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This discussion paper is/has been under review for the journal Atmospheric Chemistry and Physics (ACP). Please refer to the corresponding final paper in ACP if available. Abstract Water is a main component of atmospheric aerosols and its amount depends on the particle chemical composition. We introduce a new parameterization for the aerosol hygroscopic growth factor (HGF), based on an empirical relation between water activity (a w) and solute molality (µ s) through a single solute specific coefficient ν i. Three 5 main advantages are: (1) wide applicability, (2) simplicity and (3) analytical nature. (1) Our approach considers the Kelvin effect and covers ideal solutions at large relative humidity (RH), including CCN activation, as well as concentrated solutions with high ionic strength at low RH such as the relative humidity of deliquescence (RHD). (2) A single ν i coefficient suffices to parameterize the HGF for a wide range of particle sizes, 10 from nanometer nucleation mode to micrometer coarse mode particles. (3) In contrast to previous methods, our analytical a w parameterization depends not only on a linear correction factor for the solute molality, instead ν i also appears in the exponent in form x · a x. According to our findings, ν i can be assumed constant for the entire a w range (0–1). Thus, the ν i based method is computationally efficient. In this work we focus 15 on single solute solutions, where ν i is predetermined with the bisection method from our analytical equations using RHD measurements and the saturation molality µ sat s. The computed aerosol HGF and supersaturation (K ¨ ohler-theory) compare well with the results of the thermodynamic reference model E-AIM for the key compounds NaCl and (NH 4) 2 SO 4 relevant for CCN modeling and calibration studies. The equations in-20 troduced here provide the basis of our revised gas-liquid-solid partitioning model, i.e. version 4 of the EQuilibrium Simplified Aerosol Model (EQSAM4), described in a companion paper.


Introduction
The gas-liquid-solid partitioning of atmospheric particles and precursor gases to large degree determines the composition and water uptake of atmospheric aerosol particles, which affect human and ecosystem health, clouds and climate (e.  , 2007).The most abundant aerosol species is water.The aerosol liquid water content (AWC) governs the size distribution, the atmospheric lifetime of both particles and interacting gases, and particle optical properties.Considering thermodynamic equilibrium, the AWC depends primarily on the available water vapor, ambient temperature (T ) and relative humidity (RH).The AWC also depends on the particle hygroscopicity, which is the ability to absorb (release) water vapor from (to) the surrounding atmosphere.In particular the ability of salt solutes to dissolve causes hygroscopic growth (HG) of aerosol particles at subsaturated atmospheric conditions (RH < 100 [%]), where the equilibrium water uptake of atmospheric aerosols is generally limited by the available water vapor.For instance, sea salt particles can deliquesce at a very low RH of deliquescence (RHD) below 40 [%], because they contain a small amount of the very hygroscopic salt magnesium chloride (MgCl 2 ).Therefore, marine air is often much hazier than continental air at the same T and RH.The HG of atmospheric aerosol particles influences heterogeneous reactions, light extinction and visibility, and is important for the aerosol radiative forcing of climate (e.g.Pilinis et al., 1995).The HG and the associated AWC depends on the ionic composition of the particles, which in turn depends on the total AWC.Consequently, the HG and AWC involving gas/liquid/solid aerosol partitioning is difficult to measure or predict numerically, even though the complex thermodynamic system may be simplified by assuming phase equilibrium (e.g.Wexler and Potukuchi, 1998;Seinfeld and Pandis, 2006).
At the microscopic level two mechanisms of water uptake are important: (i) Adsorption of water on insoluble surfaces, whereby the fraction of surface area covered by adsorbed water is proposed as criterion for hydrophilicity (e.g.Naono and Nakuman, 1991).(ii) Absorption of water by soluble particulates, which are per definition hydrophilic.For atmospheric aerosols the absorption of water by soluble compounds is most important, and adsorption does not contribute much to the total AWC.Soluble and in particular hygroscopic particles take up water from the atmosphere for solute hydration.An increase in solute concentration (e.g.due to condensation of volatile compounds, coagulation, or chemical reactions) therefore either leads to additional Figures water uptake, or to solute precipitation (causing a solid phase to co-exist with the aqueous phase), while a decrease of the solute concentration (e.g.due to evaporative loss or chemical reactions) is associated with the evaporation of aerosol water, so that ultimately at equilibrium the aerosol molality of a given aerosol composition remains constant at given T , RH and water activity (a w ).
The aerosol HG can be determined for certain solutes from laboratory a w measurements (e.g.Tang and Munkelwitz, 1994), or calculated by considering the vapor pressure reduction that occurs by dissolving a salt solute in water -known as Raoult's law (Raoult, 1888) -if non-idealities of solution are taken into account (e.g.Warneck, 1988;Pruppacher and Klett, 2007).According to K öhler-theory (K öhler, 1936) and Raoult's law, the so-called Raoult-term, which considers the lowering of the equilibrium vapor pressure with increasing solute concentration at a given RH, is complemented by the so-called Kelvin-term, which accounts for the increase in the water vapor pressure due to the curvature of the particle surface.The modern K öhler equations allow to determine the equilibrium size of an aerosol droplet for a given dry size, chemical composition, RH and T , by accounting for the dissolution of gases into droplets, changes in surface tension, ion charges, or density of the droplet solutions (e.g.Reiss, 1950;Young and Warren, 1992;Konopka, 1996;Shulman et al., 1996;Laaksonen et al., 1998;Charlson et al., 2001;Russell and Ming, 2002;Mikhailov et al., 2004;Biskos et al., 2006a, b;Seinfeld and Pandis, 2006;McFiggans et al., 2006;Pruppacher and Klett, 2007;Rose et al., 2008;Mikhailov et al., 2009;Ruehl et al., 2010).
Three types of methods have been used to account for hygroscopic growth of atmospheric aerosols in general circulation models (GCMs): (i) the f (RH) method, (ii) K öhler-theory and (iii) thermodynamic equilibrium models.The f (RH) method (Charlson et al., 1992) scales particle optical properties as a function of RH.In a simplified manner, this method accounts for the hygroscopic nature of water-soluble aerosol particles and, hence, has been used for first-order estimates of aerosol HG and the corresponding radiative forcing of climate.The second method explicitly accounts for the hygroscopic nature, since the K öhler equation is based on the Raoult-term.However, Figures

