5.1.1 Ammonium sulfate particle

Figure 4a shows the change in the initial dry mobility diameter of 100.3 nm ammonium sulfate aerosol particles obtained at different drying conditions. In the range of 2 %–60 % RH the mobility diameter gradually decreases, and when RH is more than 70 % RH it becomes almost constant with $D_{b,\mathrm{H}\mathrm{\&}\mathrm{D},\min }$ observed at 80 %–90 % RH. The $D_{b,\mathrm{H}\mathrm{\&}\mathrm{D},\min }$ values obtained for all drying modes are shown in Table 1.

Table 1Microstructural rearrangement parameters for pure ammonium sulfate and glucose aerosol particles obtained in the H&D experiment for different drying conditions (Fig. 4a, b). D_b,i and $D_{b,\mathrm{H}\mathrm{\&}\mathrm{D},\min }$ are the mean plus or minus the standard deviation of 7–10 data points. The dynamic shape factor, χ, porosity, δ, and void fraction, f, together with propagated error, Δf, are calculated from Eqs. (3), (4), and (5) respectively. RT is the residence time. The χ and δ propagated uncertainty is better than ±0.002.

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Interestingly, when using only the MD-700 dryer the variation in the RT from 27 to 67 s leads to a decrease in the $D_{b,\mathrm{H}\mathrm{\&}\mathrm{D},\min }$ by only 0.4 nm (i.e., at RT=27 s, $D_{b,\mathrm{H}\mathrm{\&}\mathrm{D},\min }=\mathrm{98.4}$ nm; at RT=67 s, $D_{b,\mathrm{H}\mathrm{\&}\mathrm{D},\min }=\mathrm{98.0}$ nm). A sharp decrease in minimum mobility diameter by ∼3 nm ($D_{b,\mathrm{H}\mathrm{\&}\mathrm{D},\min }=\mathrm{95.3}$ nm) occurs when an SDD (RT=62 s) is added to the MD-700 membrane dryer (RT=27 s); that is, already effloresced aerosol particles (RH∼3 % at the outlet of the MD-700 dryer) underwent further microstructural changes inside the SDD. The maximum reduction (by ∼7 nm) of the initial DMA1 selected particles is observed when only SDD is used as a desiccant, for which $D_{b,\mathrm{H}\mathrm{\&}\mathrm{D},\min }=\mathrm{93.2}$ and 93.5 nm (first and second run with the same SDD; Fig. 4a).

Multiple Köhler model calculations based on $D_{b,\mathrm{H}\mathrm{\&}\mathrm{D},\min }$ obtained in all drying modes used are in excellent agreement with the observed hygroscopic growth curves. These findings confirm the compactness and spherical shape of dry particles, despite the fact that the absolute values of $D_{b,\mathrm{H}\mathrm{\&}\mathrm{D},\min }$ are different and strongly depend on the drying conditions (Fig. 4a).

Figure 5SEM images of initial ammonium sulfate (a) and glucose (b) aerosol particles. The samples were investigated with a high-resolution SEM (Zeiss Merlin). Operation conditions: 0.4 kV accelerating voltage, 1.5 kV ESB grid voltage, 1.8 mm working distance. Particle samples were collected directly onto 3 mm TEM copper 300 mesh grids coated with a 30–60 nm thick formvar film.

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Figure 6Possible structures of polycrystalline ammonium sulfate and amorphous glucose aerosol particles depending on drying conditions: (I) agglomerate of single crystals with fully and partly shielded cavities filled with liquid; (II, III) polycrystalline agglomerates with open and shielded cavities having a different void-to-solid ratio; (IV) amorphous glucose aerosol particle in a gel-like state. Other explanations are given in the text.

