Droplet activation of moderately surface active organic aerosol predicted with six approaches to surface activity

Surface active compounds (surfactants) found in atmospheric aerosols can decrease droplet surface tension as they adsorb to the droplet surfaces simultaneously depleting the droplet bulk. These processes may influence the activation properties of aerosols into cloud droplets and investigation of their role in cloud microphysics has been ongoing for decades. In this study, we have used six different approaches documented in the literature to represent surface activity in Köhler calculations predicting cloud droplet activation properties for particles consisting of one of three different moderately surface active 5 organics mixed with ammonium sulphate in different ratios. We find that the different models predict comparable activation properties at small organic mass fractions in the dry particles for all three moderately surface active organics tested, even with large differences in the predicted degree of bulk-to-surface partitioning of the surface active component. However, differences between the models regarding both the predicted critical diameter and supersaturation for the same dry particle size increase with the organic fraction in the particles. Comparison with available experimental data shows that assuming complete bulk10 to-surface partitioning of the organic component (total depletion of the bulk) along the full droplet growth curve does not adequately represent the activation properties of particles with high moderate surfactant mass fractions. Accounting for the surface tension depression mitigates some of the effect. Models that include the possibility for partial bulk-to-surface partitioning yield comparable results to the experimental data, even at high organic mass fractions in the particles. The study highlights the need for using thermodynamically consistent model frameworks to treat surface activity of atmospheric aerosols and for 15 firm experimental validation of model predictions across a wide range of states relevant to the atmosphere.


Introduction
The effect of atmospheric aerosols on the climate is still among the largest uncertainties to estimates and interpretations of the Earth's changing energy budget (IPCC, 2013;Seinfeld et al., 2016). An aerosol population can be composed of dozens 20 of inorganic salts mixed with hundreds of organic species. Single-particle measurements have shown mixtures of compounds from primary sources, such as soot, dust and organic carbon, mixed with sulfate, nitrate and oxidized organics from secondary give more realistic predictions of CCN activity for organic aerosol in large scale simulations than surface tension calculated without consideration of bulk-surface partitioning in droplets (Prisle et al., 2012b), but such an approximation may not be generally applicable (e.g. Nozière et al., 2014;Petters and Petters, 2016;Lowe et al., 2019).
Fatty acids and their salts are a major class of organic compounds identified in atmospheric aerosols (e.g. Mochida et al., 2002Mochida et al., , 2007Cheng et al., 2004;Li and Yu, 2005;Forestieri et al., 2018) and often are relatively strong surfactants (see e.g. 70 Prisle et al. (2008,2010) and references therein), but also surfactants of moderate strength are abundantly present in the atmosphere. In microscopic droplets, strong surfactants are predicted to be strongly depleted from the bulk phase due to bulk-surface partitioning, leading to modest effective hygroscopicity of the surface active component, while at the same time, the finite amount of surface active material is still not sufficient to significantly reduce droplet surface tension (Prisle et al., 2008(Prisle et al., , 2010(Prisle et al., , 2011Lin et al., 2020;Bzdek et al., 2020). This behavior may however be a limiting case and not representative 75 across the whole range of surfactant strength and other molecular properties found in the atmosphere. The partitioning behavior of moderate surfactants may be more complex and dependent on droplet concentration and size (governing the surface areato-bulk volume ratio). It has been shown that partitioning between droplet bulk and surface can affect activation properties of surface active aerosol: Enhancement of hygroscopicity has been reported from surface tension effects for organosulfate products (Hansen et al., 2015), secondary organic aerosol containing dicarboxylic acids (Ruehl et al., 2016), marine primary 80 organics (Ovadnevaite et al., 2017), and pollen extracts (Prisle et al., 2019). Therefore, thermodynamically consistent bulksurface partitioning models are needed to fully represent surfactant properties (strength) and mixing states in such droplet systems.
