Gas-phase oxidation of VOCs involves a large number of reactions and produces a large number of products that can condense in the particulate phase. Depending on their effective saturation concentration, they can be represented in the 1D VBS framework by 1VOC+oxidant→α1P1+α2P2+…+αnPn+volatileproducts, where n is the number of the surrogate compounds (volatility bins in the VBS), Pi is the surrogate product in the ith volatility bin, and αi is the corresponding stoichiometric mass yield. The total SOA mass yield can be then calculated as 2Y≡COAΔVOC=∑inαi1+Ci∗/COA, where COA is the total SOA concentration, ΔVOC is the consumed concentration of the VOC, and Ci∗ is the effective saturation concentration of compound i. This yield equation is an extension of the two-product model by Odum et al. (1996), replacing their semi-empirical partitioning coefficients with the assumption of a pseudo-ideal solution (Strader et al., 1999). This model assumes that the system has reached equilibrium when the yield is measured and that the differences in molecular weights are small.

The effective saturation concentrations at different temperatures are given by the Clausius–Clapeyron equation: 3Ci∗T=Ci∗TrefTrefTexp⁡ΔHvap,iR1Tref-1T, where Tref is the reference temperature in which the reference effective saturation concentration is defined (298 K in this work) and ΔHvap,i is the enthalpy of vaporization of surrogate compound i.

The time-dependent evaporation of SOA in the TD used in this work is described by the dynamic mass transfer model of Riipinen et al. (2010). The evolution of the total particle mass, mp, and the gas-phase concentration of the compound i, Ci, are given by 4dmpdt=-∑i=1nIi,5dCidt=IiNtot, where n is the number of surrogate compounds, Ntot is the total number concentration of particles (assuming a monodisperse aerosol population), and Ii is the mass flux of compound i from the gas to the particulate phase for each particle calculated by (Seinfeld and Pandis, 2016) 6Ii=2πdpMiβmiDiRTTDpi-pi0. Here dp is the particle diameter, R is the ideal gas constant, Mi is the molecular weight of compound i, Di is the diffusion coefficient of compound i in the gas phase at temperature TTD, pi and pi0 are the partial vapor pressures of i far away from the particle and at the particle surface respectively, and βmi is a factor for the correction of kinetic and transition regime effects (Fuchs and Sutugin, 1970): 7βmi=1+Kni1+43αmi+0.377Kni+43αmiKni. Here Kni is the Knudsen number of compound i and αmi is the mass accommodation coefficient of compound i on the particles. The partial vapor pressure of compound i at the particle surface is given by 8pi0=xmiCi∗RTMiexp⁡4MiσRTTDρdp, where xmi is the mass fraction of compound i in the particulate phase, Ci∗ is the effective saturation concentration, σ is the surface tension (assumed to be 0.05 N m-1 in our simulations), TTD is the particle temperature assumed to be the same as in the TD, and ρ is the particle density. The effective saturation concentrations at different TD temperatures are given by Eq. (3).

Processes other than organic aerosol evaporation may affect the TD measurements. For example, thermal decomposition may accelerate the transfer of organic compounds from the particulate to the gas phase and may lead to overestimation of the volatility of especially the least volatile components of the SOA (Epstein et al., 2010; Saha and Grieshop, 2016; Stark et al., 2017). However, the corresponding parameters for the SVOCs and the more volatile LVOCs that are important for atmospheric SOA modeling should be a lot less uncertain given that they are measured in relatively low TD temperatures. The use of isothermal dilution measurements may also help identify cases in which the model does not include a process (e.g., thermal decomposition) that dominates the behavior of the aerosol during heating. In this case, one expects that the overall algorithm will have difficulties reproducing all measurements (yields, isothermal dilution, and evaporation in the TD).

In isothermal dilution experiments, an SOA sample is injected into a reactor filled with clean air at room temperature. The concentrations of both the gas- and the particulate-phase components are lowered due to dilution leading the system out of equilibrium. The evaporation of SOA as a result of isothermal dilution is also described by Eqs. (3)–(8) (Karnezi et al., 2014), but the temperature is equal to 298 K. Evaporation in a dilution chamber depends on the initial SOA mass, time, and αm but not on ΔHvap as the particles evaporate without a change in temperature.

