Phthalic acid (PTA; 1,2 – benzenedicarboxylic acid, >99.5 %, Sigma-Aldrich^®), terephthalic acid (TPTA; 1,4 – benzenedicarboxylic acid, 98 %, Sigma-Aldrich^®) and isophthalic acid (IPTA; 1,3 – benzenedicarboxylic acid, >99 %, Fisher Scientific^®) were used as representative compounds for the aromatic acid aerosols (AAAs, hereafter). The physical properties of PTA, IPTA and TPTA are summarized in Table 1. Aqueous solutions of PTA, IPTA and TPTA were formed by mixing 30 mg of acid in 500 mL of ultrapure water (Milli-Q or Millipore^®, 18.2 MΩ cm-1). Additionally, three internally mixed solutions of PTA and IPTA were also prepared by mixing 30 mg of dry acid mixture in 500 mL ultrapure water. The internally mixed solutions were prepared for three different mass fractions of PTA and IPTA (5:1, 1:1 and 1:5 wt/wt). To facilitate the dissolution of solute in aqueous solution, all the solutions were sonicated for 2 h in a warm water bath maintained at ∼ 40 ∘C to create a uniform suspension. The solution was subsequently cooled and maintained at 20 ∘C. Polydisperse aerosols were generated using a Collison nebulizer (TSI Atomizer 3076). The wet aerosol particles were then passed through a series of two silica gel diffusion dryers (TSI 3062) to remove moisture (to RH <10 %). The dry particles were then classified for supersaturated and subsaturated measurements.

A continuous flow streamwise thermal gradient cloud condensation nuclei counter (CCNC; Droplet Measurement Technologies, DMT; ; CCN 100) was used for the droplet activation measurements (for example, but not limited to, ; ; ; ; ) of AAAs under supersaturated conditions. Briefly described here, polydisperse aerosol was generated and dried as described in Sect. 2.1. The electrical mobility aerosol size from 8 to 352 nm was measured with an electrostatic classifier (TSI 3936, DMA 3081, and CPC 3776) every 2.25 min. The size-selected aerosols exiting the DMA were then split into two streams. A condensation particle counter (CPC; TSI 3776) samples the first stream at 0.3 L min-1 to measure total dry particle concentration (CCN), and the CCNC samples the second stream at 0.5 L min-1 and constant supersaturation to measure activated particle (droplet) counts (CCCN). A sheath flow rate of 8 L min-1 was applied to maintain a sheath-to-sample ratio of 10:1 across the experimental setup. The measurements were repeated 10 times for each supersaturation. Furthermore, the measurements were performed over supersaturations ranging between 0.6 % and 1.6 %. CCNC supersaturations were calibrated using ammonium sulfate ((NH4)2SO4, AS) aerosol (Sigma-Aldrich^®, >99.9 %). AS data used for CCN calibration are provided in the Supplement (Sect. S1).

PyCAT 1.0 () was employed for data processing, analysis and visualization of the CCN measurements. CCN size-resolved activation curves were generated at a fixed supersaturation (S) as CCCNCCN across a range of dry particle diameters (Ddry). The volume-equivalent diameters were used to represent particle sizes that were obtained by combining size-resolved particle dynamic shape factor (χ) with measured electrical mobility diameters (see Supplement Fig. S4). Multiple charging errors were removed from the size-resolved activation ratio following a combination of charge correction algorithms from and . Following this, a Boltzmann sigmoidal fit expressed as 1y=(A1-A2)1+e(x-x0)/dx-A2 was applied to the size-resolved activation ratio curve. In Eq. (), y is the dependent variable CCCNCCN; A1 and A2 are the minimum and maximum of the sigmoid, respectively; dx is the slope of the sigmoid; x0 is the inflection point of the sigmoid (generally the midpoint of the sigmoid); and x is the independent variable (Ddry). The sigmoid fit is typically scaled over a range of 0.0 to 1.0, and so x0 corresponds to the critical dry diameter (Ddry,c) at the instrument supersaturation and is physically defined as the size at which 50 % of all particles are activated.

