We use the WRF-Chem model version 3.7.1 (Grell et al., 2005; Fast et al., 2006) and simulate the period from 20 May to 23 June 2018 with a spin-up period of 7 d (13–19 May). We set up two domains (Fig. S1 in the Supplement) with the horizontal resolutions of 81 and 27 km, respectively, and 18 vertical layers are applied from the surface up to 100 hPa. The meteorological initial and boundary conditions are from the National Centers for Environmental Prediction (NCEP) Global Forecast System (GFS) final (FNL) reanalysis data. The outputs of a global chemical transport model MOZART-4 (Emmons et al., 2010) provide initial and boundary conditions of chemical species over the outer domain (Fig. S1). Anthropogenic emissions are from the MIX 2010 inventory for Asia (Li et al., 2017) and the MEIC 2016 inventory for China (http://meicmodel.org.cn, last access: 1 June 2023; Zheng et al., 2018). Biogenic emissions are calculated from the Model of Emissions of Gases and Aerosols from Nature (MEGAN2.1) (Guenther et al., 2012).

The utilized physical and chemical schemes are given in Table S1 in the Supplement. We use the MOZART-4 mechanism (Emmons et al., 2010) for the gas-phase chemistry. The MOSAIC (Model for Simulating Aerosol Interactions and Chemistry) aerosol module (Zaveri et al., 2008) is applied for the aerosol chemistry, and we represent aerosol particles with four-size sections having dry diameters ranging from 0.039 to 10 µm (Knote et al., 2015). SOA formation is treated with the 1-D volatility basis set (VBS) approach (Donahue et al., 2006), which has been implemented into the MOSAIC aerosol module (Lane et al., 2008a; Ahmadov et al., 2012). Five volatility bins are considered (effective saturation mass concentrations C* of 10-4, 1, 10, 100, and 1000 µg m-3 at 298 K) in the official version 3.7.1 of the WRF-Chem model, with the enthalpy of vaporization (ΔHvap) values of 40, 131, 120, 109, and 98 kJ mol-1 used in each volatility bin (Knote et al., 2015). We follow Knote et al. (2015) with SOA mass yields adopted for four volatility bins (1, 10, 100, and 1000 µg m-3). Further gas-phase aging is simulated through OH oxidation of SOA vapors with a fixed rate of 1.0×10-11 cm3 molec.-1 s-1, with products shifted down one volatility bin (Murphy and Pandis, 2009); e.g., the condensable vapors with C* of 1 µg m-3 react with OH, forming surrogate species in the lowest volatility bin (C* of 10-4 µg m-3). The partitioning of organic compounds between the gas and particle phases is simulated based on Pankow (1994), which is implemented in MOSAIC (Shrivastava et al., 2011). We apply glass transition temperature and viscosity calculations to WRF-Chem model output for traditional SOA formed from the oxidation of volatile organic compounds (VOCs) including alkanes, alkenes, aromatics, isoprene, and monoterpenes (Lane et al., 2008b).

The glass transition temperature of SOA products in each volatility bin at dry conditions (Tg,i) is calculated as a function of the saturation mass concentration at 298 K (C0) using the parameterization (Eq. 1) developed in our previous study (Li et al., 2020). We assume ideal thermodynamic mixing where C0 is equal to C*, which is often applied in the VBS (Donahue et al., 2011). 1Tg,i=288.70-15.33×log⁡10C0-0.33×log⁡10C02 The Tg of mixtures of dry SOA compounds (Tg,org) is calculated by the Gordon–Taylor equation (Gordon and Taylor, 1952), with the Gordon–Taylor constant (kGT) assumed to be 1 (Dette et al., 2014): 2Tg,org=∑iωiTg,i, where ωi is the mass fraction of SOA products in each volatility bin simulated by the VBS module in WRF-Chem.