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Full both methods do not account for gas-liquid-solid partitioning and deliquescence that accompanies aerosol hygroscopic growth.Only models that also account for the gasliquid-solid partitioning of single and mixed solute solutions are sufficiently comprehensive to calculate the RHD based HG factor (HGF) of single and mixed solutions, which usually includes various inorganic, organic and non-soluble compounds.
Here we first introduce new single solute solutions parameterizations for the aerosol HGF, which are based on a new representation of water activity, presented in Sect. 2. The equations provide the basis of our revised gas-liquid-solid equilibrium partitioning model, i.e. version 4 of the EQuilibrium Simplified Aerosol Model (EQSAM4), which considers mixed solutions of various compounds important for atmospheric modeling.
EQSAM4 is described in the companion paper.Our HGF parameterizations require for a given temperature only one solute specific coefficient, ν i , to approximate a w for the entire range of water activity (0-1).ν i is pre-determined in Sect. 3 from our analytical equations with the bisection method using RHD measurements and applied to three cases: (a) flat surface, i.e. without Kelvin-term and sub-saturation (RH < 100 [%]), (b) curved surface, i.e. including Kelvin-term and subsaturation, and (c) supersaturation with Kelvin-term, i.e.RH ≥ 100 [%].The derived HGF and supersaturation results are compared with those derived from the thermodynamic reference model E-AIM (Wexler and Clegg, 2002;Clegg and Wexler, 2007) for two key compounds NaCl and (NH 4 ) 2 SO 4 , which are important for modeling of cloud condensation nuclei (CCN) and central for CCN calibrations (Frank et al., 2006(Frank et al., , 2007) ) and CCN studies (Dusek et al., 2006;Rose et al., 2008).A comprehensive box model inter-comparison of major inorganic aerosol thermodynamic properties of mixed solutions predicted by EQSAM4 and EQUISOLV II (Jacobson et al., 1996(Jacobson et al., , 1999) )  2 Aerosol hygroscopic growth parameterizations of single solutes

K öhler theory and models
K öhler theory relates the particle growth of a spherical droplet formed on a soluble particle to the ambient relative humidity (RH), which can be expressed as (e.g.Pruppacher and Klett, 2007;Seinfeld and Pandis, 2006;Mikhailov et al., 2004Mikhailov et al., , 2009;;Rose et al., 2008): of a compact spherical droplet).With the assumption of "volume-additivity", i.e. the volume of the solution droplet is given by the sum of the volumes of the dry solute and of the pure water contained in the droplet, D wet can be expressed in terms of the dry mass equivalent diameter D s [m] and the RH dependent mass equivalent growth factor g s [−] of the droplet.g s is defined as the ratio of wet to dry droplet diameter, and can be expressed in terms of the solute molality µ s = n s /m w [mol(solute) kg −1 (H 2 O)]: (2) is the total volume of the wet droplet with V s = m s /ρ s = n s M s /ρ s and V w = m w /ρ w = n w M w /ρ w [m 3 ], i.e. the volumes of the initially dry solute and the associated pure water, respectively.m s and m w [kg] denote the corresponding solute and water masses, M s and M w [kg mol −1 ] the molar masses, n s and n w [mol] the number of moles, and ρ s and ρ w [kg m −3 ] the densities, respectively.In the following g s will be referred to as the hygroscopic growth factor (HGF) and applied to atmospheric aerosols.

