Predictions of the glass transition temperature and viscosity of organic aerosols from volatility distributions

Volatility and viscosity are important properties of organic aerosols (OA), affecting aerosol processes such as formation, evolution, and partitioning of OA. Volatility distributions of ambient OA particles have often been measured, while viscosity measurements are scarce. We have previously developed a method to estimate the glass transition temperature (Tg) of an organic compound containing carbon, hydrogen, and oxygen. Based on analysis of over 2400 organic compounds including oxygenated organic compounds, as well as nitrogenand sulfur-containing organic compounds, we extend this method to include nitrogenand sulfur-containing compounds based on elemental composition. In addition, parameterizations are developed to predict Tg as a function of volatility and the atomic oxygento-carbon ratio based on a negative correlation between Tg and volatility. This prediction method of Tg is applied to ambient observations of volatility distributions at 11 field sites. The predicted Tg values of OA under dry conditions vary mainly from 290 to 339 K and the predicted viscosities are consistent with the results of ambient particle-phase-state measurements in the southeastern US and the Amazonian rain forest. Reducing the uncertainties in measured volatility distributions would improve predictions of viscosity, especially at low relative humidity. We also predict the Tg of OA components identified via positive matrix factorization of aerosol mass spectrometer (AMS) data. The predicted viscosity of oxidized OA is consistent with previously reported viscosity of secondary organic aerosols (SOA) derived from α-pinene, toluene, isoprene epoxydiol (IEPOX), and diesel fuel. Comparison of the predicted viscosity based on the observed volatility distributions with the viscosity simulated by a chemical transport model implies that missing low volatility compounds in a global model can lead to underestimation of OA viscosity at some sites. The relation between volatility and viscosity can be applied in the molecular corridor or volatility basis set approaches to improve OA simulations in chemical transport models by consideration of effects of particle viscosity in OA formation and evolution.

Abstract. Volatility and viscosity are important properties of organic aerosols (OA), affecting aerosol processes such as formation, evolution, and partitioning of OA. Volatility distributions of ambient OA particles have often been measured, while viscosity measurements are scarce. We have previously developed a method to estimate the glass transition temperature (T g ) of an organic compound containing carbon, hydrogen, and oxygen. Based on analysis of over 2400 organic compounds including oxygenated organic compounds, as well as nitrogen-and sulfur-containing organic compounds, we extend this method to include nitrogen-and sulfur-containing compounds based on elemental composition. In addition, parameterizations are developed to predict T g as a function of volatility and the atomic oxygento-carbon ratio based on a negative correlation between T g and volatility. This prediction method of T g is applied to ambient observations of volatility distributions at 11 field sites. The predicted T g values of OA under dry conditions vary mainly from 290 to 339 K and the predicted viscosities are consistent with the results of ambient particle-phase-state measurements in the southeastern US and the Amazonian rain forest. Reducing the uncertainties in measured volatility distributions would improve predictions of viscosity, especially at low relative humidity. We also predict the T g of OA components identified via positive matrix factorization of aerosol mass spectrometer (AMS) data. The predicted viscosity of oxidized OA is consistent with previously reported viscosity of secondary organic aerosols (SOA) derived from α-pinene, toluene, isoprene epoxydiol (IEPOX), and diesel fuel. Comparison of the predicted viscosity based on the ob-served volatility distributions with the viscosity simulated by a chemical transport model implies that missing low volatility compounds in a global model can lead to underestimation of OA viscosity at some sites. The relation between volatility and viscosity can be applied in the molecular corridor or volatility basis set approaches to improve OA simulations in chemical transport models by consideration of effects of particle viscosity in OA formation and evolution.

Introduction
Organic aerosols (OA) contribute substantially to the mass loadings of atmospheric fine particulate matter (Hallquist et al., 2009;Jimenez et al., 2009). OA formed from various anthropogenic or biogenic precursors have complex physicochemical properties (Goldstein and Galbally, 2007;Nizkorodov et al., 2011;Ditto et al., 2018), which makes predictions of their role in air quality, climate, and public health challenging (Kanakidou et al., 2005;Shrivastava et al., 2017). Volatility and viscosity are important properties of OA, both of which affect important aerosol processes such as gas-particle partitioning, new particle formation and evolution of size distribution, heterogeneous reactions, and cloud condensation and ice nucleation pathways of OA, as summarized in recent review articles (Krieger et al., 2012;Bilde 8104 Y. Li et al.: Predictions of the glass transition temperature and viscosity of organic aerosols (10 2 ≤ η ≤ 10 12 Pa s), and amorphous solid (η > 10 12 Pa s) states, depending on temperature (T ), relative humidity (RH), and chemical composition . Even though there are several particle bounce measurements to infer ambient OA phase state, there are limited ambient measurements of particle phase state or viscosity (Virtanen et al., 2010;O'Brien et al., 2014;Bateman et al., 2016Bateman et al., , 2017Pajunoja et al., 2016;Liu et al., 2017;Ditto et al., 2019;Slade et al., 2019). Viscosity can be directly converted to bulk diffusivity of organic molecules using the Stokes-Einstein equation, which has been shown to work well for organic molecules diffusing through low-viscosity materials (Price et al., 2016;Chenyakin et al., 2017). This relation is inapplicable for predicting the bulk diffusivity of water and small molecules, and it may also underestimate the diffusivity of organic molecules in a highly viscous matrix, which can be corrected using a fractional Stokes-Einstein equation (Price et al., 2016;Evoy et al., 2019).
