The simulation of the beta-caryophyllene oxidation scheme (BCARY) in MCM v3.3.1 is illustrated by a test case. Users may choose to fix oxidant concentrations, in order to reduce SOA chemical schemes of a specific VOC using the GENOA tool . For that application, the presence of RO₂ from other sources is taken into account by introducing the notion of background RO₂. Indeed, as MCM reactions may involve radical reactions, SOA may be formed as a result of RO₂ species reacting with hydroperoxy radicals (HO₂) and other RO₂ radicals. Hence, RO₂-RO₂ reactions can be treated using the concept of a “RO₂ pool” that include RO₂ from all precursors. For example, a RO₂-RO₂ reaction can be written as follows: 3.000E-01 BCBNO3 + 7.000E-01 BCAL + 7.000E-01 NO2 KINETIC RO2 1 ARR 9.200E-14]]> where ARR indicates it is an Arrhenius type of reaction, “RO2 1” indicates that the NBCO2 species reacts with the total concentration of all RO₂ species in the RO₂ pool with index “1”. The number following “RO2” is required and specifies which RO₂ pool the species reacts with. It can be adjusted to refer to a different RO₂ pool if multiple pools are considered (the numbering starts from 1). All RO₂ pool species are defined in an input RO₂ species file, where each line lists an RO₂ species name and its corresponding RO₂ pool number, separated by a space.

The RO₂-RO₂ reaction can be treated in four different ways by adopting different options (tag_RO2) in the namelist:

tag_RO2=0, compute no RO₂-RO₂ reaction.

The oxidation of BCARY for different tag_RO2 options is illustrated in Fig. . The influence of the RO₂ pool option is low for BCARY, because the RO₂ + RO₂ pathways, i.e., the part influenced by the background RO₂ pool, is not the dominant pathway for BCARY SOA formation.

In these test cases, chemical schemes of different complexities are used and compared with respect to their simulated SOA concentrations for different NO₂ levels. Initial gas concentrations are taken to be representative of atmospheric conditions simulated by over Greater Paris in July 2009. The initial NO₂ concentration is taken equal to 26 µg m⁻³, and the simulated OH concentration varies between 1.2 and 7.6×10-4 µg m⁻³. Simulations are performed with initial NO₂ concentrations multiplied by 2 (52 µg m⁻³) or divided by 2 (13 µg m⁻³). Low values are used for initial organic and inorganic aerosol concentrations, to serve as a condensation core. To keep the particle dynamic as simple as possible, only one size section (representing aerosol with diameters between 0.1585 and 0.4 µm) is used. The simulation is run for about 5 h with outputs every minute.

The different chemical schemes are combined with the CB05 mechanism for oxidant and O₃ formation. Four sets of configurations with SOA schemes of different complexity are studied here. In each setup, the scheme CB05 is supplemented with different SOA schemes for monoterpenes, sesquiterpene, toluene, and naphthalene. The SOA schemes are sumarized in Table . In the first setup, the SOA schemes represent near-explicit mechanisms. In the second setup, the SOA schemes of toluene and monoterpenes are schemes reduced using GENOA. The third and fourth setups involve SOA schemes from the H²O mechanism built from chamber experiments, with the third setup including the simple autoxidation scheme of for monoterpenes. For monoterpenes, the near-explicit scheme is from MCM and PRAM (Peroxy Radical Autoxidation Mechanism) , and the intermediate scheme is the GENOA reduced mechanism from . For toluene, the near-explicit scheme is from . It is supplemented with the molecular rearrangement with ring opening of a bicyclic peroxy radical (BPR) with an O-O bridge , as implemented in , and the GENOA reduced scheme is from . For naphthalene, the near-explicit scheme is from . For sesquiterpene (β-caryophyllene), the near-explicit scheme is from MCM. Differences in the simulated concentrations and their temporal evolution among the implicit, quasi-explicit and GENOA-reduced schemes arise from the richer formation pathways included in the quasi-explicit and GENOA-reduced mechanims.

