3.1 UNIPAR model inputs

The parameter inputs to the UNIPAR model for each precursor HC include the equations for α_i and lumped species' physicochemical parameters, such as molecular weight (MW), O / C ratios, and hydrogen bonding (HB). The model inputs associated with each individual experiment include the consumption of HC (ΔHC), the concentrations of alkyl peroxide radical (RO₂) and hydroperoxyl radical (HO₂), concentrations of ionic species (sulfate and ammonium ions), temperature, and humidity. The measured sunlight profile for each experiment is linked to the gas oxidation of each HC yielding ΔHC, [RO₂], and [HO₂]. Gas simulation was performed in the box model platform of Dynamically Simple Model of Atmospheric Chemical Complexity (DSMACC; Emmerson and Evans, 2009) interfaced with the kinetic preprocessor (KPP). The predetermined mathematical equations for stoichiometric coefficients were constructed for linear alkanes using an explicit gas mechanism, as described in the following (Sect. 3.2). For the simulation of SOA mass, HC consumption of branched alkanes was simulated with the Carbon Bond 6 mechanism (CB6r3; Emery et al., 2015). Further details about the CB6 mechanism used in this study can be found in the Supplement (Sect. S1).

3.2 Gas mechanisms

The atmospheric oxidation of alkanes begins with the reaction with an OH radical, followed by the addition of O₂ to form peroxyl radicals. In the presence of NO_x, these peroxyl radicals can form alkoxy radicals or organonitrate (Finlayson-Pitts and Pitts, 2000). In addition, peroxyl radicals can create hydroperoxides via the reaction with HO₂ radicals. In the previous literature (Crounse et al., 2013; Bianchi et al., 2019; Roldin et al., 2019), terpene peroxyl radicals are capable of undergoing autoxidation reactions to form low-volatility products. These reactions have been previously modeled (Pye Havala et al., 2019; Xavier et al., 2019). In a recent modeling study, Madhu et al. (2023) applied autoxidation mechanisms to improve the prediction of a series of linear-alkane SOA. In their study, the product distribution of linear alkanes with relatively small carbon numbers (C9–C12) were predicted using explicit gas mechanisms which consisted of their respective Master Chemical Mechanism (MCM) and newly added autoxidation mechanisms. The product distributions, associated with stoichiometric coefficients, of the α_i array were extrapolated to larger linear alkanes that do not currently have MCM mechanisms using an incremental volatility coefficient (IVC). The feasibility of the α_i arrays generated with the IVC approach was demonstrated to predict chamber-generated SOA data. For branched alkanes, explicit gas mechanisms are not currently available for the practical application to the UNIPAR SOA model, as described in Sect. 1. In this study, to produce product distributions for the variety of branched alkanes, the preexisting linear-alkane α_i arrays are extrapolated to branched alkanes, as seen in Sect. 3.4 below.

3.3 Lumping structure of the UNIPAR model

As discussed in Sect. 3.2, the α_i array of branched alkanes in the UNIPAR model is inherited from linear alkanes.