The particle phase state depends strongly on water content in particles, as water can act as a plasticizer to decrease viscosity (Mikhailov et al., 2009; Koop et al., 2011). The mass concentration of water absorbed by SOA particles under humid conditions is estimated using the effective hygroscopicity parameter (κ) (Petters and Kreidenweis, 2007) as 3mH2O=aw1-awκρwmSOAρSOA, where aw is water activity calculated as aw=RH/100 and ρw is the density of water. mSOA is the simulated total mass concentrations of traditional SOA. The density of SOA particles (ρSOA) is assumed to be 1.5 g cm-3 (Knote et al., 2015). κ is assumed to be 0.1 based on previous studies (Gunthe et al., 2009; Duplissy et al., 2011; Wu et al., 2013) and consistent with the value used in our previous global SOA phase state simulations (Shiraiwa et al., 2017).

Tg of organic–water mixtures is also calculated by the Gordon–Taylor equation (Eq. 4) with kGT suggested to be 2.5 (Koop et al., 2011). 4Tgωorg=1-ωorgTg,w+1kGTωorgTg,org1-ωorg+1kGTωorg, where ωorg is the mass fraction of the simulated SOA species in organic–water mixtures. The glass transition temperature of pure water (Tg,w) is 136 K (Kohl et al., 2005). Based on Tg(ωorg), viscosity can be calculated with the Vogel–Tammann–Fulcher (VTF) equation (Angell, 1991): η=η∞eT0DT-T0, 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 based on our previous study in DeRieux and Li et al. (2018). T0 is the Vogel temperature calculated as T0=39.17Tg(ωorg)D+39.17. We further calculate the bulk diffusion coefficient (Db) of organic molecules with a radius of 0.4 nm (Maclean et al., 2021) and water molecules in SOA particles based on predicted viscosity and the fractional Stokes–Einstein equation (Price et al., 2016; Evoy et al., 2019, 2020), which is detailed in the Supplement. The mixing timescales of molecules (τmix) within SOA particles is calculated as τmix=dp2/(4π2Db) (Seinfeld and Pandis, 2016), where dp is the particle diameter. The dp is assumed to be 200 nm (Maclean et al., 2021) when we calculate τmix.

Table 1 lists all the performed simulations. In the base case, we update the C* in the lowest volatility bin from 10-4 µg m-3 in the official WRF-Chem v3.7.1 to 0.1 µg m-3 based on the ambient volatility observations (referring to Sect. 2.4 and Fig. 1), and we calculate the ΔHvap in the lowest volatility bin using the semi-empirical parameterization in Epstein et al. (2010), leading to a value of 142 kJ mol-1. To evaluate the effects of simulated SOA volatility distributions on phase state estimations, we conduct a simulation (sensitivity case A) following the default setting in the model assuming that the lowest C* is 10-4 µg m-3 at 298 K, with ΔHvap of 40 kJ mol-1 (Knote et al., 2015). A smaller ΔHvap indicates less dependence of volatility on temperature variations. In the sensitivity case B, we increase the simulated RH by a factor of 10 % as we find that the simulated RH values are smaller than the observations (Sect. 3.1). In the base case and sensitivity cases A and B, we predict Tg for SOA–water mixtures accounting for the SOA-influenced water uptake solely, assuming that SOA particles are externally mixed with inorganic compounds such as sulfate and nitrate. In the sensitivity case C, we assume that the organic and inorganic compounds are internally mixed in one phase and include the water absorbed by inorganic compounds in viscosity calculations. The water associated with inorganics is calculated by the MOSAIC module coupled in WRF-Chem.

The observation data measured at an urban site in the Institute of Atmospheric Physics (IAP) of the Chinese Academy of Sciences (39°58′28′′ N, 116°22′16′′ E) in Beijing (Fig. S1) are used to compare with the simulation results of the WRF-Chem model. The aerosol volatility was measured from 20 May to 23 June 2018, using a thermodenuder coupled with an Aerodyne high-resolution aerosol mass spectrometer (Xu et al., 2019). The volatility distributions of oxygenated organic aerosols (OOAs) resolved from positive matrix factorization (PMF) were estimated using a dynamic mass transfer model (Riipinen et al., 2010). The volatility of OOA was found to be distributed in six logarithmically spaced C* bins including 0.001, 0.01, 0.1, 1, 10, and 100 µg m-3, based on the best fits between the measured and predicted thermograms using the methods in Karnezi et al. (2014). Chemical species including organics (Org), sulfate (SO42-), nitrate (NO3-), and ammonium (NH4+) in PM2.5 were measured using an Aerodyne time-of-flight aerosol chemical speciation monitor (Fröhlich et al., 2013) equipped with a capture vaporizer and PM2.5 lens, with the details described in Li et al. (2023). The OOA factor was identified with the PMF analysis. We obtain the mass concentrations of PM2.5 from the Olympic Center observation site (http://zx.bjmemc.com.cn, last access: 1 June 2023), which is ∼4 km from the IAP site (Fig. S1). Meteorological parameters including RH and T are from the Beijing meteorological tower at the IAP site.