Water activity representations -concepts
To clarify similarities and differences with previous work, we follow Rose et al. (2008) -in the following abbreviated as Rose08 (available at http://www.atmos-chem-phys.net/8/1153/2008/) -as they present a comprehensive overview of water activity representations on which the various present-day K öhler models are based.Rose08 have subdivided the water activity representations into 5 categories (see their  1.Activity parameterization (AP) models, e.g.Tang and Munkelwitz (1994), Tang (1996), Kreidenweis et al. (2005), which are of the type: Tang and Munkelwitz (1994) and Tang (1996) have presented parameterizations for the activity of water in aqueous solutions derived from electrodynamic balance (EDB) single particle experiments as polynomial fit functions of RH dependent solute mass percentage (100 • χ s ).The solute mass fraction χ s and the polynomial coefficients a q [−] are listed e.g. in Table A2 of Rose08 to which we refer for a further discussion (also for 2-5).
2. Osmotic coefficient (OS) models, e.g.Robinson andStokes (1959, 1965), Pitzer and Mayorga (1973), Brechtel and Kreidenweis (2000), which are of the type: According to Robinson and Stokes (1959) (the book pages are online freely accessable at http://books.google.de/books?id=6ZVkqm-J9GkC&printsec=frontcover) the water activity a w is related to the total molality of all solute species 3. Van't Hoff factor (VH) models, e.g.van't Hoff (1887), Low (1969), Young andWarren (1992), which are of the type: The van't Hoff factor i s [−] is a constant, and similar to the stoichiometric coefficient (dissociation number) ν s .deviations of i s from ν s can be attributed to solution non-idealities.The relation between i s , ν s and Φ s can be approximated by a series expansion of the exponential term in Eq. ( 4) and can be approximated as (Kreidenweis et al., 2005): 4. Effective hygrosocopicity parameter (EH) model of Petters and Kreidenweis (2007), which is of the type: Full with v s and v w [mol m −3 ] the molar volumes of the solute and of water, respectively.
Rose08 have used the Pitzer-Simonson-Clegg mole fraction based model AIM of Clegg et al. (1998a, b), Wexler and Clegg (2002) as a reference model (Clegg and Wexler, 2007), which is also based on osmotic coefficients, i.e. on Eq. ( 4), but combines the OS model with the universal functional group activity coefficient model (UNIFAC, Fredenslund et al., 1975).
The activity coefficient model describes the water activity by f w [−] denotes the rational or mole fraction scale activity coefficient of water, which is included in this water activity representation model to account for non-ideal solutions and solutes that dissociate (partly or completely).x w [−] is the mole fraction of water in the solution that at equilibrium contains the numbers of moles (amount-of-substance) n w [mol] and n s [mol] of water and solute, respectively.x w can be mathematically described as: Analogously, the mole fraction of the solute x s is given by i.e. satisfying the condition x s + x w = 1 for a binary solution (sone solute and water).
x w and x s are related to the solute molality µ s [mol(solute) kg −1 (H 2 O)] by where χ s = To consider cases for which the solution contains more than one solute, Eqs.(10a, b) is expressed in a more general form: n i is the number of moles [mol] of component i , where i = w for the solvent, or i = s for the solute; j = s 1 ,s 2 ,s 3 ,...,s n ,w is the summation over all n + 1 components in solution, so that Equation ( 9) expressed in the general form yields the activity and the activity coefficient of solutes (i = s) or the solvent water (i = w), i.e.: is the rational activity coefficients and is defined on a reference state for which f i is unity for infinite dilution (pure water), so that f i → 1 as x i → 0. f i of the solute s is related to the molal-based activity coefficients γ i by (Robinson and Stokes, 1959): with the summation in Eq. ( 14) over all solute molalities.M w is the molar mass of water [kg mol −1 )], µ i the solute molality [mol kg −1 (H 2 O)] given by Eq. ( 11).
The activity coefficients have been introduced to correct the solution molalities for non-ideality and to substitute earlier correction coefficients used in other a w representations.Following Rose08, we will use the E-AIM model version III as a reference, which is available online (http://www.aim.env.uea.ac.uk/aim/model3/mod3rhw.php) (Wexler and Clegg, 2002;Clegg and Wexler, 2007), to be consistent with the reference (AP3) of Rose08 (see their Table 3).Introduction

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Water activity parameterization as a function of ν i
To parameterize the aerosol hygroscopic growth we introduce a new parameterization for the water activity, i.e. adding to the definitions summarized above.Since the computation of all the a w representations is to some degree problematic -mainly since for all successful water activity parameterizations the correction coefficients are not independent of a w -, we suggest here a new relation of a w and the solute molality, µ s , as a central aspect of our formulations.a w and µ s are related by a single solute specific coefficient ν i : The terms ν i , A and B are dimensionless [−], while µ s is the solute molality [mol(solute) kg −1 (H 2 O)] defined by Eq. ( 11).A and B also depend only on µ s and ν i , but for certain applications they can be neglected, i.e.A = 1 and B = 0, as demonstrated in Sect.3. To match units with µ s , we multiply µ s on the rhs of Eq. ( 15) with an initial concentration of 1 mole of pure solute that dissolves in 1 kg of initially pure water, considering stoichiometry and molality scale, i.e.
Similar to the VH concept, i.e.Eq. ( 5), and the activity coefficient concept, i.e.Eq. ( 9), we express the water activity a w in terms of the solute molality µ s and a solute specific correction coefficient.However, ν i introduced here is not only a linear correction factor, which is the case in other water activity representations, but appears also as a constant in the exponent of Eq. ( 15) in the form of x • a x .According to our findings ν i can be assumed constant for the entire a w range (0 − 1).Hence, it suffices to determine ν i , e.g. with the bisection method, using any a w value if the corresponding µ s is known, Introduction

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Full e.g. at saturation.In case of saturation, a w equals the RH of deliquescence (RHD) and µ s is given through the relation Eq. ( 11) by the mass fraction χ sat s .Solubility values and the corresponding thermodynamic data are available for many compounds, e.g. in the CRC Handbook of Chemistry and Physics (2006).

RH of deliquescence (RHD) as a function of ν i
Here, we determine ν i from the compound's RH of deliquescence (RHD), since temperature dependent values are available for major compounds, which are relevant for atmospheric aerosol modeling.The RHD can be obtained as a function of ν i from Eq. ( 15) and Eq. ( 1), i.e. substitution of a w = RH K e yields: and at saturation, where RH = RHD and w s = χ sat s , Eq. ( 16a) can be expressed as: Thus, RHD values can be obtained with Eq. (16b) if w s , ν i , A and B are known.In Sect. 3 we derive these yet unknown terms first from RHD measurements, by solving Eq. (16b) once, for example with the bisection method.The temperature dependency of the RHDs is obtained by (e.g.Wexler and Potukuchi, 1998): The T −dependent RHD values (with w s at T ) can be used to determine ν i by solving Eq. (16b) e.g. with the bisection method for the compounds for which RHD and w s data are available.The procedure is described in Sect.According to our parameterization of water activity (a w ) the solute molality (µ s ) is also parameterized as a function of ν i and directly given by re-arranging Eq. ( 15): Using the relation of a w and RH, Eq. ( 1), Eq. ( 17a) can be expressed in terms of RH, ν i and K e , whereby K e also depends on µ s according to Eqs. (1-2).Thus, with a w = RH K e i.e.: Equation ( 17b) can be directly solved, e.g. with the bisection method, while Eq.(17a) can also be analytically solved, since the K e term is not relevant for flat surfaces.However, this also depends on the choice of the A and B terms.In case of a flat surface and subsaturation (RH ≤95 [%]), K e , A and B are neglected, i.e.K e = 1, A = 1 and B = 0, and Eq.(17b) reduces to: For applications up to RH ≤ 98 [%], a slightly different representation of Eq. (17c) can be used, which only differs by a term B 98 , i.e.Equations (17a-d) relate the solute molality µ s to a w and RH, depending only on the solute specific coefficient ν i .Note that B-terms are needed only for high RH > 95 [%].The A and B-terms are defined in Sect.3. Whether Eqs.(17a), (17b), (17c), or (17d) should be used depends on the desired level of complexity and the scientific questions, which determine the RH regime, i.e.only Eq. (17b) accounts for the particle size.
Equations (17a-d) can be applied in EQSAM4, while Eq. ( 17c) is equivalent to the one of EQSAM3, introduced by Metzger and Lelieveld (2007) by their Eq.( 20), but limited w.r.t.K öhler theory to the subsaturated RH regime, with RH ≤ 95 [%].For these cases, the results of the four different representations of Eq. ( 17) are, however, rather similar.This is demonstrated in the next section.