The lumping structure of the UNIPAR model, along with a dynamic α_i array that considers aging, has been developed in previous studies (Zhou et al., 2019; Han and Jang, 2020; Yu et al., 2021b). The α_i array is constructed based on products' volatility–reactivity characteristics. The α_i array consists of six different reactivity levels (very fast, VF; fast, F; medium, M; slow, S; partitioning only, P; and multi-alcohol, MA) and eight different volatility levels ($\mathrm{1}\times {\mathrm{10}}^{-\mathrm{8}}$, $\mathrm{1}\times {\mathrm{10}}^{-\mathrm{6}}$, $\mathrm{1}\times {\mathrm{10}}^{-\mathrm{5}}$, $\mathrm{1}\times {\mathrm{10}}^{-\mathrm{4}}$, $\mathrm{1}\times {\mathrm{10}}^{-\mathrm{3}}$, $\mathrm{1}\times {\mathrm{10}}^{-\mathrm{2}}$, $\mathrm{1}\times {\mathrm{10}}^{-\mathrm{1}}$, and 1.0 mm Hg) based on vapor pressure that represent 48 species. The eight volatility levels have enthalpy of vaporization values of 140×10³, 106×10³, 96×10³, 89×10³, 82×10³, 58×10³, 58×10³, and 58×10³ J mol⁻¹, respectively. Further description regarding the basis for the enthalpy of vaporization values can be found in Sect. S2. During the process, each non-radical gas oxidation product of a specific precursor is lumped into 1 of the 48 species. During SOA simulations, all oxidation products within a specific lumped group undergo partitioning, or particle-phase reactions, as single species. For example, consider the compound 3-dodecanol, a product of n-dodecane photooxidation. This compound (3-dodecanol) has a calculated vapor pressure of $\mathrm{5.1}\times {\mathrm{10}}^{-\mathrm{3}}$ mm Hg (Myrdal and Yalkowsky, 1997; Zhao et al., 1999; Stein and Brown, 1994). Because the vapor pressure is smaller than that of group 7 ($\mathrm{1}\times {\mathrm{10}}^{-\mathrm{2}}$ mm Hg) and larger than that of group 6 ($\mathrm{1}\times {\mathrm{10}}^{-\mathrm{3}}$ mm Hg), 3-dodecanol is lumped into volatility group 6. Due to a lack of functional groups that promote particle-phase reactions (i.e., ketone and aldehyde), 3-dodecanol is lumped into reactivity group P and will only participate in partitioning. This process is performed to lump each non-radical gas oxidation product, with a sufficiently low vapor pressure, into 1 of the 48 lumped groups. If group 6P contains only 3-dodecanol, then the α value of that group represents the units (i.e., µg m⁻³) of 3-dodecanol produced per unit of precursor HC consumed. If group 6P has multiple compounds, then the α value of that group represents the sum of the units produced of all products classified under 6P per unit of precursor consumed. After the initial lumping process, the amount of each gas oxidation species produced per unit of HC consumed is extracted via a standardized gas simulation. This gas simulation predicts gas oxidation product concentrations for a given initial precursor concentration at various NO_x levels under a sunlight file near the middle of summer (6 June 2018). These simulation results produce lumped arrays for four different conditions: high NO_x, fresh; high NO_x, aged; low NO_x, fresh; and high NO_x, aged. The α value of each group is represented as a polynomial equation that is a function of the HC ppbC $/$ NO_x ppb level (Table S3 in the Supplement). After α values are calculated for a specific HC ppbC $/$ NO_x ppb level, they can be scaled between fresh and aged compositions, as described below. By doing so, the UNIPAR model is able to leverage the complexity of a relatively large semi-explicit gas oxidation mechanism (generally between 200 and 500 non-radical species per precursor from MCM) while limiting the computational load. In the model, autoxidation products are typically allocated to low-volatility and low-reactivity groups (volatility group 1 in reactivity groups P and S).

Atmospheric aging can augment the product distribution, forming more reactive and less volatile products via oxidation or forming photolysis products that are more volatile but may be more reactive. The stoichiometric coefficients of the α_i array are dynamically predicted as a function of NO_x and aging, as described in previous literature (Zhou et al., 2019; Han and Jang, 2020; Yu et al., 2021b). A weighted aging factor is used to dynamically change α_i values based on fresh and highly oxidized compositions. The aging factor (f_a) at a time t, as detailed in Zhou et al. (2019), is defined as follows:

where [HO₂], [RO₂], and [HC]₀ are the concentrations of hydroperoxide radical, organic peroxyl radical, and initial HC, respectively. Generally, the amount of oxidation within a given system is correlated with the concentrations of radicals within the system, with higher concentrations of radicals in more aged systems. Thus, concentrations of major radicals, normalized by initial hydrocarbon concentration, are used to represent the amount of aging. Both fresh and highly oxidized α_i arrays are constructed for each NO_x level as well as the respective aging factors. f_a(t) is also used to generate an aging scale that ranges from zero (fresh composition) to one (highly oxidized composition) as follows (Zhou et al., 2019):

$f_{\mathrm{a}}^{\prime }\left(t\right)$ is calculated for a given NO_x level and used to dynamically calculate the α_i values for that same NO_x level as follows (Zhou et al., 2019):

Essentially, [HO₂] and [RO₂] are used to scale the oxidation product distribution of a given HC precursor (i.e., α_i array) between a fresh and aged composition. Physicochemical parameters of lumping species include MW, the O / C ratio, and HB and are used to calculate multiphase partitioning and in-particle chemistry in the UNIPAR model. Further details about lumping criteria and physicochemical parameters can be found in the study by Zhou et al. (2019).

