Estimation of rate coefficients and branching ratios for reactions of organic peroxy radicals for use in automated mechanism construction

Organic peroxy radicals (RO₂), formed from the degradation of hydrocarbons and other volatile organic compounds (VOCs), play a key role in tropospheric oxidation mechanisms. Several competing reactions may be available for a given RO₂ radical, the relative rates of which depend on both the structure of RO₂ and the ambient conditions. Published kinetics and branching ratio data are reviewed for the bimolecular reactions of RO₂ with NO, NO₂, NO₃, OH and HO₂; and for their self-reactions and cross-reactions with other RO₂ radicals. This information is used to define generic rate coefficients and structure–activity relationship (SAR) methods that can be applied to the bimolecular reactions of a series of important classes of hydrocarbon and oxygenated RO₂ radicals. Information for selected unimolecular isomerization reactions (i.e. H-atom shift and ring-closure reactions) is also summarized and discussed. The methods presented here are intended to guide the representation of RO₂ radical chemistry in the next generation of explicit detailed chemical mechanisms.

1 Introduction

Organic peroxy radicals (RO₂) are important intermediates in the tropospheric degradation of hydrocarbons and other volatile organic compounds (VOCs). It is well established that their chemistry plays a key role in the mechanisms that generate ozone (O₃), secondary organic aerosol (SOA) and other secondary pollutants (e.g. Lightfoot et al., 1992; Jenkin and Clemitshaw, 2000; Tyndall et al., 2001; Archibald et al., 2009; Orlando and Tyndall, 2012; Ehn et al., 2017), and rigorous representation of their chemistry is therefore essential for chemical mechanisms used in chemistry-transport models. As discussed in the preceding papers in this series (Jenkin et al., 2018a, b), they are formed rapidly and exclusively from the reactions of O₂ with the majority of carbon-centred organic radicals (R) (Reaction R1), these in turn being produced from the reactions that initiate VOC degradation (e.g. reaction with OH radicals), or from other routes such as decomposition of larger oxy radicals (M denotes a third body, most commonly N₂ or O₂ under atmospheric conditions):

$$\begin{array}{}\text{(R1)}& {\displaystyle }\mathrm{R}+{\mathrm{O}}_{\mathrm{2}}(+M)\to {\mathrm{RO}}_{\mathrm{2}}(+M).\end{array}$$

Under tropospheric conditions, a given RO₂ radical may have several competing reactions available, the relative rates of which are dependent both on the prevailing ambient conditions and on the structure of RO₂. These include a series of bimolecular reactions (i.e. with NO, NO₂, NO₃, OH and HO₂; and the self-reaction and cross-reactions with the multitude of other RO₂ radicals present in the atmosphere), which are generally available for all RO₂ radicals; and specific unimolecular isomerization reactions (i.e. H-atom shift or ring-closure reactions) that are potentially available for some classes of RO₂.

The propagating channel of the reaction of RO₂ with NO (Reaction R2a) plays a key role in tropospheric O₃ formation, through oxidizing NO to NO₂, and also usually represents the major reaction for RO₂ radicals under comparatively polluted conditions:

$$\begin{array}{}\text{(R2a)}& {\displaystyle }{\mathrm{RO}}_{\mathrm{2}}+\mathrm{NO}\to \mathrm{RO}+{\mathrm{NO}}_{\mathrm{2}}.\end{array}$$

The efficiency of this reaction is influenced by the relative importance of the other reactions available for a given RO₂ radical. The contribution of the terminating channel of the reaction of RO₂ with NO (forming an organic nitrate product, RONO₂) depends on the structure and size of RO₂; and the reaction of NO₂ with selected RO₂ radicals forms stable peroxynitrate products, ROONO₂. The formation, transport and degradation of these oxidized organic nitrogen reservoirs from the RO₂+NO and RO₂+NO₂ reactions has potential impacts in a number of ways, ranging from the inhibition of O₃ formation on local/regional scales to influencing the global budget and distribution of NO_x and O₃ (e.g. Perring et al., 2013). The reactions of RO₂ radicals with NO₃ primarily play a role during the nighttime in moderately polluted air, providing a radical propagation route that potentially supplements nighttime chain oxidation processes (e.g. Carslaw et al., 1997; Bey et al., 2001a, b; Geyer et al., 2003; Walker et al., 2015).

The reactions with OH, HO₂ and the pool of RO₂ radicals gain in importance as the availability of NO_x becomes more limited, and therefore also inhibit O₃ formation by competing with Reaction (R2a). In many cases, the reactions are significantly terminating and collectively make a major contribution to controlling atmospheric free radical concentrations under NO_x-limited conditions, although the branching ratios for the propagating and terminating reaction channels depend on the structure of RO₂. For some classes of RO₂, unimolecular isomerization reactions can compete with (or dominate over) the bimolecular reactions. These reactions therefore potentially play an important role in HO_x radical recycling under NO_x-limited conditions, and in rapid chain oxidation mechanisms generating highly oxidized multifunctional molecules, HOMs (e.g. Peeters et al., 2009, 2014; Crounse et al., 2013; Ehn et al., 2014, 2017; Jokinen et al., 2014; Rissanen et al., 2015; Bianchi et al., 2019). The relative contributions of the various reactions available for RO₂ thus influence the distribution and functional group content of the oxidized products formed, and their physicochemical properties (e.g. volatility and solubility), and therefore the SOA formation propensity of the chemistry.

In this paper, published data on the kinetics and branching ratios for the above bimolecular reactions of hydrocarbon and oxygenated RO₂ radicals are reviewed and discussed. Preliminary information is also presented for selected unimolecular isomerization reactions, which continue to be considered in ongoing work. The information on bimolecular reactions is used to define and document a set of rules and structure–activity relationship (SAR) methods (a chemical protocol) to guide the representation of the RO₂ reactions in future detailed chemical mechanisms (Vereecken et al., 2018). In particular, the methods presented below are being used to design the next generation of explicit mechanisms based on the Generator for Explicit Chemistry and Kinetics of Organics in the Atmosphere, GECKO-A (Aumont et al., 2005), and the Master Chemical Mechanism, MCM (Saunders et al., 2003). Application of the methods is illustrated with examples in the supporting information provided in the Supplement.

2 Bimolecular reactions of RO₂ radicals

2.1 The reactions of RO₂ with NO

2.1.1 Kinetics

Rate coefficients for the reactions of NO with a variety of specific hydrocarbon and oxygenated RO₂ radicals have been reported, as summarized in Table 1. For the vast majority of the RO₂ radicals formed in detailed mechanisms, however, kinetic data are unavailable and it is therefore necessary to assign generic rate coefficients based on the reported data.

Table 1 Kinetic data for the reactions of hydrocarbon and oxygenated peroxy radicals with NO. Where available, the temperature dependence is given by $k=A\exp (-E/RT)$.

Comments: ^a IUPAC Task Group recommendation (http://iupac.pole-ether.fr/, last access: January 2019). ^b Based on Eberhard and Howard (1996, 1997), Eberhard et al. (1996). ^c Based on Miller et al. (2004). ^d  Mixture of CH₂(OH)CH(O₂)CH₃ and CH₂(O₂)CH(OH)CH₃. ^e Mixture of CH₂(OH)CH(O₂)C₂H₅ and CH₂(O₂)CH(OH)C₂H₅. ^f Mixture of CH₂(OH)C(O₂)(CH₃)₂ and CH₂(O₂)C(OH)(CH₃)₂. ^g Mixture of CH₂(OH)CH(O₂)CH=CH₂, CH₂(OH)CH=CHCH₂O₂ and CH₂(O₂)CH(OH)CH=CH₂. ^h Mixture of CH₂(OH)C(O₂)(CH₃)CH=CH₂, CH₂(OH)C(CH₃)=CHCH₂O₂, CH₂(O₂)C(CH₃)(OH)CH=CH₂, CH₂(OH)CH(O₂)C(CH₃)=CH₂, CH₂(OH)CH=C(CH₃)CH₂O₂ and CH₂(O₂)CH(OH)C(CH₃)=CH₂. ⁱ Based on Sehested et al. (1998). ^j Based on Hsin and Elrod (2007). ^k Mixture of CH₂(OH)C(O₂)(CH₃)CHO and CH₂(O₂)C(OH)(CH₃)CHO. ^l Mixture of CH₂(OH)CH(O₂)C(=O)CH₃ and CH₂(O₂)CH(OH)C(=O)CH₃. ^m Elrod (2011). Mixture of two complex radicals of molecular formula HOC₉H₁₂[OO]O₂, although with one isomer likely dominant. ⁿ De Gouw and Howard (1997). ^o Berndt et al. (2015). Inferred to be the complex oxo-di-hydroperoxy acyl peroxy radical shown, on the basis of its molecular mass and a proposed mechanism.

For acyl peroxy radicals (i.e. of structure RC(O)O₂), a generic rate coefficient (k_APNO) is applied:

$$\begin{array}{}\text{(1)}& {\displaystyle }{k}_{\mathrm{APNO}}=\mathrm{7.5}\times {\mathrm{10}}^{-\mathrm{12}}\exp (\mathrm{290}/T){\mathrm{cm}}^{\mathrm{3}}{\mathrm{molecule}}^{-\mathrm{1}}{\mathrm{s}}^{-\mathrm{1}}.\end{array}$$

This is based on the IUPAC Task Group¹ recommendation for the reaction of NO with CH₃C(O)O₂. As shown in Table 1, this is also close to the rate coefficients recommended for the less studied acyl peroxy radicals, C₂H₅C(O)O₂ and CH₂=CH(CH₃)C(O)O₂. The 298 K value reported for C(O)(OOH)CH₂CH₂CH₂CH(OOH)C(O)O₂ (Berndt et al., 2015) is also broadly consistent with k_APNO, although further studies of highly oxygenated acyl peroxy radicals would help to establish the effects of additional substituent groups.

For other classes of hydrocarbon and oxygenated peroxy radical, a generic rate coefficient ($k_{{\mathrm{RO}}_{\mathrm{2}}\mathrm{NO}}$) is applied:

$$\begin{array}{}\text{(2)}& {\displaystyle }{k}_{{\mathrm{RO}}_{\mathrm{2}}\mathrm{NO}}=\mathrm{2.7}\times {\mathrm{10}}^{-\mathrm{12}}\exp (\mathrm{360}/T){\mathrm{cm}}^{\mathrm{3}}{\mathrm{molecule}}^{-\mathrm{1}}{\mathrm{s}}^{-\mathrm{1}}.\end{array}$$

The value of $k_{{\mathrm{RO}}_{\mathrm{2}}\mathrm{NO}}$ at 298 K ($\mathrm{9.0}\times {\mathrm{10}}^{-\mathrm{12}}$ cm³ molecule⁻¹ s⁻¹) is based on a rounded average of the 298 K rate coefficients listed for the ≥C₂ alkyl, cycloalkyl, hydroxyalkyl, hydroxyalkenyl, oxoalkyl, hydroxy-oxyalkyl and hydroxy-dioxa-bicyclo RO₂ radicals in Table 1, which show no significant trends related to the identity and structure of R. The temperature dependence is similarly based on the rounded average of the available values within this group, which are limited to those for C₂H₅O₂, n-C₃H₇O₂ and i-C₃H₇O₂. In practice, the preferred values for all the ≥C₂ (non-acyl) RO₂ radicals in Table 1 are also equivalent to $k_{{\mathrm{RO}}_{\mathrm{2}}\mathrm{NO}}$ within the reported uncertainties, such that the generic rate coefficient can reasonably be applied for simplicity in all cases except CH₃O₂. Although derived from a more extensive dataset, the expression for $k_{{\mathrm{RO}}_{\mathrm{2}}\mathrm{NO}}$ in Eq. (2) is identical to that recommended previously by Atkinson (1997).