A hygroscopicity tandem differential mobility analyzer (H-TDMA) measured droplet growth of AAAs in the subsaturated regime. The H-TDMA setup has been previously explained in detail (; ), and only a brief description is provided here. Dried polydisperse aerosol was first charged with a Kr-85 bipolar aerosol neutralizer (TSI 3081). Monodisperse charged particles with a dry diameter (Ddry) were size-selected using a differential mobility analyzer (DMA 1). The sample and the sheath flow rates were maintained at 0.3 and 3.0 L min-1, respectively (i.e., sheath-to-sample flow ratio =10:1). The size-selected particles from DMA 1 were then exposed to 95±0.46% RH using a Nafion humidification membrane (PermaPure M.H series). The humidified aerosol stream was then passed through the second DMA (DMA 2) that was equilibrated to a constant RH. DMA 2 was coupled with a condensation particle counter (CPC; TSI 3756) and operated in scanning mobility particle sizer (SMPS) mode. The median wet diameter (Dwet) of the size-resolved number concentration of the humidified aerosol stream from DMA 2 was reported. Dwet was used as the approximate final size to which the particles of size Ddry would grow under 95±0.46% RH conditions. The hygroscopic growth factors (Gf) were obtained by taking the ratio of Dwet with respective Ddry, 2Gf=DwetDdry. The RH of the H-TDMA setup was calibrated using ammonium sulfate (see Fig. S1; ). Calibration data are found in the Supplement.

A vapor sorption analyzer (VSA; TA Instruments New Castle, DE, USA) setup was used for the hygroscopicity measurements of bulk samples in the subsaturated regime. Mass change in AAAs as a function of RH (5 %–95 %) was measured at 25 ∘C. The instrument setup for the VSA has been described in detail in the literature (), and thus, the experimental procedure is briefly explained here. During each experiment, bulk samples were first dried at <1 % RH, then the RH was incremented up to 90 % with a 10 % step, followed by a 5 % step from 90 % to 95 %. A high-precision balance was used in the VSA to measure the sample mass at different RHs with a stated sensitivity of <0.1 µg. For every RH, a ≤0.1 % change in the sample mass was considered to be the standard for stabilization. The initial dry mass of AAA samples used in this measurement was typically around 1.0 mg. For each sample, a minimum of three experiments were performed. At every RH, the sample mass (m) was normalized with respect to the initial mass of the dry sample (m0). Subsequently, the mass-based growth factor was calculated as mm0.

The equilibrium supersaturation (S) can be estimated over a droplet as a function of its size (Dp) as 3S=aw,KT⋅exp⁡4σs/aMwRTρwDp, where aw is the water activity term, σs/a is the droplet surface tension at the interface, Mw and ρw are respectively the molecular weight and density of water, R is the universal gas constant (8.314 J mol-1 K-1), and T is the temperature. The water activity is mathematically expressed as aw,KT=γwxw, where γw and xw are the activity coefficient and mole fraction of water in the droplet, respectively. In traditional Köhler theory (KT), the water activity is approximated as aw,KT=xw (Raoult's law), which assumes infinite dilution and complete dissolution of the solute. Furthermore, σs/a is the surface tension of a pure water droplet. The exponential quantity is the Kelvin term that describes the curvature effect. The solute effect and curvature effect are competing effects that describe droplet growth; the solute effect accounts for the water vapor pressure drop over the droplet due to the aerosol particle, and the curvature effect accounts for the water vapor rise over the droplet due to surface tension reduction.

Traditional KT, with or without the explicit treatment of aerosol solubility, can be effectively applied for highly soluble species. However, for partially or completely insoluble species Raoult's law is substituted with adsorption isotherms to model water uptake behavior. One such isotherm is the Frenkel–Halsey–Hill (FHH) adsorption isotherm that defines water activity through multilayer water adsorption as a function of relative surface coverage (θ, or the number of adsorbed water monomolecular layers). The FHH isotherm is expressed as () 4aw,FHH=exp⁡⁡(-AFHHθ-BFHH), where AFHH and BFHH are FHH fit parameters that describe the intermolecular interactions responsible for the adsorption of water on particle surfaces. AFHH describes the interactions between the particle surface and first adsorbed water monolayer. BFHH describes the interactions between successively adsorbed monolayers. AFHH and BFHH regulate the amount of water adsorbed on the particle surface and the radial distance up to which attractive forces can contribute to adsorption of water, respectively. θ in Eq. () is expressed as Dp-Ddry2⋅Dw, where Dp and Ddry have been previously defined, and Dw is the size of the water molecule. The mathematical representation for the FHH-AT is analogous to traditional KT and combines the FHH isotherm with the Kelvin term (; ) such that 5S=aw,FHH⋅exp⁡4σwMwRTρwDp.

