2.2.1 Thermodynamic analysis of observations

We have used the thermodynamic model ISORROPIA II (Fountoukis and Nenes, 2007) to determine the liquid water content and composition (including H⁺) of an ${\mathrm{NH}}_{\mathrm{4}}^{+}$–${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$–${\mathrm{NO}}_{\mathrm{3}}^{-}$–Cl⁻–Na⁺–Ca²⁺–K⁺–Mg²⁺–water inorganic aerosol (or a subset therein) and its partitioning with corresponding gases. A molality-based definition of pH is used:

where ${\mathit{\gamma }}_{{\mathrm{H}}^{+}}$ is the hydronium ion activity coefficient (assumed = 1; note that the binary activity coefficients of ionic pairs, including H⁺, are calculated in the model), and ${\mathrm{H}}_{\mathrm{aq}}^{+}$ (mol kg⁻¹) and ${\mathrm{H}}_{\mathrm{air}}^{+}$ (µg m⁻³) are the hydronium ion concentration in particle liquid water and volume of air, respectively. W_i and W_o (µg m⁻³) are particle water concentrations associated with inorganic and organic species, respectively. The pH predicted solely with W_i is systematically lower by 0.15–0.23 units but highly correlated (r²=0.97) to pH predicted with measured total particle water (W_i+W_o) for the southeast US (which includes the SOAS study), where W_o accounted for 35 % of total particle water (Guo et al., 2015). For simplicity, we therefore use only W_i for the following pH calculations. ISORROPIA II was run in forward mode to calculate gas–particle equilibrium concentrations based on the input of total concentration of various inorganic species (e.g., NH₃ + ${\mathrm{NH}}_{\mathrm{4}}^{+}$). In all cases we also chose a metastable (not stable) solution, which assumes inorganic ions are associated with the aerosol components that are completely aqueous and contain no solid precipitate forms, other than CaSO₄ (${\mathrm{H}}_{\mathrm{aq}}^{+}$ is meaningless in a completely effloresced aerosol). Given this phase state requirement, we restrict the analysis to conditions where RH >40 %.

2.2.2 Mixing state

Because the aerosol composition data are bulk PM₁ or PM_2.5, and used as input to ISORROPIA II, the thermodynamic analysis implicitly assumes that all particle species were internally mixed, so that one value of pH represents the aerosols and governs the gas–particle partitioning. The existence of externally mixed particles may quantitatively and qualitatively affect the bulk thermodynamic analysis. To address this, we begin from the bulk analysis, then repeat the same calculation, augmenting each time the degree of external mixture of NVCs and sulfate. Direct measurements of aerosol mixing state during SOAS suggest that ambient particles indeed exhibit a range of mixing states (Bondy et al., 2018). In the external mixing analysis, the bulk aerosol is split into two subgroups: (1) species largely found in PM₁ (e.g., ${\mathrm{NH}}_{\mathrm{4}}^{+}$ and ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$) and (2) species found in PM_1−2.5, which contains mostly the NVCs, ${\mathrm{NO}}_{\mathrm{3}}^{-}$, and some ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$ and ${\mathrm{NH}}_{\mathrm{4}}^{+}$. These two external mixtures are in equilibrium with gaseous NH₃ and HNO₃ and so interact through these species (i.e., ${\mathrm{NH}}_{\mathrm{4}}^{+}$ and ${\mathrm{NO}}_{\mathrm{3}}^{-}$ can move between the two). Nonvolatile species, such as ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$ and NVCs (Na⁺), remain in the original size class assumed at the start of the analysis. To determine the composition of the two subgroups, we iteratively solve for the equilibrium conditions, by sequentially calling ISORROPIA for each subgroup. The solution is found when the composition of each group no longer changes with iteration and both are in equilibrium with the gas-phase species (in this case, NH₃, HNO₃, and H₂O (water vapor)). The mass of each species (gas plus particle) is conserved at all times and constrained by the observations. Given that pH is size dependent and generally higher at larger sizes (Fridlind and Jacobson, 2000; Young et al., 2013; Bougiatioti et al., 2016; Fang et al., 2017), bulk pH is compared against an aerosol liquid water-weighted pH:

3.1.1 SOAS dataset

We first investigate the issue of R discrepancy using PILS-IC PM_2.5 data from a 12-day period of the SOAS campaign. To test the sensitivity of ISORROPIA predictions to the level of NVCs, we ran the model with three different Na⁺ concentration inputs, with all other inputs remaining the same, including total ammonium (NH_x = ${\mathrm{NH}}_{\mathrm{4}}^{+}$ + NH₃), ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$, ${\mathrm{NO}}_{\mathrm{3}}^{-}$, and Cl⁻. Ca²⁺, Mg²⁺, and K⁺ inputs were set to zero as they were mostly below detection limits. Three sets of Na⁺ input concentrations were tested: (1) measured PM_2.5 Na⁺ from PILS-IC, including data below the LOD (data below LOD are clearly identified in the plots); (2) Na⁺ determined from an ion charge balance, Na⁺ = $\mathrm{2}{\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$ + ${\mathrm{NO}}_{\mathrm{3}}^{-}$ + Cl⁻ − ${\mathrm{NH}}_{\mathrm{4}}^{+}$ (unit: nmol m⁻³), hereafter referred to as “inferred NVCs”; and (3) Na⁺ =0, which corresponds to ignoring NVCs all together.

