Droplet hygroscopic growth and activation are calculated using a framework based on the Köhler equation in the form 1SS[100%]≡pwpw0-1=awexp⁡4σνwRTd-1. Equation () describes the equilibrium water vapor supersaturation (SS) over a spherical solution droplet as a function of its diameter (d), pw is the equilibrium partial pressure of water over the solution droplet, pw0 is the saturation vapor pressure over a planar surface of pure water, aw and σ are the water activity and surface tension of the droplet solution, νw is the partial molar volume of water in solution, approximated as the ratio of pure water molar mass and mass density Mw/ρw, R is the universal gas constant, and T is the Kelvin temperature.

The solution droplet is formed as water condenses onto an initially dry particle. Only the dry particle itself, before any water uptake has occurred, is here referred to as a “particle”, and dry-particle diameters (as well as other properties) are designated by upper case symbols, as Dp. After water uptake, the wet aerosol particle is referred to as a (solution) droplet, and corresponding diameters (and other properties) are designated by lower case symbols, as d. For each dry-particle size and composition, the critical supersaturation (SSc) is evaluated as the maximum of the equilibrium droplet growth (Köhler) curve described by Eq. (). The droplet size where SSc occurs is here referred to as the critical droplet diameter dc, or the point of activation. When droplets have grown past their respective critical point to sizes d>dc, they are described as activated cloud droplets. At earlier points d≤dc on the Köhler curves, droplets are considered to be in the process of activating.

The water activity (also called the Raoult term) describes the suppression of equilibrium water partial pressure over an aqueous solution by dissolved solutes, compared to the saturation vapor pressure of pure water, as 2pw=awpw0=γwxwpw0. Here, xw is the water (mole or mass) fraction in solution, and γw is the corresponding (mole- or mass-fraction-based) activity coefficient of water. The exponential (or Kelvin) term describes the enhancement of vapor pressure over the convex side of a curved droplet surface, compared to a planar surface of the same liquid, and depends explicitly on the droplet surface tension. Both water activity and Kelvin term are functions of droplet composition, determining xw and thus γw=γw(xw) and aw=γwxw, as well as any concentration-dependent change in droplet solution surface tension from the pure water value, Δσ=σw-σ.

The total (T) amount of solute in the growing droplets remains constant and is determined from the dry-particle compositions in terms of relative mass fractions Wp,s of each solute component s, where ∑sWp,s=1. Assuming volume additivity in spherical particles, the total dry mass of each solute component is then given from their bulk mass densities ρs as 3msT=Wp,sπ6Dp3∑jWp,sρj-1.

In calculations for NAFA–NaCl particles, values of ρNaCl=2.165 and ρNAFA=1.6gcm-3 were used. In the absence of an exact value, the NAFA density is assumed to be similar to that of Suwannee River fulvic acid, another common reference humic substance . The droplet temperature is assumed to be T=303 K, reflecting the range of effective temperatures from about 298–313 K in the CCN counter between measurements at different supersaturations (T. B. Kristensen, personal communication, 2012). The sensitivity of calculations to the assumed value of ρNAFA and variations in experimental droplet temperature is discussed in Sect. S6 of the Supplement.

At each droplet size, the total amount of water is calculated by assuming volume additivity also of water and dry-particle components within the droplet phase as 4mwT=ρwπ6(d3-Dp3). The total droplet composition {xiT}=xT is then given in terms of the mass fractions of each component i=(w,s) as 5xiT=miT∑jmjT.