Viscosity can be related to the glass transition temperature (T g ), at which a phase transition between amorphous solid and semisolid states occurs . Ambient temperature varies through 100 K throughout the troposphere, greatly influencing the viscosity of the mixture. When the ambient temperature is below T g , an amorphous particle behaves as a solid, while a particle would be semisolid or liquid when the ambient temperature is above T g . OA particles contain a number of organic compounds and also a variable amount of liquid water depending on RH, which can act as a plasticizer to reduce T g : these mixture effects can be estimated using the Gordon-Taylor relation (Mikhailov et al., 2009;Koop et al., 2011;Dette et al., 2014). In addition, ambient OA may often be internally mixed with inorganic species such as sulfate and nitrate, which would further lower T g and viscosity if they were well mixed in one phase; when the phase separation occurs, the inorganic-rich and organic-rich phases may undergo glass transition at different temperatures (Dette and Koop, 2015).
For pure organic compounds with known molecular structure, viscosity can be predicted by group contribution approaches (Cao et al., 1993;Bosse, 2005; Y. C. Rovelli et al., 2019;Gervasi et al., 2020); chemical composition of ambient OA is complex, and molecular specificity is often unavailable, which makes viscosity predictions of ambient OA challenging. We have recently developed a set of semiempirical parameterizations using molar mass (M) and atomic O : C ratio  or elemental composition (DeRieux et al., 2018) to predict T g for compounds comprised of carbon, hydrogen, and oxygen (CHO compounds). These parameterizations have been applied to high-resolution mass spectrometry measurements to estimate viscosity of organic aerosols (DeRieux et al., 2018;Schum et al., 2018;Ditto et al., 2019;Song et al., 2019) and coupled into a thermodynamic model (Gervasi et al., 2020). Note that heteroatoms and the effects of molecular structure and func-tional groups on T g are not considered in parameterizations of Shiraiwa et al. (2017) and DeRieux et al. (2018).
Viscosity of pure compounds has been found to be inversely correlated with vapor pressure (Thomas et al., 1979). The molecular-corridor-based analysis  of hundreds of secondary organic aerosol (SOA) components has shown that compounds with lower pure-compound saturation mass concentration (C 0 ) have higher T g . Rothfuss and Petters (2017) found that there is a similar trend between the sensitivity of viscosity to functional group addition and the sensitivity of vapor pressure to functional group addition. Measurements of the evaporation kinetics of maleic acid showed that decreasing particle viscosity leads to a suppression in the effective vapor pressure of maleic acid (Marshall et al., 2018). Champion et al. (2019) found that SOA with higher condensed-phase fractions of extremely low volatility organic compounds (ELVOC) and low volatility organic compounds (LVOC) showed an increased viscosity. Zhang et al. (2019) measured T g of isoprene SOA components including isoprene hydroxy hydroperoxide (ISOPOOH), isoprene-derived epoxydiol (IEPOX), 2-methyltetrols, and 2methyltetrol sulfates (2-MT-OS), observing a tight correlation between T g and vapor pressure.
Based on the above evidence showing a close relation between volatility and viscosity, in this study we develop the parameterizations predicting T g as a function of C 0 based on data from over 2000 compounds. Functional group contribution approaches are often used to predict C 0 (Capouet and Müller, 2006;Pankow and Asher, 2008;Compernolle et al., 2011;O'Meara et al., 2014); thereby, using C 0 to predict T g would include the molecular structure effect indirectly. The developed parameterizations are applied to field observations of volatility distributions to predict viscosity of ambient OA.

Dataset of glass transition temperature
The training dataset used to develop the parameterizations of T g include 2448 organic compounds classified into four classes (see the number of CH, CHO, CHON, and CHOS compounds in Table S1 in the Supplement). Measured T g values are available for 42 CH compounds, 259 CHO compounds, 35 CHON compounds, and 1 CHOS compound Rothfuss and Petters, 2017;Lessmeier et al., 2018;Zhang et al., 2019), among which there are 168 compounds with measured C 0 available (Table S1). When T g measurements are unavailable, T g is estimated from the melting temperature (T m ) by applying the Boyer-Kauzmann rule of T g = g · T m (Kauzmann, 1948;Boyer, 1954) with g = 0.70085 (±0.00375) , referred to as "estimated T g " in this study (see good agreement of measured and estimated T g in Fig. S1a in the Sup-plement). There are 1187 compounds (391 CH, 537 CHO, 241 CHON, and 18 CHOS compounds) with both measured T m and C 0 (Tables S1, S2) adopted from the MPBPWIN program test sets (http://esc.syrres.com/interkow/EpiSuiteData. htm, last access: 9 July 2020) included in the EPI Suite software version 4.1 (Estimation Programs Interface; US EPA, 2015). Measured T g , T m , or C 0 for CHOS compounds is sparse and we adopt 850 CHOS compounds included in  with their T m and C 0 estimated by the EPI Suite software (Table S2). There are estimation limitations in the EPI Suite; for example, the disagreement between measured and estimated C 0 is larger for compounds with C 0 <∼ 10 −2 µg m −3 (Fig. S1b), which may affect the T g predictions for compounds with low volatility. However, given the large number of data points with measured C 0 included in the training dataset, the estimation bias introduced by the EPI Suite may not substantially impact the accuracy of the parameterization developed in this study.