The LOD of PILS-IC Na⁺ was 0.07 µg m⁻³, which is close to the average Na⁺ concentration for the whole observation time-series. In the following, Na⁺ data below the LOD are used in the analysis. Although in most studies data below LOD are excluded, here we include them to allow a continuum in the analysis down to zero NVCs. There is no obvious discontinuity in the results for data above and below the Na⁺ LOD (e.g., see Figs. 1 and 4) and Na⁺ below the PILS LOD still roughly agrees with MARGA measurements of Na⁺, which has a lower LOD (see Fig. S2a in the Supplement).

Figure 1Time series of various measured and ISORROPIA-predicted parameters and PM_2.5 component concentrations for the SOAS study. Specific plots are as follows: (a) Na⁺ and ${\mathrm{NO}}_{\mathrm{3}}^{-}$; (b) ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$; (c) ${\mathrm{NH}}_{\mathrm{4}}^{+}$; (d) NH₃; (e) total ammonium (NH_x = ${\mathrm{NH}}_{\mathrm{4}}^{+}$ + NH₃) to sulfate molar ratio (${\mathrm{NH}}_x/{\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$); (f) ammonium-sulfate ratio (R= ${\mathrm{NH}}_{\mathrm{4}}^{+}/{\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$); (g) particle-phase fractions of total ammonium, ε(${\mathrm{NH}}_{\mathrm{4}}^{+}$); and (h) particle pH. ISORROPIA-predicted results for the base case and three different Na⁺ inputs are shown: measured Na⁺ in blue, inferred nonvolatile cations (NVCs) from an ion charge balance (where the overall NVCs are represented here by Na⁺; Na⁺ = $\mathrm{2}{\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$ + ${\mathrm{NO}}_{\mathrm{3}}^{-}$ + Cl⁻ − ${\mathrm{NH}}_{\mathrm{4}}^{+}$, µmol m⁻³) in green, and zero Na⁺ in purple. The periods with measured Na⁺ below LOD are marked with grey backgrounds.

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Figure 2Comparisons of predicted and measured particle-phase fractions of total ammonium, ε(${\mathrm{NH}}_{\mathrm{4}}^{+}$) = ${\mathrm{NH}}_{\mathrm{4}}^{+}/{\mathrm{NH}}_x$. (a) The model prediction is based on an ISORROPIA input of measured Na⁺, NH_x, ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$, ${\mathrm{NO}}_{\mathrm{3}}^{-}$, and Cl⁻. (b) Same model input, but NVCs (represented by Na⁺) are inferred from an ion charge balance and (c) Na⁺ is set to zero. Orthogonal distance regression (ODR) fits are shown and uncertainties in the fits are 1 standard deviation (SD) (the ODR fit in a is based on all the data points). The uncertainty of measured ε(${\mathrm{NH}}_{\mathrm{4}}^{+}$) is derived from error propagation of ${\mathrm{NH}}_{\mathrm{4}}^{+}$ (20 %) and NH₃ (6.8 %) measurements. The best agreement is achieved by using measured Na⁺ as input.

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The inferred NVCs, determined from the charge balance, provide an upper limit of the NVC equivalents that can affect aerosol pH and satisfy solution electroneutrality. Overall, Na⁺ is chosen as a proxy NVC in our dataset because in this case it constitutes most of the NVC mass and does not precipitate out of solution. The choice of Na⁺ as a NVC proxy, although appropriate here, may not be generally applicable, such as in regions with considerable dust contributions, as treating NVC as “equivalent Na⁺” in the thermodynamic calculations can result in large prediction errors (e.g., Fountoukis et al., 2009). Inferred NVCs have an expected high uncertainty due to error propagation of ${\mathrm{NH}}_{\mathrm{4}}^{+}$, ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$, ${\mathrm{NO}}_{\mathrm{3}}^{-}$, and Cl⁻ measurements (see Fig. S2b), and uncertainties in the dissociation state of sulfate (see Fig. S1 for the pH dependence). The concentration of H⁺ is ignored in the ion charge balance calculation for inferred NVCs, since H⁺ is at least an order of magnitude lower than the NVC ion equivalents, even for these very low pH data points (between 0 and 2). To demonstrate this, the average ion molar concentrations in PM_2.5 were ${\mathrm{NH}}_{\mathrm{4}}^{+}$ =35.4, ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$ =21.1, ${\mathrm{NO}}_{\mathrm{3}}^{-}$ =3.7, Na⁺ =2.9, and Cl⁻ =0.82 nmol m⁻³ by PILS-IC, compared to ISORROPIA-predicted H⁺ =0.31 nmol m⁻³. For the three datasets used in this study, the difference in Na⁺ predicted from an ion balance without considering H⁺ compared to including H⁺ is less than 1 % for SOAS and SEARCH CTR and 6 % for the WINTER study (see Fig. S3 in the Supplement). In the following, we have not included H⁺ in the ion balance. In SOAS, inferred NVCs are generally above zero, indicating a cation deficiency, but 8 out of 229 points (3 % of the data) were slightly below zero. In these cases, a small positive value of 0.005 µg m⁻³ was assigned to inferred NVCs. Including these data has no effect on the results because the observed R was ∼2. Figure 1a shows that the inferred NVCs were always higher than the measured Na⁺. The inferred NVCs from PILS and MARGA generally agree with each other and also agree with the total NVCs from MARGA measurements before 18 June, suggesting that the magnitude of inferred NVCs is reasonable (see Fig. S2). The larger differences after 18 June are likely from difficulties in detecting NVCs in low concentrations.