When surface-active material adsorbs at the droplet surface, it leads to a partitioning of the total mass of surface-active solute msftT between the surface (S) and bulk (B) phases of the solution droplet. This partitioning depletes the droplet bulk concentration, compared to the total concentration xsftB<xsftT, and changes the overall bulk composition {xiB}=xB, compared to the total composition of the droplet xB≠xT. To calculate the resulting amount of solute in the droplet bulk (and surface), a partitioning model is used, based on the Gibbs adsorption equation in the form 6RT∑imiTMi∂ln⁡(ai)∂msftB+A∂σ∂msftB=0, where A=πd2 is the droplet surface area, miT and Mi are the total and molar mass of each droplet component i, ai is the activity of each component in the droplet solution, and msftB is the mass of surface-active (i.e., partitioning) species in the droplet bulk. For chemically well-defined components, here water and NaCl, Mi are well-known quantities. For NAFA, an average molar mass of M‾NAFA=4266 g mol-1 is assumed, according to the experimentally based estimate of . This corresponds to an assumption of the average mass unit of surface-active solute partitioning between the bulk and surface of the solution. The sensitivity of calculations to variations in the assumed value of M‾NAFA is discussed in Sect. S6 of the Supplement.

Non-ideal (ai≠xi, γi≠1) solution effects present in the droplets are taken into account via the composition-dependent droplet water activity aw, which also enters directly into Eq. (), evaluated from an experimentally based parametrization as described in Sect.  below. Activities for non-volatile solutes are challenging to measure directly but can in binary solutions be inferred from measurements of the solvent activity via the Gibbs–Duhem equation. Since this is not possible for ternary and higher order mixtures, aNAFA and aNaCl are approximated as the corresponding mass fraction concentrations. Previous estimates for ternary surfactant–NaCl aqueous mixtures showed that this assumption had only minor effects in Köhler calculations .

The bulk–surface partitioning equilibrium is solved iteratively from Eq. () at each point along the Köhler curve, corresponding to a given droplet size d and total composition xT. Constraints are applied in terms of mass conservation for all droplet components, miT=miB+miS, and by choosing the position of the Gibbs surface to yield the droplet bulk-phase volume equal to the total droplet volume V=πd3/6 .

Evaluation of the Köhler curves from Eq. () in connection with a partitioning model requires fully continuous composition-dependent functions for surface tension and water activity of the growing droplets, with respect to independent variations in the amounts of each component in the droplets. By formulating both the Köhler (Sect. ) and partitioning (Sect. ) models on a mass basis, quantitative descriptions with respect to variations in the different solution components can be applied using mass-based concentrations, without specifically resolving the chemical species comprised within each component. This also enables continuous thermodynamic functions to be used for chemically unresolved fractions of the aerosol mixture, such as the NAFA component in the mixed NAFA–NaCl particles considered in this work. Mass-based concentrations of different aerosol fractions are often readily determined from experimental parameters related to aerosol characterization or sample preparation.

In the present implementation, both σ and aw are described as functions of the droplet bulk composition xB, which is the basis for using the Gibbs adsorption equation in the form given in Eq. (). At equilibrium, the surface and bulk compositions of a given surface-active substance are related via the concentration gradient resulting from its surface activity and mass conservation within the solution, just as the respective activities of water and all other components are by definition the same in all phases of the solution. A major advantage of using bulk-composition-based relations for solution properties is that these can be obtained from conventional measurements for macroscopic solutions, where the bulk-phase composition is readily constrained in terms of the total dissolved mass. Although the surface tension of a solution may also be given in terms of the surface composition e.g.,, the absolute surface composition is highly challenging to quantify by direct, independent measurements (e.g. ). Such a formulation is therefore less useful for self-contained, predictive modeling.

The partitioning model is the key to applying composition-dependent properties obtained for macroscopic systems to finite-sized droplets. The bulk (and surface) concentrations of surface-active droplet components change with droplet size along the Köhler growth curve in response to both dilution, as the droplet grows, and changing bulk–surface partitioning, for example expressed in terms of the surface partitioning factor msftS/msftB, with changing ratio of the droplet surface area and bulk volume A/V. When partitioning of surface-active components change, the relative mass fractions of each solute component s in the droplet bulk phase 7ws=msB∑jmjB, where msB is the mass of solute in the droplet bulk and ∑sws=1, therefore change in a continuous fashion from the corresponding dry-particle composition given by Wp,s. The effect of bulk–surface partitioning is to move the droplet bulk (and surface)-phase mixing state in the concentration domain, relative to the total mixing state of the droplet solution or to a macroscopic solution with the same composition. As a consequence, continuous parametrizations with respect to independent variations in all component concentrations are needed to evaluate droplet properties over the full range of mixing states realized during droplet growth and activation.