Several models have been developed to be used with Köhler theory to calculate bulk-surface partitioning of surface active components and the resulting effects in growing droplets, including the Gibbs surface approach by Sorjamaa et al. (2004); 85 Prisle et al. (2008Prisle et al. ( , 2010, the molecular monolayer surface model by Malila and Prisle (2018), the liquid-liquid phase separation (LLPS) approach with possible partial surface coverage by Ovadnevaite et al. (2017) and the compressed film surface model by Ruehl et al. (2016). Each of these models rely on a unique set of assumptions and requirements for application. In addition, simplified models, emulating the results of more comprehensive frameworks (Prisle et al., 2011), with further simplifying assumptions (Ovadnevaite et al., 2017), derived as analytical approximations to ease the computational load (Topping,90 2010; Raatikainen and Laaksonen, 2011) have also all been employed. A more extensive overview of different bulk-surface partitioning approaches is given by Malila and Prisle (2018). Many of the models have been shown to agree well with exper-imentally observed CCN activity for selected model aerosol systems (e.g. Ruehl et al., 2016;Lin et al., 2018;Davies et al., 2019). A few studies have presented the results from different models for the same droplet systems. Lin et al. (2018) compared the models of Prisle et al. (2010), Prisle and Mølgaard (2018) and Malila and Prisle (2018) for multiple droplet systems 95 comprising of succinic acid, sodium dodecyl sulphated (SDS) and Nordic Aquatic Fulvic Acid (NAFA) mixed with natrium chloride as well as pollenkitts mixed with ammonium sulphate. Davies et al. (2019) used the model of Ruehl et al. (2016) and different variations of the models presented in Ovadnevaite et al. (2017) for a system of ammonium sulphate particles coated with suberic acid. However, to the best of our knowledge, a comprehensive systematic comparison between multiple partitioning models has not previously been done. 100 In this study, we have used the thermodynamic and simplified partitioning models of Prisle et al. (2010), Prisle et al. (2011), Malila and Prisle (2018), Ruehl et al. (2016), Ovadnevaite et al. (2017), as well as a general bulk solution model, all implemented into the same Köhler model framework. Each of the models are used to calculate Köhler curves for particles consisting of a moderate organic surfactant mixed with ammonium sulphate. Moderate organic surfactants are represented by dicarboxylic acids due to their immediate atmospheric relevance (e.g. Shulman et al., 1996;Hori et al., 2003) and abundance (e.g. Khwaja,105 1995; Mochida et al., 2007;Jung et al., 2010). We quantify surfactant strength similarly to Prisle et al. (2010Prisle et al. ( , 2011 as the surface tension reduction from that of pure water at a given surfactant bulk phase concentration. To reduce the surface tension of water in an aqueous bulk solution by 10 % at 298.15 K, the mole fraction of malonic, succinic and glutaric acids in the solution estimated from a fit (Hyvärinen et al., 2006) to be about 0.061, 0.017 and 0.0070 respectively. In addition, the dicarboxylic acids used for this study can reduce surface tension to roughly 50 mN m −1 at most in large enough concentrations (Hyvärinen 110 et al., 2006;Booth et al., 2009). Stronger surfactants, such as fatty acid salts, can reduce surface tension to 20-30 mN m −1 at most (Prisle et al., 2010). The thermodynamic properties required for the simulations are relatively well constrained for dicarboxylic acids (e.g. Booth et al., 2009;Hyvärinen et al., 2006;Ruehl et al., 2016). We compare the Köhler growth curves predicted with the different models to investigate impacts of the significant differences between the model assumptions and the expected high sensitivity of the partitioning equilibrium for the moderately strong surfactant to the assumed droplet state.