Parameter determination
To solve our water activity parameterization, i.e.Eq. ( 15), the dimensionless terms ν i , A and B are required.Here, A and B have been empirically determined to best match the reference results of E-AIM − the explicit derivation is beyond the scope of this work and will be presented separately.A and B are defined to only depend on the solute specific coefficient, ν i , and on the solute molality µ s , i.e.: As A and B depends on µ s and ν i , we additionally define an alternative B-term for Eq. ( 17d) with RH ≤ 98 [%], which only depends on ν i : To solve Eqs.(15-20), ν i must be known.If ν i can be assumed constant for the entire range of water activity a w (0-1), it should be possible when Eq. ( 1) is used to determine ν i for any particle dry diameter (D s ) at given temperature (T ) from the saturation point, i.e. also for the case where a w equals the RH of deliquescence (RHD).In this case, we can solve Eq. (16b) directly with a root finding algorithm, notably the bisection method; see e.g.Numerical Recipes (http://www.nr.com/) in Fortran 90, Second Edition (1996), page 1185 (an online version is freely available at: http://apps.nrbook.com/fortran/index.html).To constrain the bisection method when solving Eq. ( 16b as done in the companion paper (i.e.GCM application where the RH is usually prescribed).The hygroscopic growth factor (HGF), the saturation (s) and supersaturation (S) can then be obtained from µ s with Eq. ( 2) and Eq. ( 1), respectively, by using Eq. ( 15) to solve the associated a w of Eq. ( 1).Note that this combination of Eq. ( 15) and Eq.(1) yields Eq. (16a), and if the RHD is substituted by the RH it yields Eq. (16b).
For consistency with Rose08, we also compare our results with the parametric calculations of E-AIM.Rose08 provided in their electronic supplement a w −µ s values in the a w range 0.97-1, which we have connected with the E-AIM web-output, with a w and µ s obtained by running the E-AIM model version III (http://www.aim.env.uea.ac.uk/ aim/model3/mod3rhw.php), to cover the remaining a w range from the RHD to 0.97 by keeping the large number of the Rose08 AIM data points above 0.97.We have then calculated from these E-AIM a w −µ s values our reference HGF from Eq. ( 2), and reference RH from Eq. ( 1), by assuming (for the sake of simplicity) volume additivity, and a constant pure water surface tension of the solution, σ sol = 0.076 [N m −2 ], and the pure water density ρ w = 997.1 [kg m −3 ].For a discussion of σ sol and the volume additivity of Eq. ( 1) we refer to Rose08, who provide a comprehensive sensitivity analysis to various parameters affecting Eq. (1).

Bulk particles -flat surface
To compare our water activity parameterization with the E-AIM reference data and to test the sensitivity w.r.t.different combinations of A and B, we solve Eqs.(17a-d) for two key compounds, NaCl (cr) and (NH 4 ) 2 SO 4(cr) , considering four cases: 1. Equation (17a): using the A-and B-terms, but no K e -term.
2. Equation (17b): same as (1) but considering the K e -term as our best a w estimate.
3. Equation (17c): same as (1) but without the A-and B-terms as our simplest a w estimate.Introduction

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Full   SO 4(cr) , assuming a sufficiently large dry particle diameter (geometric diameter = mass equivalent diameter for a compact spherical droplet), i.e.D s = 1 [µm], so that the Kelvin term is negligible.The HGF is shown for the deliquescence branch of the hysteresis curves, i.e. considering an initially dry solute that entirely deliquesce when the RH exceeds the solute's RHD. Figure 1 shows the HGF for the RH range, RHD ≤ RH ≤ 97 [%], while Fig. 2 shows the complementing results for the RH range, 97 ≤ RH ≤ 100 [%], to highlight possible differences in the "quasi" ideal solution range close to RH 100 [%], where the water activity approaches unity.
According to Figs. 1 and 2, the results based on Eq. (17a) and Eq. ( 17b) are almost identical for the entire RH-range, which indicates that for this case the Kelvin-effect is indeed negligible.And for RH < 95 [%], all four cases, i. RH values, and for submicron arerosol particles in general, the Kelvin-term becomes important anyhow, so that only our most complete parameterization Eq. ( 17b) can be used for further evaluation.

