The following data were used to develop a parameterization of the viscosity α-pinene SOA as a function of temperature and RH: (a) room-temperature measurements of viscosity of SOA derived from α-pinene ozonolysis by Grayson et al. (2016) (Table S1 in the Supplement), (b) low-temperature measurements of viscosity for SOA derived from α-pinene ozonolysis by Järvinen et al. (2016) (Table S2), and (c) temperature-dependent measurements of viscosity for water from Crittenden et al. (2012) (Table S3). Järvinen et al. (2016) measured the temperature and RH values at which α-pinene SOA has a viscosity of approximately 107 Pas. In these experiments, SOA was generated with a mass concentration of 707–1414 µgm-3. Grayson et al. (2016) measured viscosity of α-pinene SOA as a function of RH at 295 K. In these experiments, the SOA was generated with mass concentrations of 121 and 520 µgm-3. We use the viscosity measurements from Grayson et al. (2016) determined with a mass concentration of 520 µgm-3 to be more consistent with the mass concentrations used by Järvinen et al. (2016). Although there are other room-temperature measurements of the viscosity of α-pinene SOA (Bateman et al., 2015; Hosny et al., 2016; Kidd et al., 2014; Pajunoja et al., 2014; Renbaum-Wolff et al., 2013), we used the room-temperature measurements from Grayson et al. (2016) because (1) viscosity was measured over a range of relative humidities in this study, (2) the mass concentrations used by Grayson et al. (2016) to generate the SOA were similar to the mass concentrations used by Järvinen et al. (2016), and (3) Grayson et al. (2016) measured the viscosity of the total SOA (both the water-soluble component and the water-insoluble component).

Due to the experimental conditions used by Grayson et al. (2016) and Järvinen et al. (2016), the parameterization developed here is applicable to SOA generated using a mass concentration of ∼1000 µgm-3. We focused on ∼1000 µgm-3 because both low-temperature and room-temperature viscosity measurements have been carried out using this mass concentration. The effect of mass concentration on the viscosity α-pinene SOA is discussed in Sect. 3.5.

To develop a parameterization for viscosity as function of temperature and RH, the following equation was fit to the measurements by Grayson et al. (2016), Järvinen et al. (2016), and Crittenden et al. (2012) (Tables S1–S3): log(η)=12-C1×T-wSOATgSOA+wH2OTgH2OkGTwSOA+wH2OkGTC2+T-wSOATgSOA+wH2OTgH2OkGTwSOA+wH2OkGT, where C1 and C2 are constants, kGT is the Gordon–Taylor fitting parameter, TgSOA and TgH2O are the glass transition temperatures of dry SOA and water, and wSOA and wH2O are the weight fractions of the dry SOA and water in the particles. The weight fractions of the dry SOA and water in the particles were determined from the RH using the following equation (Koop et al., 2011): RH100=1(1+iSOAnSOAnH2O), where i is the van 't Hoff factor and n is the number of moles of dry SOA and water in the particles. We assumed a value of 1 for the van 't Hoff factor (Koop et al., 2011) and a dry molecular weight for SOA of 175 gmol-1 (Huff Hartz et al., 2005).

Since the glass transition temperature of water is known (135 K) (Corti et al., 2008), the unknowns in Eq. () (and hence fitting parameters) were C1, C2, kGT, and TgSOA. The values for these parameters retrieved by fitting the equation to the viscosity data discussed above (using a nonlinear curve fitting function in MATLAB) are reported in Table S4.

Equation () was based on the Williams–Landel–Ferry (WLF) equation and the Gordon–Taylor equation. The WLF equation provides a relationship between viscosity and temperature: log⁡ηηg=-C1(T-Tg)C2+(T-Tg), where C1 and C2 are constants, T is the temperature, Tg is the glass transition temperature, η is the viscosity, and ηg is the viscosity at the glass transition (1012 Pas). The Gordon–Taylor equation provides a relationship between the glass transition temperature of a mixture and the weight fractions of its components: Tg,mix=w1Tg1+w2Tg2kGTw1+w2kGT, where w1 and w2 are the weight fractions of the solute and water, Tg1 and Tg2 are the glass transition temperatures of the solute and water, and kGT is a fitting parameter that describes the interaction between the two species. Equations () and () can be combined to give Eq. ().