The SOAS study period investigated here includes an episode of high Na⁺ associated with a sea-salt (NaCl) aerosol event (Fig. 1a). This provided an opportunity to assess the role of NVCs on pH when concentrations were substantially above LOD. The observed Na⁺ is mainly associated with ${\mathrm{NO}}_{\mathrm{3}}^{-}$ (Fig. 1a) and to a lesser degree with Cl⁻. These ions are highly correlated (Na⁺–${\mathrm{NO}}_{\mathrm{3}}^{-}$ r²=0.82 and Na⁺–Cl⁻ r²=0.64) and indicate some level of chloride depletion as the observed ${\mathrm{Cl}}^{-}/{\mathrm{Na}}^{+}$ ratio was 0.24±0.16 (mol mol⁻¹) (mean ± SD), whereas fresh sea salts would have a molar ratio close to 1 (Tang et al., 1997). Chloride depletion occurs when an acid, such as HNO₃, is mixed with NaCl, producing HCl that evaporates owing to its higher volatility relative to HNO₃ (e.g., Katoshevski et al., 1999; Fountoukis and Nenes, 2007), resulting in a loss of aerosol Cl⁻. The chloride depletion in sea-salt aerosols during the SOAS study was discussed in detail by Bondy et al. (2017). Cl⁻ concentrations were sufficiently small (0.03±0.04 µg m⁻³, LOD = 0.01 µg m⁻³) compared to the dominant and nonvolatile anion ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$, and HCl was not included in the model input, so Cl⁻ had negligible effect on ISORROPIA predictions of pH and molar ratios. Periods where Na⁺ was closer to typical background levels and near or below the LOD lead to similar conclusions in the following analysis.

Figure 1 shows the effect of Na⁺ on ISORROPIA-predicted ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$, ${\mathrm{NH}}_{\mathrm{4}}^{+}$, NH₃, R, and pH. Figure 1b and e show that measured and predicted ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$ and NH_x are always identical. ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$ completely resides in the aerosol phase in all calculations. The model predicts the gas–particle partitioning by conserving NH_x, so the discrepancy between modeled and measured R must result from variation in the model prediction of NH_x partitioning. It is noteworthy that ${\mathrm{NH}}_x/{\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$ is practically always above 2, indicating excess NH_x compared to ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$. Under such conditions, conventional thought suggests that NH₃ must completely neutralize sulfate so that it can be in the form of ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$ (Kim et al., 2015; Silvern et al., 2017); this view, however, neglects the large difference in volatility between ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$ and NH_x, which thermodynamic models consider. Because of this, PM_2.5 can remain highly acidic, with a pH between 0 and 2 (Fig. 1h), even if there is a large amount of excess NH_x (Weber et al., 2016).

Figure 3Comparisons of PM_2.5 ammonium-sulfate molar ratios (R) between measurements and ISORROPIA predictions for the base case but with differing Na⁺ inputs. Data are from the SOAS study. Red numbers are the means and red error bars are 1 SD. Standard box–whisker plots are shown, with 100 % and 0 % data indicated by black error bars. The top and bottom of the box are the interquartile ranges (75 % and 25 %) centered around the median value (50 %). Comparisons include all data and periods when measured Na⁺ > LOD of 0.07 µg m⁻³.

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Figure 4Effect of nonvolatile cations (NVC) on the PM_2.5 ammonium-sulfate molar ratios (R) and pH as a function of measured Na⁺ concentration and organic aerosol (OA) mass fractions for the SOAS dataset studied. Plot (a) is ΔR versus measured Na⁺, (b) is ΔR versus measured OA mass fraction (OA mass divided by total particle mass reported from AMS), and (c) is ΔpH versus measured Na⁺. Grey diamonds in plots (a) and (b) are for ΔR equal to the measured R minus 2. Orange circular points are for ΔR equal to ISORROPIA-predicted R with measured Na⁺ included in the model input minus ISORROPIA-predicted R without Na⁺ in the model input. ΔpH in plot (c) is determined in a similar way. ΔR is negative since including Na⁺ in the thermodynamic model results in an R lower than 2, whereas not including Na⁺ results in an R close to 2 (see Fig. 3). ODR fits are shown and uncertainties in the fits are 1 standard deviation. A plot similar to (b), but versus OA mass concentration can be found as Fig. S6. The vertical dotted line is the Na⁺ LOD of 0.07 µg m⁻³. Regions where Na⁺ is below LOD are marked with grey backgrounds.