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A separate companion study is currently in preparation, comparing predictions with the same models for particles comprising stronger surfactants.

Theory & Methods
Six different modeling approaches are used to estimate the possible surfactant effects in Köhler calculations of droplet growth and CCN activation. Descriptions of the different model calculations are presented in the following sections. Each of the bulk-120 surface partitioning models were implemented into the same Köhler model framework running in MATLAB (2019, 2020).
Models that were not previously developed by Prisle andco-workers, were built in MATLAB (2019, 2020) with the information provided in the presenting publications and by the authors in personal communication. Köhler growth curves were simulated for dry particles consisting of one of the dicarboxylic acids, malonic acid, succinic acid or glutaric acid, mixed with ammonium the calculated Köhler curve.

Köhler theory
Cloud droplets form in the atmosphere when water vapor condenses on the surfaces of aerosol particles. The Köhler equation (Köhler, 1936) describes the process, relating the equilibrium water vapor saturation ratio (S) over a spherical solution droplet to its diameter (d) as where p w is the equilibrium partial pressure of water over the solution droplet, p 0 w is the saturation vapor pressure over a flat surface of pure water, a w is the droplet solution water activity,v w = M w /ρ w is the molar volume of water, σ is the droplet surface tension, R is the universal gas constant, and T is the temperature in Kelvin. The water vapor supersaturation is generally defined as SS = (S − 1) · 100% and droplet activation is determined in terms of the critical supersaturation (SS c ) or critical 135 saturation ratio (S c ), corresponding to the maximum value of the Köhler curve described by Eq. (1). Köhler theory is used in all cases in this study to describe the formation of cloud droplets, but the treatment of bulk-surface partitioning of surface active species and the resulting droplet water activity and surface tension varies between the models.
All the Köhler calculations are initiated by defining a dry particle size and composition which determines the total amount of solute in the growing droplets. For each dry particle size, the total amounts of ammonium sulphate salt and organic molecules 140 are calculated based on their pure solid phase densities and relative mass fractions in the particle, with the assumption of spherical dry particles and additive solid phase volumes in all cases. For each droplet size along the Köhler curve for a given dry particle, the total amount of water in the droplet phase is estimated from the droplet size via model specific methods, listed in Table 1. Here, iterative calculation refers to determining the total amount of water (n T w ) based on mass conservation using the ternary mixture density while additive calculation refers to the assumption of additive volumes of the dry particle and the condensed water (Hänel, 1976) and is calculated by subtracting the dry particle volume from the total droplet volume. The equations for calculating the total amounts of salt, organic and water are presented in the supplement section S2.2.
The droplet solution is a ternary water-inorganic-organic mixture. As each droplet grows, the surface area-to-bulk volume decreases, which in turn affects the bulk-surface partitioning of surface active species (Prisle et al., 2010;Bzdek et al., 2020).
The different partitioning models compared in this work make different assumptions regarding the partitioning of droplet 150 components. While some consider only partitioning of the surface active species into a purely organic surface phase (Prisle et al., 2011;Ruehl et al., 2016;Ovadnevaite et al., 2017), others evaluate the full composition of both droplet surface and bulk phases (Prisle et al., 2010;Malila and Prisle, 2018). Table 1 lists how the different models calculate the initial amount of water, the water activity, the surface tension of the droplets and what compounds reside in the droplet surface and bulk.

Gibbs adsorption partitioning model
In Gibbs surface thermodynamics, the gas-liquid interface is assumed to be an infinitely thin 2-dimensional surface called the Gibbs dividing surface, the location of which may vary depending on specific assumptions made for the system. The modeled idealized system is energetically and mechanically equivalent to the real system it represents (Defay and Prigogine, 1966). Several bulk-surface partitioning models for droplets have been developed based on Gibbs surface thermodynamics (e.g. Li et al., 1998;Sorjamaa et al., 2004;Prisle et al., 2010;Topping, 2010;Raatikainen and Laaksonen, 2011;Petters and 160 Kreidenweis, 2013; Prisle and Mølgaard, 2018) but the assumptions related to the position of the dividing surface and the method of solving the Gibbs adsorption equation (Gibbs, 1878) differ between the models.
In the model applied here (Prisle et al., 2010), the position of the Gibbs dividing surface is determined such that the bulkphase volume (V B ) is equal to the total (equimolar) droplet volume (V T ) of all droplet components j. Compounds adsorbed to the surface are assumed to not contribute to the total droplet volume, whence a positive surface volume of one compound 165 (surfactant) must be balanced by depletion of other compounds (water and salt) from the surface. Sorjamaa et al. (2004) combined the adsorption equation with the Gibbs-Duhem equation for the droplet bulk, resulting in where n T j is the total amount of species j in the droplet solution, k is the Boltzmann constant, n B org is the number of surfactant molecules in the droplet bulk, a B j is the activity of j in the droplet bulk solution, A is the spherical droplet surface area and 170 σ is the droplet surface tension, given as function of bulk-phase composition. Equation (2) is solved iteratively for the bulk composition with the boundary condition that the molar ratio of water and salt is the same in the bulk and surface phases, such that the only adsorbing species is the surfactant. In addition, we assume volume additivity (such that the droplet diameter is given by the sum of individual pure component molar volumes) and mass conservation (n T j = n S j + n B j ) inside the droplet (Prisle, 2006). The assumption of additive volumes is employed for calculating the total amount of water in the droplets at each 175 droplet size along the Köhler growth curve. The calculation details relating to the total amount of water are given in supplement section S2.2.

Simple complete partitioning model
The simple model of Prisle et al. (2011) was developed to emulate the more complex Gibbs model of Prisle et al. (2010) specifically for predictions of SS c . It approximates organic partitioning by simply assuming that all surface-active organics are 180 partitioned to the droplet surface into an insoluble layer. The surfactant solute therefore does not affect the water activity or surface tension of the aqueous droplet solution at the point of activation. This was shown to give very good representation of both the comprehensive model predictions and experimental results for droplet activation presented by Sorjamaa et al. (2004) and Prisle et al. (2008Prisle et al. ( , 2010. The amount of surfactant in the droplet bulk is consequently vanishing (n B org = 0), whereas neither salt nor water are present in the droplet surface. The total amount of water in the droplet is calculated assuming volume water and therefore invariant with concentrations of both salt and surfactant in the droplet. Predictions with the comprehensive Gibbs partitioning model (Prisle et al., 2010) shows that this perhaps counter-intuitive condition is closely met in many droplet states, where the very surface area-to-bulk volume ratios of microscopic droplets result in insufficient surface concentrations to significantly reduce surface tension, despite nearly all surface active material in the finite-sized droplets being partitioned to 190 the surface (Prisle et al., 2010(Prisle et al., , 2011Bzdek et al., 2020).