Submicron sized particles -curved surface
In the case of submicron sized particles (D s < 1 [µm]), surface curvature becomes important for the aerosol hygroscopic growth calculations.Equation (17b) includes the Kelvin-term and will be used to calculate the HGF of submicron sized particles with Eq. (2) using the above ν i values listed in Table 2 for Eq.(17b).Figure 3 compares the HGF obtained with Eq. ( 17b) with E-AIM for RH ≤ 97 [%] and the four different dry particle diameters D s = 0.05, D s = 0.1, D s = 0.5 and D s = 1 [µm].Figure 4 complements Fig. 3 for the high values within the subsaturated regime, i.e. 97 ≤ RH ≤ 100 [%].Note that a w , and the derived HGF results of E-AIM, do not directly depend on the Kelvinterm K e .We plot the HGF versus RH =a w •K e following Rose et al. (2008), which yields a dependency of the E-AIM results on K e , but also shifts the RHD (of E-AIM) to a higher water activity (due to a w •K e ).We can reproduce this shift in RHD if the Kelvin-term is considered in the RHD calculations using the ν i determined from the flat surface case.
The estimated size dependent RHD values are obtained by calculating the RHD with Eq. (16b) using the ν i values listed in Table 2 for Eq.(17b), which is the only equation of Eq. (17a-d) that accounts for a size dependency by including the K e -term.The size dependent RHD estimates are shown in Table 3 for the different dry particle diameters.
According to Figs. 3 and 4 the results of Eq. ( 17b) agree well for different particle sizes in the subsaturated RH regime.It also appears that the results capture the decrease of the HGF for nanometer size particles reported by measurements provided by Biskos et al. (2006a, b).Note that we have not applied a shape correction factor, and used for simplicity a constant surface tension of pure water (as mentioned above) for the K e -term calculation for both Eq. ( 17b) and E-AIM.For a discussion of these parameters we refer to Biskos et al. (2006a, b)

Supersaturation -K öhler curves
To evaluate the upper RH limit where the parameterizations may still be valid, we compare the results obtained from Eq. ( 17b) with E-AIM for the case of water vapor saturation and supersaturation, i.e.RH ≥100 [%].The supersaturation S [%] is defined as S = (s − 1) • 100, with s obtained by solving Eq. ( 1) as a function of wet diameter, D wet , using Eq.(17b).For E-AIM we again use RH= a w • K e , following Rose08.) for NaCl (cr) and (NH 4 ) 2 SO 4(cr) a shape factor of 1.15 and 1.14, respectively, which was used to calculate the Kelvinterm with Eq. ( 1) for E-AIM, Kappa and the Eq. ( 17b) method.For all other particle sizes no shape factor was applied.

Discussion
We presented aerosol hygroscopic growth parameterizations that provide the core of our revised gas-liquid-solid partitioning model, i. paper.EQSAM4 is part of our aerosol chemistry-microphysics module (GMXe), the latter being coupled to the EMAC atmospheric chemistry-climate model (Pringle et al., 2010), and will be used to study the aerosol hygroscopic growth of single and mixed solutes.Thus, a requirement is that the parameterizations are both sufficiently accurate and computationally efficient, so that large-scale, long-term and high-resolution atmospheric chemistry-climate simulations are feasible.

Simplified HGF calculations:
as demonstrated in Sect.3, our empirical representation of water activity, a w , by Eq. ( 15), and the derived equations for the solute molality µ s , Eqs. (17a-d), can help estimate the HGF of atmospheric aerosols.Our results compare well with the reference calculations of E-AIM for typical RH (a w ) values, relevant to conditions in the atmosphere.HGF estimates, as calculated from Eq. ( 2) and based on the complete a w representation, Eq. ( 17b), which includes the Kelvin-effect, also applies to submicron sized particles (e.g.Russell and Ming, 2002;Biskos et al., 2006a, b), i.e. with a dry diameter D s below 0.5 [µm].The results agree well with the κ-method of Petters and Kreidenweis (2007) for the ideal solution cases, but agree more closely with E-AIM for concentrated solutions.Both, the κ and ν i method only require one coefficient per compound, and agree relatively well w.r.t. the CCN activation of the aerosols according to K öhler theory (e.g.Charlson et al., 2001;McFiggans et al., 2006)  compared to most other representations of water activity (briefly summarized in Sect.2) our approach requires only one empirical coefficient ν i to cover a wide range of a w from concentrated solutions at low RH, around the compound's RHD, up to ideal solutions at large RH and CCN activation.The κ-method of Petters and Kreidenweis (2007), which also requires only a single parameter, is limited to ideal solutions and not optimally valid for concentrated solutions as confirmed by Figs. 3 and 4. The advantage of our parameterizations Eqs.(17a-b), compared to our Eqs.(17c-d), is that they can be applied to concentrated and ideal solutions, since they include the additional A-and B-terms.Equation ( 17b) can be further applied to the supersaturated RH regime, because it includes the Kelvin term.However, the A-, B-and K e -terms depend on µ s , which makes Eqs.(17a-b) more complex to solve in case RH or a w is prescribed.Nevertheless, despite this selfdependency, also Eq. (17a-b) can be directly solved when using ν i by a rootfinding method, such as the bisection method used in this study.4. Determining ν i : to determine ν i , any of the ν i -based equations, i.e.Eqs.(15-17d), could be solved with a root finding method, e.g.bisection, by prescribing a pair of water activity (a w ) and solute molality (µ s ) values of an a w − µ s diagram.Such a data pair can be either obtained from detailed calculations or measurements.Using for instance RHD and solubility (w s ) measurements, both values are directly known and related by ν i through Eqs.(16a-b), i.e.: a w by RHD = RH = a w • K e through Eq. ( 1), and µ s by w s through Eq. ( 11).Hence, in this work we have determined ν i by solving Eq. (16b) using a measured RHD for NaCl (cr) and (NH 4 ) 2 SO 4(cr) and assuming a flat surface, i.e.K e = 1, as described in Sect.3.