The comprehensive model evaluations were conducted in our previous studies, showing that the WRF-Chem model reasonably captured the magnitudes and spatial distributions of concentrations of major air pollutants over China (Li et al., 2011, 2014, 2015; Qu et al., 2019; Zhang et al., 2022). Here we focus on the comparison of simulations and observations at the IAP site during 20 May–23 June 2018 when the observed volatility distributions are available (Xu et al., 2019).

Figure 1 shows the average volatility distributions of observed OOA and simulated SOA at the IAP site. The C* of OOA spans from 0.001 to 100 µg m-3, with an average value of 1.16 µg m-3. The semi-volatile organic compounds (SVOCs; 0.3<C0<300 µg m-3) and the low-volatility organic compounds (LVOCs; 3×10-4<C0<0.3 µg m-3) (Donahue et al., 2012) contribute 66.3 % and 33.7 % to OOA concentrations, respectively (Xu et al., 2019). The Tg,org estimated from the observed OOA volatility distributions is 286.7 K. Figure 1a shows the simulated volatility distributions of SOA with five C* bins set to be 0.1, 1, 10, 100, and 1000 µg m-3 at 298 K and ΔHvap of 142, 131, 120, 109, and 98 kJ mol-1 used in the five C* bins, respectively (base case in Table 1). In this base case simulation, the SOA consists of 64.5 % SVOCs and 35.3 % LVOCs, and most of the SVOC species are located in the C* bin of 1 µg m-3. The simulated SOA in Fig. 1a has an average C* of 0.64 µg m-3 and Tg,org of 291.5 K, close to the values estimated from the volatility distributions of OOA. Figure 1b shows the simulated volatility distributions of SOA with the lowest C* bin set to be 0.0001 µg m-3 at 298 K with ΔHvap of 40 kJ mol-1, following the default option in the official WRF-Chem model 3.7.1 (Knote et al., 2015). In this sensitivity simulation (case A in Table 1), the SOA consists of 40.4 % extremely low-volatility organic compounds (ELVOCs; C0<3×10-4 µg m-3), which are not determined in the observed OOA, leading to a much lower average C* (0.03 µg m-3) and a higher Tg,org (309.0 K) compared to the observations. In the following we estimate the Tg,org and viscosity of SOA using the simulated volatility distributions in the base case with the lowest C* bin set as 0.1 µg m-3 at 298 K. The impacts of volatility distributions with the incorporation of ELVOCs (Fig. 1b) on viscosity estimations are evaluated in Sect. 3.3.

Figure 2 shows that the model relatively well reproduces the observed hourly variations of RH, T, mass concentrations of PM2.5, and its major inorganic components (Fig. 2a–f), with the index of agreement (IOA, defined in the Supplement) varying from ∼0.70 for inorganic components to a higher value of 0.93 for T (Table S2). The simulated values of RH are constantly lower than the observations, with the mean bias (MB) being -10.97 % (Table S2). The underestimation of RH observations was also found at other meteorological sites in the North China Plain in our previous studies (Qu et al., 2019; Zhang et al., 2022), which would affect the SOA viscosity estimations. The effects of RH on the viscosity estimations are evaluated in Sect. 3.3. Figure 2g shows that the model could generally reproduce the observed temporal variations of OOA concentrations but largely underestimates the observation peaks (MB = -5.88 µg m-3; the normalized mean bias NMB = -53.28 %, Table S2). Incorporation of the SOA formed from intermediate-volatility organic compounds (IVOCs) (Miao et al., 2021; Chang et al., 2022) would increase the simulated SOA concentrations, which is beyond the scope of this study and will be considered in our future work. The simulated SOA mean concentration is 5.15 µg  -3. Although it is lower than the observed value of 11.03 µg m-3, this difference in the simulated and observed SOA concentrations would not affect the viscosity predictions significantly. The SOA viscosity has a much closer relation with the volatility rather than its mass loadings (Champion et al., 2019; Li et al., 2020). In our previous study we estimated Tg,org and viscosities at different organic aerosol (OA) mass loadings varied from 1 to 1000 µg m-3, showing that the simulated viscosities were very similar, particularly when RH was higher than 50 % (DeRieux et al., 2018).