The dilution ratio is an important parameter, varying typically from 10 to 20 in SOA experiments (Cain et al., 2020). Low dilution ratios result in little evaporation and little signal to be explored by the parameter estimation algorithm. High dilution ratios lead to very low initial concentrations in the dilution chamber and a lot of noise in the subsequent evaporation measurements.

In order to evaluate the algorithm, we generated data using the output of SOA formation, thermodenuder and isothermal dilution models described in Sect. 2 for systems with known volatility distribution of the products and properties. Then, these data were “corrupted” with random errors to represent the “noise” observed in laboratory measurements for yields, thermograms, and areograms. As a result, there is no set of model parameters that can reproduce all the measurements. The yields were corrupted based on the variability in laboratory measurements of Pathak et al. (2007a), by assuming a normal distribution and standard deviation (σY) given by 13σY=0.1Ytrue+0.02, where Ytrue denotes the correct yields.

For TD, the errors were calculated by assuming a normal distribution and the standard deviation (σTD) suggested by Karnezi et al. (2014): 14σTD=0.51MFRTD,true-0.5MFRTD,true2, where MFRTD,true denotes the correct MFR values for each TD temperature.

For dilution, the errors were calculated by assuming a uniform distribution and standard deviation (σDil) suggested by Karnezi et al. (2014): 15σDil=0.05MFRDil,true+0.03, where MFRDil,true denotes the correct MFR values for isothermal dilution.

Based on the above methodology, we generated “pseudo-measurements” of yield, TD, and isothermal dilution for different SOA systems. The parameters used to produce the pseudo-experimental data are summarized in Table S1 in the Supplement. The “experimental” conditions assumed for the TD and isothermal dilution measurements are shown in Table S2.

In Experiment A, we test the performance of the algorithm against α-pinene ozonolysis data and examine the effect of TD and isothermal dilution data. For Experiment A, the “true” values were taken from the parametrization derived by Pathak et al. (2007b) for the ozonolysis of α-pinene under low-NOx and dark and low-RH conditions. Therefore, these results are good fits of the measurements analyzed in that study. The parametrization was derived assuming a four-volatility-bin system with saturation concentrations ranging from 1 to 103 µg m-3. The effective vaporization enthalpy estimated in that study was equal to 30 kJ mol-1. Because the effective accommodation coefficient was not part of the Pathak et al. (2007b) parametrization, we assumed a value of 0.5 in this work. We used a small number of yield measurements at atmospherically relevant SOA concentrations of 1, 5, 10, 20, and 40 µg m-3 (Fig. 1). For this SOA system, the yield at 40 µg m-3 did not exceed 20 %. The thermogram includes 10 MFR data points in the temperature range of 20 to 200 ∘C. For the highest temperature, more than 70 % of the SOA mass was evaporated. The areogram shows that the corresponding SOA evaporated by almost 70 % in the first 0.5 h and more than 90 % in less than 3 h.

For Experiment B, the true values were taken from the alternative parametrization proposed by Pathak et al. (2007b) for the same oxidation system as described before. This time, the authors used a seven-volatility-bin system with saturation concentrations ranging from 10-2 to 104 µg m-3 in their parametrization. The effective vaporization enthalpy of the parametrization was 30 kJ mol-1, while for the accommodation coefficient we assumed again a value of 0.5. The yield, TD, and isothermal dilution measurements of Experiment B are generated in the same SOA mass concentration, temperature, and dilution time range as in the previous pseudo-experiment (Fig. 2).

For Experiment C, the true values were based on the parametrization of the SOA formed during α-humulene ozonolysis by Sippial et al. (2023). The authors measured high SOA yields for α-humulene in the main smog chamber (∼ 70 % at 60 µg m-3), and their corresponding thermogram suggested that the SOA particles fully evaporated at 150 ∘C, while the areogram showed modest (20 %) evaporation in the dilution chamber after 3 h. A four-volatility-bin set with saturation concentrations ranging from 10-2 to 10 µg m-3 was used in that study to fit the measurements. The stoichiometric coefficients of the three least volatile bins (10-2, 10-1, and 1 µg m-3) were around 0.1 and of the most volatile (10 µg m-3) 0.25. The vaporization enthalpy was 115 kJ mol-1, and the accommodation coefficient was 0.01 (Table S1). We assumed five yield measurements in the SOA concentration range of 1 to 100 µg m-3 with yield values as high as 65 % at 100 µg m-3 (Fig. 3). The corresponding thermogram consisted of 10 data points, and the particles fully evaporated at TD temperatures higher than 150 ∘C. The areogram consisted of 17 data points, and only 20 % of the SOA evaporated in the dilution chamber.