2.1.2 Product branching ratios

The following channels are considered for the reactions of RO₂ with NO:

$$\begin{array}{}\text{(R2a)}& {\displaystyle }& {\displaystyle }{\mathrm{RO}}_{\mathrm{2}}+\mathrm{NO}\to \mathrm{RO}+{\mathrm{NO}}_{\mathrm{2}},\text{(R2b)}& {\displaystyle }& {\displaystyle }{\mathrm{RO}}_{\mathrm{2}}+\mathrm{NO}(+M)\to {\mathrm{RONO}}_{\mathrm{2}}(+M).\end{array}$$

It is well established that the branching ratio for alkyl peroxy radicals depends on temperature, pressure, and the size and degree of substitution of the peroxy radical (e.g. Carter and Atkinson, 1989; Arey et al., 2001; Yeh and Ziemann, 2014a). The branching ratio has also been reported to be influenced by the presence of oxygenated substituents, with most systematic information reported for β- and δ-hydroxy groups (e.g. O'Brien et al., 1998; Matsunga and Ziemann, 2009, 2010; Yeh and Ziemann, 2014b; Teng et al., 2015).

The fraction of the reaction forming a nitrate product (RONO₂) via the terminating channel, ${\mathrm{R}}_{\mathrm{2}\mathrm{b}}=k_{\mathrm{2}\mathrm{b}}/(k_{\mathrm{2}\mathrm{a}}+k_{\mathrm{2}\mathrm{b}})$, is calculated following the method originally reported for secondary alkyl peroxy radicals by Carter and Atkinson (1989), and subsequently updated by Arey et al. (2001) and Teng et al. (2015). Based on this method, the reference branching ratio for secondary alkyl peroxy radicals, $\mathrm{R}{}^{\circ }=(k_{\mathrm{2}\mathrm{b}}/k_{\mathrm{2}\mathrm{a}}){}^{\circ }$, is calculated as follows.

$$\begin{array}{}\text{(3)}& {\displaystyle }\mathrm{R}{}^{\circ }=[A/(\mathrm{1}+(A/B)\left)\right]F^z\end{array}$$

with $A=\mathrm{2}\times {\mathrm{10}}^{-\mathrm{22}}\exp \left(n_{\mathrm{CON}}\right)\left[M\right]$, $B=\mathrm{0.43}(T/\mathrm{300}{)}^{-\mathrm{8}}$, F=0.41 and $z=(\mathrm{1}+({\log }_{\mathrm{10}}(A/B){)}^{\mathrm{2}}{)}^{-\mathrm{1}}$. Variable n_CON is the number of carbon, oxygen and nitrogen atoms in the organic group (R) of the peroxy radical (i.e. excluding the peroxy radical oxygen atoms and equivalent to the carbon number in alkyl peroxy radicals), T is the temperature (in kelvin) and [M] is the gas density (in molecule cm⁻³).

The fractions of the reaction proceeding via the terminating channel, R_2b, and the propagating channel, R_2a ($=\mathrm{1}-{\mathrm{R}}_{\mathrm{2}\mathrm{b}}$), for a specific peroxy radical are then given by

$$\begin{array}{}\text{(4)}& {\displaystyle }{\mathrm{R}}_{\mathrm{2}\mathrm{b}}=f_{\mathrm{a}}{f}_{\mathrm{b}}(\mathrm{R}{}^{\circ }/(\mathrm{1}+\mathrm{R}{}^{\circ }\left)\right).\end{array}$$

The effect of the degree of substitution (i.e. whether the radical is primary, secondary or tertiary) is described by f_a, with a unity value applied to secondary peroxy radicals by definition. A further scaling factor, f_b, is used to describe systematic variations in the yields of RONO₂ resulting from the presence of oxygenated substituents (e.g. the effect of hydroxyl substituents, as indicated above), or for specific peroxy radical classes, with a value of f_b being required to account for the effect of each relevant substituent. The applied values of f_a and f_b are summarized in Tables 2 and 3, and example calculations are provided in Sect. S1 in the Supplement.

Table 2 Values of the scaling factor, f_a, applied to the branching ratio calculation for the reaction of RO₂ with NO.

Comments: ^a Applied in all cases, except for those covered by comment ^b. f_a=1 for secondary peroxy radicals by definition. The equivalent value for tertiary peroxy radicals, and the lower value for primary peroxy radicals, is based on a consensus of information from Cassanelli et al. (2007), Orlando and Tyndall (2012) and Teng et al. (2015) and on previous consideration of the OH + isoprene system (Jenkin et al., 2015); ^b Inhibition of nitrate formation has been reported for complex hydroxy-dioxa-bicyclo peroxy radicals derived from aromatics, relative to comparably sized alkyl peroxy radicals by Rickard et al. (2010) and Elrod (2011), with a particular impact from the presence of alkyl substituents reported by Elrod (2011). The reduced values of f_a for tertiary peroxy radicals, and the general reduction in f_a for peroxy radicals with a neighbouring alkyl substituent (as shown), is inferred from the trend in nitrate yields reported for benzene, toluene, p-xylene and 1,3,5-trimethylbenzene by Elrod (2011).

Table 3 Values of the scaling factor, f_b, applied to the branching ratio calculation for the reaction of RO₂ with NO^a.

Comments: ^a A value of f_b needs to be applied to account for the effect of each relevant substituent (see Sect. S1 in the Supplement for further information). ^b f_b=1 for alkyl peroxy radicals, by definition, and also used as a default in all cases other than those covered by comments ${}^{\mathrm{c}--\mathrm{h}}$. ^c Based on a compromise of information from Matsunaga and Ziemann (2009, 2010), Yeh and Ziemann (2014b) and Teng et al. (2015) for β-hydroxy substituents, but also taking account of information reported for a number of other oxygenated systems (e.g. Tuazon et al., 1998a; Crounse et al., 2012; Lee et al., 2014) and previous consideration of the OH + isoprene system (Jenkin et al., 2015). $\mathrm{OO}-\mathrm{C}-\mathrm{C}\left(\mathrm{OOH}\right)$< assumed to be in this category by analogy. ^d Based on the relative impacts of β-OH and δ-OH substituents reported by Yeh and Ziemann (2014a) and previous consideration of the OH + isoprene system (Jenkin et al., 2015). ^e f_b value for $\mathrm{OO}-\mathrm{C}-\mathrm{C}(=\mathrm{O})-$ informed by reported studies of ketone oxidation (Lightfoot et al., 1992; Praske et al., 2015); f_b values for $\mathrm{OO}-\mathrm{C}-\mathrm{C}(=\mathrm{O})-\mathrm{O}-$ and $\mathrm{OO}-\mathrm{C}-\mathrm{O}-\mathrm{C}(=\mathrm{O})-$ informed by reported studies of ester and dibasic ester oxidation (Tuazon et al., 1998b, 1999; Cavalli et al., 2001; Picquet-Varrault et al., 2001, 2002; Pimentel et al., 2010). ^f f_b=0 for $\mathrm{OO}-\mathrm{C}(=\mathrm{O})-$ is based on the general lack of observation of acyl nitrate products in systems where acyl peroxy radicals are formed. ^g Value set to recreate the hydroxy-dioxa-bicyclo nitrate yield reported for benzene by Elrod (2011); In conjunction with the values of f_a in Table 2, this also allows a consistent representation of the yields in the toluene, p-xylene and 1,3,5-trimethylbenzene systems (Elrod, 2011; Rickard et al., 2010); ^h f_b=0 for phenyl (and other aryl) peroxy radicals is based on the general lack of observation of aryl nitrate products during the oxidation of aromatic hydrocarbons.

It is also recognized that reaction channel (R2a) is significantly exothermic, such that prompt decomposition or isomerization of a fraction of the initially formed chemically activated oxy radicals has been reported to occur in some cases; with the remainder being collisionally deactivated to form thermalized RO (e.g. Orlando et al., 2003; Calvert et al., 2015). This is particularly important for β-hydroxy-oxy radicals (e.g. Orlando et al., 1998; Vereecken et al., 1999; Vereecken and Peeters, 1999; Caralp et al., 2003) and some other oxygenated oxy radicals (e.g. Christensen et al., 2000; Orlando et al., 2000a; Wallington et al., 2001). The contributions and treatment of these reactions is summarized in Sect. S2.

2.2 The reaction of RO₂ with NO₂

The reactions of RO₂ with NO₂ have generally been reported to proceed via a reversible association reaction in each case to form a peroxy nitrate (ROONO₂):

$$\begin{array}{}\text{(R3a)}& {\displaystyle }{\mathrm{RO}}_{\mathrm{2}}+{\mathrm{NO}}_{\mathrm{2}}(+M)\rightleftharpoons {\mathrm{ROONO}}_{\mathrm{2}}(+M).\end{array}$$

Rate coefficients for the forward and reverse reactions for a number of RO₂ radicals are summarized in Table 4. Those for CH₃O₂ and C₂H₅O₂ – and for the two simplest acyl peroxy radicals, CH₃C(O)O₂ and C₂H₅C(O)O₂ – are based on (or informed by) the IUPAC Task Group recommendations, and describe the pressure and temperature dependences of the reactions. In all other cases, the reactions are assumed to be at the high-pressure limit under atmospheric conditions, and generic parameters are applied. The parameters k_f PN and k_b PN (given in Table 4) can reasonably be applied to reactions involving non-acyl peroxy radicals, being based on the high-pressure limiting rate coefficients (k_∞) for the forward and reverse reactions of C₂H₅O₂ and those reported for a number of higher alkyl peroxy radicals at close to atmospheric pressure (see Table 4 comments). This assumption is also broadly consistent with the limited information available for the forward or reverse reactions of other non-acyl oxygenated peroxy radicals (e.g. Orlando and Tyndall, 2012). In practice, however, these reactions are often omitted from atmospheric chemical mechanisms, owing to the instability of the ROONO₂ products under lower tropospheric conditions (lifetime ≈0.2 s at 298 K). As a result, only the formation and decomposition of methyl peroxy nitrate, CH₃OONO₂, from the most abundant non-acyl peroxy radical, CH₃O₂, have previously been represented in the MCM (Saunders et al., 2003). This approach remains advocated here for application for lower tropospheric conditions.

Table 4 Rate coefficients for the reactions of hydrocarbon and oxygenated RO₂ radicals with NO₂ and for the reverse decomposition of the RO₂NO₂ products. Generic rate coefficients for specified RO₂ classes are shown in bold font.

Comments: ^a Rate coefficient for a pressure-dependent reaction is calculated using the expression: $k=F{k}_{\mathrm{0}}{k}_{\mathrm{\infty }}/(k_{\mathrm{0}}+k_{\mathrm{\infty }})$, where log${}_{\mathrm{10}}F={\log }_{\mathrm{10}}\left(F_{\mathrm{c}}\right)/(\mathrm{1}+[{\log }_{\mathrm{10}}(k_{\mathrm{0}}/k_{\mathrm{\infty }})/N{]}^{\mathrm{2}})$ and N=[0.75–1.27 log₁₀(F_c)] (see http://iupac.pole-ether.fr/, last access: January 2019); ^b Based on the evaluation of Golden (2005); ^c Recommended by the IUPAC Task Group (http://iupac.pole-ether.fr/, last access: January 2019); ^d Reported by McKee et al. (2016) for a mixture of n-C₄H₉O₂ and sec-C₄H₉O₂ formed from reaction of Cl with butane; ^e Pressure-independent generic rate coefficient. ^f k_∞ assumed equivalent to that for CH₃C(O)O₂+NO₂ reaction. k₀ scaled relative to that for CH₃C(O)O₂ to preserve the C₂H₅C(O)O₂+NO₂ ⇌ C₂H₅C(O)OONO₂ equilibrium constant, k_f∕k_b, over the pressure range 100–760 Torr. F_c is equivalent to that recommended for k_b. ^g Forward reaction rate coefficient, k_f PAN, is based on k_∞ for the CH₃C(O)O₂+NO₂ reaction. ^h Based on Zabel et al. (1989), as recommended by Lightfoot et al. (1992), for isomeric mixtures formed from reactions of Cl with butane, hexane or octane. Assumed to be at high-pressure limit at 800 Torr. ⁱ k_b PN is rounded average of the reported rate coefficients for C₂-C₈ alkyl peroxy radicals. ^j Based on Roberts and Bertman (1992). Assumed to be at high-pressure limit at 760 Torr. ^k k_b PAN, is based on a value of $\mathrm{3.4}\times {\mathrm{10}}^{-\mathrm{4}}$ s⁻¹ at 298 K, which is the average of those reported for n-C₃H₇C(O)OONO₂, n-C₄H₉C(O)OONO₂ and n-C₅H₁₁C(O)OONO₂ (Kabir et al., 2014) and CH₂=C(CH₃)C(O)O₂ (Roberts and Bertman, 1992) (see Fig. 1). E∕R is based on the average of the high-pressure limit values for CH₃C(O)O₂NO₂, C₂H₅C(O)O₂NO₂ and CH₂=C(CH₃)C(O)O₂, and also consistent with the approximate value for n-C₅H₁₁C(O)OONO₂ (Kabir et al., 2014). ^l Pressure-independent generic rate coefficient for thermal decomposition of ROC(O)O₂ is a factor of 2 greater, based on data for CH₃OC(O)O₂ and C₆H₅OC(O)O₂ (Kirchner et al., 1999), with reduced thermal stability also consistent with data for C₂H₅OC(O)O₂ (Bossolasco et al., 2011).