The test dataset used to validate the performance of the parameterizations predicting T g of SOA components includes 654 CHO compounds and 212 CHON compounds found in SOA oxidation products . The values of their C 0 are estimated using the EVAPORATION model (Compernolle et al., 2011). Their T m values are adopted from the EPI Suite. The T g values predicted by our parameterizations are compared with the T g estimated from the T m by applying the Boyer-Kauzmann rule in the test dataset. Figure 1a shows a dependence of T g on C 0 for 2448 organic compounds in the training dataset. The compounds with lower C 0 have higher T g , and the T g appears to level off at around 420 K at C 0 <∼ 10 −10 µg m −3 . The dependence of T g on the atomic O : C ratio is weaker (Figs. 1a and S2), in agreement with previous studies Shiraiwa et al., 2017). Note that a tight correlation between T g and the O : C ratio has been observed for oxidation products formed from specific precursors including α-pinene (Dette et al., 2014), n-heptadecane, and naphthalene (Saukko et al., 2012). Based on the trend shown in Fig. 1a, we develop a parameterization (Eq. 1) to predict T g as a function of C 0 and O : C, which are the parameters used in the two-dimensional volatility basis set (2D-VBS) framework (Donahue et al., 2011).

Parameterizations of T g as a function of volatility
The coefficients in Eq.
(1) are obtained by fitting the T g of 2448 compounds in Fig. 1a with multilinear least squares analysis with 68 % prediction and confidence intervals. The predicted T g by Eq.
(1) is plotted in Fig. 1a with the O : C ratios of 0, 0.5, and 1, showing that the predicted dependence of T g on C 0 follows the trend well in the training dataset. The O : C ratio mainly affects the predicted T g of volatile compounds or extremely low volatility compounds. Figure 1b shows that the T g values of those compounds are predicted well by Eq. (1) as indicated by a high correlation coefficient (R) of 0.92. The average absolute value of the relative error (AAVRE; Aiken et al., 2007) is 12 %.
Equation (1) is further evaluated using the test dataset for SOA components. Figure 1c compares T g predicted by Eq. (1) with estimated T g from T m by applying the Boyer-Kauzmann rule, showing that Eq. (1) also presents a good performance for predicting T g of these SOA components with R = 0.96 and AAVRE = 6 %. Note that C 0 values of SOA components were estimated using the EVAPORATION model (Compernolle et al., 2011). The T g values of individual SOA compounds can be predicted within ±20 K as indicated by the prediction band (dotted lines in Fig. 1c); however, this uncertainty may be much smaller for multicomponent SOA mixtures under ideal mixing conditions as indicated in the confidence band (dashed lines, almost overlapping with the 1 : 1 line) DeRieux et al., 2018;Song et al., 2019).
We also develop a parameterization (Eq. 2) predicting T g as a function of C 0 solely, which can be applied to the information available with the one-dimensional VBS framework (1D-VBS; Donahue et al., 2006), and this can be used when the O : C ratio is not available in measurements.

Predictions of T g and viscosity of organic aerosols
For the application of T g parameterizations in field observations of volatility distributions, T g for each volatility bin (T g,i ) is calculated by Eq. (1). The term volatility refers to the effective saturation mass concentration (C * ), and we assume ideal thermodynamic mixing in which case C * is equal to C 0 (Donahue et al., 2011). Note that there may be additional uncertainty in application of T g parameterizations (which were developed based on pure compounds) to each volatility bin representing a surrogate of complex multicomponent mixtures. The isolines in Fig. 2 show the T g,i predicted by Eq. (1) with the C * and O : C defined in the 2D-VBS framework. T g would be below ∼ 250 K for intermediate volatility organic compounds (IVOC; 300 < C 0 < 3×10 6 µg m −3 ), from ∼ 260 to 290 K for semivolatile organic compounds (SVOC; 0.3 < C 0 < 300 µg m −3 ), and higher than 300 K for low-volatility organic compounds (LVOC; 3 × 10 −4 < C 0 < 0.3 µg m −3 ) and extremely low volatility organic compounds (ELVOC; C 0 < 3 × 10 −4 µg m −3 ). The T g increases as the O : C ratio increases for SVOC and IVOC, which is consistent with previous studies Saukko et al., 2012;Berkemeier et al., 2014). The T g slightly decreases as the O : C ratio increases for LVOC and ELVOC compounds, which might be due to the uncertainties in Eq. (1) which is derived from a dataset containing fewer LVOC and ELVOC compounds as shown in Fig. 1a, which exhibits lower T g with higher O : C.
