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

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 5 for viscous flow as a function of temperature, ion activities, and ionic strength. The AIOMFAC-VISC electrolyte model’s ionspecific 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 con10 text 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 15 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. 1 20 https://doi.org/10.5194/acp-2021-836 Preprint. Discussion started: 25 October 2021 c © Author(s) 2021. CC BY 4.0 License.


Viscosity and aerosols
The dynamic viscosity of various fluids and fluid mixtures is an important material property in industrial applications, cooking, and earth system science at large and small scales. The dynamic viscosity of a fluid characterizes its resistance to flow or deformation; its inverse is known as the fluidity. In the context of aerosol phases, viscosity is also important due to its 25 relationship with the dynamics and timescales of molecular mixing and diffusion (Koop et al., 2011;Reid et al., 2018). At room temperature, liquid water has a dynamic viscosity of approximately 10 −3 Pa s; honey one of approximately 10 1 Pa s and pitch approximately 10 8 Pa s. One can intuitively understand viscosity when attempting to pour each of these fluids out of a container: Water and honey clearly move, albeit at different speeds, while pitch is imperceptibly slow. It is useful to separate the viscosity regimes encountered in aerosol particles and other amorphous solutions into three broad categories, for exam-30 ple as defined by Koop et al. (2011): liquid (< 10 2 Pa s), semi-solid (10 2 Pa s -10 12 Pa s), and amorphous solid or glassy (> 10 12 Pa s).
Aerosols impact climate and public health. Natural and anthropogenic processes introduce immense quantities of primary and secondary aerosols into the atmosphere, including organic and elemental carbon, sulfates, nitrates, chlorides, and other inorganic material. Reactive organic compounds can form secondary organic aerosol (SOA), which can homogeneously nucleate 35 or deposit onto preexisting aerosol (Hallquist et al., 2009;Heald and Kroll, 2020). Aerosol particles often contain a mixture of inorganic and organic matter (Zhang et al., 2007); this is especially true in urban environments (Fu et al., 2012). As relative humidity (RH) changes, organic-inorganic mixtures will uptake or release water, which changes the concentration of solutes such as inorganic electrolytes and can potentially induce liquid-liquid phase separation (LLPS) (Shiraiwa et al., 2013); in fact, the individual "liquid" phases could also be semi-solid or even glassy in terms of viscosity. 40 At very high RH (including water supersaturation), aerosol particles may uptake water to the point where they become or remain homogeneously mixed liquid-like solution droplets while at lower RH, phase separation can occur and one or more of these phases can become relatively viscous. Such RH-dependent viscosity transitions have been observed in laboratory experiments of surrogate aerosol mixtures and reproduced in modelling studies (Reid et al., 2018). That aerosols can be highly viscous under certain conditions has raised a number of interesting questions. Does a high viscosity in an aerosol phase 1.2 Two popular frameworks: Jones-Dole and Eyring The two main theoretical frameworks for viscosity of aqueous electrolyte solutions are the Jones-Dole and the Eyring equations. The Jones-Dole equation is one of the earliest identified relationships for relative viscosity and is expressed as 55 where η rel is the relative (dynamic) viscosity, η exp is the measured viscosity, η 0 is the pure component viscosity of the reference solvent, c is the molarity of the dissolved electrolyte before dissociation, A is a semi-empirical constant that represents the longrange electrostatic forces between ions in solution described by Debye-Hückel theory, and B is the Jones-Dole coefficient (or sometimes simply called the B-coefficient), an empirical constant that defines the contribution from short-range ion-solvent interactions to the viscosity of the solution (Jones and Dole, 1929). A and B have been calculated for many electrolytes 60 and at many temperatures, with values available in the literature. The original Jones-Dole equation is only useful for dilute electrolyte solutions, but later extensions added parameters and terms to extend the concentration range in which it is applicable (Kaminsky, 1957;Lencka et al., 1998;Wang et al., 2004). The use of a reference electrolyte assumption can also be used to solve for ionic B-coefficients. For example, KCl contains a cation and an anion of roughly the same size and charge magnitude.
Therefore, the B-coefficient for KCl can be evenly split into contributions for K + and Cl − that both equal 1 2 B KCl (Cox et al.,65 1934; Kaminsky, 1957). B-coefficents for many ions have been calculated using the same basic approach (Jenkins and Marcus, 1995). Glasstone et al. (1941) introduced another equation for viscosity based on absolute rate theory, which we call the Eyring equation: temperatures as a standalone activity coefficient model or as part of an extended equilibrium framework to compute equilibrium gas-particle partitioning, including LLPS predictions (Zuend et al., 2010;Zuend and Seinfeld, 2012;Pye et al., 2018Pye et al., , 2020. In the latter case, the fully developed AIOMFAC-VISC model, detailed in the next sections, allows for the prediction of viscosities in coexisting liquid, semi-solid, or amorphous solid phases containing water, organic compounds, and inorganic ions. where V w and ∆g * w are the average molar volume and average molar Gibbs energy of activation for viscous flow of pure water. In aqueous electrolyte solutions, knowing ∆g * w and V w allows us to solve algebraically for the ∆g * and V contributions of the dissolved ions. Goldsack and Franchetto (1977b) split these contributions as and Here, J is the number of different kinds of ions in the mixture, x i is the mole fraction of ion i defined on the basis of dissociated ions, and x w is the mole fraction of water. Ionic ∆g * contributions can be calculated, as has been previously shown for Bcoefficients (Goldsack and Franchetto, 1977b).
Equation (2) depends on the molar Gibbs energy of activation for viscous flow, a quantity that can be estimated but is not 110 directly measurable. According to Eyring, viscosity can be conceptualized as the transient formation and refilling of holes

Contributions to Gibbs energy for viscous flow
In our model, this energy is decomposed into three component-specific, additive contributions. First, the energy required for solvent molecules to move from their original locations into vacant holes, or to form new holes, is the molar Gibbs energy for viscous flow of the solvent, ∆g * w . Here we focus on water as the only solvent of ions for the purpose of this part of the AIOMFAC-VISC model for aqueous electrolyte solutions; mixtures of water, organics, and ions will be considered in Sect. 3.4. Second, the energy required for dissolved ions to move from their original locations into vacant holes is the molar Gibbs energy for viscous flow of the ions, ∆g * i . Third, in highly concentrated solutions, cations and anions can interact sufficiently frequently that they can impact the viscosity of the solution. The energy required for temporarily coupled cation-anion entities to move from their original locations into vacant holes is the molar Gibbs energy for viscous flow for cation-anion pairs, ∆g * c,a . Finally, Eq. (3) is used to define the molar activation energy for viscous flow for water.