Table 5 Kinetic data for the reactions of alkyl and oxygenated peroxy radicals with NO₃. Where available, the temperature dependence is given by $k=A\exp (-E/RT)$.

Comments: ^a IUPAC Task Group recommendation (http://iupac.pole-ether.fr/, last access: January 2019). ^b k_298 K based on an average of the values reported by Biggs et al. (1995), Ray et al. (1996), Vaughan et al. (2006) and Laversin et al. (2016). E∕R taken from Laversin et al. (2016). ^c Taken from Vaughan et al. (2006). ^d Taken from Kalalian et al. (2018). ^e Taken from Doussin et al. (2003). Canosa-Mas et al. (1996) reported a comparable value of $k=(\mathrm{4.0}\pm \mathrm{1.0})\times {\mathrm{10}}^{-\mathrm{12}}$ cm³ molecule⁻¹ s⁻¹ over the range 403–443 K.

The reactions are generally represented for acyl peroxy radicals, for which the product peroxyacyl nitrates, RC(O)OONO₂, are particularly stable (lifetime ≈40–50 min at 298 K). The generic parameters, k_f PAN and k_b PAN, are applied in the majority of cases (see Table 4). As shown in Fig. 1, larger acyl peroxy radicals have been reported to be slightly more stable than those derived from CH₃C(O)O₂ and C₂H₅C(O)O₂ (Roberts and Bertman, 1992; Kabir et al., 2014), and the assigned value of k_b PAN is consistent with the data for the larger species.

Figure 1 Reported thermal decomposition rates of selected peroxyacyl nitrates at 298 K and 760 Torr. Values for PAN, PPN and MPAN (defined in the figure) are the IUPAC Task Group recommendations (http://iupac.pole-ether.fr/, last access: January 2019). The other values are taken from the systematic study of Kabir et al. (2014), which also reports consistent values for PAN and PPN. The broken line is the generic rate coefficient, k_b PAN, for the decomposition of RC(O)OONO₂ structures (see Sect. 2.2 and Table 4).

Reported data for CH₃OC(O)O₂, C₆H₅OC(O)O₂ and C₂H₅OC(O)O₂ (Kirchner et al., 1999; Bossolasco et al., 2011) indicate a reduced thermal stability of peroxyacyl nitrates derived from formate esters, and an increased decomposition rate (2×k_b PAN) is therefore applied to ROC(O)OONO₂ species in general.

In a limited number of cases, the reaction of RO₂ with NO₂ has been reported to oxidize NO₂ to NO₃ in an irreversible reaction:

$$\begin{array}{}\text{(R3b)}& {\displaystyle }{\mathrm{RO}}_{\mathrm{2}}+{\mathrm{NO}}_{\mathrm{2}}\to \mathrm{RO}+{\mathrm{NO}}_{\mathrm{3}}.\end{array}$$

These cases include HC(O)C(O)O₂ (Orlando and Tyndall, 2001), CH₃C(O)C(O)O₂ (Jagiella and Zabel, 2008) and the phenylperoxy radical, C₆H₅O₂ (Jagiella and Zabel, 2007). Reaction (R3b) is therefore applied generally to HC(O)C(O)O₂, RC(O)C(O)O₂, C₆H₅O₂ and other aryl peroxy radicals, using the generic rate coefficient k_f PAN.

2.3 The reaction of RO₂ with NO₃

On the basis of reported information for CH₃O₂ and C₂H₅O₂ (e.g. Biggs et al., 1995; Kukui et al., 1995), the reactions of RO₂ with NO₃ are assumed to proceed via a single channel in each case as follows:

$$\begin{array}{}\text{(R4)}& {\displaystyle }{\mathrm{RO}}_{\mathrm{2}}+{\mathrm{NO}}_{\mathrm{3}}\to \mathrm{RO}+{\mathrm{NO}}_{\mathrm{2}}+{\mathrm{O}}_{\mathrm{2}}.\end{array}$$

Reported rate coefficients are summarized in Table 5. The reaction of C₂H₅O₂ with NO₃ is the most studied, with consistent 298 K rate coefficients reported in a number of studies (Biggs et al., 1995; Ray et al., 1996; Vaughan et al., 2006; Laversin et al., 2016) and with the temperature dependence systematically investigated (Laversin et al., 2016). The corresponding parameters in Table 5 therefore form the basis of a generic rate coefficient for the reactions of non-acyl peroxy radicals with NO₃:

$$\begin{array}{ll}{\displaystyle }{k}_{{\mathrm{RO}}_{\mathrm{2}}{\mathrm{NO}}_{\mathrm{3}}}=& {\displaystyle }\mathrm{8.9}\times {\mathrm{10}}^{-\mathrm{12}}\\ \text{(5)}& {\displaystyle }& {\displaystyle }\exp (-\mathrm{390}/T){\mathrm{cm}}^{\mathrm{3}}{\mathrm{molecule}}^{-\mathrm{1}}{\mathrm{s}}^{-\mathrm{1}}.\end{array}$$

Within the reported uncertainties, the value of the rate coefficient at 298 K is consistent with that for c-C₆H₁₁O₂ and with the approximate value for c-C₅H₉O₂ reported by Vaughan et al. (2006); and the temperature dependence expression for $k_{{\mathrm{RO}}_{\mathrm{2}}{\mathrm{NO}}_{\mathrm{3}}}$ is consistent with those reported for the oxygenated primary peroxy radicals, (CH₃)₂C(OH)CH₂O₂, CH₃OCH₂O₂ and CH₃C(O)CH₂O₂, by Kalalian et al. (2018). $k_{{\mathrm{RO}}_{\mathrm{2}}{\mathrm{NO}}_{\mathrm{3}}}$ is therefore currently considered appropriate for application to all ≥C₂ non-acyl peroxy radicals. For CH₃O₂, the reaction has been well studied at 298 K, and the value in Table 5 is applied in conjunction with the $k_{{\mathrm{RO}}_{\mathrm{2}}{\mathrm{NO}}_{\mathrm{3}}}$ pre-exponential factor, leading to

$$\begin{array}{ll}{\displaystyle }k({\mathrm{CH}}_{\mathrm{3}}{\mathrm{O}}_{\mathrm{2}}+{\mathrm{NO}}_{\mathrm{3}})=& {\displaystyle }\mathrm{8.9}\times {\mathrm{10}}^{-\mathrm{12}}\\ \text{(6)}& {\displaystyle }& {\displaystyle }\exp (-\mathrm{600}/T){\mathrm{cm}}^{\mathrm{3}}{\mathrm{molecule}}^{-\mathrm{1}}{\mathrm{s}}^{-\mathrm{1}}.\end{array}$$

The generic rate coefficient for acyl peroxy radicals is based on data for CH₃C(O)O₂, which have been shown to react slightly more rapidly with NO₃ (Canosa-Mas et al., 1996; Doussin et al., 2003). The value at 298 K in Table 5 (based on that reported by Doussin et al., 2003) is once again applied in conjunction with the $k_{{\mathrm{RO}}_{\mathrm{2}}{\mathrm{NO}}_{\mathrm{3}}}$ pre-exponential factor, leading to

$$\begin{array}{ll}{\displaystyle }{k}_{{\mathrm{APNO}}_{\mathrm{3}}}=& {\displaystyle }\mathrm{8.9}\times {\mathrm{10}}^{-\mathrm{12}}\\ \text{(7)}& {\displaystyle }& {\displaystyle }\exp (-\mathrm{305}/T){\mathrm{cm}}^{\mathrm{3}}{\mathrm{molecule}}^{-\mathrm{1}}{\mathrm{s}}^{-\mathrm{1}}.\end{array}$$

The resultant weak temperature dependence yields a value of $k_{{\mathrm{APNO}}_{\mathrm{3}}}$ in the range 403–443 K that is fully consistent with that reported by Canosa-Mas et al. (1996).

2.4 The reaction of RO₂ with OH

Kinetics determinations have been reported for the reactions of OH with C₁–C₄ alkyl peroxy radicals. As shown in Table 6, these reactions are reported to occur rapidly at room temperature, with the rate coefficients for all the reactions being essentially equivalent at 298 K, within the reported uncertainties. Based on the study by Yan et al. (2016), a weak temperature dependence is recommended for the reaction of CH₃O₂ with OH, and the resultant expression,

$$\begin{array}{}\text{(8)}& {\displaystyle }{k}_{{\mathrm{RO}}_{\mathrm{2}}\mathrm{OH}}=\mathrm{3.7}\times {\mathrm{10}}^{-\mathrm{11}}\exp (\mathrm{350}/T){\mathrm{cm}}^{\mathrm{3}}{\mathrm{molecule}}^{-\mathrm{1}}{\mathrm{s}}^{-\mathrm{1}},\end{array}$$

is also adopted in the present work as a generic rate coefficient for the reactions of RO₂ with OH.

Table 6 Kinetic data for the reactions of peroxy radicals with OH. Where available, the temperature dependence is given by $k=A\exp (-E/RT)$.

Comments: ^a IUPAC Task Group recommendation (http://iupac.pole-ether.fr/, last access: January 2019) based on Assaf et al. (2016) and Yan et al. (2016) with an uncertainty factor of 2 assigned to k_298 K; ^b IUPAC Task Group recommendation (http://iupac.pole-ether.fr/, last access: January 2019) based on Faragó et al. (2015) with an uncertainty factor of 1.6 assigned to k_298 K. A consistent value of $k_{\mathrm{298}\mathrm{K}}=(\mathrm{1.3}\pm \mathrm{0.3})\times {\mathrm{10}}^{-\mathrm{10}}$ cm³ molecule⁻¹ s⁻¹ has more recently been reported by Assaf et al. (2017b); ^c Global rate coefficients, $k_{\mathrm{298}\mathrm{K}}=(\mathrm{1.4}\pm \mathrm{0.3})\times {\mathrm{10}}^{-\mathrm{10}}$ and ($\mathrm{1.5}\pm \mathrm{0.3})\times {\mathrm{10}}^{-\mathrm{10}}$ cm³ molecule⁻¹ s⁻¹, reported by Assaf et al. (2017b) for isomeric mixtures of peroxy radicals formed from reactions of Cl atoms with propane and butane, respectively.