3.4 Construction of branched-alkane lumping arrays

In the current UNIPAR model, the lumping arrays of a variety of linear alkanes can be predicted using the IVC approach (Madhu et al., 2023). To extend the lumping array of linear alkanes to branched alkanes, the volatility drop caused by branched-alkyl groups was considered by matching each branched alkane to the linear alkane with the nearest vapor pressure. In addition, the lumping array in the model simulates the impact of the degree and position of branched-alkyl groups on reducing the ability of branched alkanes to undergo autoxidation reactions. Supplement Fig. S2 illustrates the ability of branched-alkyl groups to reduce the ability of branched alkanes to undergo autoxidation reactions. For example, n-heptadecane has 14 secondary carbons that can undergo autoxidation mechanism, as seen in Fig. S2 (left panel), whereas 2,2,4,4,6,8,8-heptamethylnonane has only 1 secondary carbon that can process autoxidation due to the 7 methyl branches which reduce the number of hydrogens available for abstraction. To account for this, an autoxidation reduction factor (ARF) is applied to the lumping array of branched alkane and calculated as follows:

Here, the autoxidation potential (AP) can be calculated as follows:

• a.

  Terminal, primary carbons are not included for AP.

• b.

  Carbons that are in the α or β position relative to terminal carbons on the alkane backbone are assigned an AP value of 1, and this carbon can only undergo autoxidation reactions in one direction. For example, this would apply to carbons 2, 3, 14, and 15 in Fig. S2a.

• c.

  Other carbons on the main alkane backbone are assigned an AP value of 2 due to the potential for autoxidation occurring in two directions. For example, this would apply to carbons 4 to 13 in Fig. S2a.

• d.

  If one alkyl branch is present on a carbon, then the AP value is multiplied by 0.5.

• e.

  If geminal alkyl branches are present on a carbon, then the AP value is multiplied by 0.

• f.

  The AP value of each alkane is the sum of all AP values for each carbon.

Further information on the basis for the calculation of the ARF can be found in Sect. S3. ARF in Table S1 summarizes the physical properties and ARF values for branched alkanes used in the chamber experiments of this study. Physicochemical parameters and α values of lumping arrays of compounds used for chamber experiments can be found in Sects. S5 and S6. Ultimately, this approach allows for the reduction of SOA formation found in branched alkanes compared with that in linear alkanes with same carbon numbers.

3.5 SOA formation by multiphase partitioning

The partitioning process of lumping species is fundamental to form SOA and process in-particle chemistry. Partitioning coefficients for each lumping species, i, between g and the org phase (K_org,i) or between g and the wet inorg phase (K_inorg,i) are calculated using the typical gas–particle partitioning model (Pankow, 1994):

Here, R is the gas constant (8.314 J mol⁻¹ K⁻¹) and T is temperature (K). MW_org and MW_inorg are the average molecular weights (g mol⁻¹) of the organic and inorganic phases of the aerosol, respectively. $p_{\mathrm{L},i}^o$ is the subcooled liquid vapor pressure of a species, i. The activity coefficient in the org phase for each lumping species, γ_org,i, is assumed to be unity (Jang and Kamens, 1998). The activity coefficient in the inorg phase for each lumping species, γ_inorg,i, is predicted by a semiempirical regression equation that was fit to the activity coefficients of various organic compounds as a function of the physicochemical parameters (MW, O / C ratio, and HB) and sulfate fraction (FS). The FS is an indicator of aerosol acidity, which is defined as follows:

where $\left[{\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}\right]$ and $\left[{\mathrm{NH}}_{\mathrm{4}}^{+}\right]$ are the concentration of sulfate and ammonium ions, respectively. The semiempirical equation, derived from activity coefficients, estimated using the Aerosol Inorganic–Organic Mixtures Functional Groups Activity Coefficients (AIOMFAC) model (Zuend et al., 2011) at a given RH, is as follows:

