the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Estimation of secondary organic aerosol formation parameters for the volatility basis set combining thermodenuder, isothermal dilution, and yield measurements

### Petro Uruci

### Dontavious Sippial

### Anthoula Drosatou

### Spyros N. Pandis

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.

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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 ^{∘}C) and atmospherically
relevant concentrations to constrain the *α*-pinene SOA temperature
sensitivity.

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 *α*-pinene
and cyclohexene ozonolysis SOA. The SOA in these two systems had similar
thermograms but different areograms. When only thermograms were used in the
model, the volatility distributions were quite similar. However, the
addition of areograms revealed that *α*-pinene ozonolysis SOA consists
mostly of low-volatility organic compounds (LVOCs) and the cyclohexene
ozonolysis SOA consists mostly of SVOCs.

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.

## 2.1 SOA formation

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

where *n* is the number of the surrogate compounds (volatility bins in the
VBS), *P*_{i} is the surrogate product in the *i*th volatility bin, and
*α*_{i} is the corresponding stoichiometric mass yield. The total
SOA mass yield can be then calculated as

where *C*_{OA} is the total SOA concentration, ΔVOC is the consumed
concentration of the VOC, and ${C}_{i}^{\ast}$ 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:

where *T*_{ref} is the reference temperature in which the reference effective saturation concentration is defined (298 K in this work) and Δ*H*_{vap,i} is the enthalpy of vaporization of surrogate compound *i*.

## 2.2 Thermodenuder model

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, *m*_{p}, and the gas-phase
concentration of the compound *i*, *C*_{i}, are given by

where *n* is the number of surrogate compounds, *N*_{tot} is the total number concentration of particles (assuming a monodisperse aerosol population), and *I*_{i} is the mass flux of compound *i* from the gas to the particulate phase for each particle calculated by (Seinfeld and Pandis, 2016)

Here *d*_{p} is the particle diameter, *R* is the ideal gas constant, *M*_{i} is the molecular weight of compound *i*, *D*_{i} is the diffusion coefficient of compound *i* in the gas phase at temperature *T*_{TD}, *p*_{i} and ${p}_{i}^{\mathrm{0}}$ are the partial vapor pressures of *i* far away from the particle and at the particle surface respectively, and *β*_{mi} is a factor for the correction of kinetic and transition regime effects (Fuchs and Sutugin, 1970):

Here *Kn*_{i} is the Knudsen number of compound *i* and *α*_{mi} 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

where *x*_{mi} is the mass fraction of compound *i* in the particulate phase, ${C}_{i}^{\ast}$ is the effective saturation concentration, *σ* is the surface tension (assumed to be 0.05 N m^{−1} in our simulations), *T*_{TD} 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).

## 2.3 Isothermal dilution model

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 *α*_{m} but not on Δ*H*_{vap} 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.

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 (*α*_{i}), Δ*H*_{vap},
and *α*_{m}. The yields *α*_{i} were allowed to vary from
0.0 to 0.8, with values of 0.0, 0.05, 0.1, 0.15, 0.2, 0.3, 0.4, 0.6, and
0.8. The user of the algorithm can specify an upper limit for the sum of the
yields to reduce the number of the potential solutions that the algorithm
will test. Combinations with the sum of the yields exceeding 1.0 were excluded
from the analysis originally. The sensitivity of our results to setting the
upper limit of the sum of the yields equal to 2 is examined in Sect. 4.6.
For a four-product system there are 3153 and for a six-product system 66 636
acceptable combinations. The values used for Δ*H*_{vap} were
from 20 to 200 kJ mol^{−1} with a step of 20, and for *α*_{m}, the
values used were 0.001, 0.01, 0.1, and 1. As a result 126 120 simulations
are needed (computational time of about 15 h on an office PC) for a
four-product VBS and 2 665 440 for a six-product solution.

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

where *O*_{i} represents the *i*th observed value (corresponding to a specific
SOA concentration for yield measurements, temperature for TD, or time for
isothermal dilution), *P*_{i} the corresponding model-predicted value, and
*N*_{O} is the total number of observations from each type of measurement. For each simulation (denoted as *s*), the overall error was calculated by assuming equal weight to the set of yield, TD, and dilution measurements and summing the corresponding errors:

The parameter combinations for which the overall error *E*_{s} is less than
5 % are identified. The best solution is then calculated by averaging
these solutions using the inverse error *E*_{s} as a weighting factor. The
solutions that are closer to the measurements have higher weight. Therefore,
for every combination of *α*_{i}, Δ*H*_{vap}, and
*α*_{m}, the algorithm calculates one overall NMSE following Eq. (10) and
all data points for each solution get the same weighting factor. More
specifically the best estimate $\stackrel{\mathrm{\u203e}}{x}$ is given by

where *x*_{k} is the estimated value of a property (mass yield of a
volatility bin, effective vaporization enthalpy, or effective accommodation
coefficient) and *N* is the number of combinations with error below the
threshold value. The uncertainty range of the parameters is estimated by
calculating the standard deviation (*σ*) following Karnezi et al. (2014):

## 4.1 Generation of data for evaluation

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 (*σ*_{Y})
given by

where *Y*_{true} denotes the correct yields.

