The investigated aerosols were produced by nebulizing an aqueous solution of pure ammonium sulfate (99.9 % pure, ChemCruz) and D-glucose (99.55 % pure, Fisher) at ∼0.01 % weight concentration or their mixture with AS : Gl=4:1 and 1:1 weight ratios.

The hygroscopic properties of size-selected aerosol particles were measured in the 2.0 %–99.6 % RH range with a high-humidity tandem differential mobility analyzer (HHTDMA) (see Mikhailov and Vlasenko, 2020; Mikhailov et al., 2021). Throughout the whole relative humidity range, the absolute uncertainty of RH is less than 0.5 %, and the relative growth factor uncertainty due to RH and instrumental errors does not exceed 1 %. A detailed calculation of growth factor uncertainty is described in Mikhailov and Vlasenko (2020) (Sect. 2.5). Three operation modes are available using this HHTDMA instrument: hydration and dehydration (H&D) (also called restructuring mode), hydration, and dehydration. The H&D mode was used to determine the optimal RH range in which initial irregular particles transform into compact globules.

Inorganic and organic aerosol particles as well as their mixtures restructure upon humidification below their deliquescence (Mikhailov et al., 2020). Irregular envelope shape and porosity cause a discrepancy between the mobility-equivalent and mass-equivalent particle diameters that limits precision of the mobility-diameter-based hygroscopicity tandem differential mobility analyzer (HTDMA) (Gysel et al., 2004; Mikhailov et al., 2020). To account for restructuring, we used the minimum mobility particle diameter, Db,H&D,min, obtained in the H&D HHTDMA mode as an approximation of mass-equivalent diameter of the dry solute particle, Ds (i.e., Ds=Db,H&D,min). The size-dependent restructuring factor was calculated as follows: gb,H&D=Db,RH/Db,i, where Db,i and Db,RH are the initial mobility particle diameter and that measured at RH of aerosol preconditioning, respectively (Mikhailov et al., 2021, Supplement S3). The description of the particle restructuring mechanism is beyond the scope of this work. It will be considered in detail in the next paper.

In this study, the H&D mode was in situ coupled with a conventional hydration or dehydration mode. In the combined modes, the initial monodisperse dry particles were first led through the preconditioning section where they underwent microstructural transformation, by means of humidification in the 2 %–90 % RH range (Nafion humidifier) and drying (Nafion dryer followed by a silica gel diffusion dryer). At the same time, the aerosol and sheath flow in the second differential mobility analyzer was below 3 % RH (Mikhailov et al., 2020, 2021, Supplement S3). The preconditioning RH at which the particles reached their minimum diameter, Db,H&D;min, was then kept constant throughout the standard hydration or dehydration experiment. The mobility-equivalent particle growth factor, gb, was calculated as the ratio of the mobility-equivalent diameter, Db, measured after conditioning (hydration, dehydration) to the minimum mobility diameter, Db;H&D;min, observed in the H&D mode: 1gb=DbDb,H&D,min.

The basis of the DKA method (Cheng et al., 2015) is the Köhler equation which describes the equilibrium relative humidity, sw, over a spherical droplet as a function of aw, σ, and the mass equivalent diameter Db: 2sw=RH100=awexp⁡4σνwRTDb=awexp⁡4σνwRTDsgb, where vw is the partial molar volume of water, R is the universal gas constant, and T is the temperature. Db in Eq. (2) is often replaced by the product of Ds and gb; that is, gb=Db/Ds. The logarithmic form of Eq. (2) is 3ln⁡sw=ln⁡aw(gb)+ADsσ(gb), where A=4υw/RTgb. From Eq. (3) it follows that aw and σ can be obtained as the fitting parameters for the sw(Ds) dependence with the same gb. Obviously, the larger the number of experimental values of Ds that are used for the fitting, the more accurately aw and σ can be estimated. This study utilized four sizes for ammonium sulfate and eight sizes for D-glucose and its mixture with ammonium sulfate in the Ds range 17–100 nm, where the Kelvin curvature effect has a significant impact on particle hygroscopicity. We used the Levenberg–Marquardt algorithm (implemented in Origin 9.0) to find aw and σ in the iterative procedure with standard fitting error as a measure of their uncertainty. It should be noted that fitting of the experimental sw(Ds) dependences over the entire set of dry diameters significantly improves the reliability of the DKA-derived aw and σ as compared to the original approach, where these quantities were calculated in pairs for available dry diameters (Cheng et al., 2015; Lei et al., 2023).