Compressed film surface model
The partitioning model of Ruehl et al. (2016) describes the surface as a compressed film (Jura and Harkins, 1946) and assumes phase separation in a droplet between the pure organic surface layer and droplet bulk. This framework is conceptually similar to the earlier van der Waals model of Ruehl and Wilson (2014), but applies different surface equations of state. Similarly to 195 the Gibbs adsorption and simple complete partitioning models, only the organic partitioning is considered. The partitioning of surface active organic is assumed to take place into a 2-dimensional compressed film. As the droplet grows, the surface thickness decreases and eventually reaches a single monolayer, at which point the surface undergoes a phase transition to a non-interacting "gaseous" state and the surface tension reaches that of water (Forestieri et al., 2018). Typically, the compressed film model predicts activation to take place for droplets in this state (Ruehl et al., 2016).

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Droplet growth and surfactant partitioning is treated at two different levels of iteration in the compressed film model. At the outer level, the equilibrium relative humidity (RH) is determined iteratively for each droplet diameter d via Eq. (1). Water activity of the droplet bulk is calculated as the corrected mole fraction at each step of the iteration. The initially dry particles in the compressed film model are assumed to be composed of a salt core coated by a layer of organic, where both the diameters of the salt seed (D seed ) and coated particle (D p ) are known. The total amounts of all components in the droplet phase are 205 calculated with the assumption of volume additivity, both between the different components in the dry particle, and between the dry particle and water. The calculations use a different set of relations for the total amounts of salt, organic and water than the other partitioning models used in this work, in particular relating the salt and organic solid state molar volumes to the different diameters of the seed, coated and droplet diameters. For more information, see the Ruehl et al. (2016) supplement.
The inner level iteration takes place at the beginning of the outer level and calculates the fraction of organic molecules 210 adsorbed to the droplet surface, f surf , using the isotherm for equation of state (EoS) of the compressed film model where C 0 is the bulk solution concentration at the 2-dimensional phase transition, C bulk is the bulk solution concentration, A 0 is the critical molecular area, m σ accounts for the interaction between surfactants at the interface and N A is Avogadro number.
For each value of f surf , values for C bulk and A are calculated as and The droplet surface tension is calculated via an EoS that parameterizes σ in terms of molecular area (A) and therefore relates σ to the concentration of organics at the surface as where σ min is a lower limit imposed on the surface tension.
The model parameters A 0 , C 0 , m σ and σ min are acquired through a separate fitting to experimental observations of (d, RH) using Eq. (1) in the range of droplet sizes before droplet activation (the rising part of the Köhler curve) at one organic fraction (Ruehl et al., 2016), or across several organic fractions at a fixed value of RH (Forestieri et al., 2018). The values used in 225 this work were obtained from Ruehl et al. (2016) and are provided in the supplement. These parameters are assumed to be compound specific physical constants and therefore to not be sensitive to seed or coated diameters or droplet dilution state, such that they can be applied across a range of organic fractions.

Partial organic film model
The partial organic film model used here is similar to the AIOMFAC-based simplified organic film model of Ovadnevaite compositions and volumes, to describe droplets where partial surface coverage of a hygroscopic particle core (α, bulk) by a organic rich phase (β, surface) is possible. In the model employed for this study, no water is present in the film, and it is 235 assumed to coat the bulk entirely until a minimum surface thickness (δ org ) is reached and the organic film breaks, only partially covering the core phase for larger droplets. According to Ovadnevaite et al. (2017), the value roughly corresponds to an average molecular monolayer thickness, δ org = 0.16 -0.3 nm, which is similar to the lengths scale of one to two covalent carbon-carbon bonds or van der Waals radii of carbon and oxygen atoms (Bondi, 1964). For the simulations in this study, δ org was set equal to the values given by the molecular surface monolayer model of Malila and Prisle (2018), which here predicts values between 240 0.39 -0.67 nm, depending on the droplet size and the specific organic compound (figure in the supplement section S1.3).
In the version of the model implemented here, water activity in the droplet bulk was calculated using a fit to AIOMFAC (Zuend et al., 2008(Zuend et al., , 2011; AIOMFAC-web) predictions as a function of salt mole fraction in the concentration range relevant for the growing droplets. The fit can be found in the supplement section S2.3. The initial amounts of each compound in the droplets are determined via volume additivity, same as in the simple complete partitioning model of Prisle et al. (2011).
where ϕ B j is the volume fraction of component j in the bulk phase ( j ϕ B j = 1). The surface coverage parameter c S is defined and determines whether the bulk is completely (c S = 1) or partially (c S < 1) covered by the organic film. Here, V S is the volume of the surface phase at diameter d and V δ is the corresponding volume of a spherical shell of thickness δ org . The effective surface tension of the droplet is calculated as the surface area weighted mean of the surface tensions from both phases