The FHH parameters can be empirically determined for any aerosol species from their droplet growth measurements (). For measurements in supersaturated environments, AFHH and BFHH are determined from least square minimization of the experimental data with the maxima of the FHH-AT equilibrium curves (). A higher value of AFHH implies a higher water adsorption, and a smaller value of BFHH implies stronger attractive forces over larger distances. It has been observed that BFHH has a larger influence on the shape of the adsorption isotherm and hence strongly drives CCN activation using FHH-AT (; ).

The assumptions of complete aqueous solubility or insolubility associated with KT and FHH-AT, respectively, represent two extreme possibilities of CCN activation and droplet growth. In this work, the two water activities were combined to develop a generalized “hybrid” water activity term. The droplet growth model thus obtained is called the Hybrid Activity Model, or HAM. Previous studies have discussed several other mathematical models built upon the traditional Köhler theory under different conditions. One such example is that of the solubility-partitioned Köhler theory (, ), which explicitly includes the activity coefficient (γw) of the aerosol compounds to estimate the water activity. γw≃1 in the traditional Köhler theory only under the assumption of the infinite dilution of the aqueous phase of the droplet, which holds true for several highly soluble aerosol species. For limited-water-solubility compounds, γw is calculated by treating the aqueous solubility of the compound. However, even then the contribution of the undissolved fraction of the solute to the droplet growth is not treated. Another example of a modified Köhler model is the “core–shell” model , which combines the FHH isotherm and Raoult's law in a single framework to evaluate the contribution of the insoluble and soluble component of the mixture, respectively, on droplet growth. In the core–shell model, partial water solubility is not considered for any of the mixture components. HAM builds on the concepts delineated by and and considers all particles to be a “core–shell” morphology while also treating all the components as partially water-soluble. The general mathematical representation of HAM is as follows: 6S=aw,HAMexp⁡4σwMwRTρwDp, where aw,HAM=aw,KT⋅aw,FHH, and the definitions of aw,KT and aw,FHH are provided in Sect. 3.1 and 3.2, respectively.

HAM sandwiches different phases of droplet growth for any given particle in three stages. In stage 1, HAM assumes that a particle suspended under humidified ambient conditions does not dissolve at the start of the activation process (time, t→0). That is, droplet growth at t→0 occurs entirely due to the adsorption of a water monolayer on the particle surface and can be explained using the FHH isotherm. In stage 1, 6aaw,HAM,1=aw,FHH=exp⁡-AFHHθ-BFHH. The FHH parameters (AFHH,BFHH) for any given species are determined by fitting the FHH-AT to the experimental data and can be subsequently used in the HAM framework.

Stage 2 begins as the droplet continues to grow, and more water accumulates in the aqueous phase. In this stage, the particle starts dissolving and enters the aqueous phase. The fraction of particle mass that dissolves or enters the aqueous phase depends on the solubility of the compound. Moreover, the dissolved fraction of the particle can be estimated at each step of droplet growth using the solubility partitioning concept introduced by . Briefly described here, a droplet comprises a bulk dry (undissolved) phase and an aqueous (dissolved) phase. The bulk phase can be composed of one or more internally mixed species with varying water solubility. This causes the composition and core size of the bulk phase to vary dynamically during droplet growth. The amount of water in the aqueous phase increases as the droplet grows, thereby increasing the concentration of the compounds in the aqueous phase. There is a competition for dissolution between the compounds in the bulk phase which is dependent on their solubilities. Considering a dry particle consisting of n species with limited solubility, the undissolved mass fraction of a species i (χi) during droplet growth is expressed as () 6a-1χi=1-γiχiYi,dryci,puremwmi,dryΣiχiYi,dry, where γi is the activity coefficient, ci(gH2O-1) is the solubility of the pure species, mw is the mass of water in the droplet, mi,dry is the initial mass of the pure species in the dry particle, and Yi,dry is the initial mole fraction of the pure species in the dry particle. Equation () implies that the dissolved mass fraction of the species i in the aqueous phase is given as 1-χi. A set of n coupled equations are simultaneously solved to obtain χi for all n species in the mixture. χi is then used to calculate the mole fraction of species i dissolved in the aqueous phase (xi) at any point during droplet growth. Subsequently, the KT water activity can be given as aw,KT=xw=nsns+nw, where ns and nw are respectively the number of moles of solute and water in the aqueous phase. In stage 2, the contribution of the dissolved fraction of the compound in the aqueous phase (through Raoult's law) can be combined with the undissolved fraction in the solid phase (through the FHH isotherm) to generate the overall water activity term, 6baw,HAM,2=aw,KT⋅aw,FHH=xw⋅exp⁡-AFHHθ-BFHH. Equation () highlights the main difference between the models presented by and .