According to Eq. (3), DKA assumes that water activity and surface tension depend on gb only (i.e., concentration). However, in the case of bulk–surface partitioning, both water activity and surface tension may also depend on Ds. To minimize this effect, we used surface inactive compounds. Previous studies on the example of NaCl and (NH4)2SO4 nanoparticles have shown that the size effect is negligible for such compounds (Bahadur and Russell, 2008; Cheng et al., 2015, Supplement).

In a binary system at a constant temperature and pressure, the water activity, aw, and the solute activity, as, are related by the Gibbs–Duhem equation (Prausnitz et al., 1999): 4xsdln⁡as=-xwdln⁡aw, where xw and xs are the mole fractions of water and solute, respectively. According to Robinson and Stokes (1970), the water activity and molal osmotic coefficient of solute, Φs, for the real solution are related by 5ln⁡aw=-νsmsΦsMw1000, where νs and ms are the stoichiometric dissociation number and molality of the solute (molkg-1 of water), and Mw is the molecular mass of water (gmol-1). For electrolyte solution in the molality scale, 6as=a±νs=(γ±m±)νs=γ±vsms(ν+ν+ν-ν-). ν+ and ν- are the numbers of positive and negative ions produced upon dissociation of the solute (νs=ν++ν-); and m±, γ±, and a± are the mean molality, activity coefficient, and activity of the solute. For ammonium sulfate with νs=3 and non-dissociating D-glucose with νs=1 from Eq. (6) it follows that 7aAS=41/3mASγ±3  and  aGl=γmGl. Combining Eqs. (4)–(6) gives (Prausnitz et al., 1999) 8dln⁡γ±=dΦs+1ms(Φs-1)dms. Integration from ms=0 to the solution of interest yields 9ln⁡γ±=Φs-1+∫0mΦs-1msdms. The integral in Eq. (9) was evaluated numerically by plotting values of (Φs-1)/ms against ms.

The DKA-derived σ of pure AS and Gl aqueous solutions and their mixtures were compared with the model of Li and Lu (2001) and Dutcher et al. (2010). These models were moderately successful at representing the surface tensions of aqueous solutions of some soluble salts and their mixtures with organic compounds (Topping et al., 2007). The model by Dutcher et al. (2010) expresses the surface tension (σ) in terms of the surface tension of pure water (σw) and that of a hypothetical molten salt (σs): 10ln⁡σ=xwln⁡σw+xsln⁡σs, where xw and xs are the mole fractions of water (w) and salt (s). It is assumed that the surface tension of a solution at different temperatures is described by σ=σw+xsFws(T), with Fws(T)=aws+bwsT, where aws and bws are fitted parameters. At very high salt concentrations, a similar relationship is used: σ=σs+xwFsw(T) with Fsw(T)=asw+bswT. The two above expressions can be applied to express the surface tension over the entire concentration range: 11ln⁡σ(T)=xwln⁡(σw(T)+xsFws(T))+xsln⁡(σs(T)+xwFsw(T)). Equation (11) was used to calculate σ of ammonium sulfate particles at T=298 K with parameters σw=72.0 mNm-1, σs=185.0 mNm-1, aws=asw=0, bws=0.366 mNm-1K-1, and bsw=-0.289 mNm-1K-1 as suggested by Dutcher et al. (2010). The Li and Lu (2001) surface tension model includes a Langmuir-type adsorption isotherm and a term dependent on the activity of dissolved solid. For an aqueous solution with one solute, this model yields 12σ=σw+RTΓsw0ln⁡11+Ksas, where Γsw0 is the saturated surface excess of solute (s), and Ks is the adsorption equilibrium constant of solute (s). For multi-component systems, Li and Lu (2001) proposed two approaches. The first approach, referred to as LiLu (1), implies that there is no interaction or competing adsorption between species at the interface: 13σ=σw+RT∑i-1kΓiw0ln⁡11+Kiai. The second approach, that we refer to as LiLu (2), considers the interacting and competing adsorption between species in the interface for mixed aqueous systems at higher concentrations. This yields another expression for the surface tension of mixed solutions: 14σ=σw+RT∑i-1kΓiw0ln⁡1-Kiai1+∑iKiai.