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The different values of $D_{b,\mathrm{H}\mathrm{\&}\mathrm{D},\min }$ observed upon H&D mode are a result of the different microstructure of the initial dry particles having the same D_b,i (Table 1). As previously noted the dry particle morphology depends on drying rate (Zelenyuk et al., 2006; Zhao et al., 2008; Mikhailov et al., 2009; Wang et al., 2010), since the solidification of aerosol droplets is mainly governed by kinetic rather than thermodynamic factors. Experiments with the MD-700 membrane dryer show that at the same drying rate, the residence time has little effect on ammonium sulfate particle morphology; i.e., with an increase in the RT from 27 to 67 s, the dynamic shape factor grows slightly from 1.033 to 1.040 (Table 1). The values obtained in this experiment are close to those reported by Kuwata and Kondo (2009), who used the combination of a DMA and an APM (aerosol particle mass analyzer) system. They estimated that χ of (NH₄)₂SO₄ for 50–150 nm is 1.01–1.04. Zelenyuk et al. (2006), using DMA and single-particle laser ablation time-of-flight mass spectrometer (SPLAT) measurements, showed that χ is 1.03±0.01 at 160 nm. Biskos et al. (2006) estimated χ for 6–60 nm (NH₄)₂SO₄ particles to be 1.02 based on the observed particle restructuring in the hydration HTDMA mode. Our χ values obtained with MD-700 membrane dryers and those preliminarily reported most likely reflect surface irregularities as observed by SEM (scanning electron microscope) (Fig. 5a) and TEM (transmission electronic microscope) (Dick et al., 1998; Zelenyuk et al., 2006). At the same time, an experiment with serially connected dryers (MD-700 + SDD) indicates that after the first dryer effloresced particles still contain liquid. The liquid could be located in cavities with various degrees of shielding (Cohen et al., 1987; Weis and Ewing, 1999; Colberg et al., 2004). Figure 6 outlines possible structures of aerosol particles and their microstructural rearrangements during the H&D experiment. Based on the experimental results considered above, we can assume that in membrane dryers (MD-700) the effloresced particles release surface water and water that is stored in relatively open cavities, providing an irregular aerosol particle envelope (Fig. 5a) (see also Dick et al., 1998, and Fig. 7). However, some of the liquid remains in either completely closed or partially shielded cavities. The latter can be pores, veins, and grain boundaries that retain water due to the inverse Kelvin effect on concave surfaces (Fig. 6, pattern I), thereby impeding exchange between the gas phase and the cavities. The fresh SDD added to the membrane dryer overcame the diffusion barrier created by capillary forces, most likely due to the drying rate in the SDD being much faster than that in the membrane dryer. As a result, some of partially shielded cavities released excess water. Note that this process is accompanied by further microstructural rearrangement, leading to the formation of porous aerosol particles (Fig. 6, pattern II) with $D_{b,\mathrm{H}\mathrm{\&}\mathrm{D},\min }$ smaller than that observed after the membrane dryer (∼95.3 nm vs. ∼98.4 nm; Table 1). Accordingly, the aerosol particle shape factor, χ, increases from ∼1.03 to ∼1.09 (Table 1). Finally, more porous particles are formed when solely a fresh SDD is used as a desiccant ($D_{b,\mathrm{H}\mathrm{\&}\mathrm{D},\min }=\mathrm{93.3}$ nm, χ=1.13; Table 1). As pointed out before (Mikhailov et al., 2009; Wang et al., 2010) excess charge and rate of drying are important for the microstructure of aerosol particles generated by the nebulization of aqueous solution. Most likely in the case of fresh SDD, the most porous and/or irregular particles (Fig. 6, pattern III) form due to strong kinetic limitations arising during sufficiently rapid drying inside the SDD. More pronounced multiple nucleation events could occur by increasing the number of polycrystals and accordingly the number and scale of cavities. As a result, at the same D_b,i the observed $D_{b,\mathrm{H}\mathrm{\&}\mathrm{D},\min }$ of redried particles (H&D mode) is strongly dependent on the drying conditions (Fig. 6; Table 1, last column).