Monolayer surface model
The molecular monolayer surface model of Malila and Prisle (2018) divides an aqueous droplet into a surface monolayer of thickness δ and droplet bulk with diameter d − 2δ. The monolayer is described as a pseudo-liquid phase with a distinct 260 composition from the droplet bulk and the monolayer composition is evaluated in terms of total amount of molecules for all components in the droplet, not just the surfactant. For each species j, the partitioning between the bulk and surface phases is calculated iteratively from an extension of the Laaksonen-Kulmala equation (Laaksonen and Kulmala, 1991) relating the droplet surface tension σ (as a function of bulk composition) to the surface composition as

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In Eq. (10), v j is the liquid (i.e. droplet surface) phase molecular volume, σ j the surface tension, and x S j and x B j the droplet surface and bulk mole fractions, respectively, all for component j. The condition of mass conservation (n T j = n S j + n B j ) is imposed on the calculation for each compound j. The total amount of water is determined iteratively at each droplet diameter by assuming mass conservation of all compounds in the droplet, together with the ternary mixture density, which is here a function of the droplet composition. Details of calculating the total amount of water are given in section S2.2 of the supplement.

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The thickness of the surface molecular monolayer is calculated as

The bulk solution model
Simulations with the different partitioning models described are compared to a model representing the droplet as a bulk solution, neglecting effects of surfactant partitioning between the droplet bulk and surface. All surfactant mass is assumed to 275 be even distributed in the bulk phase, which is equivalent in volume to the whole droplet (no separate surface phase). In the implementation used in this work, the amount of water in the droplet is solved iteratively from the ternary solution density, similarly to the monolayer model . The droplet surface tension is evaluated from a fit to the ternary data reported by Booth et al. (2009). Details of the fit are provided in the supplement section S3.1. Table 1. Methods of calculating the total amount of water (n T w ), the water activity (aw), the surface tension (σ) and the composition of droplet surface and bulk, when they differ from the pure surfactant, for the different models. Details of the activity and surface tension equations are given in the supplement.  Compound   Table 3 presents the droplet diameters, supersaturations and surface tensions at droplet activation predicted with the different partitioning models. The pure component properties used in the calculations are presented in Table 2. Corresponding results for simulations with succinic and glutaric acids as the organic component are presented in the supplement section S1, together with the required compound properties.  At w p,org = 0.5 in Fig. 1(b), the relative order of SS c values predicted with the different partitioning models remains the same as in Fig. 1(a). The discrepancies between the Gibbs and partial organic film model predictions decreases. The compressed film model predicts a distinctly different shape of the Köhler curve, with the droplet growing considerably less under supersaturated 310 conditions, compared to the predictions of the other models. The range of predicted SS c values increases compared to Fig. 1(a) and is now 0.19% supersaturation, while the predicted d c values are all within a range of 84 nm. Lin et al. (2018) reported the monolayer model to predict lower SS c values than the Gibbs model for mixed succinic acid-NaCl particles with the same dry particle size and organic mass fraction. In this study, the situation is reversed, also for succinic acid (results presented in the supplement section S1). This is most likely an effect of mixing with ammonium sulphate instead of NaCl, which could affect 315 both surface activity of the organic (Prisle et al., 2012a) in the Gibbs framework, as well as the density of the surface phase in the monolayer framework.