Stage 3 begins when the droplet is large enough to accommodate enough water in the aqueous phase and dissolve the particle mass entirely. This point onward, the droplet growth can be explained using traditional KT. In stage 3, 6caw,HAM,3=aw,KT=xw. Equations (), () and () were combined to describe the water activity through the three stages of droplet growth in the HAM framework. Thus HAM can effectively estimate the droplet growth across a wide range of aqueous solubilities. The HAM sandwiches two extremes represented by fully soluble and fully insoluble behavior in a single framework. Indeed one can consider it to be (H)AMbidextrous and apply the concept to improve upon the single hygroscopicity parameterization (κ).

Commonly, the CCN activity and water uptake tendencies of any given compound are expressed using a single hygroscopicity parameter (κ). A theoretical κ is derived using a simple parameterization of the solute water activity term in the droplet growth model. Additionally, critical dry particle sizes can be combined with their supersaturations to experimentally determine κ. In the following subsections, the κ parameter derived from different models is explained.

The critical dry diameters (Ddry,c) at supersaturations (S) in the range of 0.6 %–1.6 % were calculated using PyCAT 1.0. At any given supersaturation, the Ddry,c for each sample was calculated from the size-resolved activation ratio. The CCN measurements for pure AAAs over a range of supersaturations are shown in Fig. S2 (Sect. S3 in the Supplement). The activation diameters determined for every sample at applied supersaturations were corrected using their dynamic shape factor. The experimental setup for shape factor measurements and the shape factor dataset for AAAs and PTA–IPTA internal mixtures are shown in Sect. S4.1 and S4.2 (Supplement), respectively. The size-resolved shape factors were then used to transform the measured electrical mobility diameters to their respective volume-equivalent diameters (; ; ). The volume-equivalent diameters along with their corresponding supersaturations were then used to estimate the experimental hygroscopicity based on traditional KT (κKT), for all the AAA samples.

The activation properties of pure AAAs (PTA, IPTA and TPTA) along with their predicted κintrinsic and κKT are summarized in Table 2. The S values versus their corresponding Ddry,c for the samples are plotted in Fig. 2a. The experimental data are represented using individual markers. The solid and dashed lines represent the KT fits using the theoretical κintrinsic. The R2 scores are provided in Table 3. PTA is observed to have the best agreement with the KT prediction (R2≈0.99). IPTA and TPTA show poor agreement with traditional KT. The lack of agreement between measurements and traditional KT predictions for IPTA and TPTA can be attributed to their significantly low aqueous solubility compared to PTA (by an order of magnitude ∼102). In addition to the predicted and measured AAA data, (NH4)2SO4 is also shown in Fig. 2a.

For compounds that are considered “sparingly soluble” or “effectively insoluble” (; Fig. 3), an explicit treatment of the compound solubility can typically improve the agreement between predicted and measured activation properties. Based on this convention, PTA would also be considered “sparingly soluble”. However, our results suggest that an explicit treatment of PTA solubility is not required. Moreover, κintrinsic is a good representation of PTA hygroscopicity. Figure 2b shows the traditional and solubility-limited KT fits for internal mixtures of PTA and IPTA using their κintrinsic. The traditional KT predicts the CCN activity of the mixture containing excess PTA (5:1 mass ratio). This suggests that the mixture dominated by PTA must have an aqueous solubility closer to pure PTA and a κintrinsic≈κKT that can be obtained using the ZSR approximation. The agreement between traditional KT fits and experimental data reduces as the mass fraction of IPTA increases in the mixture.