Assuming that the χ value obtained in the experiment with the membrane dryer accounts for the aerosol particle envelope, i.e., $\mathit{\chi }=\mathit{\beta }=\mathrm{1.033}$ (Fig. 5a), and using Eqs. (4) and (5) we estimated the particle porosity, δ, and void fraction, f, respectively. The calculated values of the particle void fraction for the coupled (MD 700 + SDD) drying system and for the single SDD are ∼9 % and ∼14 %, respectively (Table 1). Obviously, this difference reflects the effect of drying conditions on particle morphology as discussed above. Since in the two-stage drying system, the microstructural rearrangement of particles inside the SDD occurs due to the remaining water, the obtained f≈9 % can be attributed to the volume fraction of water stored in pores and veins after first drying stage (Fig. 6, pattern I). In the mole fraction basis the water content in dry solid aerosols can be obtained from

According to Eq. (29) the H₂O:(NH₄)₂SO₄ molar ratio (n_w∕n_s) in the dry particles after the membrane dryer is ∼0.4 (0.41 and 0.39 for the first and second run). In case of one-stage aerosol drying when only the SDD is used, the volume fraction of ∼14 % corresponds to an H₂O:(NH₄)₂SO₄ molar ratio of ∼0.7 (0.69 and 0.73 for the first and second run). This value can be considered an upper limit of the water content, since it is assumed that all cavities are filled with water. The H₂O:(NH₄)₂SO₄ molar ratio range of 0.4–0.7 obtained in this study at RH<3 % is close to that reported by Weis and Ewing (1999) for submicron NaCl aerosol particles with a median diameter of 350 nm. In their FTIR spectroscopic flow tube experiment for RH of 15 %–5 %, the obtained H₂O:NaCl molar ratio in the silica-gel-dried particles varies between 0.5 and 0.7. They suggest that during the crystallization process water is present in open and shielded pockets.

5.1.2 Glucose particles

Figure 4b shows the mobility equivalent diameter observed upon the H&D mode of glucose aerosol particles. Unlike ammonium sulfate (Fig. 4a), the minimum mobility diameter of the glucose aerosol particles is already observed at ∼20 % RH. Moreover, $D_{b,\mathrm{H}\mathrm{\&}\mathrm{D},\min }$ was practically independent of drying conditions. Accordingly, the shape factor calculated from Eq. (3) is also constant (χ=1.06; Table 1). SEM images (Fig. 5b) indicate that ∼100 nm glucose particles are perfect spheres; therefore, one can assume that β=1. From Eqs. (4) and (5) it follows that δ∼1.03 and f∼10 % (Table 1). The fact that particle restructuring does not depend on the residence time and type of dryer indicates a lower energy barrier to water transport from the cavities to ambient air compared to the ammonium sulfate particles. As will be shown below, glucose aerosol particles like other monosaccharides tend toward reversible water uptake starting at very low RH, which is typical for particles with an amorphous structure (Fig. 6, pattern IV) (Mikhailov et al., 2009; Koop et al., 2011). The absorbed water acts as a plasticizer, which softens the microstructural rearrangements inside swelling particles (Fig. 6). Moreover, at low RH water uptake is facilitated by the presence of alcohol functional groups within sugars that form hydrogen bonds with water. These effects can explain why, in contrast to ammonium sulfate, restructuring of the glucose aerosol particles starts at low humidity and is practically completed at ∼20 % RH. A slight increase in D_b,H&D observed at RH above 20 % RH (Fig. 4b) is probably due to the high hygroscopicity of glucose and low diffusivity of water molecules through a (semi)solid matrix of the compacted particles (Fig. 6) (Shiraiwa et al., 2013). However, the fact that in the case of glucose aerosol particles the drying conditions do not have a significant effect on $D_{b,\mathrm{H}\mathrm{\&}\mathrm{D},\min }$ is not entirely clear. Further work is needed to clarify this effect.

One of the most important structural metrics of a porous material is the connectivity of the pore space, or the so-called pore network. This has bearing on the diffusive tortuosity and permeability of water. Most of the pores are connected to each other and to the surface via small throats (open pores), whereas some pores are shielded from the connected structure. According to estimates, the fractions of voids in the aerosol particles of ammonium sulfate and glucose obtained under the same drying conditions are comparable (Table 1). However, pore networks can be different. It is therefore possible that in amorphous glucose the pore network has more open pores than in the case of polycrystalline ammonium sulfate particles, leading to a more efficient exchange of water between filled cavities and the gas phase under the same drying conditions.