The following product channels are considered, but with their branching ratios being strongly dependent on the size of R:

$$\begin{array}{}\text{(R5a)}& {\displaystyle }& {\displaystyle }{\mathrm{RO}}_{\mathrm{2}}+\mathrm{OH}\to \mathrm{RO}+{\mathrm{HO}}_{\mathrm{2}},\text{(R5b)}& {\displaystyle }& {\displaystyle }{\mathrm{RO}}_{\mathrm{2}}+\mathrm{OH}\to \mathrm{ROH}+{\mathrm{O}}_{\mathrm{2}},\text{(R5c)}& {\displaystyle }& {\displaystyle }{\mathrm{RO}}_{\mathrm{2}}+\mathrm{OH}(+M)\to \mathrm{ROOOH}(+M).\end{array}$$

In their theoretical studies of the reaction of CH₃O₂ with OH, Bian et al. (2015), Müller et al. (2016) and Assaf et al. (2018) calculated reaction channel (R5a) to be the most favourable, with experimental confirmation of a dominant contribution from this channel reported for CH₃O₂ by Assaf et al. (2017a, 2018). A number of alternative channels have been considered in modelling assessments (e.g. Archibald et al., 2009), including formation of CH₂O₂ and H₂O or CH₃OH and O₂. However, no evidence for formation of CH₂O₂ and H₂O has been observed at room temperature, indicating that this product channel is at most minor (<5 %) (Yan et al., 2016; Assaf et al., 2017a; Caravan et al., 2018); this is also consistent with theoretical data (e.g. Müller et al., 2016). The formation of CH₃OH and O₂ via channel (R5b) has been shown to make a minor contribution (6 %– 9 %) in the experimental study of Caravan et al. (2018), consistent with the theoretical estimate of ∼7 % by Müller et al. (2016). It is noted that Caravan et al. (2018) also reported evidence for minor CH₃OOOH formation at atmospheric pressure, via channel (R5c); although this was calculated to be formed with a low yield (1.7 %) by Assaf et al. (2018). As a result, values of $k_{\mathrm{5}\mathrm{a}}/k_{\mathrm{5}}=\mathrm{0.93}$ and $k_{\mathrm{5}\mathrm{b}}/k_{\mathrm{5}}=\mathrm{0.07}$ are currently assigned to the reaction of CH₃O₂ with OH in the present work.

The experimental and theoretical study of Assaf et al. (2018) for a series of C₁–C₄ alkyl peroxy radicals has demonstrated that the reaction can more generally be regarded as proceeding by either channel (R5a) or (R5c). Formation of the thermalized hydrotrioxide, ROOOH, via channel (R5c) was found to be increasingly important for the larger RO₂. Based approximately on their theoretical calculations for 298 K and 1 atm of pressure, k_5c∕k₅ is thus currently assigned a value of 0.0 for CH₃O₂, 0.8 for RO₂ for which n_CON=2 (e.g. C₂H₅O₂ and HOCH₂O₂) and 1.0 for all other RO₂ radicals. In the n_CON=2 case, the balance of the reaction is assigned to channel (R5a), i.e. with $k_{\mathrm{5}\mathrm{b}}/k_{\mathrm{5}}=\mathrm{0}$. As discussed by Assaf et al. (2018), detailed experimental and theoretical studies of the atmospheric fate of ROOOH are therefore clearly required for the effect of the RO₂+OH reaction to be fully assessed and represented. A provisional treatment is provided in Sect. S3, based mainly on rate coefficients reported in the theoretical studies of Müller et al. (2016), Assaf et al. (2018) and Anglada and Solé (2018).

2.5 The reaction of RO₂ with HO₂

2.5.1 Kinetics

Rate coefficients for the reactions of HO₂ with a variety of specific hydrocarbon and oxygenated RO₂ radicals have been reported, as summarized in Table 7. For the vast majority of the RO₂ radicals formed in detailed mechanisms, however, kinetic data are unavailable, and it is therefore necessary to assign generic rate coefficients based on the reported data.

Table 7 Kinetic data for the reactions of hydrocarbon and oxygenated peroxy radicals with HO₂. Where available, the temperature dependence is given by $k=A\exp (-E/RT)$.

Comments: ^a IUPAC Task Group recommendation (http://iupac.pole-ether.fr/, last access: January 2019). ^b Based on Rowley et al. (1992a, b) and Boyd et al. (2003a). ^c Taken from Boyd et al. (2003a). ^d Mixture of C₁₀H₂₁O₂ radicals derived from the reaction of OH with decane. ^e Mixture of C₁₄H₂₉O₂ radicals derived from the reaction of OH with tetradecane. ^f Approximate value from Boyd et al. (1996a), based on extrapolation of higher temperature data (393–426 K) using assumed value of $E/R=-\mathrm{700}$ K. ^g Taken from Jenkin and Hayman (1995). ^h Derived from the reaction of OH with cyclohexene. RO₂ population dominated by β-hydroxy peroxy radical, HO-c-C₆H₁₀-O₂, formed from OH addition. ⁱ RO₂ population dominated by hydroxy peroxy radicals formed from OH addition to the given monoterpene. ^j Mixture of HOC₅H₈O₂ radicals derived from the reaction of OH with isoprene. ^k Approximate value from Jenkin et al. (1993a), based on steady state concentration of HO₂ formed from the self-reaction of CH₃OCH₂O₂ during modulated photolysis. ^l Based on Bridier et al. (1993). ^m k_298 K based on Groß et al. (2014), Winiberg et al. (2016) and Hui et al. (2019). E∕R based on Hui et al. (2019) (see Sect. S4). ⁿ Based on Roth et al. (2010).

As discussed previously (Jenkin et al., 1997; Saunders et al., 2003; Boyd et al., 2003a; Orlando and Tyndall, 2012; Wennberg et al., 2018), the 298 K rate coefficients tend to increase with the size of the organic group. Figure 2 shows the data plotted as a function of n_CON. The data for alkyl peroxy radicals and β-hydroxyalkyl peroxy radicals (the most systematically studied groups) show comparable values across the n_CON range. Based on optimization to these data, the following expression is derived for application to non-acyl peroxy radicals:

$$\begin{array}{ll}{\displaystyle }{k}_{{\mathrm{RO}}_{\mathrm{2}}{\mathrm{HO}}_{\mathrm{2}}}=& {\displaystyle }\mathrm{2.8}\times {\mathrm{10}}^{-\mathrm{13}}\exp (\mathrm{1300}/T)\\ \text{(9)}& {\displaystyle }& {\displaystyle }[\mathrm{1}-\exp (-\mathrm{0.23}{n}_{\mathrm{CON}}\left)\right]{\mathrm{cm}}^{\mathrm{3}}{\mathrm{molecule}}^{-\mathrm{1}}{\mathrm{s}}^{-\mathrm{1}}\end{array}$$

The temperature dependence is typical of that reported for >C₂ alkyl and β-hydroxy RO₂ radicals and remains unchanged from that applied previously by Saunders et al. (2003).

Figure 2 Rate coefficients for the reactions of various classes of RO₂ radicals with HO₂ as a function of n_CON at 298 K. The heavy broken line is the best fit to the data for alkyl and β-hydroxyalkyl RO₂ on the basis of the assumed function $k=A(\mathrm{1}-\exp (B{n}_{\mathrm{CON}}\left)\right)$. The light broken line is the same function with the 298 K value of k increased by a factor of 1.84 (see Sect. 2.5).

Based on the limited data for acyl peroxy radicals (see Fig. 2 and Table 7), and specifically that for CH₃C(O)O₂, the 298 K rate coefficients are assigned values that are almost a factor of 2 greater than those defined by Eq. (9). The temperature dependences reported for acyl peroxy radicals appear to be weaker than those for similar-sized radicals in other classes, and the temperature coefficient is again based on that recommended for CH₃C(O)O₂. The following expression is therefore assigned to acyl peroxy radicals:

$$\begin{array}{ll}{\displaystyle }{k}_{{\mathrm{APHO}}_{\mathrm{2}}}=& {\displaystyle }\mathrm{3.5}\times {\mathrm{10}}^{-\mathrm{12}}\exp (\mathrm{730}/T)\\ \text{(10)}& {\displaystyle }& {\displaystyle }[\mathrm{1}-\exp (-\mathrm{0.23}{n}_{\mathrm{CON}}\left)\right]{\mathrm{cm}}^{\mathrm{3}}{\mathrm{molecule}}^{-\mathrm{1}}{\mathrm{s}}^{-\mathrm{1}}.\end{array}$$

2.5.2 Product branching ratios

On the basis of reported information, the following channels are considered for the reactions of RO₂ with HO₂:

$$\begin{array}{}\text{(R6a)}& {\displaystyle }{\mathrm{RO}}_{\mathrm{2}}+{\mathrm{HO}}_{\mathrm{2}}& {\displaystyle }\to \mathrm{ROOH}+{\mathrm{O}}_{\mathrm{2}},\text{(R6b)}& {\displaystyle }& {\displaystyle }\to \mathrm{ROH}+{\mathrm{O}}_{\mathrm{3}},\text{(R6c)}& {\displaystyle }& {\displaystyle }\to {\mathrm{R}}_{-\mathrm{H}}=\mathrm{O}+{\mathrm{H}}_{\mathrm{2}}\mathrm{O}+{\mathrm{O}}_{\mathrm{2}},\text{(R6d)}& {\displaystyle }& {\displaystyle }\to \mathrm{RO}+\mathrm{OH}+{\mathrm{O}}_{\mathrm{2}},\text{(R6e)}& {\displaystyle }& {\displaystyle }\to {\mathrm{R}}_{-\mathrm{H}}=\mathrm{O}+\mathrm{OH}+{\mathrm{HO}}_{\mathrm{2}}.\end{array}$$

Formation of a hydroperoxide product (ROOH) and O₂ via terminating channel (R6a) is reported to be dominant for reactions of alkyl peroxy radicals, and this is also taken to be the default where no information is available (see Table 8). However, the reactions of HO₂ with oxygenated peroxy radicals have received considerable attention and evidence has been reported for several additional channels leading to both radical termination, Reactions (R6b) and (R6c), and radical propagation, Reactions (R6d) and (R6e). Table 8 summarizes the 298 K branching ratios that are applied to several classes of oxygenated peroxy radical based on reported information.

Table 8 Branching ratios assigned to reaction channels (R6a)–(R6e) for reactions of hydrocarbon and oxygenated peroxy radical classes with HO₂ at 298 K.

Comments: ^a Based on studies of CH₃O₂ and C₂H₅O₂ (as summarized by Orlando and Tyndall, 2012), and also used as a default in all cases other than those covered by comments ${}^{\mathrm{b}--\mathrm{i}}$. ^b Based on studies of CH₃C(O)O₂ (Niki et al., 1985; Horie and Moortgat, 1992; Hasson et al., 2004; Jenkin et al., 2007; Dillon and Crowley, 2008; Groß et al., 2014; Winiberg et al., 2016; Hui et al., 2019); see Sect. S4. Hasson et al. (2012) also reported broadly comparable branching ratios for C₂H₅C(O)O₂ and C₃H₇C(O)O₂. ^c Based on studies of C₆H₅C(O)O₂. k_6d∕k₆ based on Dillon and Crowley (2008) and Roth et al. (2010), with other branching ratios based on those reported by Roth et al. (2010). ^d Based on studies of CH₃C(O)CH₂O₂ (Jenkin et al., 2008; Dillon and Crowley, 2008; Hasson et al., 2012). ^e Based on studies of CH₃C(O)CH(O₂)CH₃ (Dillon and Crowley, 2008; Hasson et al., 2012) and of CH₃C(O)CH(O₂)CH₂OH Praske et al. (2015). Praske et al. (2015) also reported possible minor contribution of channel (R6e) and/or (R6c) for CH₃C(O)CH(O₂)CH₂OH. ^f Based on studies of HOCH₂O₂ (Jenkin et al., 2007) and CH₃OCH₂O₂ (Jenkin et al., 2010). Contribution of OH formation in those studies was originally attributed to channel (R6d), but is allocated here to channel (R6e) on the basis of the theoretical study of Nguyen et al. (2010). ^g Full reaction is assigned to channel (R6a), because channels (R6c) and (R6e) are unavailable for tertiary radicals owing to the absence of an α-H atom. ^h Based on study of hydroxyallylic peroxy radicals formed in isoprene system by Liu et al. (2013), with support from the studies of Paulot et al. (2009) and Navarro et al. (2013). ⁱ Based on study of nitrooxyallylic radicals formed in isoprene system by Schwantes et al. (2015).