Gibbs energy contributions from ions and cation-anion pairs
Each individual ion is assigned two coefficients, c 0,i and c 1,i , and we express Here, a ref i is considered to be the molal ion activity for the given mixture composition, e.g. computed with AIOMFAC. Initial tests indicated that the use of a single fit parameter per ion would provide inadequate flexibility for the model to fit experimental 160 data, so a second parameter was included in Eq. (13). In our approach, we have no need for a reference electrolyte assumption, as molar Gibbs energy for viscous flow is defined for each ion, not each electrolyte. Note that Eq. (13) is consistent with the functional form of Eq. (12); we can write the right-hand side of Eq. (13) equivalently as ln (a ref i ) c0,i · exp (c 1,i ) and comparison to Eq. (12) identifies a * i as a * i = (a ref i ) c0,i+1 · exp (c 1,i ). Each cation-anion pair is assigned a single coefficient, c c,a and ∆g * c,a is a function of the square root of molal ionic strength, Molal ionic strength I is defined as where m i is the molality and z i the integer (relative) electric charge of ion i. Equation (14) is inspired by similar expressions used for Pitzer-based ion activity coefficient expressions, such as those within AIOMFAC (Zuend et al., 2008).

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Here, τ c,a is a special weighting term that accounts for contributions from all possible binary cation-anion pairs in a chargeand abundance-balanced manner. This can be accomplished by treating the aqueous solution as a mixture of charge-neutral cation-anion pairs, with each cation combined with each anion proportionally to the ion amounts involved. Consider the total positive charge in the aqueous electrolyte mixture, Jc c=1 x c ·z c , which is equivalent to the total negative charge, Ja a=1 x a ·|z a |, for an overall charge-neutral solution. We can define the charge fraction ψ a as the absolute amount of charge contributed by 180 anion a relative to the sum of absolute charge contributions of all negative charges present (or alternatively, relative to the sum of all positive ones) in the mixture, and the weighting term, 185 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. This treatment is further described in the Supplemental Information (SI), in Sect. S1. Temporary cation-cation and anion-anion pairs are unlikely to form to the same extent because similarly charged ions will repel each other. Pitzer models show that to a first-order approximation, those interactions can be neglected (e.g. Zuend et al., 2008).

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Some considerations bear mentioning.
∆g * w RT is a unitless quantity related through Eq.
(3) to the viscosity of pure water. At 298.15 K, pure water has a viscosity of 8.9 × 10 −4 Pa s, and ∆g * RT = 3.44. If the total Gibbs energy for viscous flow drops below this threshold, the Eyring equation will calculate a viscosity less than that of pure water. For certain aqueous electrolyte solutions that have a local viscosity minimum in the dilute range, this is a necessary condition. When a ref i values are less than one, the ionic contribution can become negative, allowing ∆g * RT < 3.44. To avoid viscosity values that are too low, the 195 cation-anion contribution may compensate by being more positive. This interplay of viscosity contributions from water, ions, and cation-anion interactions is delicate, and requires optimized coefficients. To avoid negative ∆g * RT and nonphysical behavior, we determined that all AIOMFAC-VISC coefficients should be positive real numbers.

Representing volume, charge, and hydration effects on viscosity
The effective size of the dissolved ions impacts the amount of Gibbs energy needed to activate viscous flow. Conceptually, if 200 ions have small volumes, they slip relatively easily through the intermolecular network and into available/generated openings, displacing very few water molecules in the process; this would correspond to low viscosity. By contrast, large ions must displace more neighboring (solvent) molecules, which requires more energy temporarily and indicates higher viscosity. If an ion has a high charge density -the ratio of charge to ion volume -it will strongly attract water molecules into a temporary hydration shell, increasing the apparent size of the moving ion. Conversely, if an ion has low charge density, this hydration 205 effect is reduced, sometimes negligible. In AIOMFAC-VISC, as part of the original AIOMFAC model, hydrated volumes are used for ions, ensuring that hydration effects are included where they are important. Ions in aqueous solutions with a strong hydration effect are termed "structure-making," while those with a large ionic volume and weak hydration effect are "structurebreaking" (Marcus, 2009). This behavior is also observed in the low-concentration mixture viscosity minimum observed in the viscosity curves of aqueous solutions of structure-breaking ions like K + and NH + 4 , but not those of structure makers like Li +

Molar volume of solution
We define the effective mean molar volume of the solution, V , as the mole-fraction-weighted mean of the pure-component molar volumes of the solvent and the dissolved ions, as in Eq. (5). A volume correction, c v , is defined for the model and applied in all instances as The c v term is included to account for potential discrepancies in attributed ionic volumes, some of which include partial hydration effects. AIOMFAC uses relative van der Waals volumes, which are calculated by solving for the volume of a sphere of radius r c and dividing by 15.17 × 10 −6 m 3 mol −1 , the volume of a reference subgroup (Abrams and Prausnitz, 1975;Fredenslund et al., 1975). The reference subgroup is used to calculate relative volumes for neutral molecules as well. For  Table 1.

Available viscosity measurements
At present, AIOMFAC can predict activity coefficients for a large number of atmospherically relevant cations and anions (Yin 225 et al., 2021), including the seven cations (H + , Li + , Na + , K + , NH + 4 , Mg 2+ , Ca 2+ ) and six anions (Cl − , Br − , NO − 3 , HSO − 4 , SO 2− 4 , I − ) considered for this study. Therefore, we used viscosity measurements for aqueous electrolyte systems that included combinations of these ions. Forty-three such systems were identified and used to fit the model. Ongoing work is extending AIOMFAC for additional ions of special relevance to aerosol particles, and future versions of AIOMFAC-VISC may include these ions. Bulk measurements (i.e. those taken with a conventional viscometer or rheometer) for 28 binary systems were used 230 to fit AIOMFAC-VISC, 26 of which were previously aggregated by Laliberté (2007); detailed references for these 26 systems are included in that article and its electronic supplement. Bulk data from 15 ternary and quaternary aqueous electrolyte systems were also used. Finally, three data sets of droplet-based measurements are included from Song et al. (2021) and Baldelli et al. (2016). The aggregated data include measurements at temperatures ranging from 263.15 to 427.15 K. Points at temperatures greater than 333 K were excluded from our model fit, to avoid biasing the model toward relatively high temperatures and 235 because it is unlikely that aerosols will experience temperatures above 333 K in Earth's atmosphere. Ultimately, 6,625 data points were used to fit the AIOMFAC-VISC electrolyte model.
Data availability varies considerably across systems. The systems with the most data are aqueous KCl, NaCl, LiCl, and CaCl 2 -each with more than 500 points. The mean number of data points per data set is 144, but some systems like HCl, HNO 3 , and NaHSO 4 each contain fewer than 20 points. As shown in Figs. 4-7, most viscosity measurements are clustered in the dilute concentration range. In fact, less than 4 % of the viscosity measurements used to fit AIOMFAC-VISC are at mass fractions of water below 0.5. The highest available mass concentrations for bulk measurements were for Ca(NO 3 ) 2 , H 2 SO 4 , and NH 4 NO 3 , where measurements are available to mass fractions of water below 0.3 (solute mass fractions above 0.7). Some systems remain close to the viscosity of pure water throughout the concentration range, while others span multiple orders of magnitude -see the right two columns of Table 2. For pure water at 298 K, log 10 (η/η • ) = −3.054. Structure-245 breaking electrolytes can be identified where log 10 (η/η • ) min is less than −3.054. Ca(NO 3 ) 2 includes the greatest range, with viscosities between 10 −4 and 10 −1 Pa s, and approaching even higher values for the most concentrated solutions observed in laboratory experiments. While still in the liquid-like viscosity range, these high concentration data are of particular interest for aerosol modelling. More recently, techniques such as poke-and-flow, bead mobility, and holographic optical tweezers have enabled viscosity measurements for droplets (Reid et al., 2018). Due to their small size and absence of contact with 250 solid surfaces, aqueous droplets often attain concentrations of solute exceeding the bulk solubility limits, suggesting higher viscosities are likely to occur in nature.