Constant ν i :
in all cases we have used a solute specific coefficient, i.e. ν i , determined from the solute's RHD.According to the comparison with the E-AIM reference calculations, it appears that a constant ν i suffices to estimate the compound's HGF with Eqs.(17a-d) for a wide RH range and various particle sizes.Note that we express the water activity a w in terms of the solute molality µ s and a solute specific correction coefficient, similar to the VH concept, i.e.Eq. ( 5), and the activity coefficient concept, i.e.Eq. ( 9).However, ν i introduced here is not only a linear correction factor, which is the case in other water activity representations, but is also used in the exponent of Eq. ( 15) in form of x • a x .Apparently, even our simplest a w estimate based on Eq. (17c) yields reasonably accurate results for the subsaturated regime, RH ≤ 95 [%], but the Kelvin-term must be neglected.The advantage of using ν i is that it allows to directly solve Eqs.(17a-d), i.e. either Eqs.(17a-b) with e.g. the bisection method, or Eqs.(17c-d) analytically without iterations.In all cases, ν i allows to easily compute the HGF for a wide RH range with Eq. (2) when using Eqs.(17a-d).Thus, for our purpose, i.e. application in EQSAM4, the proposed ν i based equations suffice, and complicated calculations of solution densities, activity coefficients, or osmotic coefficients, as described in the overview article of Rose08, are not required here.24835 Introduction

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Full Thus, the comparison of our Fig.6 and their Fig. 10 shows that our best estimate based on Eq. ( 17b) lies well within the uncertainty range given by the a w representations discussed in Rose08.
7. From single to mixed solute solutions: our example calculations are shown for single solute solutions.But the ν i based equations are also applicable for mixed solutions, if e.g. the widely used additive approach of partial water masses of single solutes in mixed solutions (ZSRrelation;Zdanovskii, 1948;Stokes and Robinson, 1966), or other approaches, e.g.CSB (Hanford et al., 2008) are used to obtain the mixed solution water content.Our ν i approach even simplifies the calculation for the mixed solution molality and water content, since our Eqs.(17a-d) do not depend on the water content.Thus, the mixed solution water content and the corresponding HGF can be more easily calculated with Eqs. ( 11) and (2).Especially by considering the Kelvin term, our ν i based equations can substantially simplify the calculations, since in this case (multidimensional) iterations would be required otherwise.The mixed solution RHD, often referred to as mutual deliquescence relative humidity, MDRH, (Wexler and Seinfeld, 1991), for which all salts are simultaneously saturated with respect to all components, will be considered in our revised gas- 8. Relation to other concepts through a w : the relationship of our ν i -based a w parameterizations to the various other concepts and a w representations is given by a w itself.The water activity is the central thermodynamic property from which all other properties can be derived -the various relations, which are most important for atmospheric aerosol research, are briefly summarized in Sect. 2. The relation of ν i to the EQSAM3 concept introduced by Metzger and Lelieveld (2007) is given by our Eq.(17c) for the solute molality µ s , which equals their corresponding equation Eq. ( 20), with ν i = ν e /ν w .
Both equations are limited w.r.t. to K öhler theory to RH ≤ 95 [%] though only depend on solute specific constants.Thus, in the EQSAM3 context, ν i may be also directly related to an effective dissociation and the solute solubility.

Conclusions
We have introduced a new parameterization of the water activity (a w ) of single solute solutions, which only requires an empirical solute specific coefficient, ν i .ν i can for instance be determined from RHD measurements or reference calculations through the various relations presented.One advantage of the ν i concept is that it requires only a single coefficient to parameterize the solute molality for a wide range of water activities, while other concepts require solution dependent coefficients.The derived HGF parameterizations can be used up to high relative humidity and size ranges, from nanometer sized particles to cloud condensation nuclei (CCN).Another advantage is that the water uptake for mixed solutions can be more easily obtained from our ν i based equations, since the solute molality calculated with Eqs.(17a-d) does not depend on the water content of the solution, so that the mixed solution water content can be more easily obtained with e.g. the ZSR-relation.This work provides the equations on which the revised EQSAM4 model builds, which considers mixed solutions of various compounds important for atmospheric modeling described in a companion paper and evaluated for various mixed solution cases in the revised publication of Xu et al. (2009).Introduction

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Full  Full  Full      Full    For comparison, the results using the κ method of Petters and Kreidenweis (2007) are also included, obtained by solving Eq.A30 of Rose08 using κ = 1.28 for NaCl (cr) and κ = 0.61 for (NH 4 ) 2 SO 4(cr) .Note that it is not possible to obtain HGF results below to the RHD for NaCl (cr) with the κ−method using κ = 1.28.
22  (2007) are also included, obtained by solving Eq.A30 of Rose08 using κ = 1.28 for NaCl (cr) and κ = 0.61 for (NH 4 ) 2 SO 4(cr) .Note that it is not possible to obtain HGF results below to the RHD for NaCl (cr) with the κ-method using κ = 1.28.Introduction

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Full Wet diameter [µm] NaCl -E-AIM NaCl -Eq17b NaCl -Kappa (NH 4 ) 2 SO 4 -E-AIM (NH 4 ) 2 SO 4 -Eq17b (NH 4 ) 2 SO 4 -Kappa  1) for both our results using Eq. ( 17b) and the reference calculations using the E-AIM data of Rose 08.For comparison, the results using the κ method of Petters and Kreidenweis (2007) are also included, which are obtained by solving Eq.A30 of Rose08 using κ = 1.28 for NaCl (cr) and κ = 0.61 for (NH 4 ) 2 SO 4(cr) .(1) for both our results using Eq.(17b) and the reference calculations using the E-AIM data of Rose 08.For comparison, the results using the κ method of Petters and Kreidenweis (2007) are also included, which are obtained by solving Eq.A30 of Rose08 using κ = 1.28 for NaCl (cr) and κ = 0.61 for (NH 4 ) 2 SO 4(cr) .Introduction

