all five component fluxes should be zero. This requirement leads to the balance equations

a = quantile

N50: d bkg = 50…500 nm Responsible for 86% of coagulation sink

4. DISCUSSION 4.1. Problems The distribution of intermediate ions n 1 (d) is expected to be known as a result of measurements. However, general differential equations still contain three unknown functions GR 0 (d), GR 1 (d), n 0 (d) and don’t provide unambiguous solutions without attaching some external information. We have no measurement-based external information and the following discussion is limited with analysis of certain hypothesis-based problems. All examples are presented for the distribution n 1 (d) corresponding to the situation around the lower quartile of intermediate ion concentration and expressed with approximation at a = 2. Other fixed presumptions are p = 1013 mb, T = 0 C, c = 500 cm –3. The coagulation sink will be calculated according to approximations at selected values of N50. Some of the hypothetic situations under consideration are intentionally far of reality, and some seem to be plausible.

4.2. Perfect neutral growth and perfect charged growth If the particles grow only in the neutral state then GR 1 (d) = 0 and Equation (2b) allows one to express independent of GR 0 (d). In the reverse extreme situation, the particles grow only in the charged state, GR 0 (d) = 0, and Equation (2a) proceeds in independent of GR 1 (d). The effective factors are concentrations of small ions and background aerosol particles. A set of hypothetical diagrams is shown in Figure. Comment : We could get the same assuming that: GR 1 (d) = const & n 1 (d) = const

1

4.3. Simple kinetic growth The electric charge of a nanometer particle may assist with entrapping growth units (e.g. gas molecules or small clusters) from some distance. The simplest approximation of the effective capture cross- section is π(d + d + )2 / 4, where d + includes the effective diameter of growth units and a possible extra distance due to the electric polarization. On this occasion the growth rate is where asymptotic growth rate G is independent of d. If GR 1 (d) is known then Equation (2b) follows in

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