Further information on the derivation and statistical properties of Eq. (8) can be found in Zhou et al. (2019). The partitioning coefficients are used to calculate the concentration of each lumping species in the three phases (C_g,i, C_org,i, and C_inorg,i) from the total concentration of each lumping species (C_T,i). The total SOA mass formed by partitioning (OM_P) in both the org and inorg phases is predicted by the following equation, which was developed by Schell et al. (2001) and reconstructed to consider mass formed by particle-phase reactions (OM_AR), seen in Sect. 3.6, by Cao and Jang (2010):

where $C_g^{*}$ ($\mathrm{1}/K_{\mathrm{org},i}$) and OM₀ (mol m⁻³) represent the effective saturation concentration and preexisting OM, respectively. MW_oli,i and MW_i represent the molecular weights of oligomeric products and lumping species, respectively. Equation (9) is solved using the Newton–Raphson method, which iterates until a convergence is reached (Press et al., 1992).

3.6 SOA formation by particle-phase reactions

OM_AR is formed in both the org and inorg phases. The inclusion of particle-phase reactions has been demonstrated to significantly improve predictions for aromatic hydrocarbons (Im et al., 2014; Zhou et al., 2019). Particle-phase reactions were also included in the study by Madhu et al. (2023) that demonstrated a negligible impact on linear-alkane SOA. In the org phase, SOA formation is attributed to oligomerization, as organic species undergo self-dimerization reactions (Han and Jang, 2020; Im et al., 2014; Yu et al., 2021b; Zhou et al., 2019). In the inorg phase, the oligomerization of organic species can be accelerated by an acid catalyst (Jang et al., 2002). Oligomerization is expressed as a second-order reaction (Odian, 2004) with rate constants $k_{\mathrm{AR},\mathrm{org},i}$ and $k_{\mathrm{AR},\mathrm{inorg},i}$ (L mol⁻¹ s⁻¹) in the org and inorg phases, respectively. $k_{\mathrm{AR},\mathrm{inorg},i}$ is described as follows:

where R_i represents species reactivity, ${{\mathrm{pK}}_{{\mathrm{BH}}^{+}}}_i$ represents the protonation equilibrium constant, a_w represents the activity of water, X represents excess acidity (Cox and Yates, 1979), and [H⁺] represents the concentration of protons estimated using the extended aerosol inorganic model (E-AIM; Clegg et al., 1998). $k_{\mathrm{AR},\mathrm{org},i}$ is described as follows:

For oligomerization in the org phase, the terms related to acidity (X and a_w[H⁺]) are excluded. As explained by Zhou et al. (2019), a significant uncertainty remains in the calculation of [H⁺], specifically in low-RH and ammonia-rich environments, due to the poor performance of the E-AIM under these conditions (Li and Jang, 2012). Studies have previously demonstrated that aerosol viscosity can influence the mobility of chemical species and, thus, apparent reaction rates, which can be limited by slow bulk diffusion in the particle phase (De Schrijver and Smets, 1966; Reid et al., 2018). The molecular weight of species in the organic phase (MW_org) and the O / C ratio, which are important predictors of viscosity, are considered to calculate $k_{\mathrm{AR},\mathrm{org},i}$. Han and Jang (2022) demonstrate this method that accounts for viscosity in their application to SOA predictions from gasoline vapor composed of aromatic hydrocarbons and long-chain alkanes. Sulfuric acid can react with reactive organic compounds in the wet inorg phase of the aerosol to form dialkyl sulfate (diOS). The formed diOS can contribute to SOA mass production and leads to a reduction in [H⁺] which decreases the rate of SOA mass produced by acid-catalyzed oligomerization in the inorg phase. The formation of diOS is simulated in the UNIPAR model and reduces [H⁺] in the inorg phase, as previously reported (Im et al., 2014; Beardsley and Jang, 2016; Zhou et al., 2019).