The application of solubility-limited (modified) KT showed poor agreement with the pure AAAs and PTA–IPTA internal mixtures (Fig. S4 in the Supplement). Modified KT overpredicted the critical supersaturation for any given dry particle size for all 6 samples. Thus, the underprediction of AAAs' CCN activity is attributed to significantly low water solubility (in the range of 10-5–10-3 vol/vol water). Furthermore, a significant droplet growth is required to facilitate κsolubility=κintrinsic when solubility dependence is included in the hygroscopicity analysis (Fig. S5 in the Supplement). The AAA solubilities are 3 or more orders of magnitude smaller compared to highly soluble species such as ammonium sulfate (0.42 vol/vol water) or sucrose (1.26 vol/vol water). Quantitatively, the AAA droplets should grow to about 6.5, 23 and 45 times the dry particle size of PTA, IPTA and TPTA, respectively, when κsolubility=κintrinsic. The required droplet growth is significantly large compared to compounds like ammonium sulfate or sucrose, for which the droplet growth is 1.2 and 1.5 times the initial particle size, respectively, when κsolubility=κintrinsic (Fig. S6 in the Supplement). All of this implies that the hygroscopicity and CCN activity of AAAs and PTA–IPTA internal mixtures is more likely a consequence of water adsorption and not aqueous solubility.

FHH adsorption theory (FHH-AT) was applied for the analysis of pure and internally mixed AAAs. Figure 3 shows the measured S vs. Ddry,c data for pure AAAs and PTA–IPTA internal mixtures. The dashed lines represent FHH-AT fits for their respective CCN activity datasets. It should be noted that agreement for the FHH-AT can be obtained for every set of CCN measurements since the FHH parameters are determined by applying power law fitting to the datasets. The empirically determined FHH parameters (AFHH,BFHH) for pure compounds and internal mixtures are summarized in Table 2.

The values of AFHH and BFHH can be used to qualitatively compare the water uptake properties of the pure and internally mixed species (Kumar et al., 2009b; Hatch et al., 2019). AFHH dictates the attractive forces between the particle surface and the first adsorbed monolayer of water. A larger AFHH implies a tendency to adsorb a higher amount of water on the particle surface. For the pure compounds, AFHH decreases in the order of PTA > IPTA > TPTA (Table 2). This suggests a declining tendency to adsorb water. Additionally, AFHH values for the pure PTA, IPTA and TPTA decrease like their aqueous solubilities (Table 1). For internal mixtures, AFHH decreases with a decreasing PTA mass fraction (5:1>1:1>1:5). This also suggests a declining tendency to adsorb water with a decrease in PTA concentration.

BFHH controls the attractive forces between the particle surface and subsequently adsorbed monolayers of water. The smaller the value of BFHH, the stronger the attractive forces over a larger radial distance from the particle surface. For the pure compounds, BFHH varies in the order of IPTA > TPTA > PTA (Table 2). This suggests that the attractive force across the adsorbed monolayers is lowest in the case of the droplets formed on IPTA particles. For internal mixtures, BFHH follows a similar trend as AFHH and decreases with a decreasing PTA mass fraction (5:1>1:1>1:5). This suggests that the attractive force across the adsorbed monolayers becomes stronger with a decrease in PTA concentration.

It can be inferred that AFHH follows the trends of solubility and is most likely controlled by functional groups, and BFHH drives overall droplet growth across different compositions and molar volumes. The results here are consistent with , who showed that the AFHH values correlated with functionalized surfaces of aerosol with the same core (polystyrene latex, PSL). This suggests that the AFHH values may play a more important role with compounds of similar molar volume and highlights the importance of functionalized groups and isomeric structures in determining overall droplet growth.

One of the major factors affecting droplet growth studied in this work is the aqueous solubility of the compound. AAAs and their mixtures used in this work possess approximately equal molar mass and densities, and hence equal molar volumes. Nonetheless, they differ in terms of their water uptake. Analysis shows that the differences in their water uptake behavior could arise due to the significant variation between their aqueous solubilities. Results in the previous subsection show that either KT or an adsorption theory (FHH-AT) can be applied for the CCN analysis of moderate- and low-aqueous-solubility species, respectively. Alternatively, the Hybrid Activity Model (HAM) that sandwiches the FHH isotherm with Raoult's law through solubility partitioning may agree well with the experimental data.