Köhler curves and droplet activation
At higher organic fractions, w p,org = 0.8 in Fig. 1(c), the differences between the predictions with models where all organic is assumed to be partitioned to the surface along the entire growth curve (simple model and partial organic film model) and the models which account for evolving bulk-surface partitioning (monolayer model, Gibbs model and compressed film model) 320 becomes much more apparent. The simple partitioning model predicts a significantly higher SS c than the other models, likely reflecting the strong absence of predicted solute effect in the droplets as the organic fraction increases. The partial organic film model predicts a considerably lower SS c , and also occurring at a considerably smaller d c , than the other models. The Köhler curve predicted with the partial organic film model also includes a local minimum, matching the point where the droplet surface film breaks and no longer fully encloses the bulk phase. The Köhler curves predicted with the bulk solution, Gibbs,325 and monolayer models group together with only small differences. The curve calculated with the compressed film model has a distinct shape, and the predicted SS c value falls between the monolayer and simple model predictions, but at the largest d c .
The range of predicted SS c increases to 0.5% supersaturation and the predicted d c range also increases to 199 nm.
At w p,org = 0.95 in Fig. 1(d), the results for supersaturation show similar trends as in Fig. 1(c). The most notable difference to Fig. 1(c) is that the partial organic film model predicts comparatively higher SS c in Fig.1(d) than it did in Fig. 1(c). The 330 compressed film model again predicts the largest d c at a slightly lower SS c value than the bulk solution, Gibbs and monolayer models. In addition, a significant second local maximum is visible for the curve calculated with the compressed film model.
This maximum is also present for the other surface active acids (results presented in the supplement) and we therefore made a sensitivity analysis of the model at this mass fraction (presented in the supplement section S2.4). The sensitivity analysis shows that the compressed film model predictions are quite robust with malonic and glutaric acid simulations in that the prediction 335 of the critical point is stable with respect to relatively large variation in the model parameters. Simulations for succinic acid are more sensitive than for the other compounds. The increasing trend in the ranges of predicted SS c and d c with organic mass fraction in the dry particle continues to span ranges of 1.31% supersaturation and 282 nm, respectively. In Fig. 1(d) we

Shape of droplet growth curves
Predictions with the compressed film model are distinct from the other partitioning models (see Table 3 Fig. 1(b)). This is likely due to closely similar conditions in simulations with the two models, since in the compressed film model calculations, all organic has partitioned to the surface and the surface tension is equal to that of pure water after the point of droplet activation. For the calculations with the simple partitioning model, both these assumptions are made for the whole droplet growth curve.
In the partial organic film model, complete surface partitioning of the organic component is assumed along the whole Köhler curve, but the surface tension is calculated via Eq. (9). The partial organic film model predictions agree well with the monolayer, Gibbs and bulk solution models for particles with lower organic mass fractions, with the Köhler curve falling between those 360 predicted with the Gibbs and the other two models in Figs. 1(a) and 1(b). In Fig. 1(c), the SS c values are still similar, but the partial organic film model predicts activation at a noticeably smaller droplet size than the other models. generally predicted significantly lower SS c values compared to the Gibbs and simple partitioning models but that none of the three models were able to capture the measured pollenkitt CCN activity well over the full range of particle sizes studied.

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In Fig. 1 the situation is different, as we find the Gibbs model predicting slightly lower SS c values than the bulk solution model for w p,org = 0.2, 0.5 and 0.8 (Table 3). These differences likely reflect the surface activity of the organic component. depression. The pure surface tension of malonic acid is estimated according to Hyvärinen et al. (2006) to be about 48.24 mN m −1 at 298.15 K, which is lower than any surface tension depression reported by Hyvärinen et al. (2006). Lin et al. (2018) found that for succinic acid-NaCl particles, across D p = 50 − 150 nm for mass fraction range of w p,org = 0 − 1, the monolayer model predicts slightly lower critical supersaturation compared to the Gibbs model. In Table 3, the 380 predicted SS c of the monolayer model is slightly larger than that of the Gibbs model for w p,org = 0.2, 0.5, 0.8 of malonic acid.
For succinic acid-ammonium sulphate particles in Table S2 of the supplement, the monolayer model predicts slightly higher SS c across all calculated compositions. As mentioned in relation to Fig. 1(b), this is likely due to the different salt present in the particles.  with Dp = 48 nm is included in panel (d). Note that the vertical axis scaling changes between the panels.  Table 2). The surface tension curves predicted for particles containing succinic and glutaric acids (presented in the supplement section S1) show similar relative 400 behavior to the curves predicted for malonic acid.