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Comparing measured to ISORROPIA-predicted NH₃–${\mathrm{NH}}_{\mathrm{4}}^{+}$ partitioning (particle-phase fraction of total ammonium, ε(${\mathrm{NH}}_{\mathrm{4}}^{+}$) = ${\mathrm{NH}}_{\mathrm{4}}^{+}/{\mathrm{NH}}_x$) can be used to test the sensitivity to NVC input concentrations. Figures 1g and 2a show very good agreement between measured and observed NH₃–${\mathrm{NH}}_{\mathrm{4}}^{+}$ partitioning when measured Na⁺ is used in the model. Using inferred NVCs generally results in an underestimation of ε(${\mathrm{NH}}_{\mathrm{4}}^{+}$). This is consistent with using overestimated NVC levels – as the resulting pH is overestimated (Fig. 1h), which in turn shifts a fraction of the ${\mathrm{NH}}_{\mathrm{4}}^{+}$ to gas-phase NH₃ and biases ε(${\mathrm{NH}}_{\mathrm{4}}^{+}$) low. Zero Na⁺ shows the opposite behavior (Figs. 1g and 2c); ε(${\mathrm{NH}}_{\mathrm{4}}^{+}$) is overpredicted because neglecting NVCs biases pH low, driving more NH₃ to the particle phase and biasing ε(${\mathrm{NH}}_{\mathrm{4}}^{+}$) high.

From the above it is clear that R strongly depends on how NVCs are considered in the thermodynamic analysis. Figure 1f shows the time series comparison between R for various Na⁺ levels included in the ISORROPIA input. Figure 3 shows the summary statistics for various comparisons of R. For the SOAS analyzed time period, the predicted R using measured Na⁺ was on average 1.85±0.17. As expected, predicted R was significantly lower when inferred NVCs were used (mean $R=\mathrm{1.43}\pm \mathrm{0.32}$) and highest for zero NVC (average $R=\mathrm{1.97}\pm \mathrm{0.02}$) in the thermodynamic analysis (see Figs. 1f and 3). The average measured R was 1.70±0.23 for all PILS data and 1.61±0.19 excluding the points with Na⁺ below LOD. The MARGA-derived R is very similar, with measured $R=\mathrm{1.78}\pm \mathrm{0.18}$ for all data and 1.65±0.15 for periods when PILS Na⁺ was above LOD (see Figs. S4 and S5 in the Supplement). Note that CSN data used by other investigators (Silvern et al., 2017; Pye et al., 2018) have a much lower R (Table S1 and S2 in the Supplement) due to a known ammonium sampling artifact (Yu et al., 2006) that cannot be accounted for, and so the dataset cannot be used in this analysis. Together, the analysis shows that (1) when NVCs are well constrained by measurements, predicted R is in close agreement with measured R (t test at α=0.05 confirms no statistical difference); (2) using inferred NVCs overestimates NVC and biases R low; however, the trend in predicted R generally follows measured R (see Fig. 1f), which argues that inferred NVCs can be a useful upper limit in NVC concentrations, when not constrained by measurements; (3) when NVC levels are zero, ISORROPIA predicts R∼2, which is a consequence of having the maximum possible condensation of NH₃ to the aerosol. Even if R∼2, however, the aerosol continues to remain strongly acidic.

3.1.2 Sensitivity of R and pH to NVCs and organic species

The results until now have clearly shown that the difference between predicted and observed R for this dataset is affected by the levels of NVC. However, it is important to assess whether organic species are associated with changes in the partitioning of semivolatile inorganics and aerosol acidity (Pye et al., 2018) or other unaccounted-for effects that drive the discrepancy between observed and predicted R. To avoid any cross correlations between organics and NVC variations, we examine how the discrepancy between observed R and its theoretical limit of 2 (corresponding to when NVC =0) correlates with organic aerosol. The results in Fig. 4 clearly suggest that $\mathrm{\Delta }R=R_{\mathrm{measured}}-\mathrm{2}$ increases with measured Na⁺ but does not depend on OA mass fraction (gray points) or OA concentration (see Fig. S6 in the Supplement). This suggests that ΔR is not driven by organic aerosol effects, but instead a poor representation of NVCs in the thermodynamic model. Figure 4 also shows that ISORROPIA-predicted R also depends on Na⁺. Predicted R with Na⁺ in the model input minus predicted R without Na⁺ decreases with increasing measured Na⁺ and is remarkably correlated with Na⁺ concentration (orthogonal linear regression, $\mathrm{\Delta }R=(-\mathrm{1.74}\pm \mathrm{0.03}$) Na⁺ + (0.001±0.003), r²=0.93). The decreasing trend in R with increasing Na⁺ can be explained simply by the pH increasing with Na⁺, as shown in Fig. 4c. With increasing pH, some ${\mathrm{NH}}_{\mathrm{4}}^{+}$ shifts to the gas-phase NH₃, resulting in lower ${\mathrm{NH}}_{\mathrm{4}}^{+}$ and lower R.

From the regression slope, for the SOAS measurement period analyzed, an average measured Na⁺ level of 0.07 µg m⁻³ (very small NVC concentrations) decreases R by 0.12 units. For a Na⁺ level of 0.3 µg m⁻³, R decreases by 0.5 units, from R=2 (i.e., no NVC) to R=1.5 (i.e., with NVC). Thus, ΔR is highly correlated and sensitive to Na⁺, both of which are not seen for the organic aerosol mass fraction. Mass fraction can be used as a proxy for organic film thickness too, given that the maximum possible thickness (and delay) associated with an organic film scales with (organic volume)^1∕3 or (organic mass)^1∕3.