The toluene SOA concentrations simulated with the different configurations and with the different initial NO₂ conditions are shown in Fig. . The toluene SOA concentrations increase with time. Using the near-explicit schemes, the toluene SOA concentrations are lower when the initial NO₂ concentrations are decreased by a factor 2. If the initial NO₂ concentrations are increased by a factor 2, the toluene SOA concentrations are lower in the first 3 h of the simulations, but they are higher after. As shown in the left panel of Fig. , the toluene SOA concentrations of the reduced and near-explicit schemes are very similar. However, as shown in the right panel of Fig. , the toluene SOA concentrations of the H²O scheme are lower than those of the near-explicit schemes. This was already observed in . This is partly attributed to the impact of molecular rearrangement with ring opening of a BPR with an O-O bridge present in the near-explicit scheme, but which is missing in the H²O scheme. The evolution of the toluene SOA concentrations is similar between H²O and the near-explicit schemes when initial NO₂ concentrations are reduced: the toluene SOA concentrations decrease. SOA concentrations also decrease with the H²O scheme when initial NO₂ concentrations are doubled. This decrease persists through the whole simulation, contrary to the near-explicit scheme.

The monoterpene SOA concentrations simulated with the different schemes and in the different initial NO₂ conditions are compared in Fig. . With all mechanisms, monoterpene SOA concentrations increase as initial NO₂ levels decrease, because of the NO_x regime, as already noted in . The higher concentrations may be attributed to RO₂-HO₂ reaction rates, which tend to yield more highly oxidized monoterpene degradation products. As shown in the left panel of Fig. , the SOA concentrations simulated with the reduced schemes and their evolution with NO₂ levels are similar between the reduced and the near-explicit schemes. As shown in the lower panel of Fig. , the monoterpene SOA concentrations are significantly lower with the H²O mechanism that ignores autoxidation. Furthermore, monoterpene SOA concentrations decrease as NO₂ levels decrease in constrast to the simulation with the near-explicit mechanism. Taking into account autoxidation in the H²O mechanism leads to higher monoterpene SOA concentrations (upper right panel of Fig. ), but they are still underestimated compared to the near-explicit scheme. Similarly to the near-explicit scheme, the H²O mechanism including autoxidation gives higher SOA concentrations with lower NO₂ levels.

The naphthalene SOA concentrations simulated with the different configurations and in the different NO₂ initial conditions are shown in the left panel of Fig. . The naphtalene SOA concentrations increase as the initial NO₂ levels increase. The naphtalene SOA concentrations simulated with the H²O scheme are much lower than those simulated with the near-explicit scheme. The naphtalene SOA concentrations simulated with the H²O scheme can even be considered negligible, except when the NO₂ levels are doubled. The difference between the concentrations simulated with the near-explicit and the H²O scheme reflect large uncertainties in the H²O scheme, which was built by from a specific set of experiments from that are not representative of the full range of atmospheric conditions. Even though these experiments were corrected from wall-loss effects of particles, they were not corrected for gas wall losses. Furthermore, the behaviour of the H²O scheme may not fully extrapolate to all experimental setups, especially flow tubes where residence times are much shorter than in smog chambers.

The sesquiterpene SOA concentrations simulated with the different configurations and in the different NO₂ initial conditions are shown in the right panel of Fig. . During the first two hours of the simulations, the sesquiterpene SOA concentrations increase with increasing NO₂ levels. This is not the case in the simulation with the H²O mechanism, where the SOA concentrations are higher with the reference NO₂ levels.

These test cases highlight the importance of near-explicit chemical schemes, which incorporate various chemical pathways for SOA formation. In contrast, simplified schemes based on chamber experiments may simulate different evolution patterns under changing environmental conditions compared to near-explicit schemes. However, the reduced schemes generated by GENOA, as demonstrated in the examples presented in this study, successfully replicate the evolution predicted by near-explicit schemes.