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Full g. K ünzli et al., Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | is subject of the revised publication of Xu et al. (2009Discussion Paper | Discussion Paper | Discussion Paper | p w(g)[Pa]  and p sat w(g) [Pa] denote the partial pressures of water vapor of the ambient air at temperature T [K] and at saturation at the same T .It is common to express the dimensionless fractional relative humidity RH [−] as the saturation ratio s = RH [−], with RH (not in italics) in [%].In case of supersaturation, RH > 100 [%], it is customary to use the supersaturation S, which is defined as S = (s − 1) • 100 [%].The dimensionless term a w [−] is the water activity of the solution (droplet) and is referred to as the Raoult term (or volume contribution), since it accounts for an increase of the droplet volume (D 3 wet ) with increasing RH with a 1/D 3 wet proportionality.K e [−] is the Kelvin (or surface) term, which accounts for a compensating effect with a 1/D wet proportionality and the RH dependent surface tension σ sol [J m −2 ] of the solution droplet; R [J mol −1 K −1 ] is the ideal gas constant and T [K] the droplet temperature, D wet [m] is the ambient droplet diameter (geometric diameter = mass equivalent diameter Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | the dimensionless molal osmotic coefficient of the aqueous phase Φ w [−], where i µ i can be expressed as ν s • µ s if the molal osmotic coefficient of the solute Φ s is used in conjunction with the solute molality µ s [mol kg −1 (H 2 O)] and the solute's stoichiometric coefficient ν s .M w is the molar mass of water in SI-units [kg mol −1 ].Φ w deviates from unity as the solution becomes non-ideal.Discussion Paper | Discussion Paper | Discussion Paper | ρ s and V w = n w M w /ρ w are the volumes [m 3 ] of the initially dry solute and pure water, respectively, with M s and M w [kg mol −1 ] the molar masses of solute and water, respectively, and ρ s and ρ w [kg m −3 ] the densities of the initially dry solute and pure water, respectively.The dimensionless hygroscopicity parameter κ [−] parameterizes the composition dependent water activity of a solution droplet in analogy to the van't Hoff factor.κ and the van't Hoff factor i s are related by: Discussion Paper | Discussion Paper | Discussion Paper | +m w ) [−] is the solute mass fraction, m s and m w [kg] the masses of solute and water, with M s and M w [kg mol −1 ] the corresponding molar masses, respectively.Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | 2.1.4Solute molality (µ s ) as a function of ν i Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | ) we use the T -dependent RHD and w s values, i.e. for the left hand-side at T o = 298.15[K] RHD = 0.7528 [−] for NaCl (cr) and RHD = 0.7997 [−] for (NH 4 ) 2 SO 4(cr) , and calculate ν i as follows: 1.Given w s = χ s , we first calculate µ sat s from Eq. (11); 2. A is a function of ν i and calculated from µ s = µ sat s and M w using Eq.(18); 3. B is a function of ν i and calculated from µ s = µ sat s and M s using Eqs.(19) or (20); 4. We finally determine ν i by solving Eq. (16b) with the bisection method by using µ sat s from step 1, A and B from step 2 and 3 and assuming K e = 1.Given ν i and µ s , Eqs. (17a-d) can be solved for a w and RH, if at step 1 the HGF is additionally calculated from Eq. (2).Apart from this, the procedure of solving Eqs.(17a-d) is the same as to determine ν i .Note that the choice of A and B in Eqs.(18-20) depends on the application.At T o the following values for NaCl (cr) and (NH 4 ) 2 SO 4(cr) are: mass fraction solubility w s = 0.2647 [−] and w s = 0.4331 [−], molar masses M s = 0.05844 [kg mol −1 ] and M s = 0.1321 [kg mol −1 ], and densities ρ s = 2170 [kg m −3 ] and ρ s = 1770 [kg m −3 ], respectively.Once ν i has been determined, Eqs.(17a-d) can be either solved for RH for a given solute molality µ s , e.g. from Eq. (11) or reference model calculations, or vise versa Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | e. Eqs.(17a-d), agree well with the reference.In the RH range, 95 < RH ≤ 98 [%] the results of Eq. (17d) (using the simplified B-term) are still close to those of Eq. (17a-b) and E-AIM, while the results of Eq. (17c) without the A-and B-terms deviate noticeably.But when the RH exceeds 98 [%] the results of Eq. (17d) also start to deviate from our reference.Only the results of Eq. (17a-b) agree well with the E-AIM results close to RH 100 [%].But for these high Introduction Discussion Paper | Discussion Paper | Discussion Paper | , Rose08, Harmon et al. (2010) and Wang et al. (2010) (and references therein).In the next section we focus on the remaining RH regime of water vapor saturation and supersaturation, i.e.RH ≥ 100 [%].Discussion Paper | Discussion Paper | Discussion Paper | Figure  5completes the RH range shown in Figs.3 and 4, and shows that the results of Eq. (17b) are comparable to E-AIM from the RHD to supersaturation for pure NaCl (cr) and (NH 4 ) 2 SO 4(cr) particles and different dry particle diameters, i.e.D s = 0.05, D s = 0.1, D s = 0.5 and D s = 1 [µm].These results are also comparable to Fig.15.5 of Pandis andSeinfeld (1998), which describe their approximation for ideal solutions.Figure6shows the critical supersaturation S c , which complements Fig.5.S c is given by the maximum S. The results are comparable to Fig.15.6 of Pandis and Seinfeld (1998) and Fig.10of Rose08.Note that Fig.6covers a diameter range 5-500[nm], which is extended compared to Rose08 (their Fig.10shows20-200 [nm]), so that our maximum critical supersaturation is about 10 [%] for 5 [nm] NaCl (cr) particles.Note that these high critical supersaturations are merely shown to evaluate the Eq.(17b) method.To bring the results closer to our artificial help lines in the log−log diagram, it was necessary to assume for the 5 [nm] particles (and only for 5[nm] e. version 4 of the EQuilibrium Simplified Aerosol Model (EQSAM4), which is described in greater detail in the companion Discussion Paper | Discussion Paper | Discussion Paper | down to dry diameters D s below 0.05 [µm].Only for (NH 4 ) 2 SO 4 particles the S c obtained from the κ-method agrees more closely with E-AIM, but the steep increase in supersaturation agrees less w.r.t.E-AIM and at 0.005 [µm] also the S c obtained by the ν i method appears closer to E-AIM when the same κ and ν i values are assumed for all particles sizes.For simplicity and clarity, we have neglected here potential effects of surface tension and other size effects, though they can be included when needed.For a discussion we refer to e.g.Rose08 and references therein.Discussion Paper | Discussion Paper | Discussion Paper | 2. The advantage of Eqs.(17a-b):