We explored the performance of the algorithm for different choices of the number of volatility bins, the range of saturation concentrations, and the range of SOA mass concentration in the yield measurements. For each test, the true and the estimated properties are summarized in Table 1.

We evaluated the performance of our parameter estimation algorithm, comparing its predictions against both the measurements and the “truth” defined as the predictions of the original parametrization. In both comparisons, mean normalized error (MNE) (Emery et al., 2017) was used as the evaluation metric because it has a simpler physical meaning than NMSE.

For the evaluation against the measurements, MNEM was defined as follows: 16MNEM=100NO∑i=1NOESTi-OiOi, where ESTi is estimated by the algorithm value and corresponds to a specific measured point Oi.

For the evaluation against the truth, which includes conditions (e.g., temperatures or concentrations) for which there are no available measurements, MNET was defined as follows: 17MNET=100Nd∑j=1NdESTj-TRjTRj, where EST and TR are the estimated and the true values respectively. Nd is the total number of data points included in calculations and depends on the selected discretization of the corresponding dependent variable (e.g., SOA concentration, TD temperature, and dilution time). We used a linear discretization for the SOA concentrations (from 0.01 to 50 µg m-3 with a step of 0.01) and the TD temperatures (20 to 200 ∘C with a step of 5 ∘C but excluding zero-MFR values to avoid the division by zero). For the dilution time, the sampling time step was not constant. We used a higher resolution for the first 0.5 h (step of 2 min), in which the evaporation is usually faster, and a lower resolution afterwards (step of 10 min).

Finally, we used the average relative standard deviation (ARSD) as a metric to quantify the uncertainty in the estimates (range of good solutions) using the same discretization as in the MNET metric. The ARSD is given by 18ARSD=100Nd∑j=1NdσjESTj, where σj is the standard deviation for data point j.

In this section, we explore the performance of the algorithm for different choices of the number of volatility bins and the range of saturation concentrations. The analysis of the results of Test B1 has already quantified the effects of using a narrower volatility distribution in the parameter estimation algorithm than the one of the investigated SOA system. Additional sensitivity tests are performed here for all cases.

In Test A2, we used three volatility bins covering the 1–102 µg m-3 saturation concentration range instead of the four bins used in Test A1. The narrower assumed volatility range had a very small effect on the estimated yields at all temperatures (Table 3 and Fig. S4) compared to Test A1. The change in MNET ranged from 3 % at 5 ∘C to 0.3 % at 35 ∘C. Minor changes were detected in the predicted thermogram (change of 0.8 %) and areogram (change of 0.5 %) as well. The uncertainty in the yield estimates increased by less than 2.5 % at all temperatures. The estimated volatility distribution of the SOA products of Test A2 changed by less than 5 % in the two lower bins. The material in the 102 µg m-3 bin increased by 15 % to account for the SOA of higher volatility that could not be included otherwise in the estimated distribution. The estimated ΔHvap was in this case 32 kJ mol-1 (2.7 % decrease), and αm decreased by 12 % with respect to Test A1.

In Test A3, we shifted the assumed four-bin volatility distribution by 1 order of magnitude to lower values (from 1–1000 µg m-3 in Test A1 to 0.1–100 µg m-3 in Test A3). In this case, the algorithm distributed exactly the same material to the 1, 10, and 100 µg m-3 volatility bins as in Test A2, and it predicted correctly zero SOA in the 0.1 µg m-3 bin (Table 1). The ΔHvap and αm estimated values were also unchanged with respect to Test A2. This, in turn, led to the same estimated yields at different temperatures (no change in the error between the two tests).