The temperature dependences of the reaction channels have generally not been studied, and the branching ratios in Table 8 are thus applied independent of temperature in most cases. The only exception is the reaction of HO₂ with (non-aryl) acyl peroxy radicals. This class of reaction (in particular the reaction of HO₂ with CH₃C(O)O₂) has received the most attention and is also a class for which radical propagation is reported to be particularly important at temperatures near 298 K. As shown in Table 8, channels (R6a), (R6b) and (R6d) are reported to contribute. The temperature dependence of k_6d∕k is based on the recent study of the CH₃C(O)O₂ + HO₂ reaction reported by Hui et al. (2019). The contributions and temperature dependences of k_6a∕k and k_6b∕k also take account of the wider database for the same reaction, in particular the experimental characterization of k_6a∕k_6b reported by Horie and Moortgat (1992). This procedure (described in detail in Sect. S4) results in the following fitted Arrhenius expressions for the individual channel rate coefficients:

$$\begin{array}{ll}{\displaystyle }{k}_{\mathrm{6}\mathrm{a}{\mathrm{APHO}}_{\mathrm{2}}}=& {\displaystyle }\mathrm{3.00}\times {\mathrm{10}}^{-\mathrm{12}}\exp (\mathrm{480}/T)\\ \text{(11)}& {\displaystyle }& {\displaystyle }[\mathrm{1}-\exp (-\mathrm{0.23}{n}_{\mathrm{CON}}\left)\right],{\displaystyle }{k}_{\mathrm{6}\mathrm{b}{\mathrm{APHO}}_{\mathrm{2}}}=& {\displaystyle }\mathrm{8.83}\times {\mathrm{10}}^{-\mathrm{15}}\\ \text{(12)}& {\displaystyle }& {\displaystyle }\exp (\mathrm{1910}/T)[\mathrm{1}-\exp (-\mathrm{0.23}{n}_{\mathrm{CON}}\left)\right],{\displaystyle }{k}_{\mathrm{6}\mathrm{d}{\mathrm{APHO}}_{\mathrm{2}}}=& {\displaystyle }\mathrm{9.35}\times {\mathrm{10}}^{-\mathrm{12}}\\ \text{(13)}& {\displaystyle }& {\displaystyle }\exp (\mathrm{235}/T)[\mathrm{1}-\exp (-\mathrm{0.23}{n}_{\mathrm{CON}}\left)\right].\end{array}$$

The corresponding temperature dependences of the channel rate coefficients, derived from the CH₃C(O)O₂ data, are thus applied to all (non-aryl) acyl peroxy radicals. The variation of the branching ratios and channel rate coefficients are illustrated for the CH₃C(O)O₂ + HO₂ reaction in Figs. S2 and S3 for the 230–300 K temperature range. Summation of the channel rate coefficients given in Eqs. (11)–(13) reproduces the values of $k_{{\mathrm{APHO}}_{\mathrm{2}}}$ calculated for the overall reaction using Eq. (10) to within 5 % over this temperature range (see Sect. S4 for further details).

2.6 The permutation reactions of RO₂

The “permutation” reactions of a given RO₂ radical are its self-reactions (R7) and its cross-reactions (R8) with other peroxy radicals, R^′O₂, for which a number of product channels may occur:

$$\begin{array}{}\text{(R7a)}& {\displaystyle }{\mathrm{RO}}_{\mathrm{2}}+{\mathrm{RO}}_{\mathrm{2}}& {\displaystyle }\to \mathrm{RO}+\mathrm{RO}+{\mathrm{O}}_{\mathrm{2}},\text{(R7b)}& {\displaystyle }& {\displaystyle }\to {\mathrm{R}}_{-\mathrm{H}}=\mathrm{O}+\mathrm{ROH}+{\mathrm{O}}_{\mathrm{2}},\text{(R7c)}& {\displaystyle }& {\displaystyle }\to \mathrm{ROOR}+{\mathrm{O}}_{\mathrm{2}};\end{array}$$

$$\begin{array}{}\text{(R8a)}& {\displaystyle }{\mathrm{RO}}_{\mathrm{2}}+{\mathrm{R}}^{\prime }{\mathrm{O}}_{\mathrm{2}}& {\displaystyle }\to \mathrm{RO}+{\mathrm{R}}^{\prime }\mathrm{O}+{\mathrm{O}}_{\mathrm{2}},\text{(R8b)}& {\displaystyle }& {\displaystyle }\to {\mathrm{R}}_{-\mathrm{H}}=\mathrm{O}+{\mathrm{R}}^{\prime }\mathrm{OH}+{\mathrm{O}}_{\mathrm{2}},\text{(R8c)}& {\displaystyle }& {\displaystyle }\to {\mathrm{R}}_{-\mathrm{H}}^{\prime }=\mathrm{O}+\mathrm{ROH}+{\mathrm{O}}_{\mathrm{2}}.\text{(R8d)}& {\displaystyle }& {\displaystyle }\to {\mathrm{ROOR}}^{\prime }+{\mathrm{O}}_{\mathrm{2}}\end{array}$$

In view of the large number of RO₂ radicals generated in a detailed chemical mechanism, however, it is unrealistic to represent these reactions explicitly, and the use of simplified parameterizations is essential (Madronich and Calvert, 1990). As described in detail previously (Jenkin et al., 1997), a very simplified approach has traditionally been adopted in the MCM, in which each peroxy radical is assumed to react with all other peroxy radicals (i.e. the peroxy radical “pool”) at a single, collective rate. This is achieved by defining a parameter “∑[RO₂]” which is the sum of the concentrations of all peroxy radicals, excluding HO₂. The collective rate of all the permutation reactions of a particular peroxy radical is then represented by a single pseudo-unimolecular reaction, which has an assigned rate coefficient equal to $k_{\mathrm{9}}\times \sum$[RO₂],

$$\begin{array}{}\text{(R9)}& {\displaystyle }{\mathrm{RO}}_{\mathrm{2}}\to \text{products},\end{array}$$

with the value of k₉ depending on the structure of the reacting RO₂ radical. A similar, but more detailed, approach has been applied in GECKO-A, in which the peroxy radical population is divided into a number of reactivity classes (Aumont et al., 2005). This requires the inclusion of a pseudo-unimolecular reaction (analogous to Reaction R9) for reaction of a given peroxy radical with each peroxy radical class, but has the advantage that differential reactivity with each of those classes can be represented, as appropriate. The following subsections describe the basis for assigning rate parameters to the single parameterized permutation reactions (Reaction R9) for each peroxy radical in the more simplified MCM approach. Extension of the method to reactions with a number of reactivity classes (as traditionally applied with GECKO-A) is described in Sect. S5.

2.6.1 Kinetics of self-reactions

Rate coefficients for the self-reactions and cross-reactions of a variety of specific hydrocarbon and oxygenated RO₂ radicals have been reported (as summarized in Tables 9–11), and these form the basis of assigning rate parameters to the parameterized permutation reaction (Reaction R9) for each peroxy radical. The data show that the self-reaction reactivity, relative to that of alkyl peroxy radicals, is activated by the presence of numerous functional groups (including allyl-, benzyl-, hydroxy-, alkoxy-, oxo- and acyl-groups), and that the rate coefficients follow the general trend of decreasing reactivity, primary > secondary > tertiary, for peroxy radicals containing otherwise similar functionalities. It also appears that reactivity tends to increase with the size of the organic group towards a “plateau” value, as most clearly demonstrated by the systematic study of secondary alkyl peroxy radicals reported by Boyd et al. (1999). Based on optimization to the complete secondary alkyl peroxy radical dataset, an expression almost identical to that recommended by Boyd et al. (1999) is thus derived as a reference rate coefficient for secondary peroxy radicals at 298 K, as illustrated in Fig. 3 (units of k are cm³ molecule⁻¹ s⁻¹):

$$\begin{array}{ll}{\displaystyle }{\log }_{\mathrm{10}}\left(k{{}^{\circ }}_{{\mathrm{RO}}_{\mathrm{2}}{\mathrm{RO}}_{\mathrm{2}}\left(\sec \right)}\right)=& {\displaystyle }-\mathrm{12.9}-(\mathrm{3.2}\times \\ \text{(14)}& {\displaystyle }& {\displaystyle }\exp [-\mathrm{0.64}(n_{\mathrm{CON}}-\mathrm{2.3}\left)\right]).\end{array}$$

The data for primary alkyl peroxy radicals are more limited. Those for C₂H₅O₂, n-C₃H₇O₂, i-C₄H₉O₂ and neo-C₅H₁₁O₂ suggest a similar trend for primary alkyl peroxy radicals, and an analogous expression to Eq. (14) is therefore derived as a reference rate coefficient at 298 K (see Fig. 3):

$$\begin{array}{ll}{\displaystyle }{\log }_{\mathrm{10}}\left(k{{}^{\circ }}_{{\mathrm{RO}}_{\mathrm{2}}{\mathrm{RO}}_{\mathrm{2}}\left(\mathrm{prim}\right)}\right)=& {\displaystyle }-\mathrm{11.7}-(\mathrm{3.2}\times \\ \text{(15)}& {\displaystyle }& {\displaystyle }\exp [-\mathrm{0.55}(n_{\mathrm{CON}}-\mathrm{0.52}\left)\right]).\end{array}$$

It is noted, however, that rate coefficients for the self-reactions of n-C₄H₉O₂ and n-C₅H₁₁O₂ are reported to be comparable to that for n-C₃H₇O₂, and a factor of 2 to 3 lower than those for i-C₄H₉O₂ and neo-C₅H₁₁O₂ (see Table 9), suggesting that there may be sensitivity to whether the alkyl group is linear or branched. In the absence of additional data (and noting that the kinetics of neo-C₅H₁₁O₂ were the most directly determined of the set of C₄ and C₅ primary alkyl peroxy radicals), the above (stronger) size dependence is provisionally applied here.

Table 9 Kinetic data for the self-reactions of hydrocarbon peroxy radicals. Where available, the temperature dependence is given by $k=A\exp (-E/RT)$.

Comments: ^* Reported rate coefficient not corrected for the effects of secondary chemistry, which can lead to either an overestimate or underestimate of the rate coefficient. ^a IUPAC Task Group recommendation (http://iupac.pole-ether.fr/, last access: January 2019). ^b Taken from Glover and Miller (2005). ^c Taken from Boyd et al. (1999). ^d Based on Lightfoot et al. (1990). ^e Mixture of 2-pentyl and 3-pentyl peroxy radicals. ^f Mixture of secondary peroxy radicals of given formula. ^g Based Rowley et al. (1991, 1992c). ^h Based on Jenkin et al. (1993b) and Boyd et al. (1996a).

Figure 3 Rate coefficients for the self-reactions of alkyl (filled points), β-hydroxyalkyl (open points) and allylic (×) RO₂ at 298 K as a function of n_CON. Grey filled points indicate where the reported rate coefficient has not been corrected for secondary chemistry. Where available, data are shown for primary, secondary and tertiary radicals containing the given functionalities. Primary, secondary and tertiary alkyl and β-hydroxyalkyl radicals are shown as diamonds, circles and triangles, respectively. The “allylic” peroxy radical group contains only primary radicals and includes “δ-hydroxyallylic” peroxy radicals. The lines represent the calculated rate coefficients fitted to the data using the methods described in Sect. 2.6.

Table 10 Kinetic data for the self-reactions of oxygenated peroxy radicals. Where available, the temperature dependence is given by $k=A\exp (-E/RT)$.

Comments: ^a IUPAC Task Group recommendation (http://iupac.pole-ether.fr/, last access: January 2019). ^b Taken from on Boyd et al. (2003b). ^c Based on Jenkin and Hayman (1995) and Boyd and Lesclaux (1997). ^d Taken from Jenkin et al. (1998). ^e Based on Le Crâne and Lesclaux (2006). ^f Based on Tomas and Lesclaux (2000) and Le Crâne et al. (2004). ^g Taken from Hansen et al. (2003). The kinetics of the two peroxy radicals formed from the reaction of Cl or F with methyl formate reported to possess indistinguishable kinetics.

Table 11 Kinetic data for the cross-reactions of hydrocarbon or oxygenated peroxy radicals at 298 K. Where available, the temperature dependence expression is given in the comments.