For TD, the errors were calculated by assuming a normal distribution and the
standard deviation (*σ*_{TD}) suggested by Karnezi et al. (2014):

where MFR_{TD,true} denotes the correct MFR values for each TD temperature.

For dilution, the errors were calculated by assuming a uniform distribution
and standard deviation (*σ*_{Dil}) suggested by Karnezi et al. (2014):

where MFR_{Dil,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-NO_{x} 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 10^{3} 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 10^{4} 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.

## 4.2 Parameter estimation for Experiments A, B, and C

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, MNE_{M} was defined as follows:

where EST_{i} is estimated by the algorithm value and corresponds to a
specific measured point *O*_{i}.

For the evaluation against the truth, which includes conditions (e.g.,
temperatures or concentrations) for which there are no available
measurements, MNE_{T} was defined as follows:

where EST and TR are the estimated and the true values respectively.
*N*_{d} 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 MNE_{T} metric. The ARSD is given by

where *σ*_{j} is the standard deviation for data point *j*.

### 4.2.1 Parameter estimation for Experiment A

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^{3} µg m^{−3})
exceeded the maximum SOA concentration (40 µg m^{−3}) in the
measurement range by 1 order of magnitude.

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 ^{∘}C was quite encouraging with a small tendency of overprediction for SOA higher than 10 µg m^{−3}. The MNE_{M} of the model for the SOA yield measurements (given by Eq. 16) was equal to 25 % (Table 2). The
corresponding discrepancy between the true parametrization and the
measurements (due to the measurement error that we introduced) was 21.2 %
(Table 2). This indicates that a significant part of the algorithm error can
be explained by the uncertainty introduced into the measurements.

^{a} Calculated by $\frac{\mathrm{100}}{{N}_{\mathrm{O}}}\sum _{i=\mathrm{1}}^{{N}_{\mathrm{O}}}\frac{\left|{O}_{i}-{\mathrm{TR}}_{i}\right|}{{O}_{i}}$. ^{b} Calculated by Eq. (16).

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 µg m^{−3} SOA concentration
and at four temperatures (5, 15, 25, and 35 ^{∘}C) using the true
parameter values and the estimated parameters of Test A1 (Fig. 1a–d). At 25 ^{∘}C (Fig. 1c), the estimated yield curve is in good agreement with the true yield curve for SOA concentrations lower than 6 µg m^{−3} (error of 8 % at 6 µg m^{−3}), but the discrepancies increase at higher concentrations (error of 23 % at 50 µg m^{−3}). The average MNE_{T} error between the true parametrization and the estimated values (given by Eq. 17) was equal to 17.3 % for yields at 25 ^{∘}C (Table 3).
The uncertainties, as expected, are larger at lower temperatures. However,
the MNE_{T} error (estimated yields compared to the true value) remains less than 25 % (Table 3) even at 5 ^{∘}C, quite far from the measurement temperature. Both MNE_{T} and MNE_{M} were quite close to the introduced experimental error. Their difference can be explained by both the noise introduced to the measurements that affects MNE_{M} and the higher number
of points used to calculate MNE_{T}.

The SOA model used in this work assumes that the stoichiometric coefficients
(*α*_{i}) are temperature independent. Therefore, processes which are expected to be temperature dependent, such as formation of highly oxygenated organic molecules (HOMs) and oligomerization (Quéléver et al.,
2019; Gao et al., 2022), are not described by our algorithm.

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 ^{∘}C, the
yield range increased, as expected, at higher concentrations (yield range of
0.05 at 1 µg m^{−3} to 0.17 at 50 µg m^{−3}). The average relative standard deviation (ARSD of the estimated yields defined by Eq. 18) was equal to 26 % (Table 4) for the 25 ^{∘}C case. For the rest of the temperatures, the ARSD increased for the lower temperatures, ranging from 24 % at 35 ^{∘}C to 35 % at 5 ^{∘}C (Table 4) and including in all cases
the true solution.