3.7 Correction of intermediate organic vapor deposition to walls

Semi-volatile oxidized products derived from precursor HCs can deposit to chamber walls. As described in previous studies by Han and Jang (2020, 2022), the organic vapor deposition to the wall is kinetically treated at the given chamber with the deposition (k_on,i) and desorption (k_off,i) rate constants of each lumping species, i. k_on,i is expressed as a fractional loss rate (McMurry and Grosjean, 1985):

where D ($\mathrm{1.0}\times {\mathrm{10}}^{-\mathrm{6}}$ m² s⁻¹) and K_e (0.12 s⁻¹) are the diffusion coefficient and the coefficient of eddy diffusion applied as a fixed number, respectively. $\left(\frac{A}{V}\right)$ represents the surface-area-to-volume ratio of the chamber. ${\stackrel{\mathrm{‾}}{v}}_i$ and α_w,i represent the mean thermal speed of the gas molecules and the accommodation coefficient of i to the wall, respectively. Further information regarding the calculation of ${\stackrel{\mathrm{‾}}{v}}_i$ and α_w,i can be found a previous study (Madhu et al., 2023). K_w,i ($K_{w,i}=k_{\mathrm{on},i}/k_{\mathrm{off},i}$) is calculated as follows:

Here, OM_wall (mg m⁻³) and MW_OM are the concentration of organic matter on the wall and the molecular weight of organic matter on the wall, respectively. The activity coefficient (γ_w,i) of lumping species, i, in OM_wall is calculated using the quantitative structure–activity relationship (QSAR) approach with the physicochemical properties H_d,i, H_a,i, and P_i, which represent hydrogen bond acidity, hydrogen bond basicity, and the polarizability of each lumping group i, respectively (Abraham et al., 1991; Abraham and McGowan, 1987; Leahy et al., 1992; Platts et al., 1999; Puzyn et al., 2010). Equation (13) can be rewritten as follows:

The values of H_d,i, H_a,i, and P_i are calculated with the PaDEL-Descriptor (Yap, 2011). The value of K_w,i is used along with the k_on,i to predict lumping species' wall loss using an analytical equation from the study by Han and Jang (2020) as follows:

where C_g,i (µg m⁻³) is the gas-phase concentration of a lumping species, i, after time step t (360 s). C_T,i (µg m⁻³) is the sum of C_g,i and the concentration of lumping species i on the chamber wall (C_w,i, µg m⁻³). This method for correcting the bias originating from gas–wall partitioning has been previously demonstrated for toluene, trimethylbenzene, α-pinene (Hang and Jang, 2020), and linear alkanes (Madhu et al., 2023), as well as a composition of gasoline vapor (Han and Jang, 2022). As explained by Hang and Jang (2020), uncertainties in this method are associated with the calculation of physicochemical parameters of lumped groups. The properties of gas–wall partitioning for branched alkanes were inherited from linear alkanes.

3.8 UNIPAR procedure for SOA mass production at each time step

At each step, C_T,i is estimated using the newly produced ΔHC and α_i, and it is combined with the previous step's concentration of lumping species, except those used for the formation of OM_AR and organic vapor deposition to walls for the simulation of chamber data. Then, the updated C_T,i is applied to generate C_g,i, C_org,i, and C_inorg,i based on multiphase partitioning coefficients, as seen in Eqs. (5) and (6). C_inorg and C_org are then used to form OM_AR via oligomerization in both the inorg and org phases with the rate constants calculated in Eqs. (10) and (11), respectively. In the model, the quantity of the sulfate associated with OS in the inorg phase is also estimated and applied to recalculate [H⁺]. After the process to form OM_AR, the remaining concentration of lumping species is used to estimate the organic vapor deposition to the wall using Eq. (15). OM_P is calculated using a Newtonian approach (Eq. 9) in the presence of OM_AR and the preexisting OM₀ at the end of each time step. For the total SOA mass, OM_AR, OM_P, and OM₀ are combined.