Viscosity temperature dependence
Viscosity is strongly temperature dependent, and some viscosity models define their coefficients differently at each temperature,  Zhang and Han (1996) describe the accuracy as within 0.05 % for their viscosity measurements of aqeuous NaCl and KCl solutions. Wahab and Mahiuddin (2001) reported an error of 0.5 % for aqueous calcium chloride solutions. A proxy for viscosity error is the scatter 270 of our training data. Viscosity values measured at the same temperature and nearly identical concentrations show considerable scatter in multiple data sets (e.g., K 2 SO 4 , NaNO 3 , and KBr; see Fig. S8), likely owing to different measurement techniques and/or measurement, calibration, and transcription error. Laliberté (2007) found the standard deviation of their viscosity resid-ual to be 3.7 % of the average experimental viscosity for 74 data sets consisting of over 9,000 data points in total. Due to the wide range of reported errors for viscosity and the scatter among measurements at similar concentrations, we decided to 275 treat all measurements as if they included a 2 % error. This 2 % error is also included in the objective function used to fit the model. Displayed on a logarithmic scale, this error for bulk viscosity measurements is generally smaller than the size of plotted symbols, so error bars are mostly not shown. See Tables 2-4 for information on the temperature, concentration, and viscosity ranges of these data sets. The error for droplet-based viscosity measurements is typically larger than the error in bulk measurements, in part owing to the difficulty of precisely knowing the water content of the droplets (at a certain RH) examined 280 with these techniques.

Simultaneously fitting the AIOMFAC-VISC electrolyte model
We used a combination of global optimization methods to simultaneously fit the c v , c 0,i , c 1,i , and c c,a coefficients based on the ions and cation-anion pairs described by 33 aqueous electrolyte systems. All single-ion coefficients were fitted to data from multiple systems, e.g., c 0,K + and c 1,K + are simultaneously fitted to all data points that include the K + ion. First, we 285 used a method described by Zuend et al. (2010) called "best-of-random differential evolution" (BoRDE), which is based on the Differential Evolution algorithm by Storn and Price (1997), a robust global optimization method. To implement BoRDE, we borrowed code from Zuend et al. (2010). After honing in on the coefficients with BoRDE, we switched to the constrained global optimization method (GLOBAL) by Csendes (1988), which implements the Boender-Rinnooy Kan-Stougie-Timmer algorithm in Fortran (Boender et al., 1982). The Fortran 90 version of GLOBAL is freely available online (Miller, 2003).

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GLOBAL identifies clusters of local minima to efficiently survey the parameter space, sometimes substantially improving upon the solution found by BoRDE. Inherent in both the GLOBAL and BoRDE fitting processes is an objective function, which is used to evaluate the model performance for a given set of the adjustable coefficients, where a smaller objective function value indicates a better model fit to the data. This function often takes the form of a residual or error equation, such as root mean square error, but it can also be customized to suit the data and the intended use of the model. Our objective function 295 is described in Sect. 4.

Implementation for aqueous electrolyte systems
The AIOMFAC-VISC electrolyte model equations and coefficients have been implemented in Fortran and included as an optional module within the larger AIOMFAC model framework. The electrolyte model is incorporated alongside the aqueous organic viscosity model by Gervasi et al. (2020). AIOMFAC calculates activity coefficients for all components in a mixture 300 based on activity coefficient contributions from long-range, middle-range, and short-range molecular interactions. Those three contributions include effects from dissolved ions, so it is essential that viscosity calculations for aqueous electrolyte solutions proceed after these contributions have been calculated. A number of input quantities are needed prior to calling the aqueous electrolyte solution viscosity module within AIOMFAC, including the calculation of the pure-component viscosity of water at given temperature, for which the parameterization by Dehaoui et al. (2015) is used. The mole fractions of water and the ions, 305 the activity coefficients, and the relative ionic volumes are all available through the AIOMFAC interface, computed by various Table 2. Data set information for bulk measurements of binary aqueous electrolyte solutions used to fit the AIOMFAC-VISC electrolyte model. All data have been aggregated by Laliberté (2007Laliberté ( , 2009) unless otherwise noted. a ww,min is the minimum mass fraction of water, which corresponds to the highest solute concentration.
procedures within the AIOMFAC computer program. Equations (13) -(19) are then evaluated for the system and the mixture's dynamic viscosity is calculated via Eq. (2).  3.4 Generalizing AIOMFAC-VISC: three mixing models for organic-inorganic systems In the aerosol context, particle phases will frequently contain a mixture of water, organic compounds, and inorganic ions.
Therefore, we introduce a second extension to AIOMFAC-VISC, enabling viscosity predictions for mixtures consisting of a coupled AIOMFAC-VISC mixing model for aqueous organic-inorganic mixtures can be designed in at least three ways.
In the following sections, we introduce three approaches for combining our aqueous electrolyte and aqueous organic viscosity models and discuss their differences in terms of physicochemical justification, implementation considerations and associated computational costs. Common to our approaches is the concept of describing the organic-inorganic system in each particle phase as a combination of two distinct subsystems: (1) an aqueous organic solution free of inorganic electrolytes, and (2) an 320 organic-free aqueous electrolyte solution. Each subsystem may contain any number of components aside from water. The split into subsystems allows us to apply the appropriate organic-or electrolyte-specific viscosity model for each subsystem. For a given overall mixture composition, there is no obvious way, but several reasonable ways, by which the water content can be split into contributions to each subsystem; hence, different options emerge. Also, since water is the only common component present in the two subsystems, its modified properties (outlined in the following) can be considered to indirectly account for 325 and mediate effects from interactions among ions and organics occurring in the actual (fully mixed) system.