As the WRF-Chem model underestimates the observed RH at the IAP site, we calculate the SOA viscosity using the simulated and observed RH, respectively. Figure 2h shows that the viscosities calculated at the two conditions are similar at most times during the simulated period, ranging mainly from ∼102 to 1010 Pa s, with a median value of ∼107 Pa s, indicating that a semi-solid phase state frequently occurs. The underestimations of the observed RH by WRF-Chem mainly impact the phase state estimations at relatively high RH. For example, SOA particles occur as liquid when the observed RH is higher than ∼75 %; however, they remain in a semi-solid phase state at the simulated RH. The bulk diffusion coefficients (Db) of organic molecules range from 10-18 to 10-11 cm2 s-1 at the simulated RH (Fig. 2i), leading to the mixing timescales within 200 nm SOA particles being seconds to years, with 61 % of the time > 1 h (Fig. S2). We highlight the mixing timescale of 1 h as the time step adopted in CTMs is often ∼0.5–1 h (Maclean et al., 2021). Current CTMs usually assume that the gas–particle partitioning of SVOCs reaches equilibrium quickly within the time step (Pankow, 1994; Donahue et al., 2006). When the mixing timescales of organics within SOA particles are ≤∼1 h, the instantaneous equilibrium is a reasonable assumption. However, when the mixing timescales of organics are longer than ∼1 h, non-equilibrium between the gas phase and the particle phase, i.e., the kinetic partitioning, may need to be considered in simulating the SOA formation in CTMs (Shiraiwa and Seinfeld, 2012; Zaveri et al., 2018; Li and Shiraiwa, 2019; He et al., 2021; Jathar et al., 2021; Maclean et al., 2021; Shiraiwa and Pöschl, 2021; Shrivastava et al., 2022).

The vertical profiles of SOA viscosity exhibit diurnal variations. Figure 3a shows the median diurnal and vertical profiles of predicted SOA viscosity at the IAP site. The SOA particles remain highly viscous (∼107–108 Pa s) at the surface, with a higher viscosity occurring from late afternoon to early evening, during which the RH is less than 20 %, which is lower than the rest time of the day (Fig. S3). The SOA particles become more viscous at higher altitudes than the surface and adopt the phase transition from a semi-solid phase to a solid phase at ∼4 km at the IAP site. The predicted altitude with the phase transition is ∼2 km higher than our previous global model prediction for the region of East China, which was an average of 5 years' simulations (Shiraiwa et al., 2017). Tan et al. (2020) inferred the phase state of submicrometer particles in Beijing from the surface to an altitude of ∼1 km using a polarization lidar and found that the particle phase state exhibits a vertical variation. Further observations of SOA viscosity at high altitudes are needed to better understand the viscosity vertical profiles and validate our predictions. Figure 3b shows that the mixing timescales for organic molecules within 200 nm SOA particles are approximately 1 h at the surface and longer than ∼10 h at altitudes higher than 1 km, indicating that kinetic limitations in the gas–particle partitioning may be required to accurately predict SOA mass concentrations in summer Beijing, particularly in the upper planetary boundary layer and the free troposphere.