The glass transition temperatures of organic aerosols under dry conditions (T g,org ) are calculated by the Gordon-Taylor equation (Gordon and Taylor, 1952) by assuming the Gordon-Taylor constant (k GT ) of 1 (Dette et al., 2014): where w i is the mass fraction in the particle phase for each volatility bin. The Gordon-Taylor approach has been validated for a wide range of mixtures including SOA compounds (Dette et al., 2014;Lessmeier et al., 2018). The Gordon-Taylor approach may fail in the case of adduct or complex formation , which is highly unlikely in multicomponent mixtures with myriads of SOA compounds with very small individual mole fractions and when particular interactions between individual compounds are more likely to average out ; this aspect would need to be investigated in future studies. The phase state of aerosol particles strongly depends on their water content (Mikhailov et al., 2009;Koop et al., 2011). Under humid conditions, the water content in OA can be estimated using the effective hygroscopicity parameter (κ) (Petters and Kreidenweis, 2007). The T g of organic-water mixtures (T g (w org )) at given RH can be estimated using the Gordon-Taylor equation (Gordon and Taylor, 1952): where w org is the mass fraction of organics in particles; T g,w is the glass transition temperature of pure water (136 K; Kohl et al., 2005), and k GT is the Gordon-Taylor constant for organic-water mixtures which is suggested to be 2.5 (Zobrist et al., 2008;Koop et al., 2011). Viscosity can then be calculated by applying the Vogel-Tammann-Fulcher (VTF) equation (Angell, 1991) where η ∞ is the viscosity at infinite temperature (10 −5 Pa s; Angell, 1991), D is the fragility parameter which is assumed to be 10 (DeRieux et al., 2018), and T 0 is the Vogel temperature calculated as

Southern Oxidant and Aerosol Study (SOAS)
In this section we predict glass transition temperatures and phase state of ambient OA during the SOAS (Southern Oxidant and Aerosol Study) campaign, which took place in the southeastern United States (Centreville, Alabama) in summer 2013 (Carlton et al., 2018). The T g of organic aerosols under dry conditions (T g,org ) is calculated using Eqs.
(1) and (3) with measured volatility distributions. Figure 2 shows the calculated T g,org placed in the 2D-VBS framework against the average log 10 (C * ) calculated by i w i log 10 (C * i ) Figure 2. Predicted glass transition temperatures of organic aerosols under dry conditions (T g,org ) during the SOAS campaign placed into the 2-D VBS framework. The isopleths correspond to the T g calculated using Eq. (1) with the effective saturation mass concentration (C * ) and the O : C ratio defined in the 2D-VBS. The markers represent the T g,org of total OA (TOA) and IEPOX SOA calculated from the volatility distributions simulated by a global chemical transport model (EMAC-ORACLE; Shiraiwa et al., 2017) or measured during the SOAS campaign Saha et al., 2017;Stark et al., 2017). Three methods (Formulas, Partitioning, and Thermograms) are applied in Stark et al. (2017) to derive the C * distributions, where the Thermograms method provides the most credible volatility distributions compared to Formulas and Partitioning (marker edge lines in gray). (Kostenidou et al., 2018), and the measured O : C ratio is from Xu et al. (2015). Figure 2 shows that T g,org of total OA (TOA) range from 232 to 334 K, depending on volatility distributions measured by different methods, while the most credible predicted T g,org values span in the range of 313-330 K. The reasons are stated below by comparing the different methods deriving the C * distributions. Stark et al. (2017) used three methods (Thermograms, Partitioning, and Formulas) to derive volatility distributions by applying the measurements of organic acids (which were shown to account for about half of the total OA; Yatavelli et al., 2015) from a high-resolution chemical ionization time-of-flight mass spectrometer equipped with a filter inlet for gases and aerosols (Lopez-Hilfiker et al., 2014;Thompson et al., 2017). In the Thermograms method, C * at 298 K is estimated from the desorption temperature after calibration with known species (Faulhaber et al., 2009). This method results in 93 % of OA mass distributed in the LVOC and ELVOC , and a high T g,org of 330 K is predicted (Fig. 2). While this method may be influenced by thermal decomposition, the peak temperatures of decomposing species can be expected to relate closer to actual volatilities than any of the other two analysis methods . The result from the thermogram method is consistent with those measured by an aerosol mass spectrometer (AMS) with a thermodenuder, which also applied the thermogram method to estimate the C * distributions . Saha et al. (2017) applied an evaporation kinetic model (Lee et al., 2011) based on the VBS approach to extract the C * distributions, and the effects of enthalpy of vaporization and accommodation coefficient (α) are considered, resulting in the estimated T g,org of 313 K. This study retrieved α of ∼ 0.5, which is consistent with recent experiments Liu et al., 2019).
The lower T g,org values (< 280 K) calculated from the C * distributions estimated from the Formulas and Partitioning methods  are less atmospherically relevant. The Formulas method used SIMPOL (simple group contribution method; Pankow and Asher, 2008) to calculate vapor pressures from the composition of the identified ions. While the specific functional group distributions needed for SIMPOL are unknown from mass spectrometer measurements, some assumptions can be made, leading to limits in the volatility distributions, all of which are showing the same behavior of high volatilities . This is because many of the detected species can be thermal decomposition products rather than actual SOA molecules , which can lead to overestimations of volatilities, resulting in the unlikely low T g,org of 232 K. The Partitioning method used the measured particle-phase mass fractions of each species to estimate C * based on the partitioning theory (Pankow, 1994). The estimated C * is distributed 8108 Y. Li et al.: Predictions of the glass transition temperature and viscosity of organic aerosols mainly in the SVOC range , leading to a T g,org of 279 K (Fig. 2). This value is very close to the T g,org (281 K) simulated by a global chemical transport model, EMAC-ORACLE, in which a narrow distribution of C * (1, 10, 10 2 , and 10 3 µg m −3 ) was applied . However, Stark et al. (2017) note that the partitioning-based volatility distribution is likely too high due to an artifact of signal-to-noise limitations, confining the C * characterizable by the partitioning method to a relatively narrow range centered around the ambient OA concentration (by definition the semivolatile range). These analyses indicate that the volatility distributions derived from different methods, even when based on the same measurements, significantly affect the predicted T g,org , and the most atmospherically relevant volatility distributions should be carefully chosen to reasonably predict the glass transition temperature of ambient OA. In summary, the T g,org values during the SOAS campaign should be in the range of 313-330 K. Figure 2 also includes T g,org of isoprene-derived epoxydiol SOA (IEPOX-SOA) identified via positive matrix factorization (PMF) of AMS mass spectra (Lanz et al., 2007). IEPOX-SOA is predicted to have a T g,org of 345 K with very low volatility with the average C * lower than 10 −4 µg m −3 Lopez-Hilfiker et al., 2016;D'Ambro et al., 2019), which may be due to substantial formation of organosulfates and other oligomers (Lin et al., 2012;Hu et al., 2015;Riva et al., 2019). The predicted T g,org of IEPOX-SOA is higher than previously reported T g,org of 263-293 K for monoterpene-derived (α-pinene, 3-carene, myrcene, limonene, and ocimene) SOA .