3 .
The advantage of Eqs.(17c-d): compared to our more comprehensive parameterizations Eqs.(17a-b), our simpler parameterizations Eqs.(17c-d) are limited to the subsaturated RH regime; Eq. (17c) to approximately RH ≤95 [%]; Eq. (17d), which includes a simplified Bterm defined by Eq. (20), to approximately RH ≤ 98 [%].This actually covers the most important RH range for atmospheric aerosols, useful for many applications especially in GCMs.And for CCN activation, Eq. (17b) can be used.The advantage of using Eqs.(17c-d) is that the HGF of single or mixed compound solutions can be analytically and non-iteratively obtained for a given RH, by using a single constant ν i value per compound and various particles dry sizes, since our calculations do not depend on the aerosol liquid water content.Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | 6. Uncertainty range for HGF and S c calculations: an indication of the uncertainty of our best estimates based on Eq. (17b) is given by a comparison of Figs. 5 and 6 with the corresponding Fig. 10 of Rose08.According to our Fig. 5, a relatively large difference appears for the supersaturation estimate for D s = 0.05 (NH 4 ) 2 SO 4 particles.In the logarithmic plot of Fig. 6, these differences are less obvious, and the Fig. 10 of Rose08 even shows a larger deviation relative to the results of E-AIM for the various other a w based S c estimates.
liquid-solid partitioning model, i.e. version 4 of the EQuilibrium Simplified Aerosol Model (EQSAM4), described in the companion paper.Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | [kg m −3 ] σ sol surface tension of the solution droplet [J m −2 ] χ s solute mass fraction, referring to the solute's dry mass [−] χ sat s solute mass fraction, referring to the solute's dry mass at saturation [−Discussion Paper | Discussion Paper | Discussion Paper | vapor pressure at saturation (at given T) [Pa] RH relative humidity in percent (as used in text) [%] RH fractional relative humidity (as used in equations) , referring to the solute's dry mass required for saturation [−] x s mole fraction of solute [−] x w mole fraction of water [Discussion Paper | Discussion Paper | Discussion Paper |

Fig. 5 .
Fig. 5. Wet particle diameter, D wet , as a function of supersaturation for pure NaCl and (NH 4 ) 2 SO 4 aerosols with different dry diameters, i.e.D s = 0.05, D s = 0.1, D s = 0.5 and D s = 1 [µm] as shown in Figs. 3 and 4. S is defined as S = (s − 1) • 100 [%] and s is obtained from Eq. (1) for both our results using Eq.(17b) and the reference calculations using the E-AIM data of Rose 08.For comparison, the results using the κ method of Petters andKreidenweis (2007) are also included, which are obtained by solving Eq.A30 of Rose08 using κ = 1.28 for NaCl (cr) and κ = 0.61 for (NH 4 ) 2 SO 4(cr) .

Fig. 5 .
Fig. 5. Wet particle diameter, D wet , as a function of supersaturation for pure NaCl and (NH 4 ) 2 SO 4 aerosols with different dry diameters, i.e.D s = 0.05, D s = 0.1, D s = 0.5 and D s = 1 [µm] as shown in Figs.3 and 4. S is defined as S = (s − 1) • 100 [%] and s is obtained from Eq. (1) for both our results using Eq.(17b) and the reference calculations using the E-AIM data of Rose 08.For comparison, the results using the κ method of Petters andKreidenweis (2007) are also included, which are obtained by solving Eq.A30 of Rose08 using κ = 1.28 for NaCl (cr) and κ = 0.61 for (NH 4 ) 2 SO 4(cr) .

Fig. 6 .
Fig. 6.Critical supersaturation as a function of dry diameter, D s , for pure NaCl and (NH 4 ) 2 SO 4 particles with different diameters, i.e.D s = 0.005, D s = 0.01, D s = 0.05, D s = 0.1 and D s = 0.5 [µm], complementing Fig. 5. Note, the black solid lines are artificial help lines; the lower line and points correspond to NaCl, the upper ones to (NH 4 ) 2 SO 4 .

Fig. 6 .
Fig. 6.Critical supersaturation as a function of dry diameter, D s , for pure NaCl and (NH 4 ) 2 SO 4 particles with different diameters, i.e.D s = 0.005, D s = 0.01, D s = 0.05, D s = 0.1 and D s = 0.5 [µm], complementing Fig. 5. Note, the black solid lines are artificial help lines; the lower line and points correspond to NaCl, the upper ones to (NH 4 ) 2 SO 4 .

Table 3 )
, i.e.Figures Table1gives an overview of the equations applied for the four different cases, i.e. solute molality, µ s -models.Note that Eqs.(17a-d) can be used to calculate either a w , RH or µ s .Here we prescribe µ s and calculate a w and RH, while in the companion paper we prescribe RH and calculate µ s from Eqs. (17a-d).The RHD and the derived ν i values are listed in Table2for the four cases listed in Table1.These data, together with a Fortran 90 program to reproduce our results are provided in the Supplement.Figures1 and 2firstly compare our four RH calculations as a function of HGF for the simplest case of flat surface, i.e. particles with a sufficiently large dry diameter in the subsaturated RH regime, so that the Kelvin-term can be neglected for the aerosol HGF calculations.Then, the four cases shown in Table 1 should yield similar results if the corresponding ν i values listed in Table 2 are used to solve Eqs.(17a-d).For each case (equation), we calculate the RH by prescribing the E-AIM µ s values of two single solute solutions: (1) pure sodium chloride, NaCl (cr) , and (2) pure ammonium sulfate, (NH 4 ) 2

Table 1 .
Overview of the applied cases using four different solute molality, µ s -models.

Table 3 .
Estimated a RHD values for different D s with ν i from Table 2 using Eq.(17b) of Table 1.

Table 4 .
List of greek symbols.

Table 5 .
List of symbols.