The Köhler and partitioning models are both fully predictive and can in principle be used with any functional form and number of fit parameters used to describe the mass-based composition variation of thermodynamic properties. For this work, parametrizations of both σ and aw were obtained here by fitting continuous functions of NAFA and NaCl aqueous concentrations to the experimental surface tensions and water activities reported by and for macroscopic solutions, using the least squares method.

The framework presented in Sect.  and  enables predictive, thermodynamically consistent Köhler calculations of droplet growth and activation for chemically complex and unresolved surface-active aerosol mixtures. In this framework, droplet growth and activation is influenced by several simultaneous processes in the aqueous phase, including dilution, bulk–surface partitioning, reduced droplet surface tension, and non-ideal water activity. To highlight the interplay and relative roles of these different underlying mechanisms, Köhler calculations for NAFA–NaCl particles using the full mass-based partitioning model are compared to several simplified predictive approaches (outlined below and summarized in Table ). Two frameworks are used, which consider bulk–surface partitioning in a simplified and not thermodynamically consistent way. Of these simplified partitioning frameworks, one (labeled I, as explained below) furthermore considers surface tension reduction in the droplet, however also in a not thermodynamically consistent way, whereas one (labeled S) does not consider reduced droplet surface tension. Additionally, two frameworks are used which do not consider bulk–surface partitioning in the droplets. These bulk solution models are thermodynamically consistent, but the application of thermodynamic relations derived from macroscopic measurements to describing microscopic droplets using these models is not. Of the two bulk models, one (labeled B) considers reduced droplet surface tension, and one (labeled K) does not.

The simplified predictive frameworks are implemented using the same mass-based Köhler model (Eq. ) as the full partitioning model presented in this work (labeled P). For each NAFA–NaCl dry-particle composition and size, cloud droplet activation is therefore calculated from Eq. () according to five different representations:

(P)

The full partitioning model. The thermodynamically consistent model presented in this work (Sects. , , and ) considers both reduced droplet surface tension and non-ideal water activity. For each droplet size, the NAFA bulk–surface partitioning equilibrium is solved iteratively from Eq. () to determine the droplet bulk-phase composition {xiB}=xB, from which droplet surface tension, σ=σ(xB), and water activity, aw=aw(xB), are evaluated according to the concentration-dependent ternary parametrizations given in Eqs. () and (), respectively. Calculations using this comprehensive thermodynamic model serve as a benchmark for evaluating the effects surface activity of chemically complex, unresolved NAFA in cloud droplet activation, as well as for the performance of the simplified representations.

Modeled critical supersaturations (SSc) as functions of dry-particle diameter (Dp) are presented in Fig.  for particles with dry NAFA mass fractions (Wp,NAFA) of 0 % (blue), 20 %, (green), 50 % (red), 80 % (purple), and 100 % (black) relative to NaCl. Results of Köhler calculations with each of the four models (P), (S), (B), and (K) are shown in panels (a), (b), (c), and (d), respectively, together with the experimental values for particles with equivalent dry compositions reported by . Error bars on the experimental data are estimated as ±1 standard deviations on measured SSc, as reported by . Results for model (I) are shown in Sect. S1 of the Supplement.