We further calculate the viscosity of OA based on the T g,org of TOA predicted above in order to compare with the ambient phase-state measurements during the SOAS campaign. Figure 3a shows the predicted viscosity of total OA at different RH; T is adopted as 298 K, the average value during the SOAS campaign . The effective hygroscopicity parameter (κ) is set to 0.14 for TOA based on measurements (Cerully et al., 2015). The characteristic timescale of mass transport and mixing by molecular diffusion (τ mix ) is also calculated: τ mix = d 2 p /(4π 2 D b ) (Seinfeld and Pandis, 2006), where d p is the particle diameter, and the bulk diffusion coefficient D b is calculated from the predicted viscosity by the fractional Stokes-Einstein relation (Evoy et al., 2019). We assume a radius of the diffusing molecule of 10 −10 m and a particle diameter of 200 nm . Note that these estimated timescales represent rough estimations, as molecular interactions in complex mixtures are not considered.
The viscosity of TOA at RH of 83 % (average RH during SOAS) is predicted to be less than 10 2 Pa s with τ mix less than 1 s, which is consistent with the particle bounce measurements, suggesting that organic-dominated particles were mostly liquid during the SOAS campaign (Pajunoja et al., 2016). When RH was below ∼ 50 % in the sampling inlet, the particles were found to adopt a semisolid state (Pajunoja et al., 2016), which agrees with the predicted viscosity of 10 7 -10 11 Pa s, and τ mix can be higher than 1 h at 50 % RH (Fig. 3a). The variations in T g,org (313-330 K) due to the different measured C * distributions (Fig. 2) have a more significant impact on the predicted viscosity at low and medium RH (Fig. 3a). When RH is higher than ∼ 70 %, the predicted viscosities calculated from different T g,org values are very close; at high RH the condensed-phase water has a larger influence on the phase state than the volatility does, depending on the hygroscopicity of organic aerosols. Figure 3b shows diurnal variations of predicted viscosity of total OA using measured T and RH during the SOAS campaign . During 10:00-20:00 LT (local time) when RH < 70 % and T > 298 K, three simulations using different T g,org values predict that total OA occurs as semisolid with the predicted viscosity of 10 2 -10 7 Pa s and the mixing times of less than 1 h. Particles are predicted to have a low viscosity of < 1 Pa s, adopting a liquid phase during nighttime. The lowest viscosity occurs around 05:00-06:00 LT with RH > 95 %. Here we did not consider the effects of the diurnal variations of volatility distributions, as they did not vary dramatically over the campaign period (Saha et al., 2017). Besides T and RH, diurnal variation of ambient aerosol phase state also depends on particle chemical composition and mixing states. Organic particles in the Amazon were found to be more viscous at night than the daytime due to the influence of biomass burning that may form nonliquid particles . Particles in a mixed forest in northern Michigan, USA, were also found more viscous at night despite higher RH than the daytime, due to the formation of high molar mass organic compounds and smaller inorganic sulfate mass fractions (Slade et al., 2019). Phase-state measurements during daytime and nighttime at Atlanta, USA, suggested that the ambient particle phase state was influenced by OA composition, the presence of inorganic ions, aerosol liquid water, and particle mixing state . Figure 4 summarizes T g,org at 11 sites where the measured volatility distributions with volatility bins of four or more are available (Table S3). We did not include the data with narrower volatility ranges which may not correctly characterize the properties of ambient SOA (Bilde et al., 2015), and thus may not be appropriate for estimating volatility distributions or would result in unrealistically low T g without considering realistically low C * bins. Note that a narrow VBS may still be useful for efficiency in three-dimensional chemical transport models for SOA evaporation and condensation under a narrow range of ambient temperature variations (Kostenidou et al., 2018). Figure 4a shows the 2D-VBS framework of O : C vs. log 10 C * with the marker fill color representing T g,org , whereas panel (b) shows T g,org vs. log 10 C * with the marker  Saha et al. (2017), and the Thermograms method in Stark et al. (2017). Characteristic mixing timescales of organic molecules with a radius of 10 −10 m within 200 nm particles are also shown on the right axis. fill color representing O : C. The marker edge color represents OA components identified via positive matrix factorization of AMS mass spectra (Lanz et al., 2007), including biomass-burning OA (BBOA), hydrocarbon-like OA (HOA), cooking OA (COA), and oxygenated OA (OOA), which is sometimes further separated into more oxygenated OA (MO-OOA) and less oxygenated OA (LO-OOA) factors. Note that these different OA factors may often be internally mixed in ambient atmosphere, and predicted T g,org and particle viscosity would be irrelevant in such a case. Nevertheless, these predictions can be useful when particles are externally mixed or ambient OA are dominated by a certain OA factor.