5.2.1 Ammonium sulfate water uptake prior to deliquescence

Figure 7 shows the change in the initial mobility diameter of ammonium sulfate particles observed in hydration mode before deliquescence transition. It is seen that particles with and without preconditioning demonstrate different responses to changes in RH. The particles that bypass the preconditioning step due to irregular and/or porosity microstructure undergo a strong restructuring (D_b,i decreased by 4.3 nm), while preconditioned particles (H&D mode) exhibit a small but continuous hygroscopic growth. Assuming that initial preconditioning particles are compact and spherical (i.e., $D_{b,\mathrm{i}}=D_{b,\mathrm{H}\mathrm{\&}\mathrm{D},\min }=D_{\mathrm{m},\mathrm{s}}$) and using Eq. (27) we have converted the difference between the mobility diameters observed in hydration mode into an equivalent number of monolayers. As illustrated in Fig. 7 we obtained nearly linear growth of Θ from ∼0 to 3.5 for the range of 5 %–75 % RH and a sharp increase up to Θ∼6 over the range of 75 %–79 % RH. The findings are consistent with our earlier HTDMA studies of water adsorption on ammonium sulfate particles (Mikhailov et al., 2009) and with the literature data considered therein. The obtained Θ (RH) dependence was fitted using Eq. (6) with water activity taken from FHH isotherm Eq. (28). Calculations were performed assuming that the σ and ${\stackrel{\mathrm{‾}}{V}}_{\mathrm{w}}$ parameters are equal to those for pure water. The fit result is shown in Fig. 7 (solid line). The best-fit parameters are $A_{\mathrm{FHH}}=\mathrm{1.07}\pm \mathrm{0.08}$ and $B_{\mathrm{FHH}}=\mathrm{0.94}\pm \mathrm{0.07}$. Romakkaniemi et al. (2001) also used HTDMA measurements with NaCl and (NH₄)₂SO₄ particles between 8 and 15 nm to estimate Θ before deliquescence transition. For ammonium sulfate particles, the calculated parameters for the FHH isotherm are A_FHH=0.68 and B_FHH=0.93. The A_FHH value obtained in our work is ∼40 % higher than that reported by Romakkaniemi et al. (2001). One possible explanation is that the surface coverage is lower on surfaces of nano-sized particles compared to flat surfaces (Müller at al., 1987). Another and perhaps more important reason is that Romakkaniemi et al. (2001) used the initial mobility diameter, D_b,i, for Θ calculation without particle shape correction. Biskos et al. (2006) have shown that nano-sized ammonium sulfate particles in the range of 6–60 nm undergo a restructuring upon RH increasing. Thus, for 6–8 nm particles the minimum mobility diameter observed in the 30 %–60 % RH range is ∼2 % lower than the initial mobility diameter, which corresponds to uncertainty in Θ of about one monolayer of adsorbed water (see Eq. 27). This explains why the Romakkaniemi et al. (2001) values of Θ are lower than those obtained in this study.

The water uptake before particle deliquescence was detected in earlier studies (Weingartner et al., 2002; Gysel et al., 2002; Biskos et al., 2006), but it was observed mainly at RH>70 %. It is most likely that restructuring, which occurs upon particle hydration, masked water uptake at low RH. The experimental D_b (RH) dependence obtained for non-preconditioning particles (Fig. 7, open circles) clearly demonstrates this effect. Alternatively, the hygroscopic growth data obtained for preconditioning particles with a compact structure (Fig. 7, closed circles) show that water adsorption on the surface of a solid particle already occurs at lower humidities ranging ∼15 % RH.