Comments: ^a IUPAC Task Group recommendation (http://iupac.pole-ether.fr/, last access: January 2019). ^b Taken from Villenave and Lesclaux (1996). ^c Based on Villenave and Lesclaux (1996) and Nozière and Hanson (2017). ^d Temperature dependence expression is $\mathrm{3.8}\times {\mathrm{10}}^{-\mathrm{13}}\exp (-\mathrm{1430}/T)$. ^e Temperature dependence expression is $\mathrm{2.8}\times {\mathrm{10}}^{-\mathrm{13}}\exp (\mathrm{515}/T)$. ^f Temperature dependence expression is $\mathrm{2.0}\times {\mathrm{10}}^{-\mathrm{12}}\exp (\mathrm{500}/T)$. ^g Taken from Villenave et al. (1998). ^h Taken from Boyd et al. (2003b). The structures refer to the isomeric secondary and tertiary peroxy radicals formed from the addition of OH to 1-methylcyclohexene. ⁱ Taken from Jenkin et al. (1998). Presented values are limited to those reported for the cross-reactions of the major radicals formed from the terminal addition of OH to buta-1,3-diene and the terminal addition of OH to 2,3-dimethyl-buta-1,3-diene.

Data for tertiary alkyl peroxy radicals are currently limited to t-C₄H₉O₂, and the corresponding rate coefficient is currently applied as the reference rate coefficient at 298 K, independent of radical size (see Fig. 3):

$$\begin{array}{}\text{(16)}& {\displaystyle }k{{}^{\circ }}_{{\mathrm{RO}}_{\mathrm{2}}{\mathrm{RO}}_{\mathrm{2}}\left(\mathrm{tert}\right)}=\mathrm{2.1}\times {\mathrm{10}}^{-\mathrm{17}}{\mathrm{cm}}^{\mathrm{3}}{\mathrm{molecule}}^{-\mathrm{1}}{\mathrm{s}}^{-\mathrm{1}}.\end{array}$$

Figure 3 also shows data for allylic and β-hydroxyalkyl RO₂, demonstrating that the presence of both these functionalities has an activating effect on self-reaction reactivity. The allylic peroxy radical category includes two δ-hydroxyallylic peroxy radicals, and the assumption is made here that the δ-hydroxy group is too remote to have an influence. Table 12 summarizes a series of activation factors (defined in terms of the parameters α and β) for allylic-, β-aryl-, hydroxy-, alkoxy- and oxo-groups, optimized on the basis of the data in Tables 9 and 10. These are used in conjunction with the reference rate coefficients in Eqs. (14)–(16) to calculate the self-reaction rate coefficient for a given peroxy radical at 298 K, $k_{{\mathrm{RO}}_{\mathrm{2}}{\mathrm{RO}}_{\mathrm{2}}}$, as follows:

$$\begin{array}{ll}{\displaystyle }{k}_{{\mathrm{RO}}_{\mathrm{2}}{\mathrm{RO}}_{\mathrm{2}}}=& {\displaystyle }k{{}^{\circ }}_{{\mathrm{RO}}_{\mathrm{2}}{\mathrm{RO}}_{\mathrm{2}}}\times \mathit{\alpha }/(k{{}^{\circ }}_{{\mathrm{RO}}_{\mathrm{2}}{\mathrm{RO}}_{\mathrm{2}}}{)}^{\mathit{\beta }},\\ \text{(17)}& {\displaystyle }& {\displaystyle }=\mathit{\alpha }\times (k{{}^{\circ }}_{{\mathrm{RO}}_{\mathrm{2}}{\mathrm{RO}}_{\mathrm{2}}}{)}^{\mathrm{1}-\mathit{\beta }}.\end{array}$$

Here, $k{{}^{\circ }}_{{\mathrm{RO}}_{\mathrm{2}}{\mathrm{RO}}_{\mathrm{2}}}$ represents the appropriate reference rate coefficient (i.e. for primary, secondary or tertiary RO₂, as appropriate) as defined by Eqs. (14)–(16); and the term $\mathit{\alpha }/(k{{}^{\circ }}_{{\mathrm{RO}}_{\mathrm{2}}{\mathrm{RO}}_{\mathrm{2}}}{)}^{\mathit{\beta }}$ describes the level of activation from the given substituent. The inclusion of $k{{}^{\circ }}_{{\mathrm{RO}}_{\mathrm{2}}{\mathrm{RO}}_{\mathrm{2}}}$ within this activation term is required because the relative enhancement of reactivity resulting from a given substituent appears to decrease as the reactivity increases, as illustrated for the β-hydroxyalkyl group data in Fig. 3. Based on this method, the estimated rate coefficients correlate well with those observed for the series of peroxy radicals for which data are currently available (summarized in Tables 9 and 10), as shown in Fig. 4. It is emphasized, however, that the parameters for several of the substituent groups are based on data for very limited sets of peroxy radicals, and additional data would be valuable to test and constrain the method.

Table 12 Substituent activation factors applied to self-reaction rate coefficients, based on Eq. (17).

Comments: ^a α=1.00 and β=0 by definition for alkyl peroxy radicals. These are also used as a default for peroxy radical classes not covered by comments ^b to ⁱ, with the exception of acyl peroxy radicals (discussed in Sect. 2.6); ^b Based on data for β-hydroxyalkyl peroxy radicals in Table 10; ^c Based on data for allylic and δ-hydroxyallylic peroxy radicals in Tables 9 and 10; ^d Based on data for C₆H₅CH₂O₂ (Table 9); ^e β assumed equivalent to that for allylic substituent; ^f Based on data for CH₃OCH₂O₂ (Table 10); ^g β assumed equivalent to that for β-hydroxy substituent; ^h Based on data for CH₃C(O)CH₂O₂ (Table 10); ⁱ Based on data for γ-oxoalkyl peroxy radicals in Table 10.

Information on the effects of multiple substituents is limited to the data for the secondary and tertiary β-hydroxyallylic peroxy radicals, HOCH₂CH(O₂)CH=CH₂ and HOCH₂C(CH₃)(O₂)C(CH₃)=CH₂, given in Table 10. The reported rate coefficients are consistent with the activating impacts of the β-hydroxy and allylic substituents being approximately cumulative, suggesting that an activation factor should be applied for each relevant organic substituent. However, this would lead to unreasonably large estimated values of $k_{{\mathrm{RO}}_{\mathrm{2}}{\mathrm{RO}}_{\mathrm{2}}}$ for secondary and tertiary peroxy radicals containing two or three of the most activating substituents, such that the impact needs to be limited. In multifunctional peroxy radicals, therefore, an activating factor is only applied for the most activating oxygenated substituent in a given peroxy radical, with an additional factor also applied only for the specific cases of an allylic or a β-aryl substituent, again limited to one (i.e. the most activating) factor if the peroxy radical contains more than one allylic or β-aryl group. In these specific cases, therefore,

$$\begin{array}{}\text{(18)}& {\displaystyle }& {\displaystyle }\mathit{\alpha }=({\mathit{\alpha }}_{\mathrm{1}}\times {\mathit{\alpha }}_{\mathrm{2}}),\text{(19)}& {\displaystyle }& {\displaystyle }\mathit{\beta }=({\mathit{\beta }}_{\mathrm{1}}+{\mathit{\beta }}_{\mathrm{2}}),\end{array}$$

where α₁ and β₁ refer to the oxygenated substituent, and α₂ and β₂ refer to either the allylic substituent or the β-aryl substituent. Further information is required to allow the impacts of multiple substituents to be defined more rigorously.

2.6.2 Parameterized representation

The rate coefficients for cross-reactions of peroxy radicals (Reaction R8) have often been inferred from those for the self-reactions of the participating peroxy radicals, using a geometric mean rule as first suggested by Madronich and Calvert (1990), i.e.

$$\begin{array}{}\text{(20)}& {\displaystyle }{k}_{\mathrm{8}}=\mathrm{2}\times (k_{\mathrm{7}}\times k_{{\mathrm{7}}^{\prime }}{)}^{\mathrm{0.5}},\end{array}$$

where k₈ is the cross-reaction rate coefficient, and k₇ and $k_{{\mathrm{7}}^{\prime }}$ are the self-reaction rate coefficients for the participating peroxy radicals, RO₂ and R^′O₂. Figure 5 shows that such a correlation provides a reasonable guide in many cases (although a clear deviation from the rule occurs for the particular case of reactions involving acyl peroxy radicals). In the very simplified MCM approach, the rate coefficient for the single parameterized permutation reaction of a given peroxy radical (Reaction R9) is based on that estimated for the cross-reaction of the peroxy radical with CH₃O₂. This is regarded as a logical choice because CH₃O₂ is the most abundant organic peroxy radical in the atmosphere (and therefore most commonly the major reaction partner), and also possesses a self-reaction rate coefficient that is in the middle of the range of reported values (see Tables 9 and 10). Taking account of the correlations in Fig. 5, the rate coefficients (in cm³ molecule⁻¹ s⁻¹) for the parameterized permutation reactions at 298 K are defined as follows.

$$\begin{array}{ll}\text{(21)}& {\displaystyle }& {\displaystyle }\text{For acyl}{\mathrm{RO}}_{\mathrm{2}},k_{\mathrm{AP}\left(\mathrm{298}\mathrm{K}\right)}=\mathrm{1.1}\times {\mathrm{10}}^{-\mathrm{11}}.{\displaystyle }& {\displaystyle }\text{For other}{\mathrm{RO}}_{\mathrm{2}}\left(\text{except}{\mathrm{CH}}_{\mathrm{3}}{\mathrm{O}}_{\mathrm{2}}\right),k_{{\mathrm{RO}}_{\mathrm{2}}\left(\mathrm{298}\mathrm{K}\right)}\\ \text{(22)}& {\displaystyle }& {\displaystyle }=f_{{\mathrm{RO}}_{\mathrm{2}}}\times \mathrm{2}\times (k_{{\mathrm{RO}}_{\mathrm{2}}{\mathrm{RO}}_{\mathrm{2}}}\times k_{\mathrm{298}}({\mathrm{CH}}_{\mathrm{3}}{\mathrm{O}}_{\mathrm{2}}+{\mathrm{CH}}_{\mathrm{3}}{\mathrm{O}}_{\mathrm{2}}){)}^{\mathrm{0.5}}.\end{array}$$

Here, k₂₉₈(CH₃O₂+CH₃O₂) is the rate coefficient for the self-reaction of CH₃O₂ at 298 K ($=\mathrm{3.5}\times {\mathrm{10}}^{-\mathrm{13}}$ cm³ molecule⁻¹ s⁻¹) and $k_{{\mathrm{RO}}_{\mathrm{2}}{\mathrm{RO}}_{\mathrm{2}}}$ is the 298 K self-reaction rate coefficient, estimated as described above (Sect. 2.6.1). $f_{{\mathrm{RO}}_{\mathrm{2}}}$ is a scaling factor that is introduced to describe systematic deviations from the geometric mean rule, if required. Based on the correlations in Fig. 5, a unity value of $f_{{\mathrm{RO}}_{\mathrm{2}}}$ is considered acceptable for primary and secondary peroxy radicals (i.e. no deviation from the geometric mean rule), whereas a value of $f_{{\mathrm{RO}}_{\mathrm{2}}}=\mathrm{2}$ is applied to tertiary peroxy radicals. This elevated scaling factor is based on observation of Jenkin et al. (1998) for complex tertiary RO₂ cross-reactions.

Figure 4 Scatter plot of estimated rate coefficients ($k_{{\mathrm{RO}}_{\mathrm{2}}{\mathrm{RO}}_{\mathrm{2}}}$) for peroxy radical self-reactions with those reported (k₇), as listed in Tables 9 and 10. Those shown with reduced size symbols are where the reported value of k₇ was not corrected for secondary chemistry (see Table 9 comments). The broken lines show the factor of 3 range.

Figure 5 Scatter plot of rate coefficients for peroxy radical cross-reactions (k₈) with the geometric mean of the self-reaction rate coefficients (denoted k₇ and $k_{{\mathrm{7}}^{\prime }}$) for the participating peroxy radicals, RO₂ and R^′O₂. Open circles are reactions involving an acyl peroxy radical and a non-acyl peroxy radical; closed circles are reactions involving combinations of primary and secondary peroxy radicals; open diamonds are reactions involving a tertiary peroxy radical and a primary or secondary peroxy radical. The heavy broken line is a 1:1 relationship; the light broken line is a 2:1 relationship; the dot–dash line is $k_{\mathrm{8}}=\mathrm{1.1}\times {\mathrm{10}}^{-\mathrm{11}}$ cm³ molecule⁻¹ s⁻¹.