Equation () (and hence Eq. ) is valid only at or above the glass transition temperature. As a result, we have not used Eq. () to predict viscosities >1012 Pas (which corresponds to mixing times >5×105 h). If the temperature and RH in the PBL were such that the viscosity was greater than 1012 Pas, we assigned a viscosity of 1012 Pas and a mixing time of 5×105 hours. This assignment does not affect the conclusions in this paper since a mixing time of 5×105 hours is already well above the residence time of SOA particles in the atmosphere.

To determine the conditions in the PBL when SOA concentrations are significant we used the global chemical transport model GEOS-Chem (http://acmg.seas.harvard.edu/geos/). The version of GEOS-Chem used (v10-01) includes organic aerosol (OA) formation from SVOC and intermediate-volatility organic compounds (IVOC) (Pye and Seinfeld, 2010), plus new aerosol production from nitrate radical oxidation of isoprene and terpenes and NOx-dependent aerosol yields from terpenes (Pye et al., 2010). In this version IVOC emissions are spatially distributed based on naphthalene. To estimate SVOC emissions we scaled the default GEOS-Chem primary organic aerosol emissions inventory by 1.27 following Pye and Seinfeld (2010). GEOS-Chem was run at a horizontal grid resolution of 4∘ latitude by 5∘ longitude using GEOS-5 meteorology with 47 vertical layers with a 3-year spin-up period. Shown in Fig. 1 are the monthly averaged total OA concentrations at the surface for the months of January and July 2006. These monthly averaged total OA concentrations were used to remove times and locations where SOA concentrations are not expected to be of major importance for climate, health, or visibility.

Information on the RH and temperature distributions in the global PBL in different seasons was also needed to assess mixing times within SOA particles. First, the PBL heights were determined globally using the 6 h averaged GEOS-5 meteorology fields. Then, temperature and RH in each grid cell within the PBL were determined globally using the 6 h averaged GEOS-5 meteorology fields. To determine if a grid cell was within the PBL, the PBL heights mentioned above were used. The GEOS-5 archive provides temperature and RH at a horizontal grid resolution of 4∘ latitude by 5∘ longitude and 47 vertical layers.

Shown in Fig. 2a (contours) is the RH and temperature-dependent parameterization for α-pinene SOA viscosities based on viscosities measured at room temperature (Grayson et al., 2016) and low temperature (Järvinen et al., 2016), as well as the viscosity of water as a function of temperature (Crittenden et al., 2012). From the viscosity parameterization, the diffusion coefficients of organic molecules within α-pinene SOA particles were calculated using the Stokes–Einstein equation: D=kT6πηRH, where D is the diffusion coefficient, k is the Boltzmann constant, T is temperature in Kelvin, η is the dynamic viscosity, and RH is the hydrodynamic radius of the diffusing species. For the calculations, a hydrodynamic radius of 0.38 nm was used for the diffusing organic molecules within SOA based on an assumed molecular weight of 175 gmol-1 (Huff Hartz et al., 2005), a density of 1.3 gcm-3 (Chen and Hopke, 2009; Saathoff et al., 2009), and spherical symmetry. The Stokes–Einstein equation should give reasonable values when the radius of the diffusing molecules is roughly greater than or equal to the radius of the matrix molecules and when the viscosity of the matrix is relatively small (400 Pas) (Chenyakin et al., 2017; Price et al., 2016). When the viscosity of the matrix is large (106 Pas), the Stokes–Einstein equation can underpredict diffusion coefficients of organic molecules in organic matrices (Champion et al., 1997; Chenyakin et al., 2017; Price et al., 2016). Hence, the diffusion coefficients and mixing times estimated here should be considered lower and upper limits, respectively.