Figure 4 shows the S vs. Ddry,c measurements for AAAs and PTA–IPTA internal mixtures plotted along with their HAM fits. The dot-dashed lines represent the HAM fits for the respective CCN dataset. The calculation of the water activity term for all the samples studied in this work was done following the method described in Sect. 3.3. It was observed that KT, FHH-AT and HAM provided similar fits for samples with aqueous solubility of the order of 10-3 m3 m-3. Thus similar fits for KT, FHH-AT and HAM were observed for the samples with higher PTA mass percentage (pure PTA and the 5:1 PTA–IPTA mixture). The comparison of the goodness of fit between KT, FHH-AT and HAM can be made using the R2 scores provided in Table 3. For pure PTA and 5:1 PTA–IPTA samples, all three models provided a goodness of fit. As the aqueous solubility of the sample was decreased (1:1 and 1:5 PTA–IPTA mixtures, pure IPTA, and pure TPTA, in that order), HAM still provided an improved CCN activity prediction for the samples (R2 scores of 0.92, 0.97, 0.94 and 0.91, respectively; Table 3). FHH-AT and HAM provided similar and improved R2 scores along the decline in the aqueous solubility of the species, whereas the R2 scores corresponding to KT fits were found to decline with decreasing aqueous solubility of the samples. Moreover, the R2 scores for HAM fittings were observed to be uniformly >0.9 and generally higher than those obtained for FHH-AT.

The S vs. Ddry,c values of the AAA samples were transformed into a single hygroscopicity parameter (κ) based on KT, FHH-AT and HAM (Sect. 3.4). Figure 5 shows a closure plot between theoretical and experimental κ estimated for PTA, IPTA, TPTA and PTA–IPTA internal mixtures from KT, FHH-AT and HAM. The closure analysis provides a better understanding of the applicability of different CCN models. The shaded portion of the graph denotes a 95 % confidence interval across a 1–1 agreement line (dashed, black).

The theoretical κ for KT has been represented using size-independent κintrinsic=0.172 calculated using Eqs. (4) and (5), respectively. κKT values computed using S vs. Ddry,c measurements are plotted for each compound. For KT (solid circles), the agreement between κKT and κintrinsic decreases with a decreasing aqueous solubility of the solute. Specifically, the experimental κ lies within 95 % confidence of the theoretical κ of pure PTA, 5:1 PTA–IPTA internal mixture and 1:1 PTA–IPTA internal mixture. TPTA is the sample with the lowest aqueous solubility and hence the lowest agreement between κKT and κintrinsic.

The theoretical adsorption-based parameterization (κFHH,s) and κFHH computed from the experimental data using the FHH-AT framework are shown using solid diamond markers in Fig. 5. The κFHH,s and κFHH were estimated using Eqs. (15) and (14), respectively. It was found that κFHH values had a generally good agreement with their respective κFHH,s (R2 in the range of 0.91 to 0.99). The lowest agreement between FHH-AT κ was observed for PTA and the 5:1 PTA–IPTA internal mixture, as both likely have the highest aqueous solubilities among the studied samples. Moreover, the κFHH and κFHH,s values of IPTA and TPTA are highly consistent with each other.

The theoretical and experimental κHAM values were computed using Eqs. (19) and (18), respectively. The data points for κHAM and κHAM,s are denoted using solid squares in Fig. 5. The most important feature of the HAM-based κ framework is that it explicitly accounts for the compound solubility within the hygroscopicity parameterization. Accounting for the contribution from the solid organic phase and dissolved aqueous phase to the overall hygroscopicity of the solute generates the best agreement between the κHAM and κHAM,s values. Consequently, the R2 scores observed between κHAM and κHAM,s of the six AAA samples are >0.97. It is also important to note that κ values for AAA samples obtained from FHH-AT and HAM frameworks are smaller than those obtained using KT.

All the measurements shown in Fig. 6 were performed at a 95 % RH. Figure 6a–c show the droplet sizes (Dwet) with respect to their initial dry sizes (Ddry) for pure PTA, IPTA and TPTA. Figure 6d–f show the Dwet with respect to the Ddry for PTA–IPTA internal mixtures. The Dwet predictions based on the KT–Raoult term, FHH isotherm and hybrid water activity were derived from the parameters provided in Table 2. The Raoult model estimates (dashed black lines) for the pure and internally mixed samples were generated using their average hygroscopic growth factor (Gf; Fig. 6, Eq. ). The supersaturated average κ of 0.17 for the AAA samples was used to obtain the theoretical Dwet and Gf at given dry sizes. The R2 scores for the KT–Raoult model are summarized in Table 3. The KT–Raoult model agreed well for pure PTA and the 5:1 PTA–IPTA mixture.