Electrolyte-aware water mixed with organics
The first approach for computing the mixture viscosity, abbreviated as "aquelec", assumes that inorganic electrolytes dissolve exclusively in water as the predominant solvent for ions, which is typically a good approximation, especially under dilute aqueous solution conditions and/or in the absence of polar organic solvents. The key idea is to replace the pure component 330 viscosity of water, which is used in the prediction of the mixture viscosity of the aqueous organic subsystem, by the viscosity predicted for the aqueous electrolyte subsystem. This electrolyte-aware "pseudo-pure" water property substitute is then applied together with the properties of the organic components in the organic model, which is based in part on combinatorial-activityweighted contributions of water and organics to determine mixture viscosity . In the aquelec mixing approach, the following steps are taken: 335 1. Adjust the ion molalities, which are by default defined by the molar ion amounts relative to 1 kg of water plus organics, , to be instead redefined relative to 1 kg of pure water as solvent, where m i,aquelec = n i /W w .
In these expressions, n i is the molar amount of ion i, and W org and W w are the masses of organic and water components, respectively, present in the total mixture. This can be expressed using a conversion factor, λ, as follows: 2. Using m i,aquelec , calculate ion activities with Eq. (9) and ionic strength with Eq. (15).
which all ion molar amounts or mole fractions are summed.
4. Run the electrolyte model to calculate the viscosity of the aqueous electrolyte subsystem, ignoring organics.
5. Replace the pure-component viscosity of water in subsystem 1 with that of the aqueous electrolyte subsystem (electrolyteaware water).
6. Set the mole fractions of ions to zero to avoid double-counting their effects and renormalize the mole fractions of water 350 and organics for subsystem 1, so they become x w,aquelec = x w /(x w + x org ) and x org,aquelec = x org /(x w + x org ).
7. Run the organic model  for the established mixture of electrolyte-aware water and the organic components to compute the viscosity of the organic-inorganic mixture as a whole.

Organics-aware water mixed with ions
As opposed to aquelec, another option, "aquorg", assumes that all water mixes with organic components to create an "organics-355 aware" water component that will replace pure water as the solvent of ions in the organic-inorganic mixture. Unlike aquelec, which first computes the interactions between ions and pure water, aquorg prioritizes the calculation for aqueous organic mixture viscosity. This mixing model is similar to aquelec, but the steps proceed in a different order, as follows: 1. Run the organic model  to calculate the viscosity for the aqueous organic subsystem, ignoring ions.
2. Replace the pure-component viscosity property of water in subsystem 2 with that of the aqueous organic subsystem 360 (organics-aware water).
3. Add the mole fraction values of all organics to the mole fraction of water, and set the mole fractions of all organics to zero. Thus, the sum of moles of organics + moles of water is represented as moles of organics-aware water.
4. Run the electrolyte model for the mixture of organics-aware water and the inorganic ions to calculate viscosity of the organic-inorganic mixture.

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Note that for the aquorg mixing mode, it is not necessary to modify the ion molalities because they are computed relative to 1 kg of organics-aware water, which for this purpose is equivalent to the molality definition based on mass of water + organics in the denominator (as in the original mixture). The third mixing model is a Zdanovskii-Stokes-Robinson (ZSR) type mixing rule that preserves the organic-to-inorganic dry mass ratio (OIR). The ZSR mixing rule has been successfully used in many applications for the estimation of physical properties of a ternary mixture based on additive contributions from binary subsystems evaluated at the same water activity (i.e. RH under bulk equilibrium conditions) (Zdanovskii, 1936(Zdanovskii, , 1948Stokes and Robinson, 1966).
Unlike the other two mixing models described above, which only require a single call of the AIOMFAC program (for the 375 computation of activity coefficients), the ZSR-style approach requires an iterative numerical solution: multiple runs are needed to pinpoint the mass fraction of water of the aqueous electrolyte and aqueous organic subsystems such that they yield the same water activity as that determined for the full mixture. As water activity is an output of an AIOMFAC calculation, this requires solving a non-linear equation in one unknown (mass fraction of water) for each subsystem.
Our ZSR-style mixing rule first calculates the RH of the full organic-inorganic system. Next, we split the full system into 380 a salt-free aqueous organic subsystem (subsystem 1) and an organic-free aqueous electrolyte subsystem (subsystem 2). A modified version of Powell's hybrid method from the Fortran MINPACK library is used to calculate the water content and viscosity for the two subsystems at the target RH (Moré et al., 1980(Moré et al., , 1984. Finally, the organic-inorganic mixture viscosity is estimated using a weighted arithmetic mean of the logarithms of subsystem viscosities, which is equivalent to a weighted geometric mean of the non-log subsystem viscosities. The expression for the mixing rule, previously described in Song et al.
Expressions for f 1 and f 2 must ensure that the given OIR is preserved. Consider the mass W of the full system,

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where org, el, and w denote organic, electrolyte (salt), and water components, respectively. Subsystem 1 contains all of the organic mass and subsystem 2 contains all of the salt mass; the water content can be split in a way that preserves OIR. By defining the mass of the subsystems as OIR can be defined as contributions are defined as Combining (29) and (28), we find that 405 Note that f 1 and f 2 are not constant for constant OIR and must be recomputed at every RH step. In dynamic simulations, it is expected that ZSR mixing will be computationally expensive due to the multiple calls to AIOMFAC and the iterative approach.

Advantages and disadvantages of the different mixing models
The perfect mixing model for viscosity will be physically justifiable, efficient, and accurate. The ZSR mixing rule to determine water content is built on established thermodynamic arguments, but its implementation is computationally more expensive 410 than the other two mixing models. The "aquelec" and "aquorg" mixing models are about equally fast because neither requires an iterative approach, but the aquelec approach seems the more reasonable choice in terms of physicochemical justification.
The primary assumption of the aquelec mixing model is that ions are likely to dissolve preferentially in water. By contrast, the aquorg mixing model implicitly treats organic components similar to water in terms of acting as solvent mass for the calculations in subsystem 2, which is not always a good assumption. In terms of accuracy, recently, the ZSR mixing rule has 415 been shown to produce reasonable predictions of viscosity within an order of magnitude of measurements (Song et al., 2021). A ZSR mixing rule likely suffices for non-reactive/non-interacting mixtures that exist as Newtonian fluids over a wide RH range.
Some aqueous electrolytes and organic-inorganic mixtures, particularly those containing divalent cations, have been observed to undergo gel transitions at low RH (Cai et al., 2015;Richards et al., 2020b). Such gel phase transitions are not explicitly accounted for by AIOMFAC-VISC, and this may pose a challenge for the ZSR mixing rule. Predictions of organic-inorganic 420 mixture viscosity with the three different approaches are compared in Sect. 4.7.