From the diffusion coefficients, the mixing times of organic molecules within an α-pinene SOA particle were calculated with the following equation (Shiraiwa et al., 2011): τmix=d24π2D, where τmix is the mixing time, d is the diameter of an SOA particle, and D is the diffusion coefficient estimated from Eq. (). For these calculations, it was assumed that the α-pinene SOA particles have a diameter of 200 nm, which is roughly the median diameter in the volume distribution of ambient SOA-containing particles (Martin et al., 2010; Pöschl et al., 2010; Riipinen et al., 2011). Once the mixing time has elapsed, the concentration of the diffusing molecules at the centre of the particle is within 1/e of the equilibrium concentration (Shiraiwa et al., 2011). The calculated mixing times (Fig. 2b) illustrate that, as expected, inverse relationships exist between both mixing time and RH, as well as mixing time and temperature.

Shown in Fig. 3a and b are the normalized frequency counts of temperature and RH in the PBL for the months of January and July 2006, respectively, based on the archive of meteorological fields (GEOS-5) used in the global chemical transport model, GEOS-Chem v10-01. We only included grid points in our analysis when the grid points were within the PBL and the monthly average mass concentration of total OA was >0.5 µgm-3 at the surface, based on GEOS-Chem v10-01 (Fig. 1). In other words, we included all the grid points in a column up to the top of the PBL when determining frequency counts if the monthly averaged total OA concentration was >0.5 µgm-3 at the surface. This filtering removes cases where SOA concentrations are not expected to be of major importance for climate, health, or visibility. We chose a mass concentration of >0.5 µgm-3 for filtering because the mass concentration of total OA at the surface was >0.5 µgm-3 in all but one of the previous field measurements of OA at remote locations (Spracklen et al., 2011).

The normalized frequency counts illustrate that the temperature and RH in the PBL are often in the range of 290–300 K and >50 % RH for the month of January (Fig. 3a) and in the range of 285–300 K and >30 % RH for the month of July (Fig. 3b). For reference, shown in Figs. S1 and S2 in the Supplement are the average temperature and RH conditions at the Earth's surface and top of the planetary boundary layer, respectively, for January and July, based on the archive of meteorological fields for 2006 (GEOS-5).

Also shown in Fig. 3a and b are the mixing times within 200 nm α-pinene SOA particles predicted with our parameterization (contours). These results, together with the frequency counts of temperature and RH throughout the vertical column of the PBL, indicate that the mixing times of organic molecules within α-pinene SOA are often <1×10-1 h for conditions in the PBL.

Shown in Fig. 4 are the normalized frequency distributions of mixing times within α-pinene SOA for January and July, based on the data in Fig. 3a and b. Figure 4 suggests that the mixing times within α-pinene SOA are <1 h for 98.5 and 99.9 % of the occurrences in the PBL during January and July, respectively, when monthly average total OA concentrations are >0.5 µgm-3 at the surface.

Within the PBL, RH increases and temperature decreases with altitude, with both changes being substantial and impacting mixing times in opposite directions. Shown in Fig. 5a–c are calculated monthly average afternoon (13:00–15:00, LT) vertical profiles of temperature, RH, and mixing times within α-pinene SOA over Hyytiälä (boreal forest) and the Amazon (rainforest) for the driest month of the year at each location (the method used to calculate vertical profiles is described in the Supplement, Sect. S1). Afternoon vertical profiles were chosen since this is the time of the day when RH is typically lowest and thus mixing times are the longest. Figure 5c shows that mixing times within α-pinene SOA decrease significantly with altitude for these two locations. This is because the plasticizing effect of water on viscosity dominates the temperature effect for these conditions.

Shown in Fig. 6 are global maps of the monthly averaged mixing times of organic molecules within α-pinene SOA for conditions at the top of the PBL for the months of January and July. Figure 6 shows that 92 and 98 % of the locations for January and July, respectively, have a mixing time <0.1 h for conditions at the top of the PBL when monthly averaged total OA surface concentrations are >0.5 µgm-3. Within the PBL, vertical mixing of air masses occurs on the order of 30 min. Since the mixing times within α-pinene SOA particles for conditions at the top of the PBL are <0.1 h for most locations where the SOA concentrations are significant (total OA concentration >0.5 µgm-3 at the surface), a reasonable upper limit of the mixing time within the α-pinene SOA studied here for most locations in the PBL is 30 min. During this 30 min interval, mixing times within α-pinene SOA particles can cycle between short and long values, though rarely being >1 h (Figs. 3 and 4).