We also compared the DKA–derived σ with simpler approximations, including those describing a supersaturated solution. Thus, for a ammonium sulfate solution, we used the parameterization by Pruppacher and Klett (1997): 15σ(Nm-1)=0.072+2.34×10-2×Xs1-Xs, where Xs is the mass fraction of solute. This parameterization is expected to be valid up to Xs≈0.8. For the concentration dependence of surface tension of an aqueous D-glucose solution, the linear function given by Aumann et al. (2010) was chosen: 16σ(mNm-1)=σw+ΔσΔCC, where C is the molarity concentration (molL-1) and Δσ/ΔC=1.29 (mNm-1/molL-1) with upper limit of C being 3.5 molL-1 (mGl=6.0 molkg-1).

Figure 1 illustrates the restructuring of aerosol particles in the H&D mode. Both pure and mixed particles restructure with increasing RH, wherein this effect is strongest for the largest diameters. Interestingly, mixed AS / Gl particles with a mass ratio of 4:1 undergo strong restructuring with gb,H&D less than 1 (Fig. 1c), whereas AS / Gl particles with a mass ratio of 1:1 show no structural change (Fig. 1d). For these particles, the restructuring factor gb,H&D is larger than 1 and increases with RH, indicating that initial particles are compact and spherical (i.e., Db,i=Db,H&D,min=Ds) (Mikhailov et al., 2020). The size-dependent increase in gb,H&D (Fig. 1c) is due to a residual water film on the particle surface after dehydration in the H&D mode. The higher values of dgb,H&D/dRH observed for particles with smaller diameters support this assumption. Since the H&D mode is in situ coupled with a conventional hydration or dehydration mode, the observed Db,H&D,min values at specified RH were directly used to calculate the growth factor using Eq. (1). This has made it possible to eliminate the uncertainty in the growth factor caused by irregular particle morphology, which is as high as 5 % or more for particles with Db,i>70 nm.

Figure 2 shows the size-dependent growth factors of pure and mixed particles observed upon dehydration in the HHTDMA mode, calculated using Eq. (1). The dehydration branch was used because it allows us to determine aw and σ over a wide concentration range, including highly concentrated droplet solutions. Note that, in contrast to pure ammonium sulfate and its 4:1 mixture with D-glucose, in the case of pure D-glucose (Fig. 2b) and mixed AS / Gl particles with a mass ratio of 1:1 (Fig. 2d), the growth factors obtained in the hydration and dehydration modes are identical over the whole range of relative humidity (Fig. S1 in the Supplement). As expected, due to the Kelvin effect, for the same gb the equilibrium values of sw (or RH) for smaller particles shift towards higher values (Eq. 2). This is clearly visible for all particles measured at RH above 70 % (inserts in Fig. 2). For ammonium sulfate particles, this pattern is valid also for highly concentrated solutions up to efflorescence RH (ERH ∼30 %) (Fig. 3a). However, in the cases of D-glucose (Fig. 3b) and AS / Gl (1:1) mixed particles (Fig. 3d), this pattern is not obvious. Moreover, in the case of AS / Gl (4:1) particles, the values of sw for small particles are lower than for large particles (Fig. 3c), in contrast to the behavior of pure ammonium sulfate (Fig. 3a). The measurement data in Fig. 3b and d also demonstrate that for D-glucose and AS / Gl (1:1) particles in the vicinity of RH≈48 % the gb(RH) dependence has an inflection point (derivative dgb/dRH has extremum; see the insert in Fig. 3b and d), indicating a distinct water sorption mechanism and hence a different phase state of the particles before and after this RH point. Most likely, at low RH, D-glucose and AS / Gl (1:1) mixed particles are in the semi-solid amorphous state; the increase in humidity over glassy amorphous particles leads to a moisture-induced phase transition that occurs at a “glass transition relative humidity” of RHg≈48 % (Mikhailov et al., 2009).