Based on the reported temperature dependences of peroxy radical self- and cross-reactions (see Tables 9 and 10, and Table 11 comments), k_AP and $k_{{\mathrm{RO}}_{\mathrm{2}}}$ are assigned respective pre-exponential factors of $\mathrm{2.0}\times {\mathrm{10}}^{-\mathrm{12}}$ and $\mathrm{1.0}\times {\mathrm{10}}^{-\mathrm{13}}$ cm³ molecule⁻¹ s⁻¹. For acyl peroxy radicals, this is consistent with the temperature dependence reported for the reaction of CH₃C(O)O₂ with CH₃O₂, and results in the following temperature-dependent expression in all cases:

$$\begin{array}{}\text{(23)}& {\displaystyle }{k}_{\mathrm{AP}}=\mathrm{2.0}\times {\mathrm{10}}^{-\mathrm{12}}\exp (\mathrm{508}/T){\mathrm{cm}}^{\mathrm{3}}{\mathrm{molecule}}^{-\mathrm{1}}{\mathrm{s}}^{-\mathrm{1}}.\end{array}$$

For $k_{{\mathrm{RO}}_{\mathrm{2}}}$, the pre-exponential factor is a rounded value, based on the geometric mean of those for the self-reactions of non-acyl peroxy radicals given in Tables 9 and 10. This results in the following temperature dependence expression for non-acyl peroxy radicals (except CH₃O₂),

$$\begin{array}{ll}{\displaystyle }{k}_{{\mathrm{RO}}_{\mathrm{2}}}=& {\displaystyle }\mathrm{1.0}\times {\mathrm{10}}^{-\mathrm{13}}\\ \text{(24)}& {\displaystyle }& {\displaystyle }\exp (-(E_{{\mathrm{RO}}_{\mathrm{2}}}/R)/T){\mathrm{cm}}^{\mathrm{3}}{\mathrm{molecule}}^{-\mathrm{1}}{\mathrm{s}}^{-\mathrm{1}},\end{array}$$

with $E_{{\mathrm{RO}}_{\mathrm{2}}}/R$ having a case-dependent value of $-\mathrm{298}\times \ln (k_{{\mathrm{RO}}_{\mathrm{2}}\left(\mathrm{298}\mathrm{K}\right)}/{\mathrm{10}}^{-\mathrm{13}})$, where $k_{{\mathrm{RO}}_{\mathrm{2}}\left(\mathrm{298}\mathrm{K}\right)}$ is defined by Eq. (22). Examples of specific rate coefficients estimated using this method are given in Sect. S5 for the peroxy radicals formed from the sequential addition of OH and O₂ to isoprene. As indicated above, the collective rate of all the permutation reactions of a particular peroxy radical is then represented by a pseudo-unimolecular reaction (Reaction R9), which has an assigned rate coefficient equal to $k_{\mathrm{AP}}\times \sum$[RO₂] for acyl peroxy radicals, and $k_{{\mathrm{RO}}_{\mathrm{2}}}\times \sum$[RO₂] for all other peroxy radicals (except CH₃O₂). For the specific case of CH₃O₂, the applied rate coefficient ($k_{{\mathrm{CH}}_{\mathrm{3}}{\mathrm{O}}_{\mathrm{2}}}$) is twice the self-reaction rate coefficient given in Table 9,

$$\begin{array}{ll}{\displaystyle }{k}_{{\mathrm{CH}}_{\mathrm{3}}{\mathrm{O}}_{\mathrm{2}}}=& {\displaystyle }\mathrm{2.06}\times {\mathrm{10}}^{-\mathrm{13}}\\ \text{(25)}& {\displaystyle }& {\displaystyle }\exp (\mathrm{365}/T){\mathrm{cm}}^{\mathrm{3}}{\mathrm{molecule}}^{-\mathrm{1}}{\mathrm{s}}^{-\mathrm{1}},\end{array}$$

with the pseudo-unimolecular reaction rate coefficient equal to $k_{{\mathrm{CH}}_{\mathrm{3}}{\mathrm{O}}_{\mathrm{2}}}\times \sum$[RO₂]. This representation is therefore consistent with CH₃O₂ being lost via its self-reaction with the recommended rate coefficient when it is the dominant radical.

Each reaction potentially has up to four product channels, the branching ratios of which depend on the structure of the radical, as shown in Table 13:

$$\begin{array}{}\text{(R9a)}& {\displaystyle }{\mathrm{RO}}_{\mathrm{2}}& {\displaystyle }\to \mathrm{RO},\text{(R9b)}& {\displaystyle }& {\displaystyle }\to {\mathrm{R}}_{-\mathrm{H}}=\mathrm{O},\text{(R9c)}& {\displaystyle }& {\displaystyle }\to \mathrm{ROH},\text{(R9d)}& {\displaystyle }& {\displaystyle }[\to {\mathrm{RO}}_{\text{(peroxide)}}].\end{array}$$

Channels (R9a)–(R9c) have been considered previously in the MCM (Jenkin et al., 1997; Saunders et al., 2003). They are the pseudo-unimolecular representation of the self-reaction channels (R7a) and (R7b) and the cross-reaction channels (R8a)–(R8c), which are reported to account for most of the reaction, particularly for smaller peroxy radicals (e.g. Lightfoot et al., 1992; Orlando and Tyndall, 2012). As shown in Table 13, channels (R9a)–(R9c) continue to represent the complete reaction in the current parameterized methodology.

Table 13 Branching ratios assigned to parameterized permutation reactions of RO₂ (see text).

Comments: ^a Based on IUPAC Task Group recommendation for the CH₃O₂ self-reaction (http://iupac.pole-ether.fr/, last access: January 2019). An alternative representation using temperature-dependent channel rate coefficients is provided in Sect. S5. ^b Based on a rounded mean of the reported 298 K branching ratios for the self-reactions of C₂H₅O₂, i-C₃H₇O₂, HOCH₂CH₂O₂, (CH₃)₂C(OH)CH₂O₂, CH₃C(O)CH₂O₂, CH₃OCH₂O₂ and C₆H₅CH₂O₂ based on IUPAC Task Group recommendations (http://iupac.pole-ether.fr/, last access: January 2019); neo-C₅H₁₁O₂, c-C₆H₁₁O₂ and CH₂=CHCH₂O₂ based on Lightfoot et al. (1990), Rowley et al. (1991), Jenkin et al. (1993a, b) and Boyd et al. (1996a); and for the self- and cross-reactions of primary and secondary RO₂ formed from reactions of OH with conjugated dienes (Jenkin et al., 1998). ^c Based on a rounded mean of the reported 298 K branching ratios for the following cross-reactions: CH₃C(O)O₂ + CH₃O₂, C₂H₅C(O)O₂ + C₂H₅O₂ and CH₃C(O)O₂ + CH₃C(O)CH₂O₂ based on IUPAC Task Group recommendations (http://iupac.pole-ether.fr/, last access: January 2019); and HOCH₂C(CH₃)(O₂)C(CH₃)=CH₂ + HOCH₂C(CH₃)=C(CH₃)CH₂O₂ formed from reaction of OH with 2,3-dimethyl-buta-1,3-diene (Jenkin et al., 1998).

Although not currently included in the parameterized representation, channel (R9d) is listed to acknowledge the potential formation of peroxide products (i.e. Reactions R7c and R8d). Although these channels have generally been reported to be minor for small peroxy radicals (e.g. Lightfoot et al., 1992; Orlando and Tyndall, 2012), recent studies suggest that they may be more significant for larger peroxy radicals containing oxygenated substituents, and they have been reported to play a role in the formation of low-volatility products in a number of studies (Ziemann, 2002; Ng et al., 2008; Ehn et al., 2014; Jokinen et al., 2014; Mentel et al., 2015; Rissanen et al., 2015; Berndt et al., 2015, 2018a, b; Zhang et al., 2015; McFiggans et al., 2019). These reactions may therefore play a potentially important role in particle formation and growth in the atmosphere. The product denoted “RO_(peroxide)” in the pseudo-unimolecular approach represents the monomeric contribution the given peroxy radical makes to the total formation of (dimeric) peroxide products, but is not an independent species for which subsequent gas-phase chemistry can be rigorously defined. In principle, channel (R9d) can be included for the permutation reactions of a subset of larger peroxy radicals, with the RO_(peroxide) product assumed to transfer completely to the condensed phase (i.e. not participating in gas-phase reactions). However, there is currently insufficient information on the structural dependence of the contributions of channels (R7c) or (R8d) to the overall self- and cross-reactions to allow the branching ratio of channel (R9d) to be defined reliably. Further systematic studies of these channel contributions are therefore required as a function of peroxy radical size and functional group content.

3 Unimolecular reactions of RO₂ radicals

Unimolecular isomerization reactions are potentially available for some classes of RO₂. These generally fall into the category of either ring-closure reactions (where the peroxy radical adds intra-molecularly to an unsaturated linkage to form a peroxide-bridged radical product) or reactions involving the migration of a hydrogen atom to the peroxy radical group (e.g. forming a hydroperoxy-substituted organic radical product when abstraction from a C−H bond occurs). For some RO₂ structures, these reactions have been shown to compete with (or dominate over) the bimolecular reactions under some atmospheric conditions, as discussed further below in Sect. 3.1 and 3.2. Evidence for the operation of peroxy radical isomerization reactions has been reported in numerous theoretical and laboratory studies (e.g. Vereecken and Peeters, 2004; Peeters et al., 2009; 2014; Crounse et al., 2013; Ehn et al., 2014, 2017; Jokinen et al., 2014; Rissanen et al., 2015; Jørgensen et al., 2016; Praske et al., 2017, 2019; Otkjær et al., 2018; Mohammed et al., 2018), and new information is constantly emerging on this important aspect of peroxy radical chemistry (e.g. Bianchi et al., 2019; Xu et al., 2019; Møller et al., 2019). The present section provides a summary of selected classes of isomerization reactions that are currently being considered and represented in ongoing mechanism development work. However, it does not currently attempt to provide a full treatment of unimolecular reactions of RO₂ radicals, which will be considered further in future work as more new information becomes available.

3.1 Ring-closure reactions of RO₂

Table 14 shows representative rate coefficients for selected template ring-closure reactions. The first entry relates to the β-hydroxy cyclohexadienylperoxy radicals formed from the addition of O₂ to OH-aromatic hydrocarbon adducts. As discussed in the companion paper on the OH-initiated oxidation of aromatic VOCs (Jenkin et al., 2018b), these peroxy radicals are represented to undergo rapid and exclusive ring closure to produce a hydroxy-dioxa-bicyclo or “peroxide-bicyclic” radical. This reaction has been calculated to dominate over alternative bimolecular reactions of the peroxy radicals under atmospheric conditions (see Table 14), although evidence for competitive loss via bimolecular reactions has been characterized in experimental studies using high concentrations of NO and/or RO₂ (e.g. Birdsall et al., 2010; Birdsall and Elrod, 2011).

Table 14 Representative rate coefficients for template ring-closure reactions of peroxy radicals^a.

Comments: ^a Temperature dependence of rate coefficient given by $k=A\exp (-(E/R)/T)$. Rapid reaction of the product radical with O₂ dominates over the reverse ring-opening reaction under atmospheric conditions. Entries in bold font are based on reported data for the specific or closely related structures, with other entries inferred using assumptions given in the following comments. ^b Range of 298 K values based on the calculations of Raoult et al. (2004), Glowacki et al. (2009) and Olivella et al. (2009) for the dominant conformer of the example peroxy radical, formed during the oxidation of benzene. Based on these data, and data for other aromatic systems, analogous ring-closure reactions are assumed to be the exclusive fates of corresponding peroxy radicals formed during the oxidation of aromatic hydrocarbons (Jenkin et al., 2018b). ^c Denotes substitution of product radical. ^d Based on information reported by Vereecken and Peeters (2004) for calculations for the given peroxy radical. ^e E∕R for formation of a tertiary radical assumed to be 400 K lower than for formation of a secondary radical, corresponding to a difference in E of ≈3.3 kJ mol⁻¹. This is consistent with differences in energy barriers reported for formation of secondary and tertiary radicals (Vereecken and Peeters, 2004). ^f Based on the calculations of Vereecken and Peeters (2004) for a relevant tertiary peroxy radical formed during the oxidation of isoprene. ^g Based on the calculations of Vereecken and Peeters (2004) for a relevant tertiary peroxy radical formed during the oxidation of α-pinene. Applies specifically to anti-conformers, when the OH and peroxy radical groups on the opposite sides of the ring (as shown), which were calculated to account for 60 % of the anti- + syn-population (Vereecken and Peeters, 2004). ^h Based on the calculations of Vereecken and Peeters (2004) for a relevant tertiary peroxy radical formed during the oxidation of α-pinene. Applies specifically to syn-conformers, when the OH and peroxy radical groups on the same side of the ring (as shown), which were calculated to account for 40 % of the anti- + syn- population (Vereecken and Peeters, 2004). ⁱ Based on the calculations of Vereecken and Peeters (2004) for the a relevant tertiary peroxy radical, formed during the oxidation of β-pinene.