All representations of NAFA surface activity give similar results in the limit of pure NaCl particles, as expected. For each model, the predicted CCN activity decreases (SSc increases) with increasing Wp,NAFA for a given particle size, in agreement with the experimental trend. This shows that upon varying the dry-particle composition from pure NaCl to pure NAFA, any effect of decreased droplet surface tension at the point of activation, from the presence of surface-active NAFA in the droplet phase, cannot overcome the simultaneous increase in water activity. The droplet water activity (Eq. ) may increase from a potential combination of (i) increasing droplet non-ideality (specifically leading to increased γw>1), (ii) depletion of solute from the droplet bulk phase due to NAFA surface partitioning, and (iii) the much higher average molar mass of NAFA, compared to pure NaCl (both leading to increased xwB). Several studies have previously observed the same trend, experimentally or in model calculations, for aerosol systems comprising both simple strongly surface-active molecules, such as SDS and fatty acid sodium salts , as well as complex macromolecules and surface-active mixtures , in various mixing ratios with inorganic salts.

The overall good performance of calculations using the mass-based full partitioning model (P) with respect to measured CCN activity is reassuring in terms of the ability to capture relevant properties of the activating droplets within the comprehensive thermodynamic description. Each of the models (P), (S), and (K) describe the experimental data fairly well, except in the case of pure NAFA particles, where the partitioning models (P) and (S) underestimate CCN activity well outside the reported experimental uncertainty. In the absence of hygroscopic NaCl, predictions with model (S) correspond to condensation of water into a pure aqueous droplet phase in the presence of insoluble material, which is adsorbed at the droplet surface without attaining full coverage. Calculations with the insoluble surfactant model (I), shown in Fig. S1 in the Supplement, rely on a similar assumption, but here the reduced surface tension brings predictions somewhat closer to experimental values for pure NAFA particles, compared to (S). To reconcile predictions of model (I) with measured SSc for pure NAFA particles would however require much stronger surface tension reductions than the 5 %–20 % included in the present calculations, at the expense of increasingly poor agreement with experimental values for all other particle compositions. In general, it is clear that model (I) does not represent experimental CCN data well across the full range of NAFA–NaCl particle sizes and compositions investigated. This suggests that discrepancies observed for the other partitioning models (P) and (S) for particles with the highest mass fractions of NAFA cannot be attributed to surface tension effects alone. It is possible that relatively small amounts of hygroscopic impurities could have been present in the experimental NAFA aerosol mixture and thus enhancing measured CCN activity, as first described by . The model sensitivity analysis presented in Sect. S6 of the Supplement shows that even 3 % by mass of impurities in the NAFA mixture with hygroscopic properties corresponding to those of NaCl would be sufficient to reconcile the calculations of model (P) with experimental data for pure NAFA particles. For the simple partitioning model (S), 5 % by mass of such impurities could similarly reproduce the measured CCN activity, whereas the agreement with experiments for both bulk solution models (K) and (B) decreases when assuming impurities in the NAFA mixture.

A prominent feature of Fig.  is how Köhler calculations using the bulk solution representation (B) with reduced droplet surface tension clearly and consistently underestimate experimental critical supersaturations for all particle sizes and compositions. Similar observations have been made in several previous studies at both sub- and supersaturated conditions, for particles comprising both chemically simple and complex surfactants and for both pure surfactant particles and in various mixtures with inorganic salts . The present results confirm that bulk–surface partitioning of strongly surface-active aerosol components must be taken duly into account if the impact on droplet surface tension is considered in predictions of CCN activity. The mass-based partitioning model (P) presented in this work enables thermodynamically consistent predictions with consideration of both bulk–surface partitioning and reduced droplet surface tension, also for chemically complex and unresolved surface-active aerosol components, in addition to the simple aerosol systems that have previously been described with such comprehensive approaches. Model (K) tends to overestimate mixed NAFA–NaCl experimental CCN activity slightly more than the partitioning models (P) and (S), further suggesting that NAFA bulk depletion from surface partitioning may indeed have a more significant impact on decreasing CCN activity than surface tension reduction has on increasing it.