Continued water uptake converts the amorphous particles into concentrated solution droplets. Due to the very low molecular diffusivity of glasses, the uptake of water vapor by glassy aerosol particles is limited to surface adsorption, whereas above RHg the particles are in a viscous liquid state and absorb water in the particle bulk. Differences in water sorption mechanisms may explain why the growth factors of small particles at low RH are higher than for larger particles. At RH below RHg with the same water adsorption layer, the water film contribution to gb will be higher for small particles. For example, at a water layer thickness of 0.56 nm (i.e., two monolayers of water molecules), gb for aerosol particles of 19 and 100 nm are 1.031 and 1.006, respectively. This pattern is clearly seen for AC/Gl (1:1) particles with compact initial structures (i.e., Db,i=DH&D,min, and gb=gH&D) for which the growth factor of large particles at the same RH is lower than that for small particles (Fig. 1d). Note also that moisture-induced transformation from the semi-solid state to the solute state can occur gradually within a certain RH range (Mikhailov et al., 2009). All these features in the water uptake by amorphous particles at high concentrations (low growth factor values) lead to additional uncertainties when using the DKA method.

The DKA-derived water activity for ammonium sulfate and D-glucose aerosol particles are shown in Fig. 4a and b, respectively. More information on the DKA calculation can be found in Sect. S1 in the Supplement. The estimated uncertainty of aw for pure species is shown in Fig. S2 in the Supplement. The aw values for ammonium sulfate (Fig. 4a) agree well with electrodynamic balance (EDB) measurements by Tang and Munkelwitz (1994), Clegg et al. (1995), and Chan et al. (1992) and are almost identical to the DKA-derived aw values previously reported by Cheng et al. (2015) and Extended Aerosol Inorganic Model (E-AIM III) of Wexler and Clegg (2002), including highly concentrated aqueous solutions. In general, the DKA-derived water activities obtained for pure ammonium sulfate particles reproduce the literature data well.

Figure 4b shows the DKA-derived aw values of D-glucose aerosol particles. Our data are consistent with the EDB measurements by Peng et al. (2001) up to XGl∼0.8. The observed divergence of the water activity at XGl>0.8 may be due to kinetic limitations in highly viscous semi-solid amorphous particles (Mikhailov et al., 2009), which prevail in EDB experiments with micrometer-sized particles. As noted in Sect. 4.2 and shown in Fig. 3d at RH=RHg∼48 %, particles undergo moisture-induced phase transition where semi-solid particles become concentrated solution droplets. This RH value is very close to aw∼49 %, below which a strong deviation between DKA- and EDB-derived aw is observed (Fig. 4b). If this is the case, the DKA-derived aw here is closer to the real aw.

At the same time, the DKA-derived aw values differ from those predicted by the AIOMFAC model (Zuend et al., 2008), with a maximum deviation of 6 % at XGl∼0.7. The discrepancy between measured and AIOMFAC-predicted aw values for carbohydrates has been noted previously and is attributed to relatively strong intramolecular interactions due to several polar groups in close proximity, not accounted for by the UNIFAC model within the AIOMFAC framework (Zuend et al., 2011).