Figure 8Growth factors observed in the hydration and dehydration experiments for the preconditioned ammonium sulfate aerosol particles with $D_{b,\mathrm{H}\mathrm{\&}\mathrm{D},\min }=\mathrm{79.6}$ nm compared with the full Köhler model: (a) growth factors as a function of relative humidity measured with a capacitive RH probe (RH4; Fig. 1); (b) RH values obtained from E-AIM using experimental growth factors. Insets in panel (a): the gray area denotes the growth factor uncertainty obtained from Eq. (2); the shaded area corresponds to the growth factor uncertainty of E-AIM below the DRH obtained from EDB experimental data. Whiskers show RH uncertainty (a, b).

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5.2.2 Ammonium sulfate hydration and dehydration

Figure 8 shows the growth factors of the preconditioned ammonium sulfate particles with $D_{b,\mathrm{H}\mathrm{\&}\mathrm{D},\min }=\mathrm{79.6}$ nm obtained upon the hydration and dehydration HHTDMA modes at 298 K. The observed efflorescence RH ($\text{ERH}=\mathrm{34.8}\pm \mathrm{0.2}$ %) and deliquescence RH ($\text{DRH}=\mathrm{79.9}\pm \mathrm{0.2}$ %) are within the literature data obtained for submicron ammonium sulfate particles (Mikhailov et al., 2009; Gao et al., 2006; Ciobanu et al., 2010, and references therein). The experimental growth factors are compared to a full Köhler model with water activity parameterization derived from the E-AIM II (Clegg et al., 1998; Wexler and Clegg, 2002). As illustrated in Fig. 8a, up to 97 % RH the HHTDMA experimental data are in very good agreement with model growth factors. The insets in Fig. 8a show the experimental growth factor uncertainties, which gradually go up due to the increase in RH uncertainty (Eq. 2, terms 2${\mathit{\sigma }}_{\mathrm{RH}}/D_{b,\mathrm{RH}}$ and ΔRH(dg_b∕dRH)). Averaged over the range of 38 %–96 % RH, the mean deviation between measurement and model results is <0.5 %. The good agreement between model and measurement results confirms that the ammonium sulfate particles with $D_{b,\mathrm{H}\mathrm{\&}\mathrm{D},\min }$ are compact and spherical (i.e., $D_{b,\mathrm{H}\mathrm{\&}\mathrm{D},\min }=D_{\mathrm{m},\mathrm{s}}$). However, above ∼97 % RH due to the sharp growth of the ΔRH(dg_b∕dRH) term in Eq. (2), the observed growth factors systematically deviate from the Köhler model. Thus, at ∼98 % RH this deviation is ∼7 %, and at 99.5 % RH it is already ∼15 % (inset in Fig. 8a).

Figure 8b shows the measured growth factors, which were converted into RH using E-AIM at RH above deliquescence transition. In this case, the RH accuracy is determined by the instrumental growth factor error (Eq. 2, terms in square brackets). The insets in Fig. 8b indicate that RH accuracy progressively improves with RH increasing. Thus, at 85 % RH absolute accuracy is ±0.3 %, while at 99.5 % RH it is only ±0.03 % (Fig. 2). Thus, using experimental ammonium sulfate growth factors, it is possible to eliminate the RH uncertainty generated by capacitive and dew point sensors at RH above 80 %.

Overall, the combination of two HHTDMA operation modes (H&D and hydration–dehydration) that eliminate the effect of the particle shape factor and the precise determination of RH using ammonium sulfate are prerequisites for the accurate determination of the thermodynamic parameters of aerosol particles in a wide range of RH. In the next section, we will show the effectiveness of this approach with the example of glucose aerosol particles.

Figure 9Growth factors observed in the hydration and dehydration experiments for the preconditioned glucose aerosol particles with $D_{b,\mathrm{H}\mathrm{\&}\mathrm{D},\min }=\mathrm{99.6}$ nm in comparison with literature data (a) and relative growth factor uncertainty due to instrumental and RH errors (b).