For the TD (Fig. 1e), the model reproduced well the correspondent thermogram
with low errors compared to the measurements with an error MNE_{M} of 7 % (Table 2). The error MNE_{T} compared to the true values was 5.5 % (Table 3). The error in the TD measurements compared to the true
values was equal to 7.6 % (Table 2). Therefore, the error in the proposed
algorithm is quite similar to the experimental error. The error introduced
into the measurements was transferred, as expected, to the error metrics
of the algorithm.

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_{M} of 16.7 % (Table 2). This MNE_{M} value was
roughly 2 times higher than the corresponding error between the dilution
measurements and the true parametrization (Table 2). The error between the
estimated and the true values of MNE_{T} was 19 %. The ARSD of 24 % (Table 4) was sufficient to include the true solution.

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 (*α*_{i} absolute difference of 0.01 at 1 µg m^{−3},
0.03 at 10 µg m^{−3}, 0.07 at 10^{2} µg m^{−3}, and 0.04 at 10^{3} µg m^{−3}), and the estimated uncertainties contained the
correct values. There is a large uncertainty range for the two higher
volatility bins (standard deviation higher than 0.13), indicating that yield
values at higher SOA concentrations would be needed to better constrain
these volatility bins. The relative error in the estimated Δ*H*_{vap} is 10 %. The estimated accommodation coefficient was 0.17 compared to a true value of 0.5. The estimated uncertainty for the effective accommodation was almost 1 order of magnitude higher (from 0.06 to 0.51),
indicating the difficulty of constraining this parameter when it is close to
unity and thus the resistances to mass transfer are small.

### 4.2.2 Parameter estimation for Experiment B

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^{−2}–10^{4} µg m^{−3}). In Test B1, the algorithm was applied using a four-bin VBS with saturation concentrations ranging from 1 to 10^{3} µg m^{−3}.
In this test, we attempted to model the behavior of the system with a
narrower volatility range than the real one. The upper limit of the
saturation concentration range that we tested did not exceed 10^{3} µg m^{−3} because Experiment B took place at moderate SOA concentration levels (up to 40 µg m^{−3}), which means that it is practically impossible to constrain the 10^{4} µg m^{−3} or higher
volatile bins. Figure 2 shows the results of the fitting for the three types
of measurement in this experiment. There were 82 good solutions
under the 5 % threshold out of 126 120 simulations (Table S3), and the
density of the solutions are shown in Fig. S2. At 25 ^{∘}C, the model
performance for the yields is encouraging (MNE_{M} = 20.6 %). This is again
pretty close to the measurement error (20.5 %). By comparing the
estimated and the true yield curves at 25 ^{∘}C, the error MNE_{T} is
now 14 %. The error increases to 31 % at 5 ^{∘}C, far from the
available measurements. This is also reflected in the increase in the
uncertainty in our estimates with the ARSD increasing from 17 % at 35 ^{∘}C to 37 % at 5 ^{∘}C (Table 4). Once more the uncertainty range estimated by the algorithm includes the true values.

Both measured and true thermograms were well captured by the best
estimate (MNE_{M} of 6 % and MNE_{T} of 4 %) with an uncertainty ARSD of
20.5 %. The evaporation in the dilution chamber was a little
underestimated for the first 2 h, but then it was slightly overpredicted.
The MNE_{T} for the areogram was 13.3 %, and the true values were included within the range of the estimates (ARSD of 18 %).

Figure 5 shows the results of Test B1 for the volatility distribution of the
products. The true stoichiometric coefficient for the 1 µg m^{−3}
bin was overestimated by 0.01 by the algorithm. This overestimation actually
corresponds to the total material of the 10^{−2} and 10^{−1} µg m^{−3} bins of the true system. This indicates that the algorithm
places the material of the two lowest bins that are not part of the solution
in the bin with the lower volatility. For the 10 and 10^{2} µg m^{−3} bins, the relative errors between the estimated and true results were 58 % and 277 % respectively (Table S4), while for the 10^{3} µg m^{−3} bin, the relative error was 10 %. The Δ*H*_{vap} was predicted accurately (error of only 4 %), while *α*_{m} was underpredicted (0.1 instead of 0.5). The model compensates for
the missing volatility bins by increasing the material in the 10^{2} µg m^{−3} bin and by decreasing the accommodation coefficient.

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.

### 4.2.3 Parameter estimation for Experiment C

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^{−2}–10^{1} µg m^{−3} saturation concentration range) as the true volatility distribution.