The multicomponent surface tension model of Li and Lu (2001) is based on the activity of the solute. To take advantage of this model, we calculated the activity coefficients of single solutes by numerical integration of Eq. (9) and then, using Eq. (7), their activities (see Sect. S2 in the Supplement for more details). The calculation results are shown in Fig. 5 in comparison with literature data. One can see that the ammonium sulfate activity is in good agreement with E-AIM (Fig. 5a) up to XAS=0.75 (mAS=22.7 molkg-1). At higher XAS values, there is a discrepancy between our data and the model data. This discrepancy is most likely due to the uncertainty in DKA-derived aw in concentrated ammonium sulfate aqueous solutions (Fig. S2) and uncertainty in assessment of the integral Eq. (9) in the asymptotic region (ms→0) (Lakhanpal and Conway, 1960).

Figure 5b shows the DKA-based activity of D-glucose. One can see that these values are in agreement with bulk measurements by Miyajima et al. (1983), obtained by the isopiestic method (blue symbols) available up to XGl=0.52 (mGl=6.0 molkg-1). At the same time, a systematic deviation is observed between our results and the AIOMFAC model. As previously noted, the UNIFAC as a part of the AIOMFAC model does not match well with D-glucose and other carbohydrates, owing to strong intramolecular interactions of the polar groups in close proximity.

In general, the DKA-derived water activity and activity of ammonium sulfate and D-glucose in aqueous solutions reproduce the literature data well. For future applications, we fitted DKA-derived aw and as for single solutes with a polynomial function. The obtained fitting parameters are listed in Tables S1 and S2 in the Supplement.

Figure 6 shows the DKA-derived surface tension of ammonium sulfate and D-glucose aqueous solution droplets as a function of solute activity. For future use, we fit our data with the Li and Lu (2001) model (Eq. 12). The resulting values of Γsw0 and Ks together with the as fitting interval are listed in Table S3 in the Supplement. The negative Γsw0 values obtained for both ammonium sulfate and glucose solution droplets indicate that a negative adsorption occurs on the interface between the vapor and liquid phases, leading to surface tensions increasing with concentration, which is typical for surface-inactive compounds (Tuckermann, 2007; Aumann et al., 2010).

For comparison with the literature data, in Fig. 7 we plotted the DKA surface tension in more accessible concentration scales, i.e., in mass fraction and molality. Figure 7a shows that the DKA-derived σ for ammonium sulfate solution is in agreement, within uncertainty, with the empirical approximation of Pruppacher and Klett (1997) and also consistent with the Dutcher et al. (2010) model up to XAS≈0.5 (mAS=7.6 molkg-1). Above this value, the difference between DKA data and those by Dutcher et al. (2010) gradually increases with increasing concentration. This difference is due to the fact that model parameters used in Dutcher et al. (2010) (Sect. 3.3) are only valid for a concentration of ammonium sulfate not exceeding XAS=0.43 (mAS=5.69 molkg-1), although the model structure allows for calculation of σ in the concentration range from dilute solution to molten salt (Eq. 11). Figure 7b shows the DKA-derived σ for D-glucose solution droplets as a function of mass fraction or molality together with the literature data. It can be seen that the DKA data are consistent with bulk measurements reported by Aumann et al. (2010). In the XGl range of 0.24 to 0.52, the average deviation between DKA-derived and literature σ is 2 %.

Note that the uncertainty in the DKA-retrieval parameters is strongly influenced by of the accuracy of sw (or RH) and gb determination. At low gb (highly concentrated solutions) when Δgb/Δsw is small, the uncertainty in the DKA-derived aw and σ increases with decreasing gb. In addition, phase state ambiguity and associated kinetic limitations can provide additional uncertainties. As an example, Fig. S3 in the Supplement shows experimental sw(Ds) dependences at low and high growth factors for D-glucose aerosol particles and corresponding uncertainties in the DKA-derived aw and σ.

