Secondary organic aerosol (SOA) is a major fraction of the total organic aerosol (OA) in the atmosphere. SOA is formed by the partitioning onto pre-existent particles of low-vapor-pressure products of the oxidation of volatile, intermediate-volatility, and semivolatile organic compounds. Oxidation of the precursor molecules results in a myriad of organic products, making the detailed analysis of smog chamber experiments difficult and the incorporation of the corresponding results into chemical transport models (CTMs) challenging. The volatility basis set (VBS) is a framework that has been designed to help bridge the gap between laboratory measurements and CTMs. The parametrization of SOA formation for the VBS has been traditionally based on fitting yield measurements of smog chamber experiments. To reduce the uncertainty in this approach, we developed an algorithm to estimate the SOA product volatility distribution, effective vaporization enthalpy, and effective accommodation coefficient combining SOA yield measurements with thermograms (from thermodenuders) and areograms (from isothermal dilution chambers) from different experiments and laboratories. The algorithm is evaluated with “pseudo-data” produced from the simulation of the corresponding processes, assuming SOA with known properties and introducing experimental error. One of the novel features of our approach is that the proposed algorithm estimates the uncertainty in the predicted yields for different atmospheric conditions (temperature, SOA concentration levels, etc.). The uncertainty in these predicted yields is significantly smaller than that of the estimated volatility distributions for all conditions tested.

Submicrometer atmospheric particles are of great importance due to their negative effects on public health (Pope and Dockery, 2006; Lim et al., 2012) and their uncertain influence on Earth's climate (IPCC, 2021). Organic aerosol (OA) contributes 20 %–90 % of the submicron particulate mass (Zhang et al., 2007) and is emitted directly into the atmosphere as primary organic aerosol (POA) or formed as secondary organic aerosol (SOA). SOA constitutes a major fraction of the total OA in the atmosphere, contributing more than 60 % on average (Kanakidou et al., 2005). SOA is formed by the condensation of low-vapor-pressure products of the oxidation of volatile organic compounds (VOCs), intermediate-volatility organic compounds (IVOCs), and semivolatile organic compounds (SVOCs).

Hundreds of mostly unknown products are formed during the oxidation of each SOA precursor, making the detailed description of the corresponding reactions and eventual SOA formation extremely challenging. The volatility basis set (VBS) is one approach that has been proposed to simplify the system and to allow SOA simulation in chemical transport models (CTMs). The VBS describes the volatility distribution of OA using a set of surrogate species with effective saturation concentrations that vary by 1 order of magnitude (Donahue et al., 2006; Stanier et al., 2008). Volatility is one of the most important physical properties of SOA components as it determines to a large extent their gas–particle partitioning (Pankow, 1994a, b). The parametrization of SOA formation for the VBS requires the determination of the yields of each volatility bin (volatility distribution of products) and the corresponding enthalpies of vaporization.

The SOA parametrizations for the VBS have been traditionally based on
fitting yield measurements (Lane et al., 2008). The major weakness of this
approach is that the resulting parametrization is limited to the range of OA
concentrations and temperatures of the measurements. In most cases, the
concentration range does not include the low concentrations relevant to the
atmosphere, and usually most of the experiments take place in a relatively
narrow temperature range. Pathak et al. (2007a) needed 37 smog chamber
experiments at different temperatures (0–45

A number of approaches have been used to minimize the number of experiments needed to characterize the temperature dependence of the SOA formation. Stanier et al. (2007) developed an experimental technique with which the temperature-controlled smog chamber could be heated or cooled after the SOA formation, moving the system to new equilibrium favoring evaporation or condensation respectively. However, interactions of the SOA with the walls of the system increased the uncertainties in the approach. Stanier et al. (2008) presented an algorithm to fit the smog chamber experiments using several volatility bins. However, the number of experiments needed by the algorithm should cover a wide range of concentrations and temperatures to effectively constrain the stoichiometric mass yields and the effective vaporization enthalpy.

In an effort to cover a wider concentration and temperature range, thermodenuder measurements can be used. The thermodenuder (TD) is a common instrument developed to characterize the volatility of atmospheric aerosols by heating them and observing the resulting changes in size, mass, optical properties, etc. (Burtscher et al., 2001; Wehner et al., 2002, 2004; An et al., 2007). TDs consist of a heated tube in which the volatile particle components evaporate followed by a cooling section with activated carbon to avoid vapor recondensation. The mass changes in TDs depend on the initial SOA concentration, the residence time in the heating tube, the vaporization enthalpy, and the mass transfer resistances. The latter are described by the effective accommodation coefficient that has been traditionally used to account for resistances to mass transfer not only at the surface of the particle but also inside the particle. The evaporation rate for most particles is relatively insensitive to its value when this value is around 1. A typical way of reporting the TD measurements is by calculating the aerosol mass fraction remaining (MFR) at a given temperature after passing through the TD. The MFRs in a range of TD temperatures constitute the thermogram.