In order to represent diffusion inside viscous particles, those are discretized into layers. To explicitly represent this diffusion, a high number of layers would be necessary for the model to be accurate. To reduce the number of layers, alternative approaches involve the use of an effective mass accommodation coefficient , or a methodology that represents implicitly the diffusion inside particles . The latter method is used in SSH-aerosol v2.0. As in and in SSH-aerosol v1.1, it does not represent the exchange of compounds between layers but assumes a characteristic time to reach the interface. As illustrated by Fig. , diffusion is represented by exchange fluxes between the different layers inside the particle and the interface layer. The main consequence of this simplification is that the diffusion of a compound is computed assuming identical affinity across all layers, where this affinity represents the compound’s activity relative to the other particle-phase constituents. It may therefore miss some entrapment effect (impossibility for compounds to cross a layer because of a lack of affinity).

The flux of diffusion for a specific layer, bin and species (Jdiff,ibin,layer in µg m⁻³ of air s⁻¹) is calculated as: 10Jdiff,ibin,layer=kdiffusionbin,layer(Ap,iinterface,layerKp,ibin,layerMobin,layerKp,ibin,interfaceMobin,interface-Ap,ibin,layer) with kdiffusionbin,layer the parameter of diffusion rate from the interface to the layer (in s⁻¹), “interface” the index of the layer at the gas/particle interface, Kp,ibin,layer (in m³ µg⁻¹) the partitioning constant, Mobin,layer (in µg m⁻³ of air) the mass of the organic phase in the layer and Ap,ibin,layer the particle concentrations of the species i in the layer (in µg m⁻³ of air).

An hybrid explicit numerical method is used to solve condensation/evaporation/diffusion in and SSH-aerosol v1.1. One issue with this method is the need to distinguish cases where a compound condenses or evaporates very rapidly (typically handled with an equilibrium approach) from cases that require a dynamic resolution. The numerical method is replaced by an iterative implicit numerical (backward Euler), which is capable of simultaneously representing both types of cases: 11Bp,ibin,layer(t+Δt,iter+1)=Ap,ibin,layer(t)+Prodibin,layer(t+Δt,iter)Δt1+kloss,ibin,layerΔt With iter the iteration index, Ap,ibin,layer the initial concentration of compound i in a specific bin and layer, Bp,ibin,layer(t+Δt,iter+1) the restimated concentrations, Prodibin,layer the production rate and kloss,ibin,layer the loss kinetic rate. Prodibin,layer and kloss,ibin,layer include the production and loss due to diffusion, chemical reactions as well condensation/evaporation for the layer at the interface with air.

This method is iterative, as Prodibin,layer is determined for concentrations at time t+Δt. Iteration are performed until convergence is reached with the iterative solver described in : 12Ap,ibin,layer(t+Δt,iter+1)=f×Bp,ibin,layer(t+Δt,iter+1)+(1-f)×Ap,ibin,layer(t+Δt,iter) where Bp,ibin,layer(t+Δt) is the concentration estimated with Eq. () and f is a weighting factor ranging from 0 to 1 determined to ensure convergence. Following , f is divided by a factor 2 (therefore decreasing the variations of concentrations between two iterations) each time a non-convergence loop is detected to ensure convergence.

The implicit solver is found to be faster and more stable than the previous method based on the explicit solver. Note that the same solver is applied for aqueous-phase concentrations (but with only one layer and without representing diffusion inside the aqueous phase).

The gas/particle mass transfer is strongly affected by particle viscosity, modeled by diffusion coefficients that are used to compute kdiffusionbin,layer such as: 13kdiffusionbin,layer=1τdifαlayer14τdif=Rpπ2Dorg with Rp the radius of the particle, Dorg the diffusion coefficient inside the particle and τdif the characteristic time of diffusion. The higher the viscosity is, the lower the diffusion coefficient is. To illustrate this effect, in SSH-aerosol v1.1, the condensation of hydrophobic organic surrogates of different volatility was studied for different values of viscosity.

In SSH-aerosol v2.0, the viscosity may be estimated as a function of the particle composition, with the AIOMFAC-VISC algorithm . In this algorithm, the UNIFAC parameters are used to compute the viscosity. Therefore, the viscosity of a mixture of organic compounds can be computed as long as the decomposition in functional groups of the compounds is given. It should be noted that the aqueous-phase is considered to be inviscid. In the organic phase, by allowing the diffusion coefficient to vary with viscosity, Dorg is no longer constant and therefore Eq. () is rewritten, as a function of the apparent diffusion coefficient from the interface to the considered layer Dorg,app,ibin,layer: 15kdiffusionbin,layer=π2Dorg,app,ibin,layerRpαlayer Dorg,app,ibin,layer is computed with AIOMFAC-VISC by using cumulated concentrations between the interface and the considered layer in order to provide an average diffusion coefficient.