originally compared their measured CCN activities to two simple bulk solution Köhler models based on similar assumptions as models (B) and (K) used here, with some differences in the actual model implementation. In the present work, calculations with models (B) and (K) are for consistency made with the continuous ternary surface tension and water activity parametrizations presented in Eqs. () and  (), even if these are not strictly needed in absence of bulk–surface partitioning calculations. used simpler parametrizations, with a 1-dimensional composition domain, which are not continuous with respect to variations in the relative NAFA–NaCl mass fractions in solution (wNAFA and wNaCl) and have slightly different functional forms than the parametrizations used here, even at the lines of intersection. Furthermore, the surface tension parametrizations used by are made for data points corresponding to measurement times t=0 s after the formation of the surface and therefore based on higher surface tension values for a given solution composition, compared to the data from measurement times t=600 s used in this work (see Sect. S5). As higher macroscopic surface tensions correspond to lower surface activity of NAFA, the predictions of SSc by are similarly biased higher, incidentally partially mimicking the elevated surface tension in droplets that would occur due to surface partitioning. However, as clearly seen from Fig. 7 of , this effect still is not sufficient to bring the Köhler predictions even close to agreement with their measured CCN activities.

Similar to the model (K) used here, also find that a basic Köhler model, where effects of NAFA surface activity are completely ignored, gives good agreement with measured CCN activation for NAFA–NaCl particles with up to 50 % NAFA, as well as for pure NAFA but not for particle mixtures with 80 % NAFA. This seemingly counterintuitive result for particle mixtures comprising strongly surface-active material with significant ability to lower surface tension in macroscopic solutions has been observed also for simple, strong surfactants, such as SDS and C8–C12 fatty acid salts . On the other hand, several recent studies have shown properties for both simple and complex aerosol mixtures, including secondary organic aerosol , limonene-derived organosulfate products , marine primary organics , and water-soluble pollen extracts which are consistent with an enhancement of aerosol water uptake by surfactants, at least partly due to reduced surface tension. This suggests that model (K) is too simple to fully capture CCN activity of different types of surface-active aerosol components and in all mixing states and that, generally, a full partitioning model is needed for robust predictions. In cases where the basic Köhler model (K) gives good agreement with experimental CCN activity for NAFA–NaCl particles, it also closely traces predictions with the mass-based full partitioning model (P).

Figure a presents critical supersaturations as a function of NAFA mass fraction Wp,NAFA in 50 nm dry particles, calculated with each model described in Sect. . Experimental values from for particles with similar dry sizes are shown for reference: immediate comparison is not always possible along the particle composition dimension because used an experimental setup that scanned a set of preselected supersaturations, rather than particle sizes and therefore did not measure the exact same particle sizes for each dry-particle composition. Corresponding to each value of SSc shown in panel (a), the other panels in Fig.  show, for the same 50 nm particles, also calculated at the critical point dc of the Köhler curve, (b) the individual Kelvin and Raoult terms, (c) the droplet diameter growth factor GFc=dc/Dp, and (d) the droplet surface tension σc. Qualitatively similar results were obtained for other dry-particle sizes (not shown). Differences between the models are more clearly seen for smaller particles, which generally activate for smaller critical sizes, corresponding to smaller critical growth factors, with more concentrated droplet compositions and larger surface area-to-bulk volume ratios, introducing more pronounced effects of surface partitioning for a given total composition.

Results in Fig. a illustrate how the full partitioning model (P) traces experimental CCN activity well, but they also highlight the strong sensitivity of model predictions to NAFA mass fraction for particles with the highest Wp,NAFA. As discussed in detail below, this sensitivity is due to predicted droplet states with highly depleted bulk phases, for which the partitioning of a single mass unit of surface-active solute leads to large changes in bulk (and surface) mass concentrations of the finite-sized droplets. A similar phenomenon was noted by and is further exaggerated for the present particle compositions due to the very large average molar mass of species comprised in the NAFA mixture .