In comparison to R, pH is less sensitive to inclusion of Na⁺, or other NVCs in general. ΔpH is only 0.09 for the average Na⁺ level of 0.07 µg m⁻³ and increases to 0.38 at 0.3 µg m⁻³ Na⁺ (Fig. 4b). The magnitude of ΔpH is relatively small and consistent with our previous studies where we investigated the effects of sea salt on pH (Guo et al., 2016; Weber et al., 2016). ΔpH would be higher in regions with more abundant NVC. For instance, a ΔpH unit of 0.8 was found in Pasadena, CA, where the average PM_2.5 Na⁺ mass was 0.77 µg m⁻³ (Guo et al., 2017). Differences in sensitivity of R and pH to Na⁺ from the slope of the linear regressions (Fig. 4c) are 1.74 (ΔR − Na⁺ slope) and 1.2 (ΔpH − Na⁺), respectively. NVC effects on R and pH are studied next for a very different aerosol dataset.

3.1.3 WINTER dataset

The R discrepancy is investigated for a different season and a larger and different region by repeating the analysis using the WINTER study dataset collected from the NSF C-130 research aircraft during wintertime. The aerosol inorganic composition data used in the analysis are from an AMS and are PM₁. In this study, NVCs were generally higher than those measured during SOAS, especially when the aircraft sampled near coastlines (e.g., PM₁ Na⁺ =0.23 µg m⁻³). Also, PM₁ nitrate was comparable to sulfate, largely owing to lower temperatures (${\mathrm{NO}}_{\mathrm{3}}^{-}$ 13 nmol m⁻³ vs. ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$ 11 nmol m⁻³) (Guo et al., 2016). Therefore, $R_{{\mathrm{SO}}_{\mathrm{4}}}$ was calculated instead of R.

The base case input to ISORROPIA II in this analysis included ${\mathrm{NH}}_{\mathrm{4}}^{+}$, ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$, and total nitrate (${\mathrm{NO}}_{\mathrm{3}}^{-}$ + HNO₃). (NH₃ should be included to determine NH_x for input, but was not measured. It was found to have a small effect on predicted pH; e.g., ∼0.2 higher pH when including an NH₃ concentration of 0.10 µg m⁻³, typical of the eastern US levels, and estimated from an order-of-magnitude iteration method, Guo et al., 2016). Figure 5a shows that ISORROPIA overpredicted $R_{{\mathrm{SO}}_{\mathrm{4}}}$ for the base case (i.e., when cations are not included) and that this deviation increases as molar ratios approach 2 when inferred NVCs are smaller. Again, NVC concentrations were determined as NVCs = Na⁺ = $\mathrm{2}{\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$ + ${\mathrm{NO}}_{\mathrm{3}}^{-}$ − ${\mathrm{NH}}_{\mathrm{4}}^{+}$ (unit: nmol m⁻³), where all NVCs are assumed to be Na⁺. (Note that the predicted $R_{{\mathrm{SO}}_{\mathrm{4}}}$ should be biased low since ${\mathrm{NH}}_{\mathrm{4}}^{+}$ was underpredicted due to lack of NH₃ data, resulting in some fraction of input particle-phase ${\mathrm{NH}}_{\mathrm{4}}^{+}$ repartitioned in the model to the gas phase; thus the deviation is even worse than shown). Figure 5a shows that $R_{{\mathrm{SO}}_{\mathrm{4}}}$ is highly sensitive to a lack of inclusion of NVCs when their concentrations are very low. However, when concentrations of NVC reach zero, predicted and measured $R_{{\mathrm{SO}}_{\mathrm{4}}}$ converge to the expected value of 2 (dark blue symbols in Fig. 5a). Interestingly, as predicted NVCs increase, predicted and measured $R_{{\mathrm{SO}}_{\mathrm{4}}}$ converge to zero, because NVCs progressively dominate the cations, and force ${\mathrm{NH}}_{\mathrm{4}}^{+}$ to evaporate. On average, predicted $R_{{\mathrm{SO}}_{\mathrm{4}}}$ was 1.68±0.51 versus the measured value of 1.47±0.43.

Figure 5Comparison between PM₁ ISORROPIA-predicted $R_{{\mathrm{SO}}_{\mathrm{4}}}$ and AMS-measured $R_{{\mathrm{SO}}_{\mathrm{4}}}$ ($R_{{\mathrm{SO}}_{\mathrm{4}}}=$ (${\mathrm{NH}}_{\mathrm{4}}^{+}$ − ${\mathrm{NO}}_{\mathrm{3}}^{-}/{\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$) (mol mol⁻¹), where the ISORROPIA prediction is based on (a) ${\mathrm{NH}}_{\mathrm{4}}^{+}$, ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$, and ${\mathrm{NO}}_{\mathrm{3}}^{-}$ aerosol and (b) Na⁺, ${\mathrm{NH}}_{\mathrm{4}}^{+}$, ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$, and ${\mathrm{NO}}_{\mathrm{3}}^{-}$ aerosol, and both include HNO₃ to calculate total nitrate for the model input. All measurement data are from the WINTER study. NVCs were determined by an ion charge balance with the predicted molar concentration shown by symbol color. Error bars were determined by propagated uncertainties for $R_{{\mathrm{SO}}_{\mathrm{4}}}$ based on a 35 % AMS measurement uncertainty for ${\mathrm{NH}}_{\mathrm{4}}^{+}$, ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$, and ${\mathrm{NO}}_{\mathrm{3}}^{-}$ (Bahreini et al., 2009). Error bars are larger at higher ratios due to subtraction of higher concentrations of nitrate and so subject to greater measurement error. Data points with low ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$ levels (<0.2 µg m⁻³; 9 % of the total points) were excluded due to high uncertainties.