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5.2.3 Glucose hydration and dehydration

Figure 9a shows the mobility equivalent growth factor observed upon the hydration and dehydration of preconditioned glucose aerosol particles with $D_{b,\mathrm{H}\mathrm{\&}\mathrm{D},\min }=\mathrm{99.6}$ nm in the hydration and dehydration HHTDMA operation modes. In both modes, over the 2 %–99.6 % RH range the change in the growth factor occurred gradually. In contrast to (poly)crystalline ammonium sulfate (Fig. 8a), no stepwise changes in g_b associated with DRH and ERH phase transitions are observed. Such behavior is typical for particles with an amorphous structure as discussed in Mikhailov et al. (2009). In general, the growth factors obtained in hydration and dehydration experiments are in good agreement with those previously reported by Mochida and Kawamura (2004) and Suda and Petters (2013). Nevertheless, a slight positive deviation of ∼1 % can be traced throughout the RH range. The growth factors presented by Mochida and Kawamura (2004) and Suda and Petters (2013) were calculated without particle shape correction. As noted in Sect. 5.1.2, due to porosity the glucose aerosol particles undergo a wet restructuring, decreasing their initial mobility diameter (Fig. 4b; Table 1). Therefore, using D_b,i instead of $D_{b,\mathrm{H}\mathrm{\&}\mathrm{D},\min }$ as an approximation of the mass equivalent diameter of the dry solute particle D_m,s may lead to underestimated values of the growth factor (Eq. 8).

Figure 9b shows the change in the total relative uncertainty of the growth factor caused by RH and instrumental uncertainties. A small drop in growth factor uncertainty observed at ∼80 % RH (blue curve) is caused by replacing the RH control method (Sect. 2.4), and its decrease above ∼97 % RH is due to a sharp drop in ΔRH (g_b,E-AIM) near water saturation (Eq. 2; Fig. 2). In contrast, the instrumental g_b uncertainty increases monotonously (Fig. 9b, red curve) due to the smooth growth of the RH-dependent ${\mathit{\sigma }}_{b,\mathrm{RH}/D_{b,\mathrm{RH}}}$ ratio in Eq. (1) (Fig. S1.3).

Figure 10Growth factors observed in the hydration and dehydration experiments for the preconditioned glucose aerosol particles with $D_{b,\mathrm{H}\mathrm{\&}\mathrm{D},\min }=\mathrm{99.6}$ nm in comparison to mass equivalent growth factors calculated with the full Köhler model.

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Figure 10 shows the HHTDMA growth factors compared to the full Köhler model (Eq. 6) with a_w calculated from Eq. (15) using the bulk water activity coefficient, γ_w, from Taylor and Rowlinson (1955) (Sect. 4.1) and ${\stackrel{\mathrm{‾}}{V}}_{\mathrm{w}}$ and σ calculated from Eqs. (7) and (10), respectively. Excellent agreement between HHTDMA-based and full Köhler model data is observed: throughout the 91.0 %–99.6 % RH range the average deviation of the experimental data points from the model is 0.7 %. The observed coincidence indicates that the $D_{b,\mathrm{H}\mathrm{\&}\mathrm{D},\min }$ value obtained upon restructuring is a good approximation of the mass equivalent diameter of the dry glucose aerosol particle, i.e., $D_{b,\mathrm{H}\mathrm{\&}\mathrm{D},\min }\approx D_{\mathrm{m},\mathrm{s}}=\mathrm{99.6}$ nm. It also confirms the small growth factor uncertainty associated with RH and instrumental g_b errors, which is in the range of 0.3 %–0.9 % throughout the 2 %–99.6 % RH interval (Fig. 9b, black curve).

Table 2Parameters characterizing the hygroscopic properties of glucose aerosol particles: best-fit values (± standard errors) for the three-parameter fit (k₁, k₂, k₃; Eq. 17); n and R² are the number of data points and the coefficient of determination of the fit, respectively.

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Figure 11Growth factor (a) and hygroscopicity parameter (b) of glucose aerosol particles as a function of water activity. Black lines in panels (a) and (b) correspond to the three-parameter fit using Eqs. (17) and (19), respectively. The green line accounts for the ideal solution model (a). Insets: (a) water-activity-based growth factors at low and high a_w in comparison with model curves; (b) hygroscopicity parameter change at a_w above 0.94. The data points selected by the rectangle are used to calculate the average value of the dilute hygroscopicity parameter, κ.