Following , in the AIOMFAC-VISC algorithm, a fragility parameter Dfragility is used to compute pure compound viscosity η0: 16log⁡10(η0)=-5.0+0.434Dfragility⋅T0T-T0 with T0 the Vogel temperature, or ideal glass transition temperature: 17T0=Tg⋅39.17Dfragility+39.17 with Tg the glass transition temperature, a characteristic temperature (computed as a function of the number of N, C, and O atoms) at which the mixture changes from a glassy solid state to a semi-solid state. Although Dfragility is set to 10 in , a parameterisation of Dfragility as a function of the O/C ratio is used, following . A

The diffusion coefficient Ddiffc of the compound c is deduced from the viscosity (η), the viscosity of pure water (η0) and the diffusion coefficient in pure water Ddiff,water, following the parameterization of : 18Ddiffc=Ddiff,water×η0waterηξ where ξ is a parameter used to compare the radius of the diffusing molecule (Rdiff) to the average radius of the compound mixture: 19ξ=1-Aexp⁡BRdiffRmean with A and B equal to 0.73 and 1.79 respectively , and Rmean is calculated as the average value of Rdiff weighted by molar fractions. Rdiff is calculated from the density and molar mass of the compound: 20Rdiff=3Mc4Naρc⋅100013 with Mc the molar mass (in g mol⁻¹), Na the Avogadro number and ρc the compound density (in kg m⁻³).

This methodology allows to calculate the diffusion coefficients for each species and each particle layer and therefore to vary with particle composition. The effect of changing composition on the condensation of a semivolatile compound is illustrated in Fig. . The condensation of an organic compound (POAmP) is investigated for different particle compositions, and thus for different viscosities. POAmP is represented using the default molecular structure proposed by in the H²O mechanism for primary compounds and model species with unknown molecular structures. In this representation, the activity coefficient is set to unity, ensuring that the condensation process is not influenced by non-ideality effects. At the start of the simulation, a low concentration of POAmP in the gas phase (0.01 µg m⁻³) was set in order to not affect the viscosity of the organic phase. The time evolution of POAmP concentrations was computed for different relative humidity (RH) and for different compositions of the aerosol onto which POAmP condenses. The compositions differ by the number of alcohol groups (OH), of carbons, and of various types of functional groups (ketone, hydroxyl, acid, and nitrate). The simulations show that an aerosol composed of compounds with 16 carbons and 4 alcohol groups is very viscous at RH = 10 % and that condensation of POAmP hardly occurs. However, at RH = 70 %, the aerosol can be considered as inviscid and the condensation of POAmP is non-limited by viscosity. Adding just one OH group has a strong effect on viscosity at RH = 10 %. With 16 carbons and 2 OH groups, equilibrium between the gas and particle phases of POAmP is reached in about one hour. However, it takes more than 100 h to reach equilibrium with an aerosol composed of 16 carbons and 3 OH groups. Adding, acid, hydroperoxide or nitrate groups affects the viscosity even more significantly. Especially, just by adding one nitrate group, the aerosol switches from inviscid to extremely viscous. By comparison, the effect on viscosity of increasing the number of carbons is low. If the molecules have 2 OH groups, equilibrium is reached in less than 1 h with less than 16 carbons. Time to equilibrium increases progressively with the number of carbons. It takes about 100 h with 20 carbons.

Reactions can occur inside the particles. These reactions can be activated or deactivated by modifying the file indicated by reaction_soap_file in the section &physic_condensation of the namelist. Different types of reactions can be accounted for. Examples on how to use this reaction file are provided in the supplementary material (user manual). Currently 5 types of particle-phase reactions are taken into account: 1st order irreversible reaction, 2nd order reversible oligomerization, hydratation of aldehydes, reactions of organic compounds with inorganic ions and a specific paramterization for pinonaldehyde oligomerization.