T g,org at 11 global sites
T g,org of total OA (TOA) varies from 290 to 339 K. The lower T g,org occurs at Beijing, China, in June 2018 . OA in Beijing was found to be overall more volatile with the particle-phase semivolatile fraction of 63 %. This may be due to the higher total OA mass concentrations in Beijing , which facilitate greater partitioning of SVOC compounds into the particle phase, leading to a lower T g,org . The predicted T g,org of total OA at numerous other sites range between 300 and 320 K, including Paris (Paciga et al., 2016), Mexico City (Cappa and Jimenez, 2010), Centreville Saha et al., 2017;Stark et al., 2017), Raleigh (Saha et al., 2017), and Durham (Saha et al., 2018) in southeastern US. The T g,org value (316 K) at 220 m downwind from a highway in Durham is higher than the T g,org (309 K) at 10 m downwind from a highway due to the dilution and mixing of traffic-sourced particles with background air and evaporation of semivolatile species during downwind transport (Saha et al., 2018). The T g,org values are predicted to be high (> 320 K) at the sites in Athens (Louvaris et al., 2017), Pasadena (Ortega et al., 2016), Colorado Rocky Mountain , and the Amazon . The T g,org values for MO-OOA in Mexico City and Paris are predicted to be very high at ∼ 350 K, reflecting their very low volatility. Figure 5 shows the OA viscosity variation of OA components against RH. The hygroscopic growth is considered based on hygroscopicity (κ), which is estimated as a function of the O : C ratio (Lambe et al., 2011) when κ was not measured (Table S3). The κ values of OA factors with low O : C ratio, i.e., HOA, COA, and BBOA, are estimated to be low (< 0.08); they are predicted to undergo glass transition at RH between 25 % and 68 % and adopt a liquid phase only when RH is very high (∼ 80 %). The predicted behavior of BBOA is in line with bounce measurements observing that particles are semisolid in a biomass-burning plume . OA factors with higher O : C ratios including LO-OOA, MO-OOA, and IEPOX SOA tend to become liquid (viscosity < 10 2 Pa s) at intermediate RH (Fig. 5b).
There have been growing measurements of RH-dependent viscosity of laboratory-generated SOA formed from different precursors, e.g., isoprene (Song et al., 2015), α-pinene (Abramson et al., 2013;Renbaum-Wolff et al., 2013;Kidd et al., 2014;Pajunoja et al., 2014;Bateman et al., 2015;Zhang et al., 2015;Grayson et al., 2016;Petters et al., 2019), toluene (M. , and diesel fuel . As the OOA factors characterized from ambient AMS observations may represent ambient SOA , the predicted viscosities of OOA are compared with laboratory measurements of SOA viscosities in Fig. 5b. It shows that the majority of experimental values is well bounded by Figure 4. Predicted glass transition temperatures of organic aerosols under dry conditions (T g,org ) at 11 sites. The fill color of the markers represents T g,org (a) or the O : C ratio (b). The marker edge color indicates the OA components identified via PMF of the AMS mass spectra. The isopleths in (a) correspond to T g calculated using Eq. (1) with C * and O : C defined in the 2D-VBS. The vertical error bars correspond to uncertainties in T g,org considering parameterization uncertainties and error propagation. The horizontal error bars for the Centreville site correspond to the upper and lower limits of the average log 10 (C * ) calculated from different volatility distributions measured during the SOAS campaign Saha et al., 2017;Stark et al., 2017). the predicted viscosities of OOA, represented by the pink shaded area. One exception is the measured viscosity of isoprene SOA is lower than the predicted viscosity of IEPOX SOA at low RH (< 30 %). One possible reason is that the isoprene SOA in experiments was formed with high oxidant concentrations with a short reaction time in an oxidation flow reactor in the absence of inorganic seed particles (Song et al., 2015). In ambient environments heterogeneous reactions with acidic sulfate particles forming oligomers are suggested to be an important pathway (Surratt et al., 2010;Lin et al., 2013;Hu et al., 2015Hu et al., , 2016. These particle-phase organosulfates may contribute to a higher viscosity, as indicated by the predicted viscosity of IEPOX-derived organosulfate mixtures with their T g,org estimated to be 313 K (Riva et al., 2019).  (Song et al., 2015), α-pinene (Abramson et al., 2013;Renbaum-Wolff et al., 2013;Kidd et al., 2014;Pajunoja et al., 2014;Bateman et al., 2015;Zhang et al., 2015;Grayson et al., 2016;Petters et al., 2019), toluene (M. , and diesel fuel  are also shown. Predicted viscosity of IEPOX-derived OS mixtures (solid blue line) is from Riva et al. (2019). Note that in the case when these OA factors are internally mixed with other components, the predicted viscosity would not represent real ambient complex organic mixtures.