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5.3.1 Water activity and hygroscopicity parameter

Using Eq. (6) the experimental g_b vs. RH data points are converted into data pairs g_b vs. a_w (Sect. 4.2). Figure 11a shows the obtained activity-based growth factors, which were fitted with Eq. (17) to determine the best-fit values of the parameters k₁, k₂, and k₃ (Table 2). Figure 11a also illustrates the difference between experimental data points and the ideal solution model, which was traced throughout the water activity range (insets in Fig. 11a) with a maximal deviation of 3.6 % at a_w≈0.8. Only at a_w>0.98 did the experimental growth factors coincide with the ideal model within an uncertainty of g_b∼0.6 %. For the ideal solution model the g_b(a_w) dependence is calculated from Eq. (25) with a_w=x_w. Note that, given Eq. (20) for κ_R, Eq. (25) can be reduced to

which is an analog of Eq. (19), where κ=κ_R.

For each experimental g_b vs. a_w data pair we have calculated κ values using Eq. (19). By inserting fitted values of g_b(a_w) into Eq. (19), we have obtained the corresponding fit curve for activity-based hygroscopicity, κ. The obtained results are shown in Fig. 11b. Due to concentration effects (Mikhailov et al., 2013, and references therein) the hygroscopicity parameter decreases with g_b increasing. At a_w>0.98 the κ becomes almost constant. In this area the estimated value of κ is 0.160±0.006 (average of the 10 data points ± propagated uncertainty; see the inset in Fig. 11b), which is close to the ideal solution value of κ_R=0.154 (Eq. 20). The hygroscopicity κ obtained in this study for dilute glucose solution is in agreement with that derived by Ruehl et al. (2010) ($\mathit{\kappa }=\mathrm{0.165}\pm \mathrm{0.033}$) measured in the 99.4 %–99.9 % RH range using the continuous-flow thermal gradient column and with the HHTDMA-based value of κ=0.162 reported by Petters and Petters (2016) at RH>90 %. Note that at the same water activity range the measurement uncertainty of κ with the HHTDMA method is ∼6 times less than that in the thermal gradient column setup (0.006 vs. 0.033). As mentioned before, the optical measurements used for particle size determination (Ruehl et al., 2010; Wex et al., 2005; Suda and Petters, 2013) are subjected to limitations in accuracy resolution due to uncertainties in the refractive index and conversion from the optical to physical diameter.

Figure 12HHTDMA-based activity coefficients of water (γ_w) and glucose (γ_Gl) as a function of the mole fraction (a) and molal osmotic coefficient of glucose (Φ_Gl) vs. water activity (b) for glucose solution droplets in comparison with literature data. Bulk measurement of γ_Gl from Miajima et al. (1983) – black solid (a); the data points and error bars are from the HHTDMA experiment with hydration (blue circles) and dehydration (red circles). The Φ_Gl values from Suda and Petters (2013) – black circles (b). Model lines: (a) two-suffix Margules equation (Eq. 16, with $A=-\mathrm{1.957}$) – blue solid; (b) Rudakov and Sergievskii (2009) model (Eq. 23) with hydration number h⁰=1.88 (the best-fit parameter with standard error is 0.04) – black solid. The gray shaded area denotes the hydration number range with h⁰+0.5 (upper bound) and h⁰−0.5 (lower bound). The red dashed fit line in panel (a) is the polynomial fourth-order fit function of ln γ_Gl obtained in this study together with Miajima et al. (1983) bulk measurements. The shaded rectangle in panel (a) denotes the metastable area of the glucose solution droplet.

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5.3.2 Activity and molal osmotic coefficients

Figure 12a shows the HHTDMA-based activity coefficient of water (γ_w) and glucose (γ_Gl) in glucose aqueous solution. The activity coefficient of water was calculated from Eq. (15), where x_w was obtained based on Eqs. (8) and (9) as described in Sect. 4.4, and water activity was derived from Eq. (6) with the assumptions considered in Sect. 4.2. The activity coefficient of glucose in water solution was obtained by numerical integration of Eq. (26) (Sect. 4.4).