Once uptaken into the aqueous phase of the particles, dicarbonyls can be oxidized by OH radicals present in this aqueous phase to irreversibly form highly soluble and low-volatility compounds . Empirical reactions representing this irreversible condensation pathway for glyoxal and methylglyoxal has been added in SSH-aerosol v2.0. The term “irreversible condensation pathway” is used in contrast to the classical “reversible condensation pathway” influenced by hydration and isomerization. The irreversible condensation pathway is represented by the following reaction: 28DICARB→kirdiIRDICARB where DICARB is the semi-volatile species studied (glyoxal or methylglyoxal) and IRDICARB its non-volatile counterpart via irreversible partitioning. The kinetics kirdi of this reaction follows the parameterization of : 29kirdi=14v×γ×Asurf with: 30v=8RTπMdicarb311γ=1α+v4RTHdicarbeffklDaq×1cothq-1/q32q=RpDaqkl where γ is the reactive uptake coefficient, v and Mdicarb are the gas-phase thermal velocity and the molar mass of the DICARB species, Asurf is the aerosol surface, R is the universal gas constant, T is the temperature in Kelvins, Hdicarbeff is the effective Henry's law constant, q is a parameter that account for in-particle diffusion limitation, Rp is the particle radius, kl is the first order loss rate due to reaction with OH radicals in the aqueous phase. The aqueous-phase diffusion coefficient Daq is fixed to 10⁻⁹ m² s⁻¹, which is typical for small organics , and the mass accommodation coefficient α to 0.02, similar to that of formaldehyde uptake to water . Henry constants of DICARB are adapted from to match with observations and the Henry constant of OH used to estimate OH aqueous concentration from gaseous ones is adapted from leading to aqueous concentration in agreement with typical mean value in the atmosphere from . Further details and description of the parameters are available in .

The standard configuration of SSH-aerosol is based on two separate thermodynamic modules: SOAP and ISORROPIA. ISORROPIA is called first to compute water, pH, and inorganics. Then, SOAP is called to compute the concentrations of organics. In this system, organics do not affect the computation of pH or the concentrations of inorganics. This requires solve organics and inorganics together.

SOAP was modified in order to be able to solve the equilibrium and the dynamics of inorganic and organic aerosol simultaneously. Organics can affect the formation of inorganics by:

Influencing the condensation of water, which in turn affects the absorption of HNO₃ and NH₃.

SOAP is based on the AIOMFAC thermodynamic module that accounts for numerous ions, H⁺, OH⁻, NO3-, HSO4-, SO42-, HCO3-, CO32-, Cl⁻, Na⁺, K⁺, Mg²⁺, Ca²⁺. Inorganic aerosols are assumed to be thermodynamically metastable (inorganic compounds are always present in the aqueous phase of particles and no solid salts are formed) with the exception of CaCO₃ that is assumed to have a very low solubility. Henry's law constant are taken from . The iterative solver in Eq. () is used to solve iteratively the system of equations. f in Eq. () is constrained in order to ensure that activity coefficients of inorganics and pH do not change significantly between two iterations in order to ensure stability and convergence.

Several test cases, such as the Platt test case with isoprene and NH₃, have been run using ISORROPIA or SOAP for inorganics. Using SOAP, the simulated concentrations are very similar to the results obtained using ISORROPIA, as shown in the Supplement (user manual). To illustrate some of the capabilities of the SOAP model for both inorganic and organic thermodynamic, a test case simulates the evolution of CaCO₃ in the presence or in the absence of HNO₃ with an equilibrium approach or a dynamic approach. As shown in Fig. , when CaCO₃ is not exposed to HNO₃ the particle does not evolve. Especially, it does not absorb water. This is due to the fact that CaCO₃ is currently the only particle for which delisquescence is accounted for due to its very low saturation value that would correspond to a delisquescence relative humidity close to 100 %. When [Ca²⁺ (mol kg⁻¹ of solvents)][CO32- (mol kg⁻¹ of solvents)] > K (with K the equilibrium constant), a solid CaCO₃ is formed. When exposed to HNO₃, around 2 molec. of NO3- will replace 1 molec. of CO32- to maintain electroneutrality. It leads to the presence of the nitrate and water inside the particle, and part of Ca²⁺ that was previously associated with CO32+ is transferred to the aqueous phase. Therefore, due to the condensation of HNO₃, the particle grows bigger in size and becomes hygroscopic.