Surface tension
Figure 2(a) shows that for droplets formed on malonic acid particles with w p,org = 0.2, the surface tensions predicted along the Köhler curves with the monolayer, Gibbs and bulk solution models are similar, with the most visible differences at small droplet sizes. This indicates that the droplet compositions are similar with all three models, as surface tension has been evaluated from the same composition-dependent function (Eq. (S9) in the supplement). The smallest droplets are both most 405 concentrated and have the largest surface area-to-bulk volume ratios, so any differences in the representation of bulk-surface partitioning are expected to be more visible here. The droplet surface tensions predicted at activation (σ c ) are also very similar, as can be seen in Table 3. Quite to the contrary, the surface tension curve predicted with the compressed film model is very distinct. Droplet surface tension values start at a compound specific minimum surface tension value determined by the model parameter fitting (Ruehl et al., 2016), and then increase to the value for pure water towards the activation point. In Fig. 2(a), 410 the first droplet size at which pure water surface tension is reached in the droplets does not correspond to the d c value (see Fig.   1, Table 3). This deviation from the typical behavior of the predictions as seen in Ruehl et al. (2016) could be an artefact due to the extrapolation of the model parameters outside of their validity region. In the compressed film model, the fitted model parameters are assumed to be constant across varying organic mass fractions and dry particle sizes (concentrations), but for real droplet solutions mixing properties are likely sufficiently non-ideal (i.e. excess mixing properties are non-zero) that the 415 model parameters would be expected to show some variation across the mixing space. Forestieri et al. (2018) made a similar observation for oleic acid particles within their parameter fitting range, at w p,org = 0.8 and NaCl seed particles of 80 nm.
For the partial organic film model, droplet surface tensions start at the value of the pure organic compound (corresponding to complete surface coverage by the organic). The surface tension begins to increase as the droplet grows once the organic film breaks and the surface is no longer completely covered. In Fig. 2(a), all the σ c vales are within 3.1 mN m −1 of the surface 420 tension of water (Table 3) and the lowest σ c is predicted with the partial organic film model.
The results for predicted surface tensions shown in Figs. 2(b), 2(c) and 2(d) for particles with higher mass fractions of malonic acid have very similar trends between the different models as seen in Fig. 2 where activation occurs. In Figs. 2(b), 2(c) and 2(d), the activation point and the water surface tension are reached at the same droplet size. With the partial organic film model, the calculated droplet size at which the organic film breaks increases while σ c decreases with increasing w p,org , to eventually reach the pure organic surface tension value for the highest malonic acid mass fractions studied in Fig. 2(d). Of all the models compared here, both the partitioning models and the bulk solution model, the partial organic film model predicts the lowest surface tension at droplet activation, as can be seen in Fig. 2 and Table 3. solution model than with the Gibbs model. This is also the case here for all but the largest organic mass fraction, but the differences between the model predictions are considerably smaller (∼ 0.1 mN m −1 , Table 3). Figure 3 shows the surface partitioning factors of malonic acid calculated with the different models for D p = 50 nm particles 440 with organic mass fractions (w p,org ) of 0.2, 0.5, 0.8 and 0.95. The partitioning factor (n S org /n T org ) is defined as the fraction of the total amount of organic molecules in the droplet which are predicted to reside in the surface. The simple partitioning model and the partial organic film model calculations are made with the assumption that all organic is always partitioned to the droplet surface and therefore the partitioning factor is equal to unity for all droplet states. The bulk solution model has no partitioning and therefore the partitioning factor is zero. Between the three models that calculate droplet state dependent 445 partitioning (monolayer, Gibbs, and compressed film), large differences are seen in the predicted values of n S org /n T org for malonic acid. At the point of activation, the compressed film model predicts nearly all malonic acid is partitioned to the droplet surface, whereas the Gibbs and monolayer models both predict a moderate fraction (well below 20%) of all malonic acid solute in the surface. These significant differences between the frameworks correspond to very different solution states for the same Table 3. The critical droplet diameters (dc), supersaturations (SSc) and surface tensions (σc) predicted with the different models in simulations for mixed malonic acid-ammonium sulfate particles of Dp = 50 nm at 298.15 K. overall droplet compositions. Similar predictions were also observed for particles containing succinic and glutaric acids. More 450 details can be found in supplement section S1.

Organic bulk-surface partitioning
Figure 3(a) shows that for particles with malonic acid fractions of w p,org = 0.2, the monolayer model predicts stronger surface partitioning of malonic acid than the Gibbs model, especially at smaller droplet sizes. As the droplet grows, the partitioning factors predicted by the Gibbs model approach those of the monolayer model. Compared to the other two partitioning models, n S org /n T org predicted by the compressed film model is very pronounced, always above 0.9. We also note a decrease in 455 n S org /n T org to a minimum value before increasing towards unity, as expected. The minimum value is due to the unconstrained partitioning factor and has no physical meaning for the Köhler curve (Fig. 1) until the surface tension (Fig. 2) starts increasing from its minimum value (Eq. (6)). This is because the surface tension is the quantity being constrained by the parameter σ min in the framework. For Figs. 3(b), 3(c) and 3(d), the surface partitioning factor predicted with Gibbs model is slightly higher than with the 460 monolayer for all but the smallest droplets. The differences between the different malonic acid mass fractions is more noticeable for predictions with the monolayer model, as the amount of molecules in the surface is constrained by the volume of the molecular monolayer ) while the Gibbs model has no constraint on the extent of surface partitioning.
The partitioning factors predicted with Gibbs model are very similar across the droplet size range of the Köhler curves for malonic acid mass fractions w p,org = 0.5, 0.8 and 0.95 and the same can be seen for the compressed film model. At droplet 465 activation, the partitioning factors predicted here with the Gibbs and monolayer models are similar to the values reported by Lin et al. (2018) for mixed succinic acid and NaCl particles with the same dry size and w p,org = 0.5. With the compressed film model, the predicted partitioning factors are higher, but comparable to the results of Ruehl et al. (2016), where the values for malonic acid dry particles of 150 nm and w p,org = 0.96 are always larger than approximately 0.65.