Activity coefficient calculations
Concurrently with the viscosity calculations, a full calculation is also carried out to determine the activity coefficients of all components/ions in the mixed organic-inorganic solution, as is also done in the absence of viscosity calculations with AIOMFAC. This is necessary because the activity coefficients of the components/ions computed for the subsystems will differ 425 from those computed for the full system. water content, or temperature, makes the use of a logarithmic viscosity scale useful; hence the frequent use of log 10 (η/η • ) in this work.

Fitting AIOMFAC-VISC for aqueous electrolyte solutions
As discussed in Sect. 3.2, our method involved defining an objective function to fit the model. Our objective function is defined for each data set and takes the form where ι in the data point index, N is the number of data points, and σ is an uncertainty threshold. Summing the f obj values over all data sets and dividing by the sum gives the relative error contribution for each data set. Ca(NO 3 ) 2 , LiCl, and CaCl 2 contribute the largest shares of error, as shown in Fig. 1. Our objective function includes a 2 % uncertainty term (σ = 0.02) to characterize an approximate viscosity measurement error. However, it does not include additional consideration for the 445 asymmetric distribution of measurements across different ranges in concentrations or temperature at which the measurements were collected, which may affect the distribution of the objective function value.
Through trial and error, we arrived at a framework that includes two coefficients per ion, one per cation-anion pair, and one volume correction term that is used for all model calculations. With the 43 aqueous electrolyte systems included in our fit, 58 unique coefficients were identified, describing 13 ions and 31 cation-anion interactions. Several c c,a coefficients were 450 not covered by the measured systems but can still occur in AIOMFAC-VISC predictions; therefore, coefficients from similar cation-anion pairs were substituted in these cases, serving as approximations, e.g., c Mg 2+ ,Br − = c Mg 2+ ,Cl − . All values of these coefficients are included in Tables 5 and 6, and replacements are noted in Table S3. The fitted c v value is 1.558635.

Model design considerations
How many parameters are needed to accurately and meaningfully model the viscosity of a binary aqueous electrolyte solution?

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The answer to this question is not so simple, as some parameters are defined for the entire model, whereas others are soluteor ion-specific. The model by Lencka et al. (1998)

Comparison of AIOMFAC-VISC and the Laliberté model for aqueous electrolyte systems
Due to the substantial overlap in fitted data sets, we use the Laliberté model as a benchmark for AIOMFAC-VISC, both with 480 respect to its closeness of fit to bulk viscosity measurements and its extrapolative behavior. We fitted AIOMFAC-VISC to available bulk viscosity measurements, resulting in excellent agreement for all data sets, although that is less apparent when compared with the Laliberté model. For example, in Fig. 1, we see that in panels (b)-(d) and for all systems, AIOMFAC-VISC's error magnitude is greater than that of the Laliberté model. This is expected, because the Laliberté model is fitted to aqueous electrolyte solutions (with six specific, independent parameters for each system) as opposed to AIOMFAC-VISC, 485 which includes ion-specific coefficients shared among many electrolyte systems. The left-most panel displays mean bias error (MBE) for the binary systems defined in Table 2. MBE is defined as where N is the number of points in each data set, η calc,ι is the calculated viscosity value of either AIOMFAC-VISC or the Laliberté model, and η exp,ι is the viscosity value reported in the measurements at point ι. Overall, AIOMFAC-VISC does not exhibit systematic bias, with negative bias for 18 data sets and positive bias for 8 data sets. The magnitudes of MBE are generally larger for AIOMFAC-VISC than for the Laliberté model, which again is expected. Two systems stand out, however: NaNO 3 and Ca(NO 3 ) 2 , both of which show positive bias. These two systems include some of the highest viscosity values among the available measurements, which is a factor in their large contributions to the overall objective function error. NaNO 3 and Ca(NO 3 ) 2 also include both bulk and droplet-based measurements, and these data do not agree at low water content, 495 leading to larger fit residuals for these systems. It is also worth noting that the Laliberté model has its largest value of MBE for Ca(NO 3 ) 2 , suggesting that this system is difficult to model, even when using more adjustable parameters. Figs. 1b,c show mean absolute error (MAE), which is defined as and root mean square error (RMSE), which is defined as The most significant deviations from the measurements are for Ca(NO 3 ) 2 , NaNO 3 , LiCl, and CaCl 2 . The values of the root mean square error and the custom objective function, Eq. (32), are presented in panels (c) and (d) of Fig. 1, and reinforce the same result.   Table 2 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).
In Fig. 2, the panels are zoomed in individually to show how AIOMFAC-VISC and the Laliberté model align with the bulk 505 viscosity measurements over the covered concentration and viscosity ranges. KCl (Fig. 2a) and NH 4 Cl (Fig. 2d) show local minima in their measured viscosity curves, a characteristic of structure-breaking electrolytes that is better captured by the Laliberté model for these systems. In these panels, as well as for NaCl (Fig. 2b), it is evident that the Laliberté model has a closer fit with the measurements. Note that the panels for KCl and NH 4 Cl have extremely narrow vertical axes ranges, effectively only showing viscosities close to that of pure water. Some panels, by contrast, span more than one order of magnitude, with 510 AIOMFAC-VISC agreeing well with the highest viscosity measurements. We note that if AIOMFAC-VISC is fitted only to data for an individual binary electrolyte solution, such as that for NH 4 Cl shown in Fig. 2d, the model is capable of reproducing the local minimum in measured viscosity. This indicates that the shown deviations are a result of the simultaneous fit of the model to many data sets covering a wider range in concentrations and viscosities.

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Viscosity measurements for mixtures of water and more than one electrolyte are less common than those of binary aqueous solutions, but they better demonstrate AIOMFAC-VISC's predictive capacity. The data sets in Table 3 were used to fit AIOMFAC-VISC, but were not fitted by the Laliberté model. Therefore, we effectively compare AIOMFAC-VISC's fit for these multi-ion solutions to the Laliberté mixing model, the latter being a simple mass-fraction-weighted mixing rule. As these measurements are on the same order of magnitude as the viscosity of pure water, we change the units on the vertical axis to 520 mPa s, and include 2 % error bars.