In applications in the field (Cappa and Jimenez, 2010; Huffman et al., 2009; Lee et al., 2010; Louvaris et al., 2017a) and in the laboratory (Kalberer et al., 2004; Baltensperger et al., 2005; An et al., 2007; Lee et al., 2011; Cain et al., 2020), the particles do not reach equilibrium with the gas phase inside the TD. Therefore, dynamic aerosol evaporation models (Riipinen et al., 2010; Cappa, 2010; Fuentes and McFiggans, 2012) are needed for the interpretation of TD measurements. Karnezi et al. (2014) used the time-dependent evaporation model of Riipinen et al. (2010) to calculate the OA volatility distribution, vaporization enthalpy, and mass accommodation coefficient from TD measurements. The authors showed that a simple error minimization approach may not be appropriate for such systems as very similar thermograms can be obtained for multiple combinations of different parameters. For this reason, their approach estimates an ensemble of “good” solutions, from which the best estimate and the corresponding uncertainties are derived.

Grieshop et al. (2009) suggested the combination of TD and isothermal
dilution to constrain the volatility distribution of SOA. Karnezi et al. (2014) proposed an algorithm to include both types of measurement. The
authors concluded that the combination of the two types of measurement can
better constrain the OA volatility than each set separately. Louvaris et al. (2017b) and Cain et al. (2020) applied this algorithm to cooking OA (COA)
and SOA respectively. Louvaris et al. (2017b) showed that the use of only
TD measurements led to overestimation of the SVOC fraction of COA, while the
use of TD and isothermal dilution data reduced the uncertainty in the
volatility distribution and the effective vaporization enthalpy. Cain et al. (2020) conducted TD and isothermal dilution experiments for

To constrain the volatility product distribution of SOA and its effective vaporization enthalpy, we combine TD and isothermal dilution experiments with the SOA yield measurements. We extend here the algorithm of Karnezi et al. (2014) by introducing additional inputs (SOA yields) and by providing additional outputs (uncertainty in estimated yields in relevant atmospheric conditions). The algorithm is tested with “pseudo-experimental” data generated from the use of models simulating the corresponding measurement processes; therefore the true parameters are known. The results of the “pseudo-experiments” are corrupted so that they include experimental errors.

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

The effective saturation concentrations at different temperatures are given
by the Clausius–Clapeyron equation:

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,

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

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.

The algorithm of Karnezi et al. (2014) was first extended to include an SOA
partitioning model described by Eqs. (1)–(3) together with the TD
and isothermal dilution models in order to estimate the volatility product
distribution, vaporization enthalpy, and accommodation coefficient. We
discretized the domain of the parameters and simulated all combinations of
stoichiometric mass yields (

For each simulation and each type of measurement, we calculated the
normalized mean square error (NMSE) defined as

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 (

For TD, the errors were calculated by assuming a normal distribution and the
standard deviation (

For dilution, the errors were calculated by assuming a uniform distribution
and standard deviation (

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.

Measurements of Test A1 in Experiment A (red dots) and
true (red line) and estimated (blue line) yields at

In Experiment A, we test the performance of the algorithm against

Measurements of Test B1 in Experiment B (red dots) and
true (red line) and estimated (blue line) yields at

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

Measurements of Test C1 in Experiment C (red dots) and
true (red line) and estimated (blue line) yields at

For Experiment C, the true values were based on the parametrization
of the SOA formed during

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.

True and estimated volatility distribution of the products
for eight different tests. The uncertainty in the estimates (

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, MNE

For the evaluation against the truth, which includes conditions (e.g.,
temperatures or concentrations) for which there are no available
measurements, MNE

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 MNE

In Test A1, we applied the algorithm in the same range of saturation
concentrations and with the same number of volatility bins as those used to
produce the experimental data. The upper bin (10

Figure 1 depicts the estimated and the range of the ensemble of best
solutions for the three types of measurement for Test A1. There were
148 good solutions under the 5 % threshold out of the 126 120
simulations (Table S3). The density distribution of the solutions is
depicted in Fig. S1. The performance of the model for the yields at 25

The mean normalized error (MNE) between the measurements and true values and between the measurements and the model-estimated values for the different tests.

Our algorithm can be used to calculate the SOA yield at different
concentrations and temperatures. The yields were calculated in the
atmospherically relevant range of 0–50

The mean normalized error between the true and estimated values (MNE

The SOA model used in this work assumes that the stoichiometric coefficients
(

The algorithm provides a range of good estimates in addition to the best
estimate. The range can be defined by the lower and upper SOA yield limits
of the ensemble of the good solutions at each point. At 25

The average relative standard deviation (ARSD) for the different tests.