In the absence of solute–solute interactions (“separate solutes”), the corresponding relationship for the water activity of the mixture is aw=∏iaw,i, where aw,i is the water activity of a pure aqueous solution of i at the same concentration as in the mixture (Mikhailov et al., 2004; Clegg and Seinfeld, 2006). Within this approach, referred to as “separate solute water activity” (SSWA), the water activity of mixed particles containing ammonium sulfate and D-glucose can be expressed as 17aw=aw,ASaw,Gl.

Figure 8 shows the DKA-derived water activity of the mixed AS / Gl particles with a mass ratio of 4:1 (a) and 1:1 (b), as well as results of the AIOMFAC and SSWA models. One can see that the DKA-derived aw and both models are consistent up to Xs∼0.3. A further increase in concentration is accompanied by a significant deviation of the model aw values, both among themselves and from the DKA-derived aw. It appears that the SSWA approximation underestimates the specific interactions between ions and molecules in the droplet volume (higher aw), whereas the AIOMFAC model overestimates them (lower aw), especially in concentrated solutions.

Figure 9 shows the DKA-retrieved surface tension of the mixed AS / Gl particles with mass ratios of 4:1 (a) and 1:1 (b), compared to the Li and Lu (2001) model. Both approaches, i.e., LiLu (1) and LiLu (2), do not match well with the HHTDMA–DKA-based σ. This discrepancy is especially pronounced for the mixed particles with an AS / Gl mass ratio of 1:1 (Fig. 9b). A slight lowering of the surface tension compared to that of water is already noticeable for a 4:1 mixture (Fig. 9a). The AS : Gl=1:1 mixture (Fig. 9b) shows more significant surface tension depression, which is not a monotonic function of concentration. At Xs≈0.5, the σ reaches a minimum of 56.5 ± 3.0 mNm-1 (22 % reduction compared to pure water) and increases again with concentration. Further increase in surface tension with concentration is the result of solidification of aerosol particles. According to Eq. (10) at Xs→1, σ tends to the surface tension of molten state, σs, which for AS and Gl is 185 mNm-1 (Dutcher et al., 2010) and 150.9 mNm-1 (Docoslis et al., 2000), respectively. Thus, at Xs=1 the mole-fraction-weighted value of σ for AS : Gl=4:1 and 1:1 is 179 and 170 mNm-1, respectively, which reasonably agrees with the DKA-derived σ at high Xs values (Fig. 9a and b).

Note that the decrease in surface tension observed in Fig. 9b and its subsequent increase with concentration is reproduced in repeated measurements (see Fig. S4 in the Supplement). Most likely the reduction in surface tension at moderate concentrations is caused by the salting out effect (Setschenow, 1889; Kiss et al., 2005; Marcolli and Krieger, 2006; Frosch et al., 2011; Wang et al., 2014; Lin et al., 2020; Bzdek et al., 2020) and results from the interaction between ammonium sulfate ions and D-glucose molecules, facilitating the association of initially surface-inactive hydrophilic organic molecules into surface-active hydrophobic associates (quasi-macromolecules). Docoslis et al. (2000) analyzed the difference between polysaccharides (dextran, Ficoll) and their surface-inactive constituents (sucrose, glucose). They concluded that the contrasting results are caused by the differences in polar intermolecular reactivities of the monomeric and polymeric glucides. The singly dissolved monomeric sugars contribute strongly to the polar (Lewis acid–base) free energy of cohesion through the multiple interactions between their freely available electron donors and acceptors, through mutual interactions as well as through interactions with the surrounding water dipoles. However once covalently polymerized, these polysaccharides have lost the strong bipolarity of the monomeric sugar molecules. It can thus be assumed that, in an aqueous AS / Gl system, NH4+ and SO42- ions effectively neutralize the bipolarity of the monomeric D-glucose molecules, facilitating their association into less-polar aggregates with reduced σ values, which are more readily accommodated at the air–droplet interface. Obviously, this mechanism is more pronounced for mixed AS : Gl=1:1 particles (Fig. 9b) than for their 4:1 mixture (Fig. 9a), due to the higher D-glucose content (by a factor of 2.5).