Extrapolative behavior for binary aqueous electrolyte solutions at room temperature
In Figures 4 through 7, we compare AIOMFAC-VISC predictions with extrapolations from the Laliberté model at 295 K.
Agreement between AIOMFAC-VISC and the Laliberté model is excellent within the range of available measurements for each system, which are plotted as black circles. Outside of this range, the models diverge, sometimes to a large degree. It 545 is worth noting that crystallization is inhibited/neglected in both the AIOMFAC-VISC and the Laliberté model calculations, resulting in (mostly) smooth curves throughout the concentration range. Also, the Laliberté model occasionally depicts spurious behavior outside of the measurement range. When the Laliberté model exceeds its applicability limit, which is provided for each electrolyte in Laliberté (2007), it can sometimes produce negative viscosity values as output; on a logarithmic viscosity scale plot, these deviations are indicated by a sharp discontinuity in the viscosity curve. AIOMFAC-VISC never predicts negative 550 viscosity values, but at exceedingly low water activity, AIOMFAC by default stops its calculations when run for a single curve covering output from dilute to concentrated conditions. This is justified since the resulting water activities at low w w would be for conditions far beyond a realistic equilibrium RH in the atmosphere (or other environments). Water activity and w w vary differently for different aqueous electrolyte solutions as shown by comparing the upper and lower horizontal axis of each panel; so, the exact point at which the model output was stopped is different for each aqueous electrolyte solution, but is typically 555 below w w = 0.2.
For aqueous chloride salts/acids, (Fig. 4), AIOMFAC-VISC and the Laliberté model agree closely, generally to within one order of magnitude (even outside the concentration range of the measurements). For NaCl and LiCl (Fig. 4b,c), the Laliberté model projects a near linear increase in log 10 viscosity below the w w threshold of the measurements, while the AIOMFAC-VISC predictions include a more steep increase in viscosity below w w = 0.4, likely due to higher relative influence of the 560 ionic-strength-dependent cation-anion viscosity contributions. For KCl, NH 4 Cl, and MgCl 2 (Fig. 4a,d,e), the Laliberté model shows spurious behavior outside of the measurement range. In these cases, the AIOMFAC-VISC predictions are preferable because the curves remain smooth. In Figs. 5-7, AIOMFAC-VISC and the Laliberté model continue to agree closely. The AIOMFAC-VISC curve for H 2 SO 4 ( Fig. 5f) includes a notch below w w = 0.2, which indicates a relatively sharp change in the bisulfate dissociation degree as 565 predicted by AIOMFAC for the sulfate-bisulfate equilibrium in that system. For Ca(NO 3 ) 2 (Fig. 6g), the AIOMFAC-VISC curve closely fits the measurement points, but predicts higher viscosity than the Laliberté model below w w = 0.4, due to the influence of the droplet-based measurements used to fit this system; see comparison to droplet-based measurements in Figs. 8 and S5a. Figure S5 also shows AIOMFAC-VISC predictions for binary HI and LiBr, which are not fitted by the Laliberté model.

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Due to the lack of viscosity measurements at low mass fraction of water and the tendency for salts to crystallize at high concentration, it is difficult to determine quantitatively which model/curve, if any, is correct for any given case. What is clear, however, is that AIOMFAC-VISC provides an excellent level of accuracy in the composition range where measurement data are available and can be used in place of the Laliberté model in most instances.

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Unlike bulk viscosity measurement techniques, which determine viscosity for known composition (e.g. mass fractions), recent aerosol and/or microscopic droplet viscosity measurement techniques characterize viscosity with respect to known equilibrium defined by a 2 % change in aerosol water content, is shown in dashed curves. Bulk measurements (see Table 2) were collected at defined concentrations and converted to aw using AIOMFAC; the points collected at temperatures between 295 ± 5 K are shown. Song et al. (2021) used poke-and-flow and bead mobility techniques. Baldelli et al. (2016) used holographic optical tweezers.
water activity (RH) instead. A limited number of measurements of this type are available; we present results for three aqueous nitrate salts, Ca(NO 3 ) 2 , Mg(NO 3 ) 2 , and NaNO 3 . Figure 8 shows the predicted viscosity of aqueous nitrate salts over the full RH range with AIOMFAC-VISC model sensi-580 tivity represented by the upper and lower dashed curves. AIOMFAC-VISC model sensitivity is defined by a 2 % change in the aerosol water mass fraction, described in the supporting information of Gervasi et al. (2020). In the case of aqueous Ca(NO 3 ) 2 ( Fig. 8a), disagreement between the two measurement data sets is noticed, especially at lower water activities. AIOMFAC-VISC shows positive bias relative to the bulk measurements, and it shows better agreement with the aerosol measurements between a w = 0.65 and a w = 0.3. It is possible that these bulk measurements understate viscosity for aqueous Ca(NO 3 ) 2 , 585 which would mean that the large model deviation for this system is not necessarily so bad. In the case of Mg(NO 3 ) 2 (Fig. 8b) the aerosol measurements largely agree with the bulk measurements, and AIOMFAC-VISC correctly characterizes nearly every point. In both Fig. 8a,b there is one outlying data point at low a w with a stated viscosity value at 10 8 Pa s. In fact, Song et al. (2021) used 10 8 Pa s as the upper limit for their viscosity measurements. Such a high value reported may be best explained by the crystallization of Ca(NO 3 ) 2 or Mg(NO 3 ) 2 , but using the poke-and-flow measurement technique, it is difficult to distin-590 guish between glasses, gels, and crystallized aerosols. Crystallization is inhibited in the shown AIOMFAC-VISC predictions, likely explaining the divergence from those high-viscosity measurement points. Our AIOMFAC-based equilibrium model is capable of providing liquid-liquid and solid-liquid equilibrium calculations, but viscosity prediction would not be possible for the solid phase. In the case of NaNO 3 (Fig. 8c), there is rather poor agreement between the bulk measurements and the aerosol measurements by Baldelli et al. (2016). At a w < 0.2, the uncertainty of the AIOMFAC-VISC prediction for NaNO 3 widens 595 considerably, indicating that small changes in solution water content can greatly affect both the water activity and viscosity predictions. Indeed, a 2 % change in mass fraction of water corresponds to a much larger change in water activity for NaNO 3 (Fig. 6b) than for Ca(NO 3 ) 2 (Fig. 6e) or Mg(NO 3 ) 2 (Fig. 6g). This prediction indicates that NaNO 3 particles of semi-solid viscosity might be observed below ∼ 20 % RH, which corresponds to an observation of non-crystalline viscous NaNO 3 in particle rebound experiments by Li et al. (2017).