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In contrast to ISORROPIA-predicted $R_{{\mathrm{SO}}_{\mathrm{4}}}$ without NVCs, including NVCs (inferred NVCs) brings predicted and measured ammonium-sulfate molar ratios into agreement (Fig. 5b). Including or excluding H⁺ in the Na⁺ calculation produces similar results (Fig. S7). Findings based on other NVCs are shown in Fig. S8 in the Supplement. K⁺ and Mg²⁺ work similarly to Na⁺, while Ca²⁺ can precipitate sulfate in the form of CaSO₄ and so cannot be used. For Na⁺, the linear regression result is $R_{{\mathrm{SO}}_{\mathrm{4}},\mathrm{predicted}}=(\mathrm{1.089}\pm \mathrm{0.001})$ $R_{{\mathrm{SO}}_{\mathrm{4}},\mathrm{measured}}-(\mathrm{0.166}\pm \mathrm{0.002})$, r²=0.996. As found for the SOAS dataset, again, the molar ratio bias from the thermodynamic model appears to result from not including small amounts of NVC (e.g., in this case on average 0.15 µg m⁻³ Na⁺ or 0.26 µg m⁻³ K⁺). The average amount of inferred PM₁ Na⁺ from the ion charge balance was 0.15 µg m⁻³ – in this case smaller than what was measured offline, 0.23 µg m⁻³ (Guo et al., 2016) (in comparison, inferred NVCs are higher than the measured Na⁺ in the SOAS case). The analysis using measured PM₁ Na⁺ results in highly scattered data due to the high sensitivities of $R_{{\mathrm{SO}}_{\mathrm{4}}}$ to NVC and the significant Na⁺ measurement uncertainty at these low levels given the analytical sampling method used in this study (i.e., offline analysis).

3.2.1 Sensitivity of semivolatile species partitioning to NVCs

In our datasets, inferred NVCs group all NVCs, including K⁺ and Mg²⁺, into one species and are the upper limits of the NVCs based on the assumption of complete dissociation of all dissolved ionic species. For example, 10 % of the total sulfate is predicted to be ${\mathrm{HSO}}_{\mathrm{4}}^{-}$ and the rest as ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$ for the SOAS average pH ∼1 (see Fig. S1). Additional errors can occur if other ions are also missing, but this approach satisfies electroneutrality. Comparing ISORROPIA predictions that include the other major species, an inferred NVC input versus Na⁺ =0 input results in an average increase in pH by 0.32 for SOAS and 0.49 for WINTER. Even though the effect of NVC on pH may appear relatively small, the impact on predicted partitioning of a semivolatile species can be significant due to the highly nonlinear response of NH₃–${\mathrm{NH}}_{\mathrm{4}}^{+}$ or HNO₃–${\mathrm{NO}}_{\mathrm{3}}^{-}$ partitioning to pH (i.e., S curve). For example, as shown in Fig. S9 in the Supplement, a 0.3 unit pH bias in SOAS campaign could cause (i) ∼20 % bias in ε(${\mathrm{NH}}_{\mathrm{4}}^{+}$) or ε(${\mathrm{NO}}_{\mathrm{3}}^{-}$) prediction when ε(${\mathrm{NH}}_{\mathrm{4}}^{+}$) or ε(${\mathrm{NO}}_{\mathrm{3}}^{-}$) =50 % or (ii) no bias at all when the species are completely in one phase – ε(${\mathrm{NH}}_{\mathrm{4}}^{+}$) or ε(${\mathrm{NO}}_{\mathrm{3}}^{-}$) =0 % or 100 %. For the WINTER study, a 0.5 pH bias causes up to 30 % bias in ε(${\mathrm{NH}}_{\mathrm{4}}^{+}$) or ε(${\mathrm{NO}}_{\mathrm{3}}^{-}$). These partitioning biases may constitute a significant source of bias for aerosol nitrate formation, especially if the total nitrate present in the gas–aerosol system is significant. In fact, the bias from the NVC may completely change the predicted response of nitrate to aerosol emissions and lead to errors in the predicted vs. observed trends in pH, such as was seen in the southeastern US (Vasilakos et al., 2018).