Another reason could be the mass concentrations of isoprene SOA are much higher (100-1000 µg m −3 ; Song et al., 2015) compared to ambient OA concentrations (5 µg m −3 during SOAS; Stark et al., 2017). Higher mass concentrations can lead to lower viscosity, as more semivolatile compounds can partition into the particle phase (Grayson et al., 2016;Jain et al., 2018;Champion et al., 2019). Shiraiwa et al. (2017) simulated the global distribution of annual averages of SOA phase state using the chemical transport model EMAC (Jöckel et al., 2006) coupled with the organic aerosol module ORACLE (Tsimpidi et al., 2014). ORACLE uses the 1D-VBS framework with four C * bins (1, 10, 10 2 , and 10 3 µg m −3 ). To estimate T g , the values of molar mass and O : C ratio were assigned for each volatility bin based on molecular corridors . Note that the molar mass assigned for the volatility bin of 1 µg m −3 was assumed to have relatively high molar mass to partially compensate for the fact that ORACLE does not consider lower volatility bins with higher molar mass. As shown in Fig. 6, global distributions of T g /T presented in Shiraiwa et al. (2017) are converted to viscosity using the VTF equation. Figure 6 also includes the viscosity of total OA at the 11 sites by applying measured volatility distributions and the global model-simulated 5-year-average T and RH with κ assumed to be 0.1 (Pringle et al., 2010). Figure 6b shows that the predicted viscosities at the 11 sites generally agree with the global simulations: the amorphous solid or semisolid phase occurs over relatively dry areas, including the sites in western US, Mexico City, Beijing, and coastal sites in Greece; the lower viscosity occurs in southeastern US and Paris.

Comparison with global simulations
The global simulations show that the particles are liquid in the Amazon, while they occur as semisolid in our predictions based on measured volatility distributions (Fig. 6a). The reason for this disagreement may be mainly due to the substantial fraction of low volatility compounds observed in ambient measurements largely missing from global simulations. Hu et al. (2016) observed that 90 % of OA has volatilities lower than 1 µg m −3 , which is the lowest C * bin in the global simulations. The ambient phase-state measurements show that for background conditions of the Amazonian tropical forest particles are mostly liquid, while for the anthropogenic influence including both urban pollution and biomass burning they occur as semisolid or glassy (Bateman et al., 2016. The volatility distributions were measured in the dry season that is heavily influenced by biomass burning , which can lead to the higher predicted viscosity. Similar cases are observed in Athens and the two sites in the western US, that our predictions based on volatility distributions indicate the glassy phase state, while the global model predicts the occurrence of a semisolid phase.

Conclusions and implications
We have developed parameterizations to estimate the glass transition temperature of organic compounds using saturation mass concentration (C 0 ) and atomic O : C ratio. They Figure 6. (a) Global distributions of SOA annually averaged viscosity at the surface simulated by a global chemical transport model  with the viscosity predicted by measured volatility distributions at 11 global sites (triangle, square, and circle represent remote, forested, and urban sites, respectively, Table S3). The color code indicates viscosity on a log scale. (b) Predicted viscosity based on measured volatility distributions compared against the viscosity in global simulations. The error bars correspond to uncertainties in viscosities calculated from uncertainties in predicted T g,org shown in Fig. 4. can be applied to ambient observations of volatility distributions to estimate viscosity of ambient organic aerosols. The T g and viscosity prediction method can be applied in the volatility basis set or the molecular-corridor-based approach to improve OA simulations in chemical transport models by consideration of effects of particle viscosity on OA formation and evolution Pye et al., 2017;Schmedding et al., 2019). Most of the current chemical transport models treat particles as a homogeneously well-mixed liquid without considering particle-phase diffusion limitations, which can lead to bias in simulations of SOA mass concentrations and evolution of size distributions (Shiraiwa and Seinfeld, 2012;Zaveri et al., 2018). The SOA simulations applying the VBS framework have not yet included the effects of viscosity on SOA formation and evolution. When the gasparticle partitioning is limited by bulk diffusion, kinetic treatments of SOA partitioning may need to be applied (Perraud et al., 2012;Liu et al., 2016;Yli-Juuti et al., 2017;Li and Shiraiwa, 2019). Some chamber experiments probing the mixing timescales of SOA particles formed from isoprene, α-pinene, and limonene did not observe significant kinetic limitations at moderate and high RH under room temperature conditions (Loza et al., 2013;Ye et al., 2016), while kinetic limitations of bulk diffusion of organic molecules in β-caryophyllene SOA have been observed at 75 % RH (Ye et al., 2018), warranting further investigations on the degree of kinetic limitations in ambient tropospheric conditions. In addition, the interplay of diffusion limitations and phase separation impacts heterogeneous and multiphase chemistry (Vander Wall et al., 2018;DeRieux et al., 2019;Zhou et al., 2019) and gasparticle partitioning (Zuend and Seinfeld, 2012;Shiraiwa et al., 2013;Freedman, 2017;Pye et al., 2017;Gorkowski et al., 2019a). The particle morphology and the degree of nonideal mixing and liquid-liquid phase separation can evolve upon atmospheric aging (Gorkowski et al., 2019b). These aspects may also need to be considered for better representation of organic aerosols in future studies.