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Although viscosity measurements are not available, aqueous MgSO 4 particles have been observed as highly viscous liquids and/or (non-Newtonian) gels (Li et al., 2017;Cai et al., 2015;Richards et al., 2020a). AIOMFAC-VISC predicts high viscosity values for aqueous MgSO 4 , and a transition to a semi-solid viscosity below a w = 0.4. Richards et al. (2020a) differentiate gels from Newtonian liquids by the presence of an abrupt change in microrheology and the lack of shape relaxation (on practical experimental timescales). It is not possible to verify these findings with the present version of AIOMFAC-VISC, which does 605 not explicitly include consideration of liquid-to-gel phase transitions, and they clearly merit further study.
Nevertheless, the theory behind AIOMFAC-VISC can account to some extent for the unique behavior of aqueous MgSO 4 .
Mg 2+ and SO 2− 4 are both doubly charged ions, which likely attract water molecules into long-lasting hydration shells. As RH decreases, free water molecules evaporate from the particle, leaving behind the hydrated ions. These hydrated cations and anions agglomerate, forming chains and reducing the flow of molecules. No other aqueous electrolytes that we used to fit 610 AIOMFAC-VISC included two doubly charged ions, but we did include other electrolytes which contained either Mg 2+ or SO 2− 4 , and we have plotted them alongside MgSO 4 in Fig. 9. Below a w = 0.4, the MgSO 4 predicted viscosity is consistently higher than that of most of the other binary aqueous solutions shown by at least two orders of magnitude in viscosity, the exception being Na 2 SO 4 which is expected to effloresce above 50 % RH (Li et al., 2017;Ahn et al., 2010). Predicted viscosities for the other binary aqueous solutions remain below the semi-solid threshold (10 2 Pa s) down to RH = 10 %. MgCl 2 and Mg(NO 3 ) 2 produce nearly identical predictions, suggesting that the effects of chloride and nitrate anions are similar, or that ion interactions in MgSO 4 are more important than those of other magnesium-containing electrolytes. On the other hand, the predictions for the other sulfate-containing electrolytes differ substantially from each other. Na 2 SO 4 and Li 2 SO 4 produce higher predicted viscosities than (NH 4 ) 2 SO 4 , suggesting that the inclusion of a more charge-dense cation as the counter-ion to SO 2− 4 results in slightly higher viscosity. Sucrose is commonly used as a proxy for secondary organic aerosol because it has a similar oxygen-to-carbon ratio as highly oxidized organic aerosol components and viscosity and related diffusivity data are available in the literature (Evoy et al., 2019).
In Fig. 10, AIOMFAC-VISC viscosity predictions are tested for these systems at varying water activity, providing a comparison 625 of the three organic-inorganic mixing approaches described in Sect. 3.4. Each of these mixing approaches predicts viscosities between those of the relevant sucrose-free aqueous nitrate salt solution and the aqueous (salt-free) sucrose solution (plotted in grey). As the OIR increases, the mixture viscosity prediction approaches that of aqueous sucrose and as the OIR decreases, the prediction tends toward the viscosity of the aqueous nitrate salt. Figure 10a The aquelec mixing model predictions appear to agree best with the measurements for 1:1 sucrose-Ca(NO 3 ) 2 (Fig. 10a) and 60:40 sucrose-NaNO 3 (Fig. 10c), while the ZSR-style mixing rule performs best for 1:1 sucrose-Mg(NO 3 ) 2 (Fig. 10b) and 80:20 sucrose-NaNO 3 (Fig. 10d). In Fig. 10b, aquelec predicts values within the uncertainty of the measurements between 70 635 and 30 % RH, but underpredicts the measurements between 20 and 10 % RH. The aquorg mixing model consistently predicts lower viscosity values than the other two mixing models, and this negative bias is exacerbated at low RH. The measurements for binary solutions of some salts include abrupt increases in viscosity at low RH (5 % for Ca(NO 3 ) 2 and 35 % for Mg(NO 3 ) 2 , as shown in Fig. 8). This could be the result of crystallization of the salt, a glass transition, or a gel transition during the experiments. Regardless, when mixed with sucrose, this abrupt viscosity increase appears to be inhibited. Richards et al. 640 (2020b) found that ternary organic-inorganic mixtures containing certain doubly charged cations had higher viscosity than the corresponding salt-free aqueous organic mixture, but AIOMFAC-VISC would not be able to produce such results without further additions for organic-ion effects (a potential subject of future work). Of the three systems shown here, the viscosity of aqueous sucrose is consistently higher than the ternary mixture throughout the RH range. Additional AIOMFAC-VISC predictions are shown for the mixtures from (Richards et al., 2020b) in Figs. S6 and S7 .

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In terms of computational speed, the ZSR mixing model takes approximately five to six times longer than aquelec or aquorg (see Tables S1 and S2 in the SI. In dynamic simulations that may require repeated calls to AIOMFAC, such as kinetic multilayer diffusion models, this time difference may be an important consideration.  single phase. Furthermore, we carry out predictions of the viscosities of both particle phases during the idealized adiabatic ascent of an air parcel. Finally, we show AIOMFAC-VISC predicted particle viscosity of selected aerosols equilibrated to the thermodynamic conditions reported for an atmospheric vertical profile collected at Maniwaki Station, Quebec. In the context of meteorology, the effect of changing atmospheric pressure on viscosity is expected to be negligible and is therefore ignored. Liquid-liquid phase separation (LLPS) has been observed in aerosol particles (Bertram et al., 2011;Song et al., 2012;You 660 et al., 2014), and capturing this behavior is essential to accurately characterizing aerosol mass concentrations. Coexisting liquid phases are expected to occur when low-polarity and highly polar organics are present in a particle and/or when inorganic ions, water and moderate to lower polarity organics are present, or when weakly oxidized SOA material partitions into an existing aqueous aerosol (e.g. Huang et al., 2021). Relatively fresh SOA of moderate to lower polarity is likely to form a second, organic-rich phase that will coexist with an aqueous ion-rich phase. By contrast, aged SOA includes organic species that are 665 more highly oxygenated, and these species will more likely dissolve and mix with water and ions in a single liquid phase.
AIOMFAC-VISC can be used to predict the viscosity of any number of condensed phases, as we have done for a mixture of α-pinene-derived SOA and ammonium sulfate with an OIR = 1 exhibiting two "liquid" phases over a wide range in RH ( Fig. 11). In the LLPS case shown in Fig. 11a, two liquid phases are predicted, and the organic-rich phase attains a semi-solid viscosity for RH < 80 %. When we assume a single mixed phase (Fig. 11b), the predicted viscosity is less than that of the 670 organic-rich phase in Fig. 11a, due to the plasticizing effect of water contributed by the hygroscopic ions. A phase-separated aerosol, with a more viscous organic-rich shell, will likely be more resistant to chemical processing than a homogeneously mixed aerosol (Zhou et al., 2019). While the viscosity of the organic-rich phase becomes semi-solid (> 10 2 Pa s) below RH = 0.8, the single mixed phase remains liquid-like (< 10 2 Pa s) until RH = 0.2. The calculations for the phase viscosities in the LLPS case were carried out in two steps: first, an AIOMFAC-based coupled gas-particle and liquid-liquid equilibrium 675 computation was performed to determine the phase compositions while not computing the viscosities in the process (since it is unnecessary) and second, AIOMFAC-VISC is run for the compositions of the determined phases to provide the viscosities and associated estimations of uncertainties. The surrogate mixture representing α-pinene SOA is listed in Table S4 of the SI. Figure 12 shows the predicted viscosities for phase-separated particles consisting of α-pinene SOA and ammonium sulfate (overall OIR of 1) as a function of temperature and RH. Given an average oxygen-to-carbon ratio of about 0.5 for α-pinene 680 SOA, its mixture with aqueous ammonium sulfate is expected to result in nearly complete separation of the salt and SOA into distinct phases, except at RH levels exceeding 99.5 % (Bertram et al., 2011;Zuend and Seinfeld, 2012). Over the same ranges of RH and temperature, the organic-rich phase viscosity spans 18 orders of magnitude, whereas the ion-rich phase spans only 5 orders of magnitude, with liquid-like viscosities prevailing except at RH < 5 %. However, note that this composition and viscosity computation was conducted by assuming the ions to remain dissolved over the entire RH range. Depending on the initial 685 conditions, such as starting with dry or deliquesced particles, ammonium sulfate could potentially be present in solid-liquid equilibrium with the organic-rich phase below 80 % RH. Ammonium sulfate would be expected to be predominantly in a crystalline state for RH < 35 % (due to efflorescence), such that the predictions of the viscosity for lower RH levels in the ion-rich phase are hypothetical. Nevertheless, those viscosities are indicative of expected viscosity levels in similar aqueous ion-rich phases containing inorganic species that would less likely crystallize (such as certain nitrate salts).