Several nucleation parameterizations are implemented. They can be used separately or together. An example of their use in 3D is illustrated in . The nucleation parametrizations differ in the gas precursors (sulfuric acid, ammonia, extremely-low volatile organic compounds) and/or in the formulation of the parameterization.

binary: water and sulfuric acid with the parameterisations of or ;

To illustrate the growth of nucleated nanoparticles, the nucleation test case presented in is simulated. The initial distribution corresponds to the hazy conditions of . The sulfuric acid production rate is 0.825 µg m⁻³ h⁻¹, the temperature is 288.15 K and the relative humidity is 60 %. The particles are initially assumed to be made of 70 % sulfate and 30 % ammonium. The initial gas phase ammonia concentration is taken to be 8 µg m⁻³. The concentrations of gas-phase ammonia and particulate-phase ammonium evolve with time due to both condensation/evaporation and ternary nucleation. The simulation is run for 1 h with output every 60 s. Figure  shows the size distribution of particles at the initial time and after 1 h. The nucleated particles grow under the effect of condensation of sulfuric acid and ammonia and coagulation. This growth can be accelerated by the presence of extremely low-volatility organic compounds (ELVOCs), such as the SSH-aerosol surrogate “Monomer”, underlying the importance of an accurate modelling of the formation of ELVOCs for the growth of ultra-fine particles.

To decrease CPU time associated with the treatment of condensation and evaporation, thermodynamic equilibrium between the gas and particle phases can be assumed. In that case, the bulk equilibrium is first computed, and then the bulk aerosol concentrations need to be distributed among the different size sections.

The redistribution was changed in version 2.0 to take into account the Kelvin effect, and for inorganics, the neutralization of acids by positive inorganic ions (sodium and ammonium). The amount of ammonium that condenses to neutralize sulfate is determined separately from the amount of ammonium that condenses to neutralize other compounds, such as nitric acid. Then the amount of ammonium that neutralizes sulfate is redistributed amongst the size sections, depending on the sulfate concentration of each section. The amount of ammonium that condenses with other semi-volatile compounds such as nitric acid is redistributed amonst the size section using a formulation that depends on the Kelvin effect and the condensation rate (depending on the gas-phase diffusivity of species and the diameters of particles).

To illustrate the role of the Kelvin effect versus the sulfate neutralization, the urban condition test case of is modified by adding NH₃ in the initial conditions, and by taking into account only condensation/evaporation processes (nucleation and coagulation are not considered). The initial particle concentration is assumed to be made of sulfate only, and the initial concentration of NH₃ is assumed to be 100 µg m⁻³. The simulation is run for 12 h.

Four simulations are run: in 2 simulations condensation/evaporation is computed dynamically with or without taking into account the Kelvin effect, in the other 2 simulations thermodynamic equilibrium is assumed. The number and volume size distribution simulated with the four numerical algorithms are compared in Fig. . The four numerical algorithms give similar number concentrations, because NH₃ condenses to neutralize the sulfate independently of the Kelvin effect.

In the previous test cases, about 40 µg m⁻³ of ammonia is used to neutralize sulfate and the remaining concentration stays in the gas phase. To compare the effect of the Kelvin effect between dynamical calculation and thermodynamic assumption, nitric acid is added in the initial gas-phase concentration. Nitric acid condenses with the remaining ammonia to form ammonium nitrate, as shown in Fig. . As seen in the comparison of the number size distribution, the particles of diameters below about 0.01 µm do not grow by the condensation of ammonium nitrate if the Kelvin effect is taken into account, while these particles grow to larger diameters by condensation of ammonium nitrate if the Kelvin effect is not taken into account. The algorithm used to redistribute ammonium nitrate amongst size section is effective at limiting this condensation on particles of small diameters. This test case is also presented for organic aerosols in the supplementary material (user manual).