3.2.2 Effect of NVCs in trends in pH and $R_{{\mathrm{SO}}_{\mathrm{4}}}$ in the southeastern US

The organic aerosol impact on NH₃ equilibration (Silvern et al., 2017) was postulated to address the decreasing trend in R in the southeastern US despite the substantial drop in sulfate. Weber et al. (2016) also noted this and proposed that it could be explained by ${\mathrm{NH}}_{\mathrm{4}}^{+}$ volatility. However, the thermodynamic model predictions of $R_{{\mathrm{SO}}_{\mathrm{4}}}$ in that study did not find a comparable decreasing $R_{{\mathrm{SO}}_{\mathrm{4}}}$ rate with time (see Fig. 6a), since the SOAS study mean PILS-IC Na⁺ concentration of 0.03 µg m⁻³ was applied to all historical data. With this constant input of 0.03 µg m⁻³, predicted $R_{{\mathrm{SO}}_{\mathrm{4}}}$ was nearly constant at ∼2 for the input ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$ range (Fig. 6a) and would only rapidly decrease below 1 µg m⁻³ ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$ (See Fig. 2b in Weber et al., 2016). Repeating the calculations using Na⁺ inferred from the ion charge balance of Na⁺–${\mathrm{NH}}_{\mathrm{4}}^{+}$–${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$–${\mathrm{NO}}_{\mathrm{3}}^{-}$, determined for each daily data point in the historical dataset, results in good agreement between observed and ISORROPIA-predicted $R_{{\mathrm{SO}}_{\mathrm{4}}}$ (Figs. 6 and S10). It also predicts a decreasing $R_{{\mathrm{SO}}_{\mathrm{4}}}$ rate of −0.017 yr⁻¹, which is fairly close to the measured rate at the SOAS site (Centreville, AL) of −0.021 yr⁻¹ (see Fig. 6a) and in the range of the $R_{{\mathrm{SO}}_{\mathrm{4}}}$ trend of −0.01 to −0.03 yr⁻¹ reported by Hidy et al. (2014) for the SEARCH sites throughout the southeast. In contrast, using these different Na⁺ input concentrations did not change the trends in ISORROPIA-predicted pH; in both cases, it remained relatively constant (Fig. 6b), but as expected the pH was slightly higher with higher input Na⁺ concentrations. Thus, including daily estimates of NVC in ISORROPIA, the conclusion that PM_2.5 pH has remained largely constant over the last 15 years remains, but the unexpected decreasing $R_{{\mathrm{SO}}_{\mathrm{4}}}$ trend can be accounted for only with including NVC effects and ${\mathrm{NH}}_{\mathrm{4}}^{+}$ volatility. These observations can all be explained by volatility of ${\mathrm{NH}}_{\mathrm{4}}^{+}$ (Weber et al., 2016), without the need to invoke organic effects on the ammonia partitioning.

Figure 6Mean summer (June–August) trends in (a) measured and predicted $R_{{\mathrm{SO}}_{\mathrm{4}}}$, (b) predicted PM_2.5 pH, and (c) inferred NVCs concentration and mole fraction at the SEARCH CTR site. Na⁺ was inferred from an ion charge balance of Na⁺–${\mathrm{NH}}_{\mathrm{4}}^{+}$–${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$–${\mathrm{NO}}_{\mathrm{3}}^{-}$. ISORROPIA inputs include the measured PM_2.5 composition (${\mathrm{NH}}_{\mathrm{4}}^{+}$, ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$, ${\mathrm{NO}}_{\mathrm{3}}^{-}$) and meteorological data (RH, T) at CTR. In all cases, $R_{{\mathrm{SO}}_{\mathrm{4}}}$ and pH were estimated with ISORROPIA II run in forward mode with an assumed NH₃ level of 0.36 µg m⁻³, which is the mean concentration from the SOAS study (CTR site, summer 2013), due to limited NH₃ data before 2008. Historical NH₃ mean summer concentrations at CTR were 0.2 µg m⁻³ (2004–2007) (Blanchard et al., 2013) and 0.23±0.14 µg m⁻³ (2008–2013) (Weber et al., 2016). Error bars represent daily data ranges (SD). Linear regression fits are shown and uncertainties in the fits are 1 SD. A total of 41 data points out of 609 (7 %) with observed daily mean $R_{{\mathrm{SO}}_{\mathrm{4}}}$ above 3 were considered outliers and not shown (if included the fit slope is $-\mathrm{0.023}\pm \mathrm{0.008}$ unit yr⁻¹).

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4.1.1 Explanation for role of NVCs in R based on bulk (internal mixture) analysis