The bulk DRH of glucose varied in the range of 88 %–90 % RH (Zamora et al., 2011, and references therein), which corresponds to the saturated mole fraction of glucose aqueous solution, x_Gl, of 0.095 (μ_Gl=3.14 mol kg⁻¹). Above this value, glucose particles are metastable supersaturated droplets (selected area in Fig. 12a), which are present in an amorphous (semisolid) state. Using the bulk water pressure method, Taylor and Rowlinson (1955) obtained water activity coefficient values up to x_w of 0.195 and fitted their data using the two-suffix Margules Eq. (16) with $A=-\mathrm{1.957}$ (±0.062). Figure 12a shows that up to x_w of 0.42 our HHTDMA-derived values of γ_w are in excellent agreement with the Taylor and Rowlinson (1955) data fit, indicating that the simple two-suffix Margules equation with $A=-\mathrm{1.957}$ is also applicable for deep metastable areas. The water activity coefficients obtained in this study were also compared with those reported by Suda and Petters (2013) (Fig. 12a, green line). At x_Gl<0.07 their HTDMA-derived γ_w values are in good agreement with ours (within ∼0.2 %), while at x_Gl>0.07 a systematic deviation is observed, reaching ∼7 % at x_Gl=0.25. The observed difference can be explained by the fact that Suda and Petters (2013) used the assumption of volume additivity to calculate the water activity coefficient. Moreover, as mentioned above, in their study no shape factor correction for the dry particles was made.

In addition, Fig. 12a shows the glucose activity coefficients, which are compared to bulk measurements by Miyajima et al. (1983) obtained with the isopiestic method (black symbols). One can see that our ln λ_Gl values are in good agreement with literature data points. For future applications, we fitted our ln λ_Gl data (up to x_w=0.42), together with the Miyajima et al. (1983) bulk results, using a polynomial fourth-order fit function. The obtained fitting coefficients are listed in Table 3.

Table 3Fitted parameters (± standard deviation) of ln λ_Gl as a function of the mole fraction of glucose, x_Gl, in glucose aqueous solution; n and R² are the number of data points and the coefficient of determination of the fit, respectively.

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Figure 12b shows the HHTDMA-based molal osmotic coefficient of glucose, Φ_Gl, as a function of water activity. The molal osmotic coefficient was calculated from Eq. (21), where μ_Gl was obtained using Eq. (14). The obtained data pairs Φ_Gl vs. a_w are fitted using the theory relation (Eq. 23) proposed by Rudakov and Sergievskii (2009) (Sect. 4.3). The best-fit value of the hydration number, h⁰, is 1.88±0.04 (n=75; R²=0.858). That is close to $h^{\mathrm{0}}=\mathrm{1.7}\pm \mathrm{0.5}$ reported by Rudakov and Sergievskii (2009). Our HHTDMA-based values of Φ_Gl are within h⁰±0.5 (gray shaded area, Fig. 12b). At a_w>0.98 the Φ_Gl value is 1.034±0.025 (average ± propagated uncertainty; 11 data points). This result indicates that even in diluted glucose solution, nonideality caused by the hydration of glucose molecules still persists. Experimental values of Φ_Gl were accompanied by those obtained by Suda and Petters (2013) (black circles, Fig. 12b). In general, the Suda and Petters (2013) data points are close to our results. A noticeable deviation is observed in the water activity range of 0.85–0.95. The main reason is that the relatively small changes in the instrumental uncertainties of aerosol particle growth factors and in RH will lead to large uncertainties in the determination of their thermodynamic characteristics. Thus, for our HHTDMA system in the case of glucose aerosol particles in the RH range of 90 %–99 % the growth factor uncertainty of ∼0.6 % (Eq. 2) gives rise to uncertainty in Φ_Gl of ∼3 %.