The glass transition temperature of the dry organic phase (Tg,org) shows a geospatial gradient over China. Figure 4a shows the median surface values of Tg,org calculated in the base simulation (Table 1). Tg,org ranges from ∼287–305 K over most areas of China, with lower values occurring mainly over the southeast and higher values over the northwest. The Tg,org range simulated by the WRF-Chem model is consistent with our previous global simulations of Tg,org that varied from ∼285 to 310 K at the surface over China (Shiraiwa et al., 2017). The geospatial variation in Tg,org is related to the simulated SOA volatility distributions. Figure 4b shows the mass fractions of SOA species distributed in the lowest volatility bin (SOAX with C*=0.1 µg m-3 at 298 K). The mass fractions of SOAX are mostly 20 %–35 % in southeastern China, indicating that the majority of the simulated SOA formed from VOCs is semi-volatile. In these areas, the simulated SOA mass concentrations are higher than the other locations of China (Fig. S4) (Li et al., 2022), which is favorable for more SVOCs partitioning into the particle phase, leading to relatively low values of Tg,org (Fig. 4a). LVOCs are more frequently contained in aged SOA particles in remote areas; e.g., some areas in northwestern China where the SOA mass concentrations are very low resulted in higher Tg,org values.

The relative humidity plays an important role in regulating SOA viscosity (Koop et al., 2011). Considering the water uptake by SOA particles in the phase state estimations, the predicted geospatial patterns in the viscosity (Fig. 5a) and RH (Fig. S4) are very similar to each other, particularly in southern and northeastern China. SOA particles are predicted to mainly be liquid or with a low viscosity (<104 Pa s) in the southeast. Figure 5b shows the frequency of the liquid phase state, which is calculated as the percent time that an organic aerosol particle is in the liquid phase state during the simulated period. The frequency of liquid particles varies from ∼30 % to 70 % in southeastern China. The lowest viscosity with the highest frequency of liquid particles occurs over the southern Tibetan Plateau where RH is very high (Fig. S4), which is contributed by summer monsoons and regional moisture recycling (Dong et al., 2016). The SOA particles in central and northeastern China are predicted to be semi-solid, with the viscosity varying from 105 to 108 Pa s (Fig. 5a). Highly viscous (η>108 Pa s) or solid SOA particles are mainly found in the northwest, particularly over the northern Tibetan Plateau where the ambient temperatures are lower than other areas of China (Fig. S4). The frequency of liquid SOA particles in most areas with a latitude higher than 30° N is less than 20 % (Fig. 5b).

The simulated geospatial pattern in SOA viscosity over China agrees with previous global simulations and ambient measurements. Our previous global simulations predicted a lower viscosity (η<103 Pa s) in SOA particles in southeastern China and a higher viscosity (η>108 Pa s) in northwestern China (Shiraiwa et al., 2017; Li et al., 2020), similar to the WRF-Chem simulations in this study. Interestingly, the occurrence of liquid particles over the southern Tibetan Plateau in summer simulated by the WRF-Chem was not found in our previous global predictions, which was an average of 5 years' simulations (Shiraiwa et al., 2017). The semi-solid phase state of SOA particles simulated in Beijing is consistent with both particle bounce measurements (Liu et al., 2017) and the PM2.5 phase state determined by the poke-and-flow technique (Song et al., 2022). The simulated viscosity of SOA particles is 0.15 Pa s in Shenzhen, a coastal urban city in southeastern China, which also agrees with the findings in the previous bounce measurements indicating that the submicron particles in Shenzhen are in the liquid state (Liu et al., 2019).

The phase state of SOA particles is affected by ambient conditions and the particle chemical composition (Koop et al., 2011). Figure 6 shows the median values of viscosity as a function of RH, T, and the mass fraction of low-volatility compound (SOAX with C* of 0.1 µg m-3 at 298 K) calculated for selected regions in northern China, southern China, the northern Qinghai–Tibet Plateau, and the southern Qinghai–Tibet Plateau as specified by white boxes in Fig. 5a. There is a strong inverse relationship between SOA viscosity and relative humidity, with high RH (>∼60 %) as the dominant factor determining the phase state of SOA particles. When RH is lower than ∼60 %, the predicted viscosity is affected by both RH and T. For example, the SOA particles occur mainly as solid over the northern Qinghai–Tibet Plateau while they occur as semi-solid over northern China within similar RH ranges (20% < RH < 60 %); the reason is that the ambient T over the northern Qinghai–Tibet Plateau is much lower (∼20 K lower) than over northern China (Fig. 6b). When RH is relatively low, the viscosity of SOA particles is also influenced by particle chemical composition; i.e., the SOA particles composed of high mass fractions of low-volatility compounds tend to have higher viscosity values (Fig. 6c). RH is the main factor driving the diurnal variations of SOA viscosity in our simulations. Figure S5 shows that SOA particles have higher viscosity in the daytime than the nighttime as RH in the daytime is lower than the nighttime (Fig. S6). Compared to northern China, southern China exhibits stronger diurnal variations in SOA viscosity, where SOA particles occur mainly as semi-solid in the daytime and liquid in the nighttime. Highly viscous or solid SOA particles are found in northern China during both daytime and nighttime (Fig. S5).