In Test C2, we applied the algorithm against the Experiment C measurements using a four-volatility-bin system in the 1-to-103 µg m-3 range, which is 2 orders of magnitude higher than the actual range of the true values. Figure 7 shows the results of the fitting for the three types of measurement. Despite the significant mismatch of the volatility distributions, MNEM increased by only 2.3 % for the estimated SOA yields. The error for the TD measurements increased by 20 %, while it actually decreased a little (1.2 %) for the dilution data. The errors compared to the true values increased by less than 3 % for the temperature range 15–35 ∘C, while it increased by 12 % at 5 ∘C. These results suggest that in this case the estimated yields are quite robust to the assumed volatility range. The major effect of the mismatch in volatility ranges was evident in the predicted thermogram with overestimation of the MFR for the 60–120 ∘C temperature range and underprediction at higher temperatures. The increase in MNET for the TD MFR was 17.2 % (Table 3). The change in the predicted areogram was marginal and led to a small increase in MNET (error increase by 0.7 %) (Table 3). The algorithm not only underestimated again αm (0.004 instead of 0.01) but also recognized the high uncertainty in the corresponding estimate. The algorithm distributed significant material to the 1 µg m-3 bin (3.6 times higher than the actual amount) in an effort to account for the absence of the 10-2 and 10-1 µg m-3 bins. The ΔHvap was underestimated with an error of 21 %.

The results of the above tests indicate that a mismatch between the true and assumed volatility ranges of the SOA increases in general the estimation error but the increase is small to modest. This is reassuring for the robustness of the proposed algorithm.

During the last decade there has been a significant shift of the performed SOA smog chamber towards lower SOA concentrations. This is needed to increase the accuracy at ambient concentration levels. The high-SOA-concentration experiments that once represented the majority of performed experiments are becoming increasingly rare. In this subsection we examine the value of these high-concentration experiments for the estimation of SOA yields under ambient conditions.

To examine the effect of measurements at SOA levels much higher than the atmospheric ones, we included an extra yield measurement at 200 µg m-3 in the yield data of Experiments A and B. In Test A4 and B2, we applied the algorithm once again against the three types of measurement by using a four-volatility-bin system with saturation concentrations ranging from 1 to 103 µg m-3.

In Test A4, the additional experiment at high SOA concentration led to an MNET of 15.7 % for the yields at 25 ∘C (Table 3 and Fig. S5), which is lower by 1.6 % than that without this experiment in Test A1. The improvement was more significant at lower temperatures; e.g., MNET at 5 ∘C was reduced from 24.4 % to 20.4 %. The reduction in the ARSD for the SOA yields ranged from 3.8 % at 5 ∘C to 0.9 % at 35 ∘C (Table 4). Figure 8 depicts the results of the model for the yields and the volatility distribution of the products for Test A4. The accuracy of the predicted volatility distribution increased especially for the higher-volatility material. For example, the error for the 102 µg m-3 bin was reduced from 41 % in Test A1 to 6 % in this case (Table S3). Minor changes in the errors were detected for ΔHvap and αm between the two tests (3 % increase and 6 % decrease respectively).

Similarly to Test A4, in Test B2 we added a yield measurement at 200 µg m-3 in the Experiment B set of measurements. Figure 9 depicts the results of the model for the SOA yields at 25 ∘C and the estimated volatility distribution of the products. The use of the additional data point led to a reduction in the MNET from 13.9 % in Test B1 to 9 % in Test B2 at 25 ∘C (Table 3). Similar reductions in the MNET were observed for the other temperatures, with the highest one observed at 5 ∘C (lower error by 7 %) (Fig. 10). The reduction in the ARSD for the estimated yields ranged from 3.3 % at 5 ∘C to 1.2 % at 35 ∘C (Table 4). Minor changes were observed for the estimated thermogram (Fig. S6) (change in the MNET of 1.5 %) and the uncertainty in the estimates (change in the ARSD of 2.5 %). The error in the estimated areogram was also small, but in this case the error increased by 5 %. The additional data point helped decrease the errors for the estimated mass of the more volatile SOA products (Fig. 9) and especially for the 102 µg m-3 bin. The ΔHvap and αm estimated values were only slightly affected by the additional measurement.

By comparing the results of Tests B1 and B2 with Case A, one would expect the retrieved volatility distribution of the products to be quite similar. The differences present are due to a large extent to the different random experimental errors introduced into the two sets of measurements for Experiments A and B. A second reason for the differences is that parametrizations of the two “experiments” by Pathak et al. (2007b), even if they were derived from the same smog chamber experiments, have some differences. As a result, the true yields, thermogram, and areogram in Cases A and B are not exactly the same (Figs. 1 and 2).