Figure a shows that the simple partitioning models (S) and (I), as well as the basic Köhler bulk model (K), all predict very similar properties of activating droplets to those of the full partitioning model (P), for essentially the entire range of NAFA mass fractions in the particles. A similar close agreement between basic Köhler predictions and models that include comprehensive partitioning thermodynamics has also been observed in previous studies for particles comprising simple, well-defined, strong surfactant–salt mixtures . For NAFA particles, the agreement between models here extends to even larger surfactant fractions (up to about 80 % by mass of NAFA) than seen in the earlier studies. Furthermore, not only critical supersaturations, but also the individual Kelvin and Raoult terms at the point of droplet activation in Fig. b and predicted droplet sizes at activation, as represented by the droplet growth factors in Fig. c, agree well between these models. This was not immediately expected, since model (S) was proposed to emulate specifically SSc, not necessarily other droplet properties at the critical point , and since NAFA is seen from experimental data to give significant surface tension reductions in macroscopic solutions , contrary to the assumptions of both models (S) and (K). Figure d shows that the full partitioning model (P) predicts surface tensions of activating droplets, which are very close to that of pure water and within the range of constant values assumed in models (S), (I), and (K) for nearly the entire dry-particle composition range. The similarity between models of all activation parameters indicates that the underlying assumptions of both the simple partitioning representations and the basic Köhler model are reasonably representative of the critical droplet states for the NAFA particle systems in question, as predicted by the comprehensive model (P).

It is important to keep in mind that properties in Fig.  predicted for different Wp,NAFA correspond to critical droplets of different sizes dc. Variations of critical droplet properties with Wp,NAFA only represent states corresponding to the maxima of individual Köhler curves for each particle composition and therefore do not a priori reflect continuous variation in all underlying droplet properties. In addition to the overall variation in total solute composition given by Wp,NAFA, also the droplet dilution state, as seen by the varying activation growth factors GFc (panel c), and the ensuing size-dependent bulk–surface partitioning of NAFA, as predicted with model (P) according to the changing A/V and dilution state, vary between the critical droplets. Total composition, dilution, and partitioning each affect the droplet bulk solution composition, from which the droplet surface tension (panel d) and Kelvin and Raoult terms (panel b) are evaluated.

Figure  shows the Köhler curves for selected dry-particle compositions, Wp,NAFA, of (panel a) 0.20, (panel b) 0.50, (panel c) 0.80, and (panel d) 0.95, calculated with each model for the 50 nm particles described in Figs.  and  S2. The droplet surface tensions evaluated along the Köhler curves are shown in Fig. , and the corresponding NAFA surface partitioning factors mNAFAS/mNAFAB and water activities of the growing droplets are presented in Figs. S3 and S4 of the Supplement. All the predictive models used in this work produce meaningful Köhler curves and other solution properties for the growing droplets. For all curves representing calculated properties of the growing droplets, the respective critical points of droplet activation predicted with each model (corresponding to SSc presented in Figs.  and S1) are indicated with asterisks.

The close agreement between partitioning models (P), (S), and (I95) and the basic Köhler model (K) for calculated droplets properties at the point of activation dc (Fig. ) is evident along the full Köhler curves in Fig. , except for particles with the highest NAFA fraction Wp,NAFA=0.95 (panel d). Here, clear differences in the shapes of the calculated Köhler curves can be seen, when the representations of the surface-active NAFA component become more prominent relative to the still smaller amounts of hygroscopic NaCl in the droplets. For both models (B) and (I80), the predicted droplet surface tension in Fig.  is significantly reduced during droplet growth and well beyond the point of activation. At a given droplet size d, the equilibrium supersaturation SS is higher for representation (B) than for (I80), when the concentration-dependent droplet surface tension is higher in (B) than the fixed surface tension of σ=0.80σw in (I80), and vice versa. For models (P), (S), and (I95), the order of the Köhler curves, as well as the values of SSc at the critical points, do not simply follow the relative magnitudes of the droplet surface tensions at each d. In calculations with the full partitioning model (P), surface tension is reduced for the more concentrated droplets at the earlier stages of the Köhler growth curves, but only for the highest Wp,NAFA=0.95 in Fig. d is the reduction maintained until the critical point. Even for droplets much smaller and more concentrated than at the point of activation (d<dc), surface tension is never reduced by more than about 10 % from the pure water value in the comprehensive model (P). This reflects the very small amounts of NAFA solute remaining in the finite-sized droplet bulk phase, when bulk–surface partitioning is taken into consideration.