Recapping the bulk analysis above where ions are all assumed to be internally mixed, we have shown that the observations relating R to NVCs, and deviations in R between models and observations, can be readily explained. First, when NVCs such as Na⁺ are present in the ambient aerosol and not included in the thermodynamic model, but some fraction of the associated anion pair is, the thermodynamic model will predict higher ${\mathrm{NH}}_{\mathrm{4}}^{+}$ than observed because the model will partition greater levels of available semivolatile cations (i.e., NH₃) to the particle phase (${\mathrm{NH}}_{\mathrm{4}}^{+}$) to conserve NH_x and make up for the missing NVCs. This leads to a predicted R near 2. The trends in measured R with measured Na⁺ are also expected. As noted before, measured R becomes increasingly less than 2 as measured Na⁺ increases because at higher Na⁺ bulk aerosol pH increases (Fig. 3c), resulting in lower ε(${\mathrm{NH}}_{\mathrm{4}}^{+}$) (see ${\mathrm{NH}}_{\mathrm{4}}^{+}$ S curve in Fig. S9 in the Supplement), shifting ${\mathrm{NH}}_{\mathrm{4}}^{+}$ to gas-phase NH₃. Other NVCs have similar effects as Na⁺, as long as soluble forms of the salts are observed (e.g., NaNO₃, Na₂SO₄, KNO₃, K₂SO₄, Ca(NO₃)₂, Mg(NO₃)₂). We have shown with this bulk analysis that accurately including NVCs in the thermodynamic analysis appears to largely resolve the disparity in predicted and measured R for the datasets we analyzed. But the bulk analysis is only an approximation of the actual aerosol mixing state. We next test if assuming an internal mixture will roughly represent the behavior of externally mixed aerosols in terms of the effect of NVCs on R, pH, and partitioning of semivolatile species. To assess this, we consider the behavior of external mixing cases.

4.1.2 Explanation for the role of NVCs in R based on external mixture analysis

An extreme (and unrealistic at the timescale of aerosol lifetime Zaveri et al., 2010) external mixture is where PM₁ is composed of all the measured ${\mathrm{NH}}_{\mathrm{4}}^{+}$, ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$ and PM_1−2.5 is composed of all the measured Na⁺ (all NVCs), ${\mathrm{NO}}_{\mathrm{3}}^{-}$, and Cl⁻. NH₃, HNO₃, HCl, and H₂O (water vapor) can still equilibrate between these externally mixed particle types (see Fig. 7a), given the relatively short equilibrating timescales for these sizes of particles (Dassios and Pandis, 1999; Cruz et al., 2000; Fountoukis et al., 2009). As Fig. 7b shows, for the extreme external mixing case (i.e., 0 % sulfate in PM_1−2.5), predicted R, combined from PM₁ and PM_1−2.5, is close to 2, deviating from the lower predicted R of 1.66 ±0.13 from the internal mixture. This is due to the vastly different pH of PM₁ (0.6) and PM_1−2.5 (4.1) (Fig. 7c), where all ${\mathrm{NH}}_{\mathrm{4}}^{+}$ is predicted to be in PM₁ and all ${\mathrm{NO}}_{\mathrm{3}}^{-}$ is predicted to be in PM${}_{\mathrm{1}-\mathrm{2.5}.}$

For more realistic mixing cases, where some fraction of the sulfate is mixed with NVCs (Bondy et al., 2018), the combined R of the external mixture decreases rapidly as more ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$ is mixed with Na⁺ in PM_1−2.5. At ∼20 % ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$ fraction in PM_1−2.5, the average levels of predicted R start to converge between external and internal mixtures (Fig. 7b). The difference in pH between PM₁ and PM_1−2.5 is also reduced to within one pH unit (Fig. 7c). With these small differences in pH, ${\mathrm{NH}}_{\mathrm{4}}^{+}$ can condense on both externally mixed aerosol groups. For example, PM₁ and PM_1−2.5 ${\mathrm{NH}}_{\mathrm{4}}^{+}$ are predicted to be 0.67 and 0.04 µg m⁻³, respectively (equal to the sum of the measured PM_2.5 ${\mathrm{NH}}_{\mathrm{4}}^{+}$ of 0.71 µg m⁻³). From this analysis, based only on data when Na⁺ was above the LOD, predicted R for the bulk and external mixture are the same when on average 18±7 % (by mass) of the PM_2.5 ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$ is in the PM_1−2.5 size range (i.e., mixed with Na⁺). This is comparable to inferences of mixing based on size-resolved aerosol measurements in the southeast (e.g., Fang et al., 2017 shows ∼30 % PM_2.5 ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$ mass in PM_1−2.5). Less internal mixing of ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$ with Na⁺ is needed when Na⁺ concentrations are lower. For the SOAS 12-day Na⁺ average level of 0.07 µg m⁻³, only 5 % of the ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$ (by mass) when mixed with Na⁺ produces the same results as the bulk totally internal mixture case (see Fig. S11 in the Supplement). Note that higher Na⁺ concentrations generally require more ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$ to obtain agreement in R between external and internal mixtures (scatter plots are shown in Fig. S12 in the Supplement).

The difference between the internally and externally mixed system is not as great as may be expected, especially for particle pH and liquid water (W_i) (Fig. 7c and d). Since liquid water levels are determined as the sum of the water associated with the various salts, the bulk liquid water generally equals the sum of the two externally mixed liquid water concentrations, based on the Zdanovskii–Stokes–Robinson (ZSR) relationship (Zdanovskii, 1936; Stokes and Robinson, 1966). Because the most hygroscopic salts (i.e., ${\mathrm{NH}}_{\mathrm{4}}^{+}$ and ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$; ${\mathrm{NO}}_{\mathrm{3}}^{-}$ concentrations are low) are in PM₁, PM₁ liquid water dominates over PM_1−2.5, making the combined pH of the external mixture nearly identical to the PM₁ pH (see Eq. 4 for combined pH calculation). The combined pH of the external mixture is also similar to that of internal mixture, regardless of the ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$ fractions (see Fig. 7c).