Supplement of A predictive viscosity model for aqueous electrolytes and mixed organic– inorganic aerosol phases

Abstract. Aerosol viscosity is determined by mixture composition and temperature, with a key influence from relative humidity (RH) in modulating aerosol water content. Aerosol particles frequently contain mixtures of water, organic compounds, and inorganic ions, so we have extended the thermodynamics-based group-contribution model AIOMFAC-VISC to predict viscosity for aqueous electrolyte solutions and aqueous organic–inorganic mixtures. For aqueous electrolyte solutions, our new, semi-empirical approach uses a physical expression based on Eyring's absolute rate theory, and we define activation energy for viscous flow as a function of temperature, ion activities, and ionic strength. The AIOMFAC-VISC electrolyte model's ion-specific expressions were simultaneously fitted, which arguably makes this approach more predictive than that of other models. This also enables viscosity calculations for aqueous solutions containing an arbitrary number of cation and anion species, including mixtures that have never been studied experimentally. These predictions achieve an excellent level of accuracy while also providing physically meaningful extrapolations to extremely high electrolyte concentrations, which is essential in the context of microscopic aqueous atmospheric aerosols. For organic–inorganic mixtures, multiple mixing approaches were tested to couple the AIOMFAC-VISC electrolyte model with its existing aqueous organic model. We discuss the best-performing mixing models implemented in AIOMFAC-VISC for reproducing viscosity measurements of aerosol surrogate systems. We present advantages and drawbacks of different model design choices and associated computational costs of these methods, of importance for use of AIOMFAC-VISC in dynamic simulations. Finally, we demonstrate the capabilities of AIOMFAC-VISC predictions for phase-separated organic–inorganic particles equilibrated to observed temperature and relative humidity conditions from atmospheric balloon soundings. The predictions for the studied cases suggest liquid-like viscosities for an aqueous electrolyte-rich particle phase throughout the troposphere, yet a highly viscous or glassy organic-rich phase in the middle and upper troposphere.


define the charge fraction ψ a as the absolute amount of charge contributed by anion a relative to the sum of absolute charge contributions from all negative charges present (or alternatively, relative to the sum of all positive ones) in the mixture, and introduce a cation-anion pair contribution weighting term, 15 τ ′ c,a represents the fractional amount of the hypothetical, neutral electrolyte component el consisting of cation c and anion a, where ν c,el is the stoichiometric number of cations in a formula unit of this electrolyte. Note that there is only one such cation-anion combination per specific type of cation and anion. In Eqs. (17 -18), x c can be understood as the molar amount of cation c in solution, normalized by the total molar amount of all species. Therefore, it is clear that either using an absolute, mole-based scaling (τ ′ c,a ) or a relative mole-fraction-based scaling (τ c,a , Eq. 18) offer a description of the amounts of each When describing the viscosity contributions from cation-anions pairs using Eq. (14), in order to avoid excessive weight being attributed to a certain ion pair, we advocate that one should strive for an unbiased representation of the solution by means of accounting for all possible contributions from cation-anion pairs in a charge-equivalent manner of weighting. As a 30 counter-example, an excessive, unbalanced weighting would likely occur if one were to pair, e.g., all Mg 2+ with all SO 2− 4 present, thereby giving a relatively high weight to the c Mg 2+ ,SO 2− 4 parameter (Eq. 14) in the mixture viscosity calculation.
This may bias this model prediction toward the viscosity of aqueous MgSO 4 (at the same ionic strength) and the resulting value may be substantially different from a viscosity calculation involving a different choice of cation-anion pairing, such as if we had first combined all Na + with SO 2− 4 and only the remainder of sulfate with magnesium. Hence, the specific sequence 35 of pairing the cations with the anions into hypothetical electrolyte components will lead to different viscosity predictions by the model (if several options are possible), making the prediction dependent on seemingly arbitrary choices and thereby ambiguous. Such ambiguity can be circumvented by introducing our τ ′ c,a -based weighting, in which a fractional amount of each cation is combined with a fractional amount of each anion, proportional to the charge-weighted amounts of the anions and the stoichiometry of the electrolyte unit formed.

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In our example, Na + is paired with all anions (Cl − , SO 2+ 4 , Br − ) in such a way that the largest fractional amount of Na + is paired with SO 2+ 4 , the second-largest amount is paired with Br − , and the smallest amount is paired with Cl − . We can calculate the exact proportions for Na + by computing the charge-based fractions for each counter-ion (anion) using Eq. (S1). Here, while for each of the Na + -anion pairs, Eq. (S2) yields The τ ′ c,a value can be calculated for the other five potential charge-neutral cation-anion pairs, yielding τ ′ These values add up in a way that is stoichiometrically consistent, e.g., extracting the Na + amount from these hypothetical electrolyte component 23 mol = 5 mol Na + . In our implementation of this approach, the normalized, mole-fraction-based version of τ c,a is used directly in Eq. (16).

S2 Computational efficiency of organic-inorganic mixing approaches
We tested the speed of the three mixing approaches, finding that the ZSR-style mixing approach takes approximately five to six times longer than aquelec or aquorg. Results are shown in Tables S1 and S2.

S3 Additional ternary and quaternary aqueous electrolyte mixtures
Figures S1 to S5 show additional data for ternary and quaternary aqueous electrolyte mixtures. See Table 4 for information on 60 each data set.

S4 Cation-anion parameter substitutions
When data are unavailable for certain cation-anion interactions, substitute values are used for the related parameters; see Table   S3. Table S3. Cation-anion pair substitutions in Table 7.

Missing Pair Replacement Pair
Comparison of AIOMFAC-VISC when fitted with all data or only binary data Figure S6 shows that some ternary and quaternary mixture predictions improve when these data are included in the fit, but mostly the results are similar. This suggests that viscosity measurements at higher concentrations, whether binary or multi-ion mixtures, would be the most useful additions to the currently available measurements. Fortunately, droplet-based measurement techniques can probe concentration ranges outside of the bulk range.

S6 Additional binary aqueous electrolyte curves
Figures S7 -S10 show zoomed-in versions of Figs. 5 -8. Figure S11 demonstrates the ability of the AIOMFAC-VISC model to capture a local viscosity minimum when only fitted to a single electrolyte. Figure S12 shows AIOMFAC-predicted water activity versus mass fraction of water for the binary aqueous nitrate solutions shown in Fig. 9. Figure S13 shows how AIOMFAC-VISC and the bulk measurements used to fit our model compare to droplet-based measurements from Power et al.
(2013), which were not used to fit our model because they were not available in tabulated form. Note that the bulk measure-75 ments in Fig. S13 (aggregated by Laliberté (2007)) have significant spread between 293 and 298 K.
The scatter among similar measurement points is one reason for the inclusion of a 2 % uncertainty in viscosity applied to all bulk measurements. This is demonstrated by Fig. S14, which shows measurements and AIOMFAC-VISC predicted viscosities for temperatures between 268.15 K and 328.15 K. Figs. S15 and S16.

S7 Additional aqueous organic-inorganic viscosity predictions
S8 Mixed α-pinene SOA + ammonium sulfate aerosol components The aerosol system discussed in Sect. 5 and featured in Figs. 12-14 is defined in Table S4.
85 Table S4. Components for α-pinene SOA : ammonium sulfate containing aerosol with OIR = 1. Surrogate components for α-pinene oxidation by ozone are derived from MCM, and their fixed dry amounts are given in mol m −3 in the particulate matter (PM) phase.
Name ( Table 4 for information on number of data points, the ranges of temperature, concentration, and viscosity for each data set. η • denotes unit viscosity (1 Pa s). See also Fig. 1.