The degree of NAFA partitioning to the surface is already significant at the much higher total droplet concentrations at the early stages of the Köhler curves, before activation takes place: the amount of NAFA in the droplet surface is generally 2–3 orders of magnitude higher than the amount left in the bulk (Fig. S4). Typically, mNAFAS/mNAFAB is even higher for droplet sizes d<dc than at the critical point, reflecting that partitioning has an even greater effect on the droplet bulk concentration of NAFA than the lower degrees of dilution in the smaller droplets. As a consequence, even if NAFA is able to significantly reduce surface tension in macroscopic solutions, the same impact on surface tension is not seen in microscopic activating droplets, even at corresponding total droplet concentrations. The differences between surface tension variations predicted with Eq. () in macroscopic and microscopic droplet solutions due to changes in bulk composition from NAFA partitioning are further discussed in Sect. S4.

Figure S4 shows how the water activity increases as droplets grow and dilute along the Köhler curves. Predicted droplet aw only differs significantly between the different models when NAFA fractions in the dry particles are very large. The water activity is significantly reduced early in droplet growth but – except for the largest Wp,NAFA – is very close to 1 when droplets reach dc. As NAFA is much less hygroscopic than NaCl, and the resulting reduction of water activity is even smaller for submicron droplets due to strong bulk–surface partitioning, aw predicted with the comprehensive model (P) is governed by the hygroscopic salt, and variations along the Köhler curve mainly reflect the changing droplet dilution state.

Results of the mass-based comprehensive model presented in this work show that the cloud droplet growth and activation behavior of chemically complex and unresolved NAFA mixtures is driven by very similar mechanisms as those found in earlier works for well-defined, strong surfactants in simple binary and ternary aqueous droplet mixtures: the comprehensive thermodynamic partitioning model (P) consistently during droplet growth and at the point of activation predicts the vast majority of NAFA in the droplet to be depleted from the bulk by partitioning to the surface. Except for particles with the very highest dry mass fractions, NAFA contributes next to none of the solute in the droplet bulk phase, which governs the equilibrium surface tension and water activity of the droplet. Therefore, even if NAFA effectively reduces surface tension in macroscopic solution, the droplet surface tension predicted at activation is barely reduced at all, for particles with less than 80 % NAFA. For even higher NAFA fractions, the maximum surface tension reduction only amounts to around 10 mN m-1 from the pure water value across the investigated dry-particle size range.

The basis for this seemingly counterintuitive droplet state is the comparatively very large surface areas (A) relative to the finite volume (V) of the bulk for microscopic activating cloud droplets, compared to a macroscopic solution. This has previously been demonstrated in predictive thermodynamic modeling and experiments for chemically simple and well-defined strong surfactants. Even when essentially all the surface-active material is adsorbed at the droplet surface, the finite-sized solution droplets do not comprise enough surface-active solute altogether to generate sufficient surface (or bulk) concentrations to significantly reduce droplet surface tension. The equilibrium bulk-to-surface concentration gradient governing the adsorption of a surface-active substance is therefore established for a droplet solution state of dilute concentrations in both bulk and surface phases. The pronounced bulk-phase depletion further diminishes the low intrinsic hygroscopicity of the NAFA mixture, as seen in terms of impact on bulk water activity, to yield a nearly vanishing effective hygroscopicity in activating droplets. This leads to the strongly surface-active component effectively behaving in small droplets as an insoluble and only slightly or even non-surface-active substance.