The bulk diffusion coefficient is an important parameter determining the mass transport and mixing rates, which can be predicted by the particle viscosity through the fractional Stokes–Einstein relation (see the Method section and the Supplement). The Db of organic molecules is predicted to be >∼10-10 cm2 s-1 in southern China. The highest value is ∼10-5 cm2 s-1, occurring in liquid SOA particles in the southern Tibetan Plateau (Fig. 7a) because of the very low viscosity simulated over this region (Fig. 5a). The Db of organic molecules within semi-solid SOA particles is ∼10-18–10-10 cm2 s-1 in central and northeastern China and lower than ∼10-18 cm2 s-1 in highly viscous and solid particles in most areas of northwestern China. Figure 7b shows the percent time that the mixing timescale of organic molecules in 200 nm particles is less than 1 h in the entire simulation period. The mixing timescale is found to be nearly always less than 1 h in the southeastern region of the Hu Huanyong Line. The Hu Huanyong Line, proposed by the Chinese geographer Huanyong Hu, divides China into two parts based on contrasting population densities (Hu, 1935), which was also found useful for characterizing the drought conditions, with the northwestern region much dryer than the southeastern region (Zeng et al., 2021). The mixing timescale of organic molecules in highly viscous or solid SOA particles in the northwest of the Hu Huanyong Line is often longer than 1 h (the frequency > 70 %), indicating that in these areas kinetic limitations of bulk diffusion should be considered in SOA partitioning. Compared to the diffusion coefficients of organic molecules, the Db of water molecules in SOA particles at the surface is several orders of magnitude larger, with the values higher than 10-10 cm2 s-1 in the southeast and as low as ∼10-13 cm2 s-1 in northwestern China (Fig. 7c). The mixing timescales of water molecules in SOA particles with a diameter of 200 nm are of the order of milliseconds in the southeast and seconds in the northwest of China (Fig. S7), indicating that the activation of cloud condensation nuclei would not be inhibited, in agreement with our previous global simulations (Shiraiwa et al., 2017).

Figure 8 shows the simulated Tg,org and the phase state of SOA particles, as well as the mixing timescale of organic molecules in SOA particles at 500 hPa. The Tg,org ranges from ∼285–295 K, which is lower than the Tg,org simulated at the surface (Fig. 8a). The reason is that the mass fractions of LVOCs (SOAX with C* of 0.1 µg m-3 at 298 K) at 500 hPa (Fig. S8) are smaller than the surface values (Fig. 4b). The low temperature at 500 hPa is favorable for SVOCs partitioned into the particle phase; thus, compared to the surface conditions, there are fewer semi-volatile vapors undergoing the further gas-phase aging forming SOAX species. The percent time that an organic aerosol particle is in the liquid phase state (the frequency of liquid SOA particles) at 500 hPa in southeastern China is 20 %–35 % (Fig. 8b), which is ∼20 % lower than the surface values (Fig. 5b). In northern China, the frequency of liquid SOA particles at 500 hPa is similar to the results at the surface, which is related to the RH spatial patterns (Fig. S9). The mixing timescale of organic molecules in 200 nm SOA particles is frequently longer than 1 h at 500 hPa, with the frequency > 70 % in northern China and ∼40 % in southeastern China (Fig. 8c). The τmix is relatively short (with the frequency of τmix≤1 h being ∼80 %) in some areas of southwestern China at 500 hPa, where relatively high RH could occur (Fig. S9) in the season of summer monsoon (Huang et al., 1998).