To calculate the mixing times discussed above, we assumed that the α-pinene SOA particles have a diameter of 200 nm. We also repeated these calculations assuming a diameter of 500 nm, since aged OA can have larger diameters (Takegawa et al., 2006). Based on the viscosity parameterization shown in Fig. 2a, mixing times within 500 nm α-pinene SOA particles are <1 h for 95.9 and 99.4 % of the occurrences in the PBL during January and July, respectively (Fig. S3).

The parameterization of viscosity used above was developed using viscosity measurements by Grayson et al. (2016), Järvinen et al. (2016), and Crittenden et al. (2012). As a sensitivity analysis, we developed a second parameterization, using the same procedure as describe above but using the upper limits to the viscosities reported by Grayson et al. (2016) and the upper limits to the RH ranges reported by Järvinen et al. (2016). This should result in an upper limit to the viscosity parameterization discussed above. The uncertainties in the measurements by Crittenden et al. (2012) were not considered since they are small compared to the uncertainties reported by Grayson et al. (2016) and Järvinen et al. (2016). Based on this second parameterization, mixing times are <1 h for 96.6 and 99.5 % of the occurrences in the PBL during January and July, respectively, when the total OA was >0.5 µgm-3 at the surface (Fig. S4).

The parameterizations developed above were based on SOA generated using a mass concentration of ∼1000 µgm-3. As mentioned, we focused on ∼1000 µgm-3 because low-temperature and room-temperature viscosity measurements have been carried out using this mass concentration. However, the viscosity of some types of SOA may depend on the mass concentration used to generate the SOA. For example, Grayson et al. (2016) showed that under dry conditions, the viscosity of α-pinene SOA may increase by a factor of 5 as the production mass concentration decreased from 1200 to 120 µgm-3. In addition, mass concentrations of biogenic SOA are typically ≤10 µgm-3 in the atmosphere (Spracklen et al., 2011). As a starting point to quantify how often mixing times of organic molecules are <1 h within α-pinene SOA generated using low, atmospherically relevant mass concentrations, we developed a temperature-independent parameterization using the room-temperature viscosity data for α-pinene SOA from Zhang et al. (2015) (Table S5) and room-temperature viscosity data for water from (Crittenden et al., 2012) (Table S3). Zhang et al. (2015) measured the viscosity of α-pinene SOA over a range of relative humidities (0–60 %), and the SOA used in these experiments was generated in the laboratory using a mass concentration of ∼70 µgm-3. The median room-temperature viscosities reported by Zhang et al. are higher than the median room-temperature viscosities reported by Grayson et al. (2016) using a mass concentration of 520 µgm-3 (Fig. S5). Although not proven, a reasonable explanation for the difference in median viscosities is the difference in mass concentrations used to generate the SOA.

A temperature-independent parameterization was generated by fitting Eq. () to the room-temperature viscosity data from Zhang et al. (2015) and Crittenden et al. (2012), but with the temperature (T) in Eq. () replaced by 293 K. The values for the parameters retrieved by fitting the modified Eq. () to the viscosity data are reported in Table S6. The temperature-independent parameterization generated using this method is shown in Fig. 7a. Shown in Figs. 7b, 8a, and b (contours) is the parameterization for mixing times within 200 nm α-pinene SOA based on this temperature-independent viscosity parameterization. Also included in Fig. 8a and b are the normalized frequency counts of temperature and RH in the PBL for the months of January and July 2006, respectively, when the monthly average mass concentration of total OA was >0.5 µgm-3 at the surface. Shown in Fig. 9 are the normalized frequency distributions of mixing times within α-pinene SOA for January and July, based on the data in Fig. 8a and b. Figure 9 suggests that the mixing times within α-pinene SOA is <1 h for 27 and 19.5 % of the occurrences in the PBL during January and July, respectively, when monthly average total OA concentrations were >0.5 µgm-3 at the surface. However, several caveats need to be emphasized: (1) the parameterization was developed based on room-temperature viscosity data only. Viscosities, and hence mixing times, will increase as temperature decreases. As an illustration, the viscosity of sucrose–water mixtures can increase by 2–3 orders of magnitude as the temperature decreases by 10 K close to the glass transition temperature (Champion et al., 1997). (2) The mixing times were calculated using the Stokes–Einstein relation, which can underpredict diffusion coefficients, and hence overpredict mixing times, when the viscosity of the matrix is high. For example, the Stokes–Einstein equation can underpredict diffusion coefficients of organic molecules in sucrose–water mixtures by at least a factor of 10 to 100 at viscosities ≥106 Pas (Chenyakin et al., 2017; Price et al., 2016). (3) The viscosity data from Zhang et al. (2015) have an uncertainty of ±1 order of magnitude, which was not considered in the temperature-independent parameterization. Considering these caveats, we are unable to make strong conclusions about how often mixing times of organic molecules are <1 h within α-pinene SOA generated at low mass concentrations. To help resolve this issue, temperature-dependent studies of the viscosity of α-pinene SOA generated using low mass concentrations are needed. Also, the accuracy of the Stokes–Einstein equation for predicting diffusion coefficients of organics within α-pinene SOA needs to be determined.