These results suggest that an additional yield measurement at high SOA levels can lead to a substantial reduction in the error for the estimated yields at low temperatures (Fig. 10) and also a better estimation of the SOA products with higher volatility (102 and 103 µg m-3). These products may contribute little to the SOA concentration at 25 ∘C, but their reactions (aging) could lead to significant additional SOA in later stages.

To quantify the effect of each type of measurement on the parameter estimation and their subsequent effect on the estimated SOA yields, we repeated Tests A1, B1, and C1 withholding one set of measurements. More specifically, we provided the algorithm with the following combination of measurements: TD and isothermal dilution, SOA yields and isothermal dilution, and finally SOA yields and TD.

The use of only the TD and isothermal dilution data corresponds for all practical purposes to the previous algorithm of Karnezi et al. (2014), which was the starting point of this work. In Test A1, the absence of the yield measurements led to a significant deterioration of the ability of the algorithm to estimate SOA yields at all temperatures and concentrations (Fig. S7). The SOA yield error in the algorithm in the 5–35 ∘C temperature range increased from 14 %–24 % (when all measurements are provided) to approximately 100 % (Table S5). The corresponding uncertainty range also increased by a factor of 4–6 (Table S6). Similar results were obtained in the other tests.

Figure S8 shows the volatility distribution of the products, ΔHvap, and αm in Test A1. High discrepancies and uncertainties were observed for the estimated stoichiometric coefficients (αi), with an increase in the relative error by a factor of 3–4 for the 1 and 10 µg m-3 bins (Table S7) compared to the case when all three types of measurement are used.

Figures S9 and S10 show the results of the algorithm for Test A1 when only the SOA yields and isothermal dilution measurements are provided as inputs to the algorithm. In this case the algorithm cannot constrain well the ΔHvap (relative error of almost 270 % with respect to the true value) as a result of the missing TD measurements. This led to a significant increase in MNET for the estimated yields when moving far from the temperature of the measurements (MNE of 65 % at 15 ∘C and 122 % at 5 ∘C).

Figures S11 and S12 show the results of the algorithm for Test A1 when only yield and TD measurements are provided as inputs. In this case, there was a significant reduction in the error for ΔHvap with respect to the previous case (from 270 % to 50 %), but it was still much higher than the 10 % error when all three types of measurement were used. This led to better agreement between the true and estimated yields at lower temperatures (MNET of 23 % and ARSD of 44 %).

When comparing TD–dilution, yield–dilution, and yield–TD results, the yield–TD combination gave the best results out of the three pairs. The isothermal dilution measurements are the least valuable of the three because only a relatively small fraction of the SOA evaporates and therefore the information provided is relatively limited and focuses on the more volatile components of the particles. Also, TD measurements are important to constrain ΔHvap well and allow the more accurate extrapolation of the results to other temperatures. However, our results suggest that the combination of the three types of measurement leads to a major improvement over either the TD–dilution approach or the yield–TD approach.

The maximum sum of the VBS product yields is one of the parameters that the user of the algorithm chooses. In the analysis so far, a value of 1 has been selected to reduce the computational cost of the algorithm. Selected tests were repeated using a maximum sum of 2 to quantify the effects of this choice on the estimated parameters and more importantly on the SOA yields predicted by the parametrization. For a four-product system, there are 9191 product yield combinations, and considering the discretization of ΔHvap and αm, this leads to a total of 367 120 simulations (Table S3).

The increase in the upper limit of the sum of the yields led to an increase in the good solutions in Tests A1, A4, B1, B2, and C2. The additional solutions had different yields mostly in the 103 µg m-3 bin. This led to an increase in the mass yield of this bin by 37 % in Test A1, 47 % in Test B1, and 29 % in Test C2 (Table S8). The uncertainties were even higher, showing once again the difficulty of constraining the IVOC range where there are no SOA measurements at very high SOA concentrations. The new parametrizations had a minor effect on the estimated yields at different temperatures with maximum change in the MNET found at 5 ∘C (change of 1.8 % in Test A1 and 1.2 % in Test B2) and much smaller change otherwise (Table S9). Therefore, the use of the higher upper limit has an effect on the estimate of the 103 µg m-3 bin, which is quite uncertain in all cases, but has a minor effect on the predicted SOA yields at ambient conditions.