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Finally, we can use AIOMFAC-VISC to predict the viscosities of these two phases at different vertical levels in the atmosphere. To provide an example, we extracted temperature and relative humidity values from two sample vertical atmospheric profiles at Maniwaki, Quebec, measured on February 7, 2020, and July 1, 2020. We calculated the viscosities of the two phases for each vertical profile assuming the particles to be in equilibrium with the measured environmental conditions (T , RH) at each altitude level (which is unlike the case of an adiabatically lifted air parcel). In Fig. 13, we show the viscosities of the organic- useful information about the potential distribution of viscous (semi-solid) and glassy aerosol phase states in the atmosphere.
As expected, the viscosity of the organic-rich phase is consistently several orders of magnitude higher than the viscosity of the aqueous ion-rich phase. The predicted viscosity for the ion-rich phase remains below 10 4 Pa s throughout the vertical extent of the atmosphere. While higher liquid-state viscosities are not predicted for the ion-rich phase, freezing and efflorescence are 700 still possible -and would likely occur at the low temperatures in the upper troposphere and lower stratosphere, with associated effects on phase state. The February 7 sounding (Fig. 13a) was collected during a winter storm. A local maximum in the predicted viscosity is observed near the surface and corresponds to a local minimum in relative humidity between 200 and 300 m. There is also an abrupt change in relative humidity above 12 km, likley indicating the tropopause region. Below this altitude, there is sufficient 705 water content in the aerosols to have a diminishing effect on the viscosity (when ignoring potential freezing and/or freezeconcentration of the phase). Above this altitude, the predicted viscosity of the organic-rich phase increases by several orders of magnitude, and it is virtually certain that SOA-rich aerosol phases at this height would be glassy. However, it is important to consider assumptions made and the example character of such calculations. They provide information about the expected viscosities that equilibrated particles of similar composition would exhibit when present at different altitude levels. We do not suggest that SOA-rich aerosols are typically found in the stratosphere. Wet and dry removal processes will prevent most tropospheric aerosol particles from reaching the stratosphere (Jacobson, 2002, p. 137). The July 1 sounding (Fig. 13b) was collected during a warm, moderately dry day without clouds, and the vertical profile for viscosity of the organic-rich phase shows a more gradual increase in viscosity with height. In this case, the lack of moisture in the planetary boundary layer means that the organic-rich phase attains a glassy viscosity at approximately 5 km altitude. The presence of glassy SOA at high 715 altitudes has been previously hypothesized (Koop et al., 2011;Zobrist et al., 2008), and their utility as ice nuclei has been more recently established for isoprene-derived SOA (Wolf et al., 2020).

Conclusions
A new predictive model has been developed and parameterized to enable calculations of the viscosity of aqueous electrolyte solutions. Furthermore, the earlier framework for aqueous organic mixtures has been successfully coupled with the electrolyte 720 model, providing a more general model applicable to aqueous organic-inorganic mixtures. The AIOMFAC-VISC electrolyte model is based on Eyring's absolute rate theory for viscous flow, which has been used previously to describe the viscosity of aqueous electrolyte solutions up to approximately 10 molal. A new expression for the molar Gibbs energy of activation for viscous flow for aqueous electrolyte solutions was introduced, defining contributions from individual ions and present cation-anion pairs. Forty-three aqueous electrolyte systems comprising 6,625 data points were used to simultaneously fit the 725 AIOMFAC-VISC electrolyte model. AIOMFAC-VISC's ionic coefficients are fitted using viscosity measurements at a wide range of concentrations as opposed to classical B-coefficients, which were fitted at dilute conditions. AIOMFAC-VISC closely fits the available data and produces smooth predictive extrapolations, performing nearly as well as Laliberté's model, which is considered a benchmark. The parameterized AIOMFAC-VISC model also aligns with more recent measurements of aerosol surrogate mixtures containing aqueous nitrate salts.

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Three mixing approaches were examined; aquelec and ZSR were found to be approximately equally accurate. The aquelec mixing approach is suggested as the preferred choice for use of AIOMFAC-VISC within dynamic (kinetic) simulations of viscosity or diffusion, because it is less computationally expensive. AIOMFAC-VISC's full functionality allows predictions for aqueous organic-inorganic mixtures consisting of an arbitrary number of organic compounds and inorganic ions. For systems that undergo phase separation according to an AIOMFAC-based liquid-liquid equilibrium computation, the viscosity of each 735 phase can be computed once the equilibrium composition has been determined. Viscosity predictions for α-pinene-derived SOA were discussed in the context of idealized adiabatic ascent of an air parcel and observations from two atmospheric soundings collected in Maniwaki, Quebec. Future experimental work on a wider range of compositions and a more diverse set of multicomponent systems (presently highly data-limited) may provide data and insights that could allow further refinements of the organic-inorganic mixing model. AIOMFAC-VISC may also provide an opportunity to further explore aerosol phase 740 state and state transitions, especially gel transitions, which have become a topic of interest in laboratory aerosol studies (Song et al., 2021;Richards et al., 2020a, b).