470
We have compared Köhler model predictions for particles comprising moderately strong organic surfactants, using six different approaches to describe bulk-surface partitioning in the growing droplets. Specifically, we used the monolayer , Gibbs (Prisle et al., 2010), simple (Prisle et al., 2011), compressed film (Ruehl et al., 2016), partial organic film (Ovadnevaite et al., 2017) and a bulk solution models to predict possible bulk-surface partitioning effects during Köhler calculations for particles of D p = 50 nm consisting of atmospherically relevant dicarboxylic acids mixed with ammonium 475 sulphate over a range of compositions. From the Köhler calculations, we evaluated the droplet mixing state in terms of bulk and surface compositions, droplet surface tension and resulting equilibrium water saturation ratio, as well as the critical point of droplet activation from the Köhler growth curve maximum and corresponding diameter.
When the mass fraction of surface active organic is small (< 50 %), the predicted Köhler growth curves and critical supersaturation values for droplet activation are similar between the models. For particles with high mass fractions (> 80 %) of 480 organic, the full partitioning models (monolayer, Gibbs, and compressed film) all predict similar critical supersaturations as the bulk solution model for the investigated dicarboxylic acid systems, although the compressed film model predicts larger critical droplets. Despite these overall similarities, there are however large differences between the different models in the predicted degree of surface partitioning of the organic component. Between the full partitioning models, the degree of organic surface partitioning predicted with the compressed film model is significantly higher than with either the Gibbs or surface monolayer 485 models.
For all surface active organic mass fractions in the particles, the different models predict very different surface tension curves for the growing droplets. The surface monolayer and Gibbs partitioning models use the same surface tension parametrizations as the bulk solution model and all predict surface tension curves of similar shape, reflecting similar mixing states of the growing droplets. For the simple partitioning model, droplets are predicted to have a constant surface tension, while the compressed 490 and partial organic film models each predict distinct surface tension curves, reflecting the underlying assumptions regarding both the bulk-surface partitioning and surface tension equation of state. The predicted droplet surface tensions at the point of activation are comparable for all models at small organic fractions in the particles, but differences between the models increase with the organic fraction. The partial organic film model consistently predicts the lowest surface tension at droplet activation, for some particle compositions as low as the surface tension for the pure organic. The other models predict only moderate or 495 no surface tension depression in droplets at activation.
Among the models used in this work, the Gibbs, monolayer, and compressed film models evaluate the progression of the bulk-surface partitioning equilibrium with mixing state as the droplets grow, whereas the simple partitioning and partial organic film models rely on the simplifying assumption that the organic is completely partitioned to the droplet surface. We see that for particles where the fraction of organic is not too large, the latter models can still yield similar results as the comprehensive 500 models, but underlying assumptions may become increasingly misrepresentative as the fraction of surface active organic in the particles becomes larger. Regarding the comprehensive partitioning models, the Gibbs and monolayer models predict similar droplet properties at activation as the bulk solution model, due to the modest degree of organic surface partitioning at activation, which seems realistic for dilute solutions of a moderate surfactant, and because models use the same surface tension parametrizations based on independent measurements. The compressed film model on the other hand uses a surface 505 tension equation of state with parameters which are obtained by fitting to experimental droplet growth curves similar to those predicted and predicts very strong surface partitioning and surface tension depression in the growing droplets. Overall, this comparison of partitioning model predictions strongly highlights the need for experimental validation of the predictions from different droplet partitioning models across a wide range of particle mixtures, before either model is used as basis for broad generalizations of results to atmospheric processes. A recent development has been the reformulation of Köhler theory via a 510 Gibbs model to directly include water-soluble species and surfactants (McGraw and Wang, 2021). In future work, a similar comparison between different models will be done with more strongly surface active particle components.
Code and data availability.
Author contributions. SV adopted the models for the study, did the model simulations, and performed the analysis of model results with assistance from SMC, JM and NLP. SV wrote the original manuscript draft and made the visualizations with NLP and assistance from SMC