For the TD (Fig. 1e), the model reproduced well the correspondent thermogram
with low errors compared to the measurements with an error MNE

For the isothermal dilution (Fig. 1f), the algorithm did reasonably well for
the first 30 min and then the evaporation was slightly underpredicted,
leading to an error in MNE

The estimated volatility distribution of the products and the effective
vaporization enthalpy and accommodation coefficient using the three types of
measurement can be seen in Fig. 4 and Table 1. The estimated volatility
distribution of the products was in good agreement with the true
values (

Estimated (bars) and true (red lines) parameter values of
Experiment A in Test A1 combining yield, TD, and isothermal dilution
measurements for

In this section, we analyze the pseudo-experimental data of Experiment B,
which were obtained from the parametrization of the same smog chamber
results used in Experiment A but with more components and a much wider
range of volatilities including LVOCs, SVOCs, and IVOCs (10

Both measured and true thermograms were well captured by the best
estimate (MNE

Figure 5 shows the results of Test B1 for the volatility distribution of the
products. The true stoichiometric coefficient for the 1

Estimated (bars) and true (red lines) parameter values of
Experiment B in Test B1 combining yield, TD, and isothermal dilution
measurements for

The results of Test B1 suggest that the mismatch between the actual SOA volatility distribution and the range used for the fits can introduce significant errors into the retrieved distribution for individual volatility bins. However, despite these problems, the yields predicted by the derived parametrizations have a much lower error than the volatility distribution. This is a valuable insight for the strengths and weaknesses of this and other similar SOA parameter estimation algorithms.

In Test C1, we obtained the best fits for the pseudo-measurements of
Experiment C by applying the algorithm in the same range of saturation
concentrations and with the same number of volatility bins (four volatility
bins in the 10

Figure 3 shows the results of the fitting for the three types of
measurement. There were 3479 good solutions under the 5 %
threshold out of the 126 120 simulations (Table S3). The density
distribution of the solutions is shown in Fig. S3. The best estimate for
the SOA yields at 25

The performance of the algorithm was satisfactory compared to the TD
measurements (MNE

Figure 6 shows that the highest relative errors were calculated for the
10

Estimated (bars) and true (red lines) parameter values of
Experiment C in Test C1 combining yield, TD, and isothermal dilution
measurements for

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 1–10

In Test A3, we shifted the assumed four-bin volatility distribution by 1 order of magnitude to lower values (from 1–1000

Yields calculated using the true parameters of
Experiment C (red line) and estimated (blue line) using the parameters
of Test C2 for the following temperatures:

In Test C2, we applied the algorithm against the Experiment C
measurements using a four-volatility-bin system in the 1-to-10

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

In Test A4, the additional experiment at high SOA concentration led to an
MNE

Yields calculated using the true parameters of
Experiment B (red line) and the estimated (blue line) using the parameters
of Test B2 for the following temperatures: 5, 15, 25, and 35

Similarly to Test A4, in Test B2 we added a yield measurement at 200

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 (10

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

Figure S8 shows the volatility distribution of the products,

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

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

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

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

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 10

An algorithm was developed to estimate VBS parameters for SOA formation combining yield measurements from atmospheric simulation chambers with thermodenuder and isothermal dilution measurement chambers. An additional feature of this approach is that the algorithm estimates the uncertainty in the predicted SOA yields for different SOA concentrations and temperatures, assisting in this way in the design of future experiments.

The algorithm was evaluated against pseudo-experimental data for SOA systems
with known properties. The algorithm performed quite well at reproducing the
SOA yields at atmospherically relevant concentrations and temperatures with
errors less than 20 % for practically all cases. This was the case even at temperatures as low as 5

The errors in the retrieved SOA volatility distributions were in general
higher than those of the SOA yields. This is due to a large extent to the
existence of multiple solutions that can result in similar yields. The
accuracy of the estimated mass fractions of the more volatile SOA components
improved with an additional yield measurement at high SOA levels (e.g., at 200

In all cases the algorithm results in good estimates of the effective evaporation enthalpy. On the other hand, the estimates of the effective accommodation coefficient are usually quite uncertain. The effect of the mass accommodation coefficient on the measured quantities is relatively small compared to the other parameters (volatility distribution, effective evaporation enthalpy), making it difficult to constrain. This conclusion is consistent with the results of Karnezi et al. (2014). The addition of the SOA yields to the inputs does not make much of a difference because these are not affected by the accommodation coefficient.

The approach combining yield, TD (thermograms), and isothermal dilution (areograms) measurements is recommended for future parametrizations of SOA formation. The use of the results of these experiments that have been designed for the measurement of SOA yields in other applications (e.g., new particle formation) should be performed with caution. Our results indicate that the derived parametrizations are able to predict the SOA yields under different atmospheric conditions with errors of around 20 % or less, but the derived volatility distributions can be quite uncertain. These uncertainties are higher for the tails of the distribution (the low-volatility and the intermediate-volatility organic compounds). Different experiments should probably be performed for the derivation of the VBS distribution if for example one is interested in new particle formation and therefore the low-volatility organics focusing on low SOA concentration levels and the least volatile SOA components.

The code and simulation results are available upon request (spyros@chemeng.upatras.gr).

The supplement related to this article is available online at:

PU and SNP designed the research. PU developed the final model code. AD developed a first version of the code and performed preliminary feasibility tests. DS and SNP designed the experiments for the

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research has been supported by the Chemical evolution of gas and particulate-phase organic pollutants in the atmosphere (CHEVOPIN) project of the Hellenic Foundation for Research and Innovation (HFRI, grant agreement no. 1819) and the European Union's Horizon 2020 Framework Programme through the EUROCHAMP-2020 Infrastructure Activity (grant agreement no. 730997).

This paper was edited by Yoshiteru Iinuma and reviewed by three anonymous referees.