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 ^{∘}C was in a good agreement with the
measurements (MNE_{M} = 6.3 %) and the true values
(MNE_{T} = 9.6 %). For the rest of the temperatures, there was a decreasing
trend of the error as the temperature decreased varying from 15.5 % at 35 ^{∘}C to 6.2 % at 5 ^{∘}C. A similar decreasing trend was observed for the uncertainty ARSD of the estimates, which varied from 23 % at 35 ^{∘}C to 15 % at 5 ^{∘}C. This behavior is the opposite of what we observed in
the previous tests, in which both errors and uncertainties increased at
lower temperatures. However, the changes in both the error and the
uncertainty are small (change of around 7 % between the upper and lower
temperature for both metrics), indicating that this system is less
temperature-sensitive in this temperature range than the previous ones.

The performance of the algorithm was satisfactory compared to the TD
measurements (MNE_{M} = 12.9 %). The corresponding error in the algorithm for the true values (MNE_{T}) was 4.4 % for temperatures up to
110 ^{∘}C and equal to 10.6 % for the lower values at higher
temperatures. According to Fig. 3, the evaporation due to dilution was
initially overestimated for the first 30 min but then underestimated
(highest MFR discrepancy of 0.05), and there is a high uncertainty range of
the corresponding estimates (MFR range of 0.46 at 3 h). However, the low
dilution values resulted in low relative errors (MNE_{M} of 3.5 % and MNE_{T} of 2.7 %).

Figure 6 shows that the highest relative errors were calculated for the
10^{−1} and 10^{0} µg m^{−3} bins (23 % and 33 %
respectively), and smaller relative errors were calculated for the other two bins (less than
13 %). The uncertainties were almost of the same magnitude for all bins
with standard deviations ranging from 0.09 to 0.13. The performance of the
model was good for Δ*H*_{vap} (relative error of 7 %) but with high uncertainty for *α*_{m}.

## 4.3 Effect of the volatility range

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^{2} 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 MNE_{T} 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 10^{2} 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 Δ*H*_{vap} was in this case 32 kJ mol^{−1} (2.7 % decrease), and *α*_{m} 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 Δ*H*_{vap} and *α*_{m} 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-10^{3} 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, MNE_{M} 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 MNE_{T} for the TD MFR was 17.2 % (Table 3). The change in the predicted areogram was
marginal and led to a small increase in MNE_{T} (error increase by 0.7 %) (Table 3). The algorithm not only underestimated again *α*_{m} (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 Δ*H*_{vap} 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.

## 4.4 Effect of measurements at high SOA levels

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 10^{3} µg m^{−3}.

In Test A4, the additional experiment at high SOA concentration led to an
MNE_{T} 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., MNE_{T} 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 10^{2} 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 Δ*H*_{vap} and *α*_{m} 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 MNE_{T} from 13.9 % in Test B1 to 9 % in Test B2 at 25 ^{∘}C (Table 3). Similar reductions in the MNE_{T} 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 MNE_{T} 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 10^{2} µg m^{−3} bin. The Δ*H*_{vap} and *α*_{m} 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 (10^{2} and 10^{3} 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.

## 4.5 Significance of each type of measurement for the parametrization

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, Δ*H*_{vap}, and *α*_{m} in Test A1. High discrepancies and uncertainties were observed for the estimated stoichiometric coefficients
(*α*_{i}), 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
Δ*H*_{vap} (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 MNE_{T} 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 Δ*H*_{vap} 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 (MNE_{T} 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 Δ*H*_{vap} 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.

## 4.6 Sensitivity to the upper limit of the sum of product yields

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 Δ*H*_{vap} and *α*_{m}, 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 10^{3} 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 MNE_{T} 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 10^{3} µg m^{−3} bin, which is quite uncertain in
all cases, but has a minor effect on the predicted SOA yields at ambient
conditions.

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 ^{∘}C and also when the volatility range used for
the parameter estimation was narrower than that of the simulated SOA system.
One should note that this error was quite similar in most cases to the
experimental error assumed in the construction of the measurement
datasets.

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 µg m^{−3}). The addition of this measurement also improved the estimated
yields at low temperatures. This therefore suggests that data points at high
SOA concentrations should also be obtained experimentally, together with the
data points at atmospherically relevant atmospheric SOA levels.

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: https://doi.org/10.5194/amt-16-3155-2023-supplement.

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 *α*-humulene ozonolysis, and DS carried them out. PU performed the simulations and the formal analysis and wrote the original draft. Paper review and editing was performed by SNP.

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.

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