Recently it has been shown that the diffusion rates of organics in SOA from toluene photooxidation are slower than the diffusion rates of organics in SOA from α-pinene ozonolysis at room temperature (Liu et al., 2016; Song et al., 2016; Ye et al., 2016). These results indicate that mixing times are longer in some types of anthropogenic SOA than some types of biogenic SOA, at least at room temperature. SOA derived from anthropogenic sources can be a significant contributor to SOA over polluted regions (Hallquist et al., 2009; Spracklen et al., 2011). Viscosities or diffusion rates within toluene SOA or other types of anthropogenic SOA have yet to be measured at temperatures lower than room temperature. As a result, we have used sucrose as a proxy of anthropogenic SOA since the viscosity of sucrose is similar to the viscosity of toluene SOA at room temperature (Fig. S6) (Power and Reid, 2014; Song et al., 2016), and since a parameterization of the viscosity of sucrose as a function of temperature and RH can be developed using literature data. In the Supplement (Sect. S2, Table S7–S9, and Figs. S7–S10) we carried out a similar analysis for sucrose as for α-pinene SOA above. Assuming sucrose is a good proxy for anthropogenic SOA, the analysis suggests that 70 and 83 % of the mixing times within anthropogenic SOA in the PBL are <1 h for January and July, respectively, when SOA concentrations are significant (total OA concentration >0.5 µgm-3 at the surface). In addition, 81 and 87 % of the locations for January and July, respectively, have a mixing time <0.1 h at the top of the PBL when surface concentrations of total OA are >0.5 µgm-3. These percentages for anthropogenic SOA are likely lower limits since, as mentioned earlier, studies have shown that the Stokes–Einstein relation (which is used here to calculate diffusion coefficients of organic molecules from viscosities) can underpredict diffusion coefficients of organic molecules in sucrose–water mixtures by at least a factor of 10 to 100 at viscosities ≥106 Pas (Chenyakin et al., 2017; Price et al., 2016). Measurements of diffusion rates of organic molecules within anthropogenic SOA as a function of both temperature and RH are needed to better constrain how often mixing times are >1 h within anthropogenic SOA in the PBL.

Shiraiwa et al. (2017) recently estimated mixing times of organics within SOA in the troposphere using a global chemistry climate model and a relationship between glass transition temperatures, molar mass, and oxygen-to-carbon elemental ratios. Their results suggest mixing times of organics within SOA are short (<1 min) over the oceans, tropics, and high latitudes at the surface and 850 hPa. However, their results suggest mixing times are long (>1 h) over dry regions (i.e. major deserts) at the surface and at 850 hPa and over most continental regions at 850 hPa. The general trends observed by Shiraiwa et al. (2017) are consistent with the trends observed here. However, the mixing times predicted by Shiraiwa et al. (2017) appear to be longer than the mixing times predicted here using viscosities of α-pinene SOA generated with a mass concentration ∼1000 µgm-3. Quantitative differences between the current work and the work by Shiraiwa et al. (2017) are not surprising since Shiraiwa et al. (2017) considered both anthropogenic SOA and biogenic SOA simultaneously and since they used a very different approach to estimate viscosities of atmospheric SOA.