The remaining reactions in Table 14 are based on information presented by Vereecken and Peeters (2004) for specific peroxy radicals formed from the sequential addition of OH and O₂ to isoprene, α-pinene and β-pinene. That information has been used to assign or infer representative rate coefficients to the series of related template peroxy radical structures presented in Table 14. In these cases, the reactions are expected to occur at rates that can compete to varying extents with loss via bimolecular reactions (or other unimolecular reactions discussed below) under atmospheric conditions. It is noted that Xu et al. (2019) have also very recently reported information for a series of isomerization reactions (including ring-closure reactions) for the α- and β-pinene systems, which are being considered in ongoing work.

3.2 Hydrogen atom migration reactions of RO₂

Table 15 shows selected hydrogen atom migration reactions that are currently considered. The rate coefficient assigned generally to the 1,4-formyl H-shift reaction of α-formyl peroxy radicals is based on that determined for the methacrolein-derived peroxy radical, HOCH₂C(CH₃)(O₂)C(=O)H, in the experimental study of Crounse et al. (2012). It is noted that this is slightly higher than, but comparable with, the range of values reported for α-formyl peroxy radicals in the preliminary calculations of Peeters and Nguyen (2012).

Table 15 Representative rate coefficients for selected H-shift isomerization reactions of peroxy radicals.

Comments: ^a Based on rate coefficient reported for the methacrolein-derived peroxy radical, HOCH₂C(CH₃)(O₂)C(=O)H by Crounse et al. (2012). Applied to primary, secondary and tertiary α-formyl peroxy radicals. ^b The initially formed hydroperoxy-substituted product radical decomposes spontaneously to produce the displayed products. ^c Based on the rate coefficient estimated for CH₃CH(OH)O₂ by Hermans et al. (2005); applied to secondary α-hydroxyl peroxy radicals. ^d Based on the rate coefficient estimated for (CH₃)₂C(OH)O₂ by Hermans et al. (2005); applied to tertiary α-hydroxy peroxy radicals. ^e Based on the rate coefficient estimated for cyclo-C₆H₁₀(OH)O₂ by Hermans et al. (2005); applied generally to cyclic α-hydroxy peroxy radicals (i.e. where the OH and OO groups are substituents to a ring). ^f Based on rate coefficients reported by Peeters et al. (2014) for corresponding unsaturated secondary and tertiary β-hydroxy peroxy radicals formed in isoprene oxidation. Applied generally to unsaturated β-hydroxy peroxy radicals containing the substructures shown. ^g Based on the geometric mean of rate coefficients applied to (Z)-CH(OH)C(CH₃)=CHCH₂O₂ (CISOPAO2) and (Z)-CH(OH)CH=C(CH₃)CH₂O₂ (CISOPCO2) in MCM v3.3.1 (Jenkin et al., 2015) based on the calculations of Peeters et al. (2014) and observations of Crounse et al. (2011). Applied generally to unsaturated δ-hydroxy peroxy radicals containing the substructure shown, except for CISOPAO2 and CISOPCO2 themselves for which the species-specific rate coefficients are applied (see Table S5). Rapid reaction of the product radical with O₂ dominates over the reverse isomerization reaction under atmospheric conditions. ^h Based on the geometric mean of rate coefficients reported for (Z)-HOCH=C(CH₃)CH(O₂)CH₂OH and (Z)-HOCH=CHC(CH₃)(O₂)CH₂OH in the calculations of Peeters and Nguyen (2012). Applied to peroxy radicals containing the substructure shown. Rapid reaction of the product radical with O₂ dominates over the reverse isomerization reaction under atmospheric conditions.

The rate coefficients assigned to the 1,4-hydroxyl H-shift reactions of (thermalized) α-hydroxy peroxy radicals are based on those estimated for secondary, tertiary and cyclic peroxy radicals in the theoretical study of Hermans et al. (2005). As discussed in the companion paper on the OH-initiated oxidation of aliphatic VOCs (Jenkin et al., 2018a), thermalized α-hydroxy peroxy radicals are represented to be increasingly formed from the reactions of O₂ with larger α-hydroxy organic radicals (i.e. those with n_CON>5). At the assigned rates, the 1,4-hydroxyl H-shift reaction is likely to be the major fate of the majority of thermalized α-hydroxy peroxy radicals under atmospheric conditions, and therefore indistinguishable from that of the chemically activated α-hydroxy peroxy radical adducts that are formed predominantly from the reactions of O₂ with small α-hydroxy organic radicals (see Sect. 6.2 of Jenkin et al., 2018a). However, the rates of the 1,4-hydroxyl H-shift reactions are formalized in the present work to allow for the representation of competing rapid isomerization reactions for specific structurally complex peroxy radicals (e.g. the 1,6-enol H-shift reaction discussed below), or with bimolecular reactions under appropriate conditions. It is noted that evidence for competitive loss via bimolecular reactions has been characterized in experimental studies using high concentrations of NO (e.g. Orlando et al., 2000b; Jenkin et al., 2005; Aschmann et al., 2010), leading to the formation of organic acids.

The remaining reactions in Table 15 are inferred from information reported for specific unsaturated peroxy radicals formed during the OH-initiated oxidation of isoprene, taking particular account of the work of Peeters et al. (2009, 2014) on the Leuven isoprene mechanism (LIM1), which has been largely verified by experimental study (e.g. Wennberg et al., 2018; and references therein). The rate coefficients for the 1,5-hydroxyl H-shift reactions are those reported by Peeters et al. (2014) for the corresponding unsaturated secondary and tertiary β-hydroxy peroxy radicals formed from the sequential addition of OH and O₂ to isoprene, with these also being generally consistent with those reported by da Silva et al. (2010). The rate coefficient assigned to the 1,6-hydroxyalkyl H-shift reaction is the geometric mean of rate coefficients applied to (Z)-CH₂(OH)C(CH₃)=CHCH₂O₂ (CISOPAO2) and (Z)-CH₂(OH)CH=C(CH₃)CH₂O₂ (CISOPCO2) in MCM v3.3.1. As discussed by Jenkin et al. (2015), those rate coefficients are derived from the LIM1 calculations of Peeters et al. (2014), but with some scaling to recreate the observations of Crounse et al. (2011, 2014). The generic rate coefficient is applied generally to unsaturated δ-hydroxy peroxy radicals containing the substructure shown, but with the exceptions of CISOPAO2 and CISOPCO2 themselves, for which the species-specific rate coefficients are applied (see Sect. S6 and Table S5). Similarly, the rate coefficient for the rapid 1,6-enol H-shift reaction is the geometric mean of those calculated for (Z)-HOCH=C(CH₃)CH(O₂)CH₂OH and (Z)-HOCH=CHC(CH₃)(O₂)CH₂OH by Peeters and Nguyen (2012). Once again, the 1,6-enol H-shift reaction is likely to be the major fate of the majority of peroxy radicals containing the relevant substructure (see Table 15) under atmospheric conditions, but the rate is formalized in the present work to allow for the representation of competing rapid isomerization reactions for specific structurally complex peroxy radicals, e.g. the 1,4-hydroxyl H-shift reaction discussed above, or other reactions that may be considered and represented in future work.

As indicated above, the present paper does not attempt to provide a full treatment of unimolecular reactions of RO₂ radicals, which ideally require systematic information on the rates of a series of 1,n-H-shift reactions from C−H and O−H bonds in different environments. In this respect, it is noted that the systematic influence of a series of neighbouring functional groups and transition state sizes have been considered in theoretical studies of a number of model systems (e.g. Crounse et al., 2013; Jørgensen et al., 2016; Praske et al., 2017; Otkjær et al., 2018). Such studies provide the basis for defining systematic structure–activity methods for a wide range of RO₂ radicals and their potential isomerization reactions, and are being considered in ongoing work. A further consideration, highlighted in those studies, is that the rates of the reverse isomerization reactions are sometimes sufficiently rapid that the product radical may not be fully trapped by onward reaction (e.g. addition of O₂) under atmospheric conditions. It is noted that the explicit representation of a very large number of rapid reversible reactions in detailed mechanisms can have implications for computational efficiency, and needs to be considered carefully in method development and implementation.

4 Conclusions

Published kinetics and branching ratio data have been reviewed for the bimolecular reactions of organic peroxy radicals (RO₂), with information for selected unimolecular isomerization reactions also summarized and discussed. The information has been used to define generic rate coefficients and structure–activity relationship (SAR) methods for the reactions of a series of important classes of hydrocarbon and oxygenated RO₂ radicals, for application in the next generation of explicit detailed chemical mechanisms, based on GECKO-A and the MCM.

The availability of kinetic and mechanistic data for peroxy radical reactions has increased substantially since the appraisals of Saunders et al. (2003) and Aumont et al. (2005), on which the previous treatments of peroxy radical chemistry in the MCM and GECKO-A were mainly based. These advances have allowed improved and updated methods to be defined and summarized in the present work for an extended set of peroxy radical reactions. Nevertheless, there are still a number of specific areas (commented on in Sects. 2 and 3) where information is lacking and further studies would be beneficial. These include the following.

• Kinetics studies of the reactions with NO have only been reported for a limited number of acyl peroxy radicals. Further studies, particularly for larger and highly oxygenated acyl peroxy radicals, would help to establish whether size and/or the presence of additional substituent groups has an effect on reactivity.

• Further systematic data on RONO₂ yields from the reactions with NO are required to help improve branching ratio parameterizations. These include additional data for a variety of acyclic and cyclic oxygenated RO₂ as a function of size and structure.

• For the reactions with NO₃, studies for ≥C₂ (non-acyl) RO₂ are dominated by primary peroxy radicals. Further studies are therefore required for secondary and tertiary radicals, and product information is generally required for a variety of peroxy radical classes to test assumption that the reaction proceeds via a single channel forming RO, NO₂ and O₂.

• The reactions of ≥C₂ hydrocarbon RO₂ with OH are believed to produce a thermalized hydrotrioxide, ROOOH, as the major product. Detailed experimental and theoretical studies are therefore required to establish the atmospheric fate of these ROOOH species. Studies of the reactions of oxygenated RO₂ with OH are also required.

• The reactions of HO₂ with several oxygenated RO₂ classes have been shown to proceed via multiple channels, although the temperature dependences of the product channels have generally not been studied. Additional studies of their temperature dependences would therefore be valuable, in addition to information for larger sets of oxygenated RO₂ within some classes. Kinetics studies have only been reported for a limited number of acyl peroxy radicals. Further studies, particularly for larger and highly oxygenated acyl peroxy radicals, would help to establish whether size and/or the presence of additional substituent groups has an effect on reactivity.

• For the self- and cross-reactions of peroxy radicals, further information is required to allow the impacts of multiple substituents on the kinetics to be defined more rigorously. Further systematic studies of the formation of ROOR + O₂ (from the self-reaction of RO₂) and ${\mathrm{ROOR}}^{\prime }+{\mathrm{O}}_{\mathrm{2}}$ (from the cross-reaction of RO₂ with R^′O₂) are also required as a function of peroxy radical size and functional group content.

• For unimolecular isomerization reactions, further systematic studies are required for the rates of 1,n-H-shift reactions from C−H and O−H bonds in different chemical environments, and for the effect of ring size and substituents on ring-closure reactions, to build upon recently reported data for these reaction classes.