Appendix A: Parameterizations of T g based on elemental compositions
We recently developed a parameterization (Eq. A1) predicting T g as a function of the number of carbon (n C ), hydrogen (n H ), and oxygen (n O ) atoms (DeRieux et al., 2018), similar to the formulation used to predict C 0 (Donahue et al., 2011;: Values of the coefficients n 0 C , b C , b H , b CH , b O , and b CO are 1. 96,61.99,28.74,0,and 0 for CH compounds and 12.13,10.95,21.61,118.96, for CHO compounds. We broaden the parameterizations for CH and CHO compounds (Eq. A1) to the following equations applicable to CHON (Eq. A2) and CHOS compounds (Eq. A3): Values of the coefficients n 0  Table S1) with multilinear least squares analysis. Figure A1a shows a fair agreement between the predicted T g using Eq. (A2) and the measured or otherwise estimated T g with R of 0.55 and relatively large AAVRE of 16 % for CHON compounds in the training dataset. Figure A1b shows a better prediction performance with R of 0.83 and AAVRE of 9 % for 212 CHON compounds included in the test dataset for SOA components with their T g estimated by the Boyer-Kauzmann rule using the EPI-estimated T m . Figure A1c shows that Eq. (A3) performs well for the CHOS compounds included in the training dataset with their T g estimated by the Boyer-Kauzmann rule using the EPI-estimated T m (R = 0.87, AAVRE = 8 %). Figure S5 shows the comparison of T g predicted by the elemental composition (Eqs. A1-A3) with the T g predicted as a function of C 0 and the O : C ratio (Eq. 1). The agreement between the two sets of parameterizations for nitrogenand sulfur-containing compounds is not as good as that for CHO compounds, indicating that there are limitations of predicting T g by the elemental composition for nitrogen-and sulfur-containing compounds with complex elemental compositions and molecular structures. As volatility depends significantly on functional groups contained in a molecule (Pankow and Asher, 2008;Compernolle et al., 2011), predicting T g by volatility (Eq. 1) indirectly incorporates the molecular structure effects. As there are limited CHON and CHOS compounds with measured T g available, future experiments measuring more T g data for nitrogen-and sulfurcontaining organics would help improve the T g parameterizations by elemental composition.
Appendix B: Comparison of T g predictions with Zhang et al. (2019) Recently Zhang et al. (2019) developed a semiempirical parameterization (Eq. B1) using vapor pressure (p 0 in atm) to predict T g based on measured T g of 11 SOA compounds: p 0 can be converted to C 0 via C 0 = (10 6 Mp 0 )/(RT ), where R is the ideal gas constant (R = 8.2 × 10 −5 m 3 atm mol −1 K −1 ), M is the molar mass (g mol −1 ), and T is the temperature (K). Figure B1 compares the measured T g included in the training dataset shown in Fig. 1a to T g predicted by (a) C 0 and the atomic O : C ratio (Eq. 1), (b) elemental composition (Eqs. A1-A3), and (c) Eq. (B1) by Zhang et al. (2019). While all three methods perform reasonably well, the predictions using elemental composition (Eqs. A1-A3) show better performance (Fig. B1b) with R of 0.93 and AAVRE of 11 %.
The prediction performance is influenced by the training dataset used to develop parameterizations of T g . The compounds shown in Fig. B1 contain mostly carboxylic acid and hydroxyl functional groups Rothfuss and Petters, 2017) and are included in the training dataset used to develop Eqs. (1) and Eqs. (A1)-(A3). The training dataset used in Zhang et al. (2019) included 11 organic compounds, and their parameterization predicted T g of isoprene SOA very well , but underpredicted some low-T g compounds (Fig. B1c). For compounds with their measured T g higher than 200 K, predictions by Zhang et al. (2019) show good performance and are consistent with the predictions given by Eq. (1) as a function of C 0 and the O : C ratio. Predicted T g values of 2-MT-OS using the three methods are 297 K (Eq. 1, as a function of C 0 and the O : C ratio), 275 K (Eq. A3, as a function of the elemental composition), and 280 K (Eq. B1, Zhang et al., 2019), comparable with the measured T g of 276 ± 15 K .
Note that predictions using elemental composition (Eq. A1) overestimate the T g of phthalate compounds (the star markers in Fig. B1). For instance, the observed T g of dioctyl phthalate is 194 K , while the prediction is higher than 300 K (Fig. B1b). The reason is that ester is not an effective functional group to increase viscosity compared to carboxylic acid and hydroxyl (Rothfuss Figure A1. T g predicted by elemental composition (Eq. A2) compared to (a) measured or otherwise estimated T g by the Boyer-Kauzmann rule using measured T m for CHON compounds in the training dataset and (b) estimated T g by the Boyer-Kauzmann rule with T m estimated by the EPI Suite for CHON compounds in the test dataset for SOA components. (c) T g predicted by elemental composition (Eq. A3) compared to estimated T g by the Boyer-Kauzmann rule with T m estimated by the EPI Suite for CHOS compounds in the training dataset. The dashed and dotted lines show 68 % confidence and prediction bands, respectively. The correlation coefficient (R) and the average absolute value of the relative error (AAVRE) are included in each figure legend. Figure B1. Comparison between measured T g in the training dataset in Fig. 1a and T g predicted by (a) C 0 and O : C (Eq. 1), (b) elemental composition (Eqs. A1-A3), and (c) the parameterization (Eq. B1) in Zhang et al. (2019). The solid line shows the 1 : 1 line. The correlation coefficient (R) and the average absolute value of the relative error (AAVRE) are included in each figure legend. and Petters, 2017). Parameterizations using volatility (Eqs. 1 and B1) improve the predicted T g of phthalate compounds (Fig. B1a, c). Figure B2 shows, compared to the predictions using Eq. (B1) provided in Zhang et al. (2019), that predictions by C 0 and the atomic O : C (Eq. 1) and elemental composition (Eq. A1) agree better with the T g estimated from the Boyer-Kauzmann rule. Future experiments measuring more T g data of SOA components would help verify the T g predictions by different parameterizations. Data availability. The data used in this study are available in the Supplement.
Author contributions. YL, JLJ, and MS designed the research. YL developed the parameterizations. DAD, HS, and JLJ provided measured volatility distributions for the SOAS campaign. YL and MS wrote the article. All authors discussed the results and contributed to article editing.