Several bulk–surface partitioning models have been presented in recent years and deployed in connection with Köhler theory to evaluate CCN activity for various surface-active aerosol systems see, e.g.,. The models are each based on different assumptions and required forms of thermodynamic information. Therefore, a quantitative comparison of the performance of the different models for a common aerosol system under identical conditions is not straightforward. recently made such a comparison of Köhler model predictions for particles comprising a set of well-known moderately surface-active dicarboxylic acids, using five different partitioning models. Common for the partitioning models is that bulk and surface phases of a growing solution droplet with changing A/V are connected via a composition-dependent surface tension equation of state and an adsorption isotherm for the surface-active components. With specific choices of equations and input parameters, each model is tailored to capture certain properties of the surface-active aerosol components and has been successfully applied to describe cloud droplet activation of selected aerosol mixtures. However, altogether the different models have so far only been applied to a limited set of aerosol systems, and neither of the models fully captures all interactions of the surface-active solute across varying droplet states. Even if none of the models are currently broadly applicable to general aerosol mixtures, conditions, and modeling purposes, each provides valuable insight into the role of surface activity in growing and activating droplets.

Most of the frameworks described in the following have been presented in simplified versions, subject to various approximations. Curiously, several studies note how the simplified version of a given model performs equally well against experimental data as the full model, as was also seen here for the simple model (S).

Thermodynamically consistent, comprehensive Köhler models including bulk–surface partitioning are generally not suitable or computationally feasible for applications to large-scale frameworks . However, with simplified models, such as those described above, where the partitioning equilibrium does not require iteration for each droplet size, it is possible to explicitly consider effects of surface activity in large-scale simulations. So far, such studies have been sparse. The simple partitioning models of and have been successfully applied in a global climate model and that of in a cloud parcel model .

Results of the present work show that both the simple partitioning representation (S) and the basic Köhler model (K) may represent CCN activity well for strongly surface-active mixtures with low intrinsic hygroscopicity, such as those with low solubility or high mean molar mass similar to NAFA, in alignment with the findings of . Based on the good performance of model (S) with respect to both experimental CCN data and comprehensive calculations with the full Gibbs partitioning model for particles comprising the complex NAFA mixture, this model is a promising candidate to describe CCN potential of other strongly surface-active unresolved aerosol mixtures, including atmospheric aerosols. When reduced droplet surface tension prevails, the CLLPS will likely provide a better representation of the effects of aerosol surface activity, provided that a suitable proxy system can be identified to describe the aerosol mixture. To enable this, the applicability of various proxy mixtures across a range of atmospherically relevant aerosol compositions and ambient conditions must be carefully established. This could be achieved by comparing predictions with the CLLPS model for the chosen proxy system to self-contained, thermodynamically consistent modeling for the actual complex aerosol mixture, which is possible with the comprehensive mass-based framework presented here. For conditions where effects of intrinsic hygroscopicity and non-ideal solute interactions are more prominent than surface activity and reduced surface tension of an aerosol mixture, the basic Köhler model (K) with inclusion of a suitable thermodynamic representation of droplet non-ideality may still be preferable for large-scale applications, due to its ability to represent overall CCN activity and simplicity of implementation.

When the position of the partitioning equilibrium is highly sensitive to changes in conditions, due to moderate surface activity and non-ideal interactions in the aerosol mixtures, a comprehensive thermodynamic model is generally needed to fully capture variations in the droplet state with particle size and composition and in response to droplet growth and dilution. To explore and firmly establish which features are driving the impact of surface activity in cloud microphysics under various ambient conditions, thermodynamically consistent predictive modeling using independently derived mixture-specific model parameters will be needed for a wide range of atmospherically relevant aerosol systems. The presented mass-based framework allows this directly for chemically complex and unresolved mixtures, including actual atmospheric aerosol samples, without retrofitting of model parameters or assuming a proxy mixture to represent the actual system.