The dashed red lines in Fig. 6 show the Dwet estimated using the FHH isotherm (Eq. ). The empirical FHH parameters used here were determined by fitting the FHH-AT to the supersaturated CCNC measurements (Sect. 4.2, Table 2). FHH noticeably underpredicts the hygroscopic behavior of the AAAs except for IPTA and TPTA in the subsaturated regime (R2 estimates in Table 3). This implies that the insoluble behavior of IPTA and TPTA can be represented with high certainty in the subsaturated as well as the supersaturated regime, using the FHH theory. Moreover, the KT and FHH models (that agreed for soluble compounds, PTA and the 5:1 PTA–IPTA mixture) have different droplet growth predictions in the subsaturated regime.

The dashed blue lines in Fig. 6 show the Dwet estimated using the comprehensive hybrid water activity expressions described in Sect. 3.3 (Eq. ). Again, the hybrid water activity requires the empirical FHH parameters obtained by fitting FHH-AT to the supersaturated CCNC measurements (Table 2) and the aqueous solubility of the compound to account for the dissolved fraction of solute (Table 1). The hybrid water activity replicated the subsaturated water uptake of all six of the AAAs with high certainty (R2 estimates in Table 3). This is due to the explicit consideration of both compound solubility and water adsorption to describe the droplet growth process. Notably, the hybrid water activity is similar to either the KT–Raoult or the FHH isotherm depending on the compound solubility. For sparingly soluble samples (e.g., pure PTA), the KT–Raoult and hybrid water activity generated similar fits (R2 of 0.938 and 0.948, respectively). For effectively insoluble samples (e.g., pure TPTA), the FHH isotherm and hybrid activity generated similar fits (R2 of 0.998 and 0.999, respectively).

The sub- and supersaturated analyses are consistent with the equilibrium curves for the pure and internally mixed AAA samples. Figure 7 shows droplet growth predicted using KT, FHH-AT and HAM corresponding to one of the experimentally determined Ddry,c. The predicted critical supersaturations (Sc) are also shown in the plots. KT-predicted Sc values deviate significantly (>10 %) from the experimental Sc, as the aqueous solubility of the solute decreases. This is because KT for the structural isomers assumes similar droplet growth (κintrinsic is ∼ 0.17). However, FHH-AT and HAM require higher supersaturations and are less CCN-active, and therefore the points of activation are shifted upwards and to the left. At a given relative humidity (RH) <100 %, the KT-derived Dwet is found to be larger than those predicted using either FHH-AT or HAM. This is consistent with the models and experimental data at 95 % RH shown in Fig. 6. KT-based Dwet was found to be close to the experimental Dwet for pure PTA and the 5:1 PTA–IPTA mixture, whereas Dwet values from FHH-AT and HAM were found close to the experimental Dwet for the remaining solutes. After critical activation, there may be a jump from a water-adsorption-driven droplet growth to one driven by complete dissolution of the solute (vertical jump in green line from blue to red). This is prominently seen in PTA but is not as evident in TPTA. Furthermore, multiple transitions are observed in internal mixtures.

VSA measured the water uptake of the three AAA compounds in the subsaturated regime. None of the AAAs showed significant water uptake (with mass growth factors smaller than <1 %) even at high RH (95 %) (Fig. 8). It should be noted that the VSA measurement uses materials in the range of micrometers to millimeters. Thus, the observed κHAM in Eq. () decreases with increasing diameter and eventually approaches zero. The results across different particle measurement platforms are consistent with the hygroscopicity parameterization that is particle-size-dependent.

It should be noted that in this work, we explicitly account for particle shape morphology (dynamic shape factor) and correct the electrical mobility diameters to volume-equivalent diameters as described in . Shape factors were measured and computed for all samples studied (Fig. S3). Over the mobility diameters of interest (from 50 to 150 nm), the dynamic shape factor values were found to range from 1.00 to 1.08 and were therefore within 10 % of 1.00. This suggests that the AAA samples studied in this work are composed mainly of spherical particles. The application of the dynamic shape factor of aerosols composed of fractals and agglomerates such as black carbon to the transition from soluble to